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Метрики оценки качества моделей и анализ ошибок в машинном обучении

Одним из критически важных шагов при создании хорошей модели является правильный выбор метрики для оценки её качества, поскольку неправильный выбор может привести к неверным выводам и, как следствие, к принятию не самых оптимальных решений. Поэтому на сегодняшний день существует большое количество метрик, подходящих для самых разных задач и ситуаций.

В данном туториале будут рассмотрены популярные метрики для задач классификации, регрессии и кластеризации, а также инструмент для анализа ошибки модели, известный как bias-variance decomposition. Помимо этого, для большей части метрик будут представлены ручные расчёты и реализация с нуля на Python, а в конце вы сможете найти дополнительные источники для более глубокого ознакомления.

🔔 Несколько полезных ссылок перед тем как продолжить:
Объяснение и реализация ML-алгоритмов с нуля на Python
Ноутбук с данным материалом можно загрузить на GitHub (rus)

Содержание

Метрики классификации
Метрики регрессии
Анализ ошибок (bias-variance decomposition)
Метрики кластеризации
Дополнительные источники

Метрики классификации

Для простоты понимания рассмотрим бинарный случай. Однако, перед этим стоит тщательно ознакомиться с матрицей ошибок (confusion matrix) и её компонентами, представленными на изображении ниже.

На главной диагонали расположены правильно классифицированные положительные (TP) и отрицательные (TN) классы, а на побочной — неправильно классифицированные, которые ещё называются ошибками первого (FP) и второго (FN) рода. Теперь обучим логистическую регрессию на датасете Breast Cancer Wisconsin и построим на основе её прогнозов данную матрицу. Однако, перед этим стоит упомянуть, что в scikit-learn порядок компонентов матрицы ошибок немного отличается от изображения выше, но это не меняет сути происходящего:

$| TN \ \ FP | \\ | FN \ \ TP |$

Импорт необходимых библиотек

import numpy as np
import pandas as pd
import seaborn as sns
import matplotlib.pyplot as plt
from sklearn.model_selection import train_test_split
from sklearn.linear_model import LogisticRegression
from sklearn.metrics import confusion_matrix
from sklearn.preprocessing import LabelEncoder
from sklearn.metrics import (accuracy_score, precision_score, recall_score, f1_score,
                             fbeta_score, roc_curve, roc_auc_score, precision_recall_curve,
                             auc, average_precision_score, classification_report)

Загрузка датасета

df_path = "/content/drive/MyDrive/breast_cancer.csv"
breast_cancer = pd.read_csv(df_path)
breast_cancer.drop(columns=['id','Unnamed: 32'], inplace=True)
print(breast_cancer)

X = breast_cancer.drop(columns='diagnosis', axis=1)
y = breast_cancer['diagnosis']
y = LabelEncoder().fit_transform(y)

# 1 - Malignant, 0 - Benign
X_train, X_test, y_train, y_test = train_test_split(X, y, random_state=0)


    diagnosis  radius_mean  texture_mean  perimeter_mean  area_mean  \
0           M        17.99         10.38          122.80     1001.0   
1           M        20.57         17.77          132.90     1326.0   
2           M        19.69         21.25          130.00     1203.0   
3           M        11.42         20.38           77.58      386.1   
4           M        20.29         14.34          135.10     1297.0   
..        ...          ...           ...             ...        ...   
564         M        21.56         22.39          142.00     1479.0   
565         M        20.13         28.25          131.20     1261.0   
566         M        16.60         28.08          108.30      858.1   
567         M        20.60         29.33          140.10     1265.0   
568         B         7.76         24.54           47.92      181.0   

     smoothness_mean  compactness_mean  concavity_mean  concave points_mean  \
0            0.11840           0.27760         0.30010              0.14710   
1            0.08474           0.07864         0.08690              0.07017   
2            0.10960           0.15990         0.19740              0.12790   
3            0.14250           0.28390         0.24140              0.10520   
4            0.10030           0.13280         0.19800              0.10430   
..               ...               ...             ...                  ...   
564          0.11100           0.11590         0.24390              0.13890   
565          0.09780           0.10340         0.14400              0.09791   
566          0.08455           0.10230         0.09251              0.05302   
567          0.11780           0.27700         0.35140              0.15200   
568          0.05263           0.04362         0.00000              0.00000   

     symmetry_mean  ...  radius_worst  texture_worst  perimeter_worst  \
0           0.2419  ...        25.380          17.33           184.60   
1           0.1812  ...        24.990          23.41           158.80   
2           0.2069  ...        23.570          25.53           152.50   
3           0.2597  ...        14.910          26.50            98.87   
4           0.1809  ...        22.540          16.67           152.20   
..             ...  ...           ...            ...              ...   
564         0.1726  ...        25.450          26.40           166.10   
565         0.1752  ...        23.690          38.25           155.00   
566         0.1590  ...        18.980          34.12           126.70   
567         0.2397  ...        25.740          39.42           184.60   
568         0.1587  ...         9.456          30.37            59.16   

     area_worst  smoothness_worst  compactness_worst  concavity_worst  \
0        2019.0           0.16220            0.66560           0.7119   
1        1956.0           0.12380            0.18660           0.2416   
2        1709.0           0.14440            0.42450           0.4504   
3         567.7           0.20980            0.86630           0.6869   
4        1575.0           0.13740            0.20500           0.4000   
..          ...               ...                ...              ...   
564      2027.0           0.14100            0.21130           0.4107   
565      1731.0           0.11660            0.19220           0.3215   
566      1124.0           0.11390            0.30940           0.3403   
567      1821.0           0.16500            0.86810           0.9387   
568       268.6           0.08996            0.06444           0.0000   

     concave points_worst  symmetry_worst  fractal_dimension_worst  
0                  0.2654          0.4601                  0.11890  
1                  0.1860          0.2750                  0.08902  
2                  0.2430          0.3613                  0.08758  
3                  0.2575          0.6638                  0.17300  
4                  0.1625          0.2364                  0.07678  
..                    ...             ...                      ...  
564                0.2216          0.2060                  0.07115  
565                0.1628          0.2572                  0.06637  
566                0.1418          0.2218                  0.07820  
567                0.2650          0.4087                  0.12400  
568                0.0000          0.2871                  0.07039  

[569 rows x 31 columns]

Обучение и прогноз модели логистической регрессии

model = LogisticRegression(max_iter=10000)
model.fit(X_train, y_train)
y_pred = model.predict(X_test)
print(y_pred)


[1 0 0 0 0 0 0 0 0 0 1 0 0 1 0 1 0 1 1 1 1 1 0 0 1 0 0 1 0 1 0 1 0 1 0 1 0
 1 0 1 1 0 1 0 1 1 0 0 0 1 1 1 1 0 0 0 0 0 0 1 1 1 0 0 1 0 1 1 1 0 1 1 0 0
 1 0 0 0 0 0 1 1 1 0 1 0 0 0 1 1 0 1 1 1 0 0 1 0 0 0 0 0 0 0 1 0 1 0 1 1 0
 1 1 0 0 0 0 0 0 0 0 0 1 0 1 0 1 0 0 0 1 0 0 0 0 0 0 1 1 0 0 0 1]

Построение и визуализация confusion matrix

conf_matrix = confusion_matrix(y_test, y_pred)

sns.heatmap(conf_matrix, annot=True, fmt='d', cmap='Reds')
plt.title('Confusion Matrix')
plt.xlabel('Predicted Label')
plt.ylabel('True Label')
plt.show()

Глядя на полученную матрицу ошибок, мы имеем следующее:

(True Positive) — 52 человека классифицированы как больные и они такими являются, то есть классифицированы верно;
(True Negative) — 84 человека классифицированы как здоровые и они такими являются, то есть также классифицированы верно;
(False Positive) — 6 человек классифицированы как больные (то есть имеют злокачественную опухоль), но они здоровые (то есть опухоль доброкачественная);
(False Negative) — 1 человек определён как здоровый, но он болен (то есть имеет злокачественную опухоль).

Теперь можно переходить к самим метрикам.

Accuracy

Самым простым способом оценить качество модели является Accuracy или точность, которая отражает долю правильно спрогнозированных классов среди всех образцов:

$\text{Accuracy} = \frac{1}{N} \sum_{i=1}^{N} I [y_i = \hat y_i] = \frac{TP + TN}{TP + TN + FP + FN}$

Теперь получим значение Accuracy для нашего примера:

$\frac{52 + 84}{52 + 84 + 6 + 1} = 0.951048951048951$

Отсюда также можно определить долю ошибочных классификаций Error rate, которая показывает напрямую, как часто модель совершает ошибки:

$1 - \text{Accuracy} = 1 - 0.951048951048951 = 0.04895104895104896$

accuracy = accuracy_score(y_test, y_pred)
error_rate = 1 - accuracy
print(f'Accuracy: {accuracy}')
print(f'Error rate: {error_rate}')


Accuracy: 0.951048951048951
Error rate: 0.04895104895104896

Не смотря на свою простоту и универсальность, Accuracy имеет ряд серьёзных недостатков:

не учитывает дисбаланс классов: если один класс значительно преобладает над другим, то модель может быть смещена в его сторону, и тогда высокая точность не будет отражать истинное качество модели;
не даёт информацию о типе ошибок модели, например, о количестве ложноположительных и ложноотрицательных результатов, что не позволяет учитывать цену ошибки для разных классов;
зависит от порога классификации, изменение которого может значительно повлиять на значение точности.

Далее будут рассмотрены метрики, в которых устраняются данные недостатки.

Precision

Характеризует долю правильно предсказанных положительных классов среди всех образцов, которые модель спрогнозировала как положительный класс:

$\text{Precision} = TPR = \frac{TP}{TP + FP} = \frac{52}{52 + 6} = 0.896551724137931$

precision = precision_score(y_test, y_pred)
print(precision)


0.896551724137931

Recall (TPR)

Ещё известное как True Positive Rate, отражает долю правильно предсказанных положительных классов среди всех реальных положительных образцов:

$\text{Recall} = \frac{TP}{TP + FN} = \frac{52}{52 + 1} = 0.9811320754716981$

recall = recall_score(y_test, y_pred)
print(recall)


0.9811320754716981

Чем меньше ложноотрицательных прогнозов, тем выше Recall или TPR. В контексте рака груди, данные метрики являются ещё более важными, чем предыдущая, поскольку показывают какое число злокачественных опухолей удалось действительно выявить.

FPR

False Positive Rate характеризует долю ошибочно предсказанных положительных классов среди всех образцов, которые на самом деле являются отрицательным классом. Другими словами, это показывает, как часто модель неверно прогнозирует наличие заболевания (в случае рака груди), когда его на самом деле нет (доброкачественные образцы, ошибочно идентифицированные как раковые от всех реальных доброкачественных случаев).

Для лучшего понимания только что описанных метрик стоит ознакомиться с изображением ниже.

F1-score

Представляет собой гармоническое среднее между Precision и Recall, обеспечивая между ними баланс, что особенно полезно при неравномерном распределении классов. Использование данной метрики позволяет лучше выбрать модель, которая не только точно классифицирует злокачественные случаи, но и минимизирует количество пропущенных злокачественных опухолей.

$\text{F1-score} = 2 \cdot \frac{precision \cdot recall}{precision + recall} \approx 2 \frac{0.8966 \cdot 0.9811}{0.8966 + 0.9811} \approx 0.936948671246738$

f1 = f1_score(y_test, y_pred)
print(f1)


0.9369369369369369

Стоит добавить, что F1-score исходит из предположения, что Precision и Recall имеют одинаковую важность. Если же необходимо придать большее значение (вес) одной из метрик, то можно воспользоваться $\text{F}_{\beta}$ - score:

$\text{F}_{\beta}= (1 + \beta^2) \cdot \frac{precision \cdot recall}{\beta^2 \cdot precision + recall}$

При $\beta > 1$ большее значение придаётся Recall, а при $\beta < 1$ — Precision. Другими словами, если мы хотим сфокусироваться больше на действительно обнаруженных опухолях, то мы увеличиваем $\beta$ , если же мы хотим избежать ложноположительных прогнозов (то есть когда у человека нет рака, а модель прогнозирует, что есть), то мы уменьшаем $\beta$ . Например, при $\beta = 2$ получим следующее:

$F_2 \approx (1 + 2^2) \cdot \frac{0.8966 \cdot 0.9811}{2^2 \cdot 0.8966 + 0.9811} \approx 0.9629493814997262$

f2 = fbeta_score(y_test, y_pred, beta=2)
print(f2)


0.9629629629629629

ROC-AUC

Все предыдущие метрики позволяют оценить качество модели только при определённом пороге классификации. В случае, когда необходимо оценить качество модели при различных пороговых значениях, используется AUC-площадь (Area Under Curve) под ROC-кривой (Receiver Operating Characteristics curve), выраженной через отношение доли истинно положительных прогнозов (TPR) к доли ложноположительных (FPR).

В идеальном случае ROC-кривая будет стремиться в верхний левый угол (TPR=1 и FPR=0), а площадь под ней (AUC) будет равна единице. При значении площади 0.5 качество прогнозов модели будет сопоставимо случайному угадыванию, ну а если это значение меньше 0.5, то, модель лучше предсказывает результаты, противоположные истинным — в таком случае нужно просто поменять целевые метки местами для получения площади больше 0.5.

Для лучшего понимания построим данную метрику с нуля. В общем виде процесс состоит из следующих шагов:

1) сначала предсказанные моделью вероятности быть положительным классом сортируются в порядке убывания и принимаются как пороговые значения;
2) для каждого порога выполняется классификация (бинаризация) меток на 1 и 0, а на их основе рассчитываются FPR и TPR;
3) полученные FPR и TPR используются для расчёта AUC с помощью метода трапеций, который выглядит следующим образом:

$AUC = \sum_{i=1}^{n-1} \frac{TPR_i + TPR_{i+1}}{2} (FPR_{i+1} - FPR_i)$

Однако такой процесс является неэффективным, поскольку при построении ROC-кривой используются все пороги, большинство из которых неоптимальные. Что это означает? При построении графика между двумя точками проводится прямая и если между ними есть ещё одна точка, расположенная на том же уровне, то она не будет отображаться на построенной ROC-кривой. Именно подобного рода точки называются неоптимальными. Следовательно, для построения ROC-кривой и расчёта площади под ней достаточно знать лишь угловые точки, что помимо прочего позволит строить более лёгкие ROC-кривые. Для поиска оптимальных значений необходимо выполнить следующие шаги:

1) для каждого отсортированного в порядке убывания порога выполняется бинаризация меток, на основе которых рассчитываются FP и TP;
2) для полученных FP и TP рассчитывается вторая разница между соседними значениями, которая выступает в качестве второй производной;
3) в местах, где вторая производная не равна нулю, будут расположены угловые точки;
4) позиции в списке угловых точек используются для поиска оптимальных порогов, FP и TP;
5) к списку оптимальных FP и TP добавляются нули в начало, чтобы ROC-кривая всегда начиналась в точке (0, 0); а к списку оптимальных порогов в начало добавляется максимальное значение y_score + 1, чтобы ROC-кривая заканчивалась в единице даже если все образцы неверно классифицированы;
6) на основе полученных оптимальных значений FP и TP рассчитываются оптимальные FPR и TPR, которые в дальнейшем используются для расчёта AUC с помощью метода трапеций.

Реализация с нуля

def binary_roc_curve(y_true, y_score):
    thresholds = np.sort(y_score)[::-1]
    tps = np.array([])   # True positives
    fps = np.array([])   # False positives

    for threshold in thresholds:
        # predictions binarization by each threshold
        y_pred = (y_score >= threshold).astype(int)
        tp = np.sum((y_true == 1) & (y_pred == 1))
        fp = np.sum((y_true == 0) & (y_pred == 1))
        tps = np.append(tps, tp)
        fps = np.append(fps, fp)

    # find optimal (corner) points (thresholds)
    corner_point = True
    d2_fps = np.diff(fps, 2)   # is used as a "second derivative"
    d2_tps = np.diff(tps, 2)
    is_corner_points = np.r_[corner_point, np.logical_or(d2_fps, d2_tps), corner_point]
    optimal_indexes = np.where(is_corner_points == corner_point)[0]

    # add an extra threshold position to optimal values to make sure that the curve starts
    # at (0, 0) and also ends in 1 even if all samples are incorrectly classified
    optimal_fps = np.r_[0, fps[optimal_indexes]]
    optimal_tps = np.r_[0, tps[optimal_indexes]]
    optimal_thresholds = np.r_[max(y_score) + 1, thresholds[optimal_indexes]]

    optimal_fpr = optimal_fps / optimal_fps[-1]
    optimal_tpr = optimal_tps / optimal_tps[-1]

    return optimal_fpr, optimal_tpr, optimal_thresholds


def area_by_trapz(y, x):
    dx = np.diff(x)   # height

    return 0.5 * ((y[1:] + y[:-1]) * dx).sum()


def binary_roc_auc_score(y_test, y_pred):
    fpr, tpr, _ = binary_roc_curve(y_test, y_pred)

    return area_by_trapz(tpr, fpr)


y_pred_probas = model.predict_proba(X_test)[:, 1]
roc_auc = binary_roc_auc_score(y_test, y_pred_probas)
fpr, tpr, thresholds = binary_roc_curve(y_test, y_pred_probas)

print(f'AUC: {roc_auc}', '', sep='\n')
print('Optimal False Positive Rates:', fpr, '', sep='\n')
print('Optimal True Positive Rates:', tpr, '', sep='\n')
print('Optimal thresholds:', thresholds, sep='\n')

plt.figure(figsize=(8, 6))
plt.plot(fpr, tpr, label=f'ROC Curve (AUC = {roc_auc})', color='fuchsia')
plt.plot([0, 1], [0, 1], 'r--')  # Dashed diagonal line
plt.fill_between(fpr, tpr, color='lightblue', alpha=0.9)
plt.xlabel('False Positive Rate')
plt.ylabel('True Positive Rate')
plt.title('ROC Curve')
plt.legend(loc='lower right')
plt.show()


AUC: 0.9951781970649896

Optimal False Positive Rates:
[0.         0.         0.         0.01111111 0.01111111 0.04444444
 0.04444444 0.12222222 0.12222222 1.        ]

Optimal True Positive Rates:
[0.         0.01886792 0.86792453 0.86792453 0.94339623 0.94339623
 0.98113208 0.98113208 1.         1.        ]

Optimal thresholds:
[2.00000000e+00 1.00000000e+00 9.39251068e-01 9.28758892e-01
 8.62519599e-01 7.34708679e-01 6.73824037e-01 2.39038287e-01
 2.23663185e-01 1.80054591e-06]

Реализация scikit-learn

sk_roc_auc = roc_auc_score(y_test, y_pred_probas)
sk_fpr, sk_tpr, sk_thresholds = roc_curve(y_test, y_pred_probas)

print(f'AUC (scikit-learn): {sk_roc_auc}', '', sep='\n')
print('Optimal False Positive Rates (scikit-learn):', sk_fpr, '', sep='\n')
print('Optimal True Positive Rates (scikit-learn):', sk_tpr, '', sep='\n')
print('Optimal thresholds (scikit-learn):', sk_thresholds, sep='\n')


AUC (scikit-learn): 0.9951781970649896

Optimal False Positive Rates (scikit-learn):
[0.         0.         0.         0.01111111 0.01111111 0.04444444
 0.04444444 0.12222222 0.12222222 1.        ]

Optimal True Positive Rates (scikit-learn):
[0.         0.01886792 0.86792453 0.86792453 0.94339623 0.94339623
 0.98113208 0.98113208 1.         1.        ]

Optimal thresholds (scikit-learn):
[2.00000000e+00 1.00000000e+00 9.39251068e-01 9.28758892e-01
 8.62519599e-01 7.34708679e-01 6.73824037e-01 2.39038287e-01
 2.23663185e-01 1.80054591e-06]

ROC-AUC имеет смысл использовать в задачах, где важны не столько предсказанные классы, сколько их правильный вероятностный порядок. Например, у нас есть онлайн-кинотеатр и мы хотим узнать какие пользователи будут продлевать месячную подписку. На первый взгляд может показаться, что эта обычная бинарная классификация и нужно просто предсказать метки, однако более полезным вариантом будет упорядочивание клиентов по вероятности отмены подписки, чтобы в дальнейшем использовать различные стратегии для их удержания.

Отсюда может сложиться впечатление, что ROC-AUC является хорошей метрикой для задач ранжирования, однако не всё так просто, как может показаться на первый взгляд. Дело в том, что ROC-AUC не очень хорошо справляется с сильным дисбалансом классов, поскольку учитывает истинно отрицательные случаи (TN), что вытекает из расчётов FPR. Проще говоря, модель может показать высокий TPR, но при этом также иметь большое количество ложноположительных предсказаний (FPR).

PR-AUC и Average Precision (AP)

В таком случае можно использовать площадь в осях Precision Recall, известную как PR-AUC, которая лучше подходит для данных с сильным дисбалансом классов. Это связано с тем, что PR-AUC фокусируется на соотношении истинно положительных и ложноотрицательных результатов, что отражает лучше способность модели правильно определять положительные классы и, следовательно, лучше справляться в задачах ранжирования, где это необходимо в первую очередь. Например, если мы хотим показать пользователю наиболее релевантные фильмы, PR-AUC будет лучше учитывать действительно интересные для пользователя фильмы (TP), в то время как ROC-AUC может учесть наименее интересные фильмы (TN). Также стоит добавить, что в отличие от ROC-AUC, на графике PR-AUC будет стремиться в правый верхний угол, а её нахождение выполняется следующим образом:

precisions, recalls, _ = precision_recall_curve(y_test, y_pred_probas)
pr_auc_score = auc(recalls, precisions)
print(pr_auc_score)


0.9924452566260418

Однако и здесь не всё так просто, поскольку расчёт PR-AUC также основан на методе трапеций, который, в свою очередь, использует линейную интерполяцию. Что в этом может быть плохого? Если интерполяцию между двумя точками в ROC-пространстве можно выполнить, просто соединив их прямой линией, то в PR-пространстве интерполяция может иметь более сложную связь. При изменении уровня Recall, метрика Precision не обязательно будет изменяться линейно, поскольку FP заменяет FN в знаменателе Precision. В таком случае линейная интерполяция является ошибочной и может давать слишком оптимистичную оценку качества модели. Проще говоря, в случае PR-AUC такой подход может считать завышенную площадь под кривой.

Поэтому в scikit-learn существует альтернативная (и очень схожая) метрика, которая называется Average Precision. Её основное отличие как раз и заключается в том, что для расчёта не используется линейная интерполяция. Вместо этого кривая Precision-Recall суммируется как средневзвешенное значение Precisions, полученное для каждого порога, а в качестве веса используется увеличение Recall по сравнению с предыдущим порогом:

$\text{AP} = \sum_n (R_n - R_{n-1}) P_n$

ap_score = average_precision_score(y_test, y_pred_probas)
print(ap_score)


0.992511694643552

Когда классов больше двух

Обычно задача классификации на классов сводится к отделению класса от всех остальных с последующим расчётом consufion matrix для каждого из них. В таком случае для получения итогового значения метрики можно применить усреднённые методы:

1) Микро-усреднение (micro-averaging) является эквивалентом accuracy и подходит при сбалансированных классах. Элементы consufion matrix усредняются между бинарными прогнозами для каждого класса, после чего метрики рассчитываются на полученной матрице. На примере Precision и Recall это выглядит следующим образом:

$\text{Precision}_{\text{(micro)}} = \frac{\sum_{k=1}^{K} \text{TP}_k}{\sum_{k=1}^{K} (\text{TP}_k + \text{FP}_k)} \\ \text{Recall}_{\text{(micro)}} = \frac{\sum_{k=1}^{K} \text{TP}_k}{\sum_{k=1}^{K} (\text{TP}_k + \text{FN}_k)}$

2) Макро-усреднение (macro-averaging) представляет собой среднее арифметическое подсчитанной метрики для каждого класса и используется при дисбалансе классов, когда важен каждый класс. В таком случае все классы учитываются равномерно независимо от их размера:

$\text{Precision}_{\text{(macro)}} = \frac{1}{K} \sum_{k=1}^{K} \frac{\text{TP}_k}{\text{TP}_k + \text{FP}_k} \\ \text{Recall}_{\text{(macro)}} = \frac{1}{K} \sum_{k=1}^{K} \frac{\text{TP}_k}{\text{TP}_k + \text{FN}_k}$

3) Взвешенное усреднение (weighted averaging) рассчитывается как взвешенное среднее и также применяется в случае дисбаланса классов, но только когда важность класса учитывается в зависимости от количества объектов с таким классом, то есть когда важны наибольшие классы. При таком подходе важность каждого класса учитывается с присвоением им весов. Вес класса w_k может устанавливаться по-разному, например, как доля примеров этого класса в обучающей выборке:

$\text{Precision}_{\text{(weighted)}} = \sum_{k=1}^{K} w_k \cdot \frac{\text{TP}_k}{\text{TP}_k + \text{FP}_k} \\ \text{Recall}_{\text{(weighted)}} = \sum_{i=1}^{K} w_k \cdot \frac{\text{TP}_k}{\text{TP}_k + \text{FN}_k}$

Для лучшего понимания рассмотрим пример с подсчётом наиболее популярных метрик на данных Red Wine Quality. Для этого воспользуемся функцией classification_repot из scikit-learn.

Загрузка датасета

red_wine = pd.read_csv("/content/drive/MyDrive/winequality-red.csv")
print(red_wine)

X = red_wine.drop(columns='quality', axis=1)
y = red_wine['quality']

X_train, X_test, y_train, y_test = train_test_split(X, y, random_state=0)


      fixed acidity  volatile acidity  citric acid  residual sugar  chlorides  \
0               7.4             0.700         0.00             1.9      0.076   
1               7.8             0.880         0.00             2.6      0.098   
2               7.8             0.760         0.04             2.3      0.092   
3              11.2             0.280         0.56             1.9      0.075   
4               7.4             0.700         0.00             1.9      0.076   
...             ...               ...          ...             ...        ...   
1594            6.2             0.600         0.08             2.0      0.090   
1595            5.9             0.550         0.10             2.2      0.062   
1596            6.3             0.510         0.13             2.3      0.076   
1597            5.9             0.645         0.12             2.0      0.075   
1598            6.0             0.310         0.47             3.6      0.067   

      free sulfur dioxide  total sulfur dioxide  density    pH  sulphates  \
0                    11.0                  34.0  0.99780  3.51       0.56   
1                    25.0                  67.0  0.99680  3.20       0.68   
2                    15.0                  54.0  0.99700  3.26       0.65   
3                    17.0                  60.0  0.99800  3.16       0.58   
4                    11.0                  34.0  0.99780  3.51       0.56   
...                   ...                   ...      ...   ...        ...   
1594                 32.0                  44.0  0.99490  3.45       0.58   
1595                 39.0                  51.0  0.99512  3.52       0.76   
1596                 29.0                  40.0  0.99574  3.42       0.75   
1597                 32.0                  44.0  0.99547  3.57       0.71   
1598                 18.0                  42.0  0.99549  3.39       0.66   

      alcohol  quality  
0         9.4        5  
1         9.8        5  
2         9.8        5  
3         9.8        6  
4         9.4        5  
...       ...      ...  
1594     10.5        5  
1595     11.2        6  
1596     11.0        6  
1597     10.2        5  
1598     11.0        6  

[1599 rows x 12 columns]

Обучение модели и оценка полученных результатов

Первое, что можно заметить, что, в целом, классификатор справился не очень хорошо. Следующим интересным наблюдением является то, что из-за сильного дисбаланса классов не все классы были спрогнозированы, поэтому метрики для некоторых классов помечены нулями.

В данном случае микро-усреднение (accuracy) показало завышенные результаты, а вот макро показывает более реальную картину при условии, что все спрогнозированные классы имеют для нас одинаково значение. Взвешенное усреднение, в свою очередь, показало способность модели определять наиболее распространённые классы, которая также оказалась не очень хорошая.

ovr_model = LogisticRegression(multi_class='ovr', max_iter=1000)
ovr_model.fit(X_train, y_train)
y_pred = ovr_model.predict(X_test)

clf_report = classification_report(y_test, y_pred)
print(clf_report)


    accuracy                           0.62       400
   macro avg       0.29      0.27      0.27       400
weighted avg       0.58      0.62      0.59       400

Метрики регрессии

В данном случае целевая метка представляет собой вещественное число, поскольку это связано с природой регрессионных задач, например, прогноз погоды, стоимость акций, цена на недвижимость и так далее. Поэтому большинство метрик в задачах регрессии представляют собой среднюю оценку разности между действительными и спрогнозированными значениями, однако, с некоторыми особенностями. Рассмотрим их более подробно, обучив линейную регрессию на данных Medical Cost Personal.

Импорт необходимых библиотек

import numpy as np
import pandas as pd
from sklearn.preprocessing import LabelEncoder
from sklearn.linear_model import LinearRegression
from sklearn.model_selection import train_test_split
from sklearn.metrics import (mean_absolute_error, mean_absolute_percentage_error,
                             mean_squared_error, mean_squared_log_error, r2_score)

Загрузка датасета

df_path = "/content/drive/MyDrive/insurance.csv"
insurance_cost = pd.read_csv(df_path)
print(insurance_cost)

X = insurance_cost.drop(columns='charges', axis=1)
y = insurance_cost['charges']

cat_features_list = X.select_dtypes(include=['object']).columns
X[cat_features_list] = X[cat_features_list].apply(LabelEncoder().fit_transform)

X_train, X_test, y_train, y_test = train_test_split(X, y, random_state=0)


      age     sex     bmi  children smoker     region      charges
0      19  female  27.900         0    yes  southwest  16884.92400
1      18    male  33.770         1     no  southeast   1725.55230
2      28    male  33.000         3     no  southeast   4449.46200
3      33    male  22.705         0     no  northwest  21984.47061
4      32    male  28.880         0     no  northwest   3866.85520
...   ...     ...     ...       ...    ...        ...          ...
1333   50    male  30.970         3     no  northwest  10600.54830
1334   18  female  31.920         0     no  northeast   2205.98080
1335   18  female  36.850         0     no  southeast   1629.83350
1336   21  female  25.800         0     no  southwest   2007.94500
1337   61  female  29.070         0    yes  northwest  29141.36030

[1338 rows x 7 columns]

Обучение линейной регрессии

model = LinearRegression()
model.fit(X_train, y_train)
y_pred = model.predict(X_test)
print(y_pred)


[10947.91401491  9764.82733066 38027.18625354 16076.26656375
  7003.05093861  4162.38974052  1745.17453352 14273.5330135
  9022.7490154   7548.70107263  4742.33662827 10290.75344147
  8592.56051588  4173.37165612 27970.0324915  11026.04778351
 11286.00941429  6197.06911697  8269.51468144 27263.01056172
 33686.9512703  14247.8812616  11735.79293452 32419.5578177
  4475.57228648  9264.65728706  1336.5408973  10083.42064465
  4134.01766875 10422.0367284   9033.04363126 40177.36502272
 15327.89185262 13541.84076855 24979.41529438  5273.0794857
 12809.44891047 30538.99654744 33503.98483751  3477.84775709
  4169.03343497  4346.93367013 30642.90398321 39366.95813634
 28066.36347631  5110.98142166 10919.49675465  7870.63024919
  3790.77872548 10529.86942143  5758.50260778  3526.36470247
 32837.53966438 38431.60954739 16119.53210068  7198.88399648
  6010.47765564  9455.45703492  9323.82247057 11736.15685931
  1745.70435635 38856.13381995 15133.33969997 11575.03834933
 14071.5724629  13696.5272597  26189.53137674 32224.3994756
  1285.84165993 10192.3389615  12103.18984314 11812.13268915
 25146.32331141 15738.43743924 11190.33714978 12666.511048
  6443.77417808  9893.21714388 30312.26136507 38774.44771196
 12063.1199766  37348.74819938  4280.83232304  9194.1511214
 34855.84194627 29249.62417709  8369.44588068  4945.57095651
 11829.89576319 30234.75411865 10099.95369361 11152.51466685
  8345.39483062  9216.66152856  8456.71908548  7389.90572249
 36031.82502924 32995.87707694  7647.03076484 14761.02404168
  4335.42273688  8790.6336137   6626.82252505 31798.31619795
 33017.1906077   2094.40570673  8884.5733591   6639.27260791
 14530.96799671 36943.676041   10209.36205958 10754.28354013
 10198.39675302 27099.34704068 39995.94299687  8600.67413204
   283.08058982  8834.31659717 14929.5620573   9589.03267786
 35399.75956117  7345.60983793 16567.9444266   9694.5508457
  8068.18149856  3097.01655573 33001.67961343 31570.2503123
 39236.13757102  5563.9838186   9605.43014551  3923.56950664
  7936.19344017  8758.93978756 31529.93069335 29783.93576815
 30140.52767905  9084.18593446 32774.6651671   3447.8282725
  3714.75872713 11106.90396946 13347.95952139 12853.99458907
  5528.91820808 15704.83891849 15079.88363053  2544.67843054
   185.77752153 10942.26089435  7512.81091318 32029.66886541
 12399.16046026  2715.99117956  6398.9448798   8153.13247564
  4485.58951977  2502.88912184 11354.3765072  12473.86448172
  7378.57534425 16486.02707379 11773.13898945 13764.75456224
  3381.59967736  7414.52171201 23083.99616223  7702.25146492
  5640.00381695  5423.20666932  6771.22658553  5369.21237435
  9978.65825664  5651.47058169  5617.05819013  6992.25772557
  3927.54466569  5677.34987188 38108.35355982  1687.47789286
 12570.90785396  8990.57884784 13663.12241126  5692.74305794
  5400.09622194 36497.28911925  4499.38119881  1950.64760943
 15075.82279197 12681.99346283 35147.22262059  5131.27114327
  5547.26999983 31572.7189971   6167.37488999  2062.03461108
  8544.26381135 10074.42722051  8266.17437786  5850.88787567
 13147.1222075  38635.78272386 13659.93211399 28800.64409693
  6812.19498243 35711.06603231  3973.45030397 12098.25000178
  9304.02766868  6456.8508697  11306.30379171 14517.23252599
  5259.7484308   4261.74861629  7739.96622866  1287.11803755
  8007.7029553   4603.7389562  13174.33476059  4491.67379326
  9853.36083563  7375.34455594  9072.2189463   2578.72399596
 12933.69851212 16674.51032722 15098.07149349 10382.93145759
  5750.15941467  2634.44325287  2301.36074417 13447.00625685
 14203.53207008  5171.47470846  4159.8794872   9287.91215283
 10048.86025637 28387.85591072  7761.85208288 10586.09180311
  6147.09690682 29867.2455107  10866.38256904  7609.85428543
 10248.41393685 12190.90620035  3216.69032739 10786.13583408
  1735.08998421  7194.492167   28788.15185516 38398.20353488
  6171.96169948  8372.97420985  2665.38796458   675.24311869
 10324.23779113  4464.24740958  5084.62925272  2765.12366764
  7255.30324083 33205.81109315 38215.02085724 14677.52426265
  8294.91587575 16028.83595951 33133.59667833  9623.53886742
 33401.64850545  3686.52411934 30843.71540422  8001.71986381
 14179.94621339  4211.35722232 32497.0827103   8443.10889356
 11460.8623175   9434.21047718  4321.92599873 12464.24068476
 11701.49681152  8483.560458   13344.98515219  2972.12577494
 10675.86128124  5389.8245388  11159.3820939  31723.46535925
 10047.84307228  1472.76142561   717.98703914 39821.97375226
  9812.26823742  7166.20953879 13905.32236228 13428.33354475
 27218.10087176  7033.74308314  6785.64013095 11933.98289128
  3038.43915783  4022.38383321 25307.2029606  26414.33042405
 13250.32827696  3471.9145674   5176.44334405  9222.66486622
 12506.28838182 23844.27532952 30900.61492849  9893.41699449
 24249.49669032  3004.11445538 11542.47299099  7624.09099954
  8366.614262     502.36290308  7920.24301719 35599.63593964
  6316.23859849  6413.28197074   427.51684841 10843.82978818
  6793.9564125   9939.5724268  38715.2046951  27842.86075012
 11591.34753748 35624.43529894 14993.0317229   6934.89010657
 10983.73053465  6810.5049236  36668.7011676 ]

MAE

Mean Absolute Error показывает насколько в среднем прогнозы модели отклоняются от реальных значений по модулю:

$\text{MAE}(y, \hat y) = \frac{1}{N} \sum_{i=1}^N |y_i - \hat y_i|$

Данная метрика проста в интерпретации и устойчива к выбросам, однако не учитывает масштаб (насколько велико полученное отклонение) и направление ошибок (положительное или отрицательное отклонение от реальных значений).

def mae_score(y_true, y_pred):
    n = len(y_true)

    return 1 / n * np.sum(np.abs(y_true - y_pred))


mae = mae_score(y_test, y_pred)
sk_mae = mean_absolute_error(y_test, y_pred)
print(f'mae: {mae}')
print(f'mae(scikit-learn): {sk_mae}')


mae: 3998.2715408869726
mae(scikit-learn): 3998.2715408869726

Как можно заметить, полученная ошибка не может нам показать напрямую как сильно ошибается модель. Другими словами, в зависимости от контекста точно такое же значение MAE может быть как хорошим, так и плохим результатом. Представим следующую ситуацию: нам необходимо спрогнозировать на какую сумму сеть магазинов продаст бананы. Для этого смоделируем новые данные на основе предыдущих, добавив к ним значение 10000.

true_sales = y_test + 10_000
pred_sales = y_pred + 10_000

mae_for_sales = mean_absolute_error(true_sales, pred_sales)
print(f'mae(for sales): {mae_for_sales}')


mae(for sales): 3998.2715408869726

Нетрудно догадаться, что не смотря на одинаковые MAE, в одном случае качество модели будет лучше, чем в другом. В такой ситуации удобно рассматривать не абсолютную, а относительную ошибку на объектах.

MAPE & SMAPE

Mean Absolute Percentage Error как раз позволяет оценить в процентах насколько прогнозы модели отличаются относительно реальных значений:

$\text{MAPE}(y, \hat y) = \frac{1}{N} \sum_{i=1}^N \frac{|y_i - \hat y_i|}{|y_i|}$

def mape_score(y_true, y_pred):
    n = len(y_true)
    numerator = np.abs(y_true - y_pred)
    denominator = np.abs(y_true)

    return  1 / n * np.sum(numerator / denominator)


mape = mape_score(y_test, y_pred)
sk_mape = mean_absolute_percentage_error(y_test, y_pred)
mape_for_sales = mean_absolute_percentage_error(true_sales, pred_sales)

print(f'mape: {mape}')
print(f'mape(scikit-learn): {sk_mape}')
print(f'mape(for sales): {mape_for_sales}')


mape: 0.4138713715103651
mape(scikit-learn): 0.4138713715103651
mape(for sales): 0.15963240527305983

Теперь хорошо видно, что модель в первом случае ошибается гораздо сильнее, даже не смотря на одинаковые значения MAE. Однако MAPE не подходит для случаев, когда хотя бы одно фактическое значение равно нулю, что видно из формулы. В таком случае можно использовать симметричную MAPE, известную как Symmetric Mean Absolute Percentage Error:

$\text{SMAPE}(y, \hat y) = \frac{1}{N} \sum_{i=1}^N \frac{2 |y_i - \hat y_i|}{y_i + \hat y_i}$

def smape_score(y_true, y_pred):
    n = len(y_true)
    numerator = 2 * np.abs(y_true - y_pred)
    denominator = (y_true + y_pred)

    return  1 / n * np.sum(numerator / denominator)


smape = smape_score(y_test, y_pred)
print(f'smape: {smape}')


smape: 0.3573799807309885

WAPE

Как и другие метрики, MAPE также применима не во всех условиях. Например, она плохо справляется при анализе временных рядов с сезонными колебаниями, поскольку не учитывает относительную значимость наблюдений и плохо работает с неравномерными данными. Для такого случая лучше подходит Weighted Average Percentage Error, которая нормализует общую ошибку к общей сумме фактических значений:

$\text{WAPE}(y, \hat y) = \frac{\sum_{i=1}^N |y_i - \hat y_i|}{\sum_{i=1}^N |y_i|}$

Например, продажи мороженого имеют ярко выраженную сезонность, при этом продажи в летние месяцы значительно выше, чем в зимние. В таком случае WAPE покажет более реальную оценку, поскольку будет учитывать ошибку прогноза для каждого месяца относительно общего объёма продаж, что в итоге позволит лучше справляться с прерывистыми продажами.

def wape_score(y_true, y_pred):
    numerator = np.sum(np.abs(y_true - y_pred))
    denominator = np.sum(np.abs(y_true))

    return numerator / denominator


wape = wape_score(y_test, y_pred)
print(f'wape: {wape}')


wape: 0.29762832405759587

MSE & RMSE

Mean Squared Error показывает насколько в среднем прогнозы модели отклоняются от реальных значений в квадрате:

$\text{MSE}(y, \hat y) = \frac{1}{N} \sum_{i=1}^N (y_i - \hat y_i)^2$

Данная метрика полезна в тех случаях, когда нужно сделать акцент на больших ошибках и выбрать модель, которая допускает их в среднем меньше всего. С другой стороны, из-за квадрата разности в формуле выше, MSE чувствительна к выбросам и более сложна для восприятия по сравнению с MAE.

def mse_score(y_true, y_pred):
    n = len(y_true)

    return 1 / n * np.sum((y_true - y_pred) ** 2)


mse = mse_score(y_test, y_pred)
sk_mse = mean_squared_error(y_test, y_pred)
print(f'mse: {mse}')
print(f'mse(scikit-learn): {sk_mse}')


mse: 32073628.560109198
mse(scikit-learn): 32073628.560109198

Поэтому, чтобы придать значению MSE размерность исходных данных, из него извлекается квадратный корень. Такая метрика называется Root Mean Squared Error и её основное преимущество заключается в лучшей интерпретируемости в сравнении с MSE:

$\text{RMSE}(y, \hat y) = \sqrt{\frac{1}{N} \sum_{i=1}^N (y_i - \hat y_i)^2}$

rmse = np.sqrt(mse)
print(f'rmse: {rmse}')


rmse: 5663.358417062193

MSLE & RMSLE

Ещё одним способом перехода к относительным ошибкам является их измерение в логарифмическом масштабе, как это делается в Mean Squared Logarithmic Error:

$\text{MSLE}(y, \hat y |c) = \frac{1}{N} \sum_{i=1}^N (\text{ln}(y_i + c) - \text{ln}(\hat y_i + c))^2$

где — нормированная константа (обычно равна 1), которая вводится, чтобы избегать логарифма нуля.

Данная метрика более устойчива к выбросам, поскольку делает распределение целевых и спрогнозированных значений более однородным. С другой стороны, MSLE труднее интерпретировать, а также стоит иметь в виду, что она уделяет больше внимания заниженным прогнозам, поскольку логарифм является несимметричной функцией.

def msle_score(y_true, y_pred, c=1):
    n = len(y_true)

    return 1 / n * np.sum((np.log(y_true + c) - np.log(y_pred + c)) ** 2)


msle = msle_score(y_test, y_pred)
sk_msle = mean_squared_log_error(y_test, y_pred)
print(f'msle: {msle}')
print(f'msle(scikit-learn): {sk_msle}')


msle: 0.2966547825040618
msle(scikit-learn): 0.2966547825040618

Соответственно, чтобы оценить значение MSLE относительно размерности исходных данных, используется Root Mean Squared Logarithmic Error:

$\text{RMSLE}(y, \hat y |c) = \sqrt{\frac{1}{N} \sum_{i=1}^N (\text{ln}(y_i + c) - \text{ln}(\hat y_i + c))^2}$

rmsle = np.sqrt(msle)
print(f'rmsle {rmsle}')


rmsle 0.5446602450189125

R2 & Adjusted R2

Иногда бывает трудно понять насколько хорошее или плохое значение рассчитанной ошибки, поэтому было бы удобно иметь в каком-то смысле аналог точности в процентах, но только для задачи регрессии. К счастью, такая метрика существует и это коэффициент детерминации $\text{R}^2$ который показывает какая доля дисперсии целевых значений объясняется моделью:

$\text{R}^2 = \frac{SST - SSR}{SST} = 1 - \frac{SSR}{SST} = 1 - \frac{\sum_{i=1}^N(y_i - \hat y_i)^2}{\sum_{i=1}^N(y_i - \overline y)^2}$

где SSR — Sum of Squared Residuals, а SST — Sum of Squares Total.

В отличии от предыдущих метрик, в данном случае более высокое значение метрики говорит о лучшем качестве модели. Значения $\text{R}^2$ можно интерпретировать следующим образом:

1) $\text{R}^2 = 1$ — модель идеально предсказывает данные;
2) $\text{R}^2 = 0$ — прогнозы модели соответствуют среднему арифметическому фактических целевых значений;
3) $\text{R}^2 < 0$ — модель работает хуже, чем простое использование среднего значения фактических целевых значений (обычно это связано с тем, что модель обучалась на данных, в которые попали большие выбросы).

def manual_r2_score(y_true, y_pred):
    ssr = np.sum((y_true - y_pred) ** 2)
    sst = np.sum((y_true - y_true.mean()) ** 2)

    return 1 - ssr / sst


r2 = manual_r2_score(y_test, y_pred)
sk_r2 = r2_score(y_test, y_pred)
print(f'r2: {r2}')
print(f'r2(scikit-learn): {sk_r2}')


r2: 0.7962732059725786
r2(scikit-learn): 0.7962732059725786

Стоит отметить, что $\text{R}^2$ также имеет свои ограничения. Его основная проблема заключается в том, что он не учитывает количество признаков в модели. Другими словами, $\text{R}^2$ имеет тенденцию к увеличению при добавлении в обучающий набор новых признаков, даже если они не улучшают качество модели.

В таком случае сравнение моделей с разным количеством признаков становится некорректным, поэтому специально для этой цели используется скорректированный Adjusted $\text{R}^2$ , который использует введение штрафа за добавленные признаки:

$\text{R}_{adj}^2 = 1 - \frac{SSR \cdot (n-1)}{SST \cdot (n - k - 1)} = 1 - \frac{(1 - R^2) (n - 1)}{n - k - 1}$

где — число наблюдений, а — количество признаков.

def adjusted_r2_score(y_true, y_pred, x_shape):
    n, k = x_shape
    ssr = np.sum((y_true - y_pred) ** 2)
    sst = np.sum((y_true - y_true.mean()) ** 2)
    r2 = 1 - ssr / sst

    return 1 - (1 - r2) * (n - 1) / (n - k - 1)


r2_adj = adjusted_r2_score(y_test, y_pred, X.shape)
print(f'r2 adjusted: {r2_adj}')


r2 adjusted: 0.7953548282384204

Bias-Variance Decomposition

Не отходя далеко от темы регрессии, рассмотрим разложение ошибки модели на смещение, разброс и случайный шум в данных. Это полезный инструмент, позволяющий оценить влияние вышеописанных компонентов общей тестовой ошибки и подобрать наиболее оптимальную модель с точки зрения её сложности.

Вывод разложения ошибки для MSE

Для простоты рассмотрим задачу регрессии со среднеквадратичной функцией потерь. Предположим, что целевое значение можно выразить с помощью некоторой детерминированной функции следующим образом:

$y = f(x) + \varepsilon$

где $\varepsilon$ — случайный шум, причём $E [\varepsilon] = 0, \ E [\varepsilon^2] = \sigma^2$ .

Тогда MSE для прогноза модели $\hat f(x)$ будет выглядеть следующим образом:

$MSE = E[(y - \hat f(x))^2] = E[y^2] - 2E[y \hat f(x)] + E[\hat f(x)^2]$

где смещение и разброс для $\hat f(x)$ равны соответственно:

$Bias[\hat f(x)] = f(x) - E[\hat f(x)] \\ Var[\hat f(x)] = E[\hat f(x)^2] - E[\hat f(x)]^2$

Теперь можно расписать каждый компонент по отдельности:

$\begin{align*} E[y^2] & = E[f(x)^2] + E[\varepsilon^2] + 2E[f(x) \varepsilon] \\ &= E[f(x)^2] + \sigma^2 + 2f(x) \underbrace{E[\varepsilon]}_{=0} \\ &= f(x)^2 + \sigma^2 \end{align*}$ $\begin{align*} E[y \hat f(x)] &= E[(f(x) + \varepsilon) \hat f(x)]\\ &= E[(f(x) \hat f(x)] + E[(\varepsilon \hat f(x)] \\ &= f(x) E[\hat f(x)] + \underbrace{E[\varepsilon]}_{=0} E[\hat f(x)] \\ &= f(x) E[\hat f(x)] \end{align*}$ $E[\hat f(x)^2] = Var[\hat f(x)] + E[\hat f(x)]^2$

Таким образом, разложение MSE на компоненты приобретает вид:

$\begin{align*} MSE &= f(x)^2 + \sigma^2 - 2 f(x) E[\hat f(x)] + Var[\hat f(x)] + E[\hat f(x)]^2 \\ &= (\underbrace{f(x) - E[\hat f(x)]}_{=Bias})^2 + Var[\hat f(x)] + \sigma^2 \\ &= Bias[\hat f(x)]^2 + Var[\hat f(x)] + \sigma^2 \end{align*}$

Полученное уравнение показывает и доказывает, что общая ошибка прогноза модели состоит из квадрата смещения (насколько модель ошибается в среднем), разброса (как предсказания модели различаются при разных данных) и неустранимой ошибки из-за шума в данных. Обычно высокие значения смещения и разброса соответствуют недообучению и переобучению модели соответственно, поэтому в идеале нам хотелось бы устремить эти значения к нулю. К сожалению, на практике это практически невозможно, поэтому задача сводится к поиску оптимального баланса между смещением и разбросом. На изображении ниже представлены возможные случаи их сочетания.

Bias-Variance trade-off

В итоге остаётся два варианта:

1) создание простой модели, чтобы она была стабильной и менее чувствительной (низкий разброс), но при этом она может не улавливать все закономерности в данных (высокое смещение);
2) создание сложной модели, чтобы она могла улавливать более тонкие закономерности (низкое смещение), однако в таком случае повышается чувствительность к шуму и отдельным точкам в данных (высокий разброс).

Такое явление, как поиск компромисса между смещением и разбросом, в литературе называется bias-variance trade-off, из которого следует, что общая ошибка на тестовой выборке имеет вид U-образной кривой.

Этот график иллюстрирует, что существует оптимальная сложность модели, при которой ошибка минимальна и при этом также соблюдается баланс между переобучением (высокая точность на тренировочных данных и низкая на тестовых) и недообучением (неспособность модели улавливать закономерности в тренировочных данных, что приводит к низкой точности как на них, так и на тестовых). Ниже представлен пример для линейной регрессии.

Интересные особенности bias-variance

Стоит отметить два важных момента:

1) Полученное разложение ошибки верно только для квадратичной функции потерь. Например, это разложение нельзя применить напрямую к стандартной задаче классификации, поскольку в таком контексте обычно применяется функция потерь вида 0/1, для которой смещение и разброс уже не являются аддитивными в чистом виде. Поэтому используются более общие формы этого разложения с похожими в некотором смысле компонентами. Так для функции потерь 0/1 смещение может быть определено как 1, если прогноз не совпадает с истинной меткой y, и 0 в противном случае. Разброс, в свою очередь, можно представить как вероятность того, что предсказанная метка не соответствует прогнозу моды. Другими словами, bias-variance decomposition может быть всё ещё применим к большому числу различных функций потерь и алгоритмов, хотя и не в такой простой форме. Более подробно с этим можно ознакомиться в работах Domingos (2000) и Valentini & Dietterich (2004).
2) Bias-Variance trade-off выполняется при разложении ошибки далеко не в каждом случае. Например, тестовая ошибка глубокой нейронной сети (DNN) часто демонстрирует двойное снижение: по мере увеличения сложности модели она сначала следует классической U-образной кривой, а затем демонстрирует второе снижение. Более того, подобное явление характерно и для ансамблей над решающими деревьями, которое проявляется при одновременном росте глубины и числа деревьев. Проще говоря, с увеличением сложности модели может происходить снижение как смещения, так и разброса. Обычно это вызвано двумя факторами:
- Bias-variance decomposition в своём классическом представлении недостаточно детализирован для выявления всех основных стохастических факторов, объясняющих поведение ошибки в сложных моделях.
- Постепенное расширение DNN, которое часто приводит к колокообразному разбросу ошибок. Более того, недавно было обнаружено новое явление, известное как двойной спуск по эпохам (Nakkiran et al., 2019), возникающее при увеличении количества эпох обучения вместо увеличения сложности модели. В работе Heckel & Yilmaz (2020) показано, что двойной переход по эпохам происходит в ситуации, когда различные части DNN обучаются в разные эпохи, что может быть устранено путём правильного масштабирования размеров шага.
Обычно решение такой проблемы сводится к использованию более продвинутых методов разложения ошибки, которые включают в себя более информативное разложение разброса, например, симметричное разложение и optimization variance (OV). Поскольку это всё же статья не про нейронные сети, мы опустим реализацию этих концепций с нуля.

Метрики кластеризации

В отличии от классификации, оценка качества кластеризации представляет собой более сложную задачу, поскольку относится к методам обучения без учителя. На то есть несколько причин:

вместо точного соответствия истинных меток с предсказанными, для нас более важным является группировка похожих объектов внутри одного кластера;
также зачастую нам неизвестны истинные метки, что делает невозможным их прямое сравнение с полученными результатами.

Метрики кластеризации подразделяются на два типа:

1) Внешние — основаны на использовании заранее известной информации (например, истинных меток), которая не задействовалась в процессе кластеризации. К ним относятся: Adjusted Rand Index (ARI), Mutual Information, Homogeneity, Completeness, V-measure, Fowlkes-Mallows score.
2) Внутренние — используют информацию только из структуры обучающего набора. Среди них можно выделить следующие: Silhouette Coefficient, Calinski-Harabasz Index, Davies-Bouldin Index. Такой вариант полезен в случае, когда нет информации об истинных метках (что встречается довольно часто).

Рассмотрим более подробно каждую из этих метрик на сгенерированных данных make blobs.

Импорт необходимых библиотек

import numpy as np
from sklearn.datasets import make_blobs
from sklearn.cluster import AffinityPropagation
from sklearn.metrics import (rand_score, adjusted_rand_score, mutual_info_score,
                             normalized_mutual_info_score, adjusted_mutual_info_score,
                             homogeneity_score, completeness_score, v_measure_score,
                             homogeneity_completeness_v_measure, fowlkes_mallows_score,
                             silhouette_score, calinski_harabasz_score, davies_bouldin_score)

Загрузка датасета

X, y = make_blobs(n_samples=75, n_features=2, centers=5, random_state=0)
print(y)


[0 3 4 3 2 4 0 2 0 4 2 4 2 2 0 0 0 3 2 0 2 2 2 3 4 1 1 2 3 0 4 4 3 3 3 2 2
 0 1 1 3 1 0 2 4 1 4 4 0 4 1 0 3 0 4 1 2 1 4 3 1 1 3 0 3 4 1 2 1 4 0 3 1 1
 3]

Обучение Affinity Propagation

ap = AffinityPropagation()
y_pred = ap.fit_predict(X)
print(y_pred)


[2 0 1 2 3 1 2 3 2 1 3 1 3 4 2 2 2 0 3 2 3 3 2 0 1 4 4 3 0 3 1 1 0 0 0 3 3
 2 4 4 0 4 2 3 1 4 1 1 4 1 4 2 0 2 1 4 3 4 1 0 4 4 0 2 0 1 4 3 3 1 2 0 4 4
 0]

Визуализация полученной кластеризации

plt.scatter(X[:, 0], X[:, 1], c=y_pred, cmap='rainbow', s=10)
plt.title('AffinityPropagation')
plt.xlabel("Feature 1")
plt.ylabel("Feature 2")
plt.show()

Rand Index & Adjusted Rand Index (ARI)

Являются одними из самых простых метрик. Rand Index измеряет количество пар элементов, отнесённых к одинаковым и разным кластерам относительно общего количества возможных пар в данных, игнорируя перестановки:

$\text{RI} = \frac{a + b}{C_2^{n_{samples}}}$

Данная метрика хорошо интерпретируема, поскольку её значения расположены в диапазоне [0, 1], где 1 соответствует идеальной кластеризации. С другой стороны, Rand Index не гарантирует, что случайные присвоения меток получат значения, близкие к нулю (особенно если количество кластеров имеет тот же порядок значений, что и количество образцов). Это одна из причин почему Rand Index зачастую даёт слишком оптимистичную оценку.

ri = rand_score(y, y_pred)
print(f'rand index: {ri}')


rand index: 0.9405405405405406

Нормировав данный индекс, можно снизить ожидаемый $E[\text{RI}]$ и таким образом избавиться от негативного эффекта выше. Тогда получим скорректированную оценку, известную как Adjusted Rand Index (ARI):

$\text{ARI} = \frac{\text{RI} - E[\text{RI}]}{\max(\text{RI}) - E[\text{RI}]}$

Она симметрична, не зависит от перестановок меток и их значений, которые теперь определены в диапазоне [-1, 1]. В таком случае отрицательное значение будет указывать на то, что кластеризация хуже, чем если бы метки были присвоены случайным образом.

ari = adjusted_rand_score(y, y_pred)
print(f'adjusted rand index: {ari}')


adjusted rand index: 0.8063318846556483

Mutual Information & Adjusted Mutual Information (AMI)

Взаимная информация Mutual Information определяет меру различия между совместным распределением пары меток U, V и произведением их маргинальных распределений. Другими словами, измеряется насколько информация об одной из этих переменных уменьшает неопределенность в отношении другой, то есть между ними определяется мера взаимной зависимости.

В свою очередь, мера неопределённости — это ни что иное, как энтропия, которая определяется для разбиений следующим образом:

$H(U) = - \sum_{i=1}^{|U|}P(i)\ln(P(i)) \\ H(V) = - \sum_{j=1}^{|V|}P'(j)\ln(P'(j))$

где P(i) = |U_i| / N — вероятность попадания случайно выбранного объекта из в класс U_i . В случае с ситуация аналогична.

Тогда взаимная информация между и определяется как:

$\text{MI}(U, V) = \sum_{i=1}^{|U|}\sum_{j=1}^{|V|}P(i, j)\ln\left(\frac{P(i,j)}{P(i)P'(j)}\right)$

где $P(i, j) = |U_i \cap V_j| / N$ — вероятность попадания случайно выбранного объекта в оба класса.

mi = mutual_info_score(y, y_pred)
print(f'mutual index: {mi}')


mutual index: 1.3165387166969102

Отсюда также можно получить нормализованную версию, то есть Normalized Mutual Information (NMI):

$\text{NMI}(U, V) = \frac{\text{MI}(U, V)}{\text{mean}(H(U), H(V))}$

nmi = normalized_mutual_info_score(y, y_pred)
print(f'normalized mutual index: {nmi}')


normalized mutual index: 0.8182376204426293

Однако обе эти метрики не учитывают случайность разбиений: MI будет склонна к увеличению по мере роста числа кластеров независимо от фактического объёма взаимной информации между метками, а NMI будет давать более оптимистичную оценку в пределах [0, 1].

Для устранения этих недостатков используется Adjusted Mutual Information (AMI), значения которого также расположены в диапазоне [0, 1], где 0 указывает на независимость меток, а значение близкое к 1 на их значительное совпадение:

$\text{AMI} = \frac{\text{MI} - E[\text{MI}]}{\text{mean}(H(U), H(V)) - E[\text{MI}]}$

где ожидаемое значение взаимной информации рассчитывается как:

$E[\text{MI}(U,V)]=\sum_{i=1}^{|U|} \sum_{j=1}^{|V|} \sum_{n_{ij}=(a_i+b_j-N)^+ }^{\min(a_i, b_j)} \frac{n_{ij}}{N}\ln \left( \frac{ N.n_{ij}}{a_i b_j}\right) \cdot \\ \cdot \frac{a_i!b_j!(N-a_i)!(N-b_j)!}{N!n_{ij}!(a_i-n_{ij})!(b_j-n_{ij})! (N-a_i-b_j+n_{ij})!}$

ami = adjusted_mutual_info_score(y, y_pred)
print(f'adjusted mutual index: {ami}')


adjusted mutual index: 0.8036319585502079

Homogeneity, Completeness & V-measure

Другой, более интуитивно понятный способ применения энтропии заключается в использовании таких понятий как:

гомогенность — мера того, насколько каждый кластер содержит объекты только одного класса:

$h = 1 - \frac{H(C|K)}{H(C)} \\ H(C) = - \sum_{c=1}^{|C|} \frac{n_c}{n} \cdot \log\left(\frac{n_c}{n}\right) \\ H(C|K) = - \sum_{c=1}^{|C|} \sum_{k=1}^{|K|} \frac{n_{c,k}}{n} \cdot \ln\left(\frac{n_{c,k}}{n_k}\right)$

полнота — мера того, насколько все объекты одного класса принадлежат одному и тому же кластеру:

$c = 1 - \frac{H(K|C)}{H(K)}$

где — истинные метки классов, а — спрогнозированные.

В отличие от ARI или AMI, данные метрики не являются нормализованными, поэтому случайная кластеризация не будет давать нулевые показатели при большом числе классов и малом числе объектов. Стоит отметить, что эта проблема не касается ситуаций, когда количество образцов превышает тысячу, а число кластеров меньше 10. В противном же случае лучше использовать ARI.

Если же необходимо учитывать гомогенность и полноту одновременно, то есть обеспечить между ними баланс, то используется V-мера, представляющая собой гармоническое среднее между ними и являющаяся в каком-то смысле эквивалентом NMI:

$v = 2 \cdot \frac{h \cdot c}{h + c}$

homogeneity = homogeneity_score(y, y_pred)
completeness = completeness_score(y, y_pred)
v_measure = v_measure_score(y, y_pred)

print(f'homogeneity: {homogeneity}')
print(f'completeness: {completeness}')
print(f'v measure: {v_measure}')


homogeneity: 0.8180114973840702
completeness: 0.8184638685502272
v measure: 0.8182376204426292

Следует добавить, что все 3 метрики хорошо интерпретируемые, поскольку лежат в диапазоне [0, 1], где 1 соответствует идеальной кластеризации. Также помимо этого, в scikit-learn имеется возможность получить все три метрики сразу.

hcv = homogeneity_completeness_v_measure(y, y_pred)
print('homogeneity, completeness, v measure:', hcv, sep='\n')


homogeneity, completeness, v measure:
(0.8180114973840702, 0.8184638685502272, 0.8182376204426292)

Fowlkes-Mallows Index (FMI)

Confusion matrix также может быть использована в задачах кластеризации. В таком случае можно получить Fowlkes-Mallows Index (FMI), если взять геометрическое среднее между precision и recall:

$\text{FMI} = \frac{\text{TP}}{\sqrt{(\text{TP} + \text{FP}) (\text{TP} + \text{FN})}}$

Эта метрика также лежит в диапазоне [0, 1] и может быть полезна при сравнении различных алгоритмов кластеризации, поскольку не делает никаких предположений об их структуре и, следовательно, может дать более объективную оценку.

fmi = fowlkes_mallows_score(y, y_pred)
print(f'Fowlkes-Mallows index: {fmi}')


Fowlkes-Mallows index: 0.8430070419125244

Silhouette Coefficient

Данный коэффициент показывает насколько в среднем объекты схожи внутри одного кластера и различны с объектами других кластеров. Для одного образца силуэт определяется следующим образом:

$s = \frac{b - a}{max(a, b)}$

где:

— среднее расстояние между образцом и всеми другими точками в том же классе;
— среднее расстояние между образцом и всеми другими точками в следующем ближайшем кластере.

Обычно для выборки силуэт определяется как среднее значение силуэта объектов и лежит в диапазоне [-1, 1], где -1 соответствует неверной кластеризации, 1 указывает на высокую степень соответствия объектов своим кластерам, а 0 говорит о пересекающихся и накладывающихся друг на друга кластерах.

Как можно заметить, данная метрика особенна полезна в случае, когда необходимо подобрать оптимальное количество кластеров, которое выбирается на основе максимального значения силуэта. Однако, следует учитывать, что значение силуэта может быть склонно к завышенным оценкам для выпуклых форм кластера и занижено для сложных форм, особенно для кластеров различного размера и плотности.

silhouette = silhouette_score(X, y_pred)
print(f'silhouette: {silhouette}')


silhouette: 0.5796452132316384

Calinski-Harabasz Index (CHI)

Также известный как критерий соотношения дисперсий (VRC), представляет собой отношение сумм межкластерной и внутрикластерной дисперсий:

$s = \frac{\mathrm{tr}(B_k)}{\mathrm{tr}(W_k)} \times \frac{n_E - k}{k - 1}$

где межкластерная B_k и внутрикластерная W_k дисперсии определяются как:

$B_k = \sum_{q=1}^k n_q (c_q - c_E) (c_q - c_E)^T \\ W_k = \sum_{q=1}^k \sum_{x \in C_q} (x - c_q) (x - c_q)^T$

а также где:

— набор точек в кластере ;
— центр кластера ;
— центр в наборе данных;
— число точек в кластере .

Чем выше значение Калински-Харабаша, тем более чётко модель определяет кластеры. Хотя данная метрика и лучше с вычислительной точки зрения, но, в принципе, имеет такой же недостаток, как и коэффициент силуэта.

chi = calinski_harabasz_score(X, y_pred)
print(f'Calinski-Harabasz index: {chi}')


Calinski-Harabasz index: 305.667448634445

Davies-Bouldin Index (DBI)

Показывает среднее сходство между кластерами, которое определяется как мера, сравнивающая межкластерное расстояние с размером самих кластеров:

$DB = \frac{1}{k} \sum_{i=1}^k \max_{i \neq j} R_{ij}$

В данном случае сходство определяется как:

$R_{ij} = \frac{s_i + s_j}{d_{ij}}$

где — среднее расстояние между каждой точкой кластера и его центроидом, а $d_{ij}$ — расстояние между центроидами и .

В сравнении с метриками выше, более близкое к нулю значение DBI говорит о лучшем качестве кластеризации. Хотя данная метрика и обладает тем же недостатком, что и коэффициент силуэта, она может быть более предпочтительным вариантом из-за более простого вычисления.

dbi = davies_bouldin_score(X, y_pred)
print(f'Davies-Bouldin index: {dbi}')


Davies-Bouldin index: 0.546114132526905

Дополнительные источники

Статьи:

«An introduction to ROC analysis», Tom Fawcett;
«The Relationship Between Precision-Recall and ROC Curves», Jesse Davis, Mark Goadrich;
«Precision-Recall-Gain Curves: PR Analysis Done Right», Peter A. Flach, Meelis Kull;
«Investigation of performance metrics in regression analysis and machine learning-based prediction models», V. Plevris, G. Solorzano, N. Bakas, M. Ben Seghier;
«A Unified Bias-Variance Decomposition and its Applications», Pedro Domingos;
«Bias-Variance Analysis of Support Vector Machines for the Development of SVM-Based Ensemble Methods», Giorgio Valentini, Thomas G. Dietterich;
«Optimization Variance: Exploring Generalization Properties of DNNs», Xiao Zhang, Dongrui Wu, Haoyi Xiong, Bo Dai;
«Deep Double Descent: Where Bigger Models and More Data Hurt», Preetum Nakkiran, Gal Kaplun, Yamini Bansal, Tristan Yang, Boaz Barak, Ilya Sutskever;
«Early Stopping in Deep Networks: Double Descent and How to Eliminate it», Reinhard Heckel, Fatih Furkan Yilmaz;
«Pointwise Metrics for Clustering Evaluation», Stephan van Staden.

Документация:

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14 июн. 2024 г.

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Few months ago I experienced one of the most devastating moments of my life €300k was stolen by hackers . When my money was stolen I felt hopeless, until I was introduced to Agent Jasmine Lopez her expertise and dedication gave me peace of mind. She skillfully traced complex transactions and tracked down the scammers, which would have been impossible for me to do on my own. Thanks to her tireless efforts, I was able to recover a significant portion of my lost funds. If you've fallen victim of hackers, I highly recommend reaching out to her. You can contact her via email at Recoveryfundprovider@gmail. com WhatsApp at +{44 736-644-5035}. Her attention to detail and commitment to her work are truly impressive. Don't hesitate to reach out her expertise could be the key to recovering your lost funds.
20.08.25 20:42 vallatjosette

My Wild Ride: How a Wizard Hilton Cyber Tech Helped Me Recover Lost Bitcoin of 103,000 It all started when I had the misfortune of misplacing the digital wallet containing my substantial cryptocurrency holdings - a staggering 103,000 bitcoins. I had amassed this sizable fortune over the years through strategic investments and careful trading, but in a moment of carelessness, the private keys to my digital wallet had vanished. Panic set in as I grappled with the gravity of my situation - those bitcoins, worth millions at the time, were effectively lost to the ether, inaccessible and seemingly unrecoverable. Just when I had resigned myself to the devastating financial blow, a chance encounter with a self-proclaimed "digital wizard" Wizard Hilton Cyber Tech presented a glimmer of hope. This enigmatic individual, who operated in the shadowy corners of the crypto underworld, claimed to possess the arcane knowledge and skills necessary to retrieve my lost digital wealth. Intrigued and desperate, I put my skepticism aside and entrusted this mysterious wizard with the task. What followed was a wild, suspenseful journey into the depths of blockchain technology and cryptography, as the wizard navigated complex algorithms, exploited vulnerabilities, and pieced together the puzzle of my missing private keys. After days of intense effort, Wizard Hilton Cyber Tech emerged triumphant, having successfully recovered the entirety of my 103,000 bitcoin stash. The sheer relief and elation I felt in that moment was indescribable, as I regained access to my small digital fortune and was able to secure my financial future. This harrowing experience taught me the invaluable lesson of never taking the security of my cryptocurrency holdings for granted, and the importance of always maintaining the utmost diligence when it comes to protecting one's digital wealth. Email: wizardhiltoncybertech ( @ ) gmail (. ) com OR support ( @ ) wizardhiltoncybertech (.) com WhatsApp number +18737715701 Thanks.
21.08.25 01:54 michaeldavenport218

I was recently scammed out of $53,000 by a fraudulent Bitcoin investment scheme, which added significant stress to my already difficult health issues, as I was also facing cancer surgery expenses. Desperate to recover my funds, I spent hours researching and consulting other victims, which led me to discover the excellent reputation of Capital Crypto Recover, I came across a Google post It was only after spending many hours researching and asking other victims for advice that I discovered Capital Crypto Recovery’s stellar reputation. I decided to contact them because of their successful recovery record and encouraging client testimonials. I had no idea that this would be the pivotal moment in my fight against cryptocurrency theft. Thanks to their expert team, I was able to recover my lost cryptocurrency back. The process was intricate, but Capital Crypto Recovery's commitment to utilizing the latest technology ensured a successful outcome. I highly recommend their services to anyone who has fallen victim to cryptocurrency fraud. For assistance contact [email protected] and on Telegram OR Call Number +1 (336)390-6684 via email: [email protected] you can visit his website: https://recovercapital.wixsite.com/capital-crypto-rec-1
21.08.25 01:54 michaeldavenport218

I was recently scammed out of $53,000 by a fraudulent Bitcoin investment scheme, which added significant stress to my already difficult health issues, as I was also facing cancer surgery expenses. Desperate to recover my funds, I spent hours researching and consulting other victims, which led me to discover the excellent reputation of Capital Crypto Recover, I came across a Google post It was only after spending many hours researching and asking other victims for advice that I discovered Capital Crypto Recovery’s stellar reputation. I decided to contact them because of their successful recovery record and encouraging client testimonials. I had no idea that this would be the pivotal moment in my fight against cryptocurrency theft. Thanks to their expert team, I was able to recover my lost cryptocurrency back. The process was intricate, but Capital Crypto Recovery's commitment to utilizing the latest technology ensured a successful outcome. I highly recommend their services to anyone who has fallen victim to cryptocurrency fraud. For assistance contact [email protected] and on Telegram OR Call Number +1 (336)390-6684 via email: [email protected] you can visit his website: https://recovercapital.wixsite.com/capital-crypto-rec-1
21.08.25 01:54 michaeldavenport218

I was recently scammed out of $53,000 by a fraudulent Bitcoin investment scheme, which added significant stress to my already difficult health issues, as I was also facing cancer surgery expenses. Desperate to recover my funds, I spent hours researching and consulting other victims, which led me to discover the excellent reputation of Capital Crypto Recover, I came across a Google post It was only after spending many hours researching and asking other victims for advice that I discovered Capital Crypto Recovery’s stellar reputation. I decided to contact them because of their successful recovery record and encouraging client testimonials. I had no idea that this would be the pivotal moment in my fight against cryptocurrency theft. Thanks to their expert team, I was able to recover my lost cryptocurrency back. The process was intricate, but Capital Crypto Recovery's commitment to utilizing the latest technology ensured a successful outcome. I highly recommend their services to anyone who has fallen victim to cryptocurrency fraud. For assistance contact [email protected] and on Telegram OR Call Number +1 (336)390-6684 via email: [email protected] you can visit his website: https://recovercapital.wixsite.com/capital-crypto-rec-1
21.08.25 14:56 elainemurray

E.M.A.I.L. [email protected] W.h.a.t.s.A.p.p. +1(561)(726)(3697) It all began with an Instagram advertisement. A man presented himself as a cryptocurrency investment coach, showcasing a luxurious lifestyle and screenshots of substantial profits. He claimed his trading algorithm could double any investment in a week. Curious, I decided to reach out to him. He said the minimum investment was $50,000 and sent me a link to a platform that looked legitimate. At first, everything seemed promising. My balance increased steadily, charts showed growth, and I felt excited. When I tried to withdraw the money, the system blocked the transaction and demanded a release fee of $15,000. I paid it, but my account was soon marked under review, and the coach stopped responding. I realised I had been scammed. Desperate, I turned to Techy Force Cyber Retrieval. From the first consultation, they carefully reviewed my case and assured me the funds could be traced. They explained how they would track the wallet addresses and follow the path of the stolen money. Their clear guidance made me feel confident. Over the next few weeks, Techy Force Cyber Retrieval kept me updated on every step. They traced the transactions. Their persistence and expertise were obvious, and I could see real progress. Even when the process felt slow, Techy Force Cyber Retrieval maintained clear communication. They explained each action and why it mattered. Their careful attention made the ordeal manageable and reassuring. A month later, thanks to Techy Force Cyber Retrieval, I was able to recover 80% of my money and have it safely returned to my bank account. Relief and gratitude overwhelmed me. What started as a tempting opportunity had turned into a stressful ordeal, but Techy Force Cyber Retrieval transformed it into a successful resolution.
21.08.25 21:17 monte5757

I went through one of the most painful experiences of my life I lost over $850,000 to scammers. They had no mercy,things got so bad that I even sold some of my properties just to keep up with their demands. At one point, I felt completely hopeless and trapped. But everything changed when I came across a legit recovery agent who actually helped me recover my funds and return them safely to my wallet. It was like getting my life back. If you’ve ever been scammed, please don’t give up. Reach out to Recoveryfundprovider [at] gmail [dot] com WhatsApp at +{44 736-644-5035}.she’s professional, reliable, and truly knows how to get results. I’m genuinely grateful I found her, and I believe you’ll thank me later too.
22.08.25 08:00 wendytaylor015

My name is Wendy Taylor, I'm from Los Angeles, i want to announce to you Viewer how Capital Crypto Recover help me to restore my Lost Bitcoin, I invested with a Crypto broker without proper research to know what I was hoarding my hard-earned money into scammers, i lost access to my crypto wallet or had your funds stolen? Don’t worry Capital Crypto Recover is here to help you recover your cryptocurrency with cutting-edge technical expertise, With years of experience in the crypto world, Capital Crypto Recover employs the best latest tools and ethical hacking techniques to help you recover lost assets, unlock hacked accounts, Whether it’s a forgotten password, Capital Crypto Recover has the expertise to help you get your crypto back. a security company service that has a 100% success rate in the recovery of crypto assets, i lost wallet and hacked accounts. I provided them the information they requested and they began their investigation. To my surprise, Capital Crypto Recover was able to trace and recover my crypto assets successfully within 24hours. Thank you for your service in helping me recover my $647,734 worth of crypto funds and I highly recommend their recovery services, they are reliable and a trusted company to any individuals looking to recover lost money. Contact email [email protected] OR Telegram @Capitalcryptorecover Call/Text Number +1 (336)390-6684 his contact: [email protected] His website: https://recovercapital.wixsite.com/capital-crypto-rec-1
22.08.25 08:00 wendytaylor015

My name is Wendy Taylor, I'm from Los Angeles, i want to announce to you Viewer how Capital Crypto Recover help me to restore my Lost Bitcoin, I invested with a Crypto broker without proper research to know what I was hoarding my hard-earned money into scammers, i lost access to my crypto wallet or had your funds stolen? Don’t worry Capital Crypto Recover is here to help you recover your cryptocurrency with cutting-edge technical expertise, With years of experience in the crypto world, Capital Crypto Recover employs the best latest tools and ethical hacking techniques to help you recover lost assets, unlock hacked accounts, Whether it’s a forgotten password, Capital Crypto Recover has the expertise to help you get your crypto back. a security company service that has a 100% success rate in the recovery of crypto assets, i lost wallet and hacked accounts. I provided them the information they requested and they began their investigation. To my surprise, Capital Crypto Recover was able to trace and recover my crypto assets successfully within 24hours. Thank you for your service in helping me recover my $647,734 worth of crypto funds and I highly recommend their recovery services, they are reliable and a trusted company to any individuals looking to recover lost money. Contact email [email protected] OR Telegram @Capitalcryptorecover Call/Text Number +1 (336)390-6684 his contact: [email protected] His website: https://recovercapital.wixsite.com/capital-crypto-rec-1
22.08.25 08:00 wendytaylor015

My name is Wendy Taylor, I'm from Los Angeles, i want to announce to you Viewer how Capital Crypto Recover help me to restore my Lost Bitcoin, I invested with a Crypto broker without proper research to know what I was hoarding my hard-earned money into scammers, i lost access to my crypto wallet or had your funds stolen? Don’t worry Capital Crypto Recover is here to help you recover your cryptocurrency with cutting-edge technical expertise, With years of experience in the crypto world, Capital Crypto Recover employs the best latest tools and ethical hacking techniques to help you recover lost assets, unlock hacked accounts, Whether it’s a forgotten password, Capital Crypto Recover has the expertise to help you get your crypto back. a security company service that has a 100% success rate in the recovery of crypto assets, i lost wallet and hacked accounts. I provided them the information they requested and they began their investigation. To my surprise, Capital Crypto Recover was able to trace and recover my crypto assets successfully within 24hours. Thank you for your service in helping me recover my $647,734 worth of crypto funds and I highly recommend their recovery services, they are reliable and a trusted company to any individuals looking to recover lost money. Contact email [email protected] OR Telegram @Capitalcryptorecover Call/Text Number +1 (336)390-6684 his contact: [email protected] His website: https://recovercapital.wixsite.com/capital-crypto-rec-1
23.08.25 05:11 ramiroalcazar

A hacker skillfully impersonated one of our clients’ trusted suppliers and manipulated their finance department into transferring $180,000 worth of Bitcoin. The request appeared authentic, reinforced by falsified documents and meticulously crafted communication that mirrored the supplier’s established style. By the time the company uncovered the deception, the funds had already been funneled through numerous digital wallets. When they appealed to their bank for assistance, they were told recovery was impossible since cryptocurrency transactions are irreversible. The company suddenly faced the prospect of financial ruin. Confronted with this devastating loss, the client sought a specialized solution and turned to Techy Force Cyber Retrieval. From the initial consultation, our team demonstrated that hope remained. Unlike the bank, which dismissed their plea, Techy Force Cyber Retrieval meticulously analyzed every aspect of the fraud and assured the client that our blockchain specialists had the tools to trace and potentially reclaim the stolen assets. Our investigation began with a comprehensive audit of the fraudulent transactions. Using advanced blockchain forensics, Techy Force Cyber Retrieval’s analysts traced the stolen Bitcoin across a labyrinth of wallets and exchanges. Every development was documented with precision, and regular updates were delivered to the client, ensuring transparency and trust. This combination of technical mastery and clear communication provided reassurance during an otherwise overwhelming crisis. After persistent tracking and strategic intervention, Techy Force Cyber Retrieval achieved a remarkable outcome. From the original $180,000 that had been siphoned away, our team successfully recovered the equivalent of $160,000 in USDT. This restitution enabled the company to meet payroll obligations, stabilize its operations, and avert what had seemed like an inevitable collapse. What distinguished Techy Force Cyber Retrieval in the eyes of the client was not only our technical expertise but also our dedication and professionalism. We approached the case with urgency, maintained open communication at every stage, and treated the recovery effort with the utmost seriousness. At a moment when the client felt powerless, Techy Force Cyber Retrieval restored both their assets and their confidence. Now, victims of cryptocurrency fraud do not have to accept total loss. With the right expertise, recovery is achievable. Techy Force Cyber Retrieval stands as proof that even in situations deemed hopeless, we can deliver tangible results and provide businesses and individuals with a genuine second chance. CONSULT TECHY FORCE CYBER RETRIEVAL FOR A GREAT SOLUTION RECOVERY......... WHATSAPP.......... +15617263697 WEBSITE......... https://techyforcecyberretrieval.com
24.08.25 01:05 samshaffer

Have been on the quest for a way to recover my $195,000 from a forex broker. I was fortunate to meet Darek at ([email protected]) through a friend who was recommended as the best crypto recovery specialist he was the one who assisted me and restore my money back to my atomic wallet. I did reach him at [email protected] and it was an amazing experience. Never knew there was an expert who could trace back crypto until I met this great team. Don’t think twice, get connected with the right personnel now to avoid been fooled again..
24.08.25 06:32 amy

I have lost all hope to recover my investment in pragmatic trade which I lost $125,000 in the process of investment. I'm here to say thank you to anthony davies for being a man of your word got my funds recovered back plus a bonus within a week of working with [email protected] I hope people make use of this opportunity as well and don't give up on getting your funds back. is also available on Telegram: @anthonydavies01
24.08.25 17:13 marcushenderson624

Bitcoin Recovery Testimonial After falling victim to a cryptocurrency scam group, I lost $354,000 worth of USDT. I thought all hope was lost from the experience of losing my hard-earned money to scammers. I was devastated and believed there was no way to recover my funds. Fortunately, I started searching for help to recover my stolen funds and I came across a lot of testimonials online about Capital Crypto Recovery, an agent who helps in recovery of lost bitcoin funds, I contacted Capital Crypto Recover Service, and with their expertise, they successfully traced and recovered my stolen assets. Their team was professional, kept me updated throughout the process, and demonstrated a deep understanding of blockchain transactions and recovery protocols. They are trusted and very reliable with a 100% successful rate record Recovery bitcoin, I’m grateful for their help and highly recommend their services to anyone seeking assistance with lost crypto. Contact: [email protected] Phone CALL/Text Number: +1 (336) 390-6684 Email: [email protected] Website: https://recovercapital.wixsite.com/capital-crypto-rec-1
24.08.25 17:13 marcushenderson624

Bitcoin Recovery Testimonial After falling victim to a cryptocurrency scam group, I lost $354,000 worth of USDT. I thought all hope was lost from the experience of losing my hard-earned money to scammers. I was devastated and believed there was no way to recover my funds. Fortunately, I started searching for help to recover my stolen funds and I came across a lot of testimonials online about Capital Crypto Recovery, an agent who helps in recovery of lost bitcoin funds, I contacted Capital Crypto Recover Service, and with their expertise, they successfully traced and recovered my stolen assets. Their team was professional, kept me updated throughout the process, and demonstrated a deep understanding of blockchain transactions and recovery protocols. They are trusted and very reliable with a 100% successful rate record Recovery bitcoin, I’m grateful for their help and highly recommend their services to anyone seeking assistance with lost crypto. Contact: [email protected] Phone CALL/Text Number: +1 (336) 390-6684 Email: [email protected] Website: https://recovercapital.wixsite.com/capital-crypto-rec-1
24.08.25 17:13 marcushenderson624

Bitcoin Recovery Testimonial After falling victim to a cryptocurrency scam group, I lost $354,000 worth of USDT. I thought all hope was lost from the experience of losing my hard-earned money to scammers. I was devastated and believed there was no way to recover my funds. Fortunately, I started searching for help to recover my stolen funds and I came across a lot of testimonials online about Capital Crypto Recovery, an agent who helps in recovery of lost bitcoin funds, I contacted Capital Crypto Recover Service, and with their expertise, they successfully traced and recovered my stolen assets. Their team was professional, kept me updated throughout the process, and demonstrated a deep understanding of blockchain transactions and recovery protocols. They are trusted and very reliable with a 100% successful rate record Recovery bitcoin, I’m grateful for their help and highly recommend their services to anyone seeking assistance with lost crypto. Contact: [email protected] Phone CALL/Text Number: +1 (336) 390-6684 Email: [email protected] Website: https://recovercapital.wixsite.com/capital-crypto-rec-1
24.08.25 18:39 nelly young

Getting your USDC stolen feels like the ground has been pulled out from under you. It’s not just about the money you’ve been violated, and your security is at risk. And since crypto lives on the blockchain, there’s no bank to call and no quick refund. That’s why you need someone who knows exactly what to do. Agent Jasmine Lopez has helped countless victims trace stolen USDC and lock down their accounts before more damage happens. Time matters reach her today at [email protected] WhatsApp at +44 736-644-5035 and take the first step toward getting your funds and peace of mind back.
24.08.25 23:56 marcushenderson624

Bitcoin Recovery Testimonial After falling victim to a cryptocurrency scam group, I lost $354,000 worth of USDT. I thought all hope was lost from the experience of losing my hard-earned money to scammers. I was devastated and believed there was no way to recover my funds. Fortunately, I started searching for help to recover my stolen funds and I came across a lot of testimonials online about Capital Crypto Recovery, an agent who helps in recovery of lost bitcoin funds, I contacted Capital Crypto Recover Service, and with their expertise, they successfully traced and recovered my stolen assets. Their team was professional, kept me updated throughout the process, and demonstrated a deep understanding of blockchain transactions and recovery protocols. They are trusted and very reliable with a 100% successful rate record Recovery bitcoin, I’m grateful for their help and highly recommend their services to anyone seeking assistance with lost crypto. Contact: [email protected] Phone CALL/Text Number: +1 (336) 390-6684 Email: Recoverycapital @fastservice. com Website: https://recovercapital.wixsite.com/capital-crypto-rec-1
24.08.25 23:56 marcushenderson624

Bitcoin Recovery Testimonial After falling victim to a cryptocurrency scam group, I lost $354,000 worth of USDT. I thought all hope was lost from the experience of losing my hard-earned money to scammers. I was devastated and believed there was no way to recover my funds. Fortunately, I started searching for help to recover my stolen funds and I came across a lot of testimonials online about Capital Crypto Recovery, an agent who helps in recovery of lost bitcoin funds, I contacted Capital Crypto Recover Service, and with their expertise, they successfully traced and recovered my stolen assets. Their team was professional, kept me updated throughout the process, and demonstrated a deep understanding of blockchain transactions and recovery protocols. They are trusted and very reliable with a 100% successful rate record Recovery bitcoin, I’m grateful for their help and highly recommend their services to anyone seeking assistance with lost crypto. Contact: [email protected] Phone CALL/Text Number: +1 (336) 390-6684 Email: Recoverycapital @fastservice. com Website: https://recovercapital.wixsite.com/capital-crypto-rec-1
24.08.25 23:56 marcushenderson624

Bitcoin Recovery Testimonial After falling victim to a cryptocurrency scam group, I lost $354,000 worth of USDT. I thought all hope was lost from the experience of losing my hard-earned money to scammers. I was devastated and believed there was no way to recover my funds. Fortunately, I started searching for help to recover my stolen funds and I came across a lot of testimonials online about Capital Crypto Recovery, an agent who helps in recovery of lost bitcoin funds, I contacted Capital Crypto Recover Service, and with their expertise, they successfully traced and recovered my stolen assets. Their team was professional, kept me updated throughout the process, and demonstrated a deep understanding of blockchain transactions and recovery protocols. They are trusted and very reliable with a 100% successful rate record Recovery bitcoin, I’m grateful for their help and highly recommend their services to anyone seeking assistance with lost crypto. Contact: [email protected] Phone CALL/Text Number: +1 (336) 390-6684 Email: Recoverycapital @fastservice. com Website: https://recovercapital.wixsite.com/capital-crypto-rec-1
25.08.25 00:55 vallatjosette

My Wild Ride: How a Wizard Hilton Cyber Tech Helped Me Recover Lost Bitcoin of 103,000 It all started when I had the misfortune of misplacing the digital wallet containing my substantial cryptocurrency holdings - a staggering 103,000 bitcoins. I had amassed this sizable fortune over the years through strategic investments and careful trading, but in a moment of carelessness, the private keys to my digital wallet had vanished. Panic set in as I grappled with the gravity of my situation - those bitcoins, worth millions at the time, were effectively lost to the ether, inaccessible and seemingly unrecoverable. Just when I had resigned myself to the devastating financial blow, a chance encounter with a self-proclaimed "digital wizard" Wizard Hilton Cyber Tech presented a glimmer of hope. This enigmatic individual, who operated in the shadowy corners of the crypto underworld, claimed to possess the arcane knowledge and skills necessary to retrieve my lost digital wealth. Intrigued and desperate, I put my skepticism aside and entrusted this mysterious wizard with the task. What followed was a wild, suspenseful journey into the depths of blockchain technology and cryptography, as the wizard navigated complex algorithms, exploited vulnerabilities, and pieced together the puzzle of my missing private keys. After days of intense effort, Wizard Hilton Cyber Tech emerged triumphant, having successfully recovered the entirety of my 103,000 bitcoin stash. The sheer relief and elation I felt in that moment was indescribable, as I regained access to my small digital fortune and was able to secure my financial future. This harrowing experience taught me the invaluable lesson of never taking the security of my cryptocurrency holdings for granted, and the importance of always maintaining the utmost diligence when it comes to protecting one's digital wealth. Email: wizardhiltoncybertech ( @ ) gmail (. ) com OR support ( @ ) wizardhiltoncybertech (.) com WhatsApp number +18737715701 Thanks.
25.08.25 05:14 samshaffer

Have been on the quest for a way to recover my $195,000 from a forex broker. I was fortunate to meet Darek at ([email protected]) through a friend who was recommended as the best crypto recovery specialist he was the one who assisted me and restore my money back to my atomic wallet. I did reach him at [email protected] and it was an amazing experience. Never knew there was an expert who could trace back crypto until I met this great team. Don’t think twice, get connected with the right personnel now to avoid been fooled again..
26.08.25 10:14 marcushenderson624

Bitcoin Recovery Testimonial After falling victim to a cryptocurrency scam group, I lost $354,000 worth of USDT. I thought all hope was lost from the experience of losing my hard-earned money to scammers. I was devastated and believed there was no way to recover my funds. Fortunately, I started searching for help to recover my stolen funds and I came across a lot of testimonials online about Capital Crypto Recovery, an agent who helps in recovery of lost bitcoin funds, I contacted Capital Crypto Recover Service, and with their expertise, they successfully traced and recovered my stolen assets. Their team was professional, kept me updated throughout the process, and demonstrated a deep understanding of blockchain transactions and recovery protocols. They are trusted and very reliable with a 100% successful rate record Recovery bitcoin, I’m grateful for their help and highly recommend their services to anyone seeking assistance with lost crypto. Contact: [email protected] Phone CALL/Text Number: +1 (336) 390-6684 Email: [email protected] Website: https://recovercapital.wixsite.com/capital-crypto-rec-1
26.08.25 10:14 marcushenderson624

Bitcoin Recovery Testimonial After falling victim to a cryptocurrency scam group, I lost $354,000 worth of USDT. I thought all hope was lost from the experience of losing my hard-earned money to scammers. I was devastated and believed there was no way to recover my funds. Fortunately, I started searching for help to recover my stolen funds and I came across a lot of testimonials online about Capital Crypto Recovery, an agent who helps in recovery of lost bitcoin funds, I contacted Capital Crypto Recover Service, and with their expertise, they successfully traced and recovered my stolen assets. Their team was professional, kept me updated throughout the process, and demonstrated a deep understanding of blockchain transactions and recovery protocols. They are trusted and very reliable with a 100% successful rate record Recovery bitcoin, I’m grateful for their help and highly recommend their services to anyone seeking assistance with lost crypto. Contact: [email protected] Phone CALL/Text Number: +1 (336) 390-6684 Email: [email protected] Website: https://recovercapital.wixsite.com/capital-crypto-rec-1
26.08.25 10:14 marcushenderson624

Bitcoin Recovery Testimonial After falling victim to a cryptocurrency scam group, I lost $354,000 worth of USDT. I thought all hope was lost from the experience of losing my hard-earned money to scammers. I was devastated and believed there was no way to recover my funds. Fortunately, I started searching for help to recover my stolen funds and I came across a lot of testimonials online about Capital Crypto Recovery, an agent who helps in recovery of lost bitcoin funds, I contacted Capital Crypto Recover Service, and with their expertise, they successfully traced and recovered my stolen assets. Their team was professional, kept me updated throughout the process, and demonstrated a deep understanding of blockchain transactions and recovery protocols. They are trusted and very reliable with a 100% successful rate record Recovery bitcoin, I’m grateful for their help and highly recommend their services to anyone seeking assistance with lost crypto. Contact: [email protected] Phone CALL/Text Number: +1 (336) 390-6684 Email: [email protected] Website: https://recovercapital.wixsite.com/capital-crypto-rec-1
26.08.25 23:59 floridaberna

It began as an entirely ordinary afternoon when an unexpected message appeared in my Facebook inbox. The sender was a woman I had never encountered. She introduced herself with warmth and courtesy, explaining that her late husband had been deeply invested in Bitcoin before his sudden passing. According to her narrative, he had left behind a vast fortune in cryptocurrency. However, due to intricate international banking regulations and legal entanglements, she was unable to access the funds without external assistance. Her words carried an air of urgency yet also sincerity. She claimed she had come across my profile in a cryptocurrency discussion forum and believed I might be the right person to help. The obstacle she faced was a set of legal fees amounting to $25,000. Once those costs were paid, she assured me the Bitcoin would be unlocked and a generous portion would be mine in gratitude. Against my better instincts, I allowed myself to be persuaded. She provided what appeared to be legitimate documents and even screenshots of supposed legal correspondence. The level of detail made her story feel authentic. After several exchanges, I transferred the $25,000.Almost immediately, her demeanor shifted. Responses became sporadic and vague. Within days, communication ceased entirely. My optimism dissolved into dread as I recognized the truth. I had been deceived. Refusing to surrender, I searched for solutions and discovered TECHY FORCE CYBER RETRIEVAL, a firm specializing in tracing and reclaiming stolen cryptocurrency. Their team outlined a meticulous plan to track the blockchain trail, identify the sequence of wallets involved, and coordinate with local authorities to freeze the assets before they disappeared into the labyrinth of the crypto underworld. Time was critical. In the volatile realm of digital currency, stolen funds can be laundered through countless transactions within hours. Yet TECHY FORCE CYBER RETRIEVAL pursued each lead with relentless precision. They provided regular progress updates and explained how every crypto transaction leaves a digital footprint, no matter how deeply it is buried. Weeks later, the breakthrough came. The fraudulent account was located, the illicit funds were flagged, and to my immense relief, I recovered 100% of my stolen money. This left me with an invaluable lesson. The internet is full of opportunists who prey on trust and desperation. But should you ever fall victim, know that there are experts like TECHY FORCE CYBER RETRIEVAL who can help you reclaim what is rightfully yours. MAil support (@) techyforcecyberretrieval (.) com WhatsApp +1 561 726 369 7 Website ht tp s : / / techy force cyber retrieval . com
27.08.25 17:24 wendytaylor015

My name is Wendy Taylor, I'm from Los Angeles, i want to announce to you Viewer how Capital Crypto Recover help me to restore my Lost Bitcoin, I invested with a Crypto broker without proper research to know what I was hoarding my hard-earned money into scammers, i lost access to my crypto wallet or had your funds stolen? Don’t worry Capital Crypto Recover is here to help you recover your cryptocurrency with cutting-edge technical expertise, With years of experience in the crypto world, Capital Crypto Recover employs the best latest tools and ethical hacking techniques to help you recover lost assets, unlock hacked accounts, Whether it’s a forgotten password, Capital Crypto Recover has the expertise to help you get your crypto back. a security company service that has a 100% success rate in the recovery of crypto assets, i lost wallet and hacked accounts. I provided them the information they requested and they began their investigation. To my surprise, Capital Crypto Recover was able to trace and recover my crypto assets successfully within 24hours. Thank you for your service in helping me recover my $647,734 worth of crypto funds and I highly recommend their recovery services, they are reliable and a trusted company to any individuals looking to recover lost money. Contact email [email protected] OR Telegram @Capitalcryptorecover Call/Text Number +1 (336)390-6684 his contact: [email protected] His website: https://recovercapital.wixsite.com/capital-crypto-rec-1
27.08.25 17:24 wendytaylor015

My name is Wendy Taylor, I'm from Los Angeles, i want to announce to you Viewer how Capital Crypto Recover help me to restore my Lost Bitcoin, I invested with a Crypto broker without proper research to know what I was hoarding my hard-earned money into scammers, i lost access to my crypto wallet or had your funds stolen? Don’t worry Capital Crypto Recover is here to help you recover your cryptocurrency with cutting-edge technical expertise, With years of experience in the crypto world, Capital Crypto Recover employs the best latest tools and ethical hacking techniques to help you recover lost assets, unlock hacked accounts, Whether it’s a forgotten password, Capital Crypto Recover has the expertise to help you get your crypto back. a security company service that has a 100% success rate in the recovery of crypto assets, i lost wallet and hacked accounts. I provided them the information they requested and they began their investigation. To my surprise, Capital Crypto Recover was able to trace and recover my crypto assets successfully within 24hours. Thank you for your service in helping me recover my $647,734 worth of crypto funds and I highly recommend their recovery services, they are reliable and a trusted company to any individuals looking to recover lost money. Contact email [email protected] OR Telegram @Capitalcryptorecover Call/Text Number +1 (336)390-6684 his contact: [email protected] His website: https://recovercapital.wixsite.com/capital-crypto-rec-1
27.08.25 17:24 wendytaylor015

My name is Wendy Taylor, I'm from Los Angeles, i want to announce to you Viewer how Capital Crypto Recover help me to restore my Lost Bitcoin, I invested with a Crypto broker without proper research to know what I was hoarding my hard-earned money into scammers, i lost access to my crypto wallet or had your funds stolen? Don’t worry Capital Crypto Recover is here to help you recover your cryptocurrency with cutting-edge technical expertise, With years of experience in the crypto world, Capital Crypto Recover employs the best latest tools and ethical hacking techniques to help you recover lost assets, unlock hacked accounts, Whether it’s a forgotten password, Capital Crypto Recover has the expertise to help you get your crypto back. a security company service that has a 100% success rate in the recovery of crypto assets, i lost wallet and hacked accounts. I provided them the information they requested and they began their investigation. To my surprise, Capital Crypto Recover was able to trace and recover my crypto assets successfully within 24hours. Thank you for your service in helping me recover my $647,734 worth of crypto funds and I highly recommend their recovery services, they are reliable and a trusted company to any individuals looking to recover lost money. Contact email [email protected] OR Telegram @Capitalcryptorecover Call/Text Number +1 (336)390-6684 his contact: [email protected] His website: https://recovercapital.wixsite.com/capital-crypto-rec-1
27.08.25 20:27 maximeden68

Top Crypto Bitcoin Wallet Recovery Services Contact iFORCE HACKER RECOVERY I wish to express my sincere gratitude to iForce Hacker Recovery for their exceptional service in recovering my lost bitcoin. Their expertise and professionalism were evident throughout the process, providing me with reassurance during a challenging time. The team's dedication to client satisfaction and their effective strategies were instrumental in resolving my issue. I highly recommend their services to anyone facing similar challenges. Thank you, iForce Hacker Recovery. Contact iForce Hacker Recovery: Email: iforcehk @ consultant. com WhatsApp: +1 (240) 803-3706 Website: iforcehackers. com/
27.08.25 21:21 dbeverly

One morning I received an email that looked like it came from my cryptocurrency exchange, OKX. The subject warned: Suspicious activity detected and urged me to log in via a link. The message seemed authentic — perfect logos, formatting, even the sender’s address. My pulse spiked and I clicked. The site that opened was indistinguishable from OKX’s homepage. Convinced it was real, I entered my username and password. Moments later, dread set in: my account was emptied. $12,500 in Bitcoin was gone. I called OKX, hoping for a reversal, but the representative explained transfers were irreversible. Desperate, I turned to GRAYWARE TECH SERVICE. From the first call, their professionalism was clear. They carefully examined my case, tracing the stolen Bitcoin across the blockchain. They identified the destination wallets and mapped every move the hackers made. Their regular updates turned fear into cautious optimism. Acting swiftly, GRAYWARE coordinated with exchanges along the path of the stolen funds, freezing assets before they could vanish. Their vigilance and expertise kept me informed and reassured. After two tense weeks, the breakthrough came: my full $12,500 was recovered and returned to my OKX account. What began as a nightmare became a success thanks to GRAYWARE TECH SERVICE’s relentless determination. They didn’t just recover my money — they restored my confidence in digital security. Contact: WhatsApp: +1 858 275 9508 Website: https://graywaretechservice.com Email: [email protected]
28.08.25 10:08 wendytaylor015

My name is Wendy Taylor, I'm from Los Angeles, i want to announce to you Viewer how Capital Crypto Recover help me to restore my Lost Bitcoin, I invested with a Crypto broker without proper research to know what I was hoarding my hard-earned money into scammers, i lost access to my crypto wallet or had your funds stolen? Don’t worry Capital Crypto Recover is here to help you recover your cryptocurrency with cutting-edge technical expertise, With years of experience in the crypto world, Capital Crypto Recover employs the best latest tools and ethical hacking techniques to help you recover lost assets, unlock hacked accounts, Whether it’s a forgotten password, Capital Crypto Recover has the expertise to help you get your crypto back. a security company service that has a 100% success rate in the recovery of crypto assets, i lost wallet and hacked accounts. I provided them the information they requested and they began their investigation. To my surprise, Capital Crypto Recover was able to trace and recover my crypto assets successfully within 24hours. Thank you for your service in helping me recover my $647,734 worth of crypto funds and I highly recommend their recovery services, they are reliable and a trusted company to any individuals looking to recover lost money. Contact email [email protected] OR Telegram @Capitalcryptorecover Call/Text Number +1 (336)390-6684 his contact: [email protected] His website: https://recovercapital.wixsite.com/capital-crypto-rec-1
28.08.25 10:08 wendytaylor015

My name is Wendy Taylor, I'm from Los Angeles, i want to announce to you Viewer how Capital Crypto Recover help me to restore my Lost Bitcoin, I invested with a Crypto broker without proper research to know what I was hoarding my hard-earned money into scammers, i lost access to my crypto wallet or had your funds stolen? Don’t worry Capital Crypto Recover is here to help you recover your cryptocurrency with cutting-edge technical expertise, With years of experience in the crypto world, Capital Crypto Recover employs the best latest tools and ethical hacking techniques to help you recover lost assets, unlock hacked accounts, Whether it’s a forgotten password, Capital Crypto Recover has the expertise to help you get your crypto back. a security company service that has a 100% success rate in the recovery of crypto assets, i lost wallet and hacked accounts. I provided them the information they requested and they began their investigation. To my surprise, Capital Crypto Recover was able to trace and recover my crypto assets successfully within 24hours. Thank you for your service in helping me recover my $647,734 worth of crypto funds and I highly recommend their recovery services, they are reliable and a trusted company to any individuals looking to recover lost money. Contact email [email protected] OR Telegram @Capitalcryptorecover Call/Text Number +1 (336)390-6684 his contact: [email protected] His website: https://recovercapital.wixsite.com/capital-crypto-rec-1
28.08.25 10:08 wendytaylor015

My name is Wendy Taylor, I'm from Los Angeles, i want to announce to you Viewer how Capital Crypto Recover help me to restore my Lost Bitcoin, I invested with a Crypto broker without proper research to know what I was hoarding my hard-earned money into scammers, i lost access to my crypto wallet or had your funds stolen? Don’t worry Capital Crypto Recover is here to help you recover your cryptocurrency with cutting-edge technical expertise, With years of experience in the crypto world, Capital Crypto Recover employs the best latest tools and ethical hacking techniques to help you recover lost assets, unlock hacked accounts, Whether it’s a forgotten password, Capital Crypto Recover has the expertise to help you get your crypto back. a security company service that has a 100% success rate in the recovery of crypto assets, i lost wallet and hacked accounts. I provided them the information they requested and they began their investigation. To my surprise, Capital Crypto Recover was able to trace and recover my crypto assets successfully within 24hours. Thank you for your service in helping me recover my $647,734 worth of crypto funds and I highly recommend their recovery services, they are reliable and a trusted company to any individuals looking to recover lost money. Contact email [email protected] OR Telegram @Capitalcryptorecover Call/Text Number +1 (336)390-6684 his contact: [email protected] His website: https://recovercapital.wixsite.com/capital-crypto-rec-1
28.08.25 13:50 Maureensels

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28.08.25 13:50 Maureensels

STEPS TO RECOVERING MONEY FROM A CRYPTO INVESTMENT SCAM CONTACT WIZARD LARRY RECOVERY EXPERTS Are you trying to find a means to get your other cryptocurrency, including Bitcoin, back from scammers? Stop searching! Call or text +1 (616) - (292) - (4789) or send a mail to (WIZARDLARRY(@)MAIL. C OM) is the greatest Scammed Crypto Recovery Agency I've seen on Google Earth. I'm delighted I made the brave decision to get in touch with them. In a matter of hours, all of the cryptocurrency that I had stolen was returned to my Coin Base wallet. For assistance with their strategies, I will advise you all to get in touch with their customer service. You will thank me later. Website... Wizardlarryrecovery. com Email [email protected] o m
28.08.25 16:31 wendytaylor015

My name is Wendy Taylor, I'm from Los Angeles, i want to announce to you Viewer how Capital Crypto Recover help me to restore my Lost Bitcoin, I invested with a Crypto broker without proper research to know what I was hoarding my hard-earned money into scammers, i lost access to my crypto wallet or had your funds stolen? Don’t worry Capital Crypto Recover is here to help you recover your cryptocurrency with cutting-edge technical expertise, With years of experience in the crypto world, Capital Crypto Recover employs the best latest tools and ethical hacking techniques to help you recover lost assets, unlock hacked accounts, Whether it’s a forgotten password, Capital Crypto Recover has the expertise to help you get your crypto back. a security company service that has a 100% success rate in the recovery of crypto assets, i lost wallet and hacked accounts. I provided them the information they requested and they began their investigation. To my surprise, Capital Crypto Recover was able to trace and recover my crypto assets successfully within 24hours. Thank you for your service in helping me recover my $647,734 worth of crypto funds and I highly recommend their recovery services, they are reliable and a trusted company to any individuals looking to recover lost money. Contact email [email protected] OR Telegram @Capitalcryptorecover Call/Text Number +1 (336)390-6684 his contact: [email protected] His website: https://recovercapital.wixsite.com/capital-crypto-rec-1
28.08.25 16:31 wendytaylor015

My name is Wendy Taylor, I'm from Los Angeles, i want to announce to you Viewer how Capital Crypto Recover help me to restore my Lost Bitcoin, I invested with a Crypto broker without proper research to know what I was hoarding my hard-earned money into scammers, i lost access to my crypto wallet or had your funds stolen? Don’t worry Capital Crypto Recover is here to help you recover your cryptocurrency with cutting-edge technical expertise, With years of experience in the crypto world, Capital Crypto Recover employs the best latest tools and ethical hacking techniques to help you recover lost assets, unlock hacked accounts, Whether it’s a forgotten password, Capital Crypto Recover has the expertise to help you get your crypto back. a security company service that has a 100% success rate in the recovery of crypto assets, i lost wallet and hacked accounts. I provided them the information they requested and they began their investigation. To my surprise, Capital Crypto Recover was able to trace and recover my crypto assets successfully within 24hours. Thank you for your service in helping me recover my $647,734 worth of crypto funds and I highly recommend their recovery services, they are reliable and a trusted company to any individuals looking to recover lost money. Contact email [email protected] OR Telegram @Capitalcryptorecover Call/Text Number +1 (336)390-6684 his contact: [email protected] His website: https://recovercapital.wixsite.com/capital-crypto-rec-1
28.08.25 19:17 leticiazavala

Earlier this year, I found what looked like an unbeatable offer on first-class airline tickets through an online travel agency. They advertised heavily discounted fares for international travel for only 0.11 BTC, which was worth around 7300 USD at the time. The only condition was that payment had to be made in Bitcoin. The website looked professional, had convincing reviews, and even offered live chat support. After some hesitation, I decided to go ahead with the booking. Two days later, I received an email from the airline saying my reservation had been canceled. I went back to the agency’s website only to find it had disappeared. Their customer support was gone, and their social media accounts had vanished. That was when I realized I had been scammed. I thought my Bitcoin was lost for good. Unlike a credit card transaction, there was no way to reverse it. A friend recommended a company called Techy Force Cyber Retrieval that specializes in tracking and recovering stolen crypto. I reached out not expecting much. To my surprise, their team responded quickly. I sent them the transaction details, and they immediately began tracing the movement of the 0.11 BTC. They followed the funds through multiple wallets and discovered the Bitcoin had ended up at a large exchange known for cooperating in fraud investigations. Techy Force Cyber Retrieval worked fast. They submitted the evidence to the exchange and requested that the wallet be frozen. Over the next few weeks, they handled all communication and submitted the necessary documentation. Just under three weeks later, the exchange approved the release of my funds. I was stunned when I saw the 0.11 BTC back in my wallet. I had lost hope, but Techy Force Cyber Retrieval delivered. They were professional, honest, and transparent throughout the entire process. If you have been scammed in a crypto deal, do not assume the money is gone forever. With the right help, it is possible to trace and recover stolen assets. Techy Force Cyber Retrieval gave me a second chance, and I would not hesitate to recommend them to anyone who finds themselves in a similar situation. FOR SERVICE HELP Email: [email protected] [email protected] CALL +15617263697
29.08.25 09:20 marcushenderson624

Bitcoin Recovery Testimonial After falling victim to a cryptocurrency scam group, I lost $354,000 worth of USDT. I thought all hope was lost from the experience of losing my hard-earned money to scammers. I was devastated and believed there was no way to recover my funds. Fortunately, I started searching for help to recover my stolen funds and I came across a lot of testimonials online about Capital Crypto Recovery, an agent who helps in recovery of lost bitcoin funds, I contacted Capital Crypto Recover Service, and with their expertise, they successfully traced and recovered my stolen assets. Their team was professional, kept me updated throughout the process, and demonstrated a deep understanding of blockchain transactions and recovery protocols. They are trusted and very reliable with a 100% successful rate record Recovery bitcoin, I’m grateful for their help and highly recommend their services to anyone seeking assistance with lost crypto. Contact: [email protected] Phone CALL/Text Number: +1 (336) 390-6684 Email: [email protected] Website: https://recovercapital.wixsite.com/capital-crypto-rec-1
29.08.25 09:20 marcushenderson624

Bitcoin Recovery Testimonial After falling victim to a cryptocurrency scam group, I lost $354,000 worth of USDT. I thought all hope was lost from the experience of losing my hard-earned money to scammers. I was devastated and believed there was no way to recover my funds. Fortunately, I started searching for help to recover my stolen funds and I came across a lot of testimonials online about Capital Crypto Recovery, an agent who helps in recovery of lost bitcoin funds, I contacted Capital Crypto Recover Service, and with their expertise, they successfully traced and recovered my stolen assets. Their team was professional, kept me updated throughout the process, and demonstrated a deep understanding of blockchain transactions and recovery protocols. They are trusted and very reliable with a 100% successful rate record Recovery bitcoin, I’m grateful for their help and highly recommend their services to anyone seeking assistance with lost crypto. Contact: [email protected] Phone CALL/Text Number: +1 (336) 390-6684 Email: [email protected] Website: https://recovercapital.wixsite.com/capital-crypto-rec-1
29.08.25 09:20 marcushenderson624

Bitcoin Recovery Testimonial After falling victim to a cryptocurrency scam group, I lost $354,000 worth of USDT. I thought all hope was lost from the experience of losing my hard-earned money to scammers. I was devastated and believed there was no way to recover my funds. Fortunately, I started searching for help to recover my stolen funds and I came across a lot of testimonials online about Capital Crypto Recovery, an agent who helps in recovery of lost bitcoin funds, I contacted Capital Crypto Recover Service, and with their expertise, they successfully traced and recovered my stolen assets. Their team was professional, kept me updated throughout the process, and demonstrated a deep understanding of blockchain transactions and recovery protocols. They are trusted and very reliable with a 100% successful rate record Recovery bitcoin, I’m grateful for their help and highly recommend their services to anyone seeking assistance with lost crypto. Contact: [email protected] Phone CALL/Text Number: +1 (336) 390-6684 Email: [email protected] Website: https://recovercapital.wixsite.com/capital-crypto-rec-1
30.08.25 01:40 Moises Velez

In a single transaction, my Coinbase wallet was destroyed. I was at a loss as to what to do and was unable to contact Coinbase support Teams. This transaction resulted in the complete depletion of my Ethereum and Bitcoin. I was not the individual in question, and I required assistance after losing approximately $821,000. I visited the police station to submit a report, but they were also unable to provide any assistance. A gentleman approached me at the stiction and presented me with a note that included an email address (recoveryhacker101[at]gmail[dot]com). He informed me that this recovery company could be of assistance. Upon contacting them and submitting all necessary documentation within 72 hours, I received a letter stating that she had successfully recovered all of my funds and identified the individuals responsible for the criminal activity. I would like to inform everyone that it is feasible to retrieve their funds provided that they correspond with the appropriate individual and adhere to the prescribed procedures.
31.08.25 14:19 jack118

In 2018, Marie assisted in the recovery of stolen bitcoin. The service was excellent. The coins were stolen by hackers, creating a nightmare. Marie was suggested by a friend. Contact her at [email protected] and @MARIE_CONSULTANCY on Telegram or WhatsApp at +1 7127594675. She was responsible for recovering Bitcoin. The trading goes on.
31.08.25 23:14 robertalfred175

CRYPTO SCAM RECOVERY SUCCESSFUL – A TESTIMONIAL OF LOST PASSWORD TO YOUR DIGITAL WALLET BACK. My name is Robert Alfred, Am from Australia. I’m sharing my experience in the hope that it helps others who have been victims of crypto scams. A few months ago, I fell victim to a fraudulent crypto investment scheme linked to a broker company. I had invested heavily during a time when Bitcoin prices were rising, thinking it was a good opportunity. Unfortunately, I was scammed out of $120,000 AUD and the broker denied me access to my digital wallet and assets. It was a devastating experience that caused many sleepless nights. Crypto scams are increasingly common and often involve fake trading platforms, phishing attacks, and misleading investment opportunities. In my desperation, a friend from the crypto community recommended Capital Crypto Recovery Service, known for helping victims recover lost or stolen funds. After doing some research and reading multiple positive reviews, I reached out to Capital Crypto Recovery. I provided all the necessary information—wallet addresses, transaction history, and communication logs. Their expert team responded immediately and began investigating. Using advanced blockchain tracking techniques, they were able to trace the stolen Dogecoin, identify the scammer’s wallet, and coordinate with relevant authorities to freeze the funds before they could be moved. Incredibly, within 24 hours, Capital Crypto Recovery successfully recovered the majority of my stolen crypto assets. I was beyond relieved and truly grateful. Their professionalism, transparency, and constant communication throughout the process gave me hope during a very difficult time. If you’ve been a victim of a crypto scam, I highly recommend them with full confidence contacting: Email: [email protected] Telegram: @Capitalcryptorecover Contact: [email protected] Call/Text: +1 (336) 390-6684 Website: https://recovercapital.wixsite.com/capital-crypto-rec-1
31.08.25 23:14 robertalfred175

CRYPTO SCAM RECOVERY SUCCESSFUL – A TESTIMONIAL OF LOST PASSWORD TO YOUR DIGITAL WALLET BACK. My name is Robert Alfred, Am from Australia. I’m sharing my experience in the hope that it helps others who have been victims of crypto scams. A few months ago, I fell victim to a fraudulent crypto investment scheme linked to a broker company. I had invested heavily during a time when Bitcoin prices were rising, thinking it was a good opportunity. Unfortunately, I was scammed out of $120,000 AUD and the broker denied me access to my digital wallet and assets. It was a devastating experience that caused many sleepless nights. Crypto scams are increasingly common and often involve fake trading platforms, phishing attacks, and misleading investment opportunities. In my desperation, a friend from the crypto community recommended Capital Crypto Recovery Service, known for helping victims recover lost or stolen funds. After doing some research and reading multiple positive reviews, I reached out to Capital Crypto Recovery. I provided all the necessary information—wallet addresses, transaction history, and communication logs. Their expert team responded immediately and began investigating. Using advanced blockchain tracking techniques, they were able to trace the stolen Dogecoin, identify the scammer’s wallet, and coordinate with relevant authorities to freeze the funds before they could be moved. Incredibly, within 24 hours, Capital Crypto Recovery successfully recovered the majority of my stolen crypto assets. I was beyond relieved and truly grateful. Their professionalism, transparency, and constant communication throughout the process gave me hope during a very difficult time. If you’ve been a victim of a crypto scam, I highly recommend them with full confidence contacting: Email: [email protected] Telegram: @Capitalcryptorecover Contact: [email protected] Call/Text: +1 (336) 390-6684 Website: https://recovercapital.wixsite.com/capital-crypto-rec-1
31.08.25 23:14 robertalfred175

CRYPTO SCAM RECOVERY SUCCESSFUL – A TESTIMONIAL OF LOST PASSWORD TO YOUR DIGITAL WALLET BACK. My name is Robert Alfred, Am from Australia. I’m sharing my experience in the hope that it helps others who have been victims of crypto scams. A few months ago, I fell victim to a fraudulent crypto investment scheme linked to a broker company. I had invested heavily during a time when Bitcoin prices were rising, thinking it was a good opportunity. Unfortunately, I was scammed out of $120,000 AUD and the broker denied me access to my digital wallet and assets. It was a devastating experience that caused many sleepless nights. Crypto scams are increasingly common and often involve fake trading platforms, phishing attacks, and misleading investment opportunities. In my desperation, a friend from the crypto community recommended Capital Crypto Recovery Service, known for helping victims recover lost or stolen funds. After doing some research and reading multiple positive reviews, I reached out to Capital Crypto Recovery. I provided all the necessary information—wallet addresses, transaction history, and communication logs. Their expert team responded immediately and began investigating. Using advanced blockchain tracking techniques, they were able to trace the stolen Dogecoin, identify the scammer’s wallet, and coordinate with relevant authorities to freeze the funds before they could be moved. Incredibly, within 24 hours, Capital Crypto Recovery successfully recovered the majority of my stolen crypto assets. I was beyond relieved and truly grateful. Their professionalism, transparency, and constant communication throughout the process gave me hope during a very difficult time. If you’ve been a victim of a crypto scam, I highly recommend them with full confidence contacting: Email: [email protected] Telegram: @Capitalcryptorecover Contact: [email protected] Call/Text: +1 (336) 390-6684 Website: https://recovercapital.wixsite.com/capital-crypto-rec-1
01.09.25 04:55 Maurice Cherry

Through Hackrontech recovery, Bitcoin scam victims can retrieve their money. I recommed hackrontech to anyone who has fallen victim to a scam and has been looking for methods and techniques to recover their lost cryptocurrency or wallets. Hackrontech Recovery is a reliable cryptocurrency recovery firm that assists victims in recovering their stolen cryptocurrency and offers secure solutions to protect your wallets from online scammers. I must admit that I was deeply melancholy and had given up on life until these experts could restore my 4.1 BTC to my wallet. If you’ve lost your cryptocurrency and you are helpless about it, contact hackrontech Recovery to get your money back. Email : hackrontech @gmail.com One key aspect that makes hackrontech Recovery stand out is its focus on providing secure solutions to protect wallets from online scammers. It’s not just about recovering lost funds; it’s also about preventing future incidents and ensuring that clients’ digital assets are safeguarded against potential threats. This proactive approach demonstrates their commitment to the long-term financial security of their clients. Furthermore, for individuals who have lost their cryptocurrency and are feeling helpless, reaching out to hackrontech Recovery could be a turning point in their situation. The reassurance that they are legitimate for seeking help and recovering lost funds can provide much-needed relief and a sense of empowerment. hackrontech Recovery as a reliable cryptocurrency recovery firm is certainly well-founded. Their ability to assist scam victims in recovering stolen cryptocurrency, their focus on providing secure solutions, and their commitment to supporting clients through challenging situations make them a valuable resource for individuals navigating the complex world of digital currencies. If you or someone you know has fallen victim to a cryptocurrency scam, contacting hackrontech Recovery could be the first step towards reclaiming lost funds and regaining peace of mind. Contact info; email: hackrontech @gmail.com
01.09.25 11:06 robertalfred175

CRYPTO SCAM RECOVERY SUCCESSFUL – A TESTIMONIAL OF LOST PASSWORD TO YOUR DIGITAL WALLET BACK. My name is Robert Alfred, Am from Australia. I’m sharing my experience in the hope that it helps others who have been victims of crypto scams. A few months ago, I fell victim to a fraudulent crypto investment scheme linked to a broker company. I had invested heavily during a time when Bitcoin prices were rising, thinking it was a good opportunity. Unfortunately, I was scammed out of $120,000 AUD and the broker denied me access to my digital wallet and assets. It was a devastating experience that caused many sleepless nights. Crypto scams are increasingly common and often involve fake trading platforms, phishing attacks, and misleading investment opportunities. In my desperation, a friend from the crypto community recommended Capital Crypto Recovery Service, known for helping victims recover lost or stolen funds. After doing some research and reading multiple positive reviews, I reached out to Capital Crypto Recovery. I provided all the necessary information—wallet addresses, transaction history, and communication logs. Their expert team responded immediately and began investigating. Using advanced blockchain tracking techniques, they were able to trace the stolen Dogecoin, identify the scammer’s wallet, and coordinate with relevant authorities to freeze the funds before they could be moved. Incredibly, within 24 hours, Capital Crypto Recovery successfully recovered the majority of my stolen crypto assets. I was beyond relieved and truly grateful. Their professionalism, transparency, and constant communication throughout the process gave me hope during a very difficult time. If you’ve been a victim of a crypto scam, I highly recommend them with full confidence contacting: Email: [email protected] Telegram: @Capitalcryptorecover Contact: [email protected] Call/Text: +1 (336) 390-6684 Website: https://recovercapital.wixsite.com/capital-crypto-rec-1
01.09.25 11:07 robertalfred175

CRYPTO SCAM RECOVERY SUCCESSFUL – A TESTIMONIAL OF LOST PASSWORD TO YOUR DIGITAL WALLET BACK. My name is Robert Alfred, Am from Australia. I’m sharing my experience in the hope that it helps others who have been victims of crypto scams. A few months ago, I fell victim to a fraudulent crypto investment scheme linked to a broker company. I had invested heavily during a time when Bitcoin prices were rising, thinking it was a good opportunity. Unfortunately, I was scammed out of $120,000 AUD and the broker denied me access to my digital wallet and assets. It was a devastating experience that caused many sleepless nights. Crypto scams are increasingly common and often involve fake trading platforms, phishing attacks, and misleading investment opportunities. In my desperation, a friend from the crypto community recommended Capital Crypto Recovery Service, known for helping victims recover lost or stolen funds. After doing some research and reading multiple positive reviews, I reached out to Capital Crypto Recovery. I provided all the necessary information—wallet addresses, transaction history, and communication logs. Their expert team responded immediately and began investigating. Using advanced blockchain tracking techniques, they were able to trace the stolen Dogecoin, identify the scammer’s wallet, and coordinate with relevant authorities to freeze the funds before they could be moved. Incredibly, within 24 hours, Capital Crypto Recovery successfully recovered the majority of my stolen crypto assets. I was beyond relieved and truly grateful. Their professionalism, transparency, and constant communication throughout the process gave me hope during a very difficult time. If you’ve been a victim of a crypto scam, I highly recommend them with full confidence contacting: Email: [email protected] Telegram: @Capitalcryptorecover Contact: [email protected] Call/Text: +1 (336) 390-6684 Website: https://recovercapital.wixsite.com/capital-crypto-rec-1
01.09.25 16:41 Anitastar

Wizard Hilton Cyber Tech's strategic approach to Bitcoin recovery represents a mutually beneficial collaboration that offers valuable advantages for all involved. At the core of this innovative partnership is a shared commitment to leveraging cutting-edge technology and expertise to tackle the increasingly complex challenges of cryptocurrency theft and loss. By combining Wizard Hilton's world-class cybersecurity capabilities with the deep industry insights and recovery methodologies of Cyber Tech, this alliance is poised to revolutionize the way individuals and organizations safeguard their digital assets. Through a meticulous, multi-layered process, the team meticulously analyzes each case, employing advanced forensic techniques to trace the flow of stolen funds and identify potential recovery avenues. This rigorous approach not only maximizes the chances of successful Bitcoin retrieval, but also provides invaluable intelligence to enhance future security measures and prevention strategies. Importantly, the collaboration is built on a foundation of trust, transparency, and a genuine concern for the well-being of clients, ensuring that the recovery process is handled with the utmost care and discretion. As the cryptocurrency landscape continues to evolve, this strategic alliance between Wizard Hilton Cyber Tech stands as a shining example of how industry leaders can come together to safeguard digital assets, protect victims of cybercrime, and pave the way for a more secure and resilient cryptocurrency ecosystem. Email : wizardhiltoncybertech ( @ ) gmail (. ) com OR support ( @ ) wizardhiltoncybertech (.) com WhatsApp number +18737715701 Thanks.
01.09.25 17:03 WILLER GOODY

Don’t be misled by the countless stories you come across online many of them are unreliable and can easily steer you in the wrong direction. I personally tried several so-called recovery services, only to end up frustrated and disappointed every single time. But everything changed when I finally connected with a true tech professional who genuinely knows what they’re doing. Instead of wasting time and energy with unskilled hackers who make empty promises, it’s far better to work with a proven expert who can actually recover your stolen or lost cryptocurrency, including Bitcoins. [email protected] is, without a doubt, one of the most trustworthy and skilled blockchain specialists I’ve ever come across. If you’ve lost money to scammers, don’t give up. Reach out to their email today and take the first step toward getting your coins back. You can also contact them at +44 736-644-5035.
02.09.25 23:12 [email protected]

Sylvester Bryant has recovered over £151,000 in lost cryptocurrency. You can reach him for assistance via email at Yt7cracker@gmail. com. For direct communication, Sylvester Bryant is available on WhatsApp at +1 512 577 7957. This service proved invaluable when other recovery methods had failed. The money was recovered quickly and effectively. My financial situation was dire prior to this. This recovery has restored my financial stability. This reliable support was crucial during a difficult time. Mr. Bryant's skill in retrieving lost funds is remarkable.
03.09.25 07:17 Fabian3

Scammers have ruined the forex trading market, they have deceived many people with their fake promises on high returns. I learnt my lesson the hard way. I only pity people who still consider investing with them. Please try and make your researches, you will definitely come across better and reliable forex trading agencies that would help you yield profit and not ripping off your money. Also, if your money has been ripped-off by these scammers, you can report to a regulated crypto investigative unit who make use of software to get money back. They helped me recover my scammed funds back to me If you encounter with some issue make sure you contact ([email protected]) they're recovery expert and a very effective and reliable .
04.09.25 02:11 marcushenderson624

Bitcoin Recovery Testimonial After falling victim to a cryptocurrency scam group, I lost $354,000 worth of USDT. I thought all hope was lost from the experience of losing my hard-earned money to scammers. I was devastated and believed there was no way to recover my funds. Fortunately, I started searching for help to recover my stolen funds and I came across a lot of testimonials online about Capital Crypto Recovery, an agent who helps in recovery of lost bitcoin funds, I contacted Capital Crypto Recover Service, and with their expertise, they successfully traced and recovered my stolen assets. Their team was professional, kept me updated throughout the process, and demonstrated a deep understanding of blockchain transactions and recovery protocols. They are trusted and very reliable with a 100% successful rate record Recovery bitcoin, I’m grateful for their help and highly recommend their services to anyone seeking assistance with lost crypto. Contact: [email protected] Phone CALL/Text Number: +1 (336) 390-6684 Email: [email protected] Website: https://recovercapital.wixsite.com/capital-crypto-rec-1
04.09.25 02:11 marcushenderson624

Bitcoin Recovery Testimonial After falling victim to a cryptocurrency scam group, I lost $354,000 worth of USDT. I thought all hope was lost from the experience of losing my hard-earned money to scammers. I was devastated and believed there was no way to recover my funds. Fortunately, I started searching for help to recover my stolen funds and I came across a lot of testimonials online about Capital Crypto Recovery, an agent who helps in recovery of lost bitcoin funds, I contacted Capital Crypto Recover Service, and with their expertise, they successfully traced and recovered my stolen assets. Their team was professional, kept me updated throughout the process, and demonstrated a deep understanding of blockchain transactions and recovery protocols. They are trusted and very reliable with a 100% successful rate record Recovery bitcoin, I’m grateful for their help and highly recommend their services to anyone seeking assistance with lost crypto. Contact: [email protected] Phone CALL/Text Number: +1 (336) 390-6684 Email: [email protected] Website: https://recovercapital.wixsite.com/capital-crypto-rec-1
04.09.25 02:45 marcushenderson624

Bitcoin Recovery Testimonial After falling victim to a cryptocurrency scam group, I lost $354,000 worth of USDT. I thought all hope was lost from the experience of losing my hard-earned money to scammers. I was devastated and believed there was no way to recover my funds. Fortunately, I started searching for help to recover my stolen funds and I came across a lot of testimonials online about Capital Crypto Recovery, an agent who helps in recovery of lost bitcoin funds, I contacted Capital Crypto Recover Service, and with their expertise, they successfully traced and recovered my stolen assets. Their team was professional, kept me updated throughout the process, and demonstrated a deep understanding of blockchain transactions and recovery protocols. They are trusted and very reliable with a 100% successful rate record Recovery bitcoin, I’m grateful for their help and highly recommend their services to anyone seeking assistance with lost crypto. Contact: [email protected] Phone CALL/Text Number: +1 (336) 390-6684 Email: [email protected] Website: https://recovercapital.wixsite.com/capital-crypto-rec-1
04.09.25 16:24 Melbourne

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04.09.25 18:11 WILLER GOODY

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05.09.25 08:14 Amostampari

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05.09.25 15:39 micheal1219

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06.09.25 09:21 Young Felix

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06.09.25 09:22 Young Felix

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06.09.25 18:52 michaeldavenport218

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06.09.25 18:52 michaeldavenport218

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08.09.25 14:26 bajo171

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08.09.25 18:04 marcushenderson624

Bitcoin Recovery Testimonial After falling victim to a cryptocurrency scam group, I lost $354,000 worth of USDT. I thought all hope was lost from the experience of losing my hard-earned money to scammers. I was devastated and believed there was no way to recover my funds. Fortunately, I started searching for help to recover my stolen funds and I came across a lot of testimonials online about Capital Crypto Recovery, an agent who helps in recovery of lost bitcoin funds, I contacted Capital Crypto Recover Service, and with their expertise, they successfully traced and recovered my stolen assets. Their team was professional, kept me updated throughout the process, and demonstrated a deep understanding of blockchain transactions and recovery protocols. They are trusted and very reliable with a 100% successful rate record Recovery bitcoin, I’m grateful for their help and highly recommend their services to anyone seeking assistance with lost crypto. Contact: [email protected] Phone CALL/Text Number: +1 (336) 390-6684 Email: [email protected] Website: https://recovercapital.wixsite.com/capital-crypto-rec-1
08.09.25 18:04 marcushenderson624

Bitcoin Recovery Testimonial After falling victim to a cryptocurrency scam group, I lost $354,000 worth of USDT. I thought all hope was lost from the experience of losing my hard-earned money to scammers. I was devastated and believed there was no way to recover my funds. Fortunately, I started searching for help to recover my stolen funds and I came across a lot of testimonials online about Capital Crypto Recovery, an agent who helps in recovery of lost bitcoin funds, I contacted Capital Crypto Recover Service, and with their expertise, they successfully traced and recovered my stolen assets. Their team was professional, kept me updated throughout the process, and demonstrated a deep understanding of blockchain transactions and recovery protocols. They are trusted and very reliable with a 100% successful rate record Recovery bitcoin, I’m grateful for their help and highly recommend their services to anyone seeking assistance with lost crypto. Contact: [email protected] Phone CALL/Text Number: +1 (336) 390-6684 Email: [email protected] Website: https://recovercapital.wixsite.com/capital-crypto-rec-1
08.09.25 19:21 DAVEortega1010

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09.09.25 02:34 wendytaylor015

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09.09.25 02:34 wendytaylor015

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09.09.25 02:34 wendytaylor015

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09.09.25 11:48 Moises Velez

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09.09.25 15:05 patricialovick86

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09.09.25 15:05 patricialovick86

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09.09.25 15:05 patricialovick86

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09.09.25 20:23 Bramfloris

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09.09.25 20:37 DAVEortega1010

Be wary of fraudulent cryptocurrency recovery services. Many promise to recover your lost digital currency. These services often provide excuses or fail to deliver any results. This only adds to your distress. After several unsuccessful attempts with others, I discovered a genuine expert. This individual was highly capable and provided excellent assistance. They successfully recovered my stolen cryptocurrency. I was finally able to find peace of mind. If you have lost crypto, you should contact them. talk to [email protected] for help. You can also send a text message via WhatsApp to +447366445035, Stop losing time on scams. They can help you get your money back.
10.09.25 00:45 robertalfred175

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10.09.25 00:45 robertalfred175

CRYPTO SCAM RECOVERY SUCCESSFUL – A TESTIMONIAL OF LOST PASSWORD TO YOUR DIGITAL WALLET BACK. My name is Robert Alfred, Am from Australia. I’m sharing my experience in the hope that it helps others who have been victims of crypto scams. A few months ago, I fell victim to a fraudulent crypto investment scheme linked to a broker company. I had invested heavily during a time when Bitcoin prices were rising, thinking it was a good opportunity. Unfortunately, I was scammed out of $120,000 AUD and the broker denied me access to my digital wallet and assets. It was a devastating experience that caused many sleepless nights. Crypto scams are increasingly common and often involve fake trading platforms, phishing attacks, and misleading investment opportunities. In my desperation, a friend from the crypto community recommended Capital Crypto Recovery Service, known for helping victims recover lost or stolen funds. After doing some research and reading multiple positive reviews, I reached out to Capital Crypto Recovery. I provided all the necessary information—wallet addresses, transaction history, and communication logs. Their expert team responded immediately and began investigating. Using advanced blockchain tracking techniques, they were able to trace the stolen Dogecoin, identify the scammer’s wallet, and coordinate with relevant authorities to freeze the funds before they could be moved. Incredibly, within 24 hours, Capital Crypto Recovery successfully recovered the majority of my stolen crypto assets. I was beyond relieved and truly grateful. Their professionalism, transparency, and constant communication throughout the process gave me hope during a very difficult time. If you’ve been a victim of a crypto scam, I highly recommend them with full confidence contacting: 📧 Email: [email protected] 📱 Telegram: @Capitalcryptorecover Contact: [email protected] 📞 Call/Text: +1 (336) 390-6684 🌐 Website: https://recovercapital.wixsite.com/capital-crypto-rec-1
11.09.25 03:31 DAVEortega1010

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11.09.25 13:44 grace

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11.09.25 15:03 stevensjonas

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11.09.25 18:51 patricialovick86

How To Recover Your Bitcoin Without Falling Victim To Scams: A Testimony Experience With Capital Crypto Recover Services, Contact Telegram: @Capitalcryptorecover Dear Everyone, I would like to take a moment to share my positive experience with Capital Crypto Recover Services. Initially, I was unsure if it would be possible to recover my stolen bitcoins. However, with their expertise and professionalism, I was able to fully recover my funds. Unfortunately, many individuals fall victim to scams in the cryptocurrency space, especially those involving fraudulent investment platforms. However, I advise caution, as not all recovery services are legitimate. I personally lost $273,000 worth of Bitcoin from my Binance account due to a deceptive platform. If you have suffered a similar loss, you may be considering crypto recovery, The Capital Crypto Recover is the most knowledgeable and effective Capital Crypto Recovery Services assisted me in recovering my stolen funds within 24 hours, after getting access to my wallet. Their service was not only prompt but also highly professional and effective, and many recovery services may not be trustworthy. Therefore, I highly recommend Capital Crypto Recover to you. i do always research and see reviews about their service, For assistance finding your misplaced cryptocurrency, get in touch with them, They do their jobs quickly and excellently, Stay safe and vigilant in the crypto world. Contact: [email protected] You can reach them via email at [email protected] OR Call/Text Number +1 (336)390-6684 his contact website: https://recovercapital.wixsite.com/capital-crypto-rec-1
11.09.25 18:51 patricialovick86

How To Recover Your Bitcoin Without Falling Victim To Scams: A Testimony Experience With Capital Crypto Recover Services, Contact Telegram: @Capitalcryptorecover Dear Everyone, I would like to take a moment to share my positive experience with Capital Crypto Recover Services. Initially, I was unsure if it would be possible to recover my stolen bitcoins. However, with their expertise and professionalism, I was able to fully recover my funds. Unfortunately, many individuals fall victim to scams in the cryptocurrency space, especially those involving fraudulent investment platforms. However, I advise caution, as not all recovery services are legitimate. I personally lost $273,000 worth of Bitcoin from my Binance account due to a deceptive platform. If you have suffered a similar loss, you may be considering crypto recovery, The Capital Crypto Recover is the most knowledgeable and effective Capital Crypto Recovery Services assisted me in recovering my stolen funds within 24 hours, after getting access to my wallet. Their service was not only prompt but also highly professional and effective, and many recovery services may not be trustworthy. Therefore, I highly recommend Capital Crypto Recover to you. i do always research and see reviews about their service, For assistance finding your misplaced cryptocurrency, get in touch with them, They do their jobs quickly and excellently, Stay safe and vigilant in the crypto world. Contact: [email protected] You can reach them via email at [email protected] OR Call/Text Number +1 (336)390-6684 his contact website: https://recovercapital.wixsite.com/capital-crypto-rec-1
11.09.25 18:51 patricialovick86

How To Recover Your Bitcoin Without Falling Victim To Scams: A Testimony Experience With Capital Crypto Recover Services, Contact Telegram: @Capitalcryptorecover Dear Everyone, I would like to take a moment to share my positive experience with Capital Crypto Recover Services. Initially, I was unsure if it would be possible to recover my stolen bitcoins. However, with their expertise and professionalism, I was able to fully recover my funds. Unfortunately, many individuals fall victim to scams in the cryptocurrency space, especially those involving fraudulent investment platforms. However, I advise caution, as not all recovery services are legitimate. I personally lost $273,000 worth of Bitcoin from my Binance account due to a deceptive platform. If you have suffered a similar loss, you may be considering crypto recovery, The Capital Crypto Recover is the most knowledgeable and effective Capital Crypto Recovery Services assisted me in recovering my stolen funds within 24 hours, after getting access to my wallet. Their service was not only prompt but also highly professional and effective, and many recovery services may not be trustworthy. Therefore, I highly recommend Capital Crypto Recover to you. i do always research and see reviews about their service, For assistance finding your misplaced cryptocurrency, get in touch with them, They do their jobs quickly and excellently, Stay safe and vigilant in the crypto world. Contact: [email protected] You can reach them via email at [email protected] OR Call/Text Number +1 (336)390-6684 his contact website: https://recovercapital.wixsite.com/capital-crypto-rec-1
12.09.25 13:11 grace

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12.09.25 15:06 michaeldavenport218

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12.09.25 15:06 michaeldavenport218

I was recently scammed out of $53,000 by a fraudulent Bitcoin investment scheme, which added significant stress to my already difficult health issues, as I was also facing cancer surgery expenses. Desperate to recover my funds, I spent hours researching and consulting other victims, which led me to discover the excellent reputation of Capital Crypto Recover, I came across a Google post It was only after spending many hours researching and asking other victims for advice that I discovered Capital Crypto Recovery’s stellar reputation. I decided to contact them because of their successful recovery record and encouraging client testimonials. I had no idea that this would be the pivotal moment in my fight against cryptocurrency theft. Thanks to their expert team, I was able to recover my lost cryptocurrency back. The process was intricate, but Capital Crypto Recovery's commitment to utilizing the latest technology ensured a successful outcome. I highly recommend their services to anyone who has fallen victim to cryptocurrency fraud. For assistance contact [email protected] and on Telegram OR Call Number +1 (336)390-6684 via email: [email protected] you can visit his website: https://recovercapital.wixsite.com/capital-crypto-rec-1