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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.

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14 июн. 2024 г.

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Writing this review is a joy. Marie has provided excellent service ever since I started working with her in early 2018. I was worried I wouldn't be able to get my coins back after they were stolen by hackers. I had no idea where to begin, therefore it was a nightmare for me. However, things became easier for me after my friend sent me to [email protected] and +1 7127594675 on WhatsApp. I'm happy that she was able to retrieve my bitcoin so that I could resume trading.
13.10.25 01:11 elizabethrush89

God bless Capital Crypto Recover Services for the marvelous work you did in my life, I have learned the hard way that even the most sensible investors can fall victim to scams. When my USD was stolen, for anyone who has fallen victim to one of the bitcoin binary investment scams that are currently ongoing, I felt betrayal and upset. But then I was reading a post on site when I saw a testimony of Wendy Taylor online who recommended that Capital Crypto Recovery has helped her recover scammed funds within 24 hours. after reaching out to this cyber security firm that was able to help me recover my stolen digital assets and bitcoin. I’m genuinely blown away by their amazing service and professionalism. I never imagined I’d be able to get my money back until I complained to Capital Crypto Recovery Services about my difficulties and gave all of the necessary paperwork. I was astounded that it took them 12 hours to reclaim my stolen money back. Without a doubt, my USDT assets were successfully recovered from the scam platform, Thank you so much Sir, I strongly recommend Capital Crypto Recover for any of your bitcoin recovery, digital funds recovery, hacking, and cybersecurity concerns. You reach them Call/Text Number +1 (336)390-6684 His Email: [email protected] Contact Telegram: @Capitalcryptorecover Via Contact: [email protected] His website: https://recovercapital.wixsite.com/capital-crypto-rec-1
13.10.25 01:11 elizabethrush89

God bless Capital Crypto Recover Services for the marvelous work you did in my life, I have learned the hard way that even the most sensible investors can fall victim to scams. When my USD was stolen, for anyone who has fallen victim to one of the bitcoin binary investment scams that are currently ongoing, I felt betrayal and upset. But then I was reading a post on site when I saw a testimony of Wendy Taylor online who recommended that Capital Crypto Recovery has helped her recover scammed funds within 24 hours. after reaching out to this cyber security firm that was able to help me recover my stolen digital assets and bitcoin. I’m genuinely blown away by their amazing service and professionalism. I never imagined I’d be able to get my money back until I complained to Capital Crypto Recovery Services about my difficulties and gave all of the necessary paperwork. I was astounded that it took them 12 hours to reclaim my stolen money back. Without a doubt, my USDT assets were successfully recovered from the scam platform, Thank you so much Sir, I strongly recommend Capital Crypto Recover for any of your bitcoin recovery, digital funds recovery, hacking, and cybersecurity concerns. You reach them Call/Text Number +1 (336)390-6684 His Email: [email protected] Contact Telegram: @Capitalcryptorecover Via Contact: [email protected] His website: https://recovercapital.wixsite.com/capital-crypto-rec-1
14.10.25 01:15 tyleradams

Hi. Please be wise, do not make the same mistake I had made in the past, I was a victim of bitcoin scam, I saw a glamorous review showering praises and marketing an investment firm, I reached out to them on what their contracts are, and I invested $28,000, which I was promised to get my first 15% profit in weeks, when it’s time to get my profits, I got to know the company was bogus, they kept asking me to invest more and I ran out of patience then requested to have my money back, they refused to answer nor refund my funds, not until a friend of mine introduced me to the NVIDIA TECH HACKERS, so I reached out and after tabling my complaints, they were swift to action and within 36 hours I got back my funds with the due profit. I couldn’t contain the joy in me. I urge you guys to reach out to NVIDIA TECH HACKERS on their email: [email protected]
14.10.25 08:46 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
14.10.25 08:46 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
14.10.25 08:46 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
15.10.25 18:07 crypto

Cryptocurrency's digital realm presents many opportunities, but it also conceals complex frauds. It is quite painful to lose your cryptocurrency to scam. You can feel harassed and lost as a result. If you have been the victim of a cryptocurrency scam, this guide explains what to do ASAP. Following these procedures will help you avoid further issues or get your money back. Communication with Marie ([email protected] and WhatsApp: +1 7127594675) can make all the difference.
15.10.25 21:52 harristhomas7376

"In the crypto world, this is great news I want to share. Last year, I fell victim to a scam disguised as a safe investment option. I have invested in crypto trading platforms for about 10yrs thinking I was ensuring myself a retirement income, only to find that all my assets were either frozen, I believed my assets were secure — until I discovered that my BTC funds had been frozen and withdrawals were impossible. It was a devastating moment when I realized I had been scammed, and I thought my Bitcoin was gone forever, Everything changed when a close friend recommended the Capital Crypto Recover Service. Their professionalism, expertise, and dedication enabled me to recover my lost Bitcoin funds back — more than €560.000 DEM to my BTC wallet. What once felt impossible became a reality thanks to their support. If you have lost Bitcoin through scams, hacking, failed withdrawals, or similar challenges, don’t lose hope. I strongly recommend Capital Crypto Recover Service to anyone seeking a reliable and effective solution for recovering any wallet assets. They have a proven track record of successful reputation in recovering lost password assets for their clients and can help you navigate the process of recovering your funds. Don’t let scammers get away with your hard-earned money – contact Email: [email protected] Phone CALL/Text Number: +1 (336) 390-6684 Contact: [email protected] Website: https://recovercapital.wixsite.com/capital-crypto-rec-1
15.10.25 21:52 harristhomas7376

"In the crypto world, this is great news I want to share. Last year, I fell victim to a scam disguised as a safe investment option. I have invested in crypto trading platforms for about 10yrs thinking I was ensuring myself a retirement income, only to find that all my assets were either frozen, I believed my assets were secure — until I discovered that my BTC funds had been frozen and withdrawals were impossible. It was a devastating moment when I realized I had been scammed, and I thought my Bitcoin was gone forever, Everything changed when a close friend recommended the Capital Crypto Recover Service. Their professionalism, expertise, and dedication enabled me to recover my lost Bitcoin funds back — more than €560.000 DEM to my BTC wallet. What once felt impossible became a reality thanks to their support. If you have lost Bitcoin through scams, hacking, failed withdrawals, or similar challenges, don’t lose hope. I strongly recommend Capital Crypto Recover Service to anyone seeking a reliable and effective solution for recovering any wallet assets. They have a proven track record of successful reputation in recovering lost password assets for their clients and can help you navigate the process of recovering your funds. Don’t let scammers get away with your hard-earned money – contact Email: [email protected] Phone CALL/Text Number: +1 (336) 390-6684 Contact: [email protected] Website: https://recovercapital.wixsite.com/capital-crypto-rec-1
17.10.25 20:17 tyleradams

As time passes, there are an increasing number of frauds involving Bitcoin and other cryptocurrencies. Although there are many individuals who advertise recovering money online, people should use caution in dealing, especially when money is involved. You can trust NVIDIA TECH HACKERS [[email protected]], I promise. They are the top internet recovery company, and as their names indicate, your money is reclaimed as soon as feasible. My bitcoin was successfully retrieved in large part thanks to NVIDIA TECH HACKERS. Ensure that you get top-notch service; NVIDIA TECH HACKERS provides evidence of its work; and payment is only made when the service has been completed to your satisfaction. Reach them via email: [email protected] on google mail
17.10.25 20:20 lindseyvonn

Have you gotten yourself involved in a cryptocurrency scam or any scam at all? If yes, know that you are not alone, there are a lot of people in this same situation. I'm a Health Worker and was a victim of a cryptocurrency scam that cost me a lot of money. This happened a few weeks ago, there’s only one solution which is to talk to the right people, if you don’t do this you will end up being really depressed. I was really devastated until went on LinkedIn one evening after my work hours and i saw lots of reviews popped up on my feed about [email protected], I sent an email to the team who came highly recommended - [email protected] I started seeing some hope for myself from the moment I sent them an email. The good part is they made the entire process stress free for me, i literally sat and waited for them to finish and I received what I lost in my wallet
17.10.25 20:22 richardcharles

I would recommend NVIDIA TECH HACKERS to anyone that needs this service. I decided to get into crypto investment and I ended up getting my crypto lost to an investor late last year. The guy who was supposed to be managing my account turned out to be a scammer all along. I invested 56,000 USD and at first, my reading and profit margins were looking good. I started getting worried when I couldn’t make withdrawals and realized that I’ve been scammed. I came across some of the testimonials that people said about NVIDIA TECH HACKERS and how helpful he has been in recovering their funds. I immediately contacted him in his mail at [email protected] so I can get his assistance. One week into the recovery process the funds were traced and recovered back from the scammer. I can't appreciate him enough for his professionalism.
17.10.25 20:23 stevekalfman

If you need a hacker for scam crypto recovery or mobile spy access remotely kindly reach out to [email protected] for quick response, I hired this hacker and he did a nice job. before NVIDIA TECH HACKERS, I met with different hacker's online which turns out to be scam, this NVIDIA TECH HACKERS case was different and he is the trusted hacker I can vote and refer.
17.10.25 21:42 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
17.10.25 21:42 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
17.10.25 21:42 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
21.10.25 08:39 debby131

Given how swiftly the cryptocurrency market moves, losing USDT may be a terrifying and upsetting experience. Whether you experienced a technical problem, a transaction error, or were the victim of fraud, it is important to understand the potential recovery routes. Marie can assist you in determining the specific actions you can take to attempt to regain your lost USDT when you need guidance and clarification. You can reach her via email at [email protected] and WhatsApp at +1 7127594675.
21.10.25 11:45 harristhomas7376

"In the crypto world, this is great news I want to share. Last year, I fell victim to a scam disguised as a safe investment option. I have invested in crypto trading platforms for about 10yrs thinking I was ensuring myself a retirement income, only to find that all my assets were either frozen, I believed my assets were secure — until I discovered that my BTC funds had been frozen and withdrawals were impossible. It was a devastating moment when I realized I had been scammed, and I thought my Bitcoin was gone forever, Everything changed when a close friend recommended the Capital Crypto Recover Service. Their professionalism, expertise, and dedication enabled me to recover my lost Bitcoin funds back — more than €560.000 DEM to my BTC wallet. What once felt impossible became a reality thanks to their support. If you have lost Bitcoin through scams, hacking, failed withdrawals, or similar challenges, don’t lose hope. I strongly recommend Capital Crypto Recover Service to anyone seeking a reliable and effective solution for recovering any wallet assets. They have a proven track record of successful reputation in recovering lost password assets for their clients and can help you navigate the process of recovering your funds. Don’t let scammers get away with your hard-earned money – contact Email: [email protected] Phone CALL/Text Number: +1 (336) 390-6684 Contact: [email protected] Website: https://recovercapital.wixsite.com/capital-crypto-rec-1
21.10.25 11:45 harristhomas7376

"In the crypto world, this is great news I want to share. Last year, I fell victim to a scam disguised as a safe investment option. I have invested in crypto trading platforms for about 10yrs thinking I was ensuring myself a retirement income, only to find that all my assets were either frozen, I believed my assets were secure — until I discovered that my BTC funds had been frozen and withdrawals were impossible. It was a devastating moment when I realized I had been scammed, and I thought my Bitcoin was gone forever, Everything changed when a close friend recommended the Capital Crypto Recover Service. Their professionalism, expertise, and dedication enabled me to recover my lost Bitcoin funds back — more than €560.000 DEM to my BTC wallet. What once felt impossible became a reality thanks to their support. If you have lost Bitcoin through scams, hacking, failed withdrawals, or similar challenges, don’t lose hope. I strongly recommend Capital Crypto Recover Service to anyone seeking a reliable and effective solution for recovering any wallet assets. They have a proven track record of successful reputation in recovering lost password assets for their clients and can help you navigate the process of recovering your funds. Don’t let scammers get away with your hard-earned money – contact Email: [email protected] Phone CALL/Text Number: +1 (336) 390-6684 Contact: [email protected] Website: https://recovercapital.wixsite.com/capital-crypto-rec-1
22.10.25 04:48 MATT PHILLIP

I never imagined I’d fall for a crypto romance scam but it happened. Over the course of a few months, I sent nearly $150,000 worth of Bitcoin to someone I genuinely believed I was building a future with. When they disappeared without a trace, I was left heartbroken, humiliated, and financially devastated. For a long time, I didn’t tell anyone. I felt ashamed. But eventually, while searching for answers, I came across a Reddit thread that mentioned Agent Jasmine Lopez. I reached out, not expecting much. To my surprise, she treated me with kindness, not judgment. She used advanced tools like blockchain forensics, IP tracing, and smart contract analysis and with persistence and legal support, she was able to recover nearly 85% of what I lost. I know not everyone gets that kind of outcome, but thanks to [email protected] WhatsApp at +44 736-644-5035, I’ve started to reclaim not just my assets, but my confidence and peace of mind. If you’re going through something similar, you’re not alone and there is hope.
22.10.25 07:36 donnacollier

HOW I RECOVERED MY BITCOIN FROM CRYPTO INVESTMENT SCAM HOW I RECOVERED MY BITCOIN FROM CRYPTO INVESTMENT SCAM Hello everyone l’m by name Donna collier I live in urbandale 3 weeks ago was the darkest days of my life, i invested my hard earned money the sum of $123,000 into a crypto currency platform I was introduced to by a friend I met online everything happen so fast I was promised 300% of return of investment when it was time for me to cash out my investment and profit the platform was down I was so diversitated confused and tried to end it all then I came across upswing Ai a renowned expert in crypto currency and digital assets recovery at first it seem impossible after taking up my case within the the next 48 hours they where able to track down those fraud stars and recover my money I can’t recommend them enough to any one facing likewise change I will recommend you contact upswing Ai on Contact Details: WhatsApp +,1,2,0,2,8,1,0,1,4,0,7 E m a i l @Upswing-ai.com Website https: // u psw ing-ai. com/ platform /
22.10.25 11:59 elizabethrush89

God bless Capital Crypto Recover Services for the marvelous work you did in my life, I have learned the hard way that even the most sensible investors can fall victim to scams. When my USD was stolen, for anyone who has fallen victim to one of the bitcoin binary investment scams that are currently ongoing, I felt betrayal and upset. But then I was reading a post on site when I saw a testimony of Wendy Taylor online who recommended that Capital Crypto Recovery has helped her recover scammed funds within 24 hours. after reaching out to this cyber security firm that was able to help me recover my stolen digital assets and bitcoin. I’m genuinely blown away by their amazing service and professionalism. I never imagined I’d be able to get my money back until I complained to Capital Crypto Recovery Services about my difficulties and gave all of the necessary paperwork. I was astounded that it took them 12 hours to reclaim my stolen money back. Without a doubt, my USDT assets were successfully recovered from the scam platform, Thank you so much Sir, I strongly recommend Capital Crypto Recover for any of your bitcoin recovery, digital funds recovery, hacking, and cybersecurity concerns. You reach them Call/Text Number +1 (336)390-6684 His Email: [email protected] Contact Telegram: @Capitalcryptorecover Via Contact: [email protected] His website: https://recovercapital.wixsite.com/capital-crypto-rec-1
22.10.25 11:59 elizabethrush89

God bless Capital Crypto Recover Services for the marvelous work you did in my life, I have learned the hard way that even the most sensible investors can fall victim to scams. When my USD was stolen, for anyone who has fallen victim to one of the bitcoin binary investment scams that are currently ongoing, I felt betrayal and upset. But then I was reading a post on site when I saw a testimony of Wendy Taylor online who recommended that Capital Crypto Recovery has helped her recover scammed funds within 24 hours. after reaching out to this cyber security firm that was able to help me recover my stolen digital assets and bitcoin. I’m genuinely blown away by their amazing service and professionalism. I never imagined I’d be able to get my money back until I complained to Capital Crypto Recovery Services about my difficulties and gave all of the necessary paperwork. I was astounded that it took them 12 hours to reclaim my stolen money back. Without a doubt, my USDT assets were successfully recovered from the scam platform, Thank you so much Sir, I strongly recommend Capital Crypto Recover for any of your bitcoin recovery, digital funds recovery, hacking, and cybersecurity concerns. You reach them Call/Text Number +1 (336)390-6684 His Email: [email protected] Contact Telegram: @Capitalcryptorecover Via Contact: [email protected] His website: https://recovercapital.wixsite.com/capital-crypto-rec-1
22.10.25 16:31 MATT PHILLIP

I never imagined I’d fall for a crypto romance scam but it happened. Over the course of a few months, I sent nearly $150,000 worth of Bitcoin to someone I genuinely believed I was building a future with. When they disappeared without a trace, I was left heartbroken, humiliated, and financially devastated. For a long time, I didn’t tell anyone. I felt ashamed. But eventually, while searching for answers, I came across a Reddit thread that mentioned Agent Jasmine Lopez. I reached out, not expecting much. To my surprise, she treated me with kindness, not judgment. She used advanced tools like blockchain forensics, IP tracing, and smart contract analysis and with persistence and legal support, she was able to recover nearly 85% of what I lost. I know not everyone gets that kind of outcome, but thanks to [email protected] WhatsApp at +44 736-644-5035, I’ve started to reclaim not just my assets, but my confidence and peace of mind. If you’re going through something similar, you’re not alone and there is hope.
23.10.25 01:52 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
23.10.25 01:52 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
23.10.25 15:47 MATT PHILLIP

I never imagined I’d fall for a crypto romance scam but it happened. Over the course of a few months, I sent nearly $150,000 worth of Bitcoin to someone I genuinely believed I was building a future with. When they disappeared without a trace, I was left heartbroken, humiliated, and financially devastated. For a long time, I didn’t tell anyone. I felt ashamed. But eventually, while searching for answers, I came across a Reddit thread that mentioned Agent Jasmine Lopez. I reached out, not expecting much. To my surprise, she treated me with kindness, not judgment. She used advanced tools like blockchain forensics, IP tracing, and smart contract analysis and with persistence and legal support, she was able to recover nearly 85% of what I lost. I know not everyone gets that kind of outcome, but thanks to [email protected] WhatsApp at +44 736-644-5035, I’ve started to reclaim not just my assets, but my confidence and peace of mind. If you’re going through something similar, you’re not alone and there is hope.
23.10.25 21:43 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
23.10.25 21:43 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
24.10.25 06:08 MATT PHILLIP

I never imagined I’d fall for a crypto romance scam but it happened. Over the course of a few months, I sent nearly $150,000 worth of Bitcoin to someone I genuinely believed I was building a future with. When they disappeared without a trace, I was left heartbroken, humiliated, and financially devastated. For a long time, I didn’t tell anyone. I felt ashamed. But eventually, while searching for answers, I came across a Reddit thread that mentioned Agent Jasmine Lopez. I reached out, not expecting much. To my surprise, she treated me with kindness, not judgment. She used advanced tools like blockchain forensics, IP tracing, and smart contract analysis and with persistence and legal support, she was able to recover nearly 85% of what I lost. I know not everyone gets that kind of outcome, but thanks to [email protected] WhatsApp at +44 736-644-5035, I’ve started to reclaim not just my assets, but my confidence and peace of mind. If you’re going through something similar, you’re not alone and there is hope.
24.10.25 06:40 elizabethrush89

God bless Capital Crypto Recover Services for the marvelous work you did in my life, I have learned the hard way that even the most sensible investors can fall victim to scams. When my USD was stolen, for anyone who has fallen victim to one of the bitcoin binary investment scams that are currently ongoing, I felt betrayal and upset. But then I was reading a post on site when I saw a testimony of Wendy Taylor online who recommended that Capital Crypto Recovery has helped her recover scammed funds within 24 hours. after reaching out to this cyber security firm that was able to help me recover my stolen digital assets and bitcoin. I’m genuinely blown away by their amazing service and professionalism. I never imagined I’d be able to get my money back until I complained to Capital Crypto Recovery Services about my difficulties and gave all of the necessary paperwork. I was astounded that it took them 12 hours to reclaim my stolen money back. Without a doubt, my USDT assets were successfully recovered from the scam platform, Thank you so much Sir, I strongly recommend Capital Crypto Recover for any of your bitcoin recovery, digital funds recovery, hacking, and cybersecurity concerns. You reach them Call/Text Number +1 (336)390-6684 His Email: [email protected] Contact Telegram: @Capitalcryptorecover Via Contact: [email protected] His website: https://recovercapital.wixsite.com/capital-crypto-rec-1
24.10.25 06:40 elizabethrush89

God bless Capital Crypto Recover Services for the marvelous work you did in my life, I have learned the hard way that even the most sensible investors can fall victim to scams. When my USD was stolen, for anyone who has fallen victim to one of the bitcoin binary investment scams that are currently ongoing, I felt betrayal and upset. But then I was reading a post on site when I saw a testimony of Wendy Taylor online who recommended that Capital Crypto Recovery has helped her recover scammed funds within 24 hours. after reaching out to this cyber security firm that was able to help me recover my stolen digital assets and bitcoin. I’m genuinely blown away by their amazing service and professionalism. I never imagined I’d be able to get my money back until I complained to Capital Crypto Recovery Services about my difficulties and gave all of the necessary paperwork. I was astounded that it took them 12 hours to reclaim my stolen money back. Without a doubt, my USDT assets were successfully recovered from the scam platform, Thank you so much Sir, I strongly recommend Capital Crypto Recover for any of your bitcoin recovery, digital funds recovery, hacking, and cybersecurity concerns. You reach them Call/Text Number +1 (336)390-6684 His Email: [email protected] Contact Telegram: @Capitalcryptorecover Via Contact: [email protected] His website: https://recovercapital.wixsite.com/capital-crypto-rec-1
24.10.25 06:54 MATT PHILLIP

I never imagined I’d fall for a crypto romance scam but it happened. Over the course of a few months, I sent nearly $150,000 worth of Bitcoin to someone I genuinely believed I was building a future with. When they disappeared without a trace, I was left heartbroken, humiliated, and financially devastated. For a long time, I didn’t tell anyone. I felt ashamed. But eventually, while searching for answers, I came across a Reddit thread that mentioned Agent Jasmine Lopez. I reached out, not expecting much. To my surprise, she treated me with kindness, not judgment. She used advanced tools like blockchain forensics, IP tracing, and smart contract analysis and with persistence and legal support, she was able to recover nearly 85% of what I lost. I know not everyone gets that kind of outcome, but thanks to [email protected] WhatsApp at +44 736-644-5035, I’ve started to reclaim not just my assets, but my confidence and peace of mind. If you’re going through something similar, you’re not alone and there is hope.
24.10.25 18:18 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
24.10.25 18:18 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
25.10.25 00:49 tyleradams

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25.10.25 00:50 stevekalfman

If you ever need hacking services, look no further. I found myself in a difficult situation after losing almost $510,000 USD in bitcoin. I was distraught and had no hope of survival because i lost my life savings to this scam, I had no chance of recovering my investment. Until i came across an article on google about ([email protected]). A real-life recovery expert, everything changed completely after contacting him and explaining my ordeal to him and he quickly stepped in and assisted me in recovering 95% of my lost funds. They guarantee their clients and i got the highest level of happiness their services are highly recommended. so I’m putting this here for anyone who require their services too" Email: [email protected]
25.10.25 05:05 victoriabenny463

A few months ago, I fell for what looked like a legitimate online trading platform. The website was sleek, the support team was responsive, and they even showed me fake profit charts that seemed to grow daily. I started small, and after a few test withdrawals worked, I invested more eventually depositing 2 Bitcoin in hopes of long-term returns. But one morning, when I tried to withdraw, everything changed. The platform suddenly locked my account, claiming there were additional fees to release my funds. The support chat stopped responding, and the website started giving error messages. That’s when I realized I’d been scammed. I was devastated. Two Bitcoins wasn’t pocket change, it was my savings. I filed reports online, but everyone told me it was impossible to recover crypto once it’s sent. Then, through a discussion forum, I came across Dexdert Net Recovery, a team specializing in tracing and recovering stolen digital assets. Skeptical but desperate, I reached out. Their response was immediate and professional. They asked for transaction IDs, wallet addresses, screenshots of my correspondence with the fake platform, and any proof of payment I had. Within days, their analysts traced the stolen Bitcoin through multiple wallets using blockchain forensics, They updated me step-by-step, Working with their recovery experts and blockchain partners, Dexdert Net Recovery successfully froze the funds and verified ownership of my wallet. After 2 days of persistence and verification, I finally saw my 2 Bitcoin restored back into my own wallet. I couldn’t believe it. What stood out most wasn’t just that they recovered my funds, but how transparent and supportive the Dexdert team was throughout the process. They educated me on crypto safety, how to spot fake trading sites, and how to secure my digital assets in the future. Dexdert Net Recovery truly lives up to its name. They gave me back not just my Bitcoin, but my peace of mind. Reach Out Information Below Via: Email: ([email protected]) Telegram: (https://t.me/Dexdertprorecovery) WhatsApp: (+1 (859) 609‑4156)
25.10.25 10:13 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
25.10.25 10:13 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
26.10.25 01:03 Christopherbelle

A terrible financial shock hit me hard. I lost $860,000 in a fake crypto scheme. Things looked good until I tried to cash out my earnings. Suddenly, my account vanished into thin air. I told the police many times, but help never showed up. Despair washed over me; I felt totally beaten. Then, I found Sylvester Bryant Intelligence. Right away, they were skilled and honest about the whole thing. They looked deep into my situation. They tracked where the stolen funds went. They fought hard for me every step of the way. To my great surprise, they got my money back. I thought that was impossible. I owe them so much for their hard work and truthfulness. Sylvester Bryant Intelligence restored my peace. If scams took your crypto, contact them now. Contact Info: Yt7cracker@gmail . com WhatsApp: ‪+1 512 577 7957. or +44 7428 662701.
26.10.25 18:09 victoriabenny463

A few months ago, I fell for what looked like a legitimate online trading platform. The website was sleek, the support team was responsive, and they even showed me fake profit charts that seemed to grow daily. I started small, and after a few test withdrawals worked, I invested more eventually depositing 2 Bitcoin in hopes of long-term returns. But one morning, when I tried to withdraw, everything changed. The platform suddenly locked my account, claiming there were additional fees to release my funds. The support chat stopped responding, and the website started giving error messages. That’s when I realized I’d been scammed. I was devastated. Two Bitcoins wasn’t pocket change, it was my savings. I filed reports online, but everyone told me it was impossible to recover crypto once it’s sent. Then, through a discussion forum, I came across Dexdert Net Recovery, a team specializing in tracing and recovering stolen digital assets. Skeptical but desperate, I reached out. Their response was immediate and professional. They asked for transaction IDs, wallet addresses, screenshots of my correspondence with the fake platform, and any proof of payment I had. Within days, their analysts traced the stolen Bitcoin through multiple wallets using blockchain forensics, They updated me step-by-step, Working with their recovery experts and blockchain partners, Dexdert Net Recovery successfully froze the funds and verified ownership of my wallet. After 2 days of persistence and verification, I finally saw my 2 Bitcoin restored back into my own wallet. I couldn’t believe it. What stood out most wasn’t just that they recovered my funds, but how transparent and supportive the Dexdert team was throughout the process. They educated me on crypto safety, how to spot fake trading sites, and how to secure my digital assets in the future. Dexdert Net Recovery truly lives up to its name. They gave me back not just my Bitcoin, but my peace of mind. Reach Out Information Below Via: Email: ([email protected]) Telegram: (https://t.me/Dexdertprorecovery) WhatsApp: (+1 (859) 609‑4156)
27.10.25 11:15 harristhomas7376

"In the crypto world, this is great news I want to share. Last year, I fell victim to a scam disguised as a safe investment option. I have invested in crypto trading platforms for about 10yrs thinking I was ensuring myself a retirement income, only to find that all my assets were either frozen, I believed my assets were secure — until I discovered that my BTC funds had been frozen and withdrawals were impossible. It was a devastating moment when I realized I had been scammed, and I thought my Bitcoin was gone forever, Everything changed when a close friend recommended the Capital Crypto Recover Service. Their professionalism, expertise, and dedication enabled me to recover my lost Bitcoin funds back — more than €560.000 DEM to my BTC wallet. What once felt impossible became a reality thanks to their support. If you have lost Bitcoin through scams, hacking, failed withdrawals, or similar challenges, don’t lose hope. I strongly recommend Capital Crypto Recover Service to anyone seeking a reliable and effective solution for recovering any wallet assets. They have a proven track record of successful reputation in recovering lost password assets for their clients and can help you navigate the process of recovering your funds. Don’t let scammers get away with your hard-earned money – contact Email: [email protected] Phone CALL/Text Number: +1 (336) 390-6684 Contact: [email protected] Website: https://recovercapital.wixsite.com/capital-crypto-rec-1
27.10.25 11:15 harristhomas7376

"In the crypto world, this is great news I want to share. Last year, I fell victim to a scam disguised as a safe investment option. I have invested in crypto trading platforms for about 10yrs thinking I was ensuring myself a retirement income, only to find that all my assets were either frozen, I believed my assets were secure — until I discovered that my BTC funds had been frozen and withdrawals were impossible. It was a devastating moment when I realized I had been scammed, and I thought my Bitcoin was gone forever, Everything changed when a close friend recommended the Capital Crypto Recover Service. Their professionalism, expertise, and dedication enabled me to recover my lost Bitcoin funds back — more than €560.000 DEM to my BTC wallet. What once felt impossible became a reality thanks to their support. If you have lost Bitcoin through scams, hacking, failed withdrawals, or similar challenges, don’t lose hope. I strongly recommend Capital Crypto Recover Service to anyone seeking a reliable and effective solution for recovering any wallet assets. They have a proven track record of successful reputation in recovering lost password assets for their clients and can help you navigate the process of recovering your funds. Don’t let scammers get away with your hard-earned money – contact Email: [email protected] Phone CALL/Text Number: +1 (336) 390-6684 Contact: [email protected] Website: https://recovercapital.wixsite.com/capital-crypto-rec-1
27.10.25 17:08 raymondgonzales

Three Cheers for Wizard Hilton Cyber Tech, Bitcoin Savior Extraordinaire In the ever-evolving world of cryptocurrency, where volatility and uncertainty reign supreme, one individual has emerged as a beacon of hope and innovation - Wizard Hilton Cyber Tech, the self-proclaimed "Bitcoin Savior Extraordinaire." This enigmatic figure, with his uncanny ability to navigate the complexities of the digital asset landscape, has captivated the attention of crypto enthusiasts and skeptics alike. Wizard Hilton Cyber Tech journey began as a humble programmer, tinkering with the underlying blockchain technology that powers Bitcoin. But it was his visionary approach and unwavering commitment to the cryptocurrency's potential that propelled him into the limelight. Through a series of strategic investments, groundbreaking algorithmic trading strategies, and a knack for anticipating market shifts, Wizard Hilton Cyber Tech has managed to consistently outperform even the savviest of Wall Street traders. Wizard Hilton Cyber Tech ability to turn even the most tumultuous of Bitcoin price swings into opportunities for substantial gains has earned him the moniker "Wizard," a testament to his unparalleled mastery of the digital currency realm. But Wizard Hilton Cyber Tech impact extends far beyond personal wealth accumulation - Wizard Hilton Cyber Tech has tirelessly advocated for the widespread adoption of Bitcoin, working tirelessly to educate the public, lobby policymakers, and collaborate with industry leaders to overcome the barriers that have long hindered the cryptocurrency's mainstream acceptance. With infectious enthusiasm and boundless energy, Wizard Hilton Cyber Tech has become a true champion of the Bitcoin cause, inspiring a new generation of crypto enthusiasts to embrace the transformative power of this revolutionary technology. As the digital currency landscape continues to evolve, Wizard Hilton Cyber Tech name will undoubtedly be etched in the annals of cryptocurrency history as a visionary, a trailblazer, and, above all, the Bitcoin Savior Extraordinaire. To get your stolen bitcoin back, reach out to Wizard Hilton Cyber Tech via: Email : wizardhiltoncybertech ( @ ) gmail (. ) com WhatsApp number +18737715701 Good Day.
27.10.25 17:20 fatimanorth

The Federal Trade Commission said that more than $1 billion had been lost to cryptocurrency scams in 2023. Victims frequently lose everything, including hope, trust, and wealth. When your hard-earned money disappears into digital shadows, you feel trapped. Nowadays, cryptocurrency frauds are very common. You can get help from recovery specialist Marie at [email protected] and via WhatsApp at +1 7127594675.
28.10.25 00:55 elizabethrush89

God bless Capital Crypto Recover Services for the marvelous work you did in my life, I have learned the hard way that even the most sensible investors can fall victim to scams. When my USD was stolen, for anyone who has fallen victim to one of the bitcoin binary investment scams that are currently ongoing, I felt betrayal and upset. But then I was reading a post on site when I saw a testimony of Wendy Taylor online who recommended that Capital Crypto Recovery has helped her recover scammed funds within 24 hours. after reaching out to this cyber security firm that was able to help me recover my stolen digital assets and bitcoin. I’m genuinely blown away by their amazing service and professionalism. I never imagined I’d be able to get my money back until I complained to Capital Crypto Recovery Services about my difficulties and gave all of the necessary paperwork. I was astounded that it took them 12 hours to reclaim my stolen money back. Without a doubt, my USDT assets were successfully recovered from the scam platform, Thank you so much Sir, I strongly recommend Capital Crypto Recover for any of your bitcoin recovery, digital funds recovery, hacking, and cybersecurity concerns. You reach them Call/Text Number +1 (336)390-6684 His Email: [email protected] Contact Telegram: @Capitalcryptorecover Via Contact: [email protected] His website: https://recovercapital.wixsite.com/capital-crypto-rec-1
28.10.25 00:55 elizabethrush89

God bless Capital Crypto Recover Services for the marvelous work you did in my life, I have learned the hard way that even the most sensible investors can fall victim to scams. When my USD was stolen, for anyone who has fallen victim to one of the bitcoin binary investment scams that are currently ongoing, I felt betrayal and upset. But then I was reading a post on site when I saw a testimony of Wendy Taylor online who recommended that Capital Crypto Recovery has helped her recover scammed funds within 24 hours. after reaching out to this cyber security firm that was able to help me recover my stolen digital assets and bitcoin. I’m genuinely blown away by their amazing service and professionalism. I never imagined I’d be able to get my money back until I complained to Capital Crypto Recovery Services about my difficulties and gave all of the necessary paperwork. I was astounded that it took them 12 hours to reclaim my stolen money back. Without a doubt, my USDT assets were successfully recovered from the scam platform, Thank you so much Sir, I strongly recommend Capital Crypto Recover for any of your bitcoin recovery, digital funds recovery, hacking, and cybersecurity concerns. You reach them Call/Text Number +1 (336)390-6684 His Email: [email protected] Contact Telegram: @Capitalcryptorecover Via Contact: [email protected] His website: https://recovercapital.wixsite.com/capital-crypto-rec-1
29.10.25 03:43 Christopherbelle

A sudden money crisis struck me. I dropped $82,000 to a phony crypto scam. All seemed fine until I went to pull out my gains. Then, i was unable to get back my hard earned money. I reported it to the police several times. No aid came my way. Deep sadness hit me; I felt crushed. That's when I learned about Sylvester Bryant Intelligence. They acted quick and fair from the start. They checked my case closely. They followed the path of my lost cash. They pushed strong for me at each turn. In shock, they brought my funds back. I had figured it could not happen. I thank them for their effort and honesty. Sylvester Bryant Intelligence gave me calm again. If a scam stole your crypto, reach out to them today. Contact Info: Yt7cracker@gmail . com WhatsApp: +1 512 577 7957 or +44 7428 662701.
29.10.25 10:56 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
29.10.25 10:56 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
30.10.25 09:49 Christopherbelle

I suffered a huge money blow when $80,000 slipped away in a bogus crypto plot. It all looked great at first. Then I went to pull out my gains, and poof—my account was gone. I kept telling the cops about it, but they did zip. I felt shattered and lost. That's when I stumbled on Sylvester Bryant Intelligence. Things turned around fast. Right away, they showed clear steps, sharp know-how, and real drive to fix it. They tracked my missing cash step by step and pushed hard till they got it back. I swear, I figured it was a lost cause. Their straight-up work and trust brought back my calm. If a scam stole your crypto, reach out to them now. yt7cracker@gmail . com | WhatsApp: +1 512 577 7957 or +44 7428 662701
30.10.25 12:03 elizabethrush89

God bless Capital Crypto Recover Services for the marvelous work you did in my life, I have learned the hard way that even the most sensible investors can fall victim to scams. When my USD was stolen, for anyone who has fallen victim to one of the bitcoin binary investment scams that are currently ongoing, I felt betrayal and upset. But then I was reading a post on site when I saw a testimony of Wendy Taylor online who recommended that Capital Crypto Recovery has helped her recover scammed funds within 24 hours. after reaching out to this cyber security firm that was able to help me recover my stolen digital assets and bitcoin. I’m genuinely blown away by their amazing service and professionalism. I never imagined I’d be able to get my money back until I complained to Capital Crypto Recovery Services about my difficulties and gave all of the necessary paperwork. I was astounded that it took them 12 hours to reclaim my stolen money back. Without a doubt, my USDT assets were successfully recovered from the scam platform, Thank you so much Sir, I strongly recommend Capital Crypto Recover for any of your bitcoin recovery, digital funds recovery, hacking, and cybersecurity concerns. You reach them Call/Text Number +1 (336)390-6684 His Email: [email protected] Contact Telegram: @Capitalcryptorecover Via Contact: [email protected] His website: https://recovercapital.wixsite.com/capital-crypto-rec-1

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Bias-Variance trade-off

Интересные особенности bias-variance

Метрики кластеризации

Rand Index & Adjusted Rand Index (ARI)

Mutual Information & Adjusted Mutual Information (AMI)

Homogeneity, Completeness & V-measure

Fowlkes-Mallows Index (FMI)

Silhouette Coefficient

Calinski-Harabasz Index (CHI)

Davies-Bouldin Index (DBI)

Дополнительные источники

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Метрики оценки качества моделей и анализ ошибок в машинном обучении

Содержание

Метрики классификации

Accuracy

Precision

Recall (TPR)

FPR

F1-score

ROC-AUC

PR-AUC и Average Precision (AP)

Когда классов больше двух

Метрики регрессии

MAE

MAPE & SMAPE

WAPE

MSE & RMSE

MSLE & RMSLE

R2 & Adjusted R2

Bias-Variance Decomposition

Вывод разложения ошибки для MSE

Bias-Variance trade-off

Интересные особенности bias-variance

Метрики кластеризации

Rand Index & Adjusted Rand Index (ARI)

Mutual Information & Adjusted Mutual Information (AMI)

Homogeneity, Completeness & V-measure

Fowlkes-Mallows Index (FMI)

Silhouette Coefficient

Calinski-Harabasz Index (CHI)

Davies-Bouldin Index (DBI)

Дополнительные источники

Похожие новости

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Все главные новости мира криптовалют и технологий 30.10.2025

MegaETH по итогам токенсейла привлек $27,8 млрд

BitChat вышел в топ по скачиваниям на Ямайке из-за последствий урагана

CryptoQuant: объём торгов биткоином в октябре превысил $300 млрд

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