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Метод K-ближайших соседей (KNN). Принцип работы, разновидности и реализация с нуля на Python

0f6153f77a3a66795624792f783d0fb9.png

К-ближайших соседей (K-Nearest Neighbors или просто KNN) — алгоритм классификации и регрессии, основанный на гипотезе компактности, которая предполагает, что расположенные близко друг к другу объекты в пространстве признаков имеют схожие значения целевой переменной или принадлежат к одному классу.

Ноутбук с данными алгоритмами можно загрузить на Kaggle (eng) и GitHub (rus).

Принцип работы KNN

Алгоритм строится следующим образом:

  • 1) сначала вычисляется расстояние между тестовым и всеми обучающими образцами;

  • 2) далее из них выбирается k-ближайших образцов (соседей), где число k задаётся заранее;

  • 3) итоговым прогнозом среди выбранных k-ближайших образцов будет мода в случае классификации и среднее арифметическое в случае регрессии;

  • 4) предыдущие шаги повторяются для всех тестовых образцов.

Схема работы KNN
Схема работы KNN

Существует множество метрик для вычисления расстояния между объектами, среди которых наиболее популярными являются следующие:

  • Евклидово расстояние — это наиболее простая и общепринятая метрика, которая определяется как длина отрезка между двумя объектами a и b в пространстве с nпризнаками и вычисляется по формуле:

d(a, b) = \sqrt{\sum_{i=1}^n (a_i - b_i)^2}

  • Манхэттенское расстояние — метрика, которая определяется как сумма модулей разностей координат двух точек в пространстве между двумя объектами a и b с nпризнаками и вычисляется по формуле:

d(a, b) = \sum_{i=1}^n |a_i - b_i|

  • Косинусное расстояние — метрика, которая определяется как угол между двумя векторами a и b в пространстве с n признаками и вычисляется по формуле:

d(a, b) = 1 - \frac{\sum_{i=1}^n a_i b_i}{\sqrt{\sum_{i=1}^n a_i^2} \sqrt{\sum_{i=1}^n b_i^2}}

Более быстрые оптимизации

Описанный выше тип KNN называется Brute-Force, поскольку в нём используется метод полного перебора для поиска ближайших соседей, что делает его простым в реализации, но слишком медленным при работе с большим объемом данных. Для решения данной проблемы в реализации scikit-learn предусмотрены более продвинутые методы, основанные на бинарных деревьях, что позволяет получить значительный прирост в производительности.

BallTree

BallTree — это древовидная структура, в основе которой лежит разбиение исходного пространства данных на вложенные гиперсферы, что позволяет более эффективно отсекать большие области пространства, в которых отсутствуют ближайшие соседи для точек. В большинстве случаев такой алгоритм подходит для данных с произвольной метрикой расстояния.

Построение BallTree состоит из следующих шагов:

  • 1) из множества точек выбирается одна случайным образом и для неё находится самая дальняя точка;

  • 2) далее все точки разбиваются на сферы (узлы) по ближайшему расположению к двум точкам из шага 1, а расстояние между двумя сферами A и B в метрике M будет определяться следующим образом:

d_M(A, B) = \max(0, d_M(c_A, c_{B}) - r_A - r_{B}

где c_A и c_{B} — центры сфер, а r_Aи r_B — их радиусы.

  • 3) затем данный процесс повторяется рекурсивно для каждой сферы, пока в ней не останется определённое количество точек или не будет достигнута заданная глубина дерева.

При поиске k-ближайших соседей для новой точки, алгоритм сравнивает расстояние от заданной точки до центра каждого дочернего узла и оставляет лишь те, в которых данное расстояние меньше радиуса узлов.

KD-Tree

KD-Tree (k-dimensional tree) — ещё одна древовидная структура, отдалённо напоминающая BallTree, однако в данном случае используются гиперплоскости для разбиения точек вместо гиперсфер, что позволяет также эффективно оставлять лишь те области пространства данных, в которых могут присутствовать ближайшие соседи. Обычно KD-Tree больше подходит для данных с евклидовой или манхэттенской метрикой расстояния.

Построение KD-Tree состоит из следующих шагов:

  • 1) из множества точек выбирается одна из координат (обычно поочередно для каждого уровня дерева, но можно и случайным образом) и по ней вычисляется медиана;

  • 2) далее все точки разбиваются на два узла (подмножества) по отношению к медиане: на те, у которых значение выбранной координаты меньше либо равно медиане, и на те, у которых больше;

  • 3) данный процесс повторяется рекурсивно для каждого узла, пока в нём не останется определённое количество точек или не будет достигнута заданная глубина дерева.

При поиске ближайших соседей для новой точки, алгоритм сравнивает значение заданной точки с медианой в каждом узле, выбирая таким образом ближайшее подпространство, которое будет листом с ближайшими соседями. Возвращаясь обратно к корню, алгоритм будет сравнивать точки в текущем узле с ближайшими соседями и обновлять их значения в случае нахождения более близких к заданной точке.

Стоит также добавить, что KNN может применяться в контексте обучения без учителя, например в задачах кластеризации для вычисления расстояний между признаками, как в алгоритме DBSCAN (подробно можно ознакомится здесь). Для этого в scikit-learn существует специальный класс NearestNeighbors.

Импорт необходимых библиотек

import numpy as np
import matplotlib.pyplot as plt
from mlxtend.plotting import plot_decision_regions
from sklearn.model_selection import train_test_split
from sklearn.metrics import accuracy_score, r2_score
from sklearn.datasets import load_iris, load_diabetes
from sklearn.neighbors import KNeighborsClassifier, KNeighborsRegressor

Реализация на Python с нуля

class KNearestNeighbors:

    def __init__(self, n_neighbors=5, regression=False):
        self.n_neighbors = n_neighbors
        self.regression = regression

    def fit(self, X_train, y_train):
        self.X_train, self.y_train = X_train, y_train

    def _euclidean_distances(self, x_test_i):
        return np.sqrt(np.sum((self.X_train - x_test_i) ** 2, axis=1))

    def _make_prediction(self, x_test_i):
        distances = self._euclidean_distances(x_test_i)   # distances to all neighbors
        k_nearest_indexes = np.argsort(distances)[:self.n_neighbors]
        targets = self.y_train[k_nearest_indexes]   # k-nearest neighbors target values

        return np.mean(targets) if self.regression else np.bincount(targets).argmax()

    def predict(self, X_test):
        return np.array([self._make_prediction(x) for x in X_test])

Код для отрисовки графика

def decision_boundary_plot(X, y, X_train, y_train, clf, feature_indexes, title=None):
    feature1_name, feature2_name = X.columns[feature_indexes]
    X_feature_columns = X.values[:, feature_indexes]
    X_train_feature_columns = X_train[:, feature_indexes]
    clf.fit(X_train_feature_columns, y_train)

    plot_decision_regions(X=X_feature_columns, y=y.values, clf=clf)
    plt.xlabel(feature1_name)
    plt.ylabel(feature2_name)
    plt.title(title)

Загрузка датасетов

X1, y1 = load_iris(return_X_y=True, as_frame=True)
X1_train, X1_test, y1_train, y1_test = train_test_split(X1.values, y1.values, random_state=0)
print(X1, y1, sep='\n')


     sepal length (cm)  sepal width (cm)  petal length (cm)  petal width (cm)
0                  5.1               3.5                1.4               0.2
1                  4.9               3.0                1.4               0.2
2                  4.7               3.2                1.3               0.2
3                  4.6               3.1                1.5               0.2
4                  5.0               3.6                1.4               0.2
..                 ...               ...                ...               ...
145                6.7               3.0                5.2               2.3
146                6.3               2.5                5.0               1.9
147                6.5               3.0                5.2               2.0
148                6.2               3.4                5.4               2.3
149                5.9               3.0                5.1               1.8

[150 rows x 4 columns]
0      0
1      0
2      0
3      0
4      0
      ..
145    2
146    2
147    2
148    2
149    2
Name: target, Length: 150, dtype: int64
X2, y2 = load_diabetes(return_X_y=True, as_frame=True)
X2_train, X2_test, y2_train, y2_test = train_test_split(X2.values, y2.values, random_state=0)
print(X2, y2, sep='\n')


          age       sex       bmi        bp        s1        s2        s3  \
0    0.038076  0.050680  0.061696  0.021872 -0.044223 -0.034821 -0.043401   
1   -0.001882 -0.044642 -0.051474 -0.026328 -0.008449 -0.019163  0.074412   
2    0.085299  0.050680  0.044451 -0.005670 -0.045599 -0.034194 -0.032356   
3   -0.089063 -0.044642 -0.011595 -0.036656  0.012191  0.024991 -0.036038   
4    0.005383 -0.044642 -0.036385  0.021872  0.003935  0.015596  0.008142   
..        ...       ...       ...       ...       ...       ...       ...   
437  0.041708  0.050680  0.019662  0.059744 -0.005697 -0.002566 -0.028674   
438 -0.005515  0.050680 -0.015906 -0.067642  0.049341  0.079165 -0.028674   
439  0.041708  0.050680 -0.015906  0.017293 -0.037344 -0.013840 -0.024993   
440 -0.045472 -0.044642  0.039062  0.001215  0.016318  0.015283 -0.028674   
441 -0.045472 -0.044642 -0.073030 -0.081413  0.083740  0.027809  0.173816   

           s4        s5        s6  
0   -0.002592  0.019907 -0.017646  
1   -0.039493 -0.068332 -0.092204  
2   -0.002592  0.002861 -0.025930  
3    0.034309  0.022688 -0.009362  
4   -0.002592 -0.031988 -0.046641  
..        ...       ...       ...  
437 -0.002592  0.031193  0.007207  
438  0.034309 -0.018114  0.044485  
439 -0.011080 -0.046883  0.015491  
440  0.026560  0.044529 -0.025930  
441 -0.039493 -0.004222  0.003064  

[442 rows x 10 columns]
0      151.0
1       75.0
2      141.0
3      206.0
4      135.0
       ...  
437    178.0
438    104.0
439    132.0
440    220.0
441     57.0
Name: target, Length: 442, dtype: float64

Обучение моделей и оценка полученных результатов

KNN показал хороший результат в случае классификации. Также можно заметить, что алгоритм способен выстраивать нелинейные решающие границы, что позволяет ему также достигать хороших результатов с данными, в которых прослеживается нелинейная зависимость.

Однако, стоит обратить внимание на случай регрессии: низкий r2-score обусловлен небольшим числом ближайших соседей по умолчанию. Если увеличить число соседей до 30, то и r2-score увеличится более чем в 2 раза, что говорит о важности подбора оптимального числа ближайших соседей.

KNN classifier

knn_clf = KNearestNeighbors()
knn_clf.fit(X1_train, y1_train)
knn_clf_pred_res = knn_clf.predict(X1_test)
knn_clf_accuracy = accuracy_score(y1_test, knn_clf_pred_res)

print(f'KNN classifier accuracy: {knn_clf_accuracy:}')
print(knn_clf_pred_res)


KNN classifier accuracy: 0.9736842105263158
[2 1 0 2 0 2 0 1 1 1 2 1 1 1 1 0 1 1 0 0 2 1 0 0 2 0 0 1 1 0 2 1 0 2 2 1 0
 2]

KNN regressor 

knn_reg = KNearestNeighbors(regression=True)
knn_reg.fit(X2_train, y2_train)
knn_reg_pred_res = knn_reg.predict(X2_test)
knn_reg_r2 = r2_score(y2_test, knn_reg_pred_res)

print(f'KNN regressor R2 score: {knn_reg_r2}')
print(knn_reg_pred_res)


KNN regressor R2 score: 0.18912404854026388
[253.6 188.6 183.2 138.4 177.8 189.6 111.8 229.  178.  266.8 147.6 193.8
 136.4  55.6 297.4  73.6  97.2  83.8 130.8 214.4 173.6 115.2 167.4 101.
 186.8 175.6  97.2  75.  172.4 144.2 205.4  63.8 161.6 190.8 110.2 159.2
 199.4 141.2 121.4 140.8 155.6 173.8 140.6 175.6 134.2  84.6 110.4 127.2
 107.4 209.2 130.2  78.2 183.6 105.  227.4 160.4 155.  104.6 119.2 175.8
 159.8 141.6 150.4 100.2 279.2 128.4  91.2 269.2 183.2  88.4 118.  151.6
  74.8  97.8 126.2 140.4 127.4 223.6 236.6 191.2 111.6 219.8  69.6 169.4
  87.6  92.6 112.  145.8 117.  153.2 115.2  92.8  67.6 172.   92.4 106.6
 208.4 173.8 113.2 104.4 141.6 128.2 226.   87.  247.6 147.6 223.6 217.2
 149.   72.6 182. ]

KNN classifier (scikit-learn) 

sk_knn_clf = KNeighborsClassifier()
sk_knn_clf.fit(X1_train, y1_train)
sk_knn_clf_pred_res = sk_knn_clf.predict(X1_test)
sk_knn_clf_accuracy = accuracy_score(y1_test, sk_knn_clf_pred_res)

print(f'sk KNN classifier accuracy: {sk_knn_clf_accuracy:}')
print(sk_knn_clf_pred_res)

feature_indexes = [2, 3]
title1 = 'KNeighborsClassifier surface'
decision_boundary_plot(X1, y1, X1_train, y1_train, sk_knn_clf, feature_indexes, title1)


sk KNN classifier accuracy: 0.9736842105263158
[2 1 0 2 0 2 0 1 1 1 2 1 1 1 1 0 1 1 0 0 2 1 0 0 2 0 0 1 1 0 2 1 0 2 2 1 0
 2]
12aee3d8bc6046d6f5db3ac417f20d5d.png

KNN regressor (scikit-learn) 

sk_knn_reg = KNeighborsRegressor()
sk_knn_reg.fit(X2_train, y2_train)
sk_knn_reg_pred_res = sk_knn_reg.predict(X2_test)
sk_knn_reg_r2 = r2_score(y2_test, sk_knn_reg_pred_res)

print(f'sk KNN regressor R2 score: {sk_knn_reg_r2}')
print(sk_knn_reg_pred_res)


sk KNN regressor R2 score: 0.18912404854026388
[253.6 188.6 183.2 138.4 177.8 189.6 111.8 229.  178.  266.8 147.6 193.8
 136.4  55.6 297.4  73.6  97.2  83.8 130.8 214.4 173.6 115.2 167.4 101.
 186.8 175.6  97.2  75.  172.4 144.2 205.4  63.8 161.6 190.8 110.2 159.2
 199.4 141.2 121.4 140.8 155.6 173.8 140.6 175.6 134.2  84.6 110.4 127.2
 107.4 209.2 130.2  78.2 183.6 105.  227.4 160.4 155.  104.6 119.2 175.8
 159.8 141.6 150.4 100.2 279.2 128.4  91.2 269.2 183.2  88.4 118.  151.6
  74.8  97.8 126.2 140.4 127.4 223.6 236.6 191.2 111.6 219.8  69.6 169.4
  87.6  92.6 112.  145.8 117.  153.2 115.2  92.8  67.6 172.   92.4 106.6
 208.4 173.8 113.2 104.4 141.6 128.2 226.   87.  247.6 147.6 223.6 217.2
 149.   72.6 182. ]

Преимущества и недостатки KNN

Преимущества:

  • простота в реализации и интерпретации;

  • применяется во многих задачах, особенно в рекомендательных системах;

  • высокая точность прогнозов при правильном подборе k и метрики расстояния.

Недостатки:

  • большое потребление памяти и низкая скорость работы из-за хранения и вычисления расстояний между всеми обучающими и тестовыми образцами (имеется в виду KNN в чистом виде);

  • чувствительность к выбросам и шуму, а также к несбалансированным классам в данных;

  • при большом количестве признаков может возникнуть проблема совпадения метрической и смысловой близости объектов, что решается с помощью обучения представлений (численное описание объектов).

Дополнительные источники

Статьи:

  • «Brute-Force k-Nearest Neighbors Search on the GPU», Shengren Li and Nina Amenta;

  • «Unsupervised Space Partitioning for Nearest Neighbor Search», Abrar Fahim, Mohammed Eunus Ali and Muhammad Aamir Cheema;

  • «Ball-tree: Efficient spatial indexing for constrained nearest-neighbor search in metric spaces», Mohamad Dolatshah, Ali Hadian and Behrouz Minaei-Bidgoli;

  • «Developments in KD Tree and KNN Searches», Vijay R. Tiwari.

Документация:

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  • 09.10.25 11:05 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

  • 09.10.25 11:05 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

  • 09.10.25 11:05 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

  • 09.10.25 11:05 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

  • 11.10.25 04:41 luciajessy3

    Don’t be deceived by different testimonies online that is most likely wrong. I have made use of several recovery options that got me disappointed at the end of the day but I must confess that the tech genius I eventually found is the best out here. It’s better you devise your time to find the valid professional that can help you recover your stolen or lost crypto such as bitcoins rather than falling victim of other amateur hackers that cannot get the job done. ADAMWILSON . TRADING @ CONSULTANT COM / WHATSAPP ; +1 (603) 702 ( 4335 ) is the most reliable and authentic blockchain tech expert you can work with to recover what you lost to scammers. They helped me get back on my feet and I’m very grateful for that. Contact their email today to recover your lost coins ASAP…

  • 11.10.25 10:44 Tonerdomark

    A thief took my Dogecoin and wrecked my life. Then Mr. Sylvester stepped in and changed everything. He got back €211,000 for me, every single cent of my gains. His calm confidence and strong tech skills rebuilt my trust. Thanks to him, I recovered my cash with no issues. After months of stress, I felt huge relief. I had full faith in him. If a scam stole your money, reach out to him today at { yt7cracker@gmail . com } His help sparked my full turnaround.

  • 12.10.25 01:12 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

  • 12.10.25 01:12 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

  • 12.10.25 19:53 Tonerdomark

    A crook swiped my Dogecoin. It ruined my whole world. Then Mr. Sylvester showed up. He fixed it all. He pulled back €211,000 for me. Not one cent missing from my profits. His steady cool and sharp tech know-how won back my trust. I got my money smooth and sound. After endless worry, relief hit me hard. I trusted him completely. Lost cash to a scam? Hit him up now at { yt7cracker@gmail . com }. His aid turned my life around. WhatsApp at +1 512 577 7957.

  • 12.10.25 21:36 blessing

    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

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