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Кластеризация в ML: от теоретических основ популярных алгоритмов к их реализации с нуля на Python

Кластеризация — это набор методов без учителя для группировки данных по определённым критериям в так называемые кластеры, что позволяет выявлять сходства и различия между объектами, а также упрощать их анализ и визуализацию. Из-за частичного сходства в постановке задач с классификацией кластеризацию ещё называют unsupervised classification.

В данной статье описан не только принцип работы популярных алгоритмов кластеризации от простых к более продвинутым, но а также представлены их упрощённые реализации с нуля на Python, отражающие основную идею. Помимо этого, в конце каждого раздела указаны дополнительные источники для более глубокого ознакомления.

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

Содержание

Области применения кластеризации и её разновидности
К-средних
Агломеративная кластеризация
Спектральная кластеризация
DBSCAN
Affinity Propagation

Области применения кластеризации и её разновидности

Кластеризация широко применяется в машинном обучении для решения различного спектра задач:

классификация (определение к какому классу относится каждый объект или же выделение новых классов, которые не были известны заранее);
сегментация рынка (разделение потенциальных клиентов на группы по их характеристикам для разработки более эффективных стратегий в маркетинге и продажах);
сегментация изображений (разделение изображения на сегменты или группы пикселей);
кластеризация геоданных (группировка данных по их географическому расположению, например, разделение районов на безопасные и опасные, богатые и бедные, и так далее);
понижение размерности (уменьшение количества признаков путем объединения схожих в один кластер).

Существует множество различных типов кластеризации, которые можно разделить по следующим критериям:

По способу формирования кластеров:
- Разделительные (partitioning) — разбивают данные на заданное число кластеров, минимизируя расстояние внутри кластера и максимизируя расстояние между кластерами (например, K-means).
- Основанные на плотности (density-based) — группируют точки, которые находятся в областях с высокой плотностью и отделяют их от областей с низкой плотностью (например, DBSCAN).
- Основанные на сетке (grid-based) — разбивают пространство на ячейки сетки и анализируют плотность данных в каждой ячейке (например, STING).
- Основанные на модели (model-based) — предполагают, что данные порождены некоторой статистической моделью и пытаются подобрать параметры этой модели (например, смеси Гауссианов).
- Основанные на графах (graph-based) — используют графовое представление данных и разбивают его на подграфы, соответствующие кластерам (например, спектральная кластеризация).
- Основанные на подпространствах (subspace-based) — ищут кластеры в подпространствах признаков, а не во всём пространстве (например, CLIQUE).
- Основанные на ансамбле (ensemble-based) — комбинируют результаты различных алгоритмов кластеризации, чтобы получить более стабильное и надежное разбиение (например, CSPA).
По степени вложенности кластеров:
- Плоские (flat) — разбивают данные на один уровень кластеров, не учитывая их иерархию (например, K-means).
- Иерархические (hierarchical) — разбивают данные на несколько уровней кластеров, учитывая их иерархию. Существуют два основных подхода к иерархической кластеризации: агломеративный (начинается с того, что каждый объект является отдельным кластером, а затем постепенно наиболее близкие кластеры объединяются в более крупные) и дивизивный (начинается с того, что все объекты составляют один кластер, а затем постепенно разделяются на более мелкие кластеры).
По степени пересечения кластеров:
- Исключающие (exclusive) — каждый объект принадлежит только одному кластеру (например, K-means).
- Перекрывающие (overlapping) — каждый объект может принадлежать нескольким кластерам (например, MCOKE).
- Нечёткие (fuzzy) — каждый объект принадлежит каждому кластеру с некоторой степенью принадлежности (например, fuzzy K-means).

Алгоритмы кластеризации в scikit-learn

Далее более подробно будут рассмотрены наиболее популярные алгоритмы кластеризации, представляющие наибольший интерес с практической точки зрения.

K-Means

Кластеризация K-средних представляет собой одну из самых простых реализаций, суть которой заключается в итеративной инициализации центроидов для каждого кластера на основе среднего арифметического расположенных в нём наблюдений, а также их переопределении путём минимизации суммарного квадратичного отклонения от центроидов кластеров.

Схема кластеризации K-средних

Существуют различные вариации алгоритма K-Means, которые модифицируют его шаги или функцию потерь для улучшения производительности, а также применимости к разным типам данных. К самым популярным вариациям относятся следующие:

Lloyd's algorithm — это классический вариант K-Means, который хорошо работает для сферических кластеров с одинаковой плотностью, но может давать плохие результаты для других форм или размеров кластеров.
Elkan algorithm — более быстрый вариант классического K-Means, который использует неравенство треугольника для уменьшения количества вычислений расстояний между объектами и центроидами, что может быть эффективнее на некоторых наборах данных с хорошо определенными кластерами, однако требуется больше памяти из-за выделения дополнительного массива размера (n_samples, n_clusters).
Mini-batch K-Means — модификация классического K-Means, использующая случайные подвыборки данных на каждой итерации для обучения. Хорошо подходит для больших датасетов.
K-Medoids — вариант K-Means, который в качестве центроидов выбирает реальные точки (медоиды) из данных, а не их средние значения, что повышает устойчивость к выбросам.
K-Modes — вариант алгоритма K-Means для работы с категориальными данными, который выбирает один из объектов в кластере в качестве моды и минимизирует сумму расстояний Хэмминга между модой и объектами в кластере. Расстояние Хэмминга представляет из себя количество позиций, в которых значения векторов не совпадают.

Одним из ключевых вопросов при использовании K-Means является выбор начальных центроидов, поскольку от них зависит качество и скорость сходимости алгоритма. Существует несколько способов инициализации центроидов:

Случайный выбор: выбирается k случайных точек из данных в качестве начальных центроидов. Такой метод прост, но может привести к плохим результатам, если начальные центроиды слишком близки друг к другу или к краям распределения.
K-Means++: первый центроид выбирается случайно, а затем выбираются остальные центроиды с вероятностью, пропорциональной квадрату расстояния до ближайшего уже выбранного центроида. Данный метод улучшает качество кластеризации, уменьшая вероятность попадания в локальный минимум, но требует дополнительных вычислений.
Greedy K-Means++ — модификация K-Means++, которая ускоряет сходимость и улучшает качество кластеризации за счёт того, что на каждом шаге при выборе центра кластера производится несколько попыток и выбирается лучший (тот, который минимизирует суммарное квадратичное отклонение точек от центров кластеров).

Другим важным аспектом, влияющим на качество алгоритма, является подбор оптимального числа кластеров k, которое устанавливается заранее вручную. Существует несколько методов для выбора оптимального k:

Метод локтя, основанный на графике суммы квадратов расстояний между объектами и центроидами кластеров (SSE) в зависимости от k. Оптимальным k считается та точка, после которой SSE уменьшается незначительно.
Метод силуэта, основанный на графике среднего значения силуэта для каждого k. Силуэт — это мера того, насколько хорошо объект отнесен к своему кластеру по сравнению с другими кластерами. Оптимальным k считается та точка, где силуэт достигает максимума.
Метод комплексной оценки, основанный на анализе нескольких критериев, таких как динамика перераспределения объектов в кластерах, изменение потенциальной энергии объектов внутри кластеров и характерные точки графиков этих критериев. Оптимальным k считается точка, которая удовлетворяет набору правил, использующих данные критерии.

Принцип работы Lloyd's K-Means с инициализацией Greedy K-Means++

Алгоритм строится следующим образом:

1) изначально вычисляются центроиды кластеров с помощью алгоритма Greedy K-Means++;
2) далее рассчитывается квадрат евклидова (или другого) расстояния от каждого наблюдения до центроидов;
3) на основе полученного расстояния наблюдениям присваиваются метки кластеров, которые к ним расположены ближе всего, а также рассчитывается инерция — мера того, насколько хорошо данные были разбиты на кластеры;
4) шаги 2-3 повторяются до тех пор, пока инерция на текущей и предыдущей итерациях перестанет изменяться меньше установленного порога (пока положение центроидов перестанет изменяться в пространстве) или пока не будет достигнуто установленное количество итераций;
5) Наблюдения, расположенные ближе всего к полученным центроидам, и будут составлять кластеры.

Принцип работы алгоритма Greedy K-Means++:

1) первый центроид выбирается случайным образом из данных;
2) для каждого следующего центроида выбирается точка в данных с вероятностью, пропорциональной квадрату расстояния до ближайшего центроида среди нескольких кандидатов, выбранных случайным образом;
3) данный процесс повторяется пока не будет выбрано установленное количество центроидов.

Импорт необходимых библиотек

import numpy as np
from scipy.spatial.distance import cdist
import matplotlib.pyplot as plt
from sklearn.datasets import make_blobs
from sklearn.metrics import adjusted_rand_score
from sklearn.cluster import KMeans

Реализация на Python с нуля

class KMeansClustering:
    def __init__(self, n_clusters=8, max_iter=300, tol=0.0001, random_state=0):
        self.n_clusters = n_clusters
        self.max_iter = max_iter
        self.tol = tol
        self.random_state = random_state

    def _greedy_kmeans_plus_plus(self, X):
        np.random.seed(self.random_state)
        n_samples, n_features = X.shape
        n_local_trials = 2 + int(np.log(self.n_clusters))

        indices = np.arange(n_samples)
        first_index = np.random.choice(indices)
        centers = np.zeros((self.n_clusters, n_features))

        centers[0] = X[first_index]
        first_center = centers[0].reshape(1, -1)
        sq_distances = cdist(X, first_center, metric='sqeuclidean').ravel()

        for i in range(1, self.n_clusters):
            min_cost = np.inf
            min_new_sq_distances = []
            best_candidate_index = None

            for _ in range(n_local_trials):
                candidates_probas = sq_distances / np.sum(sq_distances)
                candidate_index = np.random.choice(indices, p=candidates_probas)
                candidate = X[candidate_index].reshape(1, -1)

                new_sq_distances = cdist(X, candidate, metric='sqeuclidean').ravel()
                new_cost = np.sum(np.minimum(sq_distances, new_sq_distances))

                if new_cost < min_cost:
                    best_candidate_index = candidate_index
                    min_new_sq_distances = new_sq_distances
                    min_cost = new_cost

            centers[i] = X[best_candidate_index]   # Choose the new center
            sq_distances = np.minimum(sq_distances, min_new_sq_distances)

        return centers

    def fit(self, X):
        n_samples, n_features = X.shape
        self.inertia_ = np.inf
        self.cluster_centers_ = self._greedy_kmeans_plus_plus(X)

        # Lloyd's algorithm
        for _ in range(self.max_iter):
            distances = cdist(X, self.cluster_centers_, metric='sqeuclidean')
            labels = np.argmin(distances, axis=1)
            new_inertia = np.sum(np.min(distances, axis=1))
            new_centers = np.zeros((self.n_clusters, n_features))

            for k in range(self.n_clusters):
                new_centers[k] = np.mean(X[labels == k], axis=0)

            if np.abs(new_inertia - self.inertia_) < self.tol:
                break

            self.inertia_ = new_inertia
            self.cluster_centers_ = new_centers

    def predict(self, X):
        distances = cdist(X, self.cluster_centers_, metric='sqeuclidean')
        predicted_labels = np.argmin(distances, axis=1)

        return predicted_labels

Загрузка датасета

X1, y1 = make_blobs(n_samples=250, n_features=2, centers=8, random_state=0)
print(y1)


[1 3 7 7 6 7 1 3 7 7 0 3 1 1 3 3 5 1 7 4 0 1 1 3 4 7 0 0 6 7 0 0 5 5 7 2 1
 1 6 5 4 7 1 2 1 1 4 3 6 4 7 3 0 2 2 1 7 2 4 0 0 0 1 4 6 5 0 4 6 6 4 4 1 4
 2 3 1 1 5 4 6 4 1 2 5 0 7 6 7 3 0 1 2 5 1 5 3 3 3 1 5 4 0 4 7 6 2 2 2 4 6
 2 5 1 6 4 0 6 5 0 0 6 3 5 1 6 0 2 5 5 6 3 3 1 5 4 5 0 2 2 3 0 4 7 5 4 2 0
 2 6 2 5 2 1 4 1 5 0 4 6 7 5 5 7 6 2 2 3 6 1 7 3 4 7 2 6 6 4 2 2 0 5 4 4 6
 3 1 7 6 7 7 0 4 5 7 2 6 6 2 5 3 3 2 7 1 7 6 6 4 3 5 7 6 3 5 0 3 3 5 5 2 0
 6 3 4 0 5 3 5 2 0 6 4 0 1 1 2 2 0 1 3 7 0 7 0 3 4 7 3 7]

Обучение моделей и оценка полученных результатов

Как можно заметить, K-Means дал достаточно хорошую, но не идеальную кластеризацию данных, что обусловлено наличием небольшого числа выбросов и более сложной формой некоторых кластеров, чем сферическая.

KMeans

kmeans = KMeansClustering(n_clusters=8, random_state=0)
kmeans.fit(X1)
kmeans_pred_res = kmeans.predict(X1)
kmeans_ari = adjusted_rand_score(y1, kmeans_pred_res)
kmeans_centroinds = kmeans.cluster_centers_
print(f'Adjusted Rand Score for KMeans: {kmeans_ari}', '', sep='\n')
print('centroids', kmeans_centroinds, '', sep='\n')
print('prediction', kmeans_pred_res, sep='\n')


Adjusted Rand Score for KMeans: 0.8082423809657193

centroids
[[ 9.20217726 -2.23709633]
 [-1.5438023   7.64224793]
 [-8.61527648 -8.32916569]
 [ 0.81231976  4.02302811]
 [-1.81106448  2.87987747]
 [ 2.3666746   1.30457024]
 [ 1.47433518  8.49698324]
 [ 5.86512606  0.19818122]]

prediction
[5 1 2 2 6 2 5 1 2 2 5 1 5 5 1 1 5 5 2 0 3 7 3 1 0 2 3 3 6 2 1 3 7 7 2 4 5
 5 6 7 0 2 5 4 5 5 0 1 6 7 2 1 3 3 4 5 2 4 0 3 3 3 5 0 6 7 3 0 6 6 0 0 5 0
 3 1 5 5 7 0 3 0 5 4 7 3 2 6 2 6 3 5 4 5 5 7 1 1 1 3 7 0 3 0 2 6 3 4 4 0 6
 4 7 5 1 0 3 6 7 3 3 6 1 7 5 6 3 4 7 7 6 1 1 5 7 0 7 3 4 4 1 3 0 2 7 0 4 3
 4 6 4 7 4 5 0 5 7 3 0 6 2 7 5 2 6 4 3 3 6 5 2 1 0 2 4 6 6 0 4 4 3 7 0 0 6
 1 5 2 6 2 2 3 0 7 2 4 6 6 4 7 1 1 4 2 5 2 6 1 0 1 5 2 6 6 7 3 1 1 7 7 4 3
 6 1 0 3 7 1 7 4 4 6 0 3 5 5 4 4 3 3 1 2 3 2 3 1 0 2 1 2]

KMeans (scikit-learn)

sk_kmeans = KMeans(n_clusters=8, n_init='auto', random_state=0)
sk_kmeans.fit(X1)
sk_kmeans_pred_res = sk_kmeans.predict(X1)
sk_kmeans_ari = adjusted_rand_score(y1, sk_kmeans_pred_res)
sk_kmeans_centroinds = sk_kmeans.cluster_centers_
print(f'Adjusted Rand Score for sk KMeans: {sk_kmeans_ari}', '', sep='\n')
print(sk_kmeans_centroinds, '', sep='\n')
print('prediction', sk_kmeans_pred_res, sep='\n')


Adjusted Rand Score for sk KMeans: 0.8082423809657193

[[ 9.20217726 -2.23709633]
 [-1.5438023   7.64224793]
 [-8.61527648 -8.32916569]
 [ 0.81231976  4.02302811]
 [-1.81106448  2.87987747]
 [ 2.3666746   1.30457024]
 [ 1.47433518  8.49698324]
 [ 5.86512606  0.19818122]]

prediction
[5 1 2 2 6 2 5 1 2 2 5 1 5 5 1 1 5 5 2 0 3 7 3 1 0 2 3 3 6 2 1 3 7 7 2 4 5
 5 6 7 0 2 5 4 5 5 0 1 6 7 2 1 3 3 4 5 2 4 0 3 3 3 5 0 6 7 3 0 6 6 0 0 5 0
 3 1 5 5 7 0 3 0 5 4 7 3 2 6 2 6 3 5 4 5 5 7 1 1 1 3 7 0 3 0 2 6 3 4 4 0 6
 4 7 5 1 0 3 6 7 3 3 6 1 7 5 6 3 4 7 7 6 1 1 5 7 0 7 3 4 4 1 3 0 2 7 0 4 3
 4 6 4 7 4 5 0 5 7 3 0 6 2 7 5 2 6 4 3 3 6 5 2 1 0 2 4 6 6 0 4 4 3 7 0 0 6
 1 5 2 6 2 2 3 0 7 2 4 6 6 4 7 1 1 4 2 5 2 6 1 0 1 5 2 6 6 7 3 1 1 7 7 4 3
 6 1 0 3 7 1 7 4 4 6 0 3 5 5 4 4 3 3 1 2 3 2 3 1 0 2 1 2]

Визуализация прогнозов

plt.figure(figsize=(12, 5))

plt.subplot(1, 2, 1)
plt.subplots_adjust(wspace=0.2)
plt.scatter(X1[:, 0], X1[:, 1], c=kmeans_pred_res, cmap="rainbow")
plt.scatter(kmeans_centroinds[:, 0], kmeans_centroinds[:, 1], marker="x", color="black", s=100)
plt.title("KMeans")
plt.xlabel("Feature 1")
plt.ylabel("Feature 2")

plt.subplot(1, 2, 2)
plt.subplots_adjust(wspace=0.2)
plt.scatter(X1[:, 0], X1[:, 1], c=sk_kmeans_pred_res, cmap="rainbow")
plt.scatter(sk_kmeans_centroinds[:, 0], sk_kmeans_centroinds[:, 1], marker="x", color="black", s=100)
plt.title("KMeans (scikit-learn)")
plt.xlabel("Feature 1")

plt.show()

Ручная реализация vs scikit-learn

Преимущества и недостатки K-Means

Преимущества:

прост в реализации и понимании;
наличие большого числа модификаций;
высокая скорость работы и точность на данных сферической формы.

Недостатки:

низкая точность на данных с несферической формой кластеров;
чувствительность к начальным значениям центроидов и выбросам;
необходимость заранее устанавливать число кластеров, что может быть сложно или не оптимально.

Дополнительные источники

Статья «Improved Guarantees for k-means++ and k-means++ Parallel», Konstantin Makarychev, Aravind Reddy and Liren Shan.

Документация: описание K-Means, K-Means (алгоритм).

Видео: один, два.

Агломеративная кластеризация

Алгломеративная кластеризация — это метод иерархической кластеризации, который объединяет объекты в кластеры на основе их близости. Процесс кластеризации начинается с того, что изначально каждый объект считается отдельным кластером, а затем на каждом шаге два наиболее близких кластера сливаются в один, пока не будет достигнуто желаемое число кластеров или один общий кластер.

Важным аспектом в алгломеративной кластеризаци является способ вычисления расстояния между кластерами, который называется связью. Можно выделить несколько методов для вычисления связи:

Метод одиночной связи (single linkage): расстояние между кластерами равно минимальному расстоянию между точками из разных кластеров. Этот метод склонен к созданию длинных и извилистых кластеров.

$d_{\min}(C_i, C_j) = \min_{x \in C_i, y \in C_j} \rho(x, y)$

Метод полной связи (complete linkage): расстояние между кластерами равно максимальному расстоянию между точками из разных кластеров. Этот метод склонен к созданию компактных и сферических кластеров.

$d_{\max}(C_i, C_j) = \max_{x \in C_i, y \in C_j} \rho(x, y)$

Метод средней связи (average linkage): расстояние между кластерами равно среднему расстоянию между всеми парами точек из разных кластеров. Этот метод является компромиссом между методами одиночной и полной связи.

$d_{\text{avg}}(C_i, C_j) = \frac{1}{n_i n_j} \sum_{x \in C_i} \sum_{y \in C_j} \rho(x, y)$

Метод Уорда (Ward's linkage): расстояние между кластерами равно приросту суммы квадратов расстояний от точек до центроидов кластеров при объединении этих кластеров. Этот метод стремится минимизировать внутрикластерную дисперсию.

$d_{\text{ward}}(C_i, C_j) = \frac{n_i n_j}{n_i + n_j} \rho^2(\bar{x}_i, \bar{x}_j)$

Принцип работы агломеративной кластеризации на основе метода Уорда

Алгоритм строится следующим образом:

1) для каждой пары кластеров рассчитывается расстояние Уорда на основе евклидова расстояния;
2) метки с минимальным расстоянием, то есть ближайшие кластеры объединяются в новый следующим образом: всем объектам одного кластера присваиваются метки другого, после чего все новые метки, которые больше меток другого кластера, уменьшаются на 1, то есть их нумерация сдвигается влево для устранения пропусков в последовательности;
3) шаги 1-2 повторяются пока количество кластеров больше целевого n_clusters.

Импорт необходимых библиотек

import numpy as np
import matplotlib.pyplot as plt
from scipy.spatial.distance import sqeuclidean
from scipy.cluster.hierarchy import linkage, dendrogram
from sklearn.datasets import make_blobs
from sklearn.metrics import adjusted_rand_score
from sklearn.cluster import AgglomerativeClustering

Реализация на Python с нуля

class HierarchicalAgglomerativeClustering:
    def __init__(self, n_clusters=2):
        self.n_clusters = n_clusters

    @staticmethod
    def _ward_distance(c1, c2):
        n1, n2 = len(c1), len(c2)
        c1_mean, c2_mean = np.mean(c1, axis=0), np.mean(c2, axis=0)
        sqeuclidean_dist = sqeuclidean(c1_mean, c2_mean)

        return (n1 * n2) / (n1 + n2) * sqeuclidean_dist

    @staticmethod
    def _update_labels(labels, min_cdist_idxs):
        # assign a cluster number to labels
        labels[labels == min_cdist_idxs[1]] = min_cdist_idxs[0]
        labels[labels > min_cdist_idxs[1]] -= 1

        return labels

    def fit_predict(self, X):
        labels = np.arange(len(X))
        clusters = [[x] for x in X]

        while len(clusters) > self.n_clusters:
            min_cdist, min_cdist_idxs = np.inf, []

            for i in range(len(clusters) - 1):
                for j in range(i + 1, len(clusters)):
                    cdist = self._ward_distance(clusters[i], clusters[j])

                    if cdist < min_cdist:
                        min_cdist = cdist
                        min_cdist_idxs = (i, j)

            labels = self._update_labels(labels, min_cdist_idxs)
            clusters[min_cdist_idxs[0]].extend(clusters.pop(min_cdist_idxs[1]))

        return np.array(labels)

Загрузка датасета

X2, y2 = make_blobs(n_samples=75, n_features=2, centers=5, random_state=0)
print(y2)


[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 моделей и оценка полученных результатов

В данном случае агломеративная кластеризация также справилась достаточно хорошо, но при сравнении графиков можно заметить, что метки ручной реализации имеют другие значения и соответственно цвет на графике. Это связано с другим порядком формирования кластеров: в scikit-learn используется структура данных, известная как linkage tree, позволяющая визуализировать иерархию кластеров с помощью древовидной диаграммы (дендограммы), что может быть полезно при выборе оптимального количества кластеров.

Стоит отметить, что порядок формирования кластеров не влияет на качество кластеризации, поскольку сами метки кластеров разделяются правильно, что видно из одинаковых значений ARI.

AgglomerativeClustering

ac = HierarchicalAgglomerativeClustering(n_clusters=5)
ac_pred_res = ac.fit_predict(X2)
ac_ari = adjusted_rand_score(y2, ac_pred_res)
print(f'Adjusted Rand Score for AgglomerativeClustering: {ac_ari}', '', sep='\n')
print('prediction', ac_pred_res, sep='\n')


Adjusted Rand Score for AgglomerativeClustering: 0.8370870157432925

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

AgglomerativeClustering (scikit-learn)

sk_ac = AgglomerativeClustering(n_clusters=5, linkage='ward')
sk_ac_pred_res = sk_ac.fit_predict(X2)
sk_ac_ari = adjusted_rand_score(y2, sk_ac_pred_res)
print(f'Adjusted Rand Score for sk AgglomerativeClustering: {sk_ac_ari}', '', sep='\n')
print('prediction', sk_ac_pred_res, sep='\n')


Adjusted Rand Score for sk AgglomerativeClustering: 0.8370870157432925

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

Визуализация прогнозов

plt.figure(figsize=(12, 5))

plt.subplot(1, 2, 1)
plt.scatter(X2[:, 0], X2[:, 1], c=ac_pred_res, cmap='rainbow')
plt.title('AgglomerativeClustering')
plt.xlabel("Feature 1")
plt.ylabel("Feature 2")

plt.subplot(1, 2, 2)
plt.scatter(X2[:, 0], X2[:, 1], c=sk_ac_pred_res, cmap='rainbow')
plt.title('AgglomerativeClustering (scikit-learn)')
plt.xlabel("Feature 1")

plt.show()

Ручная реализация vs scikit-learn

linkage_matrix = linkage(X2, method='ward', metric='euclidean')

plt.figure(figsize=(10, 6))
dendrogram(linkage_matrix, color_threshold=10)
plt.xlabel("Sample Index")
plt.ylabel("Distance")
plt.title("Dendrogram")
plt.show()

Преимущества и недостатки аглометративной кластеризации

Преимущества:

адаптация к различным ситуациям и способность обнаружения кластеров произвольной формы;
возможность формирования информативной иерархии кластеров, которая позволяет выбирать оптимальное число кластеров или лучше изучать структуру данных.

Недостатки:

использование большого количества вычислительных ресурсов и памяти из-за работы со всей матрицей расстояний между объектами;
чувствительность к выбору критерия объединения кластеров, а также неустойчивость к шуму и выбросам, что может сильно искажать иерархию кластеров.

Дополнительные источники

Статья «Modern hierarchical, agglomerative clustering algorithms», Daniel Müllner.

Документация: описание AgglomerativeClustering, AgglomerativeClustering (алгоритм).

Видео: один, два, три.

Спектральная кластеризация

Спектральная кластеризация — метод кластеризации на основе спектральных свойств матрицы сходства графа, который представляет собой набор точек данных, связанных друг с другом в зависимости от их сходства. Основная идея заключается в преобразовании матрицы сходства графа в лаплассиан для получения его собственных векторов, которые в дальнейшем используются для проекции данных в новое пространство более низкой размерности для лучшей разделимости, где затем применяется другой метод кластеризации, например, такой как K-средних.

Существуют различные способы построения матрицы сходства, среди которых наиболее популярными являются следующие:

nearest_neighbors (путем вычисления графа ближайших соседей);
rbf (с использованием радиальной базисной функции);
предварительно вычисленные, где признаки представлены как матрица сходства или граф ближайших соседей;
различные ядра: xi-квадрат, линейное, полиномиальное, сигмоидальное и другие.

Также существуют различные стратегии разложения лаплассиана на собственные вектора и значения:

ARPACK (ARnoldi PACKage) — это программный пакет на языке Fortran, который реализует итерационный метод Арнольди для нахождения нескольких собственных значений и собственных векторов большой разреженной матрицы. Метод Арнольди — это обобщение метода Ланцоша, который строит ортогональный базис подпространства Крылова, порождённого матрицей и начальным вектором, а затем проецирует исходную собственную задачу на это подпространство.
LOBPCG (Locally Optimal Block Preconditioned Conjugate Gradient) — это метод, который реализует локально оптимальный, блочный, предобусловленный, сопряжённый градиент для нахождения нескольких наименьших по модулю собственных значений и собственных векторов большой положительно определенной матрицы. Основная идея LOBPCG заключается в том, что на каждой итерации он выбирает следующее приближение к собственному вектору так, чтобы минимизировать квадратичную форму, связанную с собственной задачей на трехмерном подпространстве, порождённом текущим приближением, предобусловленным остатком и последним обновлением. Это делается с помощью метода Релея-Ритца, который находит оптимальные линейные комбинации этих векторов.
AMG (Algebraic Multigrid methods) — это класс алгоритмов для решения больших систем уравнений, возникающих при дискретизации дифференциальных уравнений. В контексте кластеризации AMG методы уменьшают размер матрицы, переводя её на грубую сетку, где она имеет меньше элементов, но сохраняет свои свойства. Это делается с помощью двух операций: сглаживания и коррекции. Сглаживание — это простой итерационный метод, который уменьшает ошибку в приближённом решении собственной задачи. Коррекция — это переход на грубую сетку, где решается упрощённая собственная задача, а затем возвращается на тонкую сетку для корректировки решения. Этот процесс повторяется несколько раз, пока не будет достигнута нужная точность.

Стоит отметить, что последние 2 метода являются более быстрыми, но менее стабильными.

Принцип работы спектральной кластеризации ARPACK с RBF ядром

Алгоритм строится следующим образом:

1) на основе матрицы сходства графа строится его нормализованный лаплассиан $L = I - D^{-\frac{1}{2}}$ и диагональная матрица степеней вершин графа, где — единичная матрица;
2) далее с помощью алгоритма ARPACK вычисляются k-собственных векторов лаплассиана с указанием сдвига для ускорения вычислений (sigma), случайного вектора (v0), а также наибольших по модулю собственных значений (which="LM") так как они будут расположены ближе всего к сдвигу, что в конечном счёте вернёт наименьшие собственные вектора;
3) полученные вектора нормализуются по степеням вершин графа, чтобы быть независимыми от весов вершин, после чего в даннных векторах изменяется знак, чтобы стать детерменированными (такой подход позволяет избежать неоднозначности в знаках собственных векторов при использовании различных реализаций алгоритма);
4) из модифицированных собственных векторов формируется матрица вложения, к которой применяется алгоритм K-Means, который спрогнозирует в конечном счёте итоговые метки.

Импорт необходимых библиотек

import numpy as np
import matplotlib.pyplot as plt
from scipy.sparse.linalg import eigsh
from scipy.sparse.csgraph import laplacian as csgraph_laplacian
from sklearn.datasets import make_moons
from sklearn.preprocessing import StandardScaler
from sklearn.cluster import KMeans, SpectralClustering
from sklearn.metrics.pairwise import rbf_kernel
from sklearn.metrics.cluster import adjusted_rand_score

Реализация на Python с нуля

class ArpackSpectralClustering:
    def __init__(self, n_clusters=8, gamma=1.0, random_state=None):
        self.n_clusters = n_clusters
        self.gamma = gamma
        self.random_state = random_state

    @staticmethod
    def _deterministic_vector_sign_flip(u):
        # Flip the sign of the vectors to make them deterministic
        max_abs_rows = np.argmax(np.abs(u), axis=1)
        signs = np.sign(u[range(u.shape[0]), max_abs_rows])

        return u * signs[:, np.newaxis]

    def _spectral_embedding(self, affinity):
        laplacian, diag_vertex_degrees = csgraph_laplacian(affinity, normed=True, return_diag=True)
        laplacian *= -1

        arpack_v0 = np.random.RandomState(self.random_state).uniform(-1, 1, laplacian.shape[0])
        _, evecs = eigsh(laplacian, k=self.n_clusters, sigma=1.0, which="LM", tol=0, v0=arpack_v0)
        norm_evecs = evecs.T[self.n_clusters::-1] / diag_vertex_degrees
        embedding = self._deterministic_vector_sign_flip(norm_evecs)

        return embedding[:self.n_clusters].T

    def fit_predict(self, X):
        affinity_matrix = rbf_kernel(X, gamma=self.gamma)
        embedding = self._spectral_embedding(affinity_matrix)
        kmeans = KMeans(n_clusters=self.n_clusters, n_init='auto', random_state=self.random_state)
        labels = kmeans.fit_predict(embedding)

        return labels

Загрузка датасета

X3, y3 = make_moons(n_samples=200, noise=0.05, random_state=0)
X3 = StandardScaler().fit_transform(X3)
print(y3)


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

Обучение моделей и оценка полученных результатов

Поскольку спектральное преобразование матрицы сходства графа позволяет выявлять скрытую геометрическую структуру данных, алгоритм спектральной кластеризации может применяться к нелинейно разделимым данным и показывать довольно хорошие результаты кластеризации сложных форм, что видно из результатов ниже.

SpectralClustering

sc = ArpackSpectralClustering(n_clusters=2, gamma=10, random_state=0)
sc_pred_res = sc.fit_predict(X3)
sc_ari = adjusted_rand_score(y3, sc_pred_res)
print(f'Adjusted Rand Score for SpectralClustering: {sc_ari}', '', sep='\n')
print('prediction', sc_pred_res, sep='\n')


Adjusted Rand Score for SpectralClustering: 1.0

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

SpectralClustering (scikit-learn)

sk_sc = SpectralClustering(n_clusters=2, gamma=10, random_state=0)
sk_sc_pred_res = sk_sc.fit_predict(X3)
sk_sc_ari = adjusted_rand_score(y3, sk_sc_pred_res)
print(f'Adjusted Rand Score for sk SpectralClustering: {sk_sc_ari}', '', sep='\n')
print('prediction', sk_sc_pred_res, sep='\n')


Adjusted Rand Score for sk SpectralClustering: 1.0

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

Визуализация прогнозов

plt.figure(figsize=(12, 5))

plt.subplot(121)
plt.scatter(X3[:, 0], X3[:, 1], c=sc_pred_res, cmap='Spectral')
plt.title('SpectralClustering')
plt.xlabel("Feature 1")
plt.ylabel("Feature 2")

plt.subplot(122)
plt.scatter(X3[:, 0], X3[:, 1], c=sk_sc_pred_res, cmap='Spectral')
plt.title('SpectralClustering (scikit-learn)')
plt.xlabel("Feature 1")

plt.show()

Ручная реализация vs scikit-learn

Преимущества и недостатки спектральной кластеризации

Преимущества:

работа с кластерами сложных форм;
возможность обработки многомерных данных из-за понижения размерности перед их кластеризацией;
устойчивость к выбросам и шуму в данных из-за учёта их глобальной структуры, а не только локальной.

Недостатки:

сложность конфигурации из-за большого количества гиперпараметров;
высокая вычислительная сложность и потребление памяти при работе с большими объемами данных, что может быть частично решено с помощью безматричных методов или случайных проекций.

Дополнительные источники

Статья «On Spectral Clustering: Analysis and an algorithm», Andrew Y. Ng, Michael I. Jordan, Yair Weiss.

Документация: описание SpectralClustering, SpectralClustering (алгоритм).

Видео: один, два, три.

DBSCAN

Более интересным алгоритмом кластеризации является DBSCAN (Density-Based Spatial Clustering of Applications with Noise), который основан на плотности точек в пространстве. Он группирует вместе точки, которые находятся близко друг к другу и отмечает как выбросы точки, которые лежат в областях с низкой плотностью. Помимо того что DBSCAN может обнаруживать кластеры произвольной формы и выбросы в данных, его главная особенность заключается в самостоятельном определении необходимого количества кластеров, что избавляет от необходимости в их подборе.

Для вычисления попарных расстояний и ближайших соседей точек в DBSCAN используется модифицированная реализация k-ближайших соседей, которая является алгоритмом обучения без учителя и представлена в scikit-learn в виде класса NearestNeighbors.

Схема образования кластера в DBSCAN

Принцип работы DBSCAN

Наиболее важными параметрами, влияющими на результат кластеризации, являются eps (максимально допустимое расстояние между точками, чтобы считаться соседями) и min_samples (минимальное количество точек в окрестности другой точки, чтобы считаться базовой точкой), а сами же точки в данных подразделяются на 3 вида:

Core points (базовые точки) — точки в кластере, имеющие min_samples соседей в своём окружении eps или более. Это означает, что точки лежат в области высокой плотности данных.
Border points (пограничные точки) — точки в кластере, которые имеют меньше, чем min_samples соседей в своём окружении eps, но лежат в окружении eps других базовых точек. Это означает, что точки лежат на границе кластеров.
Noise points (шумовые точки) — это выбросы, которые не принадлежат ни к одному кластеру, то есть точки расположены в области низкой плотности данных.

Алгоритм строится следующим образом:

1) изначально все метки кластеров помечаются как шумовые;
2) после для каждой точки в своём окружении находятся соседи и на их основе отбираются базовые точки;
3) на основе базовых точек и соседей в своём окружении, с помощью метода _dbscan_inner обновляются метки кластеров, которые и будут конечным прогнозом.

_dbscan_inner строится следующим образом:

1) непомеченным базовым точкам присваиваются текущие метки и они добавляются в очередь для дальнейшей обработки;
2) из данной очереди для каждой базовой точки находятся соседи, где каждому непомеченному соседу присваивается текущая метка и он добавляется в очередь из шага 1;
3) далее текущая метка увеличивается на 1 и шаг 2 повторяется пока очередь не станет пустой.

В результате данного процесса каждой точке присваивается метка соответствующего кластера, к которому она принадлежит. Точки, которые не принадлежат ни к одному кластеру, остаются шумовыми с меткой -1.

Импорт необходимых библиотек

import numpy as np
import matplotlib.pyplot as plt
from sklearn.cluster import DBSCAN
from sklearn.neighbors import NearestNeighbors
from sklearn.datasets import make_circles
from sklearn.metrics import adjusted_rand_score
from sklearn.preprocessing import StandardScaler

Реализация на Python с нуля

class DBSCANClustering:
    def __init__(self, eps=0.5, min_samples=5, metric='euclidean', algorithm='auto', leaf_size=30):
        self.eps = eps
        self.min_samples = min_samples
        self.metric = metric
        self.algorithm = algorithm
        self.leaf_size = leaf_size

    @staticmethod
    def _dbscan_inner(core_points, neighborhoods, labels):
        label, queue = 0, []

        for point in range(len(core_points)):
            # if the point is already assigned a label or not a core point, skip it
            if not core_points[point] or labels[point] != -1:
                continue

            labels[point] = label
            queue.append(point)

            while queue:
                current_point = queue.pop(0)
                # if the point is a core point, get it's neighbors
                if core_points[current_point]:
                    current_neighbors = neighborhoods[current_point]

                    for neighbor in current_neighbors:
                        if labels[neighbor] == -1:
                            labels[neighbor] = label
                            queue.append(neighbor)

            label += 1

    def fit_predict(self, X):
        nn = NearestNeighbors(n_neighbors=self.min_samples, radius=self.eps, metric=self.metric,
                              algorithm=self.algorithm, leaf_size=self.leaf_size)

        # find the neighbors for each point within the given radius
        neighborhoods = nn.fit(X).radius_neighbors(X, return_distance=False)
        labels = np.full(len(X), -1, dtype=np.intp)
        core_points = np.asarray([len(n) >= self.min_samples for n in neighborhoods])

        self._dbscan_inner(core_points, neighborhoods, labels)
        self.labels_ = labels

        return self.labels_

Загрузка датасета

X4, y4 = make_circles(n_samples=250, noise=0.05, factor=0.5, random_state=0)
X4 = StandardScaler().fit_transform(X4)
print(y4)


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

Обучение моделей и оценка полученных результатов

Как можно заметить, DBSCAN также отлично справился с кластеризацией данных сложной формы, однако выбор оптимальных eps и min_samples на практике может оказаться довольно трудной задачей, поскольку данные параметры существенно влияют на результаты кластеризации.

Частично данную проблему можно решить с использованием HDBSCAN — модификации DBSCAN, которая автоматически находит подходящее значение eps для каждого кластера, используя иерархический подход, что позволяет определять кластеры разной плотности и повысить устойчивость к выбросам.

Ещё одной интересной модификацией, похожей на HDBSCAN, является OPTICS (Ordering Points To Identify the Clustering Structure), где используется граф достижимости, который определяет достижимое расстояние для каждой точки, которая в дальнейшем будет относиться к ближайшему кластеру. Такой подход позволяет ещё лучше определять кластеры разной плотности, особенно если они расположены близко друг к другу, однако это увеличивает время работы алгоритма.

DBSCAN

dbscan = DBSCANClustering(eps=0.3, min_samples=3)
dbscan_pred_res = dbscan.fit_predict(X4)
dbscan_ari = adjusted_rand_score(y4, dbscan_pred_res)
print(f'Adjusted Rand Score for DBSCAN: {dbscan_ari}', '', sep='\n')
print('prediction', dbscan_pred_res, sep='\n')


Adjusted Rand Score for DBSCAN: 1.0

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

DBSCAN (scikit-learn)

sk_dbscan = DBSCAN(eps=0.3, min_samples=3)
sk_dbscan_pred_res = sk_dbscan.fit_predict(X4)
sk_dbscan_ari = adjusted_rand_score(y4, sk_dbscan_pred_res)
print(f'Adjusted Rand Score for sk DBSCAN: {sk_dbscan_ari}', '', sep='\n')
print('prediction', sk_dbscan_pred_res, sep='\n')


Adjusted Rand Score for sk DBSCAN: 1.0

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

Визуализация прогнозов

plt.figure(figsize=(12, 5))

plt.subplot(1, 2, 1)
plt.scatter(X4[:, 0], X4[:, 1], c=dbscan_pred_res, cmap='rainbow')
plt.title('DBSCAN')
plt.xlabel("Feature 1")
plt.ylabel("Feature 2")

plt.subplot(1, 2, 2)
plt.scatter(X4[:, 0], X4[:, 1], c=sk_dbscan_pred_res, cmap='rainbow')
plt.title('DBSCAN (scikit-learn)')
plt.xlabel("Feature 1")

plt.show()

Ручная реализация vs scikit-learn

Преимущества и недостатки DBSCAN

Преимущества:

устойчив к выбросам;
не требуется заранее указывать количество кластеров;
способность находить кластеры произвольной формы, а также шумные точки в данных.

Недостатки:

плохая работа с кластерами разной плотности;
требуется большой объём памяти для хранения расстояний между всеми точками;
высокая чувствительность к выбору параметров eps и min_samples, что может сильно повлиять на качество кластеризации в негативную сторону.

Дополнительные источники

Статьи:

«DBSCAN: Optimal Rates For Density-Based Cluster Estimation», Daren Wang, Xinyang Lu and Alessandro Rinaldo;
«HDBSCAN: Density based Clustering over Location Based Services», Md Farhadur Rahman, Weimo Liu Saad Bin Suhaim, Saravanan Thirumuruganathan, Nan Zhang, Gautam Das;
«OPTICS: Ordering Points To Identify the Clustering Structure», Mihael Ankerst, Markus M. Breunig, Hans-Peter Kriegel, Jörg Sander.

Документация:

описание DBSCAN ;
описание HDBSCAN ;
описание OPTICS ;
DBSCAN (алгоритм);
HDBSCAN (алгоритм);
OPTICS (алгоритм).

Видео:

DBSCAN: один, два, три.
HDBSCAN: один, два.
OPTICS: один, два.

Affinity Propagation

Ещё более продвинутым подходом относительно предыдущего алгоритма кластеризации является метод распространения близости (affinity propagation), который основан на концепции соотношения между данными и выборе из них экземпляров — наиболее репрезентативных образцов, которые представляются центроидами кластеров и группируют возле себя все остальные данные.

Соотношения между данными описываются с помощью матриц сходства S (similarity matrix), доступности A (availability matrix) и ответственности R (responsibility matrix), а наиболее важными параметрами при настройке алгоритма являются damping (фактор затухания, который не дает алгоритму слишком быстро менять своё мнение о том, какие точки данных лучше всего подходят друг другу) и preference (мера предпочтения точки быть экземпляром для себя или для других точек: чем больше это значение, тем больше вероятность быть экземпляром).

Принцип работы Affinity Propagation

Рассмотрим следующий пример для лучшего понимания сути данного алгоритма на интуитивном уровне. Представьте, что в школе есть много учеников, которые хотят найти своих друзей. Каждый ученик имеет свои предпочтения с кем он хочет дружить, исходя из общих интересов, характера, внешности и так далее. Например, один ученик любит футбол, другой ученик любит музыку, третий ученик любит математику. Эти предпочтения можно измерить числом, которое показывает насколько сильно ученик хочет дружить с другим учеником. Это называется сходством: чем оно больше, тем больше шансов, что ученики станут друзьями.

В данном случае Affinity propagation — это алгоритм, который помогает ученикам найти своих друзей на основе их предпочтений следующим образом:

Сначала каждый ученик оценивает насколько он хочет дружить с другими учениками. Например, ученик A может сказать, что он хочет дружить с учеником B на 8 из 10, с учеником C на 6 из 10, с учеником D на 4 из 10 и так далее. Эти оценки можно рассчитать с помощью отрицательного квадратичного евклидова расстояния и записать в виде матрицы сходства, где каждая строка и столбец соответствует ученику, а каждая ячейка содержит сходство между двумя учениками.

Затем каждый ученик отправляет сообщения другим ученикам, в которых говорит насколько он хочет, чтобы они были его друзьями. Это называется ответственностью. Ответственность — это степень с которой ученик выбирает другого ученика в качестве своего друга. Ответственность зависит не только от сходства, но и от того насколько другие ученики хотят быть друзьями с тем же учеником. Например, если ученик A хочет дружить с учеником B, но ученик B не хочет дружить с учеником A, то ответственность ученика A к ученику B будет низкой. Ну а если ученик A хочет дружить с учеником C, и ученик C тоже хочет дружить с учеником A, то ответственность ученика A к ученику C будет высокой. Ответственность можно вычислить по формуле:

$r(i, k) = s(i, k) - \max_{k' \neq k} \{ a(i, k') + s(i, k') \}$

где r(i, k) — это ответственность ученика i к ученику k, s(i, k) — это сходство ученика i к ученику k, a(i, k') — это доступность ученика i к ученику k', которая будет объяснена далее. Ответственность можно также записать в виде матрицы, где каждая строка и столбец соответствует ученику, а каждая ячейка содержит ответственность между двумя учениками.

Далее каждый ученик получает сообщения от других учеников, в которых они говорят насколько они хотят, чтобы он был их другом. Это называется доступностью. Доступность — это степень с которой ученик подходит для роли друга для другого ученика. Доступность зависит не только от ответственности, но и от того, насколько другие ученики подходят для роли друзей для того же ученика. Например, если ученик A хочет дружить с учеником B и ученик B хочет дружить с учеником A, то доступность ученика A к ученику B будет высокой. Ну а если ученик A хочет дружить с учеником C, но ученик C не хочет дружить с учеником A, то доступность ученика A к ученику C будет низкой. Доступность можно вычислить по формуле:

$a(i, k) = \min \{ 0, r(k, k) + \sum_{i' \neq i, k} \max \{ 0, r(i', k) \} \}$

где a(i, k) — это доступность ученика i к ученику k, r(k, k) — это ответственность ученика k к себе, r(i', k) - это ответственность другого ученика i' к ученику k. Доступность можно также записать в виде матрицы, где каждая строка и столбец соответствует ученику, а каждая ячейка содержит доступность между двумя учениками.

Алгоритм повторяет эти шаги, пока не найдет оптимальное решение, в котором каждый ученик имеет своего друга. Этот друг называется экземпляром. Экземпляры — это ученики, которые лучше всего подходят для роли друзей для других учеников. Количество экземпляров зависит от того, какие предпочтения задаются для учеников. Если предпочтения не задаются, то они равны медиане сходств. Экземпляры можно определить по формуле:

$e(i) = \arg \max_k \{ r(i, k) + a(i, k) \}$

где e(i) — это экземпляр ученика i, r(i, k) — это ответственность ученика i к ученику k, a(i, k) — это доступность ученика i к ученику k. Эта формула означает, что ученик i выбирает в качестве своего друга того ученика k, для которого сумма ответственности и доступности максимальна. Если ученик i выбирает себя в качестве своего друга, то он является экземпляром. Если ученик i выбирает другого ученика k в качестве своего друга, то он принадлежит к тому же кластеру, что и ученик k. Кластер — это группа учеников, которые имеют одного и того же друга-экземпляра.

Данный алгоритм строится следующим образом:

1) на основе отрицательного квадратичного расстояния находится матрица сходства, а также нулями инициализируются матрицы доступности и ответственности;
2) из матрицы сходства удаляются вырождения, а также добавляется небольшой шум;
3) далее итеративно обновляются значения матриц доступности и ответственности на основе временной матрицы, представленной изначально как сумма матриц сходства и доступности, чтобы найти максимальное сходство между точками данных и потенциальными экземплярами;
4) из этой матрицы вычисляются максимальные значения по столбцам, а также вторые по величине значения, которые используются для вычитания из матрицы сходства, чтобы получить новую матрицу ответственности;
5) при обновлении значений матрицы доступности, временная матрица заполняется положительными значениями матрицы ответственности, чтобы посчитать сумму сообщений между точками о том, насколько сильно они хотят быть экземплярами.
6) затем из данной матрицы вычитается её сумма по столбцам, чтобы получить отрицательную матрицу доступности, а также временная матрица обрезается по нулю для получения положительной матрицы доступности;
7) также при обновлении значений матрицы доступности и ответственности к временной матрице применяется коэффициент затухания для избегания численных колебаний при обновлении значений;
8) экземпляры с положительной суммой ответственности и доступности становятся центрами кластеров;
9) после каждой итерации проверяется условие сходимости алгоритма с использованием матрицы сходимости экземпляров: если число экземпляров не меняется в течение заданного числа итераций или если число итераций достигает максимума, то алгоритм останавливается;
10) в конце уточняется итоговый набор экземпляров и меток кластеров через выбор лучших экземпляров для каждого кластера из его членов на основе максимальной суммы сходств между ними;
11) полученные метки кластеров и будут итоговым прогнозом.

Импорт необходимых библиотек

import numpy as np
import matplotlib.pyplot as plt
from sklearn.datasets import make_blobs
from sklearn.cluster import AffinityPropagation
from sklearn.metrics import euclidean_distances, adjusted_rand_score

Реализация на Python с нуля

class AffinityPropagationClustering:
    def __init__(self, damping=0.5, max_iter=200, convergence_iter=15, preference=None, random_state=0):
        self.damping = damping
        self.max_iter = max_iter
        self.convergence_iter = convergence_iter
        self.preference = preference
        self.random_state = random_state

    @staticmethod
    def _affinity_propagation_inner(similarity_matrix, preference, convergence_iter, max_iter,
                                   damping, random_state):
        rng = np.random.RandomState(random_state)
        n_samples = similarity_matrix.shape[0]
        samples_indexes = np.arange(n_samples)

        # place preference on the diagonal of similarity matrix
        similarity_matrix.flat[:: (n_samples + 1)] = preference
        availability_matrix = np.zeros((n_samples, n_samples))
        responsibility_matrix = np.zeros((n_samples, n_samples))  # initialize messages
        exemplars_convergence_matrix = np.zeros((n_samples, convergence_iter))

        # remove degeneracies
        similarity_matrix += (np.finfo(similarity_matrix.dtype).eps * similarity_matrix +
                              np.finfo(similarity_matrix.dtype).tiny * 100) * \
                              rng.standard_normal(size=(n_samples, n_samples))

        for iter in range(max_iter):
            temp_matrix = availability_matrix + similarity_matrix   # compute responsibilities
            max_indexes = np.argmax(temp_matrix, axis=1)
            max_values = temp_matrix[samples_indexes, max_indexes]
            temp_matrix[samples_indexes, max_indexes] = -np.inf
            second_max_values = np.max(temp_matrix, axis=1)

            # temp_matrix = new_responsibility_matrix
            np.subtract(similarity_matrix, max_values[:, None], temp_matrix)
            max_responsibility = similarity_matrix[samples_indexes, max_indexes] - second_max_values
            temp_matrix[samples_indexes, max_indexes] = max_responsibility

            # damping
            temp_matrix *= 1 - damping
            responsibility_matrix *= damping
            responsibility_matrix += temp_matrix

            # temp_matrix = Rp; compute availabilities
            np.maximum(responsibility_matrix, 0, temp_matrix)
            temp_matrix.flat[:: n_samples + 1] = responsibility_matrix.flat[:: n_samples + 1]

            # temp_matrix = -new_availability_matrix
            temp_matrix -= np.sum(temp_matrix, axis=0)
            diag_availability_matrix = np.diag(temp_matrix).copy()
            temp_matrix.clip(0, np.inf, temp_matrix)
            temp_matrix.flat[:: n_samples + 1] = diag_availability_matrix

            # damping
            temp_matrix *= 1 - damping
            availability_matrix *= damping
            availability_matrix -= temp_matrix

            # check for convergence
            exemplar = (np.diag(availability_matrix) + np.diag(responsibility_matrix)) > 0
            exemplars_convergence_matrix[:, iter % convergence_iter] = exemplar
            n_exemplars = np.sum(exemplar, axis=0)

            if iter >= convergence_iter:
                exemplars_sum = np.sum(exemplars_convergence_matrix, axis=1)
                unconverged = np.sum((exemplars_sum == convergence_iter) +
                                     (exemplars_sum == 0)) != n_samples

                if (not unconverged and (n_exemplars > 0)) or (iter == max_iter):
                    break

        exemplar_indixes = np.flatnonzero(exemplar)
        n_exemplars = exemplar_indixes.size  # number of detected clusters

        if n_exemplars > 0:
            cluster_indices = np.argmax(similarity_matrix[:, exemplar_indixes], axis=1)
            cluster_indices[exemplar_indixes] = np.arange(n_exemplars)  # Identify clusters

            # refine the final set of exemplars and clusters and return results
            for k in range(n_exemplars):
                cluster_members = np.where(cluster_indices == k)[0]
                best_k = np.argmax(np.sum(similarity_matrix[cluster_members[:, np.newaxis],
                                                            cluster_members], axis=0))
                exemplar_indixes[k] = cluster_members[best_k]

            cluster_indices = np.argmax(similarity_matrix[:, exemplar_indixes], axis=1)
            cluster_indices[exemplar_indixes] = np.arange(n_exemplars)
            labels = exemplar_indixes[cluster_indices]

            # Reduce labels to a sorted, gapless, list
            cluster_centers_indices = np.unique(labels)
            labels = np.searchsorted(cluster_centers_indices, labels)
        else:
            cluster_centers_indices = []
            labels = np.array([-1] * n_samples)

        return cluster_centers_indices, labels

    def fit_predict(self, X):
        self.affinity_matrix_ = -euclidean_distances(X, squared=True)

        if self.preference is None:
            self.preference = np.median(self.affinity_matrix_)

        params = (self.affinity_matrix_, self.preference, self.convergence_iter, self.max_iter,
                  self.damping, self.random_state)

        self.cluster_centers_indices_, self.labels_ = self._affinity_propagation_inner(*params)
        self.cluster_centers_ = X[self.cluster_centers_indices_]

        return self.labels_

Код для отрисовки графика

def plot_connected_points(X, labels, centers, cmap):
    for i in range(len(X)):
        color = cmap(labels[i] / len(centers))
        plt.plot([X[i, 0], centers[labels[i], 0]], [X[i, 1], centers[labels[i], 1]], c=color, alpha=0.8)

Загрузка датасета

X5, y5 = make_blobs(n_samples=300, centers=9, cluster_std=0.5, random_state=0)
print(y5)


[1 1 1 6 5 7 0 3 2 5 8 4 1 2 0 1 6 2 1 6 6 0 6 2 5 5 5 8 8 4 7 3 8 8 0 7 4
 6 2 0 4 4 7 1 7 5 8 0 5 4 0 3 8 8 2 2 4 7 6 7 4 5 4 5 3 8 4 8 8 4 6 6 2 3
 7 7 4 8 4 1 4 8 7 3 7 8 6 1 3 7 5 0 8 3 5 2 4 2 2 2 7 4 6 2 2 7 5 5 7 2 7
 0 5 7 1 5 3 2 8 3 1 4 2 1 8 5 1 5 5 2 2 1 6 7 4 3 3 3 5 3 6 6 6 5 4 4 2 8
 6 5 7 0 8 8 8 1 6 1 6 6 0 7 7 6 0 8 0 3 6 4 7 3 1 6 0 0 4 1 5 6 1 2 1 0 8
 3 0 7 5 4 3 0 5 2 1 4 6 1 0 0 0 3 8 5 0 6 0 1 5 4 8 2 4 0 1 6 3 7 0 8 4 8
 1 8 1 5 0 0 7 5 4 4 3 0 7 3 2 4 3 5 0 7 6 1 0 7 1 3 5 1 7 3 6 7 6 2 1 3 8
 5 2 3 1 4 6 2 3 8 0 1 6 2 7 6 0 7 1 4 3 5 4 0 2 2 2 1 8 8 7 3 2 0 3 2 3 6
 5 8 2 3]

Обучение моделей и оценка полученных результатов

Affinity Propagation показал отличный результат без настройки параметров, определив верно не только количество кластеров, но и присвоив метки всем точкам данных. Данный алгоритм также способен показывать хорошие результаты для кластеров любой формы и плотности, однако может потребоваться настройка параметров, но это всё равно проще, чем в случае с DBSCAN или HDBSCAN.

AffinityPropagation

ap = AffinityPropagationClustering()
ap_pred_res = ap.fit_predict(X5)
ap_ari = adjusted_rand_score(y5, ap_pred_res)
print(f'Adjusted Rand Score for AffinityPropagation: {ap_ari}', '', sep='\n')
print('prediction', ap_pred_res, sep='\n')


Adjusted Rand Score for AffinityPropagation: 1.0

prediction
[1 1 1 7 6 0 8 5 2 6 3 4 1 2 8 1 7 2 1 7 7 8 7 2 6 6 6 3 3 4 0 5 3 3 8 0 4
 7 2 8 4 4 0 1 0 6 3 8 6 4 8 5 3 3 2 2 4 0 7 0 4 6 4 6 5 3 4 3 3 4 7 7 2 5
 0 0 4 3 4 1 4 3 0 5 0 3 7 1 5 0 6 8 3 5 6 2 4 2 2 2 0 4 7 2 2 0 6 6 0 2 0
 8 6 0 1 6 5 2 3 5 1 4 2 1 3 6 1 6 6 2 2 1 7 0 4 5 5 5 6 5 7 7 7 6 4 4 2 3
 7 6 0 8 3 3 3 1 7 1 7 7 8 0 0 7 8 3 8 5 7 4 0 5 1 7 8 8 4 1 6 7 1 2 1 8 3
 5 8 0 6 4 5 8 6 2 1 4 7 1 8 8 8 5 3 6 8 7 8 1 6 4 3 2 4 8 1 7 5 0 8 3 4 3
 1 3 1 6 8 8 0 6 4 4 5 8 0 5 2 4 5 6 8 0 7 1 8 0 1 5 6 1 0 5 7 0 7 2 1 5 3
 6 2 5 1 4 7 2 5 3 8 1 7 2 0 7 8 0 1 4 5 6 4 8 2 2 2 1 3 3 0 5 2 8 5 2 5 7
 6 3 2 5]

AffinityPropagation (scikit-learn)

sk_ap = AffinityPropagation()
sk_ap_pred_res = sk_ap.fit_predict(X5)
sk_ap_ari = adjusted_rand_score(y5, sk_ap_pred_res)
print(f'Adjusted Rand Score for sk AffinityPropagation: {sk_ap_ari}', '', sep='\n')
print('prediction', sk_ap_pred_res, sep='\n')


Adjusted Rand Score for sk AffinityPropagation: 1.0

prediction
[1 1 1 7 6 0 8 5 2 6 3 4 1 2 8 1 7 2 1 7 7 8 7 2 6 6 6 3 3 4 0 5 3 3 8 0 4
 7 2 8 4 4 0 1 0 6 3 8 6 4 8 5 3 3 2 2 4 0 7 0 4 6 4 6 5 3 4 3 3 4 7 7 2 5
 0 0 4 3 4 1 4 3 0 5 0 3 7 1 5 0 6 8 3 5 6 2 4 2 2 2 0 4 7 2 2 0 6 6 0 2 0
 8 6 0 1 6 5 2 3 5 1 4 2 1 3 6 1 6 6 2 2 1 7 0 4 5 5 5 6 5 7 7 7 6 4 4 2 3
 7 6 0 8 3 3 3 1 7 1 7 7 8 0 0 7 8 3 8 5 7 4 0 5 1 7 8 8 4 1 6 7 1 2 1 8 3
 5 8 0 6 4 5 8 6 2 1 4 7 1 8 8 8 5 3 6 8 7 8 1 6 4 3 2 4 8 1 7 5 0 8 3 4 3
 1 3 1 6 8 8 0 6 4 4 5 8 0 5 2 4 5 6 8 0 7 1 8 0 1 5 6 1 0 5 7 0 7 2 1 5 3
 6 2 5 1 4 7 2 5 3 8 1 7 2 0 7 8 0 1 4 5 6 4 8 2 2 2 1 3 3 0 5 2 8 5 2 5 7
 6 3 2 5]

Визуализация прогнозов

plt.figure(figsize=(12, 5))

plt.subplot(121)
plt.scatter(X5[:, 0], X5[:, 1], c=ap_pred_res, cmap='rainbow', s=10)
plt.scatter(ap.cluster_centers_[:, 0], ap.cluster_centers_[:, 1], c='black', s=50)
plt.title('AffinityPropagation')
plt.xlabel("Feature 1")
plt.ylabel("Feature 2")
plot_connected_points(X5, ap_pred_res, ap.cluster_centers_, plt.cm.rainbow)

plt.subplot(122)
plt.scatter(X5[:, 0], X5[:, 1], c=sk_ap_pred_res, cmap='rainbow', s=10)
plt.scatter(sk_ap.cluster_centers_[:, 0], sk_ap.cluster_centers_[:, 1], c='black', s=50)
plt.title('AffinityPropagation (scikit-learn)')
plt.xlabel("Feature 1")
plot_connected_points(X5, sk_ap_pred_res, sk_ap.cluster_centers_, plt.cm.rainbow)

plt.show()

Ручная реализация vs scikit-learn

Преимущества и недостатки Affinity Propagation

Преимущества:

высокая точность;
автоматическое определение количества кластеров;
не чувствителен к выбору начальных значений;
небольшое количество гиперпараметров для настройки.

Недостатки:

возможна чувствительность к выбросам и шуму в данных;
высокая сложность по времени выполнения и использованию памяти, что делает затруднительным его применение к данным большого размера.

Не смотря на то, что Affinity Propagation является далеко не самым быстрым алгоритмом, на данный момент существует несколько модификаций, способных значительно повысить его скорость. Среди наиболее интересных модификаций можно выделить следующие:

Hierarchical Affinity Propagation (HAP) — алгоритм кластеризации с использованием Affinity Propagation несколько раз на разных уровнях данных. Сначала находятся экземпляры на небольшом подмножестве данных, которые после применяются ко всему набору данных. Такой подход позволяет сократить время выполнения и уменьшить количество сообщений, передаваемых между точками, однако это может повлиять на качество кластеризации.
Fast Affinity Propagation — модификация на основе обрезаний сообщений, которая базируется на двух основных идеях. Во-первых, сокращается размер матрицы сходства путем выбора небольшого подмножества экземпляров в качестве кандидатов в прототипы, а также вычисления сходства только между ними и всеми остальными экземплярами. Во-вторых, используется многомерный поиск для определения параметра предпочтения путем разбиения интервала возможных значений предпочтения на несколько подынтервалов и запуска алгоритма на каждом из них параллельно. Затем выбирается тот подынтервал, который дает наилучшее качество кластеризации по некоторому установленному критерию.
Affinity Propagation на основе пиков плотности — двухэтапный алгоритм кластеризации (DDAP), который сначала определяет пики плотности в данных с помощью алгоритма DDC (Density peaks and Distance-based Clustering), а затем использует стандартный Affinity Propagation для поиска экземпляров, которые близки к этим пикам. Такой подход позволяет значительно уменьшить количество вычислений при сопоставимом качестве кластеризации.

Дополнительные источники

Статьи:

«Extended Affinity Propagation: Global Discovery and Local Insights», Rayyan Ahmad Khan, Rana Ali Amjad, Martin Kleinsteuber;
«Hierarchical Affinity Propagation», Inmar E. Givoni, Clement Chung, Brendan J. Frey;
«Fast Affinity Propagation Clustering based on Machine Learning», Shailendra Kumar Shrivastava, Dr. J.L. Rana and Dr. R.C. Jain;
«Fast Clustering by Affinity Propagation Based on Density Peaks», Yang Li, Chonghui Guo, Leilei Sun.

Документация:

описание Affinity Propagation, Affinity Propagation (алгоритм).

Видео: один, два.

The end

Вот и подошёл к концу обзор данной темы. Очень надеюсь, что вам понравилось, поскольку в дальнейшем я планирую опубликовать в подобном формате статьи по всем популярным алгоритмам машинного обучения с их реализацией с нуля на Python.

Всем успехов!

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8 мар. 2024 г.

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07.04.25 18:32 elizabethrush89

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07.04.25 18:32 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: Recovercapital AT cyberservices. com Contact Telegram: @Capitalcryptorecover
07.04.25 21:03 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 His website: https://recovercapital.wixsite.com/capital-crypto-rec-1
08.04.25 03:00 natashaohnson

Working in conflict zones means improvising. When normal banking channels failed us, our NGO relied on Bitcoin to buy medical supplies directly. It worked, until a missile strike took out our field office, along with the hardware wallet that stored $410,000 in funds. Overnight, our ability to deliver life-saving aid is paralyzed. Amidst the chaos, I reached out to contacts in the humanitarian world. A UN aid worker whispered a name: Cyber Constable Intelligence. "They're the ones who can help you recover lost crypto," he assured me. Despair and hope clashed as I dialed their team on a satellite phone in a conflict zone. What followed was nothing short of a virtual rescue mission. Cyber Constable Intelligence's blockchain forensic experts didn't simply "recover" our assets; they improvised a fix like battlefield medics performing triage. They tracked our wallet's blockchain timestamps, reconstructing lost credentials from synced backups and transaction history. Working under direst duress, we communicated information between spotty internet and backup power sources. Cyber Constable Intelligence team members improvised, rendering security protocols impenetrable as they worked through the jurisdictional nightmares of working within war zones. Every update from them was a pulse that kept our mission alive. After our last available backup failed, they instituted a complex cryptographic reconstruction technique, a process I still don't understand, but it worked. Twelve days later, my satellite device displayed a message: "Access restored. Funds secured." It was not money. It was bandages, antibiotics, clean water, and hope. Thanks to Cyber Constable Intelligence, we replenish our medical supplies, ensuring that patients, innocent victims who had been caught in the crossfire, received the treatment they deserved. More than restoration, they advised us on decentralized storage and multi-signature security for long-term durability. We don't simply utilize Bitcoin presently; we utilize it astutely. Now, each time I sign a crypto transaction, I remember that minute, receiving life-saving medication that might not have come but for this group. In times of war, not every hero wears a uniform. Some carry keyboards, hunting down lost assets and securing humanitarian aid. Cyber Constable Intelligence not only restored our crypto, they kept our mission in the battle. If you think Bitcoin is just an investment, think again. To us, it's a lifeline. CYBER CONSTABLE INTELLIGENCE INFO: WhatsApp: 1 252378-7611 Website info; www.cyberconstableintelligence.com Email Info [email protected] Telegram Info: https://t.me/cyberconstable
10.04.25 23:26 patricialovick86

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10.04.25 23:27 patricialovick86

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11.04.25 00:48 patricialovick86

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11.04.25 03:20 edwardoelliott6

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11.04.25 08:10 Beatrice Gallagher

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13.04.25 01:13 michaeldavenport218

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13.04.25 01:13 michaeldavenport218

I was recently scammed out of $53,000 by a fraudulent Bitcoin investment scheme, which added significant stress to my already difficult health issues, as I was also facing cancer surgery expenses. Desperate to recover my funds, I spent hours researching and consulting other victims, which led me to discover the excellent reputation of Capital Crypto Recover, I came across a Google post It was only after spending many hours researching and asking other victims for advice that I discovered Capital Crypto Recovery’s stellar reputation. I decided to contact them because of their successful recovery record and encouraging client testimonials. I had no idea that this would be the pivotal moment in my fight against cryptocurrency theft. Thanks to their expert team, I was able to recover my lost cryptocurrency back. The process was intricate, but Capital Crypto Recovery's commitment to utilizing the latest technology ensured a successful outcome. I highly recommend their services to anyone who has fallen victim to cryptocurrency fraud. For assistance, contact [email protected] Capital Crypto Recover on Telegram OR Call Number +1 (336)390-6684 via email: [email protected]
14.04.25 23:44 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 His website: https://recovercapital.wixsite.com/capital-crypto-rec-1
17.04.25 03:56 tylerkevin

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18.04.25 20:39 michaeldavenport218

I was recently scammed out of $53,000 by a fraudulent Bitcoin investment scheme, which added significant stress to my already difficult health issues, as I was also facing cancer surgery expenses. Desperate to recover my funds, I spent hours researching and consulting other victims, which led me to discover the excellent reputation of Capital Crypto Recover, I came across a Google post It was only after spending many hours researching and asking other victims for advice that I discovered Capital Crypto Recovery’s stellar reputation. I decided to contact them because of their successful recovery record and encouraging client testimonials. I had no idea that this would be the pivotal moment in my fight against cryptocurrency theft. Thanks to their expert team, I was able to recover my lost cryptocurrency back. The process was intricate, but Capital Crypto Recovery's commitment to utilizing the latest technology ensured a successful outcome. I highly recommend their services to anyone who has fallen victim to cryptocurrency fraud. For assistance, contact [email protected] Capital Crypto Recover on Telegram OR Call Number +1 (336)390-6684 via email: [email protected] you can visit his website: https://recovercapital.wixsite.com/capital-crypto-rec-1
18.04.25 20:39 michaeldavenport218

I was recently scammed out of $53,000 by a fraudulent Bitcoin investment scheme, which added significant stress to my already difficult health issues, as I was also facing cancer surgery expenses. Desperate to recover my funds, I spent hours researching and consulting other victims, which led me to discover the excellent reputation of Capital Crypto Recover, I came across a Google post It was only after spending many hours researching and asking other victims for advice that I discovered Capital Crypto Recovery’s stellar reputation. I decided to contact them because of their successful recovery record and encouraging client testimonials. I had no idea that this would be the pivotal moment in my fight against cryptocurrency theft. Thanks to their expert team, I was able to recover my lost cryptocurrency back. The process was intricate, but Capital Crypto Recovery's commitment to utilizing the latest technology ensured a successful outcome. I highly recommend their services to anyone who has fallen victim to cryptocurrency fraud. For assistance, contact [email protected] Capital Crypto Recover on Telegram OR Call Number +1 (336)390-6684 via email: [email protected] you can visit his website: https://recovercapital.wixsite.com/capital-crypto-rec-1
19.04.25 02:00 [email protected]

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20.04.25 00:45 khouser

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20.04.25 17:22 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. You can reach them via email at [email protected] OR Call/Text Number +1 (336)390-6684 his contact: [email protected]
20.04.25 17:22 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. You can reach them via email at [email protected] OR Call/Text Number +1 (336)390-6684 his contact: [email protected] His website: https://recovercapital.wixsite.com/capital-crypto-rec-1
21.04.25 03:51 gleerandon

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21.04.25 12:19 michaeldavenport218

I was recently scammed out of $53,000 by a fraudulent Bitcoin investment scheme, which added significant stress to my already difficult health issues, as I was also facing cancer surgery expenses. Desperate to recover my funds, I spent hours researching and consulting other victims, which led me to discover the excellent reputation of Capital Crypto Recover, I came across a Google post It was only after spending many hours researching and asking other victims for advice that I discovered Capital Crypto Recovery’s stellar reputation. I decided to contact them because of their successful recovery record and encouraging client testimonials. I had no idea that this would be the pivotal moment in my fight against cryptocurrency theft. Thanks to their expert team, I was able to recover my lost cryptocurrency back. The process was intricate, but Capital Crypto Recovery's commitment to utilizing the latest technology ensured a successful outcome. I highly recommend their services to anyone who has fallen victim to cryptocurrency fraud. For assistance, contact [email protected] Capital Crypto Recover on Telegram OR Call Number +1 (336)390-6684 via email: [email protected]
24.04.25 17:59 cynthia19morris

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24.04.25 18:04 Marcus Sandford

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03.05.25 05:01 melissaholroyd

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08.05.25 03:35 jwright70

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17.05.25 18:33 marywilliam3544

A few months ago, I became a victim of a crypto scam that cost me more than $42,000. It started with what looked like a legitimate investment opportunity on social media. The platform was well-designed, the “advisors” sounded experienced, and the returns they promised seemed realistic at the time. I did some quick research, saw fake positive reviews, and decided to give it a try.At first, everything seemed to go smoothly. I deposited small amounts and saw returns in my account almost immediately. Encouraged, I added more—until I realized I could no longer withdraw anything. The site shut down, the contact numbers went dead, and the people I had spoken to disappeared. That’s when it hit me—I had been scammed.I was embarrassed, angry, and honestly, heartbroken. I didn’t think there was any way to get the money back. My bank couldn’t help, the police didn’t have any tools for crypto tracing, and I felt completely stuck. Then, a friend referred me to PRO WIZARD GILBERT RECOVERY. I was hesitant at first, but after speaking with one of their recovery specialists, I felt a sense of hope I hadn’t felt in weeks. They were incredibly professional, didn’t make any exaggerated claims, and explained their process clearly.The team began working right away using advanced blockchain forensics to trace the stolen funds. Within a few days, they located the wallets that received my funds and started the legal steps necessary to flag and recover the crypto. I was blown away by how thorough and efficient they were After several weeks of careful investigation and coordination, PRO WIZARD GILBERT RECOVERY successfully recovered nearly 75% of my lost funds. I couldn’t believe it. I went from thinking that money was gone forever to having it returned to me—and all thanks to their dedication and expertise.If you’ve lost money to a crypto scam, I can’t recommend PRO WIZARD GILBERT RECOVERY enough. They’re the real deal. Professional, honest, and relentless in fighting for their clients. Thanks to them, I’ve not only recovered most of what I lost, but I’ve also regained peace of mind. CONTACT INFO=====WhatsApp +1 (920) 408‑1234 TELEGRAM====== http s:// t. me/Pro_Wizard_Gilbert_Recovery EMAIL========== pro wizard gilbert recovery (@) engineer. com
19.05.25 05:42 kylebro

It’s been hard losing a lot of my money to these companies. I found a crypto Recovery Agent,VIA Email:( [email protected] ) who made sure I got back everything. If your case is similar, you can consult them on how to get your money back. Whats app or text : +1 949 245 7617 Telegram : @dawsonwright Website : bestrecoveryagent.com
20.05.25 11:40 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 Call/Text: +1 (336) 390-6684 Website: https://recovercapital.wixsite.com/capital-crypto-rec-1
20.05.25 11:40 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 Call/Text: +1 (336) 390-6684 Website: https://recovercapital.wixsite.com/capital-crypto-rec-1
26.05.25 17:55 Juliuslaurence1

It started as an ordinary Tuesday. I was transferring 5 BTC my life savings at the time to a new cold wallet for better security. My fingers moved swiftly, copying the address, pasting it, and hitting Send without a second thought. Then ,the horrow struck.A split second after the transaction confirmed, I realized I had pasted the wrong address. A typo. One wrong character. My stomach twisted into knots as I watched my Bitcoin disappear into the blockchain abyss, headed to an address I didn’t recognize. Panic set in. 5 BTC. Over $300,000 at the time. Gone. I spent sleepless nights scouring forums, reaching out to blockchain experts, even pleading with miners but the immutable nature of Bitcoin meant no one could reverse it. The address was active, but the owner? Unknown. Then, I stumbled upon Washington Recovery Pro, a firm specializing in cryptocurrency transaction reversals and asset recovery. Skeptical but desperate, I reached out. Their team, led by a seasoned blockchain investigator named Marcus, took my case. They explained that while Bitcoin transactions are irreversible, if the recipient is identifiable, recovery is possible through negotiation or legal pressure. Using on-chain forensics, they traced the mistaken transaction to a crypto exchange wallet meaning the recipient had likely cashed out or moved the funds. Washington Recovery Pro contacted the exchange, providing irrefutable proof of the erroneous transfer. After weeks of pressure, the exchange froze the funds and initiated a recovery process. two weeks after my heart-stopping mistake, I received an email: Your 5 BTC has been recovered and will be returned to your wallet. I couldn’t believe it. Washington Recovery Pro had done the impossible. My Bitcoin was saved, but not by luck by skill, persistence, and legal expertise. If you ever face a similar nightmare, remember: some mistakes can be undone. Email: [email protected]: +1 (903) 249‑8633‬
30.05.25 05:03 kevinpatmore2

Good day everyone, some statements people make may appear unbelievable, yet they reflect their truth due to their personal experiences. I'm a truck driver and I enjoy playing the lottery, but it's been 8 years without winning anything significant. I became upset over this and chose to seek help online from anyone skilled in using spells to win the lottery since my grandma had faith in spells and a story about Lord Meduza spells drew my interest. I obtained his contact information, and we discussed all aspects of how he could assist me. In less than 72 hours, Lord Meduza provided me with the lottery numbers I required after he completed the spell for me. I purchased my ticket online, followed the guidance of Lord Meduza, and now my life is fortunate because I won a $42.5 million Jackpot in the lottery I participated in. I’m thankful to Lord Meduza because he truly keeps his promises and I will donate $1 million to the orphanage homes in my city. Email: [email protected] or WhatsApp +18079072687.
01.06.25 17:55 springthorpecameron5

After losing $153,000 to a fake investment platform on Telegram, I found myself under immense pressure due to both the financial loss and my ongoing cancer treatment. Desperate to recover my stolen cryptocurrency, I researched viable options and discovered Ruder Cyber Tech Sleuths, a registered Bitcoin and USDT Recovery Agency with a strong reputation. Their impressive record of successful recoveries and positive reviews gave me hope. Despite my initial doubts, reaching out to them was a pivotal moment in my recovery journey. The process was complex, but their dedication, advanced technology, and professionalism were evident. I highly recommend their services to anyone who has fallen victim to scams. Ruder Cyber Tech Sleuths are truly exceptional and provide hope in an often dishonest world. [email protected] [email protected] whatsapp: +12132801476 Telegram : @rudercybersleuths
02.06.25 16:01 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
02.06.25 16:01 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
03.06.25 22:21 [email protected]

When I lost $120,000 to obvious price manipulation on a shady derivatives platform, I felt a mix of disbelief and frustration. I had been drawn to this platform by promises of high returns and innovative trading features that seemed too good to be true. Initially, everything appeared to be going well, and I was excited about the potential profits. However, it didn’t take long for me to notice some alarming signs. As I continued to trade, I began to see unusual price fluctuations that didn’t align with the broader market trends. It was as if the platform was orchestrating these movements to manipulate prices in their favor. Despite my growing suspicions, the allure of quick profits kept me engaged. I convinced myself that I could navigate the volatility and come out ahead. Unfortunately, that was a grave mistake. I suffered a significant loss, which wiped out a substantial portion of my investment. It was a gut-wrenching moment, and I felt utterly defeated. In my desperation, I reached out to Web-site: www://trustgeekshackexpert.com/ --- (Email: [email protected]] a firm that specializes in recovering lost funds from fraudulent platforms. I hoped they could help me reclaim at least a portion of my losses.[TRUST GEEKS HACK EXPERT] was incredibly supportive and trustyworthy. They conducted a thorough investigation into the platform’s operations and quickly uncovered the truth was the exchange was fake and designed to exploit unsuspecting traders like me. [TRUST GEEKS HACK EXPERT] revealed that the operators had created a façade of legitimacy, complete with fake testimonials and misleading marketing strategies, to lure in victims.With the evidence gathered by [TRUST GEEKS HACK EXPERT], they took decisive action. They traced my lost funds to the operators’ cold wallets, which are typically used to store cryptocurrencies securely offline. This was a crucial step, as it allowed [TRUST GEEKS HACK EXPERT] to pinpoint where my stolen funds were held. Through a combination of legal pressure and technical expertise, they managed to negotiate the return of $95,000 to me a significant portion of my initial loss.This has been a stark reminder of the risks associated with trading on unregulated platforms. Thanks to [TRUST GEEKS HACK EXPERT], I learned the hard way that it’s essential to conduct thorough research before engaging with any trading service. I now recognize the signs of potential fraud, such as unrealistic profit promises and a lack of transparency. The successful recovery of my funds by [TRUST GEEKS HACK EXPERT]
06.06.25 18:43 jack67667766

Jasmine Lopez is an expert in retrieving stolen cryptocurrency, including Ethereum and USDT. Her effective methods make her a reliable ally for theft victims. One client, who lost €908,000, sought her assistance, and Jasmine managed to recover the full amount within a day, bringing immense relief to the client. Jasmine is dedicated to helping others in similar situations and is available to offer support. For help with recovering lost funds, you can contact her via email at [email protected] or on WhatsApp at ‪+44 736 644 5035‬.
08.06.25 12:49 jack67667766

Jasmine Lopez es una experta en recuperar criptomonedas robadas, como Ethereum y USDT. Ella usa métodos efectivos que la hacen una buena ayuda para quienes han sido víctimas de robo. Un cliente que perdió 908 mil euros acudió a ella, y en un día, Jasmine logró devolverle toda la cantidad. Esto trajo gran alivio al cliente. Ella tiene como objetivo ayudar a otros en situaciones similares y está lista para ofrecer apoyo. Si necesitas ayuda para recuperar fondos perdidos, puedes contactarla por correo electrónico en [email protected] o por WhatsApp al número ‪+44 736 644 5035‬.
09.06.25 19:49 garnierroux

CRYPTO RECOVERY COMPANY: CONTACT iBOLT CYBER HACKER RECOVERY COMPANY I was a victim of a cryptocurrency scam and thought I had lost everything. After doing some research, I came across iBOLT CYBER HACKER RECOVERY COMPANY and decided to give them a try. To my surprise, they were extremely professional, responsive, and knowledgeable from the start. Their team worked diligently to trace the stolen funds, and within a short time, they were able to recover a significant portion of my lost crypto. I couldn’t believe it — something I thought was gone forever was returned to me. If you've lost money to a crypto scam, I highly recommend contacting iBOLT CYBER HACKER RECOVERY COMPANY. They truly deliver on what they promise. — Satisfied Client . EMAIL: [email protected]/ . WHTSAPP: +39 351 105 3619 . TELEGRAM: t.me/iboltcyberhackservice . WEBSITE: https://iboltcyberhack.org/
21.06.25 04:37 jamesgrant

The emergence of cryptocurrencies has revolutionized the financial landscape, yet it has also given rise to significant challenges, particularly concerning security and the potential for loss of assets. In this context, individuals often seek assistance from entities called SCANNER HACKER CRYPTO RECOVERY, which purport to employ sophisticated techniques to reclaim lost Bitcoin (BTC) from compromised wallets or exchanges. However, while the allure of recovering lost assets can be compelling, it is imperative to approach these services with a critical lens. The cryptocurrency ecosystem is rife with scams and fraudulent schemes, ranging from phishing attacks to outright theft; thus, due diligence is paramount. Scholars and practitioners in the field must advocate for more robust security measures, educate stakeholders on the inherent risks of cryptocurrency ownership, and promote best practices in digital asset management to mitigate the likelihood of loss. Furthermore, effective policymaking to regulate these recovery services, hold malicious actors accountable, and protect consumers is essential to fostering a safer environment in the rapidly evolving world of digital currencies. You can communicate with SCANNER HACKER CRYPTO RECOVERY Via Email: [email protected] Web: https://scannerhacktech.com/ Whatsapp: +1 431 801 7493
23.06.25 15:44 michaeldavenport218

I was recently scammed out of $53,000 by a fraudulent Bitcoin investment scheme, which added significant stress to my already difficult health issues, as I was also facing cancer surgery expenses. Desperate to recover my funds, I spent hours researching and consulting other victims, which led me to discover the excellent reputation of Capital Crypto Recover, I came across a Google post It was only after spending many hours researching and asking other victims for advice that I discovered Capital Crypto Recovery’s stellar reputation. I decided to contact them because of their successful recovery record and encouraging client testimonials. I had no idea that this would be the pivotal moment in my fight against cryptocurrency theft. Thanks to their expert team, I was able to recover my lost cryptocurrency back. The process was intricate, but Capital Crypto Recovery's commitment to utilizing the latest technology ensured a successful outcome. I highly recommend their services to anyone who has fallen victim to cryptocurrency fraud. For assistance, contact [email protected] Capital Crypto Recover on Telegram OR Call Number +1 (336)390-6684 via email: [email protected] you can visit his website: https://recovercapital.wixsite.com/capital-crypto-rec-1
23.06.25 15:44 michaeldavenport218

I was recently scammed out of $53,000 by a fraudulent Bitcoin investment scheme, which added significant stress to my already difficult health issues, as I was also facing cancer surgery expenses. Desperate to recover my funds, I spent hours researching and consulting other victims, which led me to discover the excellent reputation of Capital Crypto Recover, I came across a Google post It was only after spending many hours researching and asking other victims for advice that I discovered Capital Crypto Recovery’s stellar reputation. I decided to contact them because of their successful recovery record and encouraging client testimonials. I had no idea that this would be the pivotal moment in my fight against cryptocurrency theft. Thanks to their expert team, I was able to recover my lost cryptocurrency back. The process was intricate, but Capital Crypto Recovery's commitment to utilizing the latest technology ensured a successful outcome. I highly recommend their services to anyone who has fallen victim to cryptocurrency fraud. For assistance, contact [email protected] Capital Crypto Recover on Telegram OR Call Number +1 (336)390-6684 via email: [email protected] you can visit his website: https://recovercapital.wixsite.com/capital-crypto-rec-1
24.06.25 04:10 Beth Wilson

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24.06.25 04:10 Beth Wilson

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24.06.25 04:10 Beth Wilson

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24.06.25 04:10 Beth Wilson

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24.06.25 04:12 Beth Wilson

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24.06.25 04:14 Beth Wilson

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24.06.25 04:14 Beth Wilson

CONTACT A LEGITIMATE CRYPTOCURRENCY RECOVERY SPECIALISTS // THE HACK ANGELS I'd like to take this opportunity to express my gratitude to A Legitimate Bitcoin Recovery Specialists THE HACK ANGELS. I wholeheartedly recommend THE HACK ANGELS to anyone dealing with lost or stolen cryptocurrency. They have deep understanding of blockchain technology, combined with advanced cyber forensic tools, makes them a powerful ally in asset recovery. THE HACK ANGELS is a trusted and reliable solution for individuals seeking to recover lost or inaccessible cryptocurrency assets, if you are experiencing similar troubles and you need help in recovering your money. I strongly advise you to contact them via details below. WhatsApp +1(520)200-2320 Email at [email protected] Website at www.thehackangels.com I am here to give glory to THE HACK ANGELS for assisting me in recovering my lost cryptocurrency.
24.06.25 04:18 Beth Wilson

CONTACT A LEGITIMATE CRYPTOCURRENCY RECOVERY SPECIALISTS // THE HACK ANGELS I'd like to take this opportunity to express my gratitude to A Legitimate Bitcoin Recovery Specialists THE HACK ANGELS. I wholeheartedly recommend THE HACK ANGELS to anyone dealing with lost or stolen cryptocurrency. They have deep understanding of blockchain technology, combined with advanced cyber forensic tools, makes them a powerful ally in asset recovery. THE HACK ANGELS is a trusted and reliable solution for individuals seeking to recover lost or inaccessible cryptocurrency assets, if you are experiencing similar troubles and you need help in recovering your money. I strongly advise you to contact them via details below. WhatsApp +1(520)200-2320 Email at [email protected] Website at www.thehackangels.com I am here to give glory to THE HACK ANGELS for assisting me in recovering my lost cryptocurrency.
24.06.25 04:23 Beth Wilson

CONTACT A LEGITIMATE CRYPTOCURRENCY RECOVERY SPECIALISTS // THE HACK ANGELS I'd like to take this opportunity to express my gratitude to A Legitimate Bitcoin Recovery Specialists THE HACK ANGELS. I wholeheartedly recommend THE HACK ANGELS to anyone dealing with lost or stolen cryptocurrency. They have deep understanding of blockchain technology, combined with advanced cyber forensic tools, makes them a powerful ally in asset recovery. THE HACK ANGELS is a trusted and reliable solution for individuals seeking to recover lost or inaccessible cryptocurrency assets, if you are experiencing similar troubles and you need help in recovering your money. I strongly advise you to contact them via details below. WhatsApp +1(520)200-2320 Email at [email protected] Website at www.thehackangels.com I am here to give glory to THE HACK ANGELS for assisting me in recovering my lost cryptocurrency.
25.06.25 22:56 rennebrandon

I am forever grateful to the amazing team at Dune Nectar Web Expert. They did what I thought was never possible. They played a crucial role in helping me recover all my lost funds from a fraudulent forex and crypto trading scheme, including the pråofits I thought I'd earned. Looking back, I realize I was a bit of a fool for trusting greedy and deceitful brokers with my hard-earned money. However, I'm overjoyed that I found Dune Nectar Web Expert. They are a team of honest and highly skilled professionals available for hire. Dune Nectar Web Expert helps individuals and organizations recover stolen cryptocurrencies and digital assets. They helped me heal every penny I lost and provided me with the right signals and a reliable platform to trade with. Thanks to Dune Nectar Web Expert, I'm earning more than ever, and I couldn't be happier. That's why I can't stop sharing my positive experience and praising their amazing services and expertise. If you're still struggling with failures in binary options, crypto, or forex trading, or if you're looking to recover your lost USDC, Bitcoin, Ethereum, or other crypto funds, I strongly advise you to reach out to: Telegram>>> ( T.me/dunenectarwebexpert ) Mail>>>> Support (@) Dunenectarwebexpert (.) Com Web>>> https://dunenectarwebexpert.com/
26.06.25 20:33 paulmargaret

Dear Wizard Hilton Cyber Tech Team, I am writing to express my heartfelt gratitude and appreciation for your exceptional service in recovering my scammed funds. After falling victim to a scam and experiencing numerous disappointments, I was amazed by your team's professionalism, expertise, and dedication to helping me retrieve my lost money. Your prompt response and efficient handling of my case were truly impressive. Within 24 hours, you successfully recovered my funds, and I was overjoyed to see the money safely returned to my account. Your team's commitment to helping individuals like me, who have been scammed, is truly commendable. Your work not only restored my financial security but also gave me peace of mind and renewed trust in the digital world. I would like to highly recommend your services to anyone who has been a victim of scams or cybercrime. Your expertise and support can make a significant difference in recovering lost funds and navigating complex digital situations. Thank you again for your outstanding work and dedication. I am grateful for your help and wish you continued success in your endeavors Email : wizardhiltoncybertech ( @ ) gmail (. ) com OR support ( @ ) wizardhiltoncybertech (.) com WhatsApp number +18737715701 . Sincerely, Kimberly Williams
27.06.25 00:46 paulmargaret

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27.06.25 22:31 rennebrandon

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28.06.25 00:46 patricialovick86

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28.06.25 00:46 patricialovick86

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29.06.25 07:39 paulmargaret

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29.06.25 11:52 wendytaylor015

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30.06.25 15:06 wendytaylor015

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