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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
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 KMeansРеализация 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_labelsЗагрузка 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).
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.
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.
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.
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.
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 AgglomerativeClusteringРеализация 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).
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
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_scoreРеализация 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 labelsЗагрузка 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).
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 StandardScalerРеализация 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:
Видео:
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:
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:
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:
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_scoreРеализация 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).
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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07.09.23 16:24
CherryTeam
Cherry Team atlyginimų skaičiavimo programa yra labai naudingas įrankis įmonėms, kai reikia efektyviai valdyti ir skaičiuoti darbuotojų atlyginimus. Ši programinė įranga, turinti išsamias funkcijas ir patogią naudotojo sąsają, suteikia daug privalumų, kurie padeda supaprastinti darbo užmokesčio skaičiavimo procesus ir pagerinti finansų valdymą. Štai keletas pagrindinių priežasčių, kodėl Cherry Team atlyginimų skaičiavimo programa yra naudinga įmonėms: Automatizuoti ir tikslūs skaičiavimai: Atlyginimų skaičiavimai rankiniu būdu gali būti klaidingi ir reikalauti daug laiko. Programinė įranga Cherry Team automatizuoja visą atlyginimų skaičiavimo procesą, todėl nebereikia atlikti skaičiavimų rankiniu būdu ir sumažėja klaidų rizika. Tiksliai apskaičiuodama atlyginimus, įskaitant tokius veiksnius, kaip pagrindinis atlyginimas, viršvalandžiai, premijos, išskaitos ir mokesčiai, programa užtikrina tikslius ir be klaidų darbo užmokesčio skaičiavimo rezultatus. Sutaupoma laiko ir išlaidų: Darbo užmokesčio valdymas gali būti daug darbo jėgos reikalaujanti užduotis, reikalaujanti daug laiko ir išteklių. Programa Cherry Team supaprastina ir pagreitina darbo užmokesčio skaičiavimo procesą, nes automatizuoja skaičiavimus, generuoja darbo užmokesčio žiniaraščius ir tvarko išskaičiuojamus mokesčius. Šis automatizavimas padeda įmonėms sutaupyti daug laiko ir pastangų, todėl žmogiškųjų išteklių ir finansų komandos gali sutelkti dėmesį į strategiškai svarbesnę veiklą. Be to, racionalizuodamos darbo užmokesčio operacijas, įmonės gali sumažinti administracines išlaidas, susijusias su rankiniu darbo užmokesčio tvarkymu. Mokesčių ir darbo teisės aktų laikymasis: Įmonėms labai svarbu laikytis mokesčių ir darbo teisės aktų, kad išvengtų baudų ir teisinių problemų. Programinė įranga Cherry Team seka besikeičiančius mokesčių įstatymus ir darbo reglamentus, užtikrindama tikslius skaičiavimus ir teisinių reikalavimų laikymąsi. Programa gali dirbti su sudėtingais mokesčių scenarijais, pavyzdžiui, keliomis mokesčių grupėmis ir įvairių rūšių atskaitymais, todėl užtikrina atitiktį reikalavimams ir kartu sumažina klaidų riziką. Ataskaitų rengimas ir analizė: Programa Cherry Team siūlo patikimas ataskaitų teikimo ir analizės galimybes, suteikiančias įmonėms vertingų įžvalgų apie darbo užmokesčio duomenis. Ji gali generuoti ataskaitas apie įvairius aspektus, pavyzdžiui, darbo užmokesčio paskirstymą, išskaičiuojamus mokesčius ir darbo sąnaudas. Šios ataskaitos leidžia įmonėms analizuoti darbo užmokesčio tendencijas, nustatyti tobulintinas sritis ir priimti pagrįstus finansinius sprendimus. Pasinaudodamos duomenimis pagrįstomis įžvalgomis, įmonės gali optimizuoti savo darbo užmokesčio strategijas ir veiksmingai kontroliuoti išlaidas. Integracija su kitomis sistemomis: Cherry Team programinė įranga dažnai sklandžiai integruojama su kitomis personalo ir apskaitos sistemomis. Tokia integracija leidžia automatiškai perkelti atitinkamus duomenis, pavyzdžiui, informaciją apie darbuotojus ir finansinius įrašus, todėl nebereikia dubliuoti duomenų. Supaprastintas duomenų srautas tarp sistemų padidina bendrą efektyvumą ir sumažina duomenų klaidų ar neatitikimų riziką. Cherry Team atlyginimų apskaičiavimo programa įmonėms teikia didelę naudą - automatiniai ir tikslūs skaičiavimai, laiko ir sąnaudų taupymas, atitiktis mokesčių ir darbo teisės aktų reikalavimams, ataskaitų teikimo ir analizės galimybės bei integracija su kitomis sistemomis. Naudodamos šią programinę įrangą įmonės gali supaprastinti darbo užmokesčio skaičiavimo procesus, užtikrinti tikslumą ir atitiktį reikalavimams, padidinti darbuotojų pasitenkinimą ir gauti vertingų įžvalgų apie savo finansinius duomenis. Programa Cherry Team pasirodo esanti nepakeičiamas įrankis įmonėms, siekiančioms efektyviai ir veiksmingai valdyti darbo užmokestį. https://cherryteam.lt/lt/
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08.10.23 01:30
davec8080
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24.06.24 04:31
tashandiarisha
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26.06.24 18:46
Jacobethannn098
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04.07.24 04:49
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13.07.24 21:13
michaelharrell825
In 2020, amidst the economic fallout of the pandemic, I found myself unexpectedly unemployed and turned to Forex trading in hopes of stabilizing my finances. Like many, I was drawn in by the promise of quick returns offered by various Forex robots, signals, and trading advisers. However, most of these products turned out to be disappointing, with claims that were far from reality. Looking back, I realize I should have been more cautious, but the allure of financial security clouded my judgment during those uncertain times. Amidst these disappointments, Profit Forex emerged as a standout. Not only did they provide reliable service, but they also delivered tangible results—a rarity in an industry often plagued by exaggerated claims. The positive reviews from other users validated my own experience, highlighting their commitment to delivering genuine outcomes and emphasizing sound financial practices. My journey with Profit Forex led to a net profit of $11,500, a significant achievement given the challenges I faced. However, my optimism was short-lived when I encountered obstacles trying to withdraw funds from my trading account. Despite repeated attempts, I found myself unable to access my money, leaving me frustrated and uncertain about my financial future. Fortunately, my fortunes changed when I discovered PRO WIZARD GIlBERT RECOVERY. Their reputation for recovering funds from fraudulent schemes gave me hope in reclaiming what was rightfully mine. With a mixture of desperation and cautious optimism, I reached out to them for assistance. PRO WIZARD GIlBERT RECOVERY impressed me from the start with their professionalism and deep understanding of financial disputes. They took a methodical approach, using advanced techniques to track down the scammers responsible for withholding my funds. Throughout the process, their communication was clear and reassuring, providing much-needed support during a stressful period. Thanks to PRO WIZARD GIlBERT RECOVERY's expertise and unwavering dedication, I finally achieved a resolution to my ordeal. They successfully traced and retrieved my funds, restoring a sense of justice and relief. Their intervention not only recovered my money but also renewed my faith in ethical financial services. Reflecting on my experience, I've learned invaluable lessons about the importance of due diligence and discernment in navigating the Forex market. While setbacks are inevitable, partnering with reputable recovery specialists like PRO WIZARD GIlBERT RECOVERY can make a profound difference. Their integrity and effectiveness have left an indelible mark on me, guiding my future decisions and reinforcing the value of trustworthy partnerships in achieving financial goals. I wholeheartedly recommend PRO WIZARD GIlBERT RECOVERY to anyone grappling with financial fraud or disputes. Their expertise and commitment to client satisfaction are unparalleled, offering a beacon of hope in challenging times. Thank you, PRO WIZARD GIlBERT RECOVERY, for your invaluable assistance in reclaiming what was rightfully mine. Your service not only recovered my funds but also restored my confidence in navigating the complexities of financial markets with greater caution and awareness. Email: prowizardgilbertrecovery(@)engineer.com Homepage: https://prowizardgilbertrecovery.xyz WhatsApp: +1 (516) 347‑9592
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17.07.24 02:26
thompsonrickey
In the vast and often treacherous realm of online investments, I was entangled in a web of deceit that cost me nearly $45,000. It all started innocuously enough with an enticing Instagram profile promising lucrative returns through cryptocurrency investment. Initially, everything seemed promising—communications were smooth, and assurances were plentiful. However, as time passed, my optimism turned to suspicion. Withdrawal requests were met with delays and excuses. The once-responsive "investor" vanished into thin air, leaving me stranded with dwindling hopes and a sinking feeling in my gut. It became painfully clear that I had been duped by a sophisticated scheme designed to exploit trust and naivety. Desperate to recover my funds, I turned to online forums where I discovered numerous testimonials advocating for Muyern Trust Hacker. With nothing to lose, I contacted them, recounting my ordeal with a mixture of skepticism and hope. Their swift response and professional demeanor immediately reassured me that I had found a lifeline amidst the chaos. Muyern Trust Hacker wasted no time in taking action. They meticulously gathered evidence, navigated legal complexities, and deployed their expertise to expedite recovery. In what felt like a whirlwind of activity, although the passage of time was a blur amidst my anxiety, they achieved the seemingly impossible—my stolen funds were returned. The relief I felt was overwhelming. Muyern Trust Hacker not only restored my financial losses but also restored my faith in justice. Their commitment to integrity and their relentless pursuit of resolution were nothing short of remarkable. They proved themselves as recovery specialists and guardians against digital fraud, offering hope to victims like me who had been ensnared by deception. My gratitude knows no bounds for Muyern Trust Hacker. Reach them at muyerntrusted @ m a i l - m e . c o m AND Tele gram @ muyerntrusthackertech
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18.07.24 20:13
austinagastya
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27.08.24 12:50
James889900
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27.08.24 13:06
James889900
All you need is to hire an expert to help you accomplish that. If there’s any need to spy on your partner’s phone. From my experience I lacked evidence to confront my husband on my suspicion on his infidelity, until I came across ETHICALAHCKERS which many commend him of assisting them in their spying mission. So I contacted him and he provided me with access into his phone to view all text messages, call logs, WhatsApp messages and even her location. This evidence helped me move him off my life . I recommend you consult ETHICALHACKERS009 @ gmail.com OR CALL/TEXT +1(716) 318-5536 or whatsapp +14106350697 if you need access to your partner’s phone
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02.09.24 20:24
[email protected]
If You Need Hacker To Recover Your Bitcoin Contact Paradox Recovery Wizard Paradox Recovery Wizard successfully recovered $123,000 worth of Bitcoin for my husband, which he had lost due to a security breach. The process was efficient and secure, with their expert team guiding us through each step. They were able to trace and retrieve the lost cryptocurrency, restoring our peace of mind and financial stability. Their professionalism and expertise were instrumental in recovering our assets, and we are incredibly grateful for their service. Email: support@ paradoxrecoverywizard.com Email: paradox_recovery @cyberservices.com Wep: https://paradoxrecoverywizard.com/ WhatsApp: +39 351 222 3051.
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06.09.24 01:35
Celinagarcia
HOW TO RECOVER MONEY LOST IN BITCOIN/USDT TRADING OR TO CRYPTO INVESTMENT !! Hi all, friends and families. I am writing From Alberton Canada. Last year I tried to invest in cryptocurrency trading in 2023, but lost a significant amount of money to scammers. I was cheated of my money, but thank God, I was referred to Hack Recovery Wizard they are among the best bitcoin recovery specialists on the planet. they helped me get every penny I lost to the scammers back to me with their forensic techniques. and I would like to take this opportunity to advise everyone to avoid making cryptocurrency investments online. If you have already lost money on forex, cryptocurrency or Ponzi schemes, please contact [email protected] or WhatsApp: +1 (757) 237–1724 at once they can help you get back the crypto you lost to scammers. BEST WISHES. Celina Garcia.
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06.09.24 01:44
Celinagarcia
HOW TO RECOVER MONEY LOST IN BITCOIN/USDT TRADING OR TO CRYPTO INVESTMENT !! Hi all, friends and families. I am writing From Alberton Canada. Last year I tried to invest in cryptocurrency trading in 2023, but lost a significant amount of money to scammers. I was cheated of my money, but thank God, I was referred to Hack Recovery Wizard they are among the best bitcoin recovery specialists on the planet. they helped me get every penny I lost to the scammers back to me with their forensic techniques. and I would like to take this opportunity to advise everyone to avoid making cryptocurrency investments online. If you have already lost money on forex, cryptocurrency or Ponzi schemes, please contact [email protected] or WhatsApp: +1 (757) 237–1724 at once they can help you get back the crypto you lost to scammers. BEST WISHES. Celina Garcia.
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16.09.24 00:10
marcusaustin
Bitcoin Recovery Services: Restoring Lost Cryptocurrency If you've lost access to your cryptocurrency and unable to make a withdrawal, I highly recommend iBolt Cyber Hacker Bitcoin Recovery Services. Their team is skilled, professional, and efficient in recovering lost Bitcoin. They provide clear communication, maintain high security standards, and work quickly to resolve issues. Facing the stress of lost cryptocurrency, iBolt Cyber Hacker is a trusted service that will help you regain access to your funds securely and reliably. Highly recommended! Email: S u p p o r t @ ibolt cyber hack . com Cont/Whtp + 3. .9 .3. .5..0. .9. 2. 9. .0 .3. 1 .8. Website: h t t p s : / / ibolt cyber hack . com /
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16.09.24 00:11
marcusaustin
Bitcoin Recovery Services: Restoring Lost Cryptocurrency If you've lost access to your cryptocurrency and unable to make a withdrawal, I highly recommend iBolt Cyber Hacker Bitcoin Recovery Services. Their team is skilled, professional, and efficient in recovering lost Bitcoin. They provide clear communication, maintain high security standards, and work quickly to resolve issues. Facing the stress of lost cryptocurrency, iBolt Cyber Hacker is a trusted service that will help you regain access to your funds securely and reliably. Highly recommended! Email: S u p p o r t @ ibolt cyber hack . com Cont/Whtp + 3. .9 .3. .5..0. .9. 2. 9. .0 .3. 1 .8. Website: h t t p s : / / ibolt cyber hack . com /
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23.09.24 18:56
matthewshimself
At first, I was admittedly skeptical about Worldcoin (ref: https://worldcoin.org/blog/worldcoin/this-is-worldcoin-video-explainer-series), particularly around the use of biometric data and the WLD token as a reward mechanism for it. However, after following the project closer, I’ve come to appreciate the broader vision and see the value in the underlying tech behind it. The concept of Proof of Personhood (ref: https://worldcoin.org/blog/worldcoin/proof-of-personhood-what-it-is-why-its-needed) has definitely caught my attention, and does seem like a crucial step towards tackling growing issues like bots, deepfakes, and identity fraud. Sam Altman’s vision is nothing short of ambitious, but I do think he & Alex Blania have the chops to realize it as mainstay in the global economy.
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01.10.24 14:54
Sinewclaudia
I lost about $876k few months ago trading on a fake binary option investment websites. I didn't knew they were fake until I tried to withdraw. Immediately, I realized these guys were fake. I contacted Sinew Claudia world recovery, my friend who has such experience before and was able to recover them, recommended me to contact them. I'm a living testimony of a successful recovery now. You can contact the legitimate recovery company below for help and assistance. [email protected] [email protected] WhatsApp: 6262645164
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02.10.24 22:27
Emily Hunter
Can those who have fallen victim to fraud get their money back? Yes, you might be able to get back what was taken from you if you fell prey to a fraud from an unregulated investing platform or any other scam, but only if you report it to the relevant authorities. With the right plan and supporting documentation, you can get back what you've lost. Most likely, the individuals in control of these unregulated platforms would attempt to convince you that what happened to your money was a sad accident when, in fact, it was a highly skilled heist. You should be aware that there are resources out there to help you if you or someone you know has experienced one of these circumstances. Do a search using (deftrecoup (.) c o m). Do not let the perpetrators of this hoaxes get away with ruining you mentally and financially.
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18.10.24 09:34
freidatollerud
The growth of WIN44 in Brazil is very interesting! If you're looking for more options for online betting and casino games, I recommend checking out Casinos in Brazil. It's a reliable platform that offers a wide variety of games and provides a safe and enjoyable experience for users. It's worth checking out! https://win44.vip
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31.10.24 00:13
ytre89
Can those who have fallen victim to fraud get their money back? Yes, you might be able to get back what was taken from you if you fell prey to a fraud from an unregulated investing platform or any other scam, but only if you report it to the relevant authorities. With the right plan and supporting documentation, you can get back what you've lost. Most likely, the individuals in control of these unregulated platforms would attempt to convince you that what happened to your money was a sad accident when, in fact, it was a highly skilled heist. You should be aware that there are resources out there to help you if you or someone you know has experienced one of these circumstances. Do a search using (deftrecoup (.) c o m). Do not let the perpetrators of this hoaxes get away with ruining you mentally and financially.
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02.11.24 14:44
diannamendoza732
In the world of Bitcoin recovery, Pro Wizard Gilbert truly represents the gold standard. My experience with Gilbert revealed just how exceptional his methods are and why he stands out as the premier authority in this critical field. When I first encountered the complexities of Bitcoin recovery, I was daunted by the technical challenges and potential risks. Gilbert’s approach immediately distinguished itself through its precision and effectiveness. His methods are meticulously designed, combining cutting-edge techniques with an in-depth understanding of the Bitcoin ecosystem. He tackled the recovery process with a level of expertise and thoroughness that was both impressive and reassuring. What sets Gilbert’s methods apart is not just their technical sophistication but also their strategic depth. He conducts a comprehensive analysis of each case, tailoring his approach to address the unique aspects of the situation. This personalized strategy ensures that every recovery effort is optimized for success. Gilbert’s transparent communication throughout the process was invaluable, providing clarity and confidence during each stage of the recovery. The results I achieved with Pro Wizard Gilbert’s methods were remarkable. His gold standard approach not only recovered my Bitcoin but did so with an efficiency and reliability that exceeded my expectations. His deep knowledge, innovative techniques, and unwavering commitment make him the definitive expert in Bitcoin recovery. For anyone seeking a benchmark in Bitcoin recovery solutions, Pro Wizard Gilbert’s methods are the epitome of excellence. His ability to blend technical prowess with strategic insight truly sets him apart in the industry. Call: for help. You may get in touch with them at ; Email: (prowizardgilbertrecovery(@)engineer.com) Telegram ; https://t.me/Pro_Wizard_Gilbert_Recovery Homepage ; https://prowizardgilbertrecovery.info
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12.11.24 00:50
TERESA
Brigadia Tech Remikeable recovery has exhibited unparalleled strength in the realm of recovery. They stand out as the premier team to collaborate with if you encounter withdrawal difficulties from the platform where you’ve invested. Recently, I engaged with them to recover over a million dollars trapped in an investment platform I’d been involved with for months. I furnished their team with every detail of the investment, including accounts, names, and wallet addresses to which I sent the funds. This decision proved to be the best I’ve made, especially after realizing the company had scammed me. Brigadia Tech Remikeable recovery ensures exemplary service delivery and ensures the perpetrators face justice. They employ advanced techniques to ensure you regain access to your funds. Understandably, many individuals who have fallen victim to investment scams may still regret engaging in online services again due to the trauma of being scammed. However, I implore you to take action. Seek assistance from Brigadia Tech Remikeable Recovery Experts today and witness their remarkable capabilities. I am grateful that I resisted their enticements, and despite the time it took me to discover brigadia tech recovery, they ultimately fulfilled my primary objective. Without Brigadia Tech Recovery's intervention, I would have remained despondent and perplexed indefinitely. Also if you are looking for the best and safest investment company you can contact them, for wallet recovery, difficult withdrawal, etc. I am so happy to keep getting my daily BTC, all I do is keep 0.1 BTC in my mining wallet with the help of Brigadia Tech. They connected me to his mining stream and I earn 0.4 btc per day with this, my daily profit. I can get myself a new house and car. I can’t believe I have thousands of dollars in my bank account. Now you can get in. ([email protected]) Telegram +1 (323)-9 1 0 -1 6 0 5
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17.11.24 09:31
Vivianlocke223
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19.11.24 03:06
[email protected]
My entire existence fell apart when a malevolent hacker recently gained access to my online accounts. I felt violated and extremely uneasy after discovering that the digital platforms I depended on for communication, employment, and finances had been compromised. Regaining control and restoring my digital security was an overwhelming task in the immediate aftermath. To help me navigate the difficult process of recovering my accounts and getting my peace of mind back, TRUST GEEKS HACK EXPERT came into my life as a ray of hope. They immediately put their highly skilled professionals to work, thoroughly examining the vulnerability and methodically preventing unwanted access. They guided me through each stage soothingly, explaining what was occurring and why, so I never felt lost or alone. They communicated with service providers to restore my legitimate access while skillfully navigating the complex labyrinth of account recovery procedures. My digital footprint was cleaned and strengthened against future attacks thanks to their equally amazing ability to remove any remaining evidence of the hacker's presence. However, TRUST GEEKS HACK EXPERT actual worth went beyond its technical aspects. They offered constant emotional support during the ordeal, understanding my fragility and sense of violation. My tense nerves were calmed by their comforting presence and kind comments, which served as a reminder that I wasn't alone in this struggle. With their help, I was able to reestablish my sense of security and control, which enabled me to return my attention to the significant areas of my life that had been upended. Ultimately, TRUST GEEKS HACK EXPERT all-encompassing strategy not only recovered my online accounts but also my general peace of mind, which is a priceless result for which I am incredibly appreciative of their knowledge and kindness. Make the approach and send a message to TRUST GEEKS HACK EXPERT Via Web site <> www://trustgeekshackexpert.com/-- E>mail: Trustgeekshackexpert(At)fastservice..com -- TeleGram,<> Trustgeekshackexpert
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19.11.24 03:07
[email protected]
My entire existence fell apart when a malevolent hacker recently gained access to my online accounts. I felt violated and extremely uneasy after discovering that the digital platforms I depended on for communication, employment, and finances had been compromised. Regaining control and restoring my digital security was an overwhelming task in the immediate aftermath. To help me navigate the difficult process of recovering my accounts and getting my peace of mind back, TRUST GEEKS HACK EXPERT came into my life as a ray of hope. They immediately put their highly skilled professionals to work, thoroughly examining the vulnerability and methodically preventing unwanted access. They guided me through each stage soothingly, explaining what was occurring and why, so I never felt lost or alone. They communicated with service providers to restore my legitimate access while skillfully navigating the complex labyrinth of account recovery procedures. My digital footprint was cleaned and strengthened against future attacks thanks to their equally amazing ability to remove any remaining evidence of the hacker's presence. However, TRUST GEEKS HACK EXPERT actual worth went beyond its technical aspects. They offered constant emotional support during the ordeal, understanding my fragility and sense of violation. My tense nerves were calmed by their comforting presence and kind comments, which served as a reminder that I wasn't alone in this struggle. With their help, I was able to reestablish my sense of security and control, which enabled me to return my attention to the significant areas of my life that had been upended. Ultimately, TRUST GEEKS HACK EXPERT all-encompassing strategy not only recovered my online accounts but also my general peace of mind, which is a priceless result for which I am incredibly appreciative of their knowledge and kindness. Make the approach and send a message to TRUST GEEKS HACK EXPERT Via Web site <> www://trustgeekshackexpert.com/-- E>mail: Trustgeekshackexpert(At)fastservice..com -- TeleGram,<> Trustgeekshackexpert
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21.11.24 04:14
ronaldandre617
Being a parent is great until your toddler figures out how to use your devices. One afternoon, I left my phone unattended for just a few minutes rookie mistake of the century. I thought I’d take a quick break, but little did I know that my curious little genius was about to embark on a digital adventure. By the time I came back, I was greeted by two shocking revelations: my toddler had somehow managed to buy a $5 dinosaur toy online and, even more alarmingly, had locked me out of my cryptocurrency wallet holding a hefty $75,000. Yes, you heard that right a dinosaur toy was the least of my worries! At first, I laughed it off. I mean, what toddler doesn’t have a penchant for expensive toys? But then reality set in. I stared at my phone in disbelief, desperately trying to guess whatever random string of gibberish my toddler had typed as a new password. Was it “dinosaur”? Or perhaps “sippy cup”? I felt like I was in a bizarre game of Password Gone Wrong. Every attempt led to failure, and soon the laughter faded, replaced by sheer panic. I was in way over my head, and my heart raced as the countdown of time ticked away. That’s when I decided to take action and turned to Digital Tech Guard Recovery, hoping they could solve the mystery that was my toddler’s handiwork. I explained my predicament, half-expecting them to chuckle at my misfortune, but they were incredibly professional and empathetic. Their confidence put me at ease, and I knew I was in good hands. Contact With WhatsApp: +1 (443) 859 - 2886 Email digital tech guard . com Telegram: digital tech guard recovery . com website link :: https : // digital tech guard . com Their team took on the challenge like pros, employing their advanced techniques to unlock my wallet with a level of skill I can only describe as magical. As I paced around, anxiously waiting for updates, I imagined my toddler inadvertently locking away my life savings forever. But lo and behold, it didn’t take long for Digital Tech Guard Recovery to work their magic. Not only did they recover the $75,000, but they also gave me invaluable tips on securing my wallet better like not leaving it accessible to tiny fingers! Who knew parenting could lead to such dramatic situations? Crisis averted, and I learned my lesson: always keep my devices out of reach of little explorers. If you ever find yourself in a similar predicament whether it’s tech-savvy toddlers or other digital disasters don’t hesitate to reach out to Digital Tech Guard Recovery. They saved my funds and my sanity, proving that no challenge is too great, even when it involves a toddler’s mischievous fingers!
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21.11.24 08:02
Emily Hunter
If I hadn't found a review online and filed a complaint via email to support@deftrecoup. com , the people behind this unregulated scheme would have gotten away with leaving me in financial ruins. It was truly the most difficult period of my life.
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22.11.24 04:41
[email protected]
I never could have imagined the nightmare of losing access to my digital wallet. All of my cryptocurrency holdings were abruptly imprisoned, inaccessible, and appeared to be lost forever following a catastrophic hardware breakdown. Years of meticulous investment and careful saving were reduced to nothing more than strings of code that I could no longer control, and I could feel the dread and sorrow that swept through me at that very instant. Thankfully, during my worst moment, I came into (TRUST GEEKS HACK EXPERT), a professional service devoted to recovering lost or inaccessible digital data. With optimism, I went out to their team of skilled technologists, laying bare the full nature of my issue. What followed was a laborious, multi-step process that required an almost surgical level of digital forensics and Bitcoin skill. In order to create a thorough profile of my wallet's contents and activities, the (TRUST GEEKS HACK EXPERT) team first thoroughly examined the transaction history and metadata connected to it. Next, they implemented a series of advanced recovery techniques, using cutting-edge software tools to bypass the access barriers that had left me locked out. The entire process was shrouded in secrecy and discretion, with the (TRUST GEEKS HACK EXPERT) team working tirelessly to protect the confidentiality of my sensitive financial information. After what felt like an eternity of nervous anticipation, the day finally arrived when I received the triumphant notification – my wallet had been successfully restored, and all of my precious digital assets had been returned to my control. The sense of relief was indescribable, as I could finally breathe easy knowing that the fruits of my financial discipline had been safeguarded. While the experience of losing access to my wallet was undoubtedly traumatic, (TRUST GEEKS HACK EXPERT) intervention allowed me to emerge from the ordeal with my cryptocurrency holdings intact, and a renewed appreciation for the importance of proactive digital asset management. You can contact Them through EMAIL: [email protected] - TELEGRAM: TRUSTGEEKSHACKEXPERT
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22.11.24 15:26
cliftonhandyman
Your Lost Bitcoins Are Not Gone Forever? Enquire From iBolt Cyber Hacker iBolt Cyber Hacker is a cybersecurity service that specializes in Bitcoin and cryptocurrency recovery. Even if your Bitcoin is locked away in a scammer inaccessible wallet, they have the tools and expertise to retrieve it. Many people, including seasoned cryptocurrency investors, face the daunting possibility of never seeing their lost funds again. iBolt cyber hacker service is a potential lifeline in these situations. I understand the concerns many people might have about trusting a third-party service to recover their Bitcoin. iBolt Cyber Hacker takes security seriously, implementing encryption and stringent privacy protocols. I was assured that no sensitive data would be compromised during the recovery process. Furthermore, their reputation in the cryptocurrency community, based on positive feedback from previous clients, gave me confidence that I was in good hands. Whtp +39, 351..105, 3619 Em.ail: ibolt @ cyber- wizard. co m
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22.11.24 23:43
teresaborja
all thanks to Tech Cyber Force Recovery expert assistance. As a novice in cryptocurrency, I had been carefully accumulating a modest amount of Bitcoin, meticulously safeguarding my digital wallet and private keys. However, as the adage goes, the best-laid plans can often go awry, and that's precisely what happened to me. Due to a series of technical mishaps and human errors, I found myself locked out of my Bitcoin wallet, unable to access the fruits of my digital labors. Panic set in as I frantically searched for a solution, scouring the internet for any glimmer of hope. That's when I stumbled upon the Tech Cyber Force Recovery team, a group of seasoned cryptocurrency specialists who had built a reputation for their ability to recover lost or inaccessible digital assets. Skeptical at first, I reached out, desperate for a miracle. To my utter amazement, the Tech Cyber Force Recovery experts quickly assessed my situation and devised a meticulous plan of attack. Through their deep technical knowledge, unwavering determination, and a keen eye for detail, they were able to navigate the complex labyrinth of blockchain technology, ultimately recovering my entire Bitcoin portfolio. What had once seemed like a hopeless endeavor was now a reality, and I found myself once again in possession of my digital wealth, all thanks to the incredible efforts of the Tech Cyber Force Recovery team. This experience has not only restored my faith in the cryptocurrency ecosystem. Still, it has also instilled in me a profound appreciation for the critical role that expert recovery services can play in safeguarding one's digital assets. ENAIL < Tech cybers force recovery @ cyber services. com > WEBSITE < ht tps : // tech cyber force recovery. info > TEXT < +1. 561. 726. 3697 >
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24.11.24 02:21
[email protected]
I never could have imagined the nightmare of losing access to my digital wallet. All of my cryptocurrency holdings were abruptly imprisoned, inaccessible, and appeared to be lost forever following a catastrophic hardware breakdown. Years of meticulous investment and careful saving were reduced to nothing more than strings of code that I could no longer control, and I could feel the dread and sorrow that swept through me at that very instant. Thankfully, during my worst moment, I came into (TRUST GEEKS HACK EXPERT), a professional service devoted to recovering lost or inaccessible digital data. With optimism, I went out to their team of skilled technologists, laying bare the full nature of my issue. What followed was a laborious, multi-step process that required an almost surgical level of digital forensics and Bitcoin skill. In order to create a thorough profile of my wallet's contents and activities, the (TRUST GEEKS HACK EXPERT) team first thoroughly examined the transaction history and metadata connected to it. Next, they implemented a series of advanced recovery techniques, using cutting-edge software tools to bypass the access barriers that had left me locked out. The entire process was shrouded in secrecy and discretion, with the (TRUST GEEKS HACK EXPERT) team working tirelessly to protect the confidentiality of my sensitive financial information. After what felt like an eternity of nervous anticipation, the day finally arrived when I received the triumphant notification – my wallet had been successfully restored, and all of my precious digital assets had been returned to my control. The sense of relief was indescribable, as I could finally breathe easy knowing that the fruits of my financial discipline had been safeguarded. While the experience of losing access to my wallet was undoubtedly traumatic, (TRUST GEEKS HACK EXPERT) intervention allowed me to emerge from the ordeal with my cryptocurrency holdings intact, and a renewed appreciation for the importance of proactive digital asset management. You can contact Them through EMAIL: [email protected] - TELEGRAM: TRUSTGEEKSHACKEXPERT
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25.11.24 02:19
briankennedy
COMMENT ON I NEED A HACKER TO RECOVER MONEY FROM BINARY TRADING. HIRE FASTFUND RECOVERY
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25.11.24 02:20
briankennedy
After countless hours of research and desperate attempts to find a solution, I stumbled upon FASTFUND RECOVERY. It was like finding an oasis in the middle of a desert. Their website promised to help victims of scams reclaim what was rightfully theirs, and I instantly knew I had to give them a shot. Before diving headfirst into the recovery process, I wanted to make sure that FASTFUND RECOVERY was the real deal. So, I did my due diligence and looked into their expertise and reputation. To my relief, I found that they had an impeccable track record, successfully assisting countless individuals in recovering their lost funds. Their team consisted of experts in cybersecurity and financial fraud, armed with the knowledge and tools needed to tackle even the most intricate scams. With their reputation preceding them, I felt a renewed sense of hope. FASTFUND RECOVERY successfully came to my aid and got back the amount I lost to these scammers and for this, I am sending this article for clarification. The info of FASTFUND RECOVERY is email: Fastfundrecovery8 (@)Gmail (.) com. Web fastfundrecovery(.)com. (W/A 1 807/500/7554)
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26.11.24 21:59
[email protected]
In a world brimming with enticing investment opportunities, it is crucial to tread carefully. The rise of digital currencies has attracted many eager investors, but along with this excitement lurk deceitful characters ready to exploit the unsuspecting. I learned this lesson the hard way, and I want to share my story in the hopes that it can save someone from making the same mistakes I did. It all began innocently enough when I came across an engaging individual on Facebook. Lured in by promises of high returns in the cryptocurrency market, I felt the electric thrill of potential wealth coursing through me. Initial investments returned some profits, and that exhilarating taste of success fueled my ambition. Encouraged by a meager withdrawal, I decided to commit even more funds. This was the moment I let my guard down, blinded by greed. As time went on, the red flags started to multiply. The moment I tried to withdraw my earnings, a cascade of unreasonable fees appeared like a thick mist, obscuring the truth. “Just a little more,” they said, “Just until the next phase.” I watched my hard-earned money slip through my fingers as I scraped together every last cent to pay those relentless fees. My trust had become my downfall. In the end, I lost not just a significant amount of cash, but my peace of mind about $1.1 million vanished into the abyss of false promises and hollow guarantees. But despair birthed hope. After a cascade of letdowns, I enlisted the help of KAY-NINE CYBER SERVICES, a team that specializes in reclaiming lost funds from scams. Amazingly, they worked tirelessly to piece together what had been ripped away, providing me with honest guidance when I felt utterly defeated. Their expertise in navigating the treacherous waters of crypto recovery was a lifeline I desperately needed. To anyone reading this, please let my story serve as a warning. High returns often come wrapped in the guise of deception. Protect your investments, scrutinize every opportunity, and trust your instincts. Remember, the allure of quick riches can lead you straight to heartbreak, but with cautious determination and support, it is possible to begin healing from such devastating loss. Stay informed, stay vigilant, and may you choose your investment paths wisely. Email: kaynine @ cyberservices . com
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26.11.24 23:12
rickrobinson8
FAST SOLUTION FOR CYPTOCURRENCY RECOVERY SPARTAN TECH GROUP RETRIEVAL
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26.11.24 23:12
rickrobinson8
Although recovering from the terrible effects of investment fraud can seem like an impossible task, it is possible to regain financial stability and go on with the correct assistance and tools. In my own experience with Wizard Web Recovery, a specialized company that assisted me in navigating the difficulties of recouping my losses following my fall prey to a sophisticated online fraud, that was undoubtedly the case. My life money had disappeared in an instant, leaving me in a state of shock when I first contacted Spartan Tech Group Retrieval through this Email: spartantechretrieval (@) g r o u p m a i l .c o m The compassionate and knowledgeable team there quickly put my mind at ease, outlining a clear and comprehensive plan of action. They painstakingly examined every aspect of my case, using their broad business contacts and knowledge to track the movement of my pilfered money. They empowered me to make knowledgeable decisions regarding the rehabilitation process by keeping me updated and involved at every stage. But what I valued most was their unrelenting commitment and perseverance; they persisted in trying every option until a sizable amount of my lost money had been successfully restored. It was a long and arduous journey, filled with ups and downs, but having Spartan Tech Group Retrieval in my corner made all the difference. Thanks to their tireless efforts, I was eventually able to rebuild my financial foundation and reclaim a sense of security and control over my life. While the emotional scars of investment fraud may never fully heal, working with this remarkable organization played a crucial role in my ability to move forward and recover. For proper talks, contact on WhatsApp:+1 (971) 4 8 7 - 3 5 3 8 and Telegram:+1 (581) 2 8 6 - 8 0 9 2 Thank you for your time reading as it will be of help.
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27.11.24 00:39
[email protected]
Although recovering lost or inaccessible Bitcoin can be difficult and unpleasant, it is frequently possible to get back access to one's digital assets with the correct help and direction. Regarding the subject at hand, the examination of Trust Geeks Hack Expert Website www://trustgeekshackexpert.com/ assistance after an error emphasizes how important specialized services may be in negotiating the difficulties of Bitcoin recovery. These providers possess the technical expertise and resources necessary to assess the situation, identify the root cause of the issue, and devise a tailored solution to retrieve the lost funds. By delving deeper into the specifics of Trust Geeks Hack Expert approach, we can gain valuable insights into the nuances of this process. Perhaps they leveraged advanced blockchain analysis tools to trace the transaction history and pinpoint the location of the missing Bitcoins. Or they may have collaborated with the relevant parties, such as exchanges or wallet providers, to facilitate the recovery process. Equally important is the level of personalized support and communication that Trust Geeks Hack Expert likely provided, guiding the affected individual through each step of the recovery effort and offering reassurance during what can be an anxious and uncertain time. The success of their efforts, as evidenced by the positive outcome, underscores the importance of seeking out reputable and experienced service providers when faced with a Bitcoin-related mishap, as they possess the specialized knowledge and resources to navigate these challenges and restore access to one's digital assets. Email.. [email protected]
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27.11.24 09:10
Michal Novotny
The biggest issue with cryptocurrency is that it is unregulated, wh ich is why different people can come up with different fake stories all the time, and it is unfortunate that platforms like Facebook and others only care about the money they make from them through ads. I saw an ad on Facebook for Cointiger and fell into the scam, losing over $30,000. I reported it to Facebook, but they did nothing until I discovered deftrecoup . c o m from a crypto community; they retrieved approximately 95% of the total amount I lost.
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01.12.24 17:21
KollanderMurdasanu
REACH OUT TO THEM WhatsApp + 156 172 63 697 Telegram (@)Techcyberforc We were in quite a bit of distress. The thrill of our crypto investments, which had once sparked excitement in our lives, was slowly turning into anxiety when my husband pointed out unusual withdrawal issues. At first, we brushed it off as minor glitches, but the situation escalated when we found ourselves facing login re-validation requests that essentially locked us out of our crypto wallet—despite entering the correct credentials. Frustrated and anxious, we sought advice from a few friends, only to hit a wall of uncertainty. Turning to the vast expanse of the internet felt daunting, but in doing so, we stumbled upon TECH CYBER FORCE RECOVERY. I approached them with a mix of skepticism and hope; after all, my understanding of these technical matters was quite limited. Yet, from our very first interaction, it was clear that they were the experts we desperately needed. They walked us through the intricacies of the recovery process, patiently explaining each mechanism—even if some of it went over my head, their reassurance was calming. Our responsibility was simple: to provide the correct information to prove our ownership of the crypto account, and thankfully, we remained on point in our responses. in a timely fashion, TECH CYBER FORCE RECOVERY delivered on their promises, addressing all our withdrawal and access issues exactly when they said they would. The relief we felt was immense, and the integrity they displayed made me confident in fully recommending their services. If you ever find yourself in a similar predicament with your crypto investments, I wholeheartedly suggest reaching out to them. You can connect with TECH CYBER FORCE RECOVERY through their contact details for assistance and valuable guidance. Remember, hope is only a reach away!
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02.12.24 23:02
ytre89
Online crypto investment can seem like a promising opportunity, but it's crucial to recognize that there are no guarantees. My experience serves as a stark reminder of this reality. I was drawn in by the allure of high returns and the persuasive marketing tactics employed by various brokers. Their polished presentations and testimonials made it seem easy to profit from cryptocurrency trading. Everything appeared to be legitimate. I received enticing messages about the potential for substantial gains, and the brokers seemed knowledgeable and professional. Driven by excitement and the fear of missing out, I invested a significant amount of my savings. The promise of quick profits overshadowed the red flags I should have noticed. I trusted these brokers without conducting proper research, which was a major mistake. As time went on, I realized that the promised returns were nothing but illusions. My attempts to withdraw funds were met with endless excuses and delays. It became painfully clear that I had fallen victim. The reality hit hard: my hard-earned money was gone, I lost my peace of mind and sanity. In my desperation, I sought help from a company called DEFTRECOUP. That was the turning point for me as I had a good conversation and eventually filed a complaint via DEFTRECOUP COM. They were quite delicate and ensured I got out of the most difficult situation of my life in one piece.
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04.12.24 22:24
andreygagloev
When I first heard about Bitcoin back in 2018, I was skeptical. The idea of a decentralized, digital currency seemed too good to be true. But I was intrigued as I learned more about the technology behind it and its potential. I started small, investing just a few hundred dollars, dipping my toes into the cryptocurrency waters. At first, it was exhilarating to watch the value of my investment grow exponentially. I felt like I was part of the future, an early adopter of this revolutionary new asset. But that euphoria was short-lived. One day, I logged into my digital wallet only to find it empty - my Bitcoin had vanished without a trace. It turned out that the online exchange I had trusted had been hacked, and my funds were stolen. I was devastated, both financially and emotionally. All the potential I had seen in Bitcoin was tainted by the harsh reality that with decentralization came a lack of regulation and oversight. My hard-earned money was gone, lost to the ether of the digital world. This experience taught me a painful lesson about the price of trust in the uncharted territory of cryptocurrency. While the technology holds incredible promise, the risks can be catastrophic if you don't approach it with extreme caution. My Bitcoin investment gamble had failed, and I was left to pick up the pieces, wiser but poorer for having placed my faith in the wrong hands. My sincere appreciation goes to MUYERN TRUST HACKER. You are my hero in recovering my lost funds. Send a direct m a i l ( muyerntrusted ( @ ) mail-me ( . )c o m ) or message on whats app : + 1 ( 4-4-0 ) ( 3 -3 -5 ) ( 0-2-0-5 )
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12.12.24 00:35
amandagregory
HOW TO HIRE A HACKER TO RECOVER STOLEN BITCOIN WITH FASTFUND RECOVERY
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12.12.24 00:35
amandagregory
HOW TO HIRE A HACKER TO RECOVER STOLEN BITCOIN WITH FASTFUND RECOVERY... A few months ago, I made a huge mistake. I invested in what seemed like a legitimate crypto opportunity, only to find out I’d been scammed. I lost a significant amount of money, and the scam platform vanished overnight. I felt completely lost.I had heard of Fastfund Recovery and decided to reach out, even though I was skeptical. From the first conversation, they made me feel heard and understood. They explained the recovery process clearly and kept me updated every step of the way.Within weeks, Fastfund Recovery successfully to recovered my lost funds—something I honestly didn’t think was possible. Their team was professional, transparent, and genuinely caring. I can’t thank them enough for turning a nightmare into a hopeful outcome. If you’re in a similar situation, don’t hesitate to contact them. They truly deliver on their promises. Gmail::: fastfundrecovery8(@)gmail com .....Whatsapp ::: 1::807::::500::::7554
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19.12.24 17:07
rebeccabenjamin
USDT RECOVERY EXPERT REVIEWS DUNAMIS CYBER SOLUTION It's great to hear that you've found a way to recover your Bitcoin and achieve financial stability, but I urge you to be cautious with services like DUNAMIS CYBER SOLUTION Recovery." While it can be tempting to turn to these companies when you’re desperate to recover lost funds, many such services are scams, designed to exploit those in vulnerable situations. Always research thoroughly before engaging with any recovery service. In the world of cryptocurrency, security is crucial. To protect your assets, use strong passwords, enable two-factor authentication, and consider using cold wallets (offline storage) for long-term storage. If you do seek professional help, make sure the company is reputable and has positive, verifiable reviews from trusted sources. While it’s good that you found a solution, it’s also important to be aware of potential scams targeting cryptocurrency users. Stay informed about security practices, and make sure you take every step to safeguard your investments. If you need help with crypto security tips or to find trustworthy resources, feel free to ask! [email protected] +13433030545 [email protected]
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24.12.24 08:33
dddana
Отличная подборка сервисов! Хотелось бы дополнить список рекомендацией: нажмите сюда - https://airbrush.com/background-remover. Этот инструмент отлично справляется с удалением фона, сохраняя при этом высокое качество изображения. Очень удобен для быстрого редактирования фото. Было бы здорово увидеть его в вашей статье!
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27.12.24 00:21
swiftdream
I lost about $475,000.00 USD to a fake cryptocurrency trading platform a few weeks back after I got lured into the trading platform with the intent of earning a 15% profit daily trading on the platform. It was a hell of a time for me as I could hardly pay my bills and got me ruined financially. I had to confide in a close friend of mine who then introduced me to this crypto recovery team with the best recovery SWIFTDREAM i contacted them and they were able to completely recover my stolen digital assets with ease. Their service was superb, and my problems were solved in swift action, It only took them 48 hours to investigate and track down those scammers and my funds were returned to me. I strongly recommend this team to anyone going through a similar situation with their investment or fund theft to look up this team for the best appropriate solution to avoid losing huge funds to these scammers. Send complaint to Email: info [email protected]
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31.12.24 04:53
Annette_Phillips
There are a lot of untrue recommendations and it's hard to tell who is legit. If you have lost crypto to scam expresshacker99@gmailcom is the best option I can bet on that cause I have seen lot of recommendations about them and I'm a witness on their capabilities. They will surely help out. Took me long to find them. The wonderful part is no upfront fee till crypto is recover successfully that's how genuine they are.
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04.01.25 04:56
florencestella
THE BEST CERTIFIED CRYPTOCURRENCY RECOVERY EXPERT DUNAMIS CYBER SOLUTION
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04.01.25 04:57
florencestella
THE BEST CERTIFIED CRYPTOCURRENCY RECOVERY EXPERT DUNAMIS CYBER SOLUTION It sounds like you went through a very frustrating experience with Cointrack, where your access to your own funds was unjustly restricted for months without clear communication or a solution. The extended periods of account freezes, lack of transparency, and vague customer support responses would make anyone anxious. It’s understandable that you suspected the issue could be related to your login activity, but it’s surprising that something as minor as using the same Wi-Fi network could trigger such severe restrictions. I’m glad to hear that DUNAMIS CYBER SOLUTION Recovery was able to help you get your account unlocked and resolve the issue. It’s unfortunate that you had to seek third-party assistance, but it’s a relief that the situation was eventually addressed. If you plan on using any platforms like this again, you might want to be extra cautious, especially when dealing with sensitive financial matters. And if you ever need to share your experience to help others avoid similar issues, feel free to reach out. It might be helpful for others to know about both the pitfalls and the eventual resolution through services like DUNAMIS CYBER SOLUTION Recovery. [email protected] +13433030545 [email protected]
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06.01.25 19:09
michaeljordan15
We now live in a world where most business transactions are conducted through Bitcoin and cryptocurrency. With the rapid growth of digital currencies, everyone seems eager to get involved in Bitcoin and cryptocurrency investments. This surge in interest has unfortunately led to the rise of many fraudulent platforms designed to exploit unsuspecting individuals. People are often promised massive profits, only to lose huge sums of money when they realize the platform they invested in was a scam. contact with WhatsApp: +1 (443) 859 - 2886 Email @ digitaltechguard.com Telegram: digitaltechguardrecovery.com website link:: https://digitaltechguard.com This was exactly what happened to me five months ago. I was excited about the opportunity to invest in Bitcoin, hoping to earn a steady return of 20%. I found a platform that seemed legitimate and made my investment, eagerly anticipating the day when I would be able to withdraw my earnings. When the withdrawal day arrived, however, I encountered an issue. My bank account was not credited, despite seeing my balance and the supposed profits in my account on the platform. At first, I assumed it was just a technical glitch. I thought, "Maybe it’s a delay in the system, and everything will be sorted out soon." However, when I tried to contact customer support, the line was either disconnected or completely unresponsive. My doubts started to grow, but I wanted to give them the benefit of the doubt and waited throughout the day to see if the situation would resolve itself. But by the end of the day, I realized something was terribly wrong. I had been swindled, and my hard-earned money was gone. The realization hit me hard. I had fallen victim to one of the many fraudulent Bitcoin platforms that promise high returns and disappear once they have your money. I knew I had to act quickly to try and recover what I had lost. I started searching online for any possible solutions, reading reviews and recommendations from others who had faced similar situations. That’s when I came across many positive reviews about Digital Tech Guard Recovery. After reading about their success stories, I decided to reach out and use their services. I can honestly say that Digital Tech Guard Recovery exceeded all my expectations. Their team was professional, efficient, and transparent throughout the process. Within a short time, they helped me recover a significant portion of my lost funds, which I thought was impossible. I am incredibly grateful to Digital Tech Guard Recovery for their dedication and expertise in helping me get my money back. If you’ve been scammed like I was, don’t lose hope. There are solutions, and Digital Tech Guard Recovery is truly one of the best. Thank you, Digital Tech Guard Recovery! You guys are the best. Good luck to everyone trying to navigate this challenging space. Stay safe.
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18.01.25 12:41
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]
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18.01.25 12:41
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]
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20.01.25 15:39
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]
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22.01.25 21:43
DoraJaimes23
Recovery expert. I lost my bitcoin to fake blockchain impostors on Facebook, they contacted me as blockchain official support and i fell stupidly for their mischievous act, this made them gain access into my blockchain wallet whereby 7.0938 btc was stolen from my wallet in total .I was almost in a comma and dumbfounded because this was all my savings i relied on . Then I made a research online and found a recovery expert , with the contact address- { RECOVERYHACKER101 (@) GMAIL . COM }... I wrote directly to the specialist explaining my loss. Hence, he helped me recover a significant part of my investment just after 2 days he helped me launch the recovery program , and the culprits were identified as well , all thanks to his expertise . I hope I have been able to help someone as well . Reach out to the recovery specialist to recover you lost funds from any form of online scam Thanks
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23.01.25 02:36
[email protected]
After falling victim to a fraudulent Bitcoin mining scam, I found myself in a desperate situation. I had invested $50,000 into a cloud mining website called Miningpool, which turned out to be a complete scam. For months, I tried reaching out to the company, but I was unable to access my funds, and I quickly realized I had been taken for a ride. In my search for help, I came across TrustGeeks Hack Expert, a service that claimed to help people recover lost funds from crypto scams. Though skeptical at first, I decided to give them a try. Here’s my experience with their service.When I initially contacted TrustGeeks Hack Expert Email.. Trustgeekshackexpert{At}fastservice{Dot}com , I was understandably hesitant. Like many others, I had been tricked into believing my Bitcoin investments were legitimate, only to discover they were locked in a non-spendable wallet with no way of accessing them. However, after sharing my story and details about the scam, the team assured me they had handled similar cases and had the expertise to help. They requested basic information about my investment and began their investigation immediately. The recovery process was nothing short of professional. Unlike many other services that promise quick fixes but fail to deliver, TrustGeeks Hack Expert kept me informed at every stage. They regularly updated me on their progress and were completely transparent about the challenges they faced. There were moments when I wondered if the process would work, but the team’s professionalism and reassurance gave me hope. They were honest about the time it would take and did not make any unrealistic promises, which I truly appreciated. After several weeks of work, TrustGeeks Hack Expert successfully recovered not just my $50,000 investment, but also the so-called profits that had been locked away in the scam's non-spendable wallet. This was a huge relief, as I had resigned myself to the idea that I had lost everything. The entire recovery process was discreet and handled with the utmost care, ensuring that the scam company remained unaware of the recovery efforts, which helped prevent further complications. TeleGram iD. Trustgeekshackexpert & What's A p p +1 7 1 9 4 9 2 2 6 9 3
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23.01.25 02:37
[email protected]
After falling victim to a fraudulent Bitcoin mining scam, I found myself in a desperate situation. I had invested $50,000 into a cloud mining website called Miningpool, which turned out to be a complete scam. For months, I tried reaching out to the company, but I was unable to access my funds, and I quickly realized I had been taken for a ride. In my search for help, I came across TrustGeeks Hack Expert, a service that claimed to help people recover lost funds from crypto scams. Though skeptical at first, I decided to give them a try. Here’s my experience with their service.When I initially contacted TrustGeeks Hack Expert Email.. Trustgeekshackexpert{At}fastservice{Dot}com , I was understandably hesitant. Like many others, I had been tricked into believing my Bitcoin investments were legitimate, only to discover they were locked in a non-spendable wallet with no way of accessing them. However, after sharing my story and details about the scam, the team assured me they had handled similar cases and had the expertise to help. They requested basic information about my investment and began their investigation immediately. The recovery process was nothing short of professional. Unlike many other services that promise quick fixes but fail to deliver, TrustGeeks Hack Expert kept me informed at every stage. They regularly updated me on their progress and were completely transparent about the challenges they faced. There were moments when I wondered if the process would work, but the team’s professionalism and reassurance gave me hope. They were honest about the time it would take and did not make any unrealistic promises, which I truly appreciated. After several weeks of work, TrustGeeks Hack Expert successfully recovered not just my $50,000 investment, but also the so-called profits that had been locked away in the scam's non-spendable wallet. This was a huge relief, as I had resigned myself to the idea that I had lost everything. The entire recovery process was discreet and handled with the utmost care, ensuring that the scam company remained unaware of the recovery efforts, which helped prevent further complications. TeleGram iD. Trustgeekshackexpert & What's A p p +1 7 1 9 4 9 2 2 6 9 3
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23.01.25 14:20
nellymargaret
DUNAM CYBER SOLUTION BTC-ETH RECOVERY EXPERT
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23.01.25 14:20
nellymargaret
DUNAM CYBER SOLUTION BTC-ETH RECOVERY EXPERT I had tried to secure my Bitcoin wallet, maybe a bit too aggressively, enabling every security feature imaginable: two-factor authentication, biometric verification, intricate passwords-the whole shebang. I wanted to make it impossible for anybody to get to my money. I tried to make this impregnable fortress of security and ended up locking myself out of my wallet with $700,000 in Bitcoin. It wasn't until I tried to access my wallet that I realized the trap I had set for myself. I was greeted with an endless series of security checks-passwords, codes, facial recognition, and more. I could remember parts of my multi-layered security setup but not enough to actually get in. In fact, my money was behind this digital fortress, and the more I tried to fix it, the worse it seemed to get. I kept tripping over my own layers of protection, unable to find a way back in. Panic quickly set in when I realized I had made it almost impossible for myself to access my own money. That is when I called DUNAMIS CYBER SOLUTION From that very first call, they reassured me that I wasn't the first person to make this kind of mistake and certainly wouldn't be the last. They listened attentively to my explanation and got to work straight away. Their team methodically began to untangle my overly complicated setup. Patience and expertise managed to crack each layer of security step by step until they had restored access to my wallet. [email protected] +13433030545 [email protected]
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26.01.25 03:54
[email protected]
Losing access to my crypto wallet account was one of the most stressful experiences ever. After spending countless hours building up my portfolio, I suddenly found myself locked out of my account without access. To make matters worse, the email address I had linked to my wallet was no longer active. When I tried reaching out, I received an error message stating that the domain was no longer in use, leaving me in complete confusion and panic. It was as though everything I had worked so hard for was gone, and I had no idea how to get it back. The hardest part wasn’t just the loss of access it was the feeling of helplessness. Crypto transactions are often irreversible, and since my wallet held significant investments, the thought that my hard-earned money could be lost forever was incredibly disheartening. I spent hours scouring forums and searching for ways to recover my funds, but most of the advice seemed either too vague or too complicated to be of any real help. With no support from the wallet provider and my email account out of reach, I was left feeling like I had no way to fix the situation.That’s when I found out about Trust Geeks Hack Expert . I was hesitant at first, but after reading about their expertise in recovering lost crypto wallets, I decided to give them a try. I reached out to their team, and from the very beginning, they were professional, understanding, and empathetic to my situation. They quickly assured me that there was a way to recover my wallet, and they got to work immediately.Thanks to Trust Geeks Hack Expert , my wallet and funds were recovered, and I couldn’t be more grateful. The process wasn’t easy, but their team guided me through each step with precision and care. The sense of relief I felt when I regained access to my crypto wallet and saw my funds safely back in place was indescribable. If you find yourself in a similar situation, I highly recommend reaching out to Trust Geeks Hack Expert. contact Them through EMAIL: [email protected] + WEBSITE. HTTPS://TRUSTGEEKSHACKEXPERT.COM + TELE GRAM: TRUSTGEEKSHACKEXPERT
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28.01.25 21:48
[email protected]
It’s unfortunate that many people have become victims of scams, and some are facing challenges accessing their Bitcoin wallets. However, there's excellent news! With Chris Wang, you can count on top-notch service that guarantees results in hacking. We have successfully helped both individuals and organizations recover lost files, passwords, funds, and more. If you need assistance, don’t hesitate—check out recoverypro247 on Google Mail! What specific methods does Chris Wang use to recover lost funds and passwords? Are there any guarantees regarding the success rate of the recovery services offered? What are the initial steps to begin the recovery process with recoverypro247? this things i tend to ask
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