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[Перевод] Оценка ценовой эластичности спроса продуктов с помощью байесовского иерархического моделирования

Детализация результатов машинного обучения с помощью одной модели

Байесовские иерархические модели в экологии (глава 6) — Стив Мидуэй (Steve Midway)

В этой статье я хочу познакомить вас с байесовским иерархическим моделированием — гибким подходом, который автоматически объединяет результаты нескольких подмоделей. Этот метод позволяет оценивать эффекты на индивидуальном уровне путем оптимального объединения информации из различных групп данных с помощью байесовского вывода. Это особенно полезно, когда данные наблюдений для некоторых объектов ограничены, но эти объекты имеют общие характеристики или поведение с другими объектами.

В последующих разделах я представлю вам концепцию, реализацию и альтернативные варианты использования этого метода.

Проблема традиционных подходов

Представьте, что мы управляем крупным продуктовым магазином и стремимся максимизировать прибыль для каждого продукта, регулируя их цены. Чтобы достичь этой цели, нам необходимо оценить кривую спроса (эластичность) для каждого товара, а затем оптимизировать функцию максимизацию прибыли. Первым шагом в этом процессе является оценка ценовой эластичности спроса, то есть определение того, насколько быстро спрос реагирует на изменение цены на 1%. Для этого у нас есть данные в динамике по i ∈ N продуктам, собранные за t ∈ T периодов. Ценовая эластичность спроса рассчитывается по следующей формуле:

Предполагая отсутствие внешних факторов, мы можем использовать для оценки интересующего нас параметра лог‑линейную регрессионную модель с фиксированным эффектом:

Где γc(i), t — набор переменных, которые учитывают средний спрос в каждой категории за определенный период времени, а δi — набор переменных, которые отражают изменения спроса, не связанные со временем, для каждого отдельного продукта. Такая форма «фиксированного эффекта» является стандартной и широко используется во многих регрессионных моделях для учета ненаблюдаемых факторов, которые могут оказывать влияние. Эта (объединенная) регрессионная модель позволяет нам определить среднюю эластичность β для всех N товарных единиц. Это означает, что магазин может установить средний уровень цен на все товары в своем ассортименте, чтобы максимизировать прибыль:

Если эти товарные единицы естественным образом группируются в категории продуктов, мы могли бы определить среднюю эластичность каждой категории, проведя отдельные регрессии (или связав ценовую эластичность с категорией продукта) для каждой категории, используя только товарные единицы из этой категории. Это позволило бы магазину ориентироваться на средние цены в каждой категории, чтобы максимизировать прибыль для конкретной категории. В результате:

Имея достаточно данных, мы могли бы даже провести отдельные регрессии для каждого отдельного продукта, чтобы получить более детализированные показатели эластичности.

Однако, когда мы имеем дело с реальными данными, не все так гладко: некоторые продукты имеют минимальные колебания цен, короткую историю продаж или несбалансированность классов категорий. В таких условиях проведение отдельных регрессий для определения эластичности продукта, вероятно, приведет к большим среднеквадратическим погрешностям и худшей оценке β. Байесовские иерархические модели решают эти проблемы, позволяя нам получать детализированные оценки интересующего нас коэффициента путем распределения статистической мощности между различными группами, сохраняя при этом неоднородность. Используя формулу байесовской иерархической модели, мы можем провести одну регрессию (как для объединенного случая), сохраняя при этом эластичность на уровне продукта, что позволяет выполнять детальную оптимизацию.

Суть байесовских иерархических моделей

Суть байесовских иерархические моделей заключается в распознавании естественной структуры наших данных. Вместо того чтобы рассматривать все наблюдения как полностью независимые (множество отдельных регрессий) или пытаться заставить их следовать одинаковым шаблонам (одна объединенная регрессия), мы признаем, что наблюдения могут собираться в группы, в рамках которых продукты имеют схожие характеристики. «Иерархический» аспект этих моделей заключается в том, что мы организуем наши параметры на разных уровнях. В самом базовом формате мы могли бы иметь:

Глобальный параметр, который применяется ко всем данным.
Групповые параметры, которые применяются к наблюдениям внутри этой группы.
Индивидуальные параметры, которые применяются к каждому отдельному объекту.

Эта методология очень гибкая: мы можем добавлять или убирать уровни иерархии в зависимости от того, насколько детализированным нам нужно сделать наше исследование. Например, если мы не видим связи между категориями, мы можем исключить глобальный параметр. Если же мы считаем, что продукты не группируются естественным образом, то можем убрать параметры на уровне групп. Если же нас интересует только влияние на уровне группы, то можно исключить параметр индивидуального уровня и использовать коэффициенты на уровне группы в качестве нашего наиболее детализированного параметра. Если же внутри групп существуют вложенные подгруппы, то мы можем добавить еще один уровень иерархии. Здесь наши возможности безграничны!

«Байесовский» аспект заключается в том, как мы корректируем наши представления об этих параметрах на основе наблюдаемых данных. Мы начинаем с предположения априорного распределения, которое отражает нашу первоначальную оценку этих параметров, а затем итеративно корректируем их, чтобы восстановить апостериорное распределение, учитывающее информацию, содержащуюся в данных. На практике это означает, что мы используем оценку глобального уровня для обоснования наших групповых оценок, а групповые параметры — для обоснования индивидуальных. Единицам с большим количеством наблюдений разрешается больше отклоняться от групповых мат ожиданий, в то время как единицы с ограниченным количеством наблюдений приближаются к мат ожиданиям.

Давайте рассмотрим это на примере ценовой эластичности. В идеале, мы хотим получить ценовую эластичность на уровне товарной единицы. Мы оцениваем:

Где:

βi ∼ Normal(βc(i),σi)
βc(i) ∼ Normal(βg,σc(i))
βg ∼ Normal(μ,σ)

Единственное отличие от первого уравнения заключается в том, что мы заменяем глобальный член β на индивидуальные продуктовые βi. Уточним, что эластичность на уровне товарных единиц βi определяется на основе нормального распределения, которое центрируется вокруг среднего значения эластичности на уровне категории βc(i), которое, в свою очередь, рассчитывается на основе общей глобальной эластичности βg для всех групп. Для дисперсий σ мы также можем предположить иерархическую структуру, но здесь мы просто установим для них стандартные априорные значения, чтобы не усложнять пример. В данном случае мы предполагаем, что априорные значения: μ =-2, σ=1, σc (i)=1, σi=1. Эта априорная формулировка предполагает, что глобальная эластичность является эластичной, 95% эластичностей находятся в диапазоне от -4 до 0, со среднеквадратическим отклонением 1 на каждом иерархическом уровне. Чтобы проверить, правильно ли мы определили эти априорные значения, мы бы провели априорные прогностические проверки (которые не рассматриваются в этой статье), чтобы увидеть, могут ли наши априорные предположения восстановить данные, которые мы наблюдаем.

Эта иерархическая структура позволяет передавать информацию как между продуктами одной категории, так и между категориями. Если количество данных о ценах для конкретного продукта ограничено, его эластичность будет приближаться к мат ожиданию эластичности по категории βc(i). Аналогичным образом, на категории с меньшим количеством товаров больше повлияет глобальная эластичность, которая определяется мат ожиданием эластичности по всем категориям. Прелесть этого подхода заключается в том, что степень «объединения» определяется автоматически на основе данных. Продукты с большим объемом данных о вариации цен будут поддерживать оценки, более близкие к их индивидуальным моделям данных, в то время как продукты с меньшим количеством данных будут черпать больше информации из своей группы.

Реализация

В этом разделе мы подробно рассмотрим процесс реализации описанной ранее модели с использованием пакета Numpyro на Python. Numpyro представляет собой легковесный вероятностный язык программирования, основанный на JAX, что обеспечивает autograd‑ и JIT‑компиляцию на графических GPU / TPU / CPU. Мы начнем с генерации наших синтетических данных, затем определим модель и подгоним ее под эти данные. В заключение мы визуализируем результаты нашей работы.

Процесс генерации данных

Мы моделируем данные о продажах, где спрос находится в лог‑линейной зависимости от цены, а эластичность на уровне продукта генерируется на основе гауссова распределения βi ∼ Normal (-2,0,7). В каждый период времени с вероятностью 50% мы добавляем случайное изменение цены, временные тренды для конкретной категории и случайный шум. Эти факторы мультипликативно суммируются для формирования нашего логарифмического ожидаемого спроса. Чтобы получить фактический спрос, мы экспонируем логарифмический ожидаемый спрос и получаем количество реализованных единиц, проданных согласно распределению Пуассона. Для повышения точности оценок мы отбираем только те единицы, которые были проданы более 100 раз. В результате получаем N = 11 798 товаров за T = 156 периодов (еженедельные данные за 3 года). Исходя из этого набора данных, истинная глобальная эластичность составляет βg=-1,6, при этом эластичности на уровне категории варьируются в пределах βc(i)∈[-1.68,-1.48].

Обратите внимание, что в этом DGP (динамическая гамма‑пуассоновская модель — методе, который мы используем для моделирования) не учитываются многие сложные аспекты реального мира. Мы не рассматриваем факторы, которые могут одновременно влиять как на цены, так и на спрос, такие как рекламные акции. Также мы не учитываем факторы, которые могут мешать работе модели. Этот пример предназначен только для того, чтобы продемонстрировать, как мы можем восстановить эластичность для конкретного продукта в рамках модели, разработанной Уэллсом. Он не ставит своей целью объяснить, как правильно определить, что цена является независимой переменной. Тем не менее, я рекомендую заинтересовавшимся читателям полистать книгу Causal Inference for the Brave and True для более глубокого понимания методов причинно‑следственного анализа.

import numpy as np
import pandas as pd
def generate_price_elasticity_data(N: int = 1000,
                                   C: int = 10,
                                   T: int = 50,
                                   price_change_prob: float = 0.2,
                                   seed = 42) -> pd.DataFrame:

    """

    Генерируем синтетические данные для анализа ценовой эластичности спроса.
    Данные генерируются с помощью
    """
    
    if seed is not None:
        np.random.seed(seed)

    
    # Спроса и тренды категории
    category_base_demand = np.random.uniform(1000, 10000, C)
    category_time_trends = np.random.uniform(0, 0.01, C)
    category_volatility = np.random.uniform(0.01, 0.05, C)  # Случайная волатильность для каждой категории
    category_demand_paths = np.zeros((C, T))
    category_demand_paths[:, 0] = 1.0
    shocks = np.random.normal(0, 1, (C, T-1)) * category_volatility[:, np.newaxis]
    trends = category_time_trends[:, np.newaxis] * np.ones((C, T-1))
    cumulative_effects = np.cumsum(trends + shocks, axis=1)
    category_demand_paths[:, 1:] = category_demand_paths[:, 0:1] + cumulative_effects

    # Эффекты продукта
    product_categories = np.random.randint(0, C, N)
    product_a = np.random.normal(-2, .7, size=N)
    product_a = np.clip(product_a, -5, -.1)

 
    # Начальные цены на каждый продукт
    initial_prices = np.random.uniform(100, 1000, N)
    prices = np.zeros((N, T))
    prices[:, 0] = initial_prices

   
    # Генерация случайных значений и определение факта изменения цен
    random_values = np.random.rand(N, T-1)
    change_mask = random_values < price_change_prob

    
    # Генерация коэффициентов изменения (от -20% до +20%)
    change_factors = 1 + np.random.uniform(-0.2, 0.2, size=(N, T-1))

    
    # Создание матрицы для хранения множителей
    multipliers = np.ones((N, T-1))

    
    # Применение коэффициентов изменения только там, где должны произойти изменения 
    multipliers[change_mask] = change_factors[change_mask]

    
    # Применение изменений кумулятивно для распространения цен
    for t in range(1, T):
        prices[:, t] = prices[:, t-1] * multipliers[:, t-1]

    
    # Генерация множители для конкретного продукта
    product_multipliers = np.random.lognormal(3, 0.5, size=N)
    
    # Получение временных эффектов для каждой категории продукта (размерность: N x T)
    time_effects = category_demand_paths[product_categories][:, np.newaxis, :].squeeze(1)

    
    # Чтобы временные эффекты не становились отрицательными
    time_effects = np.maximum(0.1, time_effects)

    
    # Генерация периодического шума для всех продуктов и периодов времени
    period_noise = 1 + np.random.uniform(-0.05, 0.05, size=(N, T))

    
    # Получение базового спроса категории для каждого продукта
    category_base = category_base_demand[product_categories]

    
    # Расчет базового спроса
    base_demand = (category_base[:, np.newaxis] *
                   product_multipliers[:, np.newaxis] *
                   time_effects *
                   period_noise)

    # Логарифм спроса
    alpha_ijt = np.log(base_demand)

    # Логарифм цены
    log_prices = np.log(prices)

    # Логарифм ожидаемого спроса
    log_lambda = alpha_ijt + product_a[:, np.newaxis] * log_prices

    # Обратное преобразование из логарифма для получения параметров интенсивности
    lambda_vals = np.exp(log_lambda)

    # Генерация проданных товарных единиц
    units_sold = np.random.poisson(lambda_vals)  # Формат: (N, T)

    
    # Создание массивов индексов для всех комбинаций продуктов и периодов времени
    product_indices, time_indices = np.meshgrid(np.arange(N), np.arange(T), indexing='ij')
    product_indices = product_indices.flatten()
    time_indices = time_indices.flatten()

    
    # Получение категорий для всех продуктов
    categories = product_categories[product_indices]

    
    # Получение всех цен и проданных единиц
    all_prices = np.round(prices[product_indices, time_indices], 2)
    all_units_sold = units_sold[product_indices, time_indices]

    
    # Вычисление эластичности
    product_elasticity = product_a[product_indices]
    df = pd.DataFrame({
        'product': product_indices,
        'category': categories,
        'time_period': time_indices,
        'price': all_prices,
        'units_sold': all_units_sold,
        'product_elasticity': product_elasticity
    })

    
    return df

# Сохраняем только единицы с >X продаж
def filter_dataframe(df, min_units = 100):
    temp = df[['product','units_sold']].groupby('product').sum().reset_index()
    unit_filter = temp[temp.units_sold>min_units]['product'].unique()
    filtered_df = df[df['product'].isin(unit_filter)].copy()
    
    # Сводка
    original_product_count = df['product'].nunique()
    remaining_product_count = filtered_df['product'].nunique()
    filtered_out = original_product_count - remaining_product_count

    
    print(f"Filtering summary:")
    print(f"- Original number of products: {original_product_count}")
    print(f"- Products with > {min_units} units: {remaining_product_count}")
    print(f"- Products filtered out: {filtered_out} ({filtered_out/original_product_count:.1%})")

    # Глобальная эластичность и эластичность по категориям
    global_elasticity = filtered_df['product_elasticity'].mean()
    filtered_df['global_elasticity'] = global_elasticity

    # Эластичность по категориям
    category_elasticities = filtered_df.groupby('category')['product_elasticity'].mean().reset_index()
    category_elasticities.columns = ['category', 'category_elasticity']
    filtered_df = filtered_df.merge(category_elasticities, on='category', how='left')

    # Сводка
    print(f"\nElasticity Information:")
    print(f"- Global elasticity: {global_elasticity:.3f}")
    print(f"- Category elasticities range: {category_elasticities['category_elasticity'].min():.3f} to {category_elasticities['category_elasticity'].max():.3f}")

    
    return filtered_df

df = generate_price_elasticity_data(N = 20000, T = 156, price_change_prob=.5, seed=42)
df = filter_dataframe(df)
df.loc[:,'cat_by_time'] = df['category'].astype(str) + '-' + df['time_period'].astype(str)
df.head()

Сводка по фильтрации:

Первоначальное количество продуктов: 20 000
Количество с более чем 100 единицами: 11 798
Количество отфильтрованных продуктов: 8202 (41,0%)

Информация об эластичности:

Глобальная эластичность: -1,598
Диапазон эластичности по категориям: от -1,681 до -1,482

product	category	time_period	price	units_sold	product_elasticity	category_elasticity	global_elasticity	cat_by_time
0	8	0	125.95	550	-1.185 907	-1.63 475	-1.597 683	8–0
0	8	1	125.95	504	-1.185 907	-1.63 475	-1.597 683	8–1
0	8	2	149.59	388	-1.185 907	-1.63 475	-1.597 683	8–2
0	8	3	149.59	349	-1.185 907	-1.63 475	-1.597 683	8–3
0	8	4	176.56	287	-1.185 907	-1.63 475	-1.597 683	8–4

Модель

Мы начинаем с создания индексов для продуктов, категорий и их сочетаний с временем, используя функцию pd.factorize(). Это позволяет нам выбрать правильный параметр для каждого наблюдения. Затем мы преобразуем ряды цен (логарифмированные) и товарных единиц в массивы JAX, затем создаем таблицы, после чего создаем таблицы для каждой из наших групп параметров. В этих таблицах хранятся значения параметров для каждого уровня иерархии, а также параметры, представляющие фиксированные эффекты.

Для определения групп параметров в модели используется таблица основных параметров NumPyro:

global_a: 1 глобальный параметр ценовой эластичности с априорным значением Normal(−2,1).
category_a: C=10 эластичностей на уровне категории с априорными значениями, центрированными по глобальному параметру, и среднеквадратическим отклонением, равным 1.
product_a: N = 11 798 эластичностей для конкретного продукта с априорными значениями, центрированными по параметрам соответствующей категории, и среднеквадратическим отклонением, равным 1.
product_effect: N=11 798 базовых эффектов спроса на конкретный продукт со среднеквадратическим отклонением 3.
time_cat_effects: (T=156)⋅(C=10) эффектов, изменяющихся во времени, для каждой комбинации категории и времени со среднеквадратическим отклонением 3.

Затем мы переназначаем параметры, используя аргумент LocScaleReparam(), чтобы повысить эффективность выборки и избежать необходимости повторной выборки. После создания параметров мы вычисляем логарифмический ожидаемый спрос, затем преобразуем его обратно в параметр интенсивности с учетом ограничений для обеспечения числовой стабильности. Наконец, мы используем табличку с данными для генерации выборок из распределения Пуассона с рассчитанным параметром интенсивности. Затем алгоритм оптимизации найдет значения параметров, которые наилучшим образом соответствуют данным, используя стохастический градиентный спуск. Ниже приведено графическое представление модели, демонстрирующее взаимосвязь между параметрами.

import jax
import jax.numpy as jnp
import numpyro
import numpyro.distributions as dist
from numpyro.infer.reparam import LocScaleReparam
def model(df: pd.DataFrame, outcome: None):
  
    # Определение индексов
    product_idx, unique_product = pd.factorize(df['product'])
    cat_idx, unique_category = pd.factorize(df['category'])
    time_cat_idx, unique_time_cat = pd.factorize(df['cat_by_time'])

    # Преобразование рядов цен и товарных единиц в массивы jax
    log_price = jnp.log(df.price.values)
    outcome = jnp.array(outcome) if outcome is not None else None

    # Генерация сопоставления
    product_to_category = jnp.array(pd.DataFrame({'product': product_idx, 'category': cat_idx}).drop_duplicates().category.values, dtype=np.int16)

    # Создание таблиц для хранения параметров
    category_plate = numpyro.plate("category", unique_category.shape[0])
    time_cat_plate = numpyro.plate("time_cat", unique_time_cat.shape[0])
    product_plate = numpyro.plate("product", unique_product.shape[0])
    data_plate = numpyro.plate("data", size=outcome.shape[0])

    # ОПРЕДЕЛЕНИЕ ПАРАМЕТРОВ МОДЕЛИ
    global_a = numpyro.sample("global_a", dist.Normal(-2, 1), infer={"reparam": LocScaleReparam()})

    with category_plate:
        category_a = numpyro.sample("category_a", dist.Normal(global_a, 1), infer={"reparam": LocScaleReparam()})

    with product_plate:
        product_a = numpyro.sample("product_a", dist.Normal(category_a[product_to_category], 2), infer={"reparam": LocScaleReparam()})
        product_effect = numpyro.sample("product_effect", dist.Normal(0, 3))

    with time_cat_plate:
        time_cat_effects = numpyro.sample("time_cat_effects", dist.Normal(0, 3))

    # Расчет ожидаемого спроса
    def calculate_demand():
        log_demand = product_a[product_idx]*log_price + time_cat_effects[time_cat_idx] + product_effect[product_idx]
        expected_demand = jnp.exp(jnp.clip(log_demand, -4, 20)) # Ограничение для стабильности 
        return expected_demand

    demand = calculate_demand()

    with data_plate:
        numpyro.sample(
            "obs",
            dist.Poisson(demand),
            obs=outcome

        )

   numpyro.render_model(
    model=model,
    model_kwargs={"df": df,"outcome": df['units_sold']},
    render_distributions=True,
    render_params=True,
)

Оценка

Хотя существует множество способов оценки данного уравнения, в этом конкретном случае мы используем стохастический вариационный вывод (SVI). SVI — это метод оптимизации на основе градиента, позволяющий минимизировать расхождение Кульбака‑Лейблера между предлагаемым апостериорным распределением и истинным апостериорным распределением путем минимизации нижней вариационной границы (ELBO). Этот метод отличается от метода Монте‑Карло с марковскими цепями (MCMC), который производит выборку непосредственно из истинного апостериорного распределения. В реальных приложениях SVI более эффективен и легко масштабируется до больших наборов данных. Для этого приложения мы устанавливаем случайное начальное значение, инициализируем guide (семейство апостериорного распределения, предполагается, что это диагональная нормаль), определяем скорость обучения и оптимизатор в Optax и запускаем оптимизацию для 1 000 000 (занимает ~ 1 час) итераций. Хотя модель, возможно, сходилась заранее, потери продолжают улучшаться на незначительную величину даже после выполнения оптимизации в течение 1 000 000 итераций. Наконец, мы отражаем потери (логарифмические) на графике.

from numpyro.infer import SVI, Trace_ELBO, autoguide, init_to_sample
import optax
import matplotlib.pyplot as plt

rng_key = jax.random.PRNGKey(42)
guide = autoguide.AutoNormal(model, init_loc_fn=init_to_sample)
# Определение скорости обучения
learning_rate_schedule = optax.exponential_decay(
    init_value=0.01,
    transition_steps=1000,
    decay_rate=0.99,
    staircase = False,
    end_value = 1e-5,
)

# Определение оптимизатора
optimizer = optax.adamw(learning_rate=learning_rate_schedule)
svi = SVI(model, guide, optimizer, loss=Trace_ELBO(num_particles=8, vectorize_particles = True))

# Запуск SVI
svi_result = svi.run(rng_key, 1_000_000, df, df['units_sold'])
plt.semilogy(svi_result.losses);

Извлечение апостериорных образцов

После обучения модели мы можем восстановить апостериорное распределение параметров, используя полученные результаты и исходный набор данных. Однако, поскольку Numpyro применяет аффинное преобразование для ненормальных распределений на серверной стороне, мы не можем напрямую вызвать параметры svi_result.params. Вместо этого мы делаем выборку из апостериорного распределения 1000 раз и вычисляем мат ожидание и среднеквадратическое отклонение каждого параметра в нашей модели. Заключительная часть следующего кода создает фрейм данных с предполагаемой эластичностью для каждого продукта на каждом иерархическом уровне, который затем мы соединяем обратно с нашим исходным фреймом данных, чтобы проверить, восстанавливает ли алгоритм истинную эластичность.

predictive = numpyro.infer.Predictive(
    autoguide.AutoNormal(model, init_loc_fn=init_to_sample),
    params=svi_result.params,
    num_samples=1000
)

samples = predictive(rng_key, df, df['units_sold'])

# Извлечение мат ожидания и среднеквадратического отклонения
results = {}
excluded_keys = ['product_effect', 'time_cat_effects']
for k, v in samples.items():
    if k not in excluded_keys:
        results[f"{k}"] = np.mean(v, axis=0)
        results[f"{k}_std"] = np.std(v, axis=0)

# Оценки эластичности продуктов
prod_elasticity_df = pd.DataFrame({
    'product': df['product'].unique(),
    'product_elasticity_svi': results['product_a'],
    'product_elasticity_svi_std': results['product_a_std'],
})
result_df = df.merge(prod_elasticity_df, on='product', how='left')

# Оценки эластичности по категориям
prod_elasticity_df = pd.DataFrame({
    'category': df['category'].unique(),
    'category_elasticity_svi': results['category_a'],
    'category_elasticity_svi_std': results['category_a_std'],
})
result_df = result_df.merge(prod_elasticity_df, on='category', how='left')

# Оценки глобальной эластичности
result_df['global_a_svi'] = results['global_a']
result_df['global_a_svi_std'] = results['global_a_std']
result_df.head()

product	category	time_period	price	units_sold	product_elasticity	category_elasticity	global_elasticity	cat_by_time	product_elasticity_svi	product_elasticity_svi_std	category_elasticity_svi	category_elasticity_svi_std	global_a_svi	global_a_svi_std
0	8	0	125.95	550	-1.185 907	-1.63 475	-1.597 683	8–0	-1.180 956	0.000 809	-1.559 872	0.027 621	-1.5 550 271	0.2 952 548
0	8	1	125.95	504	-1.185 907	-1.63 475	-1.597 683	8–1	-1.180 956	0.000 809	-1.559 872	0.027 621	-1.5 550 271	0.2 952 548
0	8	2	149.59	388	-1.185 907	-1.63 475	-1.597 683	8–2	-1.180 956	0.000 809	-1.559 872	0.027 621	-1.5 550 271	0.2 952 548
0	8	3	149.59	349	-1.185 907	-1.63 475	-1.597 683	8–3	-1.180 956	0.000 809	-1.559 872	0.027 621	-1.5 550 271	0.2 952 548
0	8	4	176.56	287	-1.185 907	-1.63 475	-1.597 683	8–4	-1.180 956	0.000 809	-1.559 872	0.027 621	-1.5 550 271	0.2 952 548

Результаты

В представленном коде показаны истинные и рассчитанные значения эластичности для каждого продукта. Каждая точка ранжирована в соответствии с их истинным значением эластичности, которое обозначено черным цветом. Также отображается расчетная эластичность, полученная с помощью модели. Можно заметить, что оцененная эластичность следует траектории истинной эластичности со средней абсолютной погрешностью, равной примерно 0.0724. Точки, окрашенные в красный цвет, представляют продукты, 95% доверительного интервала которых не включают истинную эластичность, в то время как синие точки обозначают продукты, 95% доверительного интервала которых совпадают с истинной эластичностью. Учитывая, что глобальное мат ожидание составляет -1.598, это означает, что средняя погрешность на уровне продукта составляет около 4.5%. Можно отметить, что оценки SVI точно следуют за схемой истинных эластичностей, хотя и с некоторым уровнем шума, особенно когда эластичности становятся более отрицательными. На верхней правой панели мы отображаем график зависимости между погрешностью расчетных значений эластичности и истинными значениями. С увеличением отрицательности истинной эластичности, наша модель начинает давать все более неточные результаты.

Чтобы получить оценки эластичности на уровне категории и на глобальном уровне, мы можем воспользоваться двумя методами. Первый — это увеличение эластичности на уровне каждого продукта в рамках категории. Второй метод — получение оценок на уровне категории непосредственно из апостериорных параметров. Если мы обратимся к оценкам эластичности на уровне категории, представленным в левом нижнем углу, то увидим, что как оценки, восстановленные из модели, так и оценки, полученные на основе загруженных выборок эластичности на уровне продукта, слегка смещены в сторону нуля, со значением средней абсолютной ошибки около 0.033. Однако доверительный интервал, задаваемый параметром уровня категории, охватывает истинный параметр, в отличие от оценок, полученных с использованием начальной загрузки. Это свидетельствует о том, что при определении эластичности на уровне группы следует напрямую использовать параметры на уровне группы, вместо того чтобы прибегать к начальной загрузке более детализированных оценок. При анализе на глобальном уровне оба метода дают точную оценку параметра в пределах 95% доверительного интервала. Однако глобальный параметр оказывается выше, чем первоначальная загрузка на уровне продукта, из‑за увеличения среднеквадратических погрешностей.

Соображения

Байесовский иерархический метод недооценивает апостериорную дисперсию: Одним из недостатков использования SVI для оценки является то, что он недооценивает апостериорную дисперсию. Хотя мы подробно рассмотрим эту тему в следкющей статье, целевая функция для SVI учитывает только разницу в математическом ожидании нашего предполагаемого распределения и истинного распределения. Это означает, что он не учитывает полную структуру корреляции между параметрами в апостериоре. Аппроксимация среднего поля, обычно используемая в SVI, предполагает условную (при построении предыдущей иерархии) независимость между параметрами, которая игнорирует любые ковариации между продуктами в пределах одной и той же иерархии. Это означает, что при наличии каких‑либо побочных эффектов (таких как каннибализация или перекрестная ценовая эластичность) они не будут учитываться в пределах доверительных интервалов. Из‑за этого допущения о среднем поле оценки неопределенности обычно слишком надежны. Это приводит к слишком узким доверительным интервалам, которые не могут точно отразить истинные значения параметров с необходимой интенсивностью. На верхнем левом рисунке мы видим, что только 9.7% эластичностей на уровне продукта соответствуют их истинным значениям. В следующем статье мы поговорим о возможных решениях этой проблемы.
Важность априорных значений: При использовании иерархических байесовских моделей априорные значения играют гораздо более важную роль, чем в традиционных байесовских подходах. Обычно большие наборы данных позволяют вероятностям преобладать над приоритетами при оценке глобальных параметров. Однако иерархические структуры меняют эту динамику и уменьшают эффективный размер выборки на каждом уровне. В нашей модели глобальный параметр рассматривает только 10 наблюдений на уровне категории (а не полный набор данных), категории основаны только на продуктах, которые в них входят, а продукты полагаются исключительно на свои собственные наблюдения. Это сокращение эффективного размера выборки приводит к тому, что оценки выбросов (например, очень отрицательные эластичности) приближаются к мат ожиданиям для своей категории. Это подчеркивает необходимость априорных проверок для прогнозирования, так как неправильно заданные значения могут существенно повлиять на результаты.

def elasticity_plots(result_df, results=None):
    # Создаем фигуру с сеткой 2х2
    fig = plt.figure(figsize=(12, 10))
    gs = fig.add_gridspec(2, 2)

 
    # Эластичность продукта
    ax1 = fig.add_subplot(gs[0, 0])

    
    # Подготовка данных
    df_product = result_df[['product','product_elasticity','product_elasticity_svi','product_elasticity_svi_std']].drop_duplicates()
    df_product['product_elasticity_svi_lb'] = df_product['product_elasticity_svi'] - 1.96*df_product['product_elasticity_svi_std']
    df_product['product_elasticity_svi_ub'] = df_product['product_elasticity_svi'] + 1.96*df_product['product_elasticity_svi_std']
    df_product = df_product.sort_values('product_elasticity')
    mae_product = np.mean(np.abs(df_product.product_elasticity-df_product.product_elasticity_svi))
    colors = []
    for i, row in df_product.iterrows():
        if (row['product_elasticity'] >= row['product_elasticity_svi_lb'] and 
            row['product_elasticity'] <= row['product_elasticity_svi_ub']):
            colors.append('blue')  # В пределах доверительного интервала
        else:
            colors.append('red')   # За пределами доверительного интервала

   
    # Процентное количество точек в пределах допустимых значений
    within_bounds_pct = colors.count('blue') / len(colors) * 100

    
    # Данные графика
    ax1.scatter(range(len(df_product)), df_product['product_elasticity'], 
                color='black', label='True Elasticity', s=20, zorder=3)

  
    ax1.scatter(range(len(df_product)), df_product['product_elasticity_svi'], 
                color=colors, label=f'SVI Estimate (MAE: {mae_product:.4f}, Coverage: {within_bounds_pct:.1f}%)', 
                s=3, zorder=2)
    ax1.set_xlabel('Product Index (sorted by true elasticity)')
    ax1.set_ylabel('Elasticity Value')
    ax1.set_title('SVI Estimates of True Product Elasticities')
    ax1.legend()
    ax1.grid(alpha=0.3)

   
    # Взаимосвязь между средней абсолютной ошибкой и истинной эластичностью
    ax2 = fig.add_subplot(gs[0, 1])

    
    # Расчет средней абсолютной ошибки для каждого продукта
    temp = result_df[['product','product_elasticity', 'product_elasticity_svi']].drop_duplicates().copy()
    temp['product_error'] = temp['product_elasticity'] - temp['product_elasticity_svi']
    temp['product_mae'] = np.abs(temp['product_error'])
    correlation = temp[['product_mae', 'product_elasticity']].corr()

   
    # Данные графика
    ax2.scatter(temp['product_elasticity'], temp['product_error'], alpha=0.5, s=5, color = colors)
    ax2.set_xlabel('True Elasticity')
    ax2.set_ylabel('Error (True - Estimated)')
    ax2.set_title('Relationship Between True Elasticity and Estimation Accuracy')
    ax2.grid(alpha=0.3)
    ax2.text(0.5, 0.95, f"Correlation: {correlation.iloc[0,1]:.3f}", 
                transform=ax2.transAxes, ha='center', va='top',
                bbox=dict(boxstyle='round', facecolor='white', alpha=0.7))

    
    # Эластичность категория 
    ax3 = fig.add_subplot(gs[1, 0])

    
    # Уникальные категории и эластичности
    category_data = result_df[['category', 'category_elasticity', 'category_elasticity_svi', 'category_elasticity_svi_std']].drop_duplicates()
    category_data = category_data.sort_values('category_elasticity')

    
    # Загрузка мат ожиданий и эластичностей продуктов в каждой категории
    bootstrap_means = []
    bootstrap_ci_lower = []
    bootstrap_ci_upper = []

   
    for cat in category_data['category']:
        # Получение эластичности продукта для этой категории
        prod_elasticities = result_df[result_df['category'] == cat]['product_elasticity_svi'].unique()

        
        # Загрузка мат ожиданий
        boot_means = [np.mean(np.random.choice(prod_elasticities, size=len(prod_elasticities), replace=True)) 
                     for _ in range(1000)]

       
        bootstrap_means.append(np.mean(boot_means))
        bootstrap_ci_lower.append(np.percentile(boot_means, 2.5))
        bootstrap_ci_upper.append(np.percentile(boot_means, 97.5))

    
    category_data['bootstrap_mean'] = bootstrap_means
    category_data['bootstrap_ci_lower'] = bootstrap_ci_lower
    category_data['bootstrap_ci_upper'] = bootstrap_ci_upper

    
    # Вычисление средней абсолютной ошибки
    mae_category_svi = np.mean(np.abs(category_data['category_elasticity_svi'] - category_data['category_elasticity']))
    mae_bootstrap = np.mean(np.abs(category_data['bootstrap_mean'] - category_data['category_elasticity']))

        
    # Вывод данных на график
    left_offset = -0.2
    right_offset = 0.2
    x_range = range(len(category_data))
    ax3.scatter(x_range, category_data['category_elasticity'], 
                color='black', label='True Elasticity', s=50, zorder=3)

    
    # Загрузка эластичности продукта
    ax3.scatter([x + left_offset for x in x_range], category_data['bootstrap_mean'], 
                color='green', label=f'Bootstrapped Product Estimate (MAE: {mae_bootstrap:.4f})', s=30, zorder=2)
    for i in x_range:
        ax3.plot([i + left_offset, i + left_offset], 
                [category_data['bootstrap_ci_lower'].iloc[i], category_data['bootstrap_ci_upper'].iloc[i]], 
                color='green', alpha=0.3, zorder=1)

    
    # Оценки SVI на уровне категорий
    ax3.scatter([x + right_offset for x in x_range], category_data['category_elasticity_svi'], 
                color='blue', label=f'Category SVI Estimate (MAE: {mae_category_svi:.4f})', s=30, zorder=2)
    for i in x_range:
        ci_lower = category_data['category_elasticity_svi'].iloc[i] - 1.96 * category_data['category_elasticity_svi_std'].iloc[i]
        ci_upper = category_data['category_elasticity_svi'].iloc[i] + 1.96 * category_data['category_elasticity_svi_std'].iloc[i]
        ax3.plot([i + right_offset, i + right_offset], [ci_lower, ci_upper], color='blue', alpha=0.3, zorder=1)

    ax3.set_xlabel('Category Index (sorted by true elasticity)')
    ax3.set_ylabel('Elasticity')
    ax3.set_title('Comparison with True Category Elasticity')
    ax3.legend()
    ax3.grid(alpha=0.3)

    # Глобальная эластичность
    ax4 = fig.add_subplot(gs[1, 1])
    temp = result_df[['product','product_elasticity_svi','global_elasticity']].drop_duplicates()
    bootstrap_means = [np.mean(np.random.choice(np.array(temp['product_elasticity_svi']), 100)) for i in range(10000)]
    global_means = np.random.normal(result_df['global_a_svi'].iloc[0], result_df['global_a_svi_std'].iloc[0], 10000)
    true_global = np.unique(temp.global_elasticity)[0]
    p_mae = np.abs(np.mean(bootstrap_means) - true_global)
    g_mae = np.abs(np.mean(global_means) - true_global)

   
    # Данные графика
    ax4.hist(bootstrap_means, bins=100, alpha=0.3, density=True, 
             label=f'Product Elasticity Bootstrap (MAE: {p_mae:.4f})')
    ax4.hist(global_means, bins=100, alpha=0.3, density=True, 
             label=f'Global Elasticity Distribution (MAE: {g_mae:.4f})')
    ax4.axvline(x=true_global, color='black', linestyle='--', linewidth=2, 
                label=f'Global Elasticity: {true_global:.3f}', zorder=0)
    ax4.set_xlabel('Elasticity')
    ax4.set_ylabel('Frequency')
    ax4.set_title('Global Elasticity Comparison')
    ax4.legend()
    ax4.grid(True, linestyle='--', alpha=0.7)

    # Отображение
    plt.tight_layout()
    plt.show()

elasticity_plots(result_df)

Заключение

Альтернативные варианты использования: Помимо оценки ценовой эластичности спроса, байесовские иерархические модели также находят множество других применений в Data Science. В розничной торговле они могут прогнозировать спрос на товары в уже существующих магазинах и решать проблему «холодного запуска» новых торговых точек. Для этого модели заимствуют информацию из уже открытых магазинов или сетей, которые группируются в рамках единой иерархии. В рекомендательных системах модели HB позволяют оценить предпочтения пользователей на основе сочетания характеристик как самих пользователей, так и товаров. Эта структура позволяет давать релевантные рекомендации новым пользователям, основываясь на их поведении в группах, и постепенно переходить к индивидуальным рекомендациям по мере накопления истории покупок. Если же нет возможности сформировать группы по когортам, то для группировки схожих единиц на основе их характеристик может быть использован алгоритм K‑средних.

В заключение, эти модели также могут быть полезны для объединения результатов экспериментальных и наблюдательных исследований. Ученые могут использовать исторические оценки увеличения объема наблюдений и дополнить их данными недавно проведенных A/B‑тестов, чтобы сократить необходимый объем выборки для экспериментов, используя уже имеющиеся знания. Такой подход позволяет создать основу для непрерывного обучения, где каждый новый эксперимент строится на предыдущих результатах, а не начинается с нуля. Для команд, сталкивающихся с ограниченными ресурсами, это означает более быстрое время до понимания (особенно в сочетании с суррогатными моделями) и более эффективные конвейеры экспериментов.

Заключительные замечания: В этом вводном материале мы рассмотрели несколько примеров применения иерархических байесовских моделей, но это лишь малая часть того, что можно узнать о них. Мы не касались более сложных и специфических аспектов реализации, таких как предварительные и последующие проверки прогнозов, формальные оценки соответствия, масштабирование вычислений, распределенное обучение, эффективность стратегий оценки (MCMC и SVI) и вложенные иерархические структуры. Каждая из этих тем заслуживает отдельной статьи.

Тем не менее, этот обзор должен стать практическим руководством для внедрения иерархического байесовского метода в ваш инструментарий. Эти модели предоставляют основу для обработки сложных многоуровневых структур данных, которые часто встречаются в реальных бизнес‑задачах. Когда вы начнете применять эти подходы, я с нетерпением жду возможности услышать о вашем опыте, проблемах, успехах и новых возможностях использования этого класса моделей. Пожалуйста, не стесняйтесь присылать вопросы, идеи или примеры по электронной почте или через LinkedIn. Если у вас есть отзывы об этой статье или вы хотите обсудить другую тему в разделе причинно‑следственного вывода и машинного обучения, не стесняйтесь обращаться. Спасибо за внимание!

Углубляетесь в ML и ищете что‑то мощнее стандартных моделей?

На открытом уроке 23 июня разберём байесовские иерархические модели — один из самых гибких и точных подходов в продвинутом машинном обучении. Вы поймёте, как объединять данные с разных уровней и получать устойчивые оценки даже при нехватке данных. Если интересно — записывайтесь.

Перед уроком — короткий тест: проверьте, насколько вы уже готовы к следующему шагу в ML и «потяните» ли программу продвинутого курса по ML.

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4 июн. 2025 г.

09.10.25 08:23 pHqghUme

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14.10.25 08:46 robertalfred175

CRYPTO SCAM RECOVERY SUCCESSFUL – A TESTIMONIAL OF LOST PASSWORD TO YOUR DIGITAL WALLET BACK. My name is Robert Alfred, Am from Australia. I’m sharing my experience in the hope that it helps others who have been victims of crypto scams. A few months ago, I fell victim to a fraudulent crypto investment scheme linked to a broker company. I had invested heavily during a time when Bitcoin prices were rising, thinking it was a good opportunity. Unfortunately, I was scammed out of $120,000 AUD and the broker denied me access to my digital wallet and assets. It was a devastating experience that caused many sleepless nights. Crypto scams are increasingly common and often involve fake trading platforms, phishing attacks, and misleading investment opportunities. In my desperation, a friend from the crypto community recommended Capital Crypto Recovery Service, known for helping victims recover lost or stolen funds. After doing some research and reading multiple positive reviews, I reached out to Capital Crypto Recovery. I provided all the necessary information—wallet addresses, transaction history, and communication logs. Their expert team responded immediately and began investigating. Using advanced blockchain tracking techniques, they were able to trace the stolen Dogecoin, identify the scammer’s wallet, and coordinate with relevant authorities to freeze the funds before they could be moved. Incredibly, within 24 hours, Capital Crypto Recovery successfully recovered the majority of my stolen crypto assets. I was beyond relieved and truly grateful. Their professionalism, transparency, and constant communication throughout the process gave me hope during a very difficult time. If you’ve been a victim of a crypto scam, I highly recommend them with full confidence contacting: 📧 Email: [email protected] 📱 Telegram: @Capitalcryptorecover Contact: [email protected] 📞 Call/Text: +1 (336) 390-6684 🌐 Website: https://recovercapital.wixsite.com/capital-crypto-rec-1
15.10.25 18:07 crypto

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

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

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

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17.10.25 20:20 lindseyvonn

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

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

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17.10.25 21:42 marcushenderson624

Bitcoin Recovery Testimonial After falling victim to a cryptocurrency scam group, I lost $354,000 worth of USDT. I thought all hope was lost from the experience of losing my hard-earned money to scammers. I was devastated and believed there was no way to recover my funds. Fortunately, I started searching for help to recover my stolen funds and I came across a lot of testimonials online about Capital Crypto Recovery, an agent who helps in recovery of lost bitcoin funds, I contacted Capital Crypto Recover Service, and with their expertise, they successfully traced and recovered my stolen assets. Their team was professional, kept me updated throughout the process, and demonstrated a deep understanding of blockchain transactions and recovery protocols. They are trusted and very reliable with a 100% successful rate record Recovery bitcoin, I’m grateful for their help and highly recommend their services to anyone seeking assistance with lost crypto. Contact: [email protected] Phone CALL/Text Number: +1 (336) 390-6684 Email: [email protected] Website: https://recovercapital.wixsite.com/capital-crypto-rec-1
17.10.25 21:42 marcushenderson624

Bitcoin Recovery Testimonial After falling victim to a cryptocurrency scam group, I lost $354,000 worth of USDT. I thought all hope was lost from the experience of losing my hard-earned money to scammers. I was devastated and believed there was no way to recover my funds. Fortunately, I started searching for help to recover my stolen funds and I came across a lot of testimonials online about Capital Crypto Recovery, an agent who helps in recovery of lost bitcoin funds, I contacted Capital Crypto Recover Service, and with their expertise, they successfully traced and recovered my stolen assets. Their team was professional, kept me updated throughout the process, and demonstrated a deep understanding of blockchain transactions and recovery protocols. They are trusted and very reliable with a 100% successful rate record Recovery bitcoin, I’m grateful for their help and highly recommend their services to anyone seeking assistance with lost crypto. Contact: [email protected] Phone CALL/Text Number: +1 (336) 390-6684 Email: [email protected] Website: https://recovercapital.wixsite.com/capital-crypto-rec-1
17.10.25 21:42 marcushenderson624

Bitcoin Recovery Testimonial After falling victim to a cryptocurrency scam group, I lost $354,000 worth of USDT. I thought all hope was lost from the experience of losing my hard-earned money to scammers. I was devastated and believed there was no way to recover my funds. Fortunately, I started searching for help to recover my stolen funds and I came across a lot of testimonials online about Capital Crypto Recovery, an agent who helps in recovery of lost bitcoin funds, I contacted Capital Crypto Recover Service, and with their expertise, they successfully traced and recovered my stolen assets. Their team was professional, kept me updated throughout the process, and demonstrated a deep understanding of blockchain transactions and recovery protocols. They are trusted and very reliable with a 100% successful rate record Recovery bitcoin, I’m grateful for their help and highly recommend their services to anyone seeking assistance with lost crypto. Contact: [email protected] Phone CALL/Text Number: +1 (336) 390-6684 Email: [email protected] Website: https://recovercapital.wixsite.com/capital-crypto-rec-1
21.10.25 08:39 debby131

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

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

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

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

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

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

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

I never imagined I’d fall for a crypto romance scam but it happened. Over the course of a few months, I sent nearly $150,000 worth of Bitcoin to someone I genuinely believed I was building a future with. When they disappeared without a trace, I was left heartbroken, humiliated, and financially devastated. For a long time, I didn’t tell anyone. I felt ashamed. But eventually, while searching for answers, I came across a Reddit thread that mentioned Agent Jasmine Lopez. I reached out, not expecting much. To my surprise, she treated me with kindness, not judgment. She used advanced tools like blockchain forensics, IP tracing, and smart contract analysis and with persistence and legal support, she was able to recover nearly 85% of what I lost. I know not everyone gets that kind of outcome, but thanks to [email protected] WhatsApp at +44 736-644-5035, I’ve started to reclaim not just my assets, but my confidence and peace of mind. If you’re going through something similar, you’re not alone and there is hope.
23.10.25 01:52 marcushenderson624

Bitcoin Recovery Testimonial After falling victim to a cryptocurrency scam group, I lost $354,000 worth of USDT. I thought all hope was lost from the experience of losing my hard-earned money to scammers. I was devastated and believed there was no way to recover my funds. Fortunately, I started searching for help to recover my stolen funds and I came across a lot of testimonials online about Capital Crypto Recovery, an agent who helps in recovery of lost bitcoin funds, I contacted Capital Crypto Recover Service, and with their expertise, they successfully traced and recovered my stolen assets. Their team was professional, kept me updated throughout the process, and demonstrated a deep understanding of blockchain transactions and recovery protocols. They are trusted and very reliable with a 100% successful rate record Recovery bitcoin, I’m grateful for their help and highly recommend their services to anyone seeking assistance with lost crypto. Contact: [email protected] Phone CALL/Text Number: +1 (336) 390-6684 Email: [email protected] Website: https://recovercapital.wixsite.com/capital-crypto-rec-1
23.10.25 01:52 marcushenderson624

Bitcoin Recovery Testimonial After falling victim to a cryptocurrency scam group, I lost $354,000 worth of USDT. I thought all hope was lost from the experience of losing my hard-earned money to scammers. I was devastated and believed there was no way to recover my funds. Fortunately, I started searching for help to recover my stolen funds and I came across a lot of testimonials online about Capital Crypto Recovery, an agent who helps in recovery of lost bitcoin funds, I contacted Capital Crypto Recover Service, and with their expertise, they successfully traced and recovered my stolen assets. Their team was professional, kept me updated throughout the process, and demonstrated a deep understanding of blockchain transactions and recovery protocols. They are trusted and very reliable with a 100% successful rate record Recovery bitcoin, I’m grateful for their help and highly recommend their services to anyone seeking assistance with lost crypto. Contact: [email protected] Phone CALL/Text Number: +1 (336) 390-6684 Email: [email protected] Website: https://recovercapital.wixsite.com/capital-crypto-rec-1
23.10.25 15:47 MATT PHILLIP

I never imagined I’d fall for a crypto romance scam but it happened. Over the course of a few months, I sent nearly $150,000 worth of Bitcoin to someone I genuinely believed I was building a future with. When they disappeared without a trace, I was left heartbroken, humiliated, and financially devastated. For a long time, I didn’t tell anyone. I felt ashamed. But eventually, while searching for answers, I came across a Reddit thread that mentioned Agent Jasmine Lopez. I reached out, not expecting much. To my surprise, she treated me with kindness, not judgment. She used advanced tools like blockchain forensics, IP tracing, and smart contract analysis and with persistence and legal support, she was able to recover nearly 85% of what I lost. I know not everyone gets that kind of outcome, but thanks to [email protected] WhatsApp at +44 736-644-5035, I’ve started to reclaim not just my assets, but my confidence and peace of mind. If you’re going through something similar, you’re not alone and there is hope.
23.10.25 21:43 patricialovick86

How To Recover Your Bitcoin Without Falling Victim To Scams: A Testimony Experience With Capital Crypto Recover Services, Contact Telegram: @Capitalcryptorecover Dear Everyone, I would like to take a moment to share my positive experience with Capital Crypto Recover Services. Initially, I was unsure if it would be possible to recover my stolen bitcoins. However, with their expertise and professionalism, I was able to fully recover my funds. Unfortunately, many individuals fall victim to scams in the cryptocurrency space, especially those involving fraudulent investment platforms. However, I advise caution, as not all recovery services are legitimate. I personally lost $273,000 worth of Bitcoin from my Binance account due to a deceptive platform. If you have suffered a similar loss, you may be considering crypto recovery, The Capital Crypto Recover is the most knowledgeable and effective Capital Crypto Recovery Services assisted me in recovering my stolen funds within 24 hours, after getting access to my wallet. Their service was not only prompt but also highly professional and effective, and many recovery services may not be trustworthy. Therefore, I highly recommend Capital Crypto Recover to you. i do always research and see reviews about their service, For assistance finding your misplaced cryptocurrency, get in touch with them, They do their jobs quickly and excellently, Stay safe and vigilant in the crypto world. Contact: [email protected] You can reach them via email at [email protected] OR Call/Text Number +1 (336)390-6684 his contact website: https://recovercapital.wixsite.com/capital-crypto-rec-1
23.10.25 21:43 patricialovick86

How To Recover Your Bitcoin Without Falling Victim To Scams: A Testimony Experience With Capital Crypto Recover Services, Contact Telegram: @Capitalcryptorecover Dear Everyone, I would like to take a moment to share my positive experience with Capital Crypto Recover Services. Initially, I was unsure if it would be possible to recover my stolen bitcoins. However, with their expertise and professionalism, I was able to fully recover my funds. Unfortunately, many individuals fall victim to scams in the cryptocurrency space, especially those involving fraudulent investment platforms. However, I advise caution, as not all recovery services are legitimate. I personally lost $273,000 worth of Bitcoin from my Binance account due to a deceptive platform. If you have suffered a similar loss, you may be considering crypto recovery, The Capital Crypto Recover is the most knowledgeable and effective Capital Crypto Recovery Services assisted me in recovering my stolen funds within 24 hours, after getting access to my wallet. Their service was not only prompt but also highly professional and effective, and many recovery services may not be trustworthy. Therefore, I highly recommend Capital Crypto Recover to you. i do always research and see reviews about their service, For assistance finding your misplaced cryptocurrency, get in touch with them, They do their jobs quickly and excellently, Stay safe and vigilant in the crypto world. Contact: [email protected] You can reach them via email at [email protected] OR Call/Text Number +1 (336)390-6684 his contact website: https://recovercapital.wixsite.com/capital-crypto-rec-1
24.10.25 06:08 MATT PHILLIP

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

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

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

I never imagined I’d fall for a crypto romance scam but it happened. Over the course of a few months, I sent nearly $150,000 worth of Bitcoin to someone I genuinely believed I was building a future with. When they disappeared without a trace, I was left heartbroken, humiliated, and financially devastated. For a long time, I didn’t tell anyone. I felt ashamed. But eventually, while searching for answers, I came across a Reddit thread that mentioned Agent Jasmine Lopez. I reached out, not expecting much. To my surprise, she treated me with kindness, not judgment. She used advanced tools like blockchain forensics, IP tracing, and smart contract analysis and with persistence and legal support, she was able to recover nearly 85% of what I lost. I know not everyone gets that kind of outcome, but thanks to [email protected] WhatsApp at +44 736-644-5035, I’ve started to reclaim not just my assets, but my confidence and peace of mind. If you’re going through something similar, you’re not alone and there is hope.
24.10.25 18:18 patricialovick86

How To Recover Your Bitcoin Without Falling Victim To Scams: A Testimony Experience With Capital Crypto Recover Services, Contact Telegram: @Capitalcryptorecover Dear Everyone, I would like to take a moment to share my positive experience with Capital Crypto Recover Services. Initially, I was unsure if it would be possible to recover my stolen bitcoins. However, with their expertise and professionalism, I was able to fully recover my funds. Unfortunately, many individuals fall victim to scams in the cryptocurrency space, especially those involving fraudulent investment platforms. However, I advise caution, as not all recovery services are legitimate. I personally lost $273,000 worth of Bitcoin from my Binance account due to a deceptive platform. If you have suffered a similar loss, you may be considering crypto recovery, The Capital Crypto Recover is the most knowledgeable and effective Capital Crypto Recovery Services assisted me in recovering my stolen funds within 24 hours, after getting access to my wallet. Their service was not only prompt but also highly professional and effective, and many recovery services may not be trustworthy. Therefore, I highly recommend Capital Crypto Recover to you. i do always research and see reviews about their service, For assistance finding your misplaced cryptocurrency, get in touch with them, They do their jobs quickly and excellently, Stay safe and vigilant in the crypto world. Contact: [email protected] You can reach them via email at [email protected] OR Call/Text Number +1 (336)390-6684 his contact website: https://recovercapital.wixsite.com/capital-crypto-rec-1
24.10.25 18:18 patricialovick86

How To Recover Your Bitcoin Without Falling Victim To Scams: A Testimony Experience With Capital Crypto Recover Services, Contact Telegram: @Capitalcryptorecover Dear Everyone, I would like to take a moment to share my positive experience with Capital Crypto Recover Services. Initially, I was unsure if it would be possible to recover my stolen bitcoins. However, with their expertise and professionalism, I was able to fully recover my funds. Unfortunately, many individuals fall victim to scams in the cryptocurrency space, especially those involving fraudulent investment platforms. However, I advise caution, as not all recovery services are legitimate. I personally lost $273,000 worth of Bitcoin from my Binance account due to a deceptive platform. If you have suffered a similar loss, you may be considering crypto recovery, The Capital Crypto Recover is the most knowledgeable and effective Capital Crypto Recovery Services assisted me in recovering my stolen funds within 24 hours, after getting access to my wallet. Their service was not only prompt but also highly professional and effective, and many recovery services may not be trustworthy. Therefore, I highly recommend Capital Crypto Recover to you. i do always research and see reviews about their service, For assistance finding your misplaced cryptocurrency, get in touch with them, They do their jobs quickly and excellently, Stay safe and vigilant in the crypto world. Contact: [email protected] You can reach them via email at [email protected] OR Call/Text Number +1 (336)390-6684 his contact website: https://recovercapital.wixsite.com/capital-crypto-rec-1
25.10.25 00:49 tyleradams

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25.10.25 00:50 stevekalfman

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

A few months ago, I fell for what looked like a legitimate online trading platform. The website was sleek, the support team was responsive, and they even showed me fake profit charts that seemed to grow daily. I started small, and after a few test withdrawals worked, I invested more eventually depositing 2 Bitcoin in hopes of long-term returns. But one morning, when I tried to withdraw, everything changed. The platform suddenly locked my account, claiming there were additional fees to release my funds. The support chat stopped responding, and the website started giving error messages. That’s when I realized I’d been scammed. I was devastated. Two Bitcoins wasn’t pocket change, it was my savings. I filed reports online, but everyone told me it was impossible to recover crypto once it’s sent. Then, through a discussion forum, I came across Dexdert Net Recovery, a team specializing in tracing and recovering stolen digital assets. Skeptical but desperate, I reached out. Their response was immediate and professional. They asked for transaction IDs, wallet addresses, screenshots of my correspondence with the fake platform, and any proof of payment I had. Within days, their analysts traced the stolen Bitcoin through multiple wallets using blockchain forensics, They updated me step-by-step, Working with their recovery experts and blockchain partners, Dexdert Net Recovery successfully froze the funds and verified ownership of my wallet. After 2 days of persistence and verification, I finally saw my 2 Bitcoin restored back into my own wallet. I couldn’t believe it. What stood out most wasn’t just that they recovered my funds, but how transparent and supportive the Dexdert team was throughout the process. They educated me on crypto safety, how to spot fake trading sites, and how to secure my digital assets in the future. Dexdert Net Recovery truly lives up to its name. They gave me back not just my Bitcoin, but my peace of mind. Reach Out Information Below Via: Email: ([email protected]) Telegram: (https://t.me/Dexdertprorecovery) WhatsApp: (+1 (859) 609‑4156)
25.10.25 10:13 wendytaylor015

My name is Wendy Taylor, I'm from Los Angeles, i want to announce to you Viewer how Capital Crypto Recover help me to restore my Lost Bitcoin, I invested with a Crypto broker without proper research to know what I was hoarding my hard-earned money into scammers, i lost access to my crypto wallet or had your funds stolen? Don’t worry Capital Crypto Recover is here to help you recover your cryptocurrency with cutting-edge technical expertise, With years of experience in the crypto world, Capital Crypto Recover employs the best latest tools and ethical hacking techniques to help you recover lost assets, unlock hacked accounts, Whether it’s a forgotten password, Capital Crypto Recover has the expertise to help you get your crypto back. a security company service that has a 100% success rate in the recovery of crypto assets, i lost wallet and hacked accounts. I provided them the information they requested and they began their investigation. To my surprise, Capital Crypto Recover was able to trace and recover my crypto assets successfully within 24hours. Thank you for your service in helping me recover my $647,734 worth of crypto funds and I highly recommend their recovery services, they are reliable and a trusted company to any individuals looking to recover lost money. Contact email [email protected] OR Telegram @Capitalcryptorecover Call/Text Number +1 (336)390-6684 his contact: [email protected] His website: https://recovercapital.wixsite.com/capital-crypto-rec-1
25.10.25 10:13 wendytaylor015

My name is Wendy Taylor, I'm from Los Angeles, i want to announce to you Viewer how Capital Crypto Recover help me to restore my Lost Bitcoin, I invested with a Crypto broker without proper research to know what I was hoarding my hard-earned money into scammers, i lost access to my crypto wallet or had your funds stolen? Don’t worry Capital Crypto Recover is here to help you recover your cryptocurrency with cutting-edge technical expertise, With years of experience in the crypto world, Capital Crypto Recover employs the best latest tools and ethical hacking techniques to help you recover lost assets, unlock hacked accounts, Whether it’s a forgotten password, Capital Crypto Recover has the expertise to help you get your crypto back. a security company service that has a 100% success rate in the recovery of crypto assets, i lost wallet and hacked accounts. I provided them the information they requested and they began their investigation. To my surprise, Capital Crypto Recover was able to trace and recover my crypto assets successfully within 24hours. Thank you for your service in helping me recover my $647,734 worth of crypto funds and I highly recommend their recovery services, they are reliable and a trusted company to any individuals looking to recover lost money. Contact email [email protected] OR Telegram @Capitalcryptorecover Call/Text Number +1 (336)390-6684 his contact: [email protected] His website: https://recovercapital.wixsite.com/capital-crypto-rec-1
26.10.25 01:03 Christopherbelle

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

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

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

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

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

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

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

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

A sudden money crisis struck me. I dropped $82,000 to a phony crypto scam. All seemed fine until I went to pull out my gains. Then, i was unable to get back my hard earned money. I reported it to the police several times. No aid came my way. Deep sadness hit me; I felt crushed. That's when I learned about Sylvester Bryant Intelligence. They acted quick and fair from the start. They checked my case closely. They followed the path of my lost cash. They pushed strong for me at each turn. In shock, they brought my funds back. I had figured it could not happen. I thank them for their effort and honesty. Sylvester Bryant Intelligence gave me calm again. If a scam stole your crypto, reach out to them today. Contact Info: Yt7cracker@gmail . com WhatsApp: +1 512 577 7957 or +44 7428 662701.
29.10.25 10:56 wendytaylor015

My name is Wendy Taylor, I'm from Los Angeles, i want to announce to you Viewer how Capital Crypto Recover help me to restore my Lost Bitcoin, I invested with a Crypto broker without proper research to know what I was hoarding my hard-earned money into scammers, i lost access to my crypto wallet or had your funds stolen? Don’t worry Capital Crypto Recover is here to help you recover your cryptocurrency with cutting-edge technical expertise, With years of experience in the crypto world, Capital Crypto Recover employs the best latest tools and ethical hacking techniques to help you recover lost assets, unlock hacked accounts, Whether it’s a forgotten password, Capital Crypto Recover has the expertise to help you get your crypto back. a security company service that has a 100% success rate in the recovery of crypto assets, i lost wallet and hacked accounts. I provided them the information they requested and they began their investigation. To my surprise, Capital Crypto Recover was able to trace and recover my crypto assets successfully within 24hours. Thank you for your service in helping me recover my $647,734 worth of crypto funds and I highly recommend their recovery services, they are reliable and a trusted company to any individuals looking to recover lost money. Contact email [email protected] OR Telegram @Capitalcryptorecover Call/Text Number +1 (336)390-6684 his contact: [email protected] His website: https://recovercapital.wixsite.com/capital-crypto-rec-1
29.10.25 10:56 wendytaylor015

My name is Wendy Taylor, I'm from Los Angeles, i want to announce to you Viewer how Capital Crypto Recover help me to restore my Lost Bitcoin, I invested with a Crypto broker without proper research to know what I was hoarding my hard-earned money into scammers, i lost access to my crypto wallet or had your funds stolen? Don’t worry Capital Crypto Recover is here to help you recover your cryptocurrency with cutting-edge technical expertise, With years of experience in the crypto world, Capital Crypto Recover employs the best latest tools and ethical hacking techniques to help you recover lost assets, unlock hacked accounts, Whether it’s a forgotten password, Capital Crypto Recover has the expertise to help you get your crypto back. a security company service that has a 100% success rate in the recovery of crypto assets, i lost wallet and hacked accounts. I provided them the information they requested and they began their investigation. To my surprise, Capital Crypto Recover was able to trace and recover my crypto assets successfully within 24hours. Thank you for your service in helping me recover my $647,734 worth of crypto funds and I highly recommend their recovery services, they are reliable and a trusted company to any individuals looking to recover lost money. Contact email [email protected] OR Telegram @Capitalcryptorecover Call/Text Number +1 (336)390-6684 his contact: [email protected] His website: https://recovercapital.wixsite.com/capital-crypto-rec-1
30.10.25 09:49 Christopherbelle

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

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