The phrase "контрольные карты" translates to "control cards" in English. These could refer to various types of cards used for monitoring or controlling different processes or activities. If you have a specific context in mind, feel free to provide additional details for more precise translation.

The phrase "контрольные карты Шухарта" translates to "Shuhart control charts" in English. Shuh control charts are a statistical tool used in quality control to monitor and analyze processes. These charts are used to determine if a process is in a state of statistical control or if corrective action is needed.

The Hurst exponent, named after the hydrologist Harold Edwin Hurst, is a measure of long-term memory of time series. It can be used to analyze and characterize the fractal properties of a time series, particularly in terms of its long-range dependence.

In recent years, researchers have developed innovative ways to utilize the Hurst exponent for time series segmentation. Some of the latest methods include:

1. Dynamic Segmentation: By incorporating the Hurst exponent into dynamic segmentation algorithms, researchers are able to identify and segment time series data into different regions based on their long-term dependence properties. This allows for more targeted analysis and modeling of specific segments within the time series.

2. Feature Extraction: Utilizing the Hurst exponent as a feature for machine learning algorithms, particularly for time series clustering and classification tasks. By deriving features from the Hurst exponent, researchers can improve the accuracy and robustness of machine learning models for time series segmentation.

3. Adaptive Thresholding: Incorporating the Hurst exponent to adaptively determine segmentation thresholds based on the long-term memory properties of different segments within a time series. This approach enhances the precision of identifying significant changes and patterns within the data.

4. Multiscale Segmentation: Integrating the Hurst exponent into multiscale segmentation techniques to capture and analyze the fractal properties of time series data across different time scales. This allows for a more comprehensive understanding of the underlying dynamics and patterns present in the time series.

These advanced methods showcase the growing importance and versatility of the Hurst exponent in effectively segmenting time series data, leading to more precise analysis and interpretation of complex temporal patterns.

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However, if you have access to academic databases or search engines such as PubMed, IEEE Xplore, Google Scholar, or similar platforms, you can use keywords such as "Hurst exponent time series segmentation" to find relevant, peer-reviewed articles and research papers that discuss the latest methods of using the Hurst exponent for time series segmentation. These platforms can provide you with access to up-to-date and specific scientific literature related to this topic.

The use of the Hurst exponent for time series segmentation has found various applications in recent research. Here are some of the advanced methods along with a brief description and relevant mathematical expressions:

1. Dynamic Segmentation with Hurst Exponent: This method involves the adaptive segmentation of time series based on the Hurst exponent values. Researchers have developed algorithms that incorporate the Hurst exponent to identify boundaries for segments with different long-term dependence properties. This is particularly useful for capturing changes in the underlying behavior of the time series.

   Mathematical expression: Let H(t) denote the Hurst exponent at time t. dynamic segmentation algorithm could use a criterion based on changes in H(t) to determine the optimal segmentation points. For example, a simple thresholding method could be: If |H(t) - H(t-1)| > ε, where ε is a predefined threshold, then mark a segment boundary at time t.

2. Feature Extraction using Hurst Exponent: Another approach involves using the Hurst exponent as a feature for time series segmentation. By computing the Hurst exponent for different segments of the time series, this method aims to extract features that capture the fractal properties of the data, enabling more effective segmentation.

   Mathematical expression: One technique for feature extraction could involve computing the Hurst exponent, H, for each segment s of the time series. This may involve applying the rescaled range analysis formula: R/S = (1/n) * Σ (Xi - Xmean), where R/S denotes the rescaled range and Xi denotes the values of the time series.

3. Adaptive Thresholding based on Hurst Exponent: Utilizing the Hurst exponent to adaptively determine segmentation thresholds, taking into account the long-term memory properties of different segments within a time series. This approach allows for more flexible and data-driven segmentation.

   Mathematical expression: An adaptive thresholding method might involve using the Hurst exponent to calculate a segment-specific threshold. For instance, considering a threshold based on the standard deviation of the Hurst exponent in each segment, such as: Threshold(s) = α * σ(H(s)), where α is a scaling factor, and σ(H(s)) represents the standard deviation of thest exponent in the segment s.

Please note that these methods are simplified for illustrative purposes, and specific implementations in research may involve additional considerations and variations in mathematical formulations based on the nature of the time series data and the segmentation objectives.

When calculating the Hurst exponent for a time series that contains consecutive identical values, a common method is to utilize a modified version of the rescaled range analysis, which is a technique commonly used to estimate the Hurst exponent.

The rescaled range (R/S) analysis is a method used to determine the degree of persistence, or long-term memory, in a time series. It involves calculating the range of cumulative deviations from the mean over various time scales and then rescaling these ranges. The formula typically includes dividing the range by the standard deviation of the time series.

In the case of consecutive identical values in a time series, the rescaled range analysis can still be applied with a slight modification. When calculating the mean and standard deviation for the rescaled range analysis, it's important to handle the consecutive identical values properly to avoid computational issues.

One approach to address this is to replace the consecutive identical values with a small amount of random noise. This can be achieved by adding a small random value, typically drawn from a uniform or normal distribution, to each of the identical values. This introduces minor variations in the data while preserving the general pattern and sequence. After this adjustment, the rescaled range analysis can be performed as usual to calculate the Hurst exponent.

It's important to note that the specific method for handling consecutive identical values may vary depending on the characteristics of the time series and the objectives of the analysis. Consulting relevant literature or consulting with experts in the field of time series analysis can provide additional insights into best practices for addressing this issue.

Предел отношения скользящего размаха к стандартному отклонению при устремлении к 0 сходится к бконечности.

Формула для расчета скользящего размаха (R/S) обычно выглядит следующим образом: [ R/S = \frac{1}{n} \sum_{i=1}^{n} (X_i - \overline{X}) ] где ( X_i ) - значения временного ряда, ( \overline{X} ) - среднее значение временного ряда, ( n ) - количество значений в ряду.

Стандартное отклонение (σ) вычисляется по формуле: [ \sigma = \sqrt{\frac{1}{n} \sum_{i=1}^{n} (X_i - \overline{X})^2} ]

Когда ( n ) стремится к 0, то есть при устремлении к размеру выборки равному 1 или при наличии последовательных одинаковых значений, деление суммы разностей на стандартное отклонение приводит к бесконечности. Это связано с ограничениями математических операций и указает на то, что использованиеользящего размаха для оценки временных рядов с размером выборки близким к 0 может приводить к непредсказуемым результатам.

Скользящий размах (R/S) рассчитанный на одинаковых значениях будет равен 0. Это происходит потому, что скользящий размах измеряет изменчивость или разброс значений во временном ряду относительно их среднего значения. Если все значения одинаковы, то разность каждого значения среднего значения также будет равна 0, в результате чего сумма всех таких разностей будет равна 0. Следовательно, скользящий размах на одинаковых значениях будет равен 0.

Нормированный размах (R/S) рассчитанный на одинаковых значениях будет равен 0. Для вычисления нормированного размаха необходимо рассчитать разность между максимальным и минимальным значением в ряду, а затем поделить эту разность на стандартное отклонение. Если все значения одинаковы, то разница между максимальным и минимальным значением равна 0. Следовательно, нормированный размах на одинаковых значениях также будет равен 0.

Физический смысл показателя Хёрста связан с изучением долгосрочной зависимости или памяти во временных рядах.

Когда речь идет о временных рядах, показатель Хёрстаобозначаемый как H) используется для описанияепени плавности или "фрактальности" временного ряда. Он измеряет скорость убывания или возрастания вариации или разброса данных с увеличением временного интервала.

Физический смысл показателя Хёрста может быть иллюстрирован на примере геофизических процессов. Например, при анализе долгосрочных данных о уровне воды в реке или изменении уровня моря, показатель Хёрста позволяет оценить насколько значительным образом данный процесс подвержен долгосрочным тенденциям. В случае финансовых данных, показатель Хёрста позволяет выявить наличие долгосрочных корреляций и предсказать возможные тенденции в изменениях цен на долгий период времени.

Таким образом, физический смысл показателя Хёрста состоит в том, что он помогает изучать и описывать планомерность или характерную долгосрочную динамику в различных временных рядах, что обладает большим практическим значением в различных областях, от финансов до климатических и геофизических исследований.

В научной статье на аглийском языке применяются определенные стандарты построения предложений, обеспечивающие ясность, точность и научную аргументацию. Вот несколько типичных структур предложений с примерами:

1. Описание результатов и выводов: Пример: "The analysis of the data revealed a significant correlation between the two variables, indicating a strong relationship."

2. Представление фактов или заключений: Пример:Previous studies have demonstrated the impact of environmental factors on genetic mutations."

3. Объяснение методологии или процедур: Пример: "The samples were subjected to a series of tests to determine the presence of specific biomarkers."

4. Указание на основные темы или результаты: Пример: "The main findings of the study suggest a need for further investigation into the mechanism of action."

5. Проведениеогического рассуждения или аргументации: Пример: "Based on the experimental data, it can be inferred that the proposed hypothesis is supported by the evidence."

Каждая из этих структур предложений способствует ясной и эффективной комуникации научных исследований в англоязычных научных статьях.