holt-winters：作为示例，该存储库提供了对时间序列数据集的深入分析和预测，并总结了对Holt-Winter模型有更深入...

# Holt-Winters Forecasting # Table of Contents - [Holt-Winters Forecasting](#holt-winters-forecasting) - [Table of Contents](#table-of-contents) - [Introduction](#introduction) - [Mathematical Overview](#mathematical-overview) - [Time Series](#time-series) - [Aspects of Time Series](#aspects-of-time-series) - [Exponential Smoothing](#exponential-smoothing) - [Why exponential smoothing?](#why-exponential-smoothing) - [Formula](#formula) - [Holt's Model](#holts-model) - [Drawbacks of SES](#drawbacks-of-ses) - [What is Holt's Model?](#what-is-holts-model) - [Formula](#formula-1) - [Comparison with Linear Regression](#comparison-with-linear-regression) - [Holt-Winter Model](#holt-winter-model) - [Additive Model](#additive-model) - [Formula](#formula-2) - [Multiplicative Model](#multiplicative-model) - [Formula](#formula-3) - [Anomaly Detection using Brutlag algorithm](#anomaly-detection-using-brutlag-algorithm) - [What is Brutlag algorithm?](#what-is-brutlag-algorithm) - [Formula](#formula-4) - [References](#references) - [Textbooks](#textbooks) - [Other](#other) - [Getting Started](#getting-started) - [Installation guide](#installation-guide) - [Jupyter Notebook](#jupyter-notebook) - [Python Script](#python-script) - [Generating the analysis](#generating-the-analysis) - [Anomaly Detection](#anomaly-detection) - [Author](#author) # Introduction Holt-Winters forecasting is a way to model and predict the behavior of a sequence of values over time—a time series. # Mathematical Overview Before getting into the analysis of the real-time series model let's understand a few basic concepts required to have a deeper understanding of the topic. ## Time Series A time series is a sequence of numerical data points in successive and chronological order. Generally, the x-axis or index is taken as time and the y-axis or value represents the value for the corresponding x value. ### Aspects of Time Series 1. Level - the typical value or the average 2. Trend - the slope at that instance 3. Seasonality - cyclical repeating pattern ## Exponential Smoothing Before knowing what is exponential smoothing lets understand why it's required. ### Why exponential smoothing? The real-world datasets for time series are hard to forecast and generally, it's assumed that data of recent past have higher significance compared to old data, hence more weightage is given to recent data than older data. The problem faced with [weighted moving average](https://www.fidelity.com/learning-center/trading-investing/technical-analysis/technical-indicator-guide/wma) is choosing the appropriate weights for each level. Exponential smoothing uses EWMA (Exponential Weighted Moving Average), i.e older data have lesser contribution compared to newer data. In EWMA weights reduce exponentially overtime and provide reasonable weights for each level. ### Formula If you have some time series xt, you can define a new time series st that is a smoothed version of xt.  The following graph with α=0.5 shows how weights of older data diminish over time. The black bars denote the actual value(xt) and colored bars denote the smoothened value(st).  [Image Source](https://orangematter.solarwinds.com/2014/11/25/how-exponentially-weighted-moving-averages-work/) ## Holt's Model Holt's model or Double Exponential Smoothing is an extension to simple exponential smoothing. ### Drawbacks of SES - SES assumes the time series to be [stationary](https://otexts.com/fpp2/stationarity.html),i.e it assumes the statistical properties such as the mean, variance and autocorrelation are all constant over time. - But in general, a majority of time series have either trend or seasonality component, thus forecasts made by SES are unproductive. ### What is Holt's Model? Holt's model overcomes the drawback of SES by considering both level and trend component. The term **double exponential smoothing** was coined because exponential smoothing is performed both on level and trend component. ### Formula  - **F** - the forecast at time **t**. - **D** - the actual value at time **t**. - **a** - level at time **t**. - **b** - trend/slope component at time **t**. - **α** - smoothing parameter for level. - **β** - smoothing parameter for trend. The following graph illustrates how holt's model is used to forecast time series having a trend component.  [Image Source](https://orangematter.solarwinds.com/2017/06/22/exponential-smoothing-for-time-series-forecasting/) ### Comparison with Linear Regression - Linear Regression is an effective method used to predict data points. The downside is that it considers all data with the same weightage. - This property of LR isn't suitable for time series data, hence holt's is preferred over LR for time series analysis as the model uses the exponential smoothing. ## Holt-Winter Model Holt-Winter model is an extension of Holt's model. This takes into account all the 3 time-series components (level, trend and seasonality). The seasonality component of time series is an important component since most real-world data have a seasonal period. ### Additive Model Holt-Winter additive model is preferred when the time series shows constant or roughly constant seasonality over time. #### Formula  - **F** - the forecast at time **t**. - **D** - the actual value at time **t**. - **a** - level at time **t**. - **b** - trend/slope component at time **t**. - **c** - seasonal component at time **t**. - **α** - smoothing parameter for level. - **β** - smoothing parameter for trend. - **γ** - smoothing parameter for seasonality. - **p** - period for the time series. The following graph illustrates how the Holt-Winter additive model is used to predict seasonal data having a constant seasonality of period=12 or 1 year.  [Image Source](https://orangematter.solarwinds.com/2017/06/22/exponential-smoothing-for-time-series-forecasting/) ### Multiplicative Model Holt-Winter multiplicative is useful in cases when the time series has the seasonality component changing over time. #### Formula  The following graph illustrates how the multiplicative model is used to generate forecasts to predict seasonal data having the seasonal component changes over time.  [Image So