Time Series Algorithms

(1) Introduction

Time series analysis can offer valuable insights into stock prices, sales figures, customer behaviour, and other time-dependent variables.
By leveraging these techniques, businesses can make informed decisions, optimize operations, and enhance long-term strategies.
Time series analysis offers a multitude of benefits to businesses.
The applications are also wide-ranging, whether it’s in forecasting sales to manage inventory better, identifying the seasonality in consumer behaviour to plan marketing campaigns, or even analyzing financial markets for investment strategies.
Different techniques serve distinct purposes and offer varied granularity and accuracy, making it vital for businesses to understand the methods that best suit their specific needs.

Decomposition in time series analysis refers to the process of breaking down a time series into its individual components, typically trend, seasonality, and remainder (or residual) components.
The decomposition allows us to better understand the underlying patterns and variations present in the time series data.

The most common approach for decomposition is the additive decomposition, which assumes that the observed time series can be represented as the sum of its components:
Y(t) = Trend(t) + Seasonality(t) + Remainder(t)

Trend: The trend component represents the long-term direction or movement of the time series. It captures the overall systematic increase or decrease in the series over time. The trend can be linear, polynomial, exponential, or other functional forms depending on the nature of the trend observed in the data.
Seasonality: The seasonality component represents the regular and repeating patterns that occur within a fixed period, such as daily, monthly, or yearly cycles. Seasonality can be deterministic (fixed pattern) or stochastic (varying pattern). It captures the systematic variations that occur at specific intervals.
Remainder (Residual): The remainder component, also known as the residual or error term, represents the random and irregular fluctuations that cannot be explained by the trend and seasonality components. It captures the unexplained variation, noise, or randomness in the time series.

The decomposition process involves estimating or extracting the trend, seasonality, and remainder components from the observed time series. There are different techniques to perform decomposition, including:

Moving Averages: Simple Moving Average (SMA) or Weighted Moving Average (WMA) can be used to estimate the trend component by smoothing out the short-term fluctuations.
Classical Decomposition: Methods like Classical Decomposition or the Census Bureau’s X-11 method use statistical techniques to estimate the trend and seasonal components.
Seasonal and Trend decomposition using LOESS (STL): STL is a popular non-parametric method that decomposes the time series using a local regression approach (LOESS) to estimate the trend, seasonality, and residual components.
Fourier Analysis: Fourier analysis can be used to identify and estimate the dominant frequencies in the data, which correspond to the seasonal patterns.

Decomposition helps in understanding and interpreting the various components of the time series separately.
It provides insights into the long-term trends, cyclical patterns, and random fluctuations that are present in the data.
Decomposed components can be useful for further analysis, forecasting, and anomaly detection in time series data.