Elaborate on at least two different time series decomposition methods, and state their ideal application.

Time series decomposition is a technique used to break down a time series into its constituent components, revealing the underlying patterns that might be hidden within the raw data. The primary aim is to separate a time series into several components: trend, seasonality, cyclical patterns, and residual (or irregular) noise. This decomposition can help in understanding the behavior of the data, identifying patterns, and is also important in predicting future trends. Here are two common time series decomposition methods and their ideal applications: 1. Classical Decomposition: Classical decomposition, also known as traditional decomposition, is one of the simplest and most straightforward techniques for breaking down a time series into its components. It assumes that a time series can be represented as either a sum (additive model) or a product (multiplicative model) of its components. The additive model assumes that the components are independent and their effects add up, while the multiplicative model assumes they interact and are expressed as a product. The additive model is given by Y(t) = T(t) + S(t) + R(t), and multiplicative model is Y(t) = T(t) S(t) R(t), where Y(t) is the time series value at time t, T(t) is the trend component, S(t) is the seasonal component, and R(t) is the residual or irregular component. *The Trend Component:The trend component represents the long-term direction of the data and the overall increase or decrease of the series. This can be calculated by using moving averages, which smooth out the short-term fluctuations. If a time series shows a consistent upward or downward movement over a long period, the moving average would reveal that trend. For example, in a time series showing yearly sales of a company over several years, the trend might show that the overall sales are increasing year after year. *The Seasonal Component:The seasonal component represents the repetitive short-term patterns that occur within a fixed time period. For example, retail sales may spike aroun....