Movingaverage

A moving average is a statistical technique used to analyze time series data by creating a sequence of averages of different subsets of the full data set. The method smooths short-term fluctuations and highlights longer-term trends or cycles, making patterns easier to identify in noisy data. It is widely used in statistics, signal processing, econometrics, and finance.

There are several common variants. The simple moving average (SMA) computes the arithmetic mean of a fixed number of most recent observations. The weighted moving average (WMA) assigns weights to observations, typically decreasing for older data. The exponential moving average (EMA) applies exponentially decreasing weights, giving more emphasis to recent observations. The cumulative moving average (CMA) uses all data points up to the current time point. A centered moving average shifts the window to align with the midpoint of observations.

Calculation examples. SMA for a window of n observations is the sum of the last n values divided by n. EMA follows a recursive formula, often written as EMA_t = alpha x_t + (1 - alpha) EMA_{t-1}, with alpha determined by the chosen window length. WMA uses a weighted sum of the most recent n observations divided by the sum of the weights.

Applications and limitations. Moving averages are used to smooth data, identify trends, and generate trading signals through crossovers of short-term and long-term averages. They function as simple linear filters in signal processing. Limitations include lag proportional to the window size, sensitivity to the chosen window, and reduced effectiveness when the underlying process changes rapidly or nonstationarity is present. Variants such as rolling versus expanding windows and center-aligned approaches address some edge effects and alignment concerns.