Time Series Analysis (Moving Average Filters)

Description

This article describes several filters for the analysis of time series data, including:

• Simple Moving Average (SMA)
• Exponential Moving Average (EMA)
• Linear Regression Moving Average (LRMA)

Model theory

Simple moving average

The simple moving average (SMA) at element $t$ of a data series is the unweighted mean of the previous $k$ values, or the previous $t$ values if $t<k$. That is:

${\mathit {SMA}}_{k,t}={\begin{cases}{\dfrac {1}{k}}\sum \limits _{i=t-k+1}^{t}p_{i}&t\geq k\\{\dfrac {1}{t}}\sum \limits _{i=1}^{t}p_{i}&t<k\\\end{cases}}$

where $p_{1},p_{2},\dots ,p_{t}$ are the data points to average.

Exponential moving average

The Exponential Moving Average (EMA), or exponential smoothing, of a data series at element $t$ is:

${\mathit {EMA}}_{\alpha ,t}=\alpha \cdot p_{t}+(1-\alpha )\cdot {\mathit {EMA}}_{\alpha ,t-1},\quad t>1$

where $\alpha$ is the smoothing factor.

Linear regression moving average

A Linear Regression Moving Average (LRMA) fits a straight line to a moving window of data points with width $k$.

The LRMA at element $t$ over the window of $k$ elements is calculated from:

${\mathit {LRMA}}_{k,t}=a+b\cdot t$

where $a$ (intercept) and $b$ (slope) are determined by minimising the sum of squared residuals of the data points $(t-k+1,P_{t-k+1}),(t-k+2,P_{t-k+2}),\dots ,(t,P_{t})$.

The coefficients $a$ and $b$ can be directly calculated from:

${\begin{aligned}a&={\dfrac {\sum \limits _{i=1}^{k}p_{t-k+i}-b\sum \limits _{i=1}^{k}i}{k}}\\b&={\dfrac {k\sum \limits _{i=1}^{k}ip_{t-k+i}-\sum \limits _{i=1}^{k}i\sum \limits _{i=1}^{k}p_{t-k+i}}{k\sum \limits _{i=1}^{k}i^{2}-\left(\sum \limits _{i=1}^{k}i\right)^{2}}}\end{aligned}}$

for $t>1$ and $k>2$.

The Linear Regression Moving Average method may alternatively be referred to as the Least Squares Moving Average.

Excel

Figure 1. Example showing the selection of the TimeSeriesData (blue frame), k (red frame) and Results (light blue frame) arrays in Excel, for the Simple Moving Average (SMA) case.

The Simple Moving Average (SMA) filter may be invoked from the Excel formula bar with the following function calls:

=mdTimeSeries_SMA(TimeSeriesData as Range, k as Integer)

Invoking the function with no arguments will print Help text associated with the model, including a link to this page.

The input parameters and calculation results are defined below in matrix notation, along with an example image showing the selection of the same cells and arrays in the Excel interface:

${\begin{aligned}{\mathit {TimeSeriesData}}&={\begin{bmatrix}p_{1}\\\vdots \\p_{n}\\\end{bmatrix}}\\\\{\mathit {k}}&={\big [}k{\big ]}\\\end{aligned}}$ ${\begin{aligned}{\mathit {mdTimeSeries\_SMA}}&={\begin{bmatrix}{\mathit {SMA}}_{k,1}\\\vdots \\{\mathit {SMA}}_{k,n}\\\end{bmatrix}}\end{aligned}}$ where $n$ is the number of data points.

The Exponential Moving Average (EMA) filter may be invoked from the Excel formula bar with the following function calls:

=mdTimeSeries_EMA(TimeSeriesData as Range, alpha as Double)

Invoking the function with no arguments will print Help text associated with the model, including a link to this page.

The input parameters and calculation results are defined below in matrix notation, along with an example image showing the selection of the same cells and arrays in the Excel interface:

${\begin{aligned}{\mathit {TimeSeriesData}}&={\begin{bmatrix}p_{1}\\\vdots \\p_{n}\\\end{bmatrix}}\\\\{\mathit {alpha}}&={\big [}\alpha {\big ]}\\\end{aligned}}$ ${\begin{aligned}{\mathit {mdTimeSeries\_EMA}}&={\begin{bmatrix}{\mathit {EMA}}_{\alpha ,1}\\\vdots \\{\mathit {EMA}}_{\alpha ,n}\\\end{bmatrix}}\end{aligned}}$

The Linear Regression Moving Average (LRMA) filter may be invoked from the Excel formula bar with the following function calls:

=mdTimeSeries_LRMA(TimeSeriesData as Range, k as Integer)

Invoking the function with no arguments will print Help text associated with the model, including a link to this page.

The input parameters and calculation results are defined below in matrix notation, along with an example image showing the selection of the same cells and arrays in the Excel interface:

${\begin{aligned}{\mathit {TimeSeriesData}}&={\begin{bmatrix}p_{1}\\\vdots \\p_{n}\\\end{bmatrix}}\\\\{\mathit {k}}&={\big [}k{\big ]}\\\end{aligned}}$ ${\begin{aligned}{\mathit {mdTimeSeries\_LRMA}}&={\begin{bmatrix}{\mathit {LRMA}}_{k,1}\\\vdots \\{\mathit {LRMA}}_{k,n}\\\end{bmatrix}}\end{aligned}}$