Time Series Analysis (CUSUM)

Description

This article describes a method for preparing a Cumulative Sum (CUSUM) control chart.

Model theory

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Excel

The CUSUM control chart may be invoked from the Excel formula bar with the following function calls:

=mdTimeSeries_CUSUM(TimeSeriesData as Range, T as Double, Optional k as Double = 0)

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:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \mathit{TimeSeriesData} & = \begin{bmatrix} y_1\\ \vdots\\ y_t\\ \end{bmatrix}\\ \\ \mathit{T} & = \big [ T \big ]\\ \\ \mathit{k} & = \big [ k \big ]\\ \end{align}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \mathit{mdTimeSeries\_CUSUM} & = \begin{bmatrix} \begin{bmatrix} C_{1}\\ \vdots\\ C_{t}\\ \end{bmatrix} & \begin{bmatrix} S_{\rm H,1}\\ \vdots\\ S_{\rm H,t}\\ \end{bmatrix}^* & \begin{bmatrix} S_{\rm L,1}\\ \vdots\\ S_{\rm L,t}\\ \end{bmatrix}^* \end{bmatrix} \end{align} } where: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y_t} is the original observation (data point) at time Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T} is a constant, typically a target value or the mean of historical data Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k} is the slope of the V-Mask arm, an optional parameter (V-Mask limits Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S_{\rm H}} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S_{\rm L}} are not returned if Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k} is omitted) Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C_t} is the value of the cumulative sum (CUSUM) at time Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S_{\rm H}} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S_{\rm L}} are the high and low V-Mask parameters, respectively, and the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ^*} superscripts indicate they are optionally-returned

Figure 1. Example showing the selection of the TimeSeriesData (blue frame), T (red frame), k (purple frame), and Results (light blue frame) arrays in Excel. The Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S_{\rm L}} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S_{\rm L}} parameters are returned in this example as k is specified.