StatTools : Exponentially Weighted Moving Average Explained

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Introduction Example References
Excellent descriptions of the history and basic concepts related to Exponentially Weighted Moving Average (EWMA) are available in text books, journal articles, and numerous web pages on the Internet, some of which are listed in the references section.

This page will therefore provide only a brief summary and description, suffice to quickly orientate the user, and in support of the calculations in the Exponentially Weighted Moving Average Program Page .

Only the simplest model, as described by the NIST manual (see references) is provided. A paper by Lucas and the Seccicci described further improvements, but they are not available in by StatTools.

Conceptually, EWMA is the same as averaging a number of consecutive observations, and by so doing it smooths out short term variations. This prevents outlying values from distorting the long term trend of a set of observations.

The advantages of EWMA over averaging a set number of observations are firstly the values are weighted, so that the more recent observations have greater influence on the averaging, and secondly, there is no need to maintain a database of all the numbers and no need to define the number of observations needed for the averaging.

The Exponentially Weighted Average and coefficient λ

A weighting coefficient lambda (λ) is used to assign the weight to the current observation (Yt), in relationship to the immediately previous EWMA value (EWMAt-1), and combine them to produce the current EWMA value (EWMAt), so that

    EWMAt = λYt + (1 - λ)EWMAt-1

λ is a value between 0 and 1, and governs the amount of weight the current observation has in relationship to all previous observations. A λ close to 1 means a strong dominance by the current observation, with very little averaging. A λ close to 0 means a greater level of averaging, produces a smoother trend line, but the trend is less responsive to immediate changes in the observations.

The Reference Mean and Standard Deviation These are provided by the user, either as the standards by which the new data are to be plotted against, or values obtained by previous studies.

The Standard Error of EWMA With weighted average, outlying values are brought closer to the mean, so that the original Standard Deviation is reduced. If SD is the original Standard Deviation, and λ is the weighting coefficient, the Standard Error of the resulting EWMA is SEEWMA = sqrt((λ/(2.0 - λ))SD2)

With smoothing by weighted average, and a reduced SEEWMA, a trend in the measurements that departs from the mean can be more easily identified. In quality control, an EWMA that is 3SE from the mean triggers an alert that values are drifting away from the mean

The algorithm used in the Exponentially Weighted Moving Average Program Page Plots the original values as dots, and the EWMA as lines. It also draws horizontal lines at 1 SE intervals from the mean so that departure from the mean can be identified.

Options for users : There are 4 buttons for the user to choose, representing options in two dimensions

  • Scaling : User may scale the vertical axis (Y) according to values or standard errors. Scaling by values will have greater meaning to the clinical users, but standard errors will be more familiar to statisticians. Standard error scaling will also be more useful when comparing different EWMAs, as standard error is measurement free so have the same scaling across all measurements.
  • Full output : This translates and displays each data point into its standard error values so that the user can examine in detail what the data is all about. This however becomes cumbersome if the sample size is large, as a huge table is produced. This therefore should be avoided unless the user has a need to examine each data point.