Total variation denoising

In statistics, the mean signed difference (MSD), also known as mean signed error (MSE), is a sample statistic that summarises how well an estimator ${\hat {\theta }}$ matches the quantity $\theta$ that it is supposed to estimate. It is one of a number of statistics that can be used to assess an estimation procedure, and it would often be used in conjunction with a sample version of the mean square error.

Definition

The mean signed difference is derived from a set of n pairs, $({\hat {\theta }}_{i},\theta _{i})$, where ${\hat {\theta }}_{i}$ is an estimate of the parameter $\theta$ in a case where it is known that $\theta =\theta _{i}$. In many applications, all the quantities $\theta _{i}$ will share a common value. When applied to forecasting in a time series analysis context, a forecasting procedure might be evaluated using the mean signed difference, with ${\hat {\theta }}_{i}$ being the predicted value of a series at a given lead time and $\theta _{i}$ being the value of the series eventually observed for that time-point. The mean signed difference is defined to be

$\operatorname {MSD} ({\hat {\theta }})=\sum _{i=1}^{n}{\frac {{\hat {\theta _{i}}}-\theta _{i}}{n}}.$

See also

• Mean absolute error

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