Instatistics themean squared prediction error (MSPE), also known asmean squared error of the predictions, of asmoothing,curve fitting, orregression procedure is theexpected value of thesquaredprediction errors (PE), thesquare difference between the fitted values implied by the predictive function${\displaystyle \hat{g}}$ and the values of the (unobservable)true valueg. It is an inverse measure of theexplanatory power of${\displaystyle \hat{g},}$ and can be used in the process ofcross-validation of an estimated model.Knowledge ofg would be required in order to calculate the MSPE exactly; in practice, MSPE is estimated.^[1]

If the smoothing or fitting procedure hasprojection matrix (i.e., hat matrix)L, which maps the observed values vector${\displaystyle y}$ topredicted values vector${\displaystyle \hat{y}=Ly,}$ then PE and MSPE are formulated as:

${\displaystyle \mathrm{P}{\mathrm{E}}_{\mathrm{i}}=g(x_i)-\hat{g}(x_i),}$

${\displaystyle \mathrm{MSPE}=\mathrm{E}\left[{\mathrm{PE}}_i^2\right]=\sum _{i=1}^n{\mathrm{PE}}_i^2/n.}$

The MSPE can be decomposed into two terms: the squaredbias (mean error) of the fitted values and thevariance of the fitted values:

${\displaystyle \mathrm{MSPE}={\mathrm{ME}}^2+\mathrm{VAR},}$

${\displaystyle \mathrm{ME}=\mathrm{E}\left[\hat{g}(x_i)-g(x_i)\right]}$

${\displaystyle \mathrm{VAR}=\mathrm{E}\left[{\left(\hat{g}(x_i)-\mathrm{E}\left[g(x_i)\right]\right)}^2\right].}$

The quantitySSPE=nMSPE is calledsum squared prediction error.Theroot mean squared prediction error is the square root of MSPE:RMSPE=√MSPE.

The mean squared prediction error can be computed exactly in two contexts. First, with adata sample of lengthn, thedata analyst may run theregression over onlyq of the data points (withq <n), holding back the othern – q data points with the specific purpose of using them to compute the estimated model’s MSPE out of sample (i.e., not using data that were used in the model estimation process). Since the regression process is tailored to theq in-sample points, normally the in-sample MSPE will be smaller than the out-of-sample one computed over then – q held-back points. If the increase in the MSPE out of sample compared to in sample is relatively slight, that results in the model being viewed favorably. And if two models are to be compared, the one with the lower MSPE over then – q out-of-sample data points is viewed more favorably, regardless of the models’ relative in-sample performances. The out-of-sample MSPE in this context is exact for the out-of-sample data points that it was computed over, but is merely an estimate of the model’s MSPE for the mostly unobserved population from which the data were drawn.

Second, as time goes on more data may become available to the data analyst, and then the MSPE can be computed over these new data.

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When the model has been estimated over all available data with none held back, the MSPE of the model over the entirepopulation of mostly unobserved data can be estimated as follows.

For the model${\displaystyle y_i=g(x_i)+\sigma {\epsilon }_i}$ where${\displaystyle {\epsilon }_i\sim \mathcal{N}(0,1)}$, one may write

${\displaystyle n\cdot \mathrm{MSPE}(L)=g^{\text{T}}(I-L{)}^{\text{T}}(I-L)g+{\sigma }^2\mathrm{tr}\left[L^{\text{T}}L\right].}$

Using in-sample data values, the first term on the right side is equivalent to

${\displaystyle \sum _{i=1}^n{\left(\mathrm{E}\left[g(x_i)-\hat{g}(x_i)\right]\right)}^2=\mathrm{E}\left[\sum _{i=1}^n{\left(y_i-\hat{g}(x_i)\right)}^2\right]-{\sigma }^2\mathrm{tr}\left[{\left(I-L\right)}^T\left(I-L\right)\right].}$

Thus,

${\displaystyle n\cdot \mathrm{MSPE}(L)=\mathrm{E}\left[\sum _{i=1}^n{\left(y_i-\hat{g}(x_i)\right)}^2\right]-{\sigma }^2\left(n-\mathrm{tr}\left[L\right]\right).}$

If${\displaystyle {\sigma }^2}$ is known or well-estimated by${\displaystyle {\hat{\sigma }}^2}$, it becomes possible to estimate MSPE by

${\displaystyle n\cdot \widehat{\mathrm{M}\mathrm{S}\mathrm{P}\mathrm{E}}(L)=\sum _{i=1}^n{\left(y_i-\hat{g}(x_i)\right)}^2-{\hat{\sigma }}^2\left(n-\mathrm{tr}\left[L\right]\right).}$

Colin Mallows advocated this method in the construction of his model selection statisticC_p, which is a normalized version of the estimated MSPE:

${\displaystyle C_p=\frac{\sum _{i=1}^n{\left(y_i-\hat{g}(x_i)\right)}^2}{{\hat{\sigma }}^2}-n+2p.}$

wherep the number of estimated parametersp and${\displaystyle {\hat{\sigma }}^2}$ is computed from the version of the model that includes all possible regressors.That concludes this proof.

• Akaike information criterion
• Bias-variance tradeoff
• Mean squared error
• Errors and residuals in statistics
• Law of total variance
• Mallows'sC_p
• Model selection

• Point estimation performance
• Statistical deviation and dispersion
• Loss functions

• Articles with short description
• Short description matches Wikidata
• Accuracy disputes from May 2018
• All accuracy disputes