Mean squared prediction error
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In statistics the mean squared prediction error of a smoothing procedure is the expected sum of squared deviations of the fitted values from the (unobservable) function . If the smoothing procedure has operator matrix , then
![{\displaystyle \operatorname {MSPE} (L)=\operatorname {E} \left[\sum _{i=1}^{n}\left(g(x_{i})-{\widehat {g}}(x_{i})\right)^{2}\right].}](/media/api/rest_v1/media/math/render/svg/81f3d577334ee8ee82142faace724b6b40eb5080)
The MSPE can be decomposed into two terms just like mean squared error is decomposed into bias and variance; however for MSPE one term is the sum of squared biases of the fitted values and another the sum of variances of the fitted values:
![{\displaystyle \operatorname {MSPE} (L)=\sum _{i=1}^{n}\left(\operatorname {E} \left[{\widehat {g}}(x_{i})\right]-g(x_{i})\right)^{2}+\sum _{i=1}^{n}\operatorname {var} \left[{\widehat {g}}(x_{i})\right].}](/media/api/rest_v1/media/math/render/svg/c7f41e95bfaa10c0513d2fc5d52685bde5b98a19)
Note that knowledge of is required in order to calculate MSPE exactly.
Estimation of MSPE
For the model where , one may write
![{\displaystyle \operatorname {MSPE} (L)=g'(I-L)'(I-L)g+\sigma ^{2}\operatorname {tr} \left[L'L\right].}](/media/api/rest_v1/media/math/render/svg/32dc67b998a41e0b2bee2c1c1a75607401467376)
The first term is equivalent to
![{\displaystyle \sum _{i=1}^{n}\left(\operatorname {E} \left[{\widehat {g}}(x_{i})\right]-g(x_{i})\right)^{2}=\operatorname {E} \left[\sum _{i=1}^{n}\left(y_{i}-{\widehat {g}}(x_{i})\right)^{2}\right]-\sigma ^{2}\operatorname {tr} \left[\left(I-L\right)'\left(I-L\right)\right].}](/media/api/rest_v1/media/math/render/svg/24802de08adcd428fc7c9f46883cb346669396a9)
Thus,
![{\displaystyle \operatorname {MSPE} (L)=\operatorname {E} \left[\sum _{i=1}^{n}\left(y_{i}-{\widehat {g}}(x_{i})\right)^{2}\right]-\sigma ^{2}\left(n-2\operatorname {tr} \left[L\right]\right).}](/media/api/rest_v1/media/math/render/svg/1ccf6e308053c671db94e0bdce808edb20237dd7)
If is known or well-estimated by , it becomes possible to estimate MSPE by
![{\displaystyle \operatorname {\widehat {MSPE}} (L)=\sum _{i=1}^{n}\left(y_{i}-{\widehat {g}}(x_{i})\right)^{2}-{\widehat {\sigma }}^{2}\left(n-2\operatorname {tr} \left[L\right]\right).}](/media/api/rest_v1/media/math/render/svg/5b23a1ca734192e68335b7d98d935829906b0419)
Colin Mallows advocated this method in the construction of his model selection statistic Cp, which is a normalized version of the estimated MSPE:
![{\displaystyle C_{p}={\frac {\sum _{i=1}^{n}\left(y_{i}-{\widehat {g}}(x_{i})\right)^{2}}{{\widehat {\sigma }}^{2}}}-n+2\operatorname {tr} \left[L\right].}](/media/api/rest_v1/media/math/render/svg/71c641c99e2083d9692f4bc3f099e9bb8b7ae8f5)
where comes from that fact that the number of parameters estimated for a parametric smoother is given by , and is in honor of Cuthbert Daniel.
![{\displaystyle p=\operatorname {tr} \left[L\right]}](/media/api/rest_v1/media/math/render/svg/0f01609a5f0eac27c6f6225a5559b73195a47e12)
