Editing Mean squared error (section)

===Predictor===

If a vector of <math>n</math> predictions is generated from a sample of <math>n</math> data points on all variables, and <math>Y</math> is the vector of observed values of the variable being predicted, with <math>\hat{Y}</math> being the predicted values (e.g. as from a [[least-squares fit]]), then the within-sample MSE of the predictor is computed as

:<math>\operatorname{MSE}=\frac{1}{n} \sum_{i=1}^n \left(Y_i-\hat{Y_i}\right)^2</math>

In other words, the MSE is the ''mean'' <math display="inline">\left(\frac{1}{n} \sum_{i=1}^n \right)</math> of the ''squares of the errors'' <math display="inline">\left(Y_i-\hat{Y_i}\right)^2</math>. This is an easily computable quantity for a particular sample (and hence is sample-dependent).

In [[Matrix_multiplication|matrix]] notation, 
:<math>\operatorname{MSE}=\frac{1}{n}\sum_{i=1}^n(e_i)^2=\frac{1}{n}\mathbf e^\mathsf T \mathbf e</math>
where <math>e_i</math> is <math> (Y_i-\hat{Y_i}) </math> and <math>\mathbf e</math> is a <math> n \times 1 </math> column vector.

The MSE can also be computed on ''q ''data points that were not used in estimating the model, either because they were held back for this purpose, or because these data have been newly obtained. Within this process, known as [[Cross-validation (statistics)|cross-validation]], the MSE is often called the [[test MSE]],<ref>{{cite book
|first1=James
|last1=Gareth
|first2=Daniela
|last2=Witten
|first3=Trevor
|last3=Hastie
|first4=Rob
|last4=Tibshirani
|date=2021
|title=An Introduction to Statistical Learning: with Applications in R
|url=https://www.statlearning.com/
|publisher=Springer
|isbn=978-1071614174
}}</ref> and is computed as

:<math>\operatorname{MSE} = \frac{1}{q} \sum_{i=n+1}^{n+q} \left(Y_i-\hat{Y_i}\right)^2</math>