Instatistics, themean squared error (MSE)^[1] ormean squared deviation (MSD) of anestimator (of a procedure for estimating an unobserved quantity) measures theaverage of the squares of theerrors—that is, the average squared difference between the estimated values and thetrue value. MSE is arisk function, corresponding to theexpected value of thesquared error loss.^[2] The fact that MSE is almost always strictly positive (and not zero) is because ofrandomness or because the estimatordoes not account for information that could produce a more accurate estimate.^[3] Inmachine learning, specificallyempirical risk minimization, MSE may refer to theempirical risk (the average loss on an observed data set), as an estimate of the true MSE (the true risk: the average loss on the actual population distribution).

The MSE is a measure of the quality of an estimator. As it is derived from the square ofEuclidean distance, it is always a positive value that decreases as the error approaches zero.

The MSE is the secondmoment (about the origin) of the error, and thus incorporates both thevariance of the estimator (how widely spread the estimates are from onedata sample to another) and itsbias (how far off the average estimated value is from the true value).^{[citation needed]} For anunbiased estimator, the MSE is the variance of the estimator. Like the variance, MSE has the same units of measurement as the square of the quantity being estimated. In an analogy tostandard deviation, taking the square root of MSE yields theroot-mean-square error orroot-mean-square deviation (RMSE or RMSD), which has the same units as the quantity being estimated; for an unbiased estimator, the RMSE is the square root of thevariance, known as thestandard error.

The MSE either assesses the quality of apredictor (i.e., a function mapping arbitrary inputs to a sample of values of somerandom variable), or of anestimator (i.e., amathematical function mapping asample of data to an estimate of aparameter of thepopulation from which the data is sampled). In the context of prediction, understanding theprediction interval can also be useful as it provides a range within which a future observation will fall, with a certain probability. The definition of an MSE differs according to whether one is describing a predictor or an estimator.

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

$${\displaystyle \mathrm{MSE}=\frac{1}{n}\sum _{i=1}^n{\left(Y_i-\widehat{Y_i}\right)}^2}$$

In other words, the MSE is themean$\left(\frac{1}{n}\sum _{i=1}^n\right)$ of thesquares of the errors${\left(Y_i-\widehat{Y_i}\right)}^2$. This is an easily computable quantity for a particular sample (and hence is sample-dependent).

Inmatrix notation,$${\displaystyle \mathrm{MSE}=\frac{1}{n}\sum _{i=1}^n(e_i{)}^2=\frac{1}{n}{\mathbf{e}}^{\mathsf{T}}\mathbf{e}}$$where${\displaystyle e_i}$ is${\displaystyle Y_i-\widehat{Y_i}}$ and${\displaystyle \mathbf{e}}$ is a${\displaystyle n\times 1}$ column vector.

The MSE can also be computed onqdata 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 ascross-validation, the MSE is often called thetest MSE,^[4] and is computed as

$${\displaystyle \mathrm{MSE}=\frac{1}{q}\sum _{i=n+1}^{n+q}{\left(Y_i-\widehat{Y_i}\right)}^2}$$

The MSE of an estimator${\displaystyle \hat{\theta }}$ with respect to an unknown parameter${\displaystyle \theta }$ is defined as^[1]

$${\displaystyle \mathrm{MSE}(\hat{\theta })={\mathrm{E}}_{\theta }\left[(\hat{\theta }-\theta {)}^2\right].}$$

This definition depends on the unknown parameter, therefore the MSE is apriori property of an estimator. The MSE could be a function of unknown parameters, in which case anyestimator of the MSE based on estimates of these parameters would be a function of the data (and thus a random variable). If the estimator${\displaystyle \hat{\theta }}$ is derived as a sample statistic and is used to estimate some population parameter, then the expectation is with respect to thesampling distribution of the sample statistic.

The MSE can be written as the sum of thevariance of the estimator and the squaredbias of the estimator, providing a useful way to calculate the MSE and implying that in the case of unbiased estimators, the MSE and variance are equivalent.^[5]

$${\displaystyle \mathrm{MSE}(\hat{\theta })={\mathrm{Var}}_{\theta }(\hat{\theta })+\mathrm{Bias}(\hat{\theta },\theta {)}^2.}$$

An even shorter proof can be achieved using the well-known formula that for a random variable$X$,$\mathbb{E}(X^2)=\mathrm{Var}(X)+(\mathbb{E}(X){)}^2$.^{[citation needed]} By substituting$X$ with,$\hat{\theta }-\theta$, we haveBut in real modeling case, MSE could be described as the addition of model variance, model bias, and irreducible uncertainty (seeBias–variance tradeoff). According to the relationship, the MSE of the estimators could be simply used for theefficiency comparison, which includes the information of estimator variance and bias. This is called MSE criterion.

Inregression analysis, plotting is a more natural way to view the overall trend of the whole data. The mean of the distance from each point to the predicted regression model can be calculated, and shown as the mean squared error. The squaring is critical to reduce the complexity with negative signs. To minimize MSE, the model could be more accurate, which would mean the model is closer to actual data. One example of a linear regression using this method is theleast squares method—which evaluates appropriateness of linear regression model to modelbivariate dataset,^[6] but whose limitation is related to known distribution of the data.

The termmean squared error is sometimes used to refer to the unbiased estimate of error variance: theresidual sum of squares divided by the number ofdegrees of freedom. This definition for a known, computed quantity differs from the above definition for the computed MSE of a predictor, in that a different denominator is used. The denominator is the sample size reduced by the number of model parameters estimated from the same data, (n−p) forpregressors or (n−p−1) if an intercept is used (seeerrors and residuals in statistics for more details).^[7] Although the MSE (as defined in this article) is not an unbiased estimator of the error variance, it isconsistent, given the consistency of the predictor.

In regression analysis, "mean squared error", often referred to asmean squared prediction error or "out-of-sample mean squared error", can also refer to the mean value of thesquared deviations of the predictions from the true values, over an out-of-sampletest space, generated by a model estimated over aparticular sample space. This also is a known, computed quantity, and it varies by sample and by out-of-sample test space.

In the context ofgradient descent algorithms, it is common to introduce a factor of${\displaystyle 1/2}$ to the MSE for ease of computation after taking the derivative. So a value which is technically half the mean of squared errors may be called the MSE.

Suppose we have a random sample of size${\displaystyle n}$ from a population,${\displaystyle X_1,\dots ,X_n}$. Suppose the sample units were chosenwith replacement. That is, the${\displaystyle n}$ units are selected one at a time, and previously selected units are still eligible for selection for all${\displaystyle n}$ draws. The usual estimator for the population mean${\displaystyle \mu }$ is the sample average

$${\displaystyle \overline{X}=\frac{1}{n}\sum _{i=1}^n{X}_i}$$

which has an expected value equal to the true mean${\displaystyle \mu }$ (so it is unbiased) and a mean squared error of

$${\displaystyle \mathrm{MSE}\left(\overline{X}\right)=\mathrm{E}\left[{\left(\overline{X}-\mu \right)}^2\right]={\left(\frac{\sigma }{\sqrt{n}}\right)}^2=\frac{{\sigma }^2}{n}}$$

where${\displaystyle {\sigma }^2}$ is thepopulation variance.

For aGaussian distribution this is thebest unbiased estimator of the population mean, that is the one with the lowest MSE (and hence variance) among all unbiased estimators. One can check that the MSE above equals the inverse of theFisher information (seeCramér–Rao bound). But the same sample mean is not the best estimator of the population mean, say, for auniform distribution.

The usual estimator for the variance is thecorrectedsample variance:

$${\displaystyle S_{n-1}^2=\frac{1}{n-1}\sum _{i=1}^n{\left(X_i-\overline{X}\right)}^2=\frac{1}{n-1}\left(\sum _{i=1}^n{X}_i^2-n{\overline{X}}^2\right).}$$

This is unbiased (its expected value is${\displaystyle {\sigma }^2}$), hence also called theunbiased sample variance, and its MSE is^[8]

$${\displaystyle \mathrm{MSE}(S_{n-1}^2)=\frac{1}{n}\left({\mu }_4-\frac{n-3}{n-1}{\sigma }^4\right)=\frac{1}{n}\left({\gamma }_2+\frac{2n}{n-1}\right){\sigma }^4,}$$

where${\displaystyle {\mu }_4}$ is the fourthcentral moment of the distribution or population, and${\displaystyle {\gamma }_2={\mu }_4/{\sigma }^4-3}$ is theexcess kurtosis.

However, one can use other estimators for${\displaystyle {\sigma }^2}$ which are proportional to${\displaystyle S_{n-1}^2}$, and an appropriate choice can always give a lower mean squared error. If we define

$${\displaystyle S_a^2=\frac{n-1}{a}{S}_{n-1}^2=\frac{1}{a}\sum _{i=1}^n{\left(X_i-\overline{X}\right)}^2}$$

then we calculate:

This is minimized when

$${\displaystyle a=\frac{(n-1){\gamma }_2+n^2+n}{n}=n+1+\frac{n-1}{n}{\gamma }_2.}$$

For aGaussian distribution, where${\displaystyle {\gamma }_2=0}$, this means that the MSE is minimized when dividing the sum by${\displaystyle a=n+1}$. The minimum excess kurtosis is${\displaystyle {\gamma }_2=-2}$,^[a] which is achieved by aBernoulli distribution withp = 1/2 (a coin flip), and the MSE is minimized for${\displaystyle a=n-1+\frac{2}{n}.}$ Hence regardless of the kurtosis, we get a "better" estimate (in the sense of having a lower MSE) by scaling down the unbiased estimator a little bit; this is a simple example of ashrinkage estimator: one "shrinks" the estimator towards zero (scales down the unbiased estimator).

Further, while the corrected sample variance is thebest unbiased estimator (minimum mean squared error among unbiased estimators) of variance for Gaussian distributions, if the distribution is not Gaussian, then even among unbiased estimators, the best unbiased estimator of the variance may not be${\displaystyle S_{n-1}^2.}$

The following table gives several estimators of the true parameters of the population, μ and σ², for the Gaussian case.^[9]

True value	Estimator	Mean squared error
${\displaystyle \theta =\mu }$	${\displaystyle \hat{\theta }}$ = the unbiased estimator of thepopulation mean,${\displaystyle \overline{X}=\frac{1}{n}\sum _{i=1}^n(X_i)}$	${\displaystyle \mathrm{MSE}(\overline{X})=\mathrm{E}[(\overline{X}-\mu {)}^2]=\frac{{\sigma }^2}{n}}$
${\displaystyle \theta ={\sigma }^2}$	${\displaystyle \hat{\theta }}$ = the unbiased estimator of thepopulation variance,${\displaystyle S_{n-1}^2=\frac{1}{n-1}\sum _{i=1}^n{\left(X_i-\overline{X}\right)}^2}$	${\displaystyle \mathrm{MSE}(S_{n-1}^2)=\mathrm{E}[(S_{n-1}^2-{\sigma }^2{)}^2]=\frac{2}{n-1}{\sigma }^4}$
${\displaystyle \theta ={\sigma }^2}$	${\displaystyle \hat{\theta }}$ = the biased estimator of thepopulation variance,${\displaystyle S_n^2=\frac{1}{n}\sum _{i=1}^n{\left(X_i-\overline{X}\right)}^2}$	${\displaystyle \mathrm{MSE}(S_n^2)=\mathrm{E}[(S_n^2-{\sigma }^2{)}^2]=\frac{2n-1}{n^2}{\sigma }^4}$
${\displaystyle \theta ={\sigma }^2}$	${\displaystyle \hat{\theta }}$ = the biased estimator of thepopulation variance,${\displaystyle S_{n+1}^2=\frac{1}{n+1}\sum _{i=1}^n{\left(X_i-\overline{X}\right)}^2}$	${\displaystyle \mathrm{MSE}(S_{n+1}^2)=\mathrm{E}([S_{n+1}^2-{\sigma }^2{)}^2]=\frac{2}{n+1}{\sigma }^4}$

An MSE of zero, meaning that the estimator${\displaystyle \hat{\theta }}$ predicts observations of the parameter${\displaystyle \theta }$ with perfect accuracy, is ideal (but typically not possible).

Values of MSE may be used for comparative purposes. Two or morestatistical models may be compared using their MSEs—as a measure of how well they explain a given set of observations: An unbiased estimator (estimated from a statistical model) with the smallest variance among all unbiased estimators is thebest unbiased estimator or MVUE (Minimum-Variance Unbiased Estimator).

Bothanalysis of variance andlinear regression techniques estimate the MSE as part of the analysis and use the estimated MSE to determine thestatistical significance of the factors or predictors under study. The goal ofexperimental design is to construct experiments in such a way that when the observations are analyzed, the MSE is close to zero relative to the magnitude of at least one of the estimated treatment effects.

Inone-way analysis of variance, MSE can be calculated by the division of the sum of squared errors and the degree of freedom. Also, the f-value is the ratio of the mean squared treatment and the MSE.

MSE is also used in severalstepwise regression techniques as part of the determination as to how many predictors from a candidate set to include in a model for a given set of observations.

Minimizing MSE is a key criterion in selecting estimators; seeminimum mean-square error. Among unbiased estimators, minimizing the MSE is equivalent to minimizing the variance, and the estimator that does this is theminimum variance unbiased estimator. However, a biased estimator may have lower MSE; seeestimator bias.

Instatistical modelling the MSE can represent the difference between the actual observations and the observation values predicted by the model. In this context, it is used to determine the extent to which the model fits the data as well as whether removing some explanatory variables is possible without significantly harming the model's predictive ability.

Inforecasting andprediction, theBrier score is a measure offorecast skill based on MSE.

Squared error loss is one of the most widely usedloss functions in statistics, though its widespread use stems more from mathematical convenience than considerations of actual loss in applications.Carl Friedrich Gauss, who introduced the use of mean squared error, was aware of its arbitrariness and was in agreement with objections to it on these grounds.^[3] The mathematical benefits of mean squared error are particularly evident in its use at analyzing the performance oflinear regression, as it allows one to partition the variation in a dataset into variation explained by the model and variation explained by randomness.

The use of mean squared error without question has been criticized by thedecision theoristJames Berger. Mean squared error is the negative of the expected value of one specificutility function, the quadratic utility function, which may not be the appropriate utility function to use under a given set of circumstances. There are, however, some scenarios where mean squared error can serve as a good approximation to a loss function occurring naturally in an application.^[10]

Likevariance, mean squared error has the disadvantage of heavily weightingoutliers.^[11] This is a result of the squaring of each term, which effectively weights large errors more heavily than small ones. This property, undesirable in many applications, has led researchers to use alternatives such as themean absolute error, or those based on themedian.

• Bias–variance tradeoff
• Hodges' estimator
• James–Stein estimator
• Mean percentage error
• Mean square quantization error
• Reduced chi-squared statistic
• Mean squared displacement
• Mean squared prediction error
• Minimum mean square error
• Overfitting
• Peak signal-to-noise ratio

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

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