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Mean squared prediction error
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In statistics the mean squared prediction error (MSPE), also known as mean squared error of the predictions, of a smoothing, curve fitting, or regression procedure is the expected value of the squared prediction errors (PE), the square difference between the fitted values implied by the predictive function and the values of the (unobservable) true value g. It is an inverse measure of the explanatory power of and can be used in the process of cross-validation of an estimated model. Knowledge of g would be required in order to calculate the MSPE exactly; in practice, MSPE is estimated.[1]
Formulation
[edit]If the smoothing or fitting procedure has projection matrix (i.e., hat matrix) L, which maps the observed values vector to predicted values vector then PE and MSPE are formulated as:
![{\displaystyle \operatorname {MSPE} =\operatorname {E} \left[\operatorname {PE} _{i}^{2}\right]=\sum _{i=1}^{n}\operatorname {PE} _{i}^{2}/n.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e4a5bceae972e0557d12efefe37c384d8b22e190)
The MSPE can be decomposed into two terms: the squared bias (mean error) of the fitted values and the variance of the fitted values:
![{\displaystyle \operatorname {ME} =\operatorname {E} \left[{\widehat {g}}(x_{i})-g(x_{i})\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/54834f1e0d444f8bbda18b726e98dd157a8803ab)
![{\displaystyle \operatorname {VAR} =\operatorname {E} \left[\left({\widehat {g}}(x_{i})-\operatorname {E} \left[{g}(x_{i})\right]\right)^{2}\right].}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e41dc0a2d2ae468dcd6e5fbc6e9bfce56a5abf4e)
The quantity SSPE=nMSPE is called sum squared prediction error. The root mean squared prediction error is the square root of MSPE: RMSPE=√MSPE.
Computation of MSPE over out-of-sample data
[edit]The mean squared prediction error can be computed exactly in two contexts. First, with a data sample of length n, the data analyst may run the regression over only q of the data points (with q < n), holding back the other n – 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 the q in-sample points, normally the in-sample MSPE will be smaller than the out-of-sample one computed over the n – 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 the n – 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.
Estimation of MSPE over the population
[edit] | This article's factual accuracy is disputed. (May 2018) |
When the model has been estimated over all available data with none held back, the MSPE of the model over the entire population of mostly unobserved data can be estimated as follows.
For the model where , one may write
![{\displaystyle n\cdot \operatorname {MSPE} (L)=g^{\text{T}}(I-L)^{\text{T}}(I-L)g+\sigma ^{2}\operatorname {tr} \left[L^{\text{T}}L\right].}](https://wikimedia.org/api/rest_v1/media/math/render/svg/afa4a2de4f014dc1c2cf31bc803f5d642c68a26f)
Using in-sample data values, the first term on the right side is equivalent to
![{\displaystyle \sum _{i=1}^{n}\left(\operatorname {E} \left[g(x_{i})-{\widehat {g}}(x_{i})\right]\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)^{T}\left(I-L\right)\right].}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e9f60fa601829826d71a7d40c74ee73564dfab9c)
Thus,
![{\displaystyle n\cdot \operatorname {MSPE} (L)=\operatorname {E} \left[\sum _{i=1}^{n}\left(y_{i}-{\widehat {g}}(x_{i})\right)^{2}\right]-\sigma ^{2}\left(n-\operatorname {tr} \left[L\right]\right).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b6da6d150e8f3f19e291b2e7fd536f366690eff3)
If is known or well-estimated by , it becomes possible to estimate MSPE by
![{\displaystyle n\cdot \operatorname {\widehat {MSPE}} (L)=\sum _{i=1}^{n}\left(y_{i}-{\widehat {g}}(x_{i})\right)^{2}-{\widehat {\sigma }}^{2}\left(n-\operatorname {tr} \left[L\right]\right).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d4890e1a2ff444ff87af3a9a80dfd00ee733f1bf)
Colin Mallows advocated this method in the construction of his model selection statistic Cp, which is a normalized version of the estimated MSPE:
where p the number of estimated parameters p and is computed from the version of the model that includes all possible regressors. That concludes this proof.
See also
[edit]References
[edit]- ^ Pindyck, Robert S.; Rubinfeld, Daniel L. (1991). "Forecasting with Time-Series Models". Econometric Models & Economic Forecasts (3rd ed.). New York: McGraw-Hill. pp. 516–535. ISBN 0-07-050098-3.
Mean squared prediction error
View on Grokipediafrom Grokipedia
The mean squared prediction error (MSPE) is a fundamental statistical metric used to evaluate the predictive accuracy of a model by quantifying the expected squared difference between actual outcomes and model predictions for new, unseen data.[1] Formally, it is defined as , where represents the true value and the predicted value, with the expectation taken over the joint distribution of inputs and outputs.[2] This measure emphasizes out-of-sample performance, distinguishing it from in-sample error estimates that may overestimate accuracy due to overfitting.[1]
In practice, MSPE is estimated using a test dataset by averaging the squared residuals: , providing a direct assessment of how closely predictions align with reality on average.[1] It serves as a cornerstone for model selection and validation in fields such as regression analysis, forecasting, and machine learning, where minimizing MSPE guides choices between simpler and more complex models.[3] A related metric, the root mean squared prediction error (RMSPE), takes the square root of MSPE to express error in the original units of the target variable, aiding interpretability.[1]
The MSPE decomposes into three components—squared bias (systematic prediction error), variance of the predictor (sensitivity to training data fluctuations), and irreducible noise—illustrating the inherent trade-off between underfitting and overfitting in predictive modeling.[4] For instance, in ordinary least squares regression, the optimal predictor under MSPE is the conditional expectation , which minimizes the risk function .[2] This decomposition underscores MSPE's role in balancing model flexibility with generalization, influencing techniques like cross-validation to reliably estimate it.[3]
This computes the sample MSPE as referenced earlier.[12]
Basic Concepts
Definition
The mean squared prediction error (MSPE) serves as a key measure of predictive accuracy in statistical modeling, representing the expected value of the squared difference between a model's predicted value and the actual outcome for a new or unseen observation. This metric quantifies how well a model generalizes beyond the data used to train it, capturing both bias and variance in predictions. In contrast to the mean squared error (MSE), which evaluates the performance of an estimator by computing the expected squared deviation from the true underlying parameter, MSPE specifically emphasizes the quality of forecasts in a prediction setting, where the focus is on future responses rather than fitted values from the training data. This distinction highlights MSPE's role in assessing out-of-sample performance, making it particularly valuable for model selection and validation in regression and forecasting tasks. For instance, consider a model predicting house prices based on variables such as square footage and neighborhood characteristics; here, MSPE measures the average squared deviation between the model's price forecasts and actual transaction prices for new listings, offering a direct gauge of the forecasts' reliability on a squared scale.Interpretation
The mean squared prediction error (MSPE) serves as a key metric for evaluating the average inaccuracy in a model's predictions, representing the expected value of the squared differences between actual and predicted outcomes. In practical terms, it captures how closely a model's forecasts align with observed data on average, with smaller MSPE values signaling superior predictive accuracy and reliability for future observations.[5] Because MSPE involves squaring the errors, it is reported in units that are the square of the target variable's units, rendering it inherently scale-dependent and challenging to interpret directly in the context of the original data scale. For instance, if the target is measured in dollars, MSPE would be in dollars squared, which may obscure intuitive understanding without additional normalization.[5] Assessing whether an MSPE value is "good" remains highly context-dependent, varying by field, data scale, and baseline expectations; there is no universal threshold, but in domains like financial forecasting, an MSPE substantially below the unconditional variance of the target variable is often deemed acceptable, with relative reductions of 10-20% compared to simple benchmarks (such as random walks) highlighting meaningful improvements.[6][5] A notable limitation of MSPE is its heightened sensitivity to outliers, as the squaring process disproportionately penalizes large errors relative to smaller ones, potentially skewing assessments in noisy datasets. Furthermore, its squared-unit nature limits direct interpretability, prompting frequent use of the root mean squared prediction error (RMSPE), the square root of MSPE, as a variant that restores the original scale for more accessible analysis.[5]Mathematical Formulation
Population MSPE
The population mean squared prediction error (MSPE) represents the theoretical average squared deviation between true outcomes and predictions across the entire population, assuming access to infinite data from a stationary joint distribution. This metric serves as an ideal benchmark for model performance, capturing the minimal achievable error under perfect estimation conditions. It is particularly useful for understanding the fundamental limits of prediction in statistical models.[7] Formally, the population MSPE is given by where denotes the true outcome variable, is the predicted value (typically for a predictor function and covariates ), and the expectation is over the population joint distribution of . This formulation assumes a fixed true underlying model, where predictions are deterministic functions of the covariates, and the population distribution remains invariant over time or draws. These assumptions ensure that the MSPE reflects intrinsic model limitations rather than sampling variability.[7] The MSPE can be intuitively decomposed as where is the irreducible error arising from stochastic noise in conditional on , quantifies the average deviation of the predictor from the true conditional expectation, and measures the predictor's variability across possible realizations (which diminishes to zero in the infinite-data population limit for consistent estimators). This breakdown highlights how prediction error stems from inherent data noise, systematic model mismatch, and predictor instability, providing an intuitive entry point to error sources without exhaustive analysis.[7] In the context of linear regression over a population, suppose the true model is , with and . Using the population least-squares coefficients (attainable with infinite data), the predictor incurs no bias or variance, yielding , the irreducible error. If the linear form is misspecified relative to the true , the MSPE includes an additional bias term, manifesting as estimation error from the model's inability to capture nonlinearity, though variance remains negligible in this idealized setting.[7]Sample MSPE
The sample mean squared prediction error (MSPE) adapts the theoretical population MSPE to finite datasets, providing an empirical measure of prediction accuracy based on observed data. Unlike the population version, which represents an expected value over infinite data, the sample MSPE is calculated directly from a limited number of observations, making it susceptible to variability and bias in small datasets. This empirical formulation serves as the theoretical target for evaluating model performance in practice.[8] The standard formula for sample MSPE is where denotes the observed values in the dataset, the corresponding predictions from the model, and the number of observations used in the computation.[8] Here, the predictions may be generated on the same data used for model fitting (in-sample) or on a separate held-out portion (out-of-sample test set). In sample contexts, a critical distinction arises between training and test sets: computations on the training set often yield optimistically low MSPE values due to overfitting, whereas test set evaluations better reflect generalization to unseen data.[9] In small samples, the unadjusted sample MSPE can underestimate the true prediction error by failing to account for model complexity. To mitigate this, adjustments incorporating degrees of freedom are applied, such as dividing the sum of squared errors by (where is the number of estimated parameters) to obtain an unbiased estimate of the error variance, akin to the standard MSE in linear regression. This correction helps prevent downward bias, particularly when is close to .[10] For illustration, consider computing sample MSPE in a forecasting model applied to a dataset of 100 observations, such as time series data on economic indicators. The model might be fitted to the first 80 observations (training set) to generate predictions, with the remaining 20 held out as the test set. The sample MSPE is then the average of the squared differences between the 20 test observations and their one-step-ahead forecasts, yielding a scalar value that quantifies the model's predictive fidelity on this finite holdout sample.[9]Computation Methods
Out-of-Sample Computation
Out-of-sample mean squared prediction error (MSPE) is computed by first partitioning the dataset into a training set, used for model fitting, and a separate test set reserved for evaluation. The model is trained solely on the training data to generate parameter estimates, after which predictions are produced for each observation in the test set based on its features. The MSPE is then obtained by averaging the squared residuals between the observed test values and these predictions, providing a direct measure of predictive accuracy on unseen data.[9] This method delivers an unbiased assessment of the model's ability to generalize to new data, distinct from training performance, and is essential for identifying overfitting where a model performs well on familiar data but poorly on novel instances. For time-series data, out-of-sample computation requires careful handling to maintain temporal dependencies and prevent the use of future information in training. A common strategy involves a hold-out period, where the final portion of the series (e.g., the last 20% of observations) serves as the test set, with the model fitted to all preceding data.[9] Alternatively, rolling windows are employed, in which the training window slides forward: for each step, the model is refitted on a contiguous block of past observations to forecast the next one or more periods, and prediction errors are aggregated across these steps to compute the overall MSPE. This respects the chronological order and simulates real-world forecasting scenarios.[11] The following Python pseudocode illustrates a basic implementation using scikit-learn for out-of-sample MSPE in a non-time-series context; for time-series, the split would use sequential indexing instead of random partitioning:python
from sklearn.model_selection import train_test_split
from sklearn.linear_model import LinearRegression
from sklearn.metrics import mean_squared_error
# Assume X (features) and y (target) are defined
X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.2, random_state=42)
model = LinearRegression()
model.fit(X_train, y_train)
y_pred = model.predict(X_test)
mspe = mean_squared_error(y_test, y_pred)
print(f"Out-of-sample MSPE: {mspe}")
from sklearn.model_selection import train_test_split
from sklearn.linear_model import LinearRegression
from sklearn.metrics import mean_squared_error
# Assume X (features) and y (target) are defined
X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.2, random_state=42)
model = LinearRegression()
model.fit(X_train, y_train)
y_pred = model.predict(X_test)
mspe = mean_squared_error(y_test, y_pred)
print(f"Out-of-sample MSPE: {mspe}")