# Cross-validation (statistics)

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Cross-validation,    sometimes called rotation estimation    or out-of-sample testing, is any of various similar model validation techniques for assessing how the results of a statistical analysis will generalize to an independent data set. Cross-validation is a resampling method that uses different portions of the data to test and train a model on different iterations. It is mainly used in settings where the goal is prediction, and one wants to estimate how accurately a predictive model will perform in practice. In a prediction problem, a model is usually given a dataset of known data on which training is run (training dataset), and a dataset of unknown data (or first seen data) against which the model is tested (called the validation dataset or testing set).   The goal of cross-validation is to test the model's ability to predict new data that was not used in estimating it, in order to flag problems like overfitting or selection bias  and to give an insight on how the model will generalize to an independent dataset (i.e., an unknown dataset, for instance from a real problem).

## Contents

One round of cross-validation involves partitioning a sample of data into complementary subsets, performing the analysis on one subset (called the training set), and validating the analysis on the other subset (called the validation set or testing set). To reduce variability, in most methods multiple rounds of cross-validation are performed using different partitions, and the validation results are combined (e.g. averaged) over the rounds to give an estimate of the model's predictive performance.

In summary, cross-validation combines (averages) measures of fitness in prediction to derive a more accurate estimate of model prediction performance. 

## Motivation

Assume a model with one or more unknown parameters, and a data set to which the model can be fit (the training data set). The fitting process optimizes the model parameters to make the model fit the training data as well as possible. If an independent sample of validation data is taken from the same population as the training data, it will generally turn out that the model does not fit the validation data as well as it fits the training data. The size of this difference is likely to be large especially when the size of the training data set is small, or when the number of parameters in the model is large. Cross-validation is a way to estimate the size of this effect.

In linear regression, there exist real response valuesy1, ..., yn, and np-dimensional vector covariatesx1, ..., xn. The components of the vector xi are denoted xi1, ..., xip. If least squares is used to fit a function in the form of a hyperplane ŷ = a + βTx to the data (xi, yi) 1  i  n, then the fit can be assessed using the mean squared error (MSE). The MSE for given estimated parameter values a and β on the training set (xi, yi) 1  i  n is defined as:

{\begin{aligned}{\text{MSE}}&={\frac {1}{n}}\sum _{i=1}^{n}(y_{i}-{\hat {y}}_{i})^{2}={\frac {1}{n}}\sum _{i=1}^{n}(y_{i}-a-{\boldsymbol {\beta }}^{T}\mathbf {x} _{i})^{2}\\&={\frac {1}{n}}\sum _{i=1}^{n}(y_{i}-a-\beta _{1}x_{i1}-\dots -\beta _{p}x_{ip})^{2}\end{aligned}} If the model is correctly specified, it can be shown under mild assumptions that the expected value of the MSE for the training set is (n  p  1)/(n + p + 1) < 1 times the expected value of the MSE for the validation set  [ irrelevant citation ] (the expected value is taken over the distribution of training sets). Thus, a fitted model and computed MSE on the training set will result in an optimistically biased assessment of how well the model will fit an independent data set. This biased estimate is called the in-sample estimate of the fit, whereas the cross-validation estimate is an out-of-sample estimate.

Since in linear regression it is possible to directly compute the factor (n  p  1)/(n + p + 1) by which the training MSE underestimates the validation MSE under the assumption that the model specification is valid, cross-validation can be used for checking whether the model has been overfitted, in which case the MSE in the validation set will substantially exceed its anticipated value. (Cross-validation in the context of linear regression is also useful in that it can be used to select an optimally regularized cost function.) In most other regression procedures (e.g. logistic regression), there is no simple formula to compute the expected out-of-sample fit. Cross-validation is, thus, a generally applicable way to predict the performance of a model on unavailable data using numerical computation in place of theoretical analysis.

## Types

Two types of cross-validation can be distinguished: exhaustive and non-exhaustive cross-validation.

### Exhaustive cross-validation

Exhaustive cross-validation methods are cross-validation methods which learn and test on all possible ways to divide the original sample into a training and a validation set.

#### Leave-p-out cross-validation

Leave-p-out cross-validation (LpO CV) involves using p observations as the validation set and the remaining observations as the training set. This is repeated on all ways to cut the original sample on a validation set of p observations and a training set. 

LpO cross-validation require training and validating the model $C_{p}^{n}$ times, where n is the number of observations in the original sample, and where $C_{p}^{n}$ is the binomial coefficient. For p > 1 and for even moderately large n, LpO CV can become computationally infeasible. For example, with n = 100 and p = 30, $C_{30}^{100}\approx 3\times 10^{25}.$ A variant of LpO cross-validation with p=2 known as leave-pair-out cross-validation has been recommended as a nearly unbiased method for estimating the area under ROC curve of binary classifiers. 

#### Leave-one-out cross-validation Illustration of leave-one-out cross-validation (LOOCV) when n = 8 observations. A total of 8 models will be trained and tested.

Leave-one-out cross-validation (LOOCV) is a particular case of leave-p-out cross-validation with p = 1. The process looks similar to jackknife; however, with cross-validation one computes a statistic on the left-out sample(s), while with jackknifing one computes a statistic from the kept samples only.

LOO cross-validation requires less computation time than LpO cross-validation because there are only $C_{1}^{n}=n$ passes rather than $C_{p}^{n}$ . However, $n$ passes may still require quite a large computation time, in which case other approaches such as k-fold cross validation may be more appropriate. 

Pseudo-code algorithm:

Input:

x, {vector of length N with x-values of incoming points}

y, {vector of length N with y-values of the expected result}

interpolate( x_in, y_in, x_out ), { returns the estimation for point x_out after the model is trained with x_in-y_in pairs}

Output:

err, {estimate for the prediction error}

Steps:

 err ← 0  for i ← 1, ..., N do    // define the cross-validation subsets    x_in ← (x, ..., x[i − 1], x[i + 1], ..., x[N])    y_in ← (y, ..., y[i − 1], y[i + 1], ..., y[N])    x_out ← x[i]    y_out ← interpolate(x_in, y_in, x_out)    err ← err + (y[i] − y_out)^2  end for  err ← err/N

### Non-exhaustive cross-validation

Non-exhaustive cross validation methods do not compute all ways of splitting the original sample. These methods are approximations of leave-p-out cross-validation.

#### k-fold cross-validation Illustration of k-fold cross-validation when n = 12 observations and k = 3. After data is shuffled, a total of 3 models will be trained and tested.

[ citation needed ]

In k-fold cross-validation, the original sample is randomly partitioned into k equal sized subsamples. Of the k subsamples, a single subsample is retained as the validation data for testing the model, and the remaining k  1 subsamples are used as training data. The cross-validation process is then repeated k times, with each of the k subsamples used exactly once as the validation data. The k results can then be averaged to produce a single estimation. The advantage of this method over repeated random sub-sampling (see below) is that all observations are used for both training and validation, and each observation is used for validation exactly once. 10-fold cross-validation is commonly used,  but in general k remains an unfixed parameter.

For example, setting k = 2 results in 2-fold cross-validation. In 2-fold cross-validation, we randomly shuffle the dataset into two sets d0 and d1, so that both sets are equal size (this is usually implemented by shuffling the data array and then splitting it in two). We then train on d0 and validate on d1, followed by training on d1 and validating on d0.

When k = n (the number of observations), k-fold cross-validation is equivalent to leave-one-out cross-validation. 

In stratifiedk-fold cross-validation, the partitions are selected so that the mean response value is approximately equal in all the partitions. In the case of binary classification, this means that each partition contains roughly the same proportions of the two types of class labels.

In repeated cross-validation the data is randomly split into k partitions several times. The performance of the model can thereby be averaged over several runs, but this is rarely desirable in practice. 

When many different statistical or machine learning models are being considered, greedyk-fold cross-validation can be used to quickly identify the most promising candidate models. 

#### Holdout method

In the holdout method, we randomly assign data points to two sets d0 and d1, usually called the training set and the test set, respectively. The size of each of the sets is arbitrary although typically the test set is smaller than the training set. We then train (build a model) on d0 and test (evaluate its performance) on d1.

In typical cross-validation, results of multiple runs of model-testing are averaged together; in contrast, the holdout method, in isolation, involves a single run. It should be used with caution because without such averaging of multiple runs, one may achieve highly misleading results. One's indicator of predictive accuracy (F*) will tend to be unstable since it will not be smoothed out by multiple iterations (see below). Similarly, indicators of the specific role played by various predictor variables (e.g., values of regression coefficients) will tend to be unstable.

While the holdout method can be framed as "the simplest kind of cross-validation",  many sources instead classify holdout as a type of simple validation, rather than a simple or degenerate form of cross-validation.  

#### Repeated random sub-sampling validation

This method, also known as Monte Carlo cross-validation,  creates multiple random splits of the dataset into training and validation data.  For each such split, the model is fit to the training data, and predictive accuracy is assessed using the validation data. The results are then averaged over the splits. The advantage of this method (over k-fold cross validation) is that the proportion of the training/validation split is not dependent on the number of iterations (i.e., the number of partitions). The disadvantage of this method is that some observations may never be selected in the validation subsample, whereas others may be selected more than once. In other words, validation subsets may overlap. This method also exhibits Monte Carlo variation, meaning that the results will vary if the analysis is repeated with different random splits.

As the number of random splits approaches infinity, the result of repeated random sub-sampling validation tends towards that of leave-p-out cross-validation.

In a stratified variant of this approach, the random samples are generated in such a way that the mean response value (i.e. the dependent variable in the regression) is equal in the training and testing sets. This is particularly useful if the responses are dichotomous with an unbalanced representation of the two response values in the data.

A method that applies repeated random sub-sampling is RANSAC. 

## Nested cross-validation

When cross-validation is used simultaneously for selection of the best set of hyperparameters and for error estimation (and assessment of generalization capacity), a nested cross-validation is required. Many variants exist. At least two variants can be distinguished:

### k*l-fold cross-validation

This is a truly nested variant which contains an outer loop of k sets and an inner loop of l sets. The total data set is split into k sets. One by one, a set is selected as the (outer) test set and the k - 1 other sets are combined into the corresponding outer training set. This is repeated for each of the k sets. Each outer training set is further sub-divided into l sets. One by one, a set is selected as inner test (validation) set and the l - 1 other sets are combined into the corresponding inner training set. This is repeated for each of the l sets. The inner training sets are used to fit model parameters, while the outer test set is used as a validation set to provide an unbiased evaluation of the model fit. Typically, this is repeated for many different hyperparameters (or even different model types) and the validation set is used to determine the best hyperparameter set (and model type) for this inner training set. After this, a new model is fit on the entire outer training set, using the best set of hyperparameters from the inner cross-validation. The performance of this model is then evaluated using the outer test set.

### k-fold cross-validation with validation and test set

This is a type of k*l-fold cross-validation when l = k - 1. A single k-fold cross-validation is used with both a validation and test set. The total data set is split into k sets. One by one, a set is selected as test set. Then, one by one, one of the remaining sets is used as a validation set and the other k - 2 sets are used as training sets until all possible combinations have been evaluated. Similar to the k*l-fold cross validation, the training set is used for model fitting and the validation set is used for model evaluation for each of the hyperparameter sets. Finally, for the selected parameter set, the test set is used to evaluate the model with the best parameter set. Here, two variants are possible: either evaluating the model that was trained on the training set or evaluating a new model that was fit on the combination of the training and the validation set.

## Measures of fit

The goal of cross-validation is to estimate the expected level of fit of a model to a data set that is independent of the data that were used to train the model. It can be used to estimate any quantitative measure of fit that is appropriate for the data and model. For example, for binary classification problems, each case in the validation set is either predicted correctly or incorrectly. In this situation the misclassification error rate can be used to summarize the fit, although other measures like positive predictive value could also be used. When the value being predicted is continuously distributed, the mean squared error, root mean squared error or median absolute deviation could be used to summarize the errors.

## Using prior information

When users apply cross-validation to select a good configuration $\lambda$ , then they might want to balance the cross-validated choice with their own estimate of the configuration. In this way, they can attempt to counter the volatility of cross-validation when the sample size is small and include relevant information from previous research. In a forecasting combination exercise, for instance, cross-validation can be applied to estimate the weights that are assigned to each forecast. Since a simple equal-weighted forecast is difficult to beat, a penalty can be added for deviating from equal weights.  Or, if cross-validation is applied to assign individual weights to observations, then one can penalize deviations from equal weights to avoid wasting potentially relevant information.  Hoornweg (2018) shows how a tuning parameter $\gamma$ can be defined so that a user can intuitively balance between the accuracy of cross-validation and the simplicity of sticking to a reference parameter $\lambda _{R}$ that is defined by the user.

If $\lambda _{i}$ denotes the $i^{th}$ candidate configuration that might be selected, then the loss function that is to be minimized can be defined as

$L_{\lambda _{i}}=(1-\gamma ){\mbox{ Relative Accuracy}}_{i}+\gamma {\mbox{ Relative Simplicity}}_{i}.$ Relative accuracy can be quantified as ${\mbox{MSE}}(\lambda _{i})/{\mbox{MSE}}(\lambda _{R})$ , so that the mean squared error of a candidate $\lambda _{i}$ is made relative to that of a user-specified $\lambda _{R}$ . The relative simplicity term measures the amount that $\lambda _{i}$ deviates from $\lambda _{R}$ relative to the maximum amount of deviation from $\lambda _{R}$ . Accordingly, relative simplicity can be specified as ${\frac {(\lambda _{i}-\lambda _{R})^{2}}{(\lambda _{\max }-\lambda _{R})^{2}}}$ , where $\lambda _{\max }$ corresponds to the $\lambda$ value with the highest permissible deviation from $\lambda _{R}$ . With $\gamma \in [0,1]$ , the user determines how high the influence of the reference parameter is relative to cross-validation.

One can add relative simplicity terms for multiple configurations $c=1,2,...,C$ by specifying the loss function as

$L_{\lambda _{i}}={\mbox{ Relative Accuracy}}_{i}+\sum _{c=1}^{C}{\frac {\gamma _{c}}{1-\gamma _{c}}}{\mbox{ Relative Simplicity}}_{i,c}.$ Hoornweg (2018) shows that a loss function with such an accuracy-simplicity tradeoff can also be used to intuitively define shrinkage estimators like the (adaptive) lasso and Bayesian / ridge regression.  Click on the lasso for an example.

## Statistical properties

Suppose we choose a measure of fit F, and use cross-validation to produce an estimate F* of the expected fit EF of a model to an independent data set drawn from the same population as the training data. If we imagine sampling multiple independent training sets following the same distribution, the resulting values for F* will vary. The statistical properties of F* result from this variation.

The cross-validation estimator F* is very nearly unbiased for EF.  [ citation needed ] The reason that it is slightly biased is that the training set in cross-validation is slightly smaller than the actual data set (e.g. for LOOCV the training set size is n  1 when there are n observed cases). In nearly all situations, the effect of this bias will be conservative in that the estimated fit will be slightly biased in the direction suggesting a poorer fit. In practice, this bias is rarely a concern.

The variance of F* can be large.   For this reason, if two statistical procedures are compared based on the results of cross-validation, the procedure with the better estimated performance may not actually be the better of the two procedures (i.e. it may not have the better value of EF). Some progress has been made on constructing confidence intervals around cross-validation estimates,  but this is considered a difficult problem.

## Computational issues

Most forms of cross-validation are straightforward to implement as long as an implementation of the prediction method being studied is available. In particular, the prediction method can be a "black box" – there is no need to have access to the internals of its implementation. If the prediction method is expensive to train, cross-validation can be very slow since the training must be carried out repeatedly. In some cases such as least squares and kernel regression, cross-validation can be sped up significantly by pre-computing certain values that are needed repeatedly in the training, or by using fast "updating rules" such as the Sherman–Morrison formula. However one must be careful to preserve the "total blinding" of the validation set from the training procedure, otherwise bias may result. An extreme example of accelerating cross-validation occurs in linear regression, where the results of cross-validation have a closed-form expression known as the prediction residual error sum of squares (PRESS).

## Limitations and misuse

Cross-validation only yields meaningful results if the validation set and training set are drawn from the same population and only if human biases are controlled.

In many applications of predictive modeling, the structure of the system being studied evolves over time (i.e. it is "non-stationary"). Both of these can introduce systematic differences between the training and validation sets. For example, if a model for predicting stock values is trained on data for a certain five-year period, it is unrealistic to treat the subsequent five-year period as a draw from the same population. As another example, suppose a model is developed to predict an individual's risk for being diagnosed with a particular disease within the next year. If the model is trained using data from a study involving only a specific population group (e.g. young people or males), but is then applied to the general population, the cross-validation results from the training set could differ greatly from the actual predictive performance.

In many applications, models also may be incorrectly specified and vary as a function of modeler biases and/or arbitrary choices. When this occurs, there may be an illusion that the system changes in external samples, whereas the reason is that the model has missed a critical predictor and/or included a confounded predictor. New evidence is that cross-validation by itself is not very predictive of external validity, whereas a form of experimental validation known as swap sampling that does control for human bias can be much more predictive of external validity.  As defined by this large MAQC-II study across 30,000 models, swap sampling incorporates cross-validation in the sense that predictions are tested across independent training and validation samples. Yet, models are also developed across these independent samples and by modelers who are blinded to one another. When there is a mismatch in these models developed across these swapped training and validation samples as happens quite frequently, MAQC-II shows that this will be much more predictive of poor external predictive validity than traditional cross-validation.

The reason for the success of the swapped sampling is a built-in control for human biases in model building. In addition to placing too much faith in predictions that may vary across modelers and lead to poor external validity due to these confounding modeler effects, these are some other ways that cross-validation can be misused:

• By performing an initial analysis to identify the most informative features using the entire data set – if feature selection or model tuning is required by the modeling procedure, this must be repeated on every training set. Otherwise, predictions will certainly be upwardly biased.  If cross-validation is used to decide which features to use, an inner cross-validation to carry out the feature selection on every training set must be performed. 
• By allowing some of the training data to also be included in the test set – this can happen due to "twinning" in the data set, whereby some exactly identical or nearly identical samples are present in the data set. To some extent twinning always takes place even in perfectly independent training and validation samples. This is because some of the training sample observations will have nearly identical values of predictors as validation sample observations. And some of these will correlate with a target at better than chance levels in the same direction in both training and validation when they are actually driven by confounded predictors with poor external validity. If such a cross-validated model is selected from a k-fold set, human confirmation bias will be at work and determine that such a model has been validated. This is why traditional cross-validation needs to be supplemented with controls for human bias and confounded model specification like swap sampling and prospective studies.

## Cross validation for time-series models

Since the order of the data is important, cross-validation might be problematic for time-series models. A more appropriate approach might be to use rolling cross-validation. 

However, if performance is described by a single summary statistic, it is possible that the approach described by Politis and Romano as a stationary bootstrap  will work. The statistic of the bootstrap needs to accept an interval of the time series and return the summary statistic on it. The call to the stationary bootstrap needs to specify an appropriate mean interval length.

## Applications

Cross-validation can be used to compare the performances of different predictive modeling procedures. For example, suppose we are interested in optical character recognition, and we are considering using either a Support Vector Machine (SVM) or k-nearest neighbors (KNN) to predict the true character from an image of a handwritten character. Using cross-validation, we could objectively compare these two methods in terms of their respective fractions of misclassified characters. If we simply compared the methods based on their in-sample error rates, one method would likely appear to perform better, since it is more flexible and hence more prone to overfitting [ citation needed ] compared to the other method.

Cross-validation can also be used in variable selection.  Suppose we are using the expression levels of 20 proteins to predict whether a cancer patient will respond to a drug. A practical goal would be to determine which subset of the 20 features should be used to produce the best predictive model. For most modeling procedures, if we compare feature subsets using the in-sample error rates, the best performance will occur when all 20 features are used. However under cross-validation, the model with the best fit will generally include only a subset of the features that are deemed truly informative.

A recent development in medical statistics is its use in meta-analysis. It forms the basis of the validation statistic, Vn which is used to test the statistical validity of meta-analysis summary estimates.  It has also been used in a more conventional sense in meta-analysis to estimate the likely prediction error of meta-analysis results. 

## Notes and references

1. Piryonesi S. Madeh; El-Diraby Tamer E. (2020-03-01). "Data Analytics in Asset Management: Cost-Effective Prediction of the Pavement Condition Index". Journal of Infrastructure Systems. 26 (1): 04019036. doi:10.1061/(ASCE)IS.1943-555X.0000512. S2CID   213782055.
2. Allen, David M (1974). "The Relationship between Variable Selection and Data Agumentation and a Method for Prediction". Technometrics. 16 (1): 125–127. doi:10.2307/1267500. JSTOR   1267500.
3. Stone, M (1974). "Cross-Validatory Choice and Assessment of Statistical Predictions". Journal of the Royal Statistical Society, Series B (Methodological). 36 (2): 111–147. doi:10.1111/j.2517-6161.1974.tb00994.x. S2CID   62698647.
4. Stone, M (1977). "An Asymptotic Equivalence of Choice of Model by Cross-Validation and Akaike's Criterion". Journal of the Royal Statistical Society, Series B (Methodological). 39 (1): 44–47. doi:10.1111/j.2517-6161.1977.tb01603.x. JSTOR   2984877.
5. Geisser, Seymour (1993). Predictive Inference. New York, NY: Chapman and Hall. ISBN   978-0-412-03471-8.
6. Kohavi, Ron (1995). "A study of cross-validation and bootstrap for accuracy estimation and model selection". Proceedings of the Fourteenth International Joint Conference on Artificial Intelligence. San Mateo, CA: Morgan Kaufmann. 2 (12): 1137–1143. CiteSeerX  .
7. Devijver, Pierre A.; Kittler, Josef (1982). Pattern Recognition: A Statistical Approach. London, GB: Prentice-Hall. ISBN   0-13-654236-0.
8. Galkin, Alexander (November 28, 2011). "What is the difference between test set and validation set?" . Retrieved 10 October 2018.
9. "Newbie question: Confused about train, validation and test data!". Archived from the original on 2015-03-14. Retrieved 2013-11-14.{{cite web}}: CS1 maint: bot: original URL status unknown (link)
10. Cawley, Gavin C.; Talbot, Nicola L. C. (2010). "On Over-fitting in Model Selection and Subsequent Selection Bias in Performance Evaluation" (PDF). 11. Journal of Machine Learning Research: 2079–2107.{{cite journal}}: Cite journal requires |journal= (help)
11. Grossman, Robert; Seni, Giovanni; Elder, John; Agarwal, Nitin; Liu, Huan (2010). "Ensemble Methods in Data Mining: Improving Accuracy Through Combining Predictions". Synthesis Lectures on Data Mining and Knowledge Discovery. Morgan & Claypool. 2: 1–126. doi:10.2200/S00240ED1V01Y200912DMK002.
12. Trippa, Lorenzo; Waldron, Levi; Huttenhower, Curtis; Parmigiani, Giovanni (March 2015). "Bayesian nonparametric cross-study validation of prediction methods". The Annals of Applied Statistics. 9 (1): 402–428. arXiv:. Bibcode:2015arXiv150600474T. doi:10.1214/14-AOAS798. ISSN   1932-6157. S2CID   51943497.
13. Celisse, Alain (1 October 2014). "Optimal cross-validation in density estimation with the $L^{2}$-loss". The Annals of Statistics. 42 (5): 1879–1910. arXiv:. doi:10.1214/14-AOS1240. ISSN   0090-5364. S2CID   17833620.
14. Airola, A.; Pahikkala, T.; Waegeman, W.; De Baets, Bernard; Salakoski, T. (2011-04-01). "An experimental comparison of cross-validation techniques for estimating the area under the ROC curve". Computational Statistics & Data Analysis. 55 (4): 1828–1844. doi:10.1016/j.csda.2010.11.018.
15. Molinaro, A. M.; Simon, R.; Pfeiffer, R. M. (2005-08-01). "Prediction error estimation: a comparison of resampling methods". Bioinformatics. 21 (15): 3301–3307. doi:. ISSN   1367-4803. PMID   15905277.
16. McLachlan, Geoffrey J.; Do, Kim-Anh; Ambroise, Christophe (2004). Analyzing microarray gene expression data. Wiley.
17. "Elements of Statistical Learning: data mining, inference, and prediction. 2nd Edition". web.stanford.edu. Retrieved 2019-04-04.
18. Vanwinckelen, Gitte (2 October 2019). On Estimating Model Accuracy with Repeated Cross-Validation. lirias.kuleuven. pp. 39–44. ISBN   9789461970442.
19. Soper, Daniel S. (2021). "Greed Is Good: Rapid Hyperparameter Optimization and Model Selection Using Greedy k-Fold Cross Validation" (PDF). Electronics. 10 (16): 1973.
20. "Cross Validation" . Retrieved 11 November 2012.
21. Arlot, Sylvain; Celisse, Alain (2010). "A survey of cross-validation procedures for model selection". Statistics Surveys. 4: 40–79. arXiv:. doi:10.1214/09-SS054. S2CID   14332192. In brief, CV consists in averaging several hold-out estimators of the risk corresponding to different data splits.
22. Dubitzky, Werner; Granzow, Martin; Berrar, Daniel (2007). Fundamentals of data mining in genomics and proteomics. Springer Science & Business Media. p. 178.
23. Kuhn, Max; Johnson, Kjell (2013). Applied Predictive Modeling. New York, NY: Springer New York. doi:10.1007/978-1-4614-6849-3. ISBN   9781461468486.
24. Cantzler, H. "Random sample consensus (ransac)." Institute for Perception, Action and Behaviour, Division of Informatics, University of Edinburgh (1981). http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.106.3035&rep=rep1&type=pdf
25. Hoornweg, Victor (2018). Science: Under Submission. Hoornweg Press. ISBN   978-90-829188-0-9.
26. Christensen, Ronald (May 21, 2015). "Thoughts on prediction and cross-validation" (PDF). Department of Mathematics and Statistics University of New Mexico. Retrieved May 31, 2017.
27. Efron, Bradley; Tibshirani, Robert (1997). "Improvements on cross-validation: The .632 + Bootstrap Method". Journal of the American Statistical Association . 92 (438): 548–560. doi:10.2307/2965703. JSTOR   2965703. MR   1467848.
28. Stone, Mervyn (1977). "Asymptotics for and against cross-validation". Biometrika . 64 (1): 29–35. doi:10.1093/biomet/64.1.29. JSTOR   2335766. MR   0474601.
29. Consortium, MAQC (2010). "The Microarray Quality Control (MAQC)-II study of common practices for the development and validation of microarray-based predictive models". Nature Biotechnology. London: Nature Publishing Group. 28 (8): 827–838. doi:10.1038/nbt.1665. PMC  . PMID   20676074.
30. Bermingham, Mairead L.; Pong-Wong, Ricardo; Spiliopoulou, Athina; Hayward, Caroline; Rudan, Igor; Campbell, Harry; Wright, Alan F.; Wilson, James F.; Agakov, Felix; Navarro, Pau; Haley, Chris S. (2015). "Application of high-dimensional feature selection: evaluation for genomic prediction in man". Sci. Rep. 5: 10312. Bibcode:2015NatSR...510312B. doi:10.1038/srep10312. PMC  . PMID   25988841.
31. Varma, Sudhir; Simon, Richard (2006). "Bias in error estimation when using cross-validation for model selection". BMC Bioinformatics. 7: 91. doi:10.1186/1471-2105-7-91. PMC  . PMID   16504092.
32. Bergmeir, Christopher; Benitez, Jose (2012). "On the use of cross-validation for time series predictor evaluation". Information Sciences. 191: 192–213. doi:10.1016/j.ins.2011.12.028 via Elsevier Science Direct.
33. Politis, Dimitris N.; Romano, Joseph P. (1994). "The Stationary Bootstrap". Journal of the American Statistical Association. 89 (428): 1303–1313. doi:10.1080/01621459.1994.10476870. hdl:.
34. Picard, Richard; Cook, Dennis (1984). "Cross-Validation of Regression Models". Journal of the American Statistical Association. 79 (387): 575–583. doi:10.2307/2288403. JSTOR   2288403.
35. Willis BH, Riley RD (2017). "Measuring the statistical validity of summary meta-analysis and meta-regression results for use in clinical practice". Statistics in Medicine. 36 (21): 3283–3301. doi:10.1002/sim.7372. PMC  . PMID   28620945.
36. Riley RD, Ahmed I, Debray TP, Willis BH, Noordzij P, Higgins JP, Deeks JJ (2015). "Summarising and validating test accuracy results across multiple studies for use in clinical practice". Statistics in Medicine. 34 (13): 2081–2103. doi:10.1002/sim.6471. PMC  . PMID   25800943.

## Related Research Articles Supervised learning (SL) is a machine learning paradigm for problems where the available data consists of labelled examples, meaning that each data point contains features (covariates) and an associated label. The goal of supervised learning algorithms is learning a function that maps feature vectors (inputs) to labels (output), based on example input-output pairs. It infers a function from labeled training data consisting of a set of training examples. In supervised learning, each example is a pair consisting of an input object (typically a vector) and a desired output value (also called the supervisory signal). A supervised learning algorithm analyzes the training data and produces an inferred function, which can be used for mapping new examples. An optimal scenario will allow for the algorithm to correctly determine the class labels for unseen instances. This requires the learning algorithm to generalize from the training data to unseen situations in a "reasonable" way (see inductive bias). This statistical quality of an algorithm is measured through the so-called generalization error.

In statistics, the mean squared error (MSE) or mean squared deviation (MSD) of an estimator measures the average of the squares of the errors—that is, the average squared difference between the estimated values and the actual value. MSE is a risk function, corresponding to the expected value of the squared error loss. The fact that MSE is almost always strictly positive is because of randomness or because the estimator does not account for information that could produce a more accurate estimate. In machine learning, specifically empirical risk minimization, MSE may refer to the empirical risk, as an estimate of the true MSE. In statistics, the logistic model is a statistical model that models the probability of an event taking place by having the log-odds for the event be a linear combination of one or more independent variables. In regression analysis, logistic regression is estimating the parameters of a logistic model. Formally, in binary logistic regression there is a single binary dependent variable, coded by an indicator variable, where the two values are labeled "0" and "1", while the independent variables can each be a binary variable or a continuous variable. The corresponding probability of the value labeled "1" can vary between 0 and 1, hence the labeling; the function that converts log-odds to probability is the logistic function, hence the name. The unit of measurement for the log-odds scale is called a logit, from logistic unit, hence the alternative names. See § Background and § Definition for formal mathematics, and § Example for a worked example.

Forecasting is the process of making predictions based on past and present data. Later these can be compared (resolved) against what happens. For example, a company might estimate their revenue in the next year, then compare it against the actual results. Prediction is a similar but more general term. Forecasting might refer to specific formal statistical methods employing time series, cross-sectional or longitudinal data, or alternatively to less formal judgmental methods or the process of prediction and resolution itself. Usage can vary between areas of application: for example, in hydrology the terms "forecast" and "forecasting" are sometimes reserved for estimates of values at certain specific future times, while the term "prediction" is used for more general estimates, such as the number of times floods will occur over a long period. In statistical modeling, regression analysis is a set of statistical processes for estimating the relationships between a dependent variable and one or more independent variables. The most common form of regression analysis is linear regression, in which one finds the line that most closely fits the data according to a specific mathematical criterion. For example, the method of ordinary least squares computes the unique line that minimizes the sum of squared differences between the true data and that line. For specific mathematical reasons, this allows the researcher to estimate the conditional expectation of the dependent variable when the independent variables take on a given set of values. Less common forms of regression use slightly different procedures to estimate alternative location parameters or estimate the conditional expectation across a broader collection of non-linear models. Random forests or random decision forests is an ensemble learning method for classification, regression and other tasks that operates by constructing a multitude of decision trees at training time. For classification tasks, the output of the random forest is the class selected by most trees. For regression tasks, the mean or average prediction of the individual trees is returned. Random decision forests correct for decision trees' habit of overfitting to their training set. Random forests generally outperform decision trees, but their accuracy is lower than gradient boosted trees. However, data characteristics can affect their performance.

In statistics, multicollinearity is a phenomenon in which one predictor variable in a multiple regression model can be linearly predicted from the others with a substantial degree of accuracy. In this situation, the coefficient estimates of the multiple regression may change erratically in response to small changes in the model or the data. Multicollinearity does not reduce the predictive power or reliability of the model as a whole, at least within the sample data set; it only affects calculations regarding individual predictors. That is, a multivariate regression model with collinear predictors can indicate how well the entire bundle of predictors predicts the outcome variable, but it may not give valid results about any individual predictor, or about which predictors are redundant with respect to others. In statistics, the coefficient of determination, denoted R2 or r2 and pronounced "R squared", is the proportion of the variation in the dependent variable that is predictable from the independent variable(s). In mathematics, statistics, finance, computer science, particularly in machine learning and inverse problems, regularization is a process that changes the result answer to be "simpler". It is often used to obtain results for ill-posed problems or to prevent overfitting.

In statistics, a generalized additive model (GAM) is a generalized linear model in which the linear response variable depends linearly on unknown smooth functions of some predictor variables, and interest focuses on inference about these smooth functions.

In statistics, resampling is the creation of new samples based on one observed sample. Resampling methods are:

1. Permutation tests
2. Bootstrapping
3. Cross validation

Omnibus tests are a kind of statistical test. They test whether the explained variance in a set of data is significantly greater than the unexplained variance, overall. One example is the F-test in the analysis of variance. There can be legitimate significant effects within a model even if the omnibus test is not significant. For instance, in a model with two independent variables, if only one variable exerts a significant effect on the dependent variable and the other does not, then the omnibus test may be non-significant. This fact does not affect the conclusions that may be drawn from the one significant variable. In order to test effects within an omnibus test, researchers often use contrasts.

In statistics, Mallows's Cp, named for Colin Lingwood Mallows, is used to assess the fit of a regression model that has been estimated using ordinary least squares. It is applied in the context of model selection, where a number of predictor variables are available for predicting some outcome, and the goal is to find the best model involving a subset of these predictors. A small value of Cp means that the model is relatively precise.

Structural risk minimization (SRM) is an inductive principle of use in machine learning. Commonly in machine learning, a generalized model must be selected from a finite data set, with the consequent problem of overfitting – the model becoming too strongly tailored to the particularities of the training set and generalizing poorly to new data. The SRM principle addresses this problem by balancing the model's complexity against its success at fitting the training data. This principle was first set out in a 1974 paper by Vladimir Vapnik and Alexey Chervonenkis and uses the VC dimension.

In statistics, confirmatory factor analysis (CFA) is a special form of factor analysis, most commonly used in social science research. It is used to test whether measures of a construct are consistent with a researcher's understanding of the nature of that construct. As such, the objective of confirmatory factor analysis is to test whether the data fit a hypothesized measurement model. This hypothesized model is based on theory and/or previous analytic research. CFA was first developed by Jöreskog (1969) and has built upon and replaced older methods of analyzing construct validity such as the MTMM Matrix as described in Campbell & Fiske (1959).

In statistics, regression validation is the process of deciding whether the numerical results quantifying hypothesized relationships between variables, obtained from regression analysis, are acceptable as descriptions of the data. The validation process can involve analyzing the goodness of fit of the regression, analyzing whether the regression residuals are random, and checking whether the model's predictive performance deteriorates substantially when applied to data that were not used in model estimation.

In statistics, principal component regression (PCR) is a regression analysis technique that is based on principal component analysis (PCA). More specifically, PCR is used for estimating the unknown regression coefficients in a standard linear regression model. In statistics, maximum spacing estimation (MSE or MSP), or maximum product of spacing estimation (MPS), is a method for estimating the parameters of a univariate statistical model. The method requires maximization of the geometric mean of spacings in the data, which are the differences between the values of the cumulative distribution function at neighbouring data points.

In statistics and machine learning, lasso is a regression analysis method that performs both variable selection and regularization in order to enhance the prediction accuracy and interpretability of the resulting statistical model. It was originally introduced in geophysics, and later by Robert Tibshirani, who coined the term. In statistics and machine learning, the bias–variance tradeoff is the property of a model that the variance of the parameter estimated across samples can be reduced by increasing the bias in the estimated parameters. The bias–variance dilemma or bias–variance problem is the conflict in trying to simultaneously minimize these two sources of error that prevent supervised learning algorithms from generalizing beyond their training set: