Excluding irrelevant variables from a multiple regression model. Can be done automatically or manually. 
Important because irrelevant variables decrease the accuracy of your data.

Feature Selection

Reducing an initial set of raw data down to smaller groups so that they are more managable. Usually used when looking at large data sets with multiple variables. 

Feature Extraction

Feature selection: Use feature selection as a form of dimensionality reduction when your data has dimensions that are not relevant to the problem you are trying to solve or if you have repeating information within dimensions. In these cases, simply remove the unnecessary dimensions.

Feature extraction: Use feature extraction for dimensionality reduction when you want to use most of the information from your data set, but need the information explained in fewer dimensions. Also, feature extraction will take care of some of the repeat information from your data. 

When to use feature selection vs. feature extraction

- Principal Component Analysis (PCA)
- Linear Discriminant Analysis (LDA)
- Generalized Discriminant Analysis (GDA)
- Singular Value Decomposition (SVD)

Methods of dimensionality reduction

PCA 
-identifies the axis that accounts for the largest amount of variance
- tries to preserve global structure of data 
t-SNE
- A powerful manifold learning method that finds a 2D projection preserving information about neighbors
- Gives much more weight to preserving information about distances between points that are neighbors
- Work better on datasets that have more well defined local structure


Types of dimension reduction


Process of reducing the number of random variables under consideration by obtaining a set of principal variables. Dimension reduction methods transform the variables before fitting a least squares model. The two types of dimension reduction are
1. Feature selection, 
which selects a subset of variables from the original dataset, making regression or classification more accurate.
2. Feature extraction, 
which combines variables into features, reducing the amount of data processed while maintaining the accuracy of the original dataset. 

University of Michigan - Ann Arbor

Methods:
- Subset selection
- Shrinkage
- Dimension reduction

Criteria:
(a) Indirectly estimate test error by making adjustment to the training error:
- Adjusted R-Squared
- AIC
- BIC
- Mallow’s $C_p$
(b) Directly estimate the test error:
- Cross-validation

Methods of choosing important predictors to improve interpretability

James, G., Witten, D., Hastie, T., & Tibshirani, R. (2013). An introduction to statistical learning (Vol. 112, pp. 3-7). New York: springer.

An Introduction to Statistical Learning with Applications in R

In most basic terms, subset selection has two steps:
1. Identify a subset of predictors related to response
2. Use least squares on our reduced set of variables to fit a new model

Subset Selection

Dimension Reduction

$R_{adj}^2 = 1-(1-R^2)*(n-1)/(n-p-1)$   

Adjusted R-Squared is a version of R-Squared that adjusted for the number of predictors (independent variables) in a model. As we know, $R^2$ always increase if we adding more predictors. This Adjusted R-Squared has an advantage over the normal R-Squared metric because it accounts for statistical shrinkage and the normal R-Squared meter tends to hurt more when more independent variables occur in the system. This adjusted R-square allows us to do predictors selection and can be a metrics for forward/backward selection.


Adjusted R-Square Explained 

AIC = $ \frac{RSS+2d \hat{\sigma}}{n \hat{\sigma ^2}}$

The AIC (Akaike Information Criterion) Metric describes the quality of the model with the data that is given. AIC is applicable in a board range of modeling frameworks as it only requires large sample properties of maximum likelihood estimator. It uses candidate models to manipulate the data but does not require the assumption that these models be true or correct. AIC is the trade-off between goodness of fit and complexity of the variables that are considered in the problem. R-Squared changes relative to the complexity of the system (variables) but AIC does not. 

Note: AIC cannot be used to compare models with different dataset. For example, if one model involves transformations on response variable (like log transformations) and another one does not, one cannot use AIC to compare two models. 

AIC Explained

BIC = $$\frac{RSS+log(n)d \hat{\sigma^2}}{n \hat{\sigma^2}}$$

Bayesian information criterion (BIC) is a criterion for model selection among a finite set of models. The models can be tested using corresponding BIC values. Lower BIC value indicates lower penalty terms hence a better model. It is worth to know BIC closely relates to AIC. The only difference is BIC considers the number of observations in the formula, which AIC does not.

More details: https://www.immagic.com/eLibrary/ARCHIVES/GENERAL/WIKIPEDI/W120607B.pdf

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