Regressions

Regression models are used to predict a target variable based on predictor variables. Typically these take the form of a generalized linear model (GLM) with a linear combination of predictors and a link function to transform the linear combination into the expected value of the target variable.

\[\mathbb{E}[Y|X] = g(X\beta)\]

Where:

• \(Y\) is the normalized target variable

• \(X\) is the normalized predictor variables

• \(g\) is the link function

• \(\beta\) are the linear coefficients

Contextualized Linear Regression

Linear regression is the simplest form of the GLM above, where the link function is the identity function. As with all models in Contextualized, we can fit a linear regression model with context-specific parameters by learning a context encoder, which maps context variables to context-specific parameters.

\[\mathbb{E}[Y|X, C] = (X - \mu(C))\beta(C)\]

Where:

• \(C\) are the context variables

• \(\beta(\cdot)\) is the context encoder for the linear coefficients, which outputs context-specific linear coefficients.

• \(\mu(\cdot)\) is the context encoder for the offsets (re-introduced to account for the mean of the target variable given \(C\)), which outputs context-specific offsets.

This model is implemented by the ContextualizedRegressor.

Contextualized Logistic Regression

Logistic regression is a GLM where the link function is the logistic function. Now the expected value of the target variable is the probability of the target variable being 1.

\[\mathbb{E}[Y = 1|X, C] = \frac{1}{1 + \exp(-(X - \mu(C))\beta(C))}\]

This model is implemented by the ContextualizedClassifier.

Next, we provide some basic code examples using each of these models.