In statistics and machine learning, understanding the relationships between variables is crucial for constructing predictive models and analyzing data. One in every of the essential techniques for exploring these relationships is the bivariate projection, which relies on the concept of the bivariate normal distribution. This method allows for the examination and prediction of the behavior of 1 variable by way of one other, utilizing the dependency structure between them.
Bivariate projection helps determining the expected value of 1 random variable given a selected value of one other variable. For example, in linear regression, projection helps estimate how a dependent variable changes with respect to an independent variable.
This text is split into 3 parts: within the first part, I’ll explore the basics of bivariate projection, deriving its formulation and demonstrating its application in regression models. Within the second part, I’ll provide some intuition behind the projection and a few plots to raised understand its implications. Within the third part, I’ll use the projection to derive the parameters for a linear regression.
In my derivation of the bivariate projection formula, I’ll use some well-known results. So as to not be too heavy on the reader, I’ll provide the proofs…
