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== Regression Effect ==
== Regression Effect ==
 
[[File:Regression Effect Scatter Plot.png|thumb|Regression effect demonstrated by SD line and Regression line]]
The '''regression effect''' is the phenomenon that the best prediction
The '''regression effect''' is the phenomenon that the best prediction
of <math>Y</math> given <math>X = x</math> is less rare for
of <math>Y</math> given <math>X = x</math> is less rare for

Revision as of 18:28, 18 March 2024

Consider two numerica random variables and . We can measure their covariance.

The correlation of two random variables measures the line dependent between and

Correlation is always between -1 and 1

Bivariate Normal

The bivariate normal (aka. bivariate gaussian) is one special type of continuous random variable.

is bivariate normal if

  1. The marginal PDF of both X and Y are normal
  2. For any , the condition PDF of given is Normal
    • Works the other way around: Bivariate gaussian means that condition is satisfied

Predicting Y given X

Given bivariate normal, we can predict one variable given another. Let us try estimating the expected Y given X is x

There are three main methods

  • Scatter plot approximation
  • Joint PDF
  • 5 statistics

5 Parameters

We need to know 5 parameters about and

If follows bivariate normal distribution, then we have

The left side is the predicted Z-score for Y, and the right side is the product of correlation and Z-score of X = x

The variance is given by

Due to the range of , the variance of Y given X is always smaller than the actual variance. The standard deviation is just rooted that.

Regression Effect

Regression effect demonstrated by SD line and Regression line

The regression effect is the phenomenon that the best prediction of given is less rare for than ; Future predictions regress to mediocrity.

When you plot all the predicted , you get the linear regression line. The regression effect can be demonstrated by also plotting the SD line (where the correlation is not applied).