Using Correlation to Predict Values

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The discussion focuses on predicting the expected value of a variable Y based on multiple correlated variables A, B, and C. Participants clarify the distinction between predicting a value and calculating the expected value given known correlations and variances. The conversation highlights the application of formulas typically used for bivariate normal distributions and questions whether the assumption of a joint multinormal distribution holds for the variables in question. One suggestion involves using principal component analysis to transform correlated variables into independent ones for multiple regression analysis. The need for careful examination of distribution assumptions is emphasized, as marginal normality does not ensure joint multivariate normality.
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I've searched the forums but am unable to find an answer to this:

Given two variables with a correlation, you can predict one from the other using the familiar
E(Y|X) = EY + r * s_y * (X - EX) / s_x

What I want to know is how to predict values from multiple variables, especially when these variables themselves are correlated.

E(Y | A B C) = ??
 
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You example shows computation for the expected value of a random variable, but you are using the word "predict" to phrase your question. Are you trying to "predict" the value of a random variable Y given the values of other random variables? Or is your goal to compute the expected value of Y give the distribution functions for other random variables?
 
Stephen Tashi said:
You example shows computation for the expected value of a random variable, but you are using the word "predict" to phrase your question. Are you trying to "predict" the value of a random variable Y given the values of other random variables? Or is your goal to compute the expected value of Y give the distribution functions for other random variables?

You are correct, I am looking to calculate the expected value of Y given A, B, C and known correlations YA, YB, YC, AB, AC, BC (and necessary variances, etc...)
 
Soveraign said:
E(Y|X) = EY + r * s_y * (X - EX) / s_x

I've only seen that formula applied to random variables that have a joint bivariate normal distribution. Are you assuming all the random variables in your question have a joint multinormal distribution?
 
Stephen Tashi said:
I've only seen that formula applied to random variables that have a joint bivariate normal distribution. Are you assuming all the random variables in your question have a joint multinormal distribution?

If I understand the definition correctly, then I think so. Y, A, B, C are normally distributed about a mean, but not necessarily independent (i.e. covariance != 0).

A thought I had was to perform principle component analysis on A, B, C so I then would have some new (independent) eigenvectors to work with. Perhaps then I could do multiple regression with my new A', B', C' working out an n-1 dimensional "plane" through my n space, thus working out E(Y|A', B', C')?

But I assume this is a solved problem and I'm just not looking in the right places.
 
Soveraign said:
But I assume this is a solved problem and I'm just not looking in the right places.

I looked too. I think this page (in the section called "The Multivariate Normal Distribution") gives the answer, but I haven't deciphered all the matrix notation.

As I recall, the fact that the marginal distributions are normal does not guarantee that the joint distribution is a multivariate normal. So you need to examine this assumption.
 
I was reading a Bachelor thesis on Peano Arithmetic (PA). PA has the following axioms (not including the induction schema): $$\begin{align} & (A1) ~~~~ \forall x \neg (x + 1 = 0) \nonumber \\ & (A2) ~~~~ \forall xy (x + 1 =y + 1 \to x = y) \nonumber \\ & (A3) ~~~~ \forall x (x + 0 = x) \nonumber \\ & (A4) ~~~~ \forall xy (x + (y +1) = (x + y ) + 1) \nonumber \\ & (A5) ~~~~ \forall x (x \cdot 0 = 0) \nonumber \\ & (A6) ~~~~ \forall xy (x \cdot (y + 1) = (x \cdot y) + x) \nonumber...
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