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The Regression Analysis
The figure shows a scatter diagram showing the relationship between two variables X & Y. It is obvious that the two variables have some relationship as the higher values of X are associated with higher values of Y, however it is also visible that the relationship cannot be modeled by a deterministic function of X. |
Therefore we assume that the relationship between X and Y is sum of deterministic and stochastic relationship which can be written as follows: yi=α+βxi+εi Where α+βxi is deterministic andεi is stochastic part. The deterministic part we hope to be able to predict, however the stochastic part cannot be predicted. |
Obviously, it is desirable to have minimum of the stochastic i.e. unpredictable part. If we ignore the stochastic part we get: α+βxi=y This is equation of straight line. This straight lines joins the values on X axis with corresponding predictions on Y axis. We want such a straight line which minimizes the unpredicted part.
Instead of focusing on any one εi we want to minimize the aggregate error. This is a problem which should be solvable by calculus (differentiation].
Let us see how it can be done.
Minimizing aggregate error:
We have yi-α-βxi=εi
There are three options to minimize the aggregate error:
Option I: minimize εi
This option is not appropriate because εi=yi-yi Here yi is the prediction of yi lying on the straight line. The actual yi are on both sides of straight line. If yi is above straight line => actual value of yi is greater than yi => εi is positive. On the other hand, if yi is below the straight line => actual value of yi is smaller than yi => εi is negative. These negative and positive values cancel each other and total result we get is zero.
Consider a small data of only two points as shown in fig. These two points show that X & Y have positive

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