Читать книгу Artificial Intelligence and Quantum Computing for Advanced Wireless Networks - Savo G. Glisic - Страница 15

Regression analysis deals with the problem of fitting straight lines to patterns of data. In a linear regression model, the variable of interest (the so‐called “dependent” variable) is predicted from k other variables (the so‐called “independent” variables) using a linear equation. If Y denotes the dependent variable and X₁, …, X_k, are the independent variables, then the assumption is that the value of Y at time t is determined by the linear equation

(2.1)

The corresponding equation for predicting Y_t from the corresponding values of the X’s is therefore

(2.2)

where the b’s are estimates of the betas obtained by least squares, that is, minimizing the squared prediction error within the sample. This is about the simplest possible model for predicting one variable from a group of others, and it rests on the assumption that the expected value of Y is a linear function of the X variables. More precisely, the following is assumed:

1 The expected value of Y is a linear function of the X variables. This means: (i) If Xi changes by an amount ∆Xi, holding other variables fixed, then the expected value of Y changes by a proportional amount βi∆Xi, for some constant βi (which in general could be a positive or negative number). (ii) The value of βi is always the same, regardless of values of the other X’s. (iii) The total effect of the X’s on the expected value of Y is the sum of their separate effects.

2 The unexplained variations of Y are independent random variables (in particular, not “autocorrelated” if the variables are time series).

3 They all have the same variance (“homoscedasticity”).

4 They are normally distributed.

These assumptions will never be exactly satisfied by real data, but you hope that they are not badly wrong. For proper regression modeling, we need to collect data that are relevant and informative with respect to our decision problem, and then define the variables and construct the model in such a way that the assumptions listed above are plausible, at least as a first‐order approximation to reality.

If we normalize the values of Y and X as

with the correlation function defined as

the phenomenon that Galton noted was that the regression line for predicting Y* from X* passes through the origin and has a slope equal to the correlation between Y and X; that is, the regression equation in normalized units is

Figures 2.1 and 2.2 illustrate this equation [1]. When the units of X and Y are standardized and both are also normally distributed, their values are distributed in an elliptical pattern that is symmetric around the 45° line, which has a slope equal to 1.

However, the regression line for predicting Y* from X* is not the 45° line. Rather, it is a line passing through the origin whose slope is r_XY, the dashed gray line in the picture below, which is tilted toward the horizontal because the correlation is less than 1 in magnitude. In other words, it is a line that “regresses” (i.e. moves backward) toward the X‐axis.