Theory of Regression Lines and Associated Linear Systems and Channels

I have summarized the results of a theory regarding linear regression, previously scattered in several articles, with several applications to real-world data. The linear relationship between a dependent variable $y$ and an independent variable $x$ is interpretable as describing the input-output characteristic of a linear system, or a linear channel. The conviction that the theory is applicable to any scientific discipline—provided a linear input-output relationship exists—has led me to summarize the main results in a general form easily applicable to any experimental problem. Defining the regression noise-to-signal power ratio (NSR) $R_m$ and the correlation noise-to-signal power ratio, $R_r$ , the channel noise-to-signal ratio $R$ is the sum of the two. A direct and insightful analysis is achieved by using the NSR, instead of the reciprocal signal-to-noise ratio (SNR), because it makes easier to study which addend determines $R$ and allows a useful graphical representation. The SNR can be considered an indicator of the degree of deterministic nature of a channel, because the higher it is, the greater the “degree” of the channel’s deterministic nature. I have considered various connections of single channels (single, cross, deterministic, parallel, series) that are likely found in applications. In conclusion, the theory and its mathematical results concerning the input-output characteristics of linear channels, described by experimental regression lines, should be useful in any scientific discipline.