Abstract (15.0: Least squares method and Regression analysis )

Abstract: Although regression analysis has a great history, we consider that it has always continued being confused. For example, the fundamental terms in regression analysis (e.g., "regression", "least-squares method", "explanatory variable", "response variable", etc.) seem to be historically conventional, that is, these words do not express the essence of regression analysis. In this chapter, we show that the least squares method acquires a quantum linguistic story as follows. \begin{align} & \underset{\mbox{(Section 15.1)}}{\fbox{The least squares method}} \xrightarrow[\mbox{quantum language}]{\mbox{describe by}} \underset{\mbox{(Section 15.2)}}{\fbox{Regression analysis}} \nonumber \\ & \qquad \qquad \qquad \xrightarrow[\mbox{generalization}]{\mbox{natural}} \underset{\mbox{(Section 15.4)}}{\fbox{Generalized linear model}} \tag{$\sharp$} \end{align}

In this story, the terms "explanatory variable" and "response variable" are clarified in terms of quantum language. As the general theory of regression analysis, it suffices to devote ourselves to Theorem 13.4. However, from the practical point of view, we have to add the above story $(\sharp)$

Again recall that, as mentioned in $\S$1.1, the main purpose of this book is to assert the following figure 1.1:
Fig.1.1: the location of "quantum language" in the world-views
This(particularly, ⑦--⑨) implies that quantum language has the following three aspects: $$ \left\{\begin{array}{l} \mbox{ ⑦ :the standard interpretation of quantum mechanics} \\ \mbox{ $\qquad$ (i.e., the true colors of the Copenhagen interpretation) } \\ \\ \mbox{ ⑧ : the final goal of the dualistic idealism (Descartes=Kant philosophy) } \\ \\ \mbox{ ⑨ : theoretical statistics of the future } \end{array}\right. $$