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doc/src/week34/week34.do.txt

@@ -1553,13 +1553,13 @@ Similarly, "Mehta et al's article":"https://arxiv.org/abs/1803.08823" is also re
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 Regression modeling deals with the description of  the sampling distribution of a given random variable $y$ and how it varies as function of another variable or a set of such variables $\bm{x} =[x_0, x_1,\dots, x_{n-1}]^T$. 
-The first variable is called the _dependent_, the _outcome_ or the _response_ variable while the set of variables $\bm{x}$ is called the independent variable, or the predictor variable or the explanatory variable. 
+The first variable is called the _dependent_, the _outcome_ or the _response_ variable while the set of variables $\bm{x}$ is called the independent variable, or the predictor variable or the explanatory variable, or simply just the _inputs_. 
  
-A regression model aims at finding a likelihood function $p(\bm{y}\vert \bm{x})$, that is the conditional distribution for $\bm{y}$ with a given $\bm{x}$. The estimation of  $p(\bm{y}\vert \bm{x})$ is made using a data set with 
+A regression model aims at finding a likelihood function $p(\bm{y}\vert \bm{x})$ or in the more traditional sense a function $\bm{y}(\bm{x})$, that is the conditional distribution for $\bm{y}$ with a given $\bm{x}$. The estimation of  $p(\bm{y}\vert \bm{x})$ is made using a data set with 
 * $n$ cases $i = 0, 1, 2, \dots, n-1$ 
 * Response (target, dependent or outcome) variable $y_i$ with $i = 0, 1, 2, \dots, n-1$ 
-* $p$ so-called explanatory (independent or predictor) variables $\bm{x}_i=[x_{i0}, x_{i1}, \dots, x_{ip-1}]$ with $i = 0, 1, 2, \dots, n-1$ and explanatory variables running from $0$ to $p-1$. See below for more explicit examples.   
- The goal of the regression analysis is to extract/exploit relationship between $\bm{y}$ and $\bm{x}$ in or to infer causal dependencies, approximations to the likelihood functions, functional relationships and to make predictions, making fits and many other things.
+* $p$ so-called explanatory (independent or predictor or feature) variables $\bm{x}_i=[x_{i0}, x_{i1}, \dots, x_{ip-1}]$ with $i = 0, 1, 2, \dots, n-1$ and explanatory variables running from $0$ to $p-1$. See below for more explicit examples.   
+ The goal of the regression analysis is to extract/exploit relationship between $\bm{y}$ and $\bm{x}$ in order to infer specific dependencies, approximations to the likelihood functions, functional relationships and to make predictions, making fits and many other things.
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@@ -1567,7 +1567,7 @@ A regression model aims at finding a likelihood function $p(\bm{y}\vert \bm{x})$
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-Consider an experiment in which $p$ characteristics of $n$ samples are
+Consider an experiment in which $p$ characteristics/features of $n$ samples are
 measured. The data from this experiment, for various explanatory variables $p$ are normally represented by a matrix  
 $\mathbf{X}$.
 
@@ -1616,9 +1616,9 @@ so-called "credit card default data from Taiwan":"https://www.sciencedirect.com/
 
 
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-===== General linear models  =====
+===== General linear models and linear algebra =====
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-Before we proceed let us study a case from linear algebra where we aim at fitting a set of data $\bm{y}=[y_0,y_1,\dots,y_{n-1}]$. We could think of these data as a result of an experiment or a complicated numerical experiment. These data are functions of a series of variables $\bm{x}=[x_0,x_1,\dots,x_{n-1}]$, that is $y_i = y(x_i)$ with $i=0,1,2,\dots,n-1$. The variables $x_i$ could represent physical quantities like time, temperature, position etc. We assume that $y(x)$ is a smooth function. 
+Before we proceed let us study a case where we aim at fitting a set of data $\bm{y}=[y_0,y_1,\dots,y_{n-1}]$. We could think of these data as a result of an experiment or a complicated numerical experiment. These data are functions of a series of variables $\bm{x}=[x_0,x_1,\dots,x_{n-1}]$, that is $y_i = y(x_i)$ with $i=0,1,2,\dots,n-1$. The variables $x_i$ could represent physical quantities like time, temperature, position etc. We assume that $y(x)$ is a smooth function. 
 
 Since obtaining these data points may not be trivial, we want to use these data to fit a function which can allow us to make predictions for values of $y$ which are not in the present set. The perhaps simplest approach is to assume we can parametrize our function in terms of a polynomial of degree $n-1$ with $n$ points, that is
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