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 \( \boldsymbol{x} =[x_0, x_1,\dots, x_{n-1}]^T \). -The first variable is called the dependent, the outcome or the response variable while the set of variables \( \boldsymbol{x} \) 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 \( \boldsymbol{x} \) 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 \( p(\boldsymbol{y}\vert \boldsymbol{x}) \), that is the conditional distribution for \( \boldsymbol{y} \) with a given \( \boldsymbol{x} \). The estimation of \( p(\boldsymbol{y}\vert \boldsymbol{x}) \) is made using a data set with
+A regression model aims at finding a likelihood function \( p(\boldsymbol{y}\vert \boldsymbol{x}) \) or in the more traditional sense a function \( \boldsymbol{y}(\boldsymbol{x}) \), that is the conditional distribution for \( \boldsymbol{y} \) with a given \( \boldsymbol{x} \). The estimation of \( p(\boldsymbol{y}\vert \boldsymbol{x}) \) is made using a data set with
The goal of the regression analysis is to extract/exploit relationship between \( \boldsymbol{y} \) and \( \boldsymbol{x} \) in or to infer causal dependencies, approximations to the likelihood functions, functional relationships and to make predictions, making fits and many other things.
+The goal of the regression analysis is to extract/exploit relationship between \( \boldsymbol{y} \) and \( \boldsymbol{x} \) in order to infer specific dependencies, approximations to the likelihood functions, functional relationships and to make predictions, making fits and many other things.
diff --git a/doc/pub/week34/html/._week34-bs040.html b/doc/pub/week34/html/._week34-bs040.html index ac6bf4af6..409f7c936 100644 --- a/doc/pub/week34/html/._week34-bs040.html +++ b/doc/pub/week34/html/._week34-bs040.html @@ -155,7 +155,10 @@ doconce format html week34.do.txt --html_style=bootstrap --pygments_html_style=d None, 'regression-analysis-overarching-aims-ii'), ('Examples', 2, None, 'examples'), - ('General linear models', 2, None, 'general-linear-models'), + ('General linear models and linear algebra', + 2, + None, + 'general-linear-models-and-linear-algebra'), ('Rewriting the fitting procedure as a linear algebra problem', 2, None, @@ -316,7 +319,7 @@ MathJax.Hub.Config({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 \( \mathbf{X} \).
diff --git a/doc/pub/week34/html/._week34-bs041.html b/doc/pub/week34/html/._week34-bs041.html index 53f3de582..2fba55f2c 100644 --- a/doc/pub/week34/html/._week34-bs041.html +++ b/doc/pub/week34/html/._week34-bs041.html @@ -155,7 +155,10 @@ doconce format html week34.do.txt --html_style=bootstrap --pygments_html_style=d None, 'regression-analysis-overarching-aims-ii'), ('Examples', 2, None, 'examples'), - ('General linear models', 2, None, 'general-linear-models'), + ('General linear models and linear algebra', + 2, + None, + 'general-linear-models-and-linear-algebra'), ('Rewriting the fitting procedure as a linear algebra problem', 2, None, @@ -316,7 +319,7 @@ MathJax.Hub.Config({
-
Before we proceed let us study a case from linear algebra where we aim at fitting a set of data \( \boldsymbol{y}=[y_0,y_1,\dots,y_{n-1}] \). 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 \( \boldsymbol{x}=[x_0,x_1,\dots,x_{n-1}] \), 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 \( \boldsymbol{y}=[y_0,y_1,\dots,y_{n-1}] \). 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 \( \boldsymbol{x}=[x_0,x_1,\dots,x_{n-1}] \), 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
$$ diff --git a/doc/pub/week34/html/._week34-bs043.html b/doc/pub/week34/html/._week34-bs043.html index 8c533ee7a..0e8acfbab 100644 --- a/doc/pub/week34/html/._week34-bs043.html +++ b/doc/pub/week34/html/._week34-bs043.html @@ -155,7 +155,10 @@ doconce format html week34.do.txt --html_style=bootstrap --pygments_html_style=d None, 'regression-analysis-overarching-aims-ii'), ('Examples', 2, None, 'examples'), - ('General linear models', 2, None, 'general-linear-models'), + ('General linear models and linear algebra', + 2, + None, + 'general-linear-models-and-linear-algebra'), ('Rewriting the fitting procedure as a linear algebra problem', 2, None, @@ -316,7 +319,7 @@ MathJax.Hub.Config({
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 \( \boldsymbol{x} =[x_0, x_1,\dots, x_{n-1}]^T \). -The first variable is called the dependent, the outcome or the response variable while the set of variables \( \boldsymbol{x} \) 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 \( \boldsymbol{x} \) 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 \( p(\boldsymbol{y}\vert \boldsymbol{x}) \), that is the conditional distribution for \( \boldsymbol{y} \) with a given \( \boldsymbol{x} \). The estimation of \( p(\boldsymbol{y}\vert \boldsymbol{x}) \) is made using a data set with
+A regression model aims at finding a likelihood function \( p(\boldsymbol{y}\vert \boldsymbol{x}) \) or in the more traditional sense a function \( \boldsymbol{y}(\boldsymbol{x}) \), that is the conditional distribution for \( \boldsymbol{y} \) with a given \( \boldsymbol{x} \). The estimation of \( p(\boldsymbol{y}\vert \boldsymbol{x}) \) is made using a data set with
-
The goal of the regression analysis is to extract/exploit relationship between \( \boldsymbol{y} \) and \( \boldsymbol{x} \) in or to infer causal dependencies, approximations to the likelihood functions, functional relationships and to make predictions, making fits and many other things.
+The goal of the regression analysis is to extract/exploit relationship between \( \boldsymbol{y} \) and \( \boldsymbol{x} \) in order to infer specific dependencies, approximations to the likelihood functions, functional relationships and to make predictions, making fits and many other things.
-
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 \( \mathbf{X} \).
@@ -2663,11 +2663,11 @@ so-called General linear models +-
Before we proceed let us study a case from linear algebra where we aim at fitting a set of data \( \boldsymbol{y}=[y_0,y_1,\dots,y_{n-1}] \). 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 \( \boldsymbol{x}=[x_0,x_1,\dots,x_{n-1}] \), 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 \( \boldsymbol{y}=[y_0,y_1,\dots,y_{n-1}] \). 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 \( \boldsymbol{x}=[x_0,x_1,\dots,x_{n-1}] \), 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
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 \( \boldsymbol{x} =[x_0, x_1,\dots, x_{n-1}]^T \).
-The first variable is called the dependent, the outcome or the response variable while the set of variables \( \boldsymbol{x} \) 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 \( \boldsymbol{x} \) 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 \( p(\boldsymbol{y}\vert \boldsymbol{x}) \), that is the conditional distribution for \( \boldsymbol{y} \) with a given \( \boldsymbol{x} \). The estimation of \( p(\boldsymbol{y}\vert \boldsymbol{x}) \) is made using a data set with A regression model aims at finding a likelihood function \( p(\boldsymbol{y}\vert \boldsymbol{x}) \) or in the more traditional sense a function \( \boldsymbol{y}(\boldsymbol{x}) \), that is the conditional distribution for \( \boldsymbol{y} \) with a given \( \boldsymbol{x} \). The estimation of \( p(\boldsymbol{y}\vert \boldsymbol{x}) \) is made using a data set with The goal of the regression analysis is to extract/exploit relationship between \( \boldsymbol{y} \) and \( \boldsymbol{x} \) in or to infer causal dependencies, approximations to the likelihood functions, functional relationships and to make predictions, making fits and many other things. The goal of the regression analysis is to extract/exploit relationship between \( \boldsymbol{y} \) and \( \boldsymbol{x} \) in order to infer specific dependencies, approximations to the likelihood functions, functional relationships and to make predictions, making fits and many other things.
diff --git a/doc/pub/week34/html/week34-solarized.html b/doc/pub/week34/html/week34-solarized.html
index 55c7c83d6..426fedb11 100644
--- a/doc/pub/week34/html/week34-solarized.html
+++ b/doc/pub/week34/html/week34-solarized.html
@@ -182,7 +182,10 @@ div.toc p,a {
None,
'regression-analysis-overarching-aims-ii'),
('Examples', 2, None, 'examples'),
- ('General linear models', 2, None, 'general-linear-models'),
+ ('General linear models and linear algebra',
+ 2,
+ None,
+ 'general-linear-models-and-linear-algebra'),
('Rewriting the fitting procedure as a linear algebra problem',
2,
None,
@@ -2570,16 +2573,16 @@ Similarly, Mehta et a
-
-
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 \( \mathbf{X} \).
@@ -2639,11 +2642,11 @@ so-called General linear models +-
Before we proceed let us study a case from linear algebra where we aim at fitting a set of data \( \boldsymbol{y}=[y_0,y_1,\dots,y_{n-1}] \). 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 \( \boldsymbol{x}=[x_0,x_1,\dots,x_{n-1}] \), 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 \( \boldsymbol{y}=[y_0,y_1,\dots,y_{n-1}] \). 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 \( \boldsymbol{x}=[x_0,x_1,\dots,x_{n-1}] \), 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
$$ diff --git a/doc/pub/week34/html/week34.html b/doc/pub/week34/html/week34.html index f343b6147..03e53dfdc 100644 --- a/doc/pub/week34/html/week34.html +++ b/doc/pub/week34/html/week34.html @@ -259,7 +259,10 @@ div.toc p,a { None, 'regression-analysis-overarching-aims-ii'), ('Examples', 2, None, 'examples'), - ('General linear models', 2, None, 'general-linear-models'), + ('General linear models and linear algebra', + 2, + None, + 'general-linear-models-and-linear-algebra'), ('Rewriting the fitting procedure as a linear algebra problem', 2, None, @@ -2647,16 +2650,16 @@ Similarly, Mehta et a
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 \( \boldsymbol{x} =[x_0, x_1,\dots, x_{n-1}]^T \). -The first variable is called the dependent, the outcome or the response variable while the set of variables \( \boldsymbol{x} \) 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 \( \boldsymbol{x} \) 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 \( p(\boldsymbol{y}\vert \boldsymbol{x}) \), that is the conditional distribution for \( \boldsymbol{y} \) with a given \( \boldsymbol{x} \). The estimation of \( p(\boldsymbol{y}\vert \boldsymbol{x}) \) is made using a data set with
+A regression model aims at finding a likelihood function \( p(\boldsymbol{y}\vert \boldsymbol{x}) \) or in the more traditional sense a function \( \boldsymbol{y}(\boldsymbol{x}) \), that is the conditional distribution for \( \boldsymbol{y} \) with a given \( \boldsymbol{x} \). The estimation of \( p(\boldsymbol{y}\vert \boldsymbol{x}) \) is made using a data set with
The goal of the regression analysis is to extract/exploit relationship between \( \boldsymbol{y} \) and \( \boldsymbol{x} \) in or to infer causal dependencies, approximations to the likelihood functions, functional relationships and to make predictions, making fits and many other things.
+The goal of the regression analysis is to extract/exploit relationship between \( \boldsymbol{y} \) and \( \boldsymbol{x} \) in order to infer specific dependencies, approximations to the likelihood functions, functional relationships and to make predictions, making fits and many other things.
-
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 \( \mathbf{X} \).
@@ -2716,11 +2719,11 @@ so-called General linear models +-
Before we proceed let us study a case from linear algebra where we aim at fitting a set of data \( \boldsymbol{y}=[y_0,y_1,\dots,y_{n-1}] \). 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 \( \boldsymbol{x}=[x_0,x_1,\dots,x_{n-1}] \), 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 \( \boldsymbol{y}=[y_0,y_1,\dots,y_{n-1}] \). 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 \( \boldsymbol{x}=[x_0,x_1,\dots,x_{n-1}] \), 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
$$ diff --git a/doc/pub/week34/ipynb/ipynb-week34-src.tar.gz b/doc/pub/week34/ipynb/ipynb-week34-src.tar.gz index d4fa5e8dd..8ee339b28 100644 Binary files a/doc/pub/week34/ipynb/ipynb-week34-src.tar.gz and b/doc/pub/week34/ipynb/ipynb-week34-src.tar.gz differ diff --git a/doc/pub/week34/ipynb/week34.ipynb b/doc/pub/week34/ipynb/week34.ipynb index 165ea68e0..f76c4c40e 100644 --- a/doc/pub/week34/ipynb/week34.ipynb +++ b/doc/pub/week34/ipynb/week34.ipynb @@ -2,7 +2,7 @@ "cells": [ { "cell_type": "markdown", - "id": "13e4dc19", + "id": "5fea11f8", "metadata": { "editable": true }, @@ -14,7 +14,7 @@ }, { "cell_type": "markdown", - "id": "8cf0e82d", + "id": "74c076d9", "metadata": { "editable": true }, @@ -27,7 +27,7 @@ }, { "cell_type": "markdown", - "id": "83d2a9f7", + "id": "6e318d36", "metadata": { "editable": true }, @@ -51,7 +51,7 @@ }, { "cell_type": "markdown", - "id": "16b14357", + "id": "dc22f763", "metadata": { "editable": true }, @@ -67,7 +67,7 @@ }, { "cell_type": "markdown", - "id": "4ad19e95", + "id": "3e1eeb38", "metadata": { "editable": true }, @@ -91,7 +91,7 @@ }, { "cell_type": "markdown", - "id": "26223efa", + "id": "edc30d48", "metadata": { "editable": true }, @@ -105,7 +105,7 @@ }, { "cell_type": "markdown", - "id": "c8ab147f", + "id": "2fb4e114", "metadata": { "editable": true }, @@ -127,7 +127,7 @@ }, { "cell_type": "markdown", - "id": "ecea0b07", + "id": "37fe4be6", "metadata": { "editable": true }, @@ -157,7 +157,7 @@ }, { "cell_type": "markdown", - "id": "3566a03e", + "id": "facf99a0", "metadata": { "editable": true }, @@ -175,7 +175,7 @@ }, { "cell_type": "markdown", - "id": "2083f434", + "id": "5737d42c", "metadata": { "editable": true }, @@ -189,7 +189,7 @@ }, { "cell_type": "markdown", - "id": "c7566646", + "id": "f62fc9b0", "metadata": { "editable": true }, @@ -209,7 +209,7 @@ }, { "cell_type": "markdown", - "id": "773a106a", + "id": "36d93f21", "metadata": { "editable": true }, @@ -221,7 +221,7 @@ }, { "cell_type": "markdown", - "id": "e2a072b9", + "id": "a0d7f707", "metadata": { "editable": true }, @@ -241,7 +241,7 @@ }, { "cell_type": "markdown", - "id": "44ddceb6", + "id": "147b7186", "metadata": { "editable": true }, @@ -259,7 +259,7 @@ }, { "cell_type": "markdown", - "id": "2552abd3", + "id": "9cb9f456", "metadata": { "editable": true }, @@ -284,7 +284,7 @@ }, { "cell_type": "markdown", - "id": "4af5e853", + "id": "2fc8fc95", "metadata": { "editable": true }, @@ -320,7 +320,7 @@ }, { "cell_type": "markdown", - "id": "17d334e6", + "id": "db17a873", "metadata": { "editable": true }, @@ -336,7 +336,7 @@ }, { "cell_type": "markdown", - "id": "3e0a0db6", + "id": "76c4fd7b", "metadata": { "editable": true }, @@ -358,7 +358,7 @@ }, { "cell_type": "markdown", - "id": "08a6b7dd", + "id": "b0487b17", "metadata": { "editable": true }, @@ -376,7 +376,7 @@ }, { "cell_type": "markdown", - "id": "af8219ed", + "id": "1c8e6d7a", "metadata": { "editable": true }, @@ -420,7 +420,7 @@ }, { "cell_type": "markdown", - "id": "19840fa1", + "id": "ba570612", "metadata": { "editable": true }, @@ -463,7 +463,7 @@ }, { "cell_type": "markdown", - "id": "c584b1b3", + "id": "c865e519", "metadata": { "editable": true }, @@ -483,7 +483,7 @@ }, { "cell_type": "markdown", - "id": "21fe8057", + "id": "f3e1fb8a", "metadata": { "editable": true }, @@ -556,7 +556,7 @@ }, { "cell_type": "markdown", - "id": "187046e7", + "id": "e7ebcd32", "metadata": { "editable": true }, @@ -585,7 +585,7 @@ }, { "cell_type": "markdown", - "id": "a871f439", + "id": "ebc932ed", "metadata": { "editable": true }, @@ -603,7 +603,7 @@ }, { "cell_type": "markdown", - "id": "c4e55931", + "id": "58f61af4", "metadata": { "editable": true }, @@ -615,7 +615,7 @@ }, { "cell_type": "markdown", - "id": "1c3112c6", + "id": "c8c94169", "metadata": { "editable": true }, @@ -650,7 +650,7 @@ }, { "cell_type": "markdown", - "id": "f5e967c7", + "id": "bf117e84", "metadata": { "editable": true }, @@ -680,7 +680,7 @@ }, { "cell_type": "markdown", - "id": "c1e1f63b", + "id": "367496b3", "metadata": { "editable": true }, @@ -711,7 +711,7 @@ }, { "cell_type": "markdown", - "id": "55f45375", + "id": "1cf4bb34", "metadata": { "editable": true }, @@ -750,7 +750,7 @@ }, { "cell_type": "markdown", - "id": "8d0c1d60", + "id": "ed5a4ac5", "metadata": { "editable": true }, @@ -783,7 +783,7 @@ }, { "cell_type": "markdown", - "id": "2d31f4f0", + "id": "916f7b88", "metadata": { "editable": true }, @@ -818,7 +818,7 @@ }, { "cell_type": "markdown", - "id": "655c6cd0", + "id": "ad9a3913", "metadata": { "editable": true }, @@ -842,7 +842,7 @@ }, { "cell_type": "markdown", - "id": "2181bae1", + "id": "a01dabc0", "metadata": { "editable": true }, @@ -869,7 +869,7 @@ }, { "cell_type": "markdown", - "id": "0a842681", + "id": "4551ca08", "metadata": { "editable": true }, @@ -879,7 +879,7 @@ }, { "cell_type": "markdown", - "id": "f2f46dfc", + "id": "5b46693e", "metadata": { "editable": true }, @@ -891,7 +891,7 @@ }, { "cell_type": "markdown", - "id": "1511e979", + "id": "0ec791e7", "metadata": { "editable": true }, @@ -912,7 +912,7 @@ }, { "cell_type": "markdown", - "id": "155a5117", + "id": "1676ba00", "metadata": { "editable": true }, @@ -924,7 +924,7 @@ { "cell_type": "code", "execution_count": 1, - "id": "a62df053", + "id": "984f2d67", "metadata": { "collapsed": false, "editable": true @@ -936,7 +936,7 @@ }, { "cell_type": "markdown", - "id": "d8c0e2d9", + "id": "90dc846d", "metadata": { "editable": true }, @@ -947,7 +947,7 @@ { "cell_type": "code", "execution_count": 2, - "id": "de4fc767", + "id": "05dfd736", "metadata": { "collapsed": false, "editable": true @@ -961,7 +961,7 @@ }, { "cell_type": "markdown", - "id": "3a1b5476", + "id": "46923c55", "metadata": { "editable": true }, @@ -973,7 +973,7 @@ { "cell_type": "code", "execution_count": 3, - "id": "cc752f30", + "id": "ea5024fd", "metadata": { "collapsed": false, "editable": true @@ -987,7 +987,7 @@ }, { "cell_type": "markdown", - "id": "07879742", + "id": "89be4a45", "metadata": { "editable": true }, @@ -999,7 +999,7 @@ { "cell_type": "code", "execution_count": 4, - "id": "5702e358", + "id": "49dd8890", "metadata": { "collapsed": false, "editable": true @@ -1013,7 +1013,7 @@ }, { "cell_type": "markdown", - "id": "d906520a", + "id": "dc6f39e5", "metadata": { "editable": true }, @@ -1030,7 +1030,7 @@ { "cell_type": "code", "execution_count": 5, - "id": "126f5860", + "id": "67ab2ddc", "metadata": { "collapsed": false, "editable": true @@ -1047,7 +1047,7 @@ }, { "cell_type": "markdown", - "id": "d14718ab", + "id": "d3ad1870", "metadata": { "editable": true }, @@ -1059,7 +1059,7 @@ { "cell_type": "code", "execution_count": 6, - "id": "b3643674", + "id": "82d69255", "metadata": { "collapsed": false, "editable": true @@ -1073,7 +1073,7 @@ }, { "cell_type": "markdown", - "id": "caa7a368", + "id": "44b00fa6", "metadata": { "editable": true }, @@ -1084,7 +1084,7 @@ { "cell_type": "code", "execution_count": 7, - "id": "0ae5832d", + "id": "465584fa", "metadata": { "collapsed": false, "editable": true @@ -1098,7 +1098,7 @@ }, { "cell_type": "markdown", - "id": "dc90824d", + "id": "5dd71446", "metadata": { "editable": true }, @@ -1109,7 +1109,7 @@ { "cell_type": "code", "execution_count": 8, - "id": "02fd5e51", + "id": "065040e7", "metadata": { "collapsed": false, "editable": true @@ -1123,7 +1123,7 @@ }, { "cell_type": "markdown", - "id": "9cc3d81d", + "id": "860e3937", "metadata": { "editable": true }, @@ -1138,7 +1138,7 @@ { "cell_type": "code", "execution_count": 9, - "id": "13730b22", + "id": "f2c32ae7", "metadata": { "collapsed": false, "editable": true @@ -1152,7 +1152,7 @@ }, { "cell_type": "markdown", - "id": "e19beccd", + "id": "99ab0db1", "metadata": { "editable": true }, @@ -1163,7 +1163,7 @@ { "cell_type": "code", "execution_count": 10, - "id": "ad0c506f", + "id": "5a2bb9d8", "metadata": { "collapsed": false, "editable": true @@ -1178,7 +1178,7 @@ }, { "cell_type": "markdown", - "id": "5f7e9328", + "id": "16e71369", "metadata": { "editable": true }, @@ -1189,7 +1189,7 @@ { "cell_type": "code", "execution_count": 11, - "id": "cc6790e5", + "id": "b0996873", "metadata": { "collapsed": false, "editable": true @@ -1204,7 +1204,7 @@ }, { "cell_type": "markdown", - "id": "a399fb61", + "id": "18524cbd", "metadata": { "editable": true }, @@ -1215,7 +1215,7 @@ { "cell_type": "code", "execution_count": 12, - "id": "c57dea61", + "id": "2e285fb0", "metadata": { "collapsed": false, "editable": true @@ -1231,7 +1231,7 @@ }, { "cell_type": "markdown", - "id": "12283acc", + "id": "6de8a151", "metadata": { "editable": true }, @@ -1242,7 +1242,7 @@ { "cell_type": "code", "execution_count": 13, - "id": "ef5dd9ca", + "id": "3c9a5073", "metadata": { "collapsed": false, "editable": true @@ -1258,7 +1258,7 @@ }, { "cell_type": "markdown", - "id": "657698e4", + "id": "dcc6c32e", "metadata": { "editable": true }, @@ -1269,7 +1269,7 @@ { "cell_type": "code", "execution_count": 14, - "id": "68893311", + "id": "79854096", "metadata": { "collapsed": false, "editable": true @@ -1285,7 +1285,7 @@ }, { "cell_type": "markdown", - "id": "97339ea6", + "id": "32c14a22", "metadata": { "editable": true }, @@ -1297,7 +1297,7 @@ }, { "cell_type": "markdown", - "id": "55b04e16", + "id": "62346cc6", "metadata": { "editable": true }, @@ -1312,7 +1312,7 @@ }, { "cell_type": "markdown", - "id": "54fafab5", + "id": "76492d25", "metadata": { "editable": true }, @@ -1322,7 +1322,7 @@ }, { "cell_type": "markdown", - "id": "96067385", + "id": "b1142258", "metadata": { "editable": true }, @@ -1334,7 +1334,7 @@ }, { "cell_type": "markdown", - "id": "478ac609", + "id": "8647c1bf", "metadata": { "editable": true }, @@ -1345,7 +1345,7 @@ }, { "cell_type": "markdown", - "id": "73b1c7e9", + "id": "34105c78", "metadata": { "editable": true }, @@ -1360,7 +1360,7 @@ }, { "cell_type": "markdown", - "id": "eb8aefd8", + "id": "fb66c986", "metadata": { "editable": true }, @@ -1375,7 +1375,7 @@ { "cell_type": "code", "execution_count": 15, - "id": "11f952f0", + "id": "af2f7c0e", "metadata": { "collapsed": false, "editable": true @@ -1402,7 +1402,7 @@ { "cell_type": "code", "execution_count": 16, - "id": "77d1ec9b", + "id": "9f715559", "metadata": { "collapsed": false, "editable": true @@ -1426,7 +1426,7 @@ }, { "cell_type": "markdown", - "id": "27e94f61", + "id": "b8f5afdf", "metadata": { "editable": true }, @@ -1452,7 +1452,7 @@ { "cell_type": "code", "execution_count": 17, - "id": "3d4016b7", + "id": "8937ddb8", "metadata": { "collapsed": false, "editable": true @@ -1472,7 +1472,7 @@ }, { "cell_type": "markdown", - "id": "3ce5543c", + "id": "5785f86c", "metadata": { "editable": true }, @@ -1486,7 +1486,7 @@ { "cell_type": "code", "execution_count": 18, - "id": "fc83838c", + "id": "fdfd0e65", "metadata": { "collapsed": false, "editable": true @@ -1499,7 +1499,7 @@ }, { "cell_type": "markdown", - "id": "c1101885", + "id": "76206b90", "metadata": { "editable": true }, @@ -1510,7 +1510,7 @@ { "cell_type": "code", "execution_count": 19, - "id": "0adf83b9", + "id": "8fed0b92", "metadata": { "collapsed": false, "editable": true @@ -1522,7 +1522,7 @@ }, { "cell_type": "markdown", - "id": "0273453d", + "id": "764440ce", "metadata": { "editable": true }, @@ -1533,7 +1533,7 @@ { "cell_type": "code", "execution_count": 20, - "id": "c6667fcc", + "id": "b365e3d5", "metadata": { "collapsed": false, "editable": true @@ -1551,7 +1551,7 @@ }, { "cell_type": "markdown", - "id": "fd2a6345", + "id": "a18dbb39", "metadata": { "editable": true }, @@ -1563,7 +1563,7 @@ { "cell_type": "code", "execution_count": 21, - "id": "391489c8", + "id": "a2d23f69", "metadata": { "collapsed": false, "editable": true @@ -1587,7 +1587,7 @@ }, { "cell_type": "markdown", - "id": "669e19e7", + "id": "aa2ebba0", "metadata": { "editable": true }, @@ -1598,7 +1598,7 @@ { "cell_type": "code", "execution_count": 22, - "id": "7f45c0a1", + "id": "2ae06134", "metadata": { "collapsed": false, "editable": true @@ -1627,7 +1627,7 @@ }, { "cell_type": "markdown", - "id": "30e6ca1b", + "id": "211d39f4", "metadata": { "editable": true }, @@ -1638,7 +1638,7 @@ { "cell_type": "code", "execution_count": 23, - "id": "f74df0b7", + "id": "63c91afd", "metadata": { "collapsed": false, "editable": true @@ -1653,7 +1653,7 @@ }, { "cell_type": "markdown", - "id": "a34f4c55", + "id": "a2edfba5", "metadata": { "editable": true }, @@ -1670,7 +1670,7 @@ }, { "cell_type": "markdown", - "id": "7fdae851", + "id": "55e52d20", "metadata": { "editable": true }, @@ -1701,7 +1701,7 @@ }, { "cell_type": "markdown", - "id": "d1a7d8dc", + "id": "e376e0aa", "metadata": { "editable": true }, @@ -1713,7 +1713,7 @@ }, { "cell_type": "markdown", - "id": "046daa16", + "id": "216a5a58", "metadata": { "editable": true }, @@ -1741,7 +1741,7 @@ { "cell_type": "code", "execution_count": 24, - "id": "b411bbc3", + "id": "5e4fb206", "metadata": { "collapsed": false, "editable": true @@ -1771,7 +1771,7 @@ }, { "cell_type": "markdown", - "id": "bb84c9ee", + "id": "6106a3da", "metadata": { "editable": true }, @@ -1788,7 +1788,7 @@ }, { "cell_type": "markdown", - "id": "93d8bff9", + "id": "a26032d7", "metadata": { "editable": true }, @@ -1800,7 +1800,7 @@ }, { "cell_type": "markdown", - "id": "fc1fc072", + "id": "165f7264", "metadata": { "editable": true }, @@ -1821,7 +1821,7 @@ }, { "cell_type": "markdown", - "id": "b3874212", + "id": "44950ad4", "metadata": { "editable": true }, @@ -1834,7 +1834,7 @@ }, { "cell_type": "markdown", - "id": "64f32e9a", + "id": "972a3ce6", "metadata": { "editable": true }, @@ -1865,7 +1865,7 @@ }, { "cell_type": "markdown", - "id": "3ce7a544", + "id": "82b560a5", "metadata": { "editable": true }, @@ -1877,7 +1877,7 @@ }, { "cell_type": "markdown", - "id": "86a2d8e3", + "id": "bdbf7634", "metadata": { "editable": true }, @@ -1895,7 +1895,7 @@ { "cell_type": "code", "execution_count": 25, - "id": "99747fda", + "id": "b6093722", "metadata": { "collapsed": false, "editable": true @@ -1922,7 +1922,7 @@ }, { "cell_type": "markdown", - "id": "a44358f8", + "id": "7d39373d", "metadata": { "editable": true }, @@ -1944,7 +1944,7 @@ { "cell_type": "code", "execution_count": 26, - "id": "8cd9ade5", + "id": "9157b75c", "metadata": { "collapsed": false, "editable": true @@ -1982,7 +1982,7 @@ }, { "cell_type": "markdown", - "id": "eda7f6eb", + "id": "29cabbf7", "metadata": { "editable": true }, @@ -1993,7 +1993,7 @@ }, { "cell_type": "markdown", - "id": "4dfb373a", + "id": "6a1761fe", "metadata": { "editable": true }, @@ -2006,7 +2006,7 @@ }, { "cell_type": "markdown", - "id": "4c6071e2", + "id": "e0b06635", "metadata": { "editable": true }, @@ -2027,7 +2027,7 @@ }, { "cell_type": "markdown", - "id": "37c15c5e", + "id": "3e9c7e7b", "metadata": { "editable": true }, @@ -2039,7 +2039,7 @@ }, { "cell_type": "markdown", - "id": "98bf96ff", + "id": "88927109", "metadata": { "editable": true }, @@ -2049,7 +2049,7 @@ }, { "cell_type": "markdown", - "id": "cb94c959", + "id": "826d841b", "metadata": { "editable": true }, @@ -2061,7 +2061,7 @@ }, { "cell_type": "markdown", - "id": "cfb0846f", + "id": "d83d5e8d", "metadata": { "editable": true }, @@ -2073,7 +2073,7 @@ }, { "cell_type": "markdown", - "id": "b4ef89ab", + "id": "2202ead5", "metadata": { "editable": true }, @@ -2085,7 +2085,7 @@ }, { "cell_type": "markdown", - "id": "ce79d2b0", + "id": "b45560c9", "metadata": { "editable": true }, @@ -2096,7 +2096,7 @@ }, { "cell_type": "markdown", - "id": "0af059e0", + "id": "e00b8018", "metadata": { "editable": true }, @@ -2108,7 +2108,7 @@ }, { "cell_type": "markdown", - "id": "83e3869d", + "id": "14f161a1", "metadata": { "editable": true }, @@ -2130,7 +2130,7 @@ }, { "cell_type": "markdown", - "id": "fd921a53", + "id": "60b46420", "metadata": { "editable": true }, @@ -2142,7 +2142,7 @@ }, { "cell_type": "markdown", - "id": "d3794154", + "id": "3fd6e89a", "metadata": { "editable": true }, @@ -2155,7 +2155,7 @@ }, { "cell_type": "markdown", - "id": "519f2922", + "id": "df079e6a", "metadata": { "editable": true }, @@ -2172,7 +2172,7 @@ }, { "cell_type": "markdown", - "id": "b69829a4", + "id": "16e438b3", "metadata": { "editable": true }, @@ -2184,7 +2184,7 @@ }, { "cell_type": "markdown", - "id": "1e61c72b", + "id": "86d04aea", "metadata": { "editable": true }, @@ -2194,7 +2194,7 @@ }, { "cell_type": "markdown", - "id": "99da35d5", + "id": "f208235e", "metadata": { "editable": true }, @@ -2206,7 +2206,7 @@ }, { "cell_type": "markdown", - "id": "d1e3fc5d", + "id": "8cd1a4e3", "metadata": { "editable": true }, @@ -2216,7 +2216,7 @@ }, { "cell_type": "markdown", - "id": "aa74ead3", + "id": "0cad3f2f", "metadata": { "editable": true }, @@ -2228,7 +2228,7 @@ }, { "cell_type": "markdown", - "id": "2d30da67", + "id": "eb7aed32", "metadata": { "editable": true }, @@ -2238,7 +2238,7 @@ }, { "cell_type": "markdown", - "id": "e5795983", + "id": "0ee263af", "metadata": { "editable": true }, @@ -2250,7 +2250,7 @@ }, { "cell_type": "markdown", - "id": "369318ab", + "id": "520d10c7", "metadata": { "editable": true }, @@ -2266,7 +2266,7 @@ }, { "cell_type": "markdown", - "id": "ae96e8af", + "id": "3f684dfa", "metadata": { "editable": true }, @@ -2278,7 +2278,7 @@ }, { "cell_type": "markdown", - "id": "551aba56", + "id": "b914c6e7", "metadata": { "editable": true }, @@ -2289,7 +2289,7 @@ }, { "cell_type": "markdown", - "id": "21959b27", + "id": "83d5bb15", "metadata": { "editable": true }, @@ -2301,7 +2301,7 @@ }, { "cell_type": "markdown", - "id": "fe01d57a", + "id": "c23c3e18", "metadata": { "editable": true }, @@ -2315,7 +2315,7 @@ }, { "cell_type": "markdown", - "id": "0c170c8c", + "id": "a1cec49a", "metadata": { "editable": true }, @@ -2327,7 +2327,7 @@ }, { "cell_type": "markdown", - "id": "c0bef848", + "id": "209b99cb", "metadata": { "editable": true }, @@ -2352,7 +2352,7 @@ }, { "cell_type": "markdown", - "id": "3e2c00bf", + "id": "7c377fa3", "metadata": { "editable": true }, @@ -2369,7 +2369,7 @@ { "cell_type": "code", "execution_count": 27, - "id": "5026698f", + "id": "941d1cc8", "metadata": { "collapsed": false, "editable": true @@ -2413,7 +2413,7 @@ }, { "cell_type": "markdown", - "id": "d36a4109", + "id": "3f28ffa6", "metadata": { "editable": true }, @@ -2424,7 +2424,7 @@ { "cell_type": "code", "execution_count": 28, - "id": "1764c75a", + "id": "fa2002d0", "metadata": { "collapsed": false, "editable": true @@ -2446,7 +2446,7 @@ }, { "cell_type": "markdown", - "id": "c3fa7451", + "id": "2451049d", "metadata": { "editable": true }, @@ -2463,7 +2463,7 @@ { "cell_type": "code", "execution_count": 29, - "id": "55b8fe6c", + "id": "f69e3d7f", "metadata": { "collapsed": false, "editable": true @@ -2484,7 +2484,7 @@ }, { "cell_type": "markdown", - "id": "aaa6d47a", + "id": "79fd53c7", "metadata": { "editable": true }, @@ -2498,7 +2498,7 @@ { "cell_type": "code", "execution_count": 30, - "id": "1147c023", + "id": "fcc1ed96", "metadata": { "collapsed": false, "editable": true @@ -2527,7 +2527,7 @@ }, { "cell_type": "markdown", - "id": "1181a733", + "id": "5ed8023e", "metadata": { "editable": true }, @@ -2547,7 +2547,7 @@ { "cell_type": "code", "execution_count": 31, - "id": "891da8e4", + "id": "88c9b2bd", "metadata": { "collapsed": false, "editable": true @@ -2564,7 +2564,7 @@ }, { "cell_type": "markdown", - "id": "7b05d48d", + "id": "969390da", "metadata": { "editable": true }, @@ -2576,7 +2576,7 @@ { "cell_type": "code", "execution_count": 32, - "id": "232d3d41", + "id": "7eaf8ba7", "metadata": { "collapsed": false, "editable": true @@ -2594,7 +2594,7 @@ }, { "cell_type": "markdown", - "id": "da661c72", + "id": "f6e6873f", "metadata": { "editable": true }, @@ -2605,7 +2605,7 @@ { "cell_type": "code", "execution_count": 33, - "id": "7fc26f20", + "id": "88e7e33d", "metadata": { "collapsed": false, "editable": true @@ -2618,7 +2618,7 @@ }, { "cell_type": "markdown", - "id": "76253467", + "id": "d3eeb1f5", "metadata": { "editable": true }, @@ -2630,7 +2630,7 @@ { "cell_type": "code", "execution_count": 34, - "id": "41a2ec8e", + "id": "311b7212", "metadata": { "collapsed": false, "editable": true @@ -2661,7 +2661,7 @@ }, { "cell_type": "markdown", - "id": "955874f6", + "id": "84d65db4", "metadata": { "editable": true }, @@ -2675,7 +2675,7 @@ { "cell_type": "code", "execution_count": 35, - "id": "55f7d79b", + "id": "ba27d96d", "metadata": { "collapsed": false, "editable": true @@ -2715,7 +2715,7 @@ }, { "cell_type": "markdown", - "id": "b7c4105f", + "id": "ec92009b", "metadata": { "editable": true }, @@ -2735,7 +2735,7 @@ }, { "cell_type": "markdown", - "id": "47a0a326", + "id": "05f9fdbf", "metadata": { "editable": true }, @@ -2767,7 +2767,7 @@ }, { "cell_type": "markdown", - "id": "71fae485", + "id": "1f0e98dc", "metadata": { "editable": true }, @@ -2775,28 +2775,28 @@ "## Regression analysis, overarching aims\n", "\n", "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 $\\boldsymbol{x} =[x_0, x_1,\\dots, x_{n-1}]^T$. \n", - "The first variable is called the **dependent**, the **outcome** or the **response** variable while the set of variables $\\boldsymbol{x}$ is called the independent variable, or the predictor variable or the explanatory variable. \n", + "The first variable is called the **dependent**, the **outcome** or the **response** variable while the set of variables $\\boldsymbol{x}$ is called the independent variable, or the predictor variable or the explanatory variable, or simply just the **inputs**. \n", "\n", - "A regression model aims at finding a likelihood function $p(\\boldsymbol{y}\\vert \\boldsymbol{x})$, that is the conditional distribution for $\\boldsymbol{y}$ with a given $\\boldsymbol{x}$. The estimation of $p(\\boldsymbol{y}\\vert \\boldsymbol{x})$ is made using a data set with \n", + "A regression model aims at finding a likelihood function $p(\\boldsymbol{y}\\vert \\boldsymbol{x})$ or in the more traditional sense a function $\\boldsymbol{y}(\\boldsymbol{x})$, that is the conditional distribution for $\\boldsymbol{y}$ with a given $\\boldsymbol{x}$. The estimation of $p(\\boldsymbol{y}\\vert \\boldsymbol{x})$ is made using a data set with \n", "* $n$ cases $i = 0, 1, 2, \\dots, n-1$ \n", "\n", "* Response (target, dependent or outcome) variable $y_i$ with $i = 0, 1, 2, \\dots, n-1$ \n", "\n", - "* $p$ so-called explanatory (independent or predictor) variables $\\boldsymbol{x}_i=[x_{i0}, x_{i1}, \\dots, x_{ip-1}]$ with $i = 0, 1, 2, \\dots, n-1$ and explanatory variables running from $0$ to $p-1$. See below for more explicit examples. \n", + "* $p$ so-called explanatory (independent or predictor or feature) variables $\\boldsymbol{x}_i=[x_{i0}, x_{i1}, \\dots, x_{ip-1}]$ with $i = 0, 1, 2, \\dots, n-1$ and explanatory variables running from $0$ to $p-1$. See below for more explicit examples. \n", "\n", - " The goal of the regression analysis is to extract/exploit relationship between $\\boldsymbol{y}$ and $\\boldsymbol{x}$ in or to infer causal dependencies, approximations to the likelihood functions, functional relationships and to make predictions, making fits and many other things." + " The goal of the regression analysis is to extract/exploit relationship between $\\boldsymbol{y}$ and $\\boldsymbol{x}$ in order to infer specific dependencies, approximations to the likelihood functions, functional relationships and to make predictions, making fits and many other things." ] }, { "cell_type": "markdown", - "id": "e9e37440", + "id": "58952519", "metadata": { "editable": true }, "source": [ "## Regression analysis, overarching aims II\n", "\n", - "Consider an experiment in which $p$ characteristics of $n$ samples are\n", + "Consider an experiment in which $p$ characteristics/features of $n$ samples are\n", "measured. The data from this experiment, for various explanatory variables $p$ are normally represented by a matrix \n", "$\\mathbf{X}$.\n", "\n", @@ -2817,7 +2817,7 @@ }, { "cell_type": "markdown", - "id": "5fcc508e", + "id": "d11074d1", "metadata": { "editable": true }, @@ -2832,7 +2832,7 @@ }, { "cell_type": "markdown", - "id": "5c14a70a", + "id": "d48c4819", "metadata": { "editable": true }, @@ -2844,7 +2844,7 @@ }, { "cell_type": "markdown", - "id": "367cb3ba", + "id": "ae1f2872", "metadata": { "editable": true }, @@ -2859,20 +2859,20 @@ }, { "cell_type": "markdown", - "id": "e82f61b0", + "id": "4aa84fcb", "metadata": { "editable": true }, "source": [ - "## General linear models\n", - "Before we proceed let us study a case from linear algebra where we aim at fitting a set of data $\\boldsymbol{y}=[y_0,y_1,\\dots,y_{n-1}]$. 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 $\\boldsymbol{x}=[x_0,x_1,\\dots,x_{n-1}]$, 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. \n", + "## General linear models and linear algebra\n", + "Before we proceed let us study a case where we aim at fitting a set of data $\\boldsymbol{y}=[y_0,y_1,\\dots,y_{n-1}]$. 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 $\\boldsymbol{x}=[x_0,x_1,\\dots,x_{n-1}]$, 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. \n", "\n", "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" ] }, { "cell_type": "markdown", - "id": "e3491387", + "id": "b50d1301", "metadata": { "editable": true }, @@ -2884,7 +2884,7 @@ }, { "cell_type": "markdown", - "id": "345f9532", + "id": "12f6e265", "metadata": { "editable": true }, @@ -2894,7 +2894,7 @@ }, { "cell_type": "markdown", - "id": "18712eb3", + "id": "9d960511", "metadata": { "editable": true }, @@ -2905,7 +2905,7 @@ }, { "cell_type": "markdown", - "id": "c4948d05", + "id": "65af2b76", "metadata": { "editable": true }, @@ -2923,7 +2923,7 @@ }, { "cell_type": "markdown", - "id": "7c451b02", + "id": "e4ae5f1a", "metadata": { "editable": true }, @@ -2934,7 +2934,7 @@ }, { "cell_type": "markdown", - "id": "98f70047", + "id": "573ff2a0", "metadata": { "editable": true }, @@ -2946,7 +2946,7 @@ }, { "cell_type": "markdown", - "id": "df73ac61", + "id": "f3e4a4ea", "metadata": { "editable": true }, @@ -2956,7 +2956,7 @@ }, { "cell_type": "markdown", - "id": "ca990e01", + "id": "33bcde09", "metadata": { "editable": true }, @@ -2968,7 +2968,7 @@ }, { "cell_type": "markdown", - "id": "1f3e8fad", + "id": "62f3ec32", "metadata": { "editable": true }, @@ -2978,7 +2978,7 @@ }, { "cell_type": "markdown", - "id": "2c604296", + "id": "c5dabb91", "metadata": { "editable": true }, @@ -2990,7 +2990,7 @@ }, { "cell_type": "markdown", - "id": "b9a79bf2", + "id": "2295527a", "metadata": { "editable": true }, @@ -3000,7 +3000,7 @@ }, { "cell_type": "markdown", - "id": "feeff2ea", + "id": "5edd0f94", "metadata": { "editable": true }, @@ -3019,7 +3019,7 @@ }, { "cell_type": "markdown", - "id": "a99ab920", + "id": "92340ca4", "metadata": { "editable": true }, @@ -3029,7 +3029,7 @@ }, { "cell_type": "markdown", - "id": "a5255c53", + "id": "91ae19a4", "metadata": { "editable": true }, @@ -3041,7 +3041,7 @@ }, { "cell_type": "markdown", - "id": "8c4390e8", + "id": "14ce8216", "metadata": { "editable": true }, @@ -3051,7 +3051,7 @@ }, { "cell_type": "markdown", - "id": "f2f0f45a", + "id": "50548ea3", "metadata": { "editable": true }, @@ -3067,7 +3067,7 @@ }, { "cell_type": "markdown", - "id": "00670503", + "id": "04dbaffa", "metadata": { "editable": true }, @@ -3087,7 +3087,7 @@ }, { "cell_type": "markdown", - "id": "b9c80356", + "id": "031622ad", "metadata": { "editable": true }, @@ -3097,7 +3097,7 @@ }, { "cell_type": "markdown", - "id": "2220f064", + "id": "5320f8a7", "metadata": { "editable": true }, @@ -3108,7 +3108,7 @@ }, { "cell_type": "markdown", - "id": "7774868a", + "id": "80b3b6dd", "metadata": { "editable": true }, @@ -3127,7 +3127,7 @@ }, { "cell_type": "markdown", - "id": "ae309c3c", + "id": "0c84182b", "metadata": { "editable": true }, @@ -3137,7 +3137,7 @@ }, { "cell_type": "markdown", - "id": "46c3fcdb", + "id": "e9efbf8f", "metadata": { "editable": true }, @@ -3149,7 +3149,7 @@ }, { "cell_type": "markdown", - "id": "ced0b619", + "id": "fb2fcb41", "metadata": { "editable": true }, @@ -3159,7 +3159,7 @@ }, { "cell_type": "markdown", - "id": "57e5afc1", + "id": "9cdea5e9", "metadata": { "editable": true }, @@ -3170,7 +3170,7 @@ }, { "cell_type": "markdown", - "id": "d1ae20b4", + "id": "137effde", "metadata": { "editable": true }, @@ -3190,7 +3190,7 @@ }, { "cell_type": "markdown", - "id": "0b856846", + "id": "608d2a86", "metadata": { "editable": true }, @@ -3202,7 +3202,7 @@ }, { "cell_type": "markdown", - "id": "b91cdaa6", + "id": "b1bdcfbd", "metadata": { "editable": true }, @@ -3217,7 +3217,7 @@ { "cell_type": "code", "execution_count": 36, - "id": "35acb4d2", + "id": "08229782", "metadata": { "collapsed": false, "editable": true @@ -3297,7 +3297,7 @@ }, { "cell_type": "markdown", - "id": "911d6b16", + "id": "9cb7a0b4", "metadata": { "editable": true }, @@ -3307,7 +3307,7 @@ }, { "cell_type": "markdown", - "id": "a5c57c5f", + "id": "c0bcdac1", "metadata": { "editable": true }, @@ -3319,7 +3319,7 @@ }, { "cell_type": "markdown", - "id": "6cc8a09e", + "id": "63cbaac5", "metadata": { "editable": true }, @@ -3329,7 +3329,7 @@ }, { "cell_type": "markdown", - "id": "bf7ed8b8", + "id": "74f85824", "metadata": { "editable": true }, @@ -3340,7 +3340,7 @@ }, { "cell_type": "markdown", - "id": "dd11ece6", + "id": "9ab59a60", "metadata": { "editable": true }, @@ -3352,7 +3352,7 @@ }, { "cell_type": "markdown", - "id": "89374ffa", + "id": "7e23ea68", "metadata": { "editable": true }, @@ -3362,7 +3362,7 @@ }, { "cell_type": "markdown", - "id": "478d97e1", + "id": "a8ee6941", "metadata": { "editable": true }, @@ -3374,7 +3374,7 @@ }, { "cell_type": "markdown", - "id": "4d984d77", + "id": "68f87375", "metadata": { "editable": true }, @@ -3384,7 +3384,7 @@ }, { "cell_type": "markdown", - "id": "bb763652", + "id": "9f64f73e", "metadata": { "editable": true }, @@ -3396,7 +3396,7 @@ }, { "cell_type": "markdown", - "id": "33b0ad7f", + "id": "dfce87a2", "metadata": { "editable": true }, @@ -3409,7 +3409,7 @@ }, { "cell_type": "markdown", - "id": "f31dc107", + "id": "4c3343b6", "metadata": { "editable": true }, @@ -3421,7 +3421,7 @@ }, { "cell_type": "markdown", - "id": "739d3ee5", + "id": "956816e4", "metadata": { "editable": true }, @@ -3431,7 +3431,7 @@ }, { "cell_type": "markdown", - "id": "f4c3d450", + "id": "a5246238", "metadata": { "editable": true }, @@ -3443,7 +3443,7 @@ }, { "cell_type": "markdown", - "id": "85e57e55", + "id": "25c4566c", "metadata": { "editable": true }, @@ -3455,7 +3455,7 @@ }, { "cell_type": "markdown", - "id": "80c87b30", + "id": "88ae3dc6", "metadata": { "editable": true }, @@ -3466,7 +3466,7 @@ }, { "cell_type": "markdown", - "id": "fe86415d", + "id": "f644b957", "metadata": { "editable": true }, @@ -3478,7 +3478,7 @@ }, { "cell_type": "markdown", - "id": "f3bd065c", + "id": "c2aa461a", "metadata": { "editable": true }, @@ -3497,7 +3497,7 @@ }, { "cell_type": "markdown", - "id": "142d1884", + "id": "b8ae7152", "metadata": { "editable": true }, @@ -3510,7 +3510,7 @@ }, { "cell_type": "markdown", - "id": "08bdfc90", + "id": "2b57b079", "metadata": { "editable": true }, @@ -3520,7 +3520,7 @@ }, { "cell_type": "markdown", - "id": "275c2d7b", + "id": "6cf7b5bb", "metadata": { "editable": true }, @@ -3532,7 +3532,7 @@ }, { "cell_type": "markdown", - "id": "517a5e0a", + "id": "6b262186", "metadata": { "editable": true }, @@ -3542,7 +3542,7 @@ }, { "cell_type": "markdown", - "id": "e5fa80cc", + "id": "791f86dd", "metadata": { "editable": true }, @@ -3554,7 +3554,7 @@ }, { "cell_type": "markdown", - "id": "cf9845ec", + "id": "90636f3b", "metadata": { "editable": true }, @@ -3564,7 +3564,7 @@ }, { "cell_type": "markdown", - "id": "7cb30642", + "id": "9e5f03de", "metadata": { "editable": true }, @@ -3576,7 +3576,7 @@ }, { "cell_type": "markdown", - "id": "0559ecfb", + "id": "ca6b01ab", "metadata": { "editable": true }, @@ -3587,7 +3587,7 @@ }, { "cell_type": "markdown", - "id": "8033a39d", + "id": "b5d6a41f", "metadata": { "editable": true }, @@ -3599,7 +3599,7 @@ }, { "cell_type": "markdown", - "id": "e3e3bdaf", + "id": "9ea69f7c", "metadata": { "editable": true }, @@ -3609,7 +3609,7 @@ }, { "cell_type": "markdown", - "id": "9607b664", + "id": "a596bdf0", "metadata": { "editable": true }, @@ -3621,7 +3621,7 @@ }, { "cell_type": "markdown", - "id": "5613e0a5", + "id": "d038383d", "metadata": { "editable": true }, @@ -3631,7 +3631,7 @@ }, { "cell_type": "markdown", - "id": "8c24d5f8", + "id": "6137c167", "metadata": { "editable": true }, @@ -3643,7 +3643,7 @@ }, { "cell_type": "markdown", - "id": "ddb396ba", + "id": "74e5b5a8", "metadata": { "editable": true }, @@ -3664,7 +3664,7 @@ }, { "cell_type": "markdown", - "id": "846e4adc", + "id": "7aa68eb3", "metadata": { "editable": true }, @@ -3678,7 +3678,7 @@ }, { "cell_type": "markdown", - "id": "21bcf6d3", + "id": "771ee3d3", "metadata": { "editable": true }, @@ -3689,7 +3689,7 @@ }, { "cell_type": "markdown", - "id": "c816f910", + "id": "800f111b", "metadata": { "editable": true }, @@ -3701,7 +3701,7 @@ }, { "cell_type": "markdown", - "id": "7a1c3965", + "id": "2d552dcc", "metadata": { "editable": true }, @@ -3711,7 +3711,7 @@ }, { "cell_type": "markdown", - "id": "7f0388c3", + "id": "fdbbbb87", "metadata": { "editable": true }, @@ -3723,7 +3723,7 @@ }, { "cell_type": "markdown", - "id": "eeedb987", + "id": "b80deabc", "metadata": { "editable": true }, @@ -3733,7 +3733,7 @@ }, { "cell_type": "markdown", - "id": "2571cf6a", + "id": "5508e7e7", "metadata": { "editable": true }, @@ -3745,7 +3745,7 @@ }, { "cell_type": "markdown", - "id": "a7cdebb6", + "id": "5f310441", "metadata": { "editable": true }, @@ -3757,7 +3757,7 @@ }, { "cell_type": "markdown", - "id": "6377e79e", + "id": "a187b4ff", "metadata": { "editable": true }, @@ -3771,7 +3771,7 @@ { "cell_type": "code", "execution_count": 37, - "id": "09eab794", + "id": "e31defa4", "metadata": { "collapsed": false, "editable": true @@ -3786,7 +3786,7 @@ }, { "cell_type": "markdown", - "id": "eca8752a", + "id": "e5ce034c", "metadata": { "editable": true }, @@ -3797,7 +3797,7 @@ { "cell_type": "code", "execution_count": 38, - "id": "77fc1624", + "id": "fa13598b", "metadata": { "collapsed": false, "editable": true @@ -3810,7 +3810,7 @@ }, { "cell_type": "markdown", - "id": "503985dc", + "id": "77910e5a", "metadata": { "editable": true }, @@ -3821,7 +3821,7 @@ { "cell_type": "code", "execution_count": 39, - "id": "0cb88234", + "id": "e78dbfa7", "metadata": { "collapsed": false, "editable": true @@ -3844,7 +3844,7 @@ }, { "cell_type": "markdown", - "id": "1fd42ca1", + "id": "5dc4c1d9", "metadata": { "editable": true }, @@ -3858,7 +3858,7 @@ { "cell_type": "code", "execution_count": 40, - "id": "2d3cbcc4", + "id": "1171479c", "metadata": { "collapsed": false, "editable": true @@ -3871,7 +3871,7 @@ }, { "cell_type": "markdown", - "id": "30f21ee1", + "id": "ca538584", "metadata": { "editable": true }, @@ -3882,7 +3882,7 @@ { "cell_type": "code", "execution_count": 41, - "id": "888e603b", + "id": "21e61b2b", "metadata": { "collapsed": false, "editable": true @@ -3894,7 +3894,7 @@ }, { "cell_type": "markdown", - "id": "d8bb934f", + "id": "b433d290", "metadata": { "editable": true }, @@ -3905,7 +3905,7 @@ { "cell_type": "code", "execution_count": 42, - "id": "23592914", + "id": "00fc6d4f", "metadata": { "collapsed": false, "editable": true @@ -3921,7 +3921,7 @@ }, { "cell_type": "markdown", - "id": "1047ad9e", + "id": "346105a7", "metadata": { "editable": true }, @@ -3932,7 +3932,7 @@ { "cell_type": "code", "execution_count": 43, - "id": "c30fb298", + "id": "d4d7f29f", "metadata": { "collapsed": false, "editable": true @@ -3946,7 +3946,7 @@ }, { "cell_type": "markdown", - "id": "e05ba885", + "id": "2f474e9d", "metadata": { "editable": true }, @@ -3968,7 +3968,7 @@ }, { "cell_type": "markdown", - "id": "829c005a", + "id": "cc5a8753", "metadata": { "editable": true }, @@ -3980,7 +3980,7 @@ }, { "cell_type": "markdown", - "id": "c6b4a5d0", + "id": "817ec172", "metadata": { "editable": true }, @@ -3990,7 +3990,7 @@ }, { "cell_type": "markdown", - "id": "8121128a", + "id": "35087384", "metadata": { "editable": true }, @@ -4002,7 +4002,7 @@ }, { "cell_type": "markdown", - "id": "a53f53e3", + "id": "d7c6153a", "metadata": { "editable": true }, @@ -4014,7 +4014,7 @@ }, { "cell_type": "markdown", - "id": "bf2ced2c", + "id": "a6192d74", "metadata": { "editable": true }, @@ -4024,7 +4024,7 @@ }, { "cell_type": "markdown", - "id": "90896931", + "id": "d3a969e9", "metadata": { "editable": true }, @@ -4036,7 +4036,7 @@ }, { "cell_type": "markdown", - "id": "79350e80", + "id": "eece2a7a", "metadata": { "editable": true }, @@ -4046,7 +4046,7 @@ }, { "cell_type": "markdown", - "id": "62cd8cdf", + "id": "7a1c8dd9", "metadata": { "editable": true }, @@ -4058,7 +4058,7 @@ }, { "cell_type": "markdown", - "id": "a9b5ec4d", + "id": "fb89a4ee", "metadata": { "editable": true }, @@ -4068,7 +4068,7 @@ }, { "cell_type": "markdown", - "id": "9f5dc948", + "id": "a03fc257", "metadata": { "editable": true }, @@ -4080,7 +4080,7 @@ }, { "cell_type": "markdown", - "id": "22e24150", + "id": "429c9050", "metadata": { "editable": true }, @@ -4092,7 +4092,7 @@ }, { "cell_type": "markdown", - "id": "56cc0b29", + "id": "6ba27a86", "metadata": { "editable": true }, @@ -4102,7 +4102,7 @@ }, { "cell_type": "markdown", - "id": "e77e7ce1", + "id": "428e4e6f", "metadata": { "editable": true }, @@ -4114,7 +4114,7 @@ }, { "cell_type": "markdown", - "id": "3eee5223", + "id": "637c08b5", "metadata": { "editable": true }, @@ -4124,7 +4124,7 @@ }, { "cell_type": "markdown", - "id": "fa5065cf", + "id": "a1f7b83b", "metadata": { "editable": true }, @@ -4136,7 +4136,7 @@ }, { "cell_type": "markdown", - "id": "2f4ff28e", + "id": "4f3b999f", "metadata": { "editable": true }, @@ -4148,7 +4148,7 @@ }, { "cell_type": "markdown", - "id": "c669940c", + "id": "b7bc19dd", "metadata": { "editable": true }, @@ -4160,7 +4160,7 @@ }, { "cell_type": "markdown", - "id": "7bf1aac0", + "id": "c2d453e0", "metadata": { "editable": true }, @@ -4170,7 +4170,7 @@ }, { "cell_type": "markdown", - "id": "20d32a52", + "id": "9ae38bc3", "metadata": { "editable": true }, @@ -4182,7 +4182,7 @@ }, { "cell_type": "markdown", - "id": "00911983", + "id": "1e3acfcb", "metadata": { "editable": true }, @@ -4192,7 +4192,7 @@ }, { "cell_type": "markdown", - "id": "47c8b687", + "id": "2293d407", "metadata": { "editable": true }, @@ -4204,7 +4204,7 @@ }, { "cell_type": "markdown", - "id": "06a69921", + "id": "50faad34", "metadata": { "editable": true }, @@ -4214,7 +4214,7 @@ }, { "cell_type": "markdown", - "id": "8bde8a19", + "id": "3e4b126b", "metadata": { "editable": true }, @@ -4226,7 +4226,7 @@ }, { "cell_type": "markdown", - "id": "4b626d74", + "id": "77bb4e7b", "metadata": { "editable": true }, @@ -4237,7 +4237,7 @@ }, { "cell_type": "markdown", - "id": "fef0bbcb", + "id": "f06fd70f", "metadata": { "editable": true }, @@ -4249,7 +4249,7 @@ }, { "cell_type": "markdown", - "id": "c48101b4", + "id": "720111d3", "metadata": { "editable": true }, @@ -4259,7 +4259,7 @@ }, { "cell_type": "markdown", - "id": "2a441f4b", + "id": "286cb5a4", "metadata": { "editable": true }, @@ -4271,7 +4271,7 @@ }, { "cell_type": "markdown", - "id": "4123e410", + "id": "ccbe28a2", "metadata": { "editable": true }, @@ -4281,7 +4281,7 @@ }, { "cell_type": "markdown", - "id": "f7321f2b", + "id": "2b134bed", "metadata": { "editable": true }, @@ -4293,7 +4293,7 @@ }, { "cell_type": "markdown", - "id": "3d4bc6e6", + "id": "70960a78", "metadata": { "editable": true }, @@ -4306,7 +4306,7 @@ }, { "cell_type": "markdown", - "id": "8fd31f19", + "id": "c80082e2", "metadata": { "editable": true }, @@ -4318,7 +4318,7 @@ }, { "cell_type": "markdown", - "id": "8dc97d57", + "id": "b6a9b109", "metadata": { "editable": true }, @@ -4330,7 +4330,7 @@ }, { "cell_type": "markdown", - "id": "aba6ba16", + "id": "873aac1e", "metadata": { "editable": true }, @@ -4342,7 +4342,7 @@ }, { "cell_type": "markdown", - "id": "0472cd8a", + "id": "19c04081", "metadata": { "editable": true }, @@ -4354,7 +4354,7 @@ }, { "cell_type": "markdown", - "id": "105c27eb", + "id": "9de8f48f", "metadata": { "editable": true }, @@ -4366,7 +4366,7 @@ }, { "cell_type": "markdown", - "id": "a2465810", + "id": "4756f2c0", "metadata": { "editable": true }, @@ -4376,7 +4376,7 @@ }, { "cell_type": "markdown", - "id": "9b236518", + "id": "1fc2c41c", "metadata": { "editable": true }, @@ -4388,7 +4388,7 @@ }, { "cell_type": "markdown", - "id": "fb483987", + "id": "ccdc9480", "metadata": { "editable": true }, @@ -4400,7 +4400,7 @@ }, { "cell_type": "markdown", - "id": "a4170454", + "id": "08bdebae", "metadata": { "editable": true }, @@ -4413,7 +4413,7 @@ }, { "cell_type": "markdown", - "id": "f8636dd8", + "id": "2cc4955e", "metadata": { "editable": true }, @@ -4440,7 +4440,7 @@ }, { "cell_type": "markdown", - "id": "d073f49e", + "id": "1f47fd2f", "metadata": { "editable": true }, @@ -4451,7 +4451,7 @@ { "cell_type": "code", "execution_count": 44, - "id": "4a766e49", + "id": "55e38cd0", "metadata": { "collapsed": false, "editable": true @@ -4547,7 +4547,7 @@ }, { "cell_type": "markdown", - "id": "36ed596d", + "id": "eb68d2ed", "metadata": { "editable": true }, @@ -4562,7 +4562,7 @@ }, { "cell_type": "markdown", - "id": "ef500e43", + "id": "69638678", "metadata": { "editable": true }, @@ -4584,7 +4584,7 @@ { "cell_type": "code", "execution_count": 45, - "id": "eced4003", + "id": "fd08370e", "metadata": { "collapsed": false, "editable": true @@ -4659,7 +4659,7 @@ }, { "cell_type": "markdown", - "id": "e424dea3", + "id": "54f228d1", "metadata": { "editable": true }, @@ -4671,7 +4671,7 @@ }, { "cell_type": "markdown", - "id": "56fb881b", + "id": "6b512cee", "metadata": { "editable": true }, @@ -4740,7 +4740,7 @@ }, { "cell_type": "markdown", - "id": "a6425973", + "id": "12c51537", "metadata": { "editable": true }, @@ -4754,7 +4754,7 @@ { "cell_type": "code", "execution_count": 46, - "id": "4cd16835", + "id": "d4fa50f9", "metadata": { "collapsed": false, "editable": true @@ -4767,7 +4767,7 @@ }, { "cell_type": "markdown", - "id": "1c9ee6aa", + "id": "13325a4c", "metadata": { "editable": true }, @@ -4781,7 +4781,7 @@ }, { "cell_type": "markdown", - "id": "cd6b4600", + "id": "5ca90cc6", "metadata": { "editable": true }, @@ -4794,7 +4794,7 @@ }, { "cell_type": "markdown", - "id": "be133461", + "id": "a49a74e2", "metadata": { "editable": true }, @@ -4805,7 +4805,7 @@ }, { "cell_type": "markdown", - "id": "a82b5812", + "id": "6638cc26", "metadata": { "editable": true }, @@ -4817,7 +4817,7 @@ }, { "cell_type": "markdown", - "id": "57563908", + "id": "1bcafec3", "metadata": { "editable": true }, @@ -4827,7 +4827,7 @@ }, { "cell_type": "markdown", - "id": "f2d998a9", + "id": "d224b25e", "metadata": { "editable": true }, @@ -4839,7 +4839,7 @@ }, { "cell_type": "markdown", - "id": "8ab4ba25", + "id": "3ce3600c", "metadata": { "editable": true }, @@ -4850,7 +4850,7 @@ }, { "cell_type": "markdown", - "id": "b3a469fc", + "id": "1a158560", "metadata": { "editable": true }, @@ -4869,7 +4869,7 @@ { "cell_type": "code", "execution_count": 47, - "id": "2603515a", + "id": "335572c4", "metadata": { "collapsed": false, "editable": true @@ -4885,7 +4885,7 @@ }, { "cell_type": "markdown", - "id": "03b27779", + "id": "277c90a6", "metadata": { "editable": true }, @@ -4895,7 +4895,7 @@ }, { "cell_type": "markdown", - "id": "90f20952", + "id": "65b66e40", "metadata": { "editable": true }, @@ -4906,7 +4906,7 @@ }, { "cell_type": "markdown", - "id": "b4cc4809", + "id": "e55cdedd", "metadata": { "editable": true }, @@ -4918,7 +4918,7 @@ }, { "cell_type": "markdown", - "id": "d4f5d26c", + "id": "9ecf1920", "metadata": { "editable": true }, diff --git a/doc/src/week34/week34.do.txt b/doc/src/week34/week34.do.txt index 4081156e2..c50f80c1f 100644 --- a/doc/src/week34/week34.do.txt +++ b/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 !bblock 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. !eblock !split @@ -1567,7 +1567,7 @@ A regression model aims at finding a likelihood function $p(\bm{y}\vert \bm{x})$ !bblock -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/ !split -===== General linear models ===== +===== General linear models and linear algebra ===== !bblock -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 !bt