Data Analysis and Machine Learning: Linear Regression and more Advanced Regression Analysis
Contents
Why Linear Regression (aka Ordinary Least Squares and family)
Regression analysis, overarching aims
Regression analysis, overarching aims II
Examples
General linear models
Rewriting the fitting procedure as a linear algebra problem
Rewriting the fitting procedure as a linear algebra problem, more details
Generalizing the fitting procedure as a linear algebra problem
Generalizing the fitting procedure as a linear algebra problem
Optimizing our parameters
Our model for the nuclear binding energies
Optimizing our parameters, more details
Interpretations and optimizing our parameters
Interpretations and optimizing our parameters
Some useful matrix and vector expressions
Interpretations and optimizing our parameters
Own code for Ordinary Least Squares
Adding error analysis and training set up
The \( \chi^2 \) function
The \( \chi^2 \) function
The \( \chi^2 \) function
The \( \chi^2 \) function
The \( \chi^2 \) function
The \( \chi^2 \) function
Fitting an Equation of State for Dense Nuclear Matter
The code
Splitting our Data in Training and Test data
The Boston housing data example
Housing data, the code
Reducing the number of degrees of freedom, overarching view
Preprocessing our data
More preprocessing
Simple preprocessing examples, Franke function and regression
The singular value decomposition
Linear Regression Problems
Fixing the singularity
Basic math of the SVD
The SVD, a Fantastic Algorithm
Economy-size SVD
Codes for the SVD
Mathematical Properties
Ridge and LASSO Regression
More on Ridge Regression
Interpreting the Ridge results
More interpretations
A better understanding of regularization
Decomposing the OLS and Ridge expressions
Introducing the Covariance and Correlation functions
Correlation Function and Design/Feature Matrix
Covariance Matrix Examples
Correlation Matrix
Correlation Matrix with Pandas
Correlation Matrix with Pandas and the Franke function
Rewriting the Covariance and/or Correlation Matrix
Linking with SVD
Where are we going?
Resampling methods
Resampling approaches can be computationally expensive
Why resampling methods ?
Statistical analysis
Linking the regression analysis with a statistical interpretation
Assumptions made
Expectation value and variance
Expectation value and variance for \( \boldsymbol{\beta} \)
Resampling methods
Resampling methods: Jackknife and Bootstrap
Resampling methods: Jackknife
Jackknife code example
Resampling methods: Bootstrap
Resampling methods: Bootstrap background
Resampling methods: More Bootstrap background
Resampling methods: Bootstrap approach
Resampling methods: Bootstrap steps
Code example for the Bootstrap method
Various steps in cross-validation
How to set up the cross-validation for Ridge and/or Lasso
Cross-validation in brief
Code Example for Cross-validation and \( k \)-fold Cross-validation
The bias-variance tradeoff
Example code for Bias-Variance tradeoff
Understanding what happens
Summing up
Another Example from Scikit-Learn's Repository
More examples on bootstrap and cross-validation and errors
The same example but now with cross-validation
Cross-validation with Ridge
The Ising model
Reformulating the problem to suit regression
Linear regression
Singular Value decomposition
The one-dimensional Ising model
Ridge regression
LASSO regression
Performance as function of the regularization parameter
Finding the optimal value of \( \lambda \)
Linking with SVD
See lecture september 11. More text to be added here soon.
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