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NOTES ON STATISTICS, PROBABILITY and MATHEMATICS

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DISTRIBUTIONS:

Ditributions and their stories

Gosset’s t Distribution

Chi Square Distribution

CTL

Uniform Distribution and the Probability Integral Transform

Moments

Moment Generating Functions (MGF)

Probability Generating Functions (PGF)

Characteristic Functions (CF)

Multivariate Gaussian Distribution

Multivariate Gaussian Distribution: Random Sampling and the Cholesky

Dirichlet distribution

Gamma Function

Gamma Distribution

Power-law, Heavy & Fat-Tailed Distributions

Copulas

Mahalanobis

INFERENCE:

What test to use & how to code it in R

One-Sample Mean compared to Population & Confidence Intervals

Independent Samples Means Test

Paired t Test

Statistical Power

Calculating Power and Sample Size

Maximum Likelihood estimation

Method of moments

Distribution of the sample variance

Extreme Value Theory

PROPORTIONS:

Proportion Standard Error

Confidence Interval and Margin of Error

Comparing Sample Proportion to Population: Z-test proportions, CI and margin of error

Comparing Sample Proportions to Population: Exact Multinomial (Goodness of Fit)

Comparing Sample Proportions: Fisher Exact; Chi square; G-test; Z-test

Chi Square: Goodness of Fit

Chi Square: Test of Independence and Homogeneity

Chi Square: Testing Standard Deviation (Variance)

Chi Square Test in R: Monte Carlo Option

Grading on the curve: detectable with GoF test

Monte Carlo Validation of the Z Test of Proportions

Contingency Tables

Three Way Contingency Tables

Monte Carlo alternative to Chi Square

Chi Square of Proportions is the Same as a z-test

RR and OR

ANOVA:

ANOVA Results Interpretation

ANOVA Is Just a form of Regression

ANOVA Regression Equivalents

ANOVA with Summary Statistics

ANOVA Contrasts

ANOVA Alphabet Soup

REGRESSION:

OLS

Global and individual predictors p-values of explained variance in a linear model

Model Selection

OLS without Linear Algebra

Linear Algebra of OLS

Optimization: Cost Function

R Squared, SST, SSR, SSE

Calculating Sts. Error, t value and Pr(>|t|) of coefficients

Manual calculation ANOVA OLS lm Models

Confidence Intervals & p-values of OLS

OLS Diagnostics

OLS Diagnostics: VIF

Dummy Coding in Regression

Interpretation of Output in Linear Model with Interactions

Logistic Regression

ROC Curves and AUC

Survival Analysis

Random Effects Simulation

Comparing Models with Mixed Effects

Predict Function in Mixed Effects

Model Matrices in Mixed Effects

Omitted Variable Bias (OLS)

Confounders, Control Variables, Endogeneity

Variance of Estimated Parameters

Ridge and Lasso regression

Interrrupted Time Series (ITS) & Poisson Regression

Conditional Logistic Regression

Gaussian Process Regression

CO2 and Global Temperature: Model analysis, autocorrelation and time series

NON-PARAMETRIC:

On the meaning of Parametric v Nonparametric

Wikoxon Rank-Sum Test or Mann-Whitney U

Wikoxon Signed Rank Test

MACHINE LEARNING:

Perceptron

Neural Nets

Backpropagation

Neural Nets and Transistor Gates

Lagrange review

Support Vector Machines (SVM)

Softmax

Unconstrained Optimization

Backpropagation in LSTM

TIME SERIES:

Stochastic processes - general concepts

Random walk - Return to origin

Time Series

Time Series: Variance, Covariance and Correlation

How to calculate manually ACF and PACFs

Reading ACF and PACF plots

Time series: Climate Change

Time series: Financial

Time series: VTI

Stationarity and the Characteristic Polynomial

Time Series Simulations, ACF, PACF Plotting Practice

PROBABILITY TOPICS:

Independence, Correlation and Orthogonality

Conditional Probability

Counting with Generating Functions

Correlation and Mutual Information

BAYES:

Bayesian Paradigm

Using Bayes to Assess Fairness of a Coin

OTHER:

Alternate formula variance (raw moments)

Measure Theory

Lebesgue Integral

PCA

PCA linear algebra derivation

Convolution or pdf of Z = X + Y

Simpson’s Paradox and Random Effects

Natural phenomena are not Gaussian

Markov Chains (Stochastic Matrices) & Ergodicity

A Practical Example of Markov Chain Monte Carlo (MCMC)

Order statistics

Discrete Fourier transform (DFT)

Fast Fourier transform (FFT)

Newton Raphson Method

Manifolds and distributions

Graphs

Conjugation of Permutations

The Bernoulli Numbers

Modular Forms and Möbius transformation

Notes on Galois theory

Zeta Function: Functional Equation and Analytical Continuation

Calculations with p-adics

Birch Swinnerton Dyer (BSD) Conjecture

LINEAR ALGEBRA:

Systems of linear equations and basis for the four spaces

It’s all about the MATRIX

Ax is a change of basis

LU, LDL’ and Cholesky

Fun wih linear algebra

Variance-Covariance Matrix and A transpose A

A transpose A; covariance matrix; projection matrix

Householder decomposition

TENSORS & DIFFERENTIAL GEOMETRY:

Riemannian geometry

Understanding curvature

Tensors Linearization of Multilinear map

Motivation for tensors

Tensors as multilinear maps

Exterior algebra

dx as a covector (linear functional)

covectors, 1-forms and gradient

Examples of tensor products

Tensors in physics

Covariant gradient field transformation

Covariant, Contravariant Forms and Metric Tensor

Tensors in physics: The Metric Tensor

Riemann Tensor for a Sphere and Schwarzschild Christoffel Symbols

Tensors in physics: Lorentz transformations

The curse of dimensionality

Manifold Learning

The Pushforward and the Pullback

Calculating the Lie Algebra of SL(2,C)

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NOTE: These are tentative notes on different topics for personal use - expect mistakes and misunderstandings.