Overview

G. Jogesh Babu

Probability theory

Probability is all about flip of a coinConditional probability & Bayes theorem (Bayesian analysis)Expectation, variance, standard deviation (units free estimates)density of a continuous random variable (as opposed to density defined in physics)Normal (Gaussian) distribution, Chi-square distribution (not Chi-square statistic)Probability inequalities and the CLT

R Programming environment

Introduction to R programming language

R is an integrated suite of software facilities for data manipulation, calculation and graphical display.

Commonly used techniques such as, graphical description, tabular description, and summary statistics, are illustrated through R.

Exploratory Data Analysis

An approach/philosophy for data analysis that employs a variety of techniques (mostly graphical) to:– maximize insight into a data set– uncover underlying structure– extract important variables– detect outliers and anomalies– formulate hypotheses worth testing– develop parsimonious models– provide a basis for further data collection through

surveys or experiments

Statistical Inference

While Exploratory Data Analysis provides tools to understand what the data shows, the statistical inference helps in reaching conclusions that extend beyond the immediate data alone. Statistical inference helps in making judgments of an observed difference between groups is a dependable one or one that might have happened by chance in a study. Topics include:– Point estimation– Confidence intervals for unknown parameters– Principles of testing of hypotheses

Maximum Likelihood Estimation

Likelihood - differs from that of a probability– Probability refers to the occurrence of future events

– while a likelihood refers to past events with known outcomes

MLE is used for fitting a mathematical model to data.

Modeling real world data by estimating maximum likelihood offers a way of tuning the free parameters of the model to provide a good fit.

Regression

Basic Concepts in Regression

Bias-Variance Tradeoff

Linear Regression

Nonparametric Regression

Local Polynomial Regression

Confidence Bands

Splines

Linear regression issues in astronomy

Compares different regression lines used in astronomy

Illustrates them with Faber-Jackson relation.

Measurement Error models are also discussed

Multivariate analysis

Analysis of data on two or more attributes (variables) that may depend on each other– Principle components analysis, to reduce the

number of variables– Canonical correlation– Tests of hypotheses– Confidence regions – Multivariate regression– Discriminant analysis (supervised learning).

Computational aspects are covered in the lab

Cluster Analysis

Data mining techniques

Classifying data into clusters – k-means

– Model clustering

– Single linkage (friends of friends)

– Complete linkage clustering algorithm

Nonparametric Statistics

These statistical procedures make no assumptions about the probability distributions of the population. The model structure is not specified a priori but is instead determined from data. As non-parametric methods make fewer assumptions, their applicability is much wider Procedures described include:– Sign test– Mann-Whitney two sample test – Kruskal-Wallis test for comparing several samples

Density Estimation

Bootstrap

How to get most out of repeated use of the data. Bootstrap is similar to Monte Carlo method but the `simulation' is carried out from the data itself. A very general, mostly non-parametric procedure, and is widely applicable. Applications to regression, cases where the procedure fails, and where it outperforms traditional procedures will be also discussed

Model selection

Chi-square test

Wald Test

Rao's score test

Likelihood ratio test

AIC, BIC

Goodness of Fit

Curve (model) fitting or goodness of fit using bootstrap procedure. Procedure like Kolmogorov-Smirnov does not work in multidimensional case, or when the parameters of the curve are estimated. Bootstrap comes to rescueSome of these procedures are illustrated using R in a lab session on Hypothesis testing and bootstrapping

Bayesian Inference

As evidence accumulates, the degree of belief in a hypothesis ought to changeBayesian inference takes prior knowledge into account The quality of Bayesian analysis depends on how best one can convert the prior information into mathematical prior probabilityMethods for parameter estimation, model assessment etcIllustrations with examples from astronomy

Spatial Statistics

Spatial Point ProcessesGaussian Processes (Inference and computational aspects)Modeling Lattice DataHomogeneous and inhomogeneous Poisson processesEstimation of Ripley's K function (useful for point pattern analysis)Cox Process (doubly stochastic Poisson Process)Markov Point Processes

Time Series

Time domain procedures

State space models

Kernel smoothing

Poisson processes

Spectral methods for inference

A brief discussion of Kalman filter

Illustrations with examples from astronomy

Monte Carlo Markov Chain

MCMC methods are a collection of techniques that use pseudo-random (computer simulated) values to estimate solutions to mathematical problems

MCMC for Bayesian inference

Illustration of MCMC for the evaluation of expectations with respect to a distribution

MCMC for estimation of maxima or minima of functions

MCMC procedures are successfully used in the search for extra-solar planets