Overview
G. Jogesh Babu
Probability theory
Probability is all about flip of a coin
Conditional 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 rescue
Some 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 change
Bayesian inference takes prior knowledge
into account
The quality of Bayesian analysis depends on
how best one can convert the prior
information into mathematical prior probability
Methods for parameter estimation, model
assessment etc
Illustrations with examples from astronomy
Spatial Statistics
Spatial Point Processes
Gaussian Processes (Inference and computational
aspects)
Modeling Lattice Data
Homogeneous and inhomogeneous Poisson
processes
Estimation 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