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STATISTICAL COMPUTING

The interface between statistics and computer science is known as computational statistics. It is an area of scientific computing specific to the mathematical science of statistics. It can also be defined as providing the computational tools for statistics by using various tools and techniques from computer science.

Computational statistics is used to refer computationally intensive statistical methods including re-sampling and local regression etc. Important topics included are as follows:

• Simulation techniques
• Computational techniques
• Markov Chain technique
• Re-sampling
• Annealing techniques

While dealing with computational statistics we must not forget various concepts like smoothing and function estimation, Monte Carlo technique, linear and non- linear regression problems, computational problems, robust linear regression, tree-structured regression and its classification, robust multivariate analysis, neutral networks their associated rules and algorithms.

Computational statistics consists of three main components:

• Deepen computational reasoning and literacy
• Compute with data in practice of statistics

Computing is an important aspect of a statistician’s work and must be incorporated into statistics. The nature of statistics changes with many opportunities and impact of science and policies. Computational literacy and programming are fundamental to statistical practice so for this purpose information technologies should be added to the curriculum.

Software used by scientists for computational statistics is given below:

• R: It is an integrated suite of software facilities for calculation and graphical display, data manipulation. It has its own latex documentation format for supplying comprehensive documentation.
• SPSS: It’s a software package which involves the study of batched and non-batched statistical analysis. It includes data mining, predictive modeling, decision management, reporting, deployment, analysis of data.
• SAS: Statistical Analysis System is a software suite developed for predictive analysis, business intelligence, multivariate analysis, advanced analytics, data management etc.
• STATA: Stata being a general-purpose statistical software package which finds its application in the fields of economics, sociology, political science, biomedical and epidemiology

Clustering is an important technique in which rows are grouped together in such a manner that similar values occur across a number of variables. The technique is helpful in understanding clumping structures of your data. Its main tasks are data mining, analyzing data, data compression, image analysis, machine learning, pattern recognition, information retrieval, bioinformatics and computer graphics.

Normal mixtures, k-means and hierarchical methods are the three different clustering methods provided by JMP. Grouping a set of objects in a way that objects kept in same group are more familiar with each other than those in other group and this is called clustering. Various algorithms included in clustering are:

• Connectivity-based clustering
• Centroid-based clustering
• Distribution-based clustering
• Density-based clustering

High-performance computing (HPC) is used in parallel processing for fast and efficient running of advanced application programs. For solving complex computation problems we are using super computers and parallel processing techniques. It focuses on developing parallel processing algorithms and systems. HPC can analyze and handle large amount of data at high speed very effectively and efficiently which a normal computer cannot do properly. We can use it for the purpose of solving highly complex problems across a wide range.

Beside this, various applications of HPC are detailed below:

• Redesigning of products
• Optimized production
• Analyzing large datasheets
• Conducting research projects
• Storage of large amount of data
• Modeling of complex processes

There are more advantages for the purpose of which we are using HPC such as performing consumer trend monitoring, searching and profiling, creating computer visualizations for research results and carrying out simulations etc.

Dynamic programming is a process in which complex problems can be solved by breaking it into simple sub problems, solving them and storing their solutions using a memory-based data structure. It is also known as dynamic optimization. It mainly includes:

• Matrix-chain multiplication
• Knapsack
• Longest Common Subsequence

Gibbs Sampling: It is a common method of probabilistic interference and is well suited for copying with incomplete information. It comes at computational cost for many applications but this method is less efficient. It provides some valuable perception of statistical interference.

Monte Carlo Integration: Its one of the best method of computing complex integrals using probabilistic techniques. It’s a game of mathematics involved in this integration. Some methods are involved in Monte Carlo Integration that is- they are sampling methods based on the theories of probability, for revealing information they rely on trials, they are based on the central limit theorem, they are capable of handling complex problems, experiments results in random numbers, there is probabilistic distribution among the structures.

Graphical Lasso is an advanced algorithm for the estimation of precision matrix from the observations made from multivariate Gaussian distribution. It’s an algorithm for learning the structure in an undirected Gaussian graphical method.

Various advanced topics included in statistical computing are given as:

• Metropolis-hasting algorithm
• Romberg integration
• Parallel programming
• Monte Carlo integration
• Sampling distribution
• Visualization

We can further include advanced topics such as simulation of random variables, power functions, numerical algorithms, numerical methods (Bootstrap, EM, MCMC, Baye’s).

MCMC is Markov-Chain Monte Carlo which consists of various algorithms and models such as:

• Dijkstra’s algorithm
• Hidden Markov models
• Cloud computing
• Conditional random fields
• Message passing algorithms

Furthermore, we focus on the theories and applications of common algorithms used in statistical computing including numerical analysis, numerical integration, simulation, smoothing and density estimation, Bootstrapping, Newton Raphson,  Gaussian quadrature, sampling and its importance, Monte  Carlo methods, Kernel densities, simplex algorithms, root finding, optimization, random number generation, sorting etc.

• Programming languages, data structures, programming using R, C/C++, and Perl, HTML and CGI., Calling C/C++ from R., Writing R extensions (called packages)., Grid graphics; interactive and dynamic graphics., Static and dynamic memory management
• numerical libraries and optimization in R, Parallel computing in R., Massive data challanges and solutions., Image processing., Hierarchical models., cluster analysis, Random Number Generation, Pseudo random numbers, linear congruential generator, properties of PRNGs, inverse transform method, basic rejection sampling, envelope rejection sampling, transformation of random variables.
• Monte Carlo Methods, Monte Carlo estimates, bias, mean square error, estimating probabilities, choice of sample size,, importance sampling, antithetic variables, variance reduction, applications in statistical inference, Markov Chain Monte-Carlo Methods, Markov Chains with discrete/continuous state space, random walk, transition matrices, transition densities, stationary distributions
• Metropolis-Hastings algorithm, convergence of MCMC methods,, burn-in period, application to Bayesian inference, MH with discrete state space, detailed balance condition, Metropolis algorithm, random-walk Metropolis, independence sampler., Bootstrap Methods, Empirical distributions, the bootstrap principle, bootstrap estimate for the bias,, bootstrap estimate of the standard error
• Introduction to statistical computing, Least squares (regression), Penalized and weighted least squares, Density estimation and smoothing, Matrix computations, Optimization (likelihood estimation), Newton-Raphson, Fisher scoring, Combinatorial optimization, Integration (probabilities), Quadrature, Laplace approximation, Resampling and Monte Carlo inferences, Jackknife and Bootstrap, Permutation procedures, Monte Carlo simulation, Statistical graphics

Statistical computing

• Statistical simulation;, monte carlo simulation, Random number generation: continuous rv’s, Density estimation: kernel density estimators, Unconstrained optimization, Constrained optimization, Missing data: em algorithm
• Missing data: augmented em algorithm, Missing data: multiple imputation, Resampling: jackknife, Resampling: jackknife, Resampling: bootstrap, scatterplot smoothing: splines, Gam, Mcmc, Metropolis-hastings algorithm , gibbs sampler, Gibbs sampler and winbugs, Statistical computing packages , statistical computing packages , importing text and excel files into sas , Statistical analysis concepts thursday
• importing text and excel files into r , random sample, stratification, survey weights , descriptive statistics, graphical analysis, exploratory analysis , confirmatory analysis , Univariate analysis in sas, exploratory analysis , O descriptive statistics , O graphical analysis , O group comparisons , D univariate analysis in r , univariate analysis concepts , confirmatory analysis , O hypothesis testing , univariate methods in r
• Bivariate analysis in sas , exploratory analysis , cross-tabulations , regression concepts , correlation , Bivariate analysis , confirmatory analysis , hypothesis testing , more on regression, correlation, and cross-tabulation , bivariate methods in r, Multivariate analysis thursday, march 8th, exploratory analysis , O descriptive statistics & graphical analysis , O cross-tabulations , multivariate regression
• begin project discussions , Multivariate analysis , confirmatory analysis , O hypothesis testing , O more on regression and correlation , finalize project assignments , Multivariate analysis, R: data types , R: graphics , R: programming , latex , R: programming & debug , R: statistical functions , building r packages

Topics for STAT 5304 - Statistical Computing

• Computational methods for statistical computing problems
• Orthogonal transformations
• sweep operators
•  maximum likelihood
• least squares estimation problems
• SAS programming language