Statistics

University of Calcutta (informally known as Calcutta University or CU) is a collegiate public state university located in Kolkata, West Bengal, India.&nbsp;Within India it is recognized as a &quot;Five-Star University&quot; and accredited &quot;A&quot; Grade by National Assessment and Accreditation Council.

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The University of Calcutta has secured the Fifth position among the Universities of India in the prestigious &quot;Indian University Ranking 2019&quot; list, released by the NIRF of the Ministry of Human Resource Development, Government of India.

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It was established on 24 January 1857, and was one of the first institutions in Asia to be established as a multidisciplinary and Western-style university. Its alumni and faculty include several heads of state, heads of government, social reformers, prominent artists, only academy award winner and Dirac medal winner in India, many Fellows of the Royal Society and five Nobel laureates as of 2019 which is highest in South Asia.

University Of Calcutta

For multivariate analysis with p variables the problem that often arises is the ambiguous nature of the correlation or covariance matrix. When p is moderately or very large it is generally difficult to identify the true nature of relationship among the variables as well as observations from the covariance or correlation matrix. Under such situations a very common way to simplify the matter is to reduce the dimension by considering only those variables (actual or derived) which are truly responsible for the overall variation. Important and useful dimension reduction techniques are Principal Component Analysis (PCA), Factor Analysis, Multidimensional Scaling, Independent Component Analysis (ICA), etc. Among them PCA is the most popular one. One may look at this method in three different ways. It may be considered as a method of transforming correlated variables into uncorrelated one or a method of finding linear combinations with relatively small or large variability or a tool for data reduction. The third criterion is more data oriented. In PCA primarily it is not necessary to make any assumption regarding the underlying multivariate distribution but if we are interested in some inference problems related to PCA then assumption of multivariate normality is necessary. The eigen values and eigen vectors of the covariance or correlation matrix are the main contributors of a PCA. The eigen vectors determine the directions of maximum variability whereas the eigen values specify the variances. In practice, decisions regarding the quality of the principal component approximation should be made on the basis of eigen value–eigen vector pairs. © Springer Science+Business Media New York 2014.

Applied Mathematics

Springer Series in Astrostatistics

Dimension reduction and clustering

The name “Monte Carlo” emerged from the name of the city “Monte Carlo”, famous for its Casino. In the casino there was a roulette where a small button was fixed at the centre of a wheel and the numbers 0–9 were marked at the end of the ten spokes of the wheel. When the button was pressed, the wheel starts rotating. When the wheel stops, a number within 0–9 was marked by a marker. Depending upon the number near the marker, one had to play the game. Thus this mechanical device was the device, first constructed to generate random numbers. The method was theoretically came through the work “The Monte Carlo Method” published by American mathematicians John Von Neumann and Stanslav Ulam in 1949. In spite of the theoretical background this method could not be used in significant scale until the invention of electronic computers. There are two aspects of this method. First it is used to simulate any process whose growth is affected by random factors and second to solve mathematical problems not affected by random factors but can be connected to artificially constructed probabilistic model giving rise to solution, otherwise unavailable. For example, suppose we are to calculate the area A within a unit square in Fig. 10.1. © Springer Science+Business Media New York 2014.

Monte Carlo simulation

A time series is a sequential collection of data indexed over time. In most cases the data are continuous but they are recorded at a discrete and finite set of equally spaced points. If a time series has N-observations (x0, x1, …, xN), then the time indexed distance between any two successive observations is referred to as the sampling interval. Time series lets one to explore, analyse and forecast univariate time series. Moreover, autocorrelations and partial auto correlations of the series indicate how and to what degree each point in the series is correlated with earlier values in the series. The analysis of time series is of immense significance not only for economist and businessman but also for scientist, geologist, biologist, research worker, etc. for various reasons. © Springer Science+Business Media New York 2014.

Time series analysis

Spatial clustering nature of galaxies has been studied previously through auto correlation function. The same type of cross-correlation function has been used in the present work to investigate parametric clustering nature of galaxies with respect to masses and sizes of galaxies. Here, formation and evolution of several components of nearby massive early type galaxies (M ∗ ≥ 1.3 × 1011 M⊙ have been envisaged through cross-correlations, in the mass-size parametric plane, with high redshift (0.2 ≤ Ζ ≤ 7) ETGs. It is found that the inner most components of nearby ETGs have significant correlation (~ 0.5 ± (0.02-0.07)) with ETGs in the highest redshift range (2 ≤ Ζ ≤ 7) called 'red nuggets' whereas intermediate components are highly correlated (~ 0.65 ± (0.03-0.07)) with ETGs in the redshift range 0.5 ≤ Ζ ≤ 0.75. The outermost part has no correlation in any range, suggesting a scenario through in situ accretion. The above formation scenario is consistent with the previous results obtained for NGC5128 and to some extent for nearby elliptical galaxies after considering a sample of ETGs at high redshift with stellar masses greater than or equal to 108.73 M⊙. So the present work indicates a three phase formation instead of two as discussed in previous works. © 2014 Astronomical Society of Australia.

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Fulltext

Publications of the Astronomical Society of Australia

Use of Cross-Correlation Function to Study Formation Mechanism of Massive Elliptical Galaxies

In this chapter, we review the theory of optimum regression designs. Concept of continuous design and different optimality criteria are introduced. The role of de la Garza phenomenon and Loewner order domination are discussed. Equivalence theorems for different optimality criteria, which play an important role in checking the optimality of a given otherwise prospective design, are presented. These results are repeatedly used in later chapters in the search for optimal mixture designs. We present standard optimality results for single variable polynomial regression model and multivariate linear and quadratic regression model. Kronecker product representation of the model(s) and related optimality results are also discussed. © 2014, Springer India.

GENERAL

Lecture Notes in Statistics

Optimal regression designs

This chapter examines the optimum designs for estimating the optimum mixing proportions in Scheffé’s quadratic mixture model with respect to the A-optimality criterion. By optimum mixing proportion, we refer to the one that maximizes the mean response. Since the dispersion matrix of the estimate depends on the unknown model parameters, a pseudo-Bayesian approach is used in defining the optimality criterion. The optimum designs under this criterion have been obtained for two- and three-component mixtures. Further, using Kiefer’s equivalence theorem, it has been shown that under invariant assumption on prior moments, the optimum design for a q -component mixture is a (q, 2) simplex lattice design for q= 3, 4. © 2014, Springer India.

In this chapter, we review the optimality results for the estimation of parameters and subset of parameters of the Scheffé’s mixture models while the factor space is the entire simplex, i.e., while we are in the framework of an unconstrained factor space. Though most of the results are related to quadratic model, optimality results for linear- and higher-degree polynomials are also discussed. Kiefer’s equivalence theorem plays an important role in characterizing optimum designs. It has been observed that the support points of an optimum design under different optimality criteria belong to the subclass of union of barycenters. © 2014, Springer India.

Statistical tools and techniques play a major role in many areas of science like biology, economics, physics etc. In particular, a systematic development in statistical methodology for biological problems started long back and constituted a new area of biostatistics. Compared to biology, the rate of progress in the development of statistical techniques for physical science is very slow. The reason may be due to lack of interaction between Physicists and Statisticians. In areas like Astronomy and Astrophysics the application of sophisticated statistical analysis is a comparatively recent phenomenon. Astronomy is the science to study the different features of planets, stars, galaxies and the universe as a whole. Astrophysicists try to model observed astronomical properties by using laws governing physical process. The problem is to make inference for the underlying properties on the basis of observations related to a few external characteristics. During the past two decades the inter-disciplinary field of astrostatistics has newly emerged in order to study important astrophysical issues through appropriate statistical analysis. With the advancement of technology, at present several data archives have been prepared which contain tera bytes of astronomical data. Statistical analysis of these high dimensional large data sets is a challenging problem and a good solution can be obtained only through interaction among Astronomers, Statisticians and Computer Scientists. © Springer Science+Business Media New York 2014.

Introduction to statistics

The most important carrier of information from all kinds of heavenly bodies is light. Light is an electromagnetic wave which is characterized by an electric field E and magnetic field H, and the direction of its propagation is at right angles to both these fields. Unlike mechanical waves, e.g. sound waves on the ocean, it can propagate through empty space. So, if Z-direction is the direction of its propagation (Fig. 1.1), then Y and X directions are the directions of electric and magnetic fields. The wave motion can be described by a sine curve and the distance between two consecutive peaks measures the wavelength λ, which is an important characterization of the propagating light. © Springer Science+Business Media New York 2014.