Modelling and Analysis of Some Time Series

The thesis deals with some of the non-linear Gaussian and non-Gaussian time models and mainly concentrated in studying the properties and application of a first order autoregressive process with Cauchy marginal distribution. In this thesis some of the non-linear Gaussian and non-Gaussian time series models and mainly concentrated in studying the properties and application of a order autoregressive process with Cauchy marginal distribution. Time series relating to prices, consumptions, money in circulation, bank deposits and bank clearing, sales and profit in a departmental store, national income and foreign exchange reserves, prices and dividend of shares in a stock exchange etc. are examples of economic and business time series. The thesis discuses the application of a threshold autoregressive(TAR) model, try to fit this model to a time series data. Another important non-linear model is the ARCH model, and the third model is the TARCH model. The main objective here is to identify an appropriate model to a given set of data. The data considered are the daily coconut oil prices for a period of three years. Since it is a price data the consecutive prices may not be independent and hence a time series based model is more appropriate. In this study the properties like ergodicity, mixing property and time reversibility and also various estimation procedures used to estimate the unknown parameters of the process.

Analysis of Some Single and Two Commodity Inventory Problems

This thesis is devoted to the study of some stochastic models in inventories. An inventory system is a facility at which items of materials are stocked. In order to promote smooth and efficient running of business, and to provide adequate service to the customers, an inventory materials is essential for any enterprise. When uncertainty is present, inventories are used as a protection against risk of stock out. It is advantageous to procure the item before it is needed at a lower marginal cost. Again, by bulk purchasing, the advantage of price discounts can be availed. All these contribute to the formation of inventory. Maintaining inventories is a major expenditure for any organization. For each inventory, the fundamental question is how much new stock should be ordered and when should the orders are replaced. In the present study, considered several models for single and two commodity stochastic inventory problems. The thesis discusses two models. In the first model, examined the case in which the time elapsed between two consecutive demand points are independent and identically distributed with common distribution function F(.) with mean  (assumed finite) and in which demand magnitude depends only on the time elapsed since the previous demand epoch. The time between disasters has an exponential distribution with parameter . In Model II, the inter arrival time of disasters have general distribution (F.) with mean  ( ) and the quantity destructed depends on the time elapsed between disasters. Demands form compound poison processes with inter arrival times of demands having mean 1/. It deals with linearly correlated bulk demand two Commodity inventory problem, where each arrival demands a random number of items of each commodity C1 and C2, the maximum quantity demanded being a (< S1) and b(of linearly correlated demand is also discussed

Mean Squared Residue Based Biclustering Algorithms for the Analysis of Gene Expression Data

Computational Biology is the research are that contributes to the analysis of biological data through the development of algorithms which will address significant research problems.The data from molecular biology includes DNA,RNA ,Protein and Gene expression data.Gene Expression Data provides the expression level of genes under different conditions.Gene expression is the process of transcribing the DNA sequence of a gene into mRNA sequences which in turn are later translated into proteins.The number of copies of mRNA produced is called the expression level of a gene.Gene expression data is organized in the form of a matrix. Rows in the matrix represent genes and columns in the matrix represent experimental conditions.Experimental conditions can be different tissue types or time points.Entries in the gene expression matrix are real values.Through the analysis of gene expression data it is possible to determine the behavioral patterns of genes such as similarity of their behavior,nature of their interaction,their respective contribution to the same pathways and so on. Similar expression patterns are exhibited by the genes participating in the same biological process.These patterns have immense relevance and application in bioinformatics and clinical research.Theses patterns are used in the medical domain for aid in more accurate diagnosis,prognosis,treatment planning.drug discovery and protein network analysis.To identify various patterns from gene expression data,data mining techniques are essential.Clustering is an important data mining technique for the analysis of gene expression data.To overcome the problems associated with clustering,biclustering is introduced.Biclustering refers to simultaneous clustering of both rows and columns of a data matrix. Clustering is a global whereas biclustering is a local model.Discovering local expression patterns is essential for identfying many genetic pathways that are not apparent otherwise.It is therefore necessary to move beyond the clustering paradigm towards developing approaches which are capable of discovering local patterns in gene expression data.A biclusters is a submatrix of the gene expression data matrix.The rows and columns in the submatrix need not be contiguous as in the gene expression data matrix.Biclusters are not disjoint.Computation of biclusters is costly because one will have to consider all the combinations of columans and rows in order to find out all the biclusters.The search space for the biclustering problem is 2 m+n where m and n are the number of genes and conditions respectively.Usually m+n is more than 3000.The biclustering problem is NP-hard.Biclustering is a powerful analytical tool for the biologist.The research reported in this thesis addresses the problem of biclustering.Ten algorithms are developed for the identification of coherent biclusters from gene expression data.All these algorithms are making use of a measure called mean squared residue to search for biclusters.The objective here is to identify the biclusters of maximum size with the mean squared residue lower than a given threshold. All these algorithms begin the search from tightly coregulated submatrices called the seeds.These seeds are generated by K-Means clustering algorithm.The algorithms developed can be classified as constraint based,greedy and metaheuristic.Constarint based algorithms uses one or more of the various constaints namely the MSR threshold and the MSR difference threshold.The greedy approach makes a locally optimal choice at each stage with the objective of finding the global optimum.In metaheuristic approaches particle Swarm Optimization(PSO) and variants of Greedy Randomized Adaptive Search Procedure(GRASP) are used for the identification of biclusters.These algorithms are implemented on the Yeast and Lymphoma datasets.Biologically relevant and statistically significant biclusters are identified by all these algorithms which are validated by Gene Ontology database.All these algorithms are compared with some other biclustering algorithms.Algorithms developed in this work overcome some of the problems associated with the already existing algorithms.With the help of some of the algorithms which are developed in this work biclusters with very high row variance,which is higher than the row variance of any other algorithm using mean squared residue, are identified from both Yeast and Lymphoma data sets.Such biclusters which make significant change in the expression level are highly relevant biologically.