Eric S. Fung

On construction of stochastic genetic networks based on gene expression sequences.

Reconstruction of genetic regulatory networks from time series data of gene expression patterns is an important research topic in bioinformatics. Probabilistic Boolean Networks (PBNs) have been proposed as an effective model for gene regulatory networks. PBNs are able to cope with uncertainty, corporate rule-based dependencies between genes and discover the sensitivity of genes in their interactions with other genes. However, PBNs are unlikely to use directly in practice because of huge amount of computational cost for obtaining predictors and their corresponding probabilities. In this paper, we propose a multivariate Markov model for approximating PBNs and describing the dynamics of a genetic network for gene expression sequences. The main contribution of the new model is to preserve the strength of PBNs and reduce the complexity of the networks. The number of parameters of our proposed model is O(n2) where n is the number of genes involved. We also develop efficient estimation methods for solving the model parameters. Numerical examples on synthetic data sets and practical yeast data sequences are given to demonstrate the effectiveness of the proposed model.

On a multivariate Markov chain model for credit risk measurement

In this paper, we use credibility theory to estimate credit transition matrices in a multivariate Markov chain model for credit rating. A transition matrix is estimated by a linear combination of the prior estimate of the transition matrix and the empirical transition matrix. These estimates can be easily computed by solving a set of linear programming (LP) problems. The estimation procedure can be implemented easily on Excel spreadsheets without requiring much computational effort and time. The number of parameters is O(s2 m2 ), where s is the dimension of the categorical time series for credit ratings and m is the number of possible credit ratings for a security. Numerical evaluations of credit risk measures based on our model are presented.

A higher-order Markov model for the Newsboy's problem

Markov models are commonly used in modelling many practical systems such as telecommunication systems, manufacturing systems and inventory systems. However, higher-order Markov models are not commonly used in practice because of their huge number of states and parameters that lead to computational difficulties. In this paper, we propose a higher-order Markov model whose number of states and parameters are linear with respect to the order of the model. We also develop efficient estimation methods for the model parameters. We then apply the model and method to solve the generalised Newsboys problem. Numerical examples with applications to production planning are given to illustrate the power of our proposed model.

A high-order Markov-switching model for risk measurement

In this paper, we introduce a High-order Markov-Switching (HMS) model for measuring the risk of a portfolio. We suppose that the rate of return from a risky portfolio follows an HMS model with the drift and the volatility modulated by a discrete-time weak Markov chain. The states of the weak Markov chain are interpreted as observable states of an economy. We adopt the Value-at-Risk (VaR) as a metric for market risk quantification and examine the high-order effect of the underlying Markov chain on the risk measures via backtesting.

On sparse Fisher discriminant method for microarray data analysis.

One of the applications of the discriminant analysis on microarray data is to classify patient and normal samples based on gene expression values. The analysis is especially important in medical trials and diagnosis of cancer subtypes. The main contribution of this paper is to propose a simple Fisher-type discriminant method on gene selection in microarray data. In the new algorithm, we calculate a weight for each gene and use the weight values as an indicator to identify the subsets of relevant genes that categorize patient and normal samples. A l2 - l1 norm minimization method is implemented to the discriminant process to automatically compute the weights of all genes in the samples. The experiments on two microarray data sets have shown that the new algorithm can generate classification results as good as other classification methods, and effectively determine relevant genes for classification purpose. In this study, we demonstrate the gene selections ability and the computational effectiveness of the proposed algorithm. Experimental results are given to illustrate the usefulness of the proposed model.

Higher-order hidden markov models with applications to DNA sequences

Hidden Markov models (HMMs) have been applied to many real-world applications. Very often HMMs only deal with the first order transition probability distribution among the hidden states. In this paper we develop higher-order HMMs. We study the evaluation of the probability of a sequence of observations based on higher-order HMMs and determination of a best sequence of model states.

Option Valuation under a Multivariate Markov Chain Model

In this paper, we develop an option valuation model in the context of a discrete-time multivariate Markov chain model using the Esscher transform. The multivariate Markov chain provides a flexible way to incorporate the dependency of the underlying asset price processes and price multi-state options written on several dependent underlying assets. In our model, the price of an individual asset can take finitely many values. The market described by our model is incomplete in general, hence there are more than one equivalent martingale pricing measures. We adopt conditional Esscher transform to determine an equivalent martingale measure for option valuation. We also document consequences for option prices of the dependency of the underlying asset prices described by the multivariate Markov chain model.

Building higher-order Markov chain models with EXCEL

Categorical data sequences occur in many applications such as forecasting, data mining and bioinformatics. In this note, we present higher-order Markov chain models for modelling categorical data sequences with an efficient algorithm for solving the model parameters. The algorithm can be implemented easily in a Microsoft EXCEL worksheet. We give a detailed description for the implementation which is accessible and useful to anyone who is interested in the applications of higher-order Markov chain models and has some knowledge of EXCEL.

Risk measures and behaviors for bonds under stochastic interest rate models

Abstract This paper develops a model for measuring the risk inherent from trading a bond position under some important stochastic interest rate models. We employ the value at risk (VaR) and expected shortfall (ES) as proxies for the extreme risk inherent from trading a bond position. In particular, we concern ourselves with the average tail behavior of the real-world profit/loss distribution for a bond position. We investigate the risk behaviors of a bond position under some stochastic interest rate models including the Merton model, the Vasicek model, and the Cox–Ingersoll–Ross (CIR) model.

Unidimensional nonnegative scaling for genome-wide Linkage Disequilibrium maps

The main aim of this paper is to propose and develop a unidimensional nonnegative scaling model to construct Linkage Disequilibrium (LD) maps. The proposed constrained scaling model can be efficiently solved by transforming it to an unconstrained model. The method is implemented in PC Clusters at Hong Kong Baptist University. The LD maps are constructed for four populations from Hapmap data sets with chromosomes of several ten thousand Single Nucleotide Polymorphisms (SNPs). The similarities and dissimilarities of the LD maps are studied and analysed. Computational results are also reported to show the effectiveness of the method using parallel computation.