arXiv subjects

Pritam Ranjan

Publications and source records attributed to Pritam Ranjan.

31 records · Page 2Linked to original sources

arXiv2016

Mean-correction and Higher Order Moments for a Stochastic Volatility Model with Correlated Errors

In an efficient stock market, the log-returns and their time-dependent variances are often jointly modelled by stochastic volatility models (SVMs). Many SVMs assume that errors in log-return and latent volatility process are uncorrelated, which is unrealistic. It turns out that if a non-zero correlation is included in the SVM (e.g., Shephard (2005)), then the expected log-return at time t conditional on the past returns is non-zero, which is not a desirable feature of an efficient stock market. In this paper, we propose a mean-correction for such an SVM for discrete-time returns with non-zero correlation. We also find closed form analytical expressions for higher moments of log-return and its lead-lag correlations with the volatility process. We compare the performance of the proposed and classical SVMs on S&P 500 index returns obtained from NYSE.

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arXiv2016

Design of Computer Experiments for Optimization, Estimation of Function Contours, and Related Objectives

A computer code or simulator is a mathematical representation of a physical system, for example a set of differential equations. Running the code with given values of the vector of inputs, x, leads to an output y(x) or several such outputs. For instance, one application we use for illustration simulates the average tidal power, y, generated as a function of the turbine location, x = (x1, x2), in the Bay of Fundy, Nova Scotia, Canada (Ranjan et al. 2011). Performing scientific or engineering experiments via such a computer code is often more time and cost effective than running a physical experiment. Choosing new runs sequentially for optimization, moving y to a target, etc. has been formalized using the concept of expected improvement (Jones et al. 1998). The next experimental run is made where the expected improvement in the function of interest is largest. This expectation is with respect to the predictive distribution of y from a statistical model relating y to x. By considering a set of possible inputs x for the new run, we can choose that which gives the largest expectation.

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arXiv2015

Modeling an Augmented Lagrangian for Blackbox Constrained Optimization

Constrained blackbox optimization is a difficult problem, with most approaches coming from the mathematical programming literature. The statistical literature is sparse, especially in addressing problems with nontrivial constraints. This situation is unfortunate because statistical methods have many attractive properties: global scope, handling noisy objectives, sensitivity analysis, and so forth. To narrow that gap, we propose a combination of response surface modeling, expected improvement, and the augmented Lagrangian numerical optimization framework. This hybrid approach allows the statistical model to think globally and the augmented Lagrangian to act locally. We focus on problems where the constraints are the primary bottleneck, requiring expensive simulation to evaluate and substantial modeling effort to map out. In that context, our hybridization presents a simple yet effective solution that allows existing objective-oriented statistical approaches, like those based on Gaussian process surrogates and expected improvement heuristics, to be applied to the constrained setting with minor modification. This work is motivated by a challenging, real-data benchmark problem from hydrology where, even with a simple linear objective function, learning a nontrivial valid region complicates the search for a global minimum.

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arXiv2014

Space-filling Latin Hypercube Designs based on Randomization Restrictions in Factorial Experiments

Latin hypercube designs (LHDs) with space-filling properties are widely used for emulating computer simulators. Over the last three decades, a wide spectrum of LHDs have been proposed with space-filling criteria like minimum correlation among factors, maximin interpoint distance, and orthogonality among the factors via orthogonal arrays (OAs). Projective geometric structures like spreads, covers and stars of PG(p-1,q) can be used to characterize the randomization restriction of multistage factorial experiments. These geometric structures can also be used for constructing OAs and nearly OAs (NOAs). In this paper, we present a new class of space-filling LHDs based on NOAs derived from stars of PG(p-1, 2).

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arXiv2013

Efficient Optimization of the Likelihood Function in Gaussian Process Modelling

Gaussian Process (GP) models are popular statistical surrogates used for emulating computationally expensive computer simulators. The quality of a GP model fit can be assessed by a goodness of fit measure based on optimized likelihood. Finding the global maximum of the likelihood function for a GP model is typically very challenging as the likelihood surface often has multiple local optima, and an explicit expression for the gradient of the likelihood function is typically unavailable. Previous methods for optimizing the likelihood function (e.g. MacDonald et al. (2013)) have proven to be robust and accurate, though relatively inefficient. We propose several likelihood optimization techniques, including two modified multi-start local search techniques, based on the method implemented by MacDonald et al. (2013), that are equally as reliable, and significantly more efficient. A hybridization of the global search algorithm Dividing Rectangles (DIRECT) with the local optimization algorithm BFGS provides a comparable GP model quality for a fraction of the computational cost, and is the preferred optimization technique when computational resources are limited. We use several test functions and a real application from an oil reservoir simulation to test and compare the performance of the proposed methods with the one implemented by MacDonald et al. (2013) in the R library GPfit. The proposed method is implemented in a Matlab package, GPMfit.

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arXiv2013

A new Bayesian ensemble of trees classifier for identifying multi-class labels in satellite images

Classification of satellite images is a key component of many remote sensing applications. One of the most important products of a raw satellite image is the classified map which labels the image pixels into meaningful classes. Though several parametric and non-parametric classifiers have been developed thus far, accurate labeling of the pixels still remains a challenge. In this paper, we propose a new reliable multiclass-classifier for identifying class labels of a satellite image in remote sensing applications. The proposed multiclass-classifier is a generalization of a binary classifier based on the flexible ensemble of regression trees model called Bayesian Additive Regression Trees (BART). We used three small areas from the LANDSAT 5 TM image, acquired on August 15, 2009 (path/row: 08/29, L1T product, UTM map projection) over Kings County, Nova Scotia, Canada to classify the land-use. Several prediction accuracy and uncertainty measures have been used to compare the reliability of the proposed classifier with the state-of-the-art classifiers in remote sensing.

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arXiv2013

GPfit: An R package for Gaussian Process Model Fitting using a New Optimization Algorithm

Gaussian process (GP) models are commonly used statistical metamodels for emulating expensive computer simulators. Fitting a GP model can be numerically unstable if any pair of design points in the input space are close together. Ranjan, Haynes, and Karsten (2011) proposed a computationally stable approach for fitting GP models to deterministic computer simulators. They used a genetic algorithm based approach that is robust but computationally intensive for maximizing the likelihood. This paper implements a slightly modified version of the model proposed by Ranjan et al. (2011), as the new R package GPfit. A novel parameterization of the spatial correlation function and a new multi-start gradient based optimization algorithm yield optimization that is robust and typically faster than the genetic algorithm based approach. We present two examples with R codes to illustrate the usage of the main functions in GPfit. Several test functions are used for performance comparison with a popular R package mlegp. GPfit is a free software and distributed under the general public license, as part of the R software project (R Development Core Team 2012).

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arXiv2012

A Short Note on Gaussian Process Modeling for Large Datasets using Graphics Processing Units

The graphics processing unit (GPU) has emerged as a powerful and cost effective processor for general performance computing. GPUs are capable of an order of magnitude more floating-point operations per second as compared to modern central processing units (CPUs), and thus provide a great deal of promise for computationally intensive statistical applications. Fitting complex statistical models with a large number of parameters and/or for large datasets is often very computationally expensive. In this study, we focus on Gaussian process (GP) models -- statistical models commonly used for emulating expensive computer simulators. We demonstrate that the computational cost of implementing GP models can be significantly reduced by using a CPU+GPU heterogeneous computing system over an analogous implementation on a traditional computing system with no GPU acceleration. Our small study suggests that GP models are fertile ground for further implementation on CPU+GPU systems.

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arXiv2012

Sequential Design for Computer Experiments with a Flexible Bayesian Additive Model

In computer experiments, a mathematical model implemented on a computer is used to represent complex physical phenomena. These models, known as computer simulators, enable experimental study of a virtual representation of the complex phenomena. Simulators can be thought of as complex functions that take many inputs and provide an output. Often these simulators are themselves expensive to compute, and may be approximated by "surrogate models" such as statistical regression models. In this paper we consider a new kind of surrogate model, a Bayesian ensemble of trees (Chipman et al. 2010), with the specific goal of learning enough about the simulator that a particular feature of the simulator can be estimated. We focus on identifying the simulator's global minimum. Utilizing the Bayesian version of the Expected Improvement criterion (Jones et al. 1998), we show that this ensemble is particularly effective when the simulator is ill-behaved, exhibiting nonstationarity or abrupt changes in the response. A number of illustrations of the approach are given, including a tidal power application.

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arXiv2012

A Computationally Stable Approach to Gaussian Process Interpolation of Deterministic Computer Simulation Data

For many expensive deterministic computer simulators, the outputs do not have replication error and the desired metamodel (or statistical emulator) is an interpolator of the observed data. Realizations of Gaussian spatial processes (GP) are commonly used to model such simulator outputs. Fitting a GP model to $n$ data points requires the computation of the inverse and determinant of $n \times n$ correlation matrices, $R$, that are sometimes computationally unstable due to near-singularity of $R$. This happens if any pair of design points are very close together in the input space. The popular approach to overcome near-singularity is to introduce a small nugget (or jitter) parameter in the model that is estimated along with other model parameters. The inclusion of a nugget in the model often causes unnecessary over-smoothing of the data. In this paper, we propose a lower bound on the nugget that minimizes the over-smoothing and an iterative regularization approach to construct a predictor that further improves the interpolation accuracy. We also show that the proposed predictor converges to the GP interpolator.

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arXiv2010

Branch and Bound Algorithms for Maximizing Expected Improvement Functions

Deterministic computer simulations are often used as a replacement for complex physical experiments. Although less expensive than physical experimentation, computer codes can still be time-consuming to run. An effective strategy for exploring the response surface of the deterministic simulator is the use of an approximation to the computer code, such as a Gaussian process (GP) model, coupled with a sequential sampling strategy for choosing design points that can be used to build the GP model. The ultimate goal of such studies is often the estimation of specific features of interest of the simulator output, such as the maximum, minimum, or a level set (contour). Before approximating such features with the GP model, sufficient runs of the computer simulator must be completed. Sequential designs with an expected improvement (EI) function can yield good estimates of the features with a minimal number of runs. The challenge is that the expected improvement function itself is often multimodal and difficult to maximize. We develop branch and bound algorithms for efficiently maximizing the EI function in specific problems, including the simultaneous estimation of a minimum and a maximum, and in the estimation of a contour. These branch and bound algorithms outperform other optimization strategies such as genetic algorithms, and over a number of sequential design steps can lead to dramatically superior accuracy in estimation of features of interest.

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arXiv2009

Existence and construction of randomization defining contrast subspaces for regular factorial designs

Regular factorial designs with randomization restrictions are widely used in practice. This paper provides a unified approach to the construction of such designs using randomization defining contrast subspaces for the representation of randomization restrictions. We use finite projective geometry to determine the existence of designs with the required structure and develop a systematic approach for their construction. An attractive feature is that commonly used factorial designs with randomization restrictions are special cases of this general representation. Issues related to the use of these designs for particular factorial experiments are also addressed.

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arXiv2009

$\mathcal{G}$-SELC: Optimization by sequential elimination of level combinations using genetic algorithms and Gaussian processes

Identifying promising compounds from a vast collection of feasible compounds is an important and yet challenging problem in the pharmaceutical industry. An efficient solution to this problem will help reduce the expenditure at the early stages of drug discovery. In an attempt to solve this problem, Mandal, Wu and Johnson [Technometrics 48 (2006) 273--283] proposed the SELC algorithm. Although powerful, it fails to extract substantial information from the data to guide the search efficiently, as this methodology is not based on any statistical modeling. The proposed approach uses Gaussian Process (GP) modeling to improve upon SELC, and hence named $\mathcal{G}$-SELC. The performance of the proposed methodology is illustrated using four and five dimensional test functions. Finally, we implement the new algorithm on a real pharmaceutical data set for finding a group of chemical compounds with optimal properties.

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