2015 IEEE International Conference on Big Data (Big Data), 2015
We provide an algorithm to build quantile regression trees in O(N log N) time, where N is the num... more We provide an algorithm to build quantile regression trees in O(N log N) time, where N is the number of instances in the training set. Quantile regression trees are regression trees that model conditional quantiles of the response variable, rather than the conditional expectation as in standard regression trees. We build quantile regression trees by using the quantile loss function in our node splitting criterion. The performance of our algorithm stems from new online update procedures for both the quantile function and the quantile loss function. We test the quantile tree algorithm in three ways, comparing its running time against implementations of standard regression trees, demonstrating its ability to recover a known set of nonlinear quantile functions, and showing that quantile trees yield smaller test set errors (computed using mean absolute deviation) than standard regression trees. The tests include training sets with up to 16 million instances. Overall, our results enable future use of quantile regression trees for large-scale data mining.
We develop a spectral method for computing the probability density function for delayed random wa... more We develop a spectral method for computing the probability density function for delayed random walks; for such problems, the method is exact to machine precision and faster than existing approaches. In conjunction with step function approximation and the weak Euler-Maruyama discretization, the spectral method can be applied to nonlinear stochastic delay differential equations (SDDE). In essence, this means approximating the SDDE by a delayed random walk, which is then solved using the spectral method. We carry out tests for a particular nonlinear SDDE that shows that this method captures the solution without the need for Monte Carlo sampling.
We examine a Markov tree (MT) model for option pricing in which the dynamics of the underlying as... more We examine a Markov tree (MT) model for option pricing in which the dynamics of the underlying asset are modeled by a non-IID process. We show that the discrete probability mass function of log returns generated by the tree is closely approximated by a continuous mixture of two normal distributions. Using this normal mixture distribution and risk-neutral pricing, we derive a closed-form expression for European call option prices. We also suggest a regression tree-based method for estimating three volatility parameters σ, σ + , and σ − required to apply the MT model. We apply the MT model to price call options on 89 non-dividend paying stocks from the S&P 500 index. For each stock symbol on a given day, we use the same parameters to price options across all strikes and expiries. Comparing against the Black-Scholes model, we find that the MT model's prices are closer to market prices.
We use an exact Bayesian calculation to design classifiers that distinguish whether a finite sequ... more We use an exact Bayesian calculation to design classifiers that distinguish whether a finite sequence drawn from a finite alphabet is a sample path of a Markov chain of order k = 0 or of order k > 0. Three exact Bayes (EB) classifiers are derived, each corresponding to a different prior. We also include a classifier based on the Bayesian Information Criterion (BIC), a popular technique for Markov order estimation. Using thousands of random Markov chains of known order, we test the performance of the classifiers. In both average accuracy and ROC analyses, we find that EB classifiers with informative priors perform better than the BIC classifier, with the difference becoming strikingly large when either the size of the alphabet is large or the length of the sequence is small. We also test the classifiers on five real-world data sets and find that the EB classifications, unlike the BIC classifications, match the orders of the models with highest out-of-sample predictive accuracies.
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Papers by nitesh kumar