Adaptive Lanczos methods for recursive condition estimation

Estimates for the condition number of a matrix are useful in many areas of scientific computing, including: recursive least squares computations, optimization, eigenanalysis, and general nonlinear problems solved by linearization techniques where matrix modification techniques are used. The purpose of this paper is to propose anadaptiveLanczosestimator scheme, which we callale, for tracking the condition number of the modified matrix over time. Applications to recursive least squares (RLS) computations using the covariance method with sliding data windows are considered.ale is fast for relatively smalln-parameter problems arising in RLS methods in control and signal processing, and is adaptive over time, i.e., estimates at timet are used to produce estimates at timet+1. Comparisons are made with other adaptive and non-adaptive condition estimators for recursive least squares problems. Numerical experiments are reported indicating thatale yields a very accurate recursive condition estimator.Research supported by the US Air Force under grant no. AFOSR-88-0285.Research supported by the US Army under grant no. DAAL03-90-G-105.Research supported by the US Air Force under grant no. AFOSR-88-0285.     

A genetic filtering problem ∗

Members of a population of fixed size N can be in any one of n states. In discrete time the individuals jump from one state to another, independently of each other, and with probabilities described by a homogeneous Markov chain. At each time a sample of size M is withdrawn, (with replacement). Based on these observations, and using the techniques of Hidden Markov Models, recursive estimates for the distribution of the population are obtained     

The structure and distances in Yule recursive trees

Based on uniform recursive trees, we introduce random trees with the factor of time, which are named Yule recursive trees. The structure and the distance between the vertices in Yule recursive trees are investigated in this paper. For arbitrary time t > 0, we first give the probability that a Yule recursive tree Y_t is isomorphic to a given rooted tree γ; and then prove that the asymptotic distribution of ζ_t,m, the number of the branches of size m, is the Poisson distribution with parameter λ = 1/m. Finally, two types of distance between vertices in Yule recursive trees are studied, and some limit theorems for them are established.© 2007 Wiley Periodicals, Inc. Random Struct. Alg., 2007     

Analysis of economic time series: effects of extremal observations on testing heteroscedastic components

Macroeconomic and financial time series are often tested for the presence of non‐linearity effects. Sometimes, small patches of extremal observations may wrongly influence non‐linearity tests. In this paper, a robust analysis of the Lagrange multiplier (LM) test for GARCH components is suggested. With Monte‐Carlo simulation we show that extreme observations might cause over‐estimation of the number of GARCH components, with the main contribution consisting by introducing the forward search method into the GARCH model family. Using robust estimators of regression coefficients and graphical displays of results, the effect of influential observations on estimates can be efficiently monitored. Analysing macroeconomic and financial time series we show that identifying the order of a GARCH model can be unduly influenced by a few isolated large values, and extremal observations affect p‐values and t‐statistics in an unexpected manner. Copyright © 2004 John Wiley & Sons, Ltd.     

Stability of Regression‐Based Monte Carlo Methods for Solving Nonlinear PDEs

The regression‐based Monte Carlo methods for backward stochastic differential equations (BSDEs) have been the object of considerable research, particularly for solving nonlinear partial differential equations (PDEs). Unfortunately, such methods often become unstable when implemented with small time steps because the variance of gradient estimates is inversely proportional to the time step (σ²∼ 1/Δ t). Recently new variance reduction techniques were introduced to address this problem in~a paper by the author and Avellaneda. The purpose of this paper is to provide a rigorous justification for these techniques in the context of the discrete‐time BSDE scheme of Bouchard and Touzi. We also suggest a new higher‐order scheme that makes the variance proportional to the time step (σ²∼Δ t). These techniques are easy to implement. Numerical examples strongly indicate that they render the regression‐based Monte Carlo methods stable for small time steps and thus viable for numerical solution of nonlinear PDEs.© 2016 Wiley Periodicals, Inc.     

Asymptotic equivalence for nonparametric generalized linear models

Summary. We establish that a non-Gaussian nonparametric regression model is asymptotically equivalent to a regression model with Gaussian noise. The approximation is in the sense of Le Cam's deficiency distance Δ; the models are then asymptotically equivalent for all purposes of statistical decision with bounded loss. Our result concerns a sequence of independent but not identically distributed observations with each distribution in the same real-indexed exponential family. The canonical parameter is a value f(t _i ) of a regression function f at a grid point t _i (nonparametric GLM). When f is in a H?lder ball with exponent we establish global asymptotic equivalence to observations of a signal Γ(f(t)) in Gaussian white noise, where Γ is related to a variance stabilizing transformation in the exponential family. The result is a regression analog of the recently established Gaussian approximation for the i.i.d. model. The proof is based on a functional version of the Hungarian construction for the partial sum process. Received: 4 February 1997     

Treed Regression

Abstract

Given a data set consisting of n observations on p independent variables and a single dependent variable, treed regression creates a binary tree with a simple linear regression function at each of the leaves. Each node of the tree consists of an inequality condition on one of the independent variables. The tree is generated from the training data by a recursive partitioning algorithm. Treed regression models are more parsimonious than CART models because there are fewer splits. Additionally, monotonicity in some or all of the variables can be imposed.     

Control problems with time-lag. II

Consider a continuous time Markov chain with stationary transition probabilities. A function of the state is observed. A regular conditional probability distribution for the trajectory of the chain, given observations up to time t, is obtained. This distribution also corresponds to a Markov chain, but the conditional chain has nonstationary transition probabilities. In particular, computation of the conditional distribution of the state at time s is discussed. For s > t, we have prediction (extrapolation), while s < t corresponds to smoothing (interpolation). Equations for the conditional state distribution are given on matrix form and as recursive differential equations with varying s or t. These differential equations are closely related to Kolmogorov's forward and backward equations. Markov chains with one observed and one unobserved component are treated as a special case. In an example, the conditional distribution of the change-point is derived for a Poisson process with a changing intensity, given observations of the Poisson process.     

Strong Universal Pointwise Consistency of Recursive Regression Estimates

For semi-recursive and recursive kernel estimates of a regression of Y on X (d-dimensional random vector X, integrable real random variable Y), introduced by Devroye and Wagner and by Révész, respectively, strong universal pointwise consistency is shown, i.e. strong consistency P _X -almost everywhere for general distribution of (X, Y). Similar results are shown for the corresponding partitioning estimates.     

Simple Non-trivial Designs with an Arbitrary Automorphism Group

We give a new recursive construction of simple non-trivial designs. Using this construction, we show that given a natural number t and a finite group G, a simple non-trivial t-design admitting an automorphism group isomorphic to G exists. Further, we apply our construction to get a recursive construction of large sets.     

The tangent approximation to one-sided Brownian exit densities

Summary Let p(t) be the density of the first-exit time of a Brownian motion over a one-sided moving boundary, and let p ₁(t) be the density at t of the first-exit time over the tangent to the boundary at t. When is p ₁(t) a good approximation to p(t)? We investigate this question by means of a new integral equation for p(t) which makes possible explicit estimates for the error of the approximation.Work supported by the Deutsche Forschungsgemeinschaft at the Sonderforschungsbereich 123, Universität Heidelberg     

Adaptive estimates for autoregressive processes

Let {X _t :t=0, ±1, ±2, ...} be a stationaryrth order autoregressive process whose generating disturbances are independent identically distributed random variables with marginal distribution functionF. Adaptive estimates for the parameters of {X _t } are constructed from the observed portion of a sample path. The asymptotic efficiency of these estimates relative to the least squares estimates is greater than or equal to one for all regularF. The nature of the adaptive estimates encourages stable behavior for moderate sample sizes. A similar approach can be taken to estimation problems in the general linear model. This research was partially supported by National Science Foundation Grant GP-31091X. American Mathematical Society 1970 subject classification. Primary 62N10; Secondary 62G35. Key words and phrases: autoregressive process, adaptive estimates, robust estimates.     

Perfect difference families,perfect difference matrices,and related combinatorial structures

The existence problems of perfect difference families with block size k, k=4,5, and additive sequences of permutations of length n, n=3,4, are two outstanding open problems in combinatorial design theory for more than 30 years. In this article, we mainly investigate perfect difference families with block size k=4 and additive sequences of permutations of length n=3. The necessary condition for the existence of a perfect difference family with block size 4 and order v, or briefly (v, 4,1)‐PDF, is v≡1(mod12), and that of an additive sequence of permutations of length 3 and order m, or briefly ASP (3, m), is m≡1(mod2). So far, (12t+1,4,1)‐PDFs with t<50 are known only for t=1,4−36,41,46 with two definiteexceptions of t=2,3, and ASP (3, m)'s with odd 3<m<200 are known only for m=5,7,13−29,35,45,49,65,75,85,91,95,105,115,119,121,125,133,135,145,147,161,169,175,189,195 with two definite exceptions of m=9,11. In this article, we show that a (12t+1,4,1)‐PDF exists for any t⩽1,000 except for t=2,3, and an ASP (3, m) exists for any odd 3<m<200 except for m=9,11 and possibly for m=59. The main idea of this article is to use perfect difference families and additive sequences of permutations with “holes”. We first introduce the concepts of an incomplete perfect difference matrix with a regular hole and a perfect difference packing with a regular difference leave, respectively. We show that an additive sequence of permutations is in fact equivalent to a perfect difference matrix, then describe an important recursive construction for perfect difference matrices via perfect difference packings with a regular difference leave. Plenty of perfect difference packings with a desirable difference leave are constructed directly. We also provide a general recursive construction for perfect difference packings, and as its applications, we obtain extensive recursive constructions for perfect difference families, some via incomplete perfect difference matrices with a regular hole. Examples of perfect difference packings directly constructed are used as ingredients in these recursive constructions to produce vast numbers of perfect difference families with block size 4. © 2010 Wiley Periodicals, Inc. J Combin Designs 18: 415–449, 2010     

The coolest way to generate combinations

We present a practical and elegant method for generating all (s,t)-combinations (binary strings with s zeros and t ones): Identify the shortest prefix ending in 010 or 011 (or the entire string if no such prefix exists), and rotate it by one position to the right. This iterative rule gives an order to (s,t)-combinations that is circular and genlex. Moreover, the rotated portion of the string always contains at most four contiguous runs of zeros and ones, so every iteration can be achieved by transposing at most two pairs of bits. This leads to an efficient loopless and branchless implementation that consists only of two variables and six assignment statements. The order also has a number of striking similarities to colex order, especially its recursive definition and ranking algorithm. In the light of these similarities we have named our order cool-lex!     

General M-estimates for contaminated p th-order autoregressive processes: Consistency and asymptotic normality

Summary This work is concerned with simultaneous estimation of coefficients and a scale parameter of a p th-order autoregressive process (X _t ). The observations are Y _t =V _t Z _t +(1-V _t )X _t where (Z _t) is a contaminating process and (V _t ) represents the proportion of contamination. If (X _t ) or (Z _t ) have heavy tails both least squares estimates and ordinary M-estimates are seriously affected. Under general conditions we prove consistency and asymptotic normality of a general class of M-estimates which contains some M-estimates studied by Denby and Martin [6].     

Domains of t-Functions

A nonempty and nonconstant partial recursive function such that any function resembling it is recursively isomorphic to it is called a t-function. It is proved that the domain of any t-function is neither a simple nor a pseudosimple set.     

Products of mixed covering arrays of strength two

A covering array CA(N;t,k, v is an N × k array such that every N × t subarray contains all t‐tuples from v symbols at least once, where t is the strength of the array. Covering arrays are used to generate software test suites to cover all t‐sets of component interactions. The particular case when t = 2 (pairwise coverage) has been extensively studied, both to develop combinatorial constructions and to provide effective algorithmic search techniques. In this paper, a simple “cut‐and‐paste” construction is extended to covering arrays in which different columns (factors) admit different numbers of symbols (values); in the process an improved recursive construction for covering arrays with t = 2 is derived. © 2005 Wiley Periodicals, Inc. J Combin Designs 14: 124–138, 2006     

Some limit results for probabilities estimates of Brownian motion with polynomial drift

Let (B _t + f(t)) _t∈[0,+∞) be a Brownian motion with polynomial drift f(t), where f(t) is a polynomial. Some Limit Results for Lower tail and large deviation probabilities estimates, and Level crossing probabilities estimates of (B _t + f(t)) _t∈[0,+∞) are given in this paper.     

Stable Determination of X-Ray Transforms of Time Dependent Potentials from Partial Boundary Data

We consider compact smooth Riemmanian manifolds with boundary of dimension greater than or equal to two. For the initial-boundary value problem for the wave equation with a lower order term q(t, x), we can recover the X-ray transform of time dependent potentials q(t, x) from the dynamical Dirichlet-to-Neumann map in a stable way. We derive conditional Hölder stability estimates for the X-ray transform of q(t, x). The essential technique involved is the Gaussian beam Ansatz, and the proofs are done with the minimal assumptions on the geometry for the Ansatz to be well-defined.