Vladimir Pozdnyakov

On modeling animal movements using Brownian motion with measurement error

Modeling animal movements with Brownian motion (or more generally by a Gaussian process) has a long tradition in ecological studies. The recent Brownian bridge movement model (BBMM), which incorporates measurement errors, has been quickly adopted by ecologists because of its simplicity and tractability. We discuss some nontrivial properties of the discrete-time stochastic process that results from observing a Brownian motion with added normal noise at discrete times. In particular, we demonstrate that the observed sequence of random variables is not Markov. Consequently the expected occupation time between two successively observed locations does not depend on just those two observations; the whole path must be taken into account. Nonetheless, the exact likelihood function of the observed time series remains tractable; it requires only sparse matrix computations. The likelihood-based estimation procedure is described in detail and compared to the BBMM estimation.

Waiting Times for Patterns and a Method of Gambling Teams

We flip a fair coin five times. Which pattern is “more difficult” to get: HHHHH or HTHTH? If we posed this question to the typical man on the street, the most likely answer would be: the first one. Of course, we know that this answer is not correct, for both patterns have the same probability of occurring, namely, 1/32. However, there is a sense in which a street-smart person is, in fact, correct. If we flip the coin without stopping, then the average waiting time until the first occurrence of the pattern HHHHH is 62, whereas for the pattern HTHTH it is 42. From this perspective, the pattern HHHHH is indeed “more difficult” to get. Now, if we ask a person familiar with probability theory (but unfamiliar with this particular topic) to rank the average waiting times until the patterns HHHHH, HHHHT , HHHTH, and HTHTH occur, then most likely the first pattern would get rank 1 (the longest average waiting time), the second—2, the third—3, and the last one—4 (the shortest average waiting time). This ranking is based on the “intuitive” idea that long runs of the same outcomes, such as HHHH or HHHHH, require more time until they occur. In fact, the average waiting times are 62, 32, 34, and 42, respectively. All the foregoing waiting times are easily and elegantly obtained by using martingale theory and the “optional stopping theorem,” as shown in the classical paper by Li [16] and briefly described in section 2. Our focus in the present paper is the average time until we observe the first of several different patterns. Suppose, for instance, that Melanie flips a coin until she observes either HHHTH or HTHTH, while Kyle flips another coin until he observes either HHHHT or HHHTH. Since Kyle was assigned the two patterns with the shortest waiting times, 32 and 34 versus 34 and 42, one would expect him to have a shorter average waiting time. In fact, the averages are the same—22 for both Melanie and Kyle. Let us present another counterintuitive fact. Consider the two patterns HHHHT and HHHTH. What is the probability that in a stochastic sequence of heads and tails the pattern HHHHT appears earlier than HHHTH? Since the average waiting times (32 and 34, respectively) are close to each other, one might guess that the probability would be reasonably close to 1/2. However, the exact answer is 2/3! As we will see, this probability is determined by the relationship between patterns rather than by their individual average waiting times. Finally, consider two special patterns: a run of Hs of length r and a run of T s of length ρ . The expected waiting time until the run of Hs is 2 − 2, while for the run of T s it is 2 − 2. We can ask: what is the expected time until either of these two runs happens for the first time? Using results presented in this article

A Repeated Significance Test With Applications To Sequential Detection In Sensor Networks

In this paper we introduce a randomly truncated sequential hypothesis test. Using the framework of a repeated significance test (RST), we study a sequential test with truncation time based on a random stopping time. Using the functional central limit theorem (FCLT) for a sequence of statistics, we derive a general result that can be employed in developing a repeated significance test with random sample size. We present effective methods for evaluating accurate approximations for the probability of type I error and the power function. Numerical results are presented to evaluate the accuracy of these approximations. We apply the proposed test to a decentralized sequential detection problem in sensor networks (SNs) with communication constraints. Finally, a sequential detection problem with measurements at random times is investigated.

A note on functional CLT for truncated sums

Let {X,Xi}i[greater-or-equal, slanted]1 be i.i.d. random variables with a symmetric continuous distribution and EX2=[infinity], and {bn}n[greater-or-equal, slanted]1 be a sequence of increasing positive numbers. When X belongs to the Feller class, and nP(X>bn)~[gamma]n[short up arrow][infinity], a functional CLT for the truncated sums Sn=[summation operator]i=1nXiIXi[less-than-or-equals, slant]bn is proved.

Bugs on a budget: Distributed sensing with cost for reporting and nonreporting

We consider a simple model of sequential decisions made by a fusion agent that receives binary-passive reports from distributed sensors. The main result is an explicit formula for the probability of making a decision before a fixed budget is exhausted. These results depend on the relationship between a special ruin problem for a “lazy random walk” and a traditional biased walk.

Martingale Methods for Patterns and Scan Statistics

We show how martingale techniques (both old and new) can be used to obtain otherwise hard-to-get information for the moments and distributions of waiting times for patterns in independent or Markov sequences. In particular, we show how these methods provide moments and distribution approximations for certain scan statistics, including variable length scan statistics. Each general problem that is considered is also illustrated with a concrete example confirming the computational tractability of the method.

A moving-resting process with an embedded Brownian motion for animal movements

Animal movements are of great importance in studying home ranges, migration routes, resource selection, and social interactions. The Global Positioning System provides relatively continuous animal tracking over time and long distances. Nevertheless, the continuous trajectory of an animal’s movement is usually only observed at discrete time points. Brownian bridge models have been used to model movement of an animal between two observed locations within a reasonably short time interval. Assuming that animals are in perpetual motion, these models ignore inactivity such as resting or sleeping. Using the latest developments in applied probability, we propose a moving–resting process model where an animal is assumed to alternate between a moving state, during which it moves in a Brownian motion, and a resting state, during which it does not move. Theoretical properties of the process are studied as a first step towards more realistic models for animal movements. Analytic expressions are derived for the distribution of one increment and two consecutive increments, and are validated with simulations. The induced bridge model conditioning on the starting and end points is used to compute an animal’s probability of occurrence in an observation area during the time of observation, which has wide applications in wildlife behavior research.

A note on occurrence of gapped patterns in i.i.d. sequences

A new martingale technique is developed to find formulas for the expected value and generating function of the waiting time until one observes a gapped pattern (or a structured motif) in an i.i.d. sequence of random letters from a finite alphabet.

A Repeated Significance Test for Distributions with Heavy Tails

ABSTRACT Repeated significance tests are frequently used in areas of science and technology in which the data is accumulated sequentially over time. In this article we develop a repeated significance test for independent and identically distributed observations from a continuous symmetric distribution with heavy tails, infinite variance, and possibly no mean. This repeated significance test is nonparametric in nature, as it is applicable to a class of distributions with a specified tail behavior. We present an algorithm for selecting the continuation region associated with the repeated significance test that achieves specified significance level and power at a given alternative. Moreover, we derive approximations for the power function and expected sample size. Numerical results are presented to evaluate the performance of these approximations.

Convexity Bias in the Pricing of Eurodollar Swaps

The traditional use of LIBOR futures prices to obtain surrogates for the Eurodollar forward rates is proved to yield a systematic bias in the pricing of Eurodollar swaps when one assumes that the yield curve is well described by the Heath-Jarrow-Morton model. The resulting theoretical inequality is consistent with the empirical observations of Burghardt and Hoskins (1995), and it provide a theoretical basis for price anomalies that are suggested by more recent empirical data.