Classical Occupancy Problem Research Articles

The Classical Occupancy Distribution: Computation and Approximation

We examine the discrete distributional form that arises from the “classical occupancy problem,” which looks at the behavior of the number of occupied bins when we allocate a given number of balls uniformly at random to a given number of bins. We review the mass function and moments of the classical occupancy distribution and derive exact and asymptotic results for the mean, variance, skewness and kurtosis. We develop an algorithm to compute a cubic array of log-probabilities from the classical occupancy distribution. This algorithm allows the computation of large blocks of values while avoiding underflow problems in computation. Using this algorithm, we compute the classical occupancy distribution for a large block of values of balls and bins, and we measure the accuracy of its asymptotic approximation using the normal distribution. We analyze the accuracy of the normal approximation with respect to the variance, skewness and kurtosis of the distribution. Based on this analysis, we give some practical guidance on the feasibility of computing large blocks of values from the occupancy distribution, and when approximation is required.

Degree sequences of random digraphs and bipartite graphs

We investigate the joint distribution of the vertex degrees in three models of random bipartite graphs. Namely, we can choose each edge with a specified probability, choose a specified number of edges, or specify the vertex degrees in one of the two colour classes. This problem can alternatively be described in terms of the row and sum columns of random binary matrix or the in-degrees and out-degrees of a random digraph, in which case we can optionally forbid loops. It can also be cast as a problem in random hypergraphs, or as a classical occupancy, allocation, or coupon collection problem. In each case, provided the two colour classes are not too different in size nor the number of edges too low, we define a probability space based on independent binomial variables and show that its probability masses asymptotically equal those of the degrees in the graph model almost everywhere. The accuracy is sufficient to asymptotically determine the expectation of any joint function of the degrees whose maximum is at most polynomially greater than its expectation.

The size of the character state space affects the occurrence and detection of homoplasy: modelling the probability of incompatibility for unordered phylogenetic characters.

This study models the probability of incompatibility versus compatibility for binary or unordered multistate phylogenetic characters, by treating the allocation of taxa to character states as a classical occupancy problem in probability. It is shown that, under this model, the number of character states has a non-linear effect on the probability of character incompatibility, which is also affected by the number of taxa. Effects on homoplasy from the number of character states are further explored using evolutionary computer simulations. The results indicate that the character state space affects both the known levels of homoplasy (recorded during simulated evolution) and those inferred from parsimony analysis of the resulting character data, with particular relevance for morphological phylogenetic analyses which generally use the parsimony method. When the evolvable state space is large (more potential states per character) there is a reduction in the known occurrence of homoplasy (as reported previously). However, this is not always reflected in the levels of homoplasy detected in a parsimony analysis, because higher numbers of states per character can lead to an increase in the probability of character incompatibility (as well as the maximum homoplasy measurable with some indices). As a result, inferred trends in homoplasy can differ markedly from the underlying trend (that recorded during evolutionary simulation). In such cases, inferred homoplasy can be entirely misleading with regard to tree quality (with higher levels of homoplasy inferred for better quality trees). When rates of evolution are low, commonly used indices such as the number of extra steps (H) and the consistency index (CI) provide relatively good measures of homoplasy. However, at higher rates, estimates may be improved by using the retention index (RI), and particularly by accounting for homoplasy measured among randomised character data using the homoplasy excess ratio (HER).

Estimation of the Population Size in the Classical Occupancy Problem

In the classical occupancy problem where the random variable X is the number of N elements selected by K individuals when each element is equally likely to be chosen by any of the individuals, it is desired to estimate N. Three estimators given in the literature are compared with three estimators derived in this article, two of which are based on Bayesian methods, utilizing a simulation study. One of the Bayes estimators appears to perform the best along with one proposed in the literature in 1968. The estimators are also compared using data obtained from a cemetery in Ohio.

On the asymptotic behavior of a sequence of random variables of interest in the classical occupancy problem

In the classical occupancy problem one puts balls in boxes, and each ball is independently assigned to any fixed box with probability . It is well known that, if we consider the random number of balls required to have all the boxes filled with at least one ball, the sequence converges to 1 in probability. Here we present the large deviation principle associated to this convergence. We also discuss the use of the Gartner Ellis Theorem for the proof of some parts of this large deviation principle

Usefulness of Asymptotic Distributions in the Classical Occupancy Problem

Approximations to the distribution of a discrete random variable originating from the classical occupancy problem are explored. The random variable X of interest is defined to be how many of N elements selected by or assigned to K individuals when each of the N elements is equally likely to be chosen by or assigned to any of the K individuals. Assuming that N represents the number of cells and each of the K individuals is placed in exactly one of the cells, X can also be defined as the number of cells occupied by the Kindividuals. In the literature, various asymptotic results for the distributions of X and (N − X) are given; however, no guidelines are specified with respect to their utilization. In this article, these approximations are explored for various values of K and N, and rules of thumb are given for their appropriate use.

Cross-situational and supervised learning in the emergence of communication

Scenarios for the emergence or bootstrap of a lexicon involve the repeated interaction between at least two agents who must reach a consensus on how to name N objects using H words. Here we consider minimal models of two types of learning algorithms: cross-situational learning, in which the individuals determine the meaning of a word by looking for something in common across all observed uses of that word, and supervised operant conditioning learning, in which there is strong feedback between individuals about the intended meaning of the words. Despite the stark differences between these learning schemes, we show that they yield the same communication accuracy in the limits of large N and H, which coincides with the result of the classical occupancy problem of randomly assigning N objects to H words. Key words: lexicon bootstrapping; cross-situational learning; supervised learning; random occupancy problems

Competition between discrete random variables, with applications to occupancy problems

Consider n players whose “scores” are independent and identically distributed values { X i } i = 1 n from some discrete distribution F. We pay special attention to the cases where (i) F is geometric with parameter p → 0 and (ii) F is uniform on { 1 , 2 , … , N } ; the latter case clearly corresponds to the classical occupancy problem. The quantities of interest to us are, first, the U-statistic W which counts the number of “ties” between pairs i, j; second, the univariate statistic Y r , which counts the number of strict r-way ties between contestants, i.e., episodes of the form X i 1 = X i 2 = ⋯ = X i r ; X j ≠ X i 1 ; j ≠ i 1 , i 2 , … , i r ; and, last but not least, the multivariate vector Z AB =( Y A , Y A+1 ,…, Y B ). We provide Poisson approximations for the distributions of W, Y r and Z AB under some general conditions. New results on the joint distribution of cell counts in the occupancy problem are derived as a corollary.

AVERAGE-CASE "MESSY" BROADCASTING

Current studies of "messy" broadcasting have so far concentrated on finding worst-case times. However, such worst-case scenarios are extremely unlikely to occur in general. Hence, determining average-case times or tight upper bounds for completing "messy" broadcasting in various network topologies is both necessary and meaningful in practice. In this paper, we focus on seeking the average-case "messy" broadcast times of stars, paths, cycles, and d-ary trees, and finding good upper bounds for hypercubes. Finally, we derive a recursive formula to express the average-case time for a specific "messy" broadcast model on a complete graph using a classical occupancy problem in probability theory, and provide a nice simulation result which indicates that this model behaves like classical broadcasting.

Refined Large Deviation Asymptotics for the Classical Occupancy Problem

In this paper refined large deviation asymptotics are derived for the classical occupancy problem. The asymptotics are established for a sequential filling experiment and an occupancy experiment. In the first case the random variable of interest is the number of balls required to fill a given fraction of the urns, while in the second a fixed number of balls are thrown and the random variable is the fraction of nonempty urns.

Size of the protein-coding genome and rate of molecular evolution

In diploid populations of size N, there will be 2 Nmu mutations per nucleotide (nt) site (or per locus) per generation (mu stands for mutation rate). If either the population or the coding genome double in size, one expects 4 Nmu mutations. What is important is not the population size per se but the number of genes (coding sites), the two being often interconverted. Here we compared the total physical length of protein-coding genomes (n) with the corresponding absolute rates of synonymous substitution (K(S)), an empirical neutral reference. In the classical occupancy problem and in the coupons collector (CC) problem, n was expressed as the mean rate of change (K(CC)). Despite inherently very low power of the approaches involving averaging of rates, the mode of molecular evolution of the total size phenotype of the coding genome could be evidenced through differences between the genomic estimates of K(CC) [K(CC)=1/(ln n + 0.57721) n] and rate of molecular evolution, K(S). We found that (1) the estimates of n and K(S) are reciprocally correlated across taxa (r=0.812; p<< 0.001); (2) the gamete-cell division hypothesis (Chang et al. Proc Natl Acad Sci USA 91:827-831, 1994) can be confirmed independently in terms of K(CC)/K(S) ratios; (3) the time scale of molecular evolution changes with change in mutation rate, as previously shown by Takahata (Proc Natl Acad Sci USA 87:2419-2423, 1990), Takahata et al. (Genetics 130:925-938, 1992), and Vekemans and Slatkin (Genetics 137:1157-1165, 1994); (4) the generation time and population size (Lynch and Conery, Science 302:1401-1404, 2003) effects left their "signatures" at the level of the size phenotype of the protein-coding genome.

Statistical Weakness of Multiplexed Sequences

It is proved that the out-of-phase autocorrelation of a well-known class of multiplexed sequences for cryptographic and spread spectrum applications is necessarily high for relatively small values of the phase shifts. Related design criteria for the multiplexer generator including the use of full positive difference sets are proposed. By generalizing the classical occupancy problem, higher order statistical weaknesses of multiplexed sequences are also established.

Statistical Weakness of Multiplexed Sequences,

It is proved that the out-of-phase autocorrelation of a well-known class of multiplexed sequences for cryptographic and spread spectrum applications is necessarily high for relatively small values of the phase shifts. Related design criteria for the multiplexer generator including the use of full positive difference sets are proposed. By generalizing the classical occupancy problem, higher order statistical weaknesses of multiplexed sequences are also established.

Optical switch dimensioning and the classical occupancy problem

AbstractResults for optical switch dimensioning are obtained by analysing an urn occupancy problem in which a random number of balls is used. This analysis is applied to a high speed bufferless optical switch which uses tuneable wavelength converters to resolve contention between packets at the output fibres. Under symmetric packet routing the urn problem reduces to the classical occupancy problem. Since the problem is large scale and the loss probabilities are small, exact analysis by combinatorial methods is problematic. As an alternative, we outline a large deviations approximation which may be generalised in various ways. Copyright © 2002 John Wiley &amp; Sons, Ltd.

Occupancy Numbers in Testing Random Number Generators

The classical occupancy problem where n balls are placed in N cells is used for testing of random number generators. We show that the statistics of appropriately chosen occupancy numbers are incompatible with the statistics of many pseudorandom number generators (PRNGs) evenif they are trun cated. More thanthat, the in compatibility shows up onrelatively small samples long before the period of the PRNG is reached. We introduce generalized Fermi-Dirac models as idealized models for PRNGs. These models are used to make educated guesses of how large the sample sizes should be to detect the deficiencies of the PRNGs under study. We use the developed occupancy tests together with some other ones to examine the performance of several widely used random number generators, including random() of the UNIX C library. We found that random() failed two of the conducted tests rather badly. We also tested a true random number generator based on α-decay which passed all our relevant tests successfully.

Measures of Distinctness for Random Partitions and Compositions of an Integer

Partitions and compositions of integers are, besides their intrinsic interests, usually used as theoretical models for evolutionary processes in different contexts: statistical mechanics, theory of quantum strings, populaw x tion biology, nonparametric statistics, etc.; cf. 1, 4, 8, 10, 12, 30, 49, 54 . Also parameters in partitions often have natural interpretations in terms w x Ž of characters in symmetric groups; cf. 15, 47 . Thus properties statistical, . algebraic, analytic, . . . of these objects received constant attention in the literature. In many situations, the notion of ‘‘degree of distinctness’’ naturally arises. The classical birthday paradox states that one needs on the average ) 23 people to discover two that have the same birthday with probability w x ) 1r2, assuming all birth dates to be equally likely; cf. 16 . The coupon collector problem is similar: what is the expected number of coupons one needs to gather before a full collection, under suitable probability assumptions on the issuing of the coupons? In applications in which only the first Ž . product element, particle, . . . is ‘‘expensive’’ and the ‘‘cost’’ of the remaining reproductions is negligible, the study of the measures of distinctness becomes meaningful and important. The number of distinct outcomes Ž . in a sequence of multinomial trials the classical occupancy problem has w x w x wide applications; see, for example, Knuth 34 , Johnson and Kotz 28 , w x w x w x Kolchin et al. 35 , Arato and Benczur 5 , and Vitter and Chen 50 . The number of distinct sites visited by a random walk plays an important role

Tail bounds for occupancy and the satisfiability threshold conjecture

AbstractThe classical occupancy problem is concerned with studying the number of empty bins resulting from a random allocation of m balls to n bins. We provide a series of tail bounds on the distribution of the number of empty bins. These tail bounds should find application in randomized algorithms and probabilistic analysis. Our motivating application is the following well‐known conjecture on threshold phenomenon for the satisfiability problem. Consider random 3‐SAT formulas with cn clauses over n variables, where each clause is chosen uniformly and independently from the space of all clauses of size 3. It has been conjectured that there is a sharp threshold for satisfiability at c* ≈︁ 4.2. We provide a strong upper bound on the value of c*, showing that for c &gt; 4.758 a random 3‐SAT formula is unsatisfiable with high probability. This result is based on a structural property, possibly of independent interest, whose proof needs several applications of the occupancy tail bounds.

A class of population genetic questions formulated as the generalized occupancy problem.

In categorical genetic data analysis when the sampling units are classified into an arbitrary number of distinct classes, sometimes the sample size may not be large enough to apply large sample approximations for hypothesis testing purposes. Exact sampling distributions of several statistics are derived here, using combinatorial approaches parallel to the classical occupancy problem to help overcome this difficulty. Since the multinomial probabilities can be unequal, this situation is described as a generalized occupancy problem. The sampling properties derived are used to examine nonrandomness of occurrence of mutagen-induced mutations across loci, to devise tests of Hardy-Weinberg proportions of genotype frequencies in the presence of a large number of alleles, and to provide a global test of gametic phase disequilibrium of several restriction site polymorphisms.

Occupancy with two types of balls

The classical occupancy problem is extended to the case where two types of balls are thrown. In particular, the probability that no urn contains both types of balls is studied. This is a birthday problem in two groups of boys and girls to consider the coincidence of a boy's and a girl's birthday. Let N~ and N2 denote the numbers of balls of each type thrown one by one when the first collision between the two types occurs in one ofm urns. Then NIN2/m is asymptotically exponentially distributed as m tends to infinity. This problem is related to the security evaluation of authentication procedures in electronic message communication.

Properties of Bootstrap Samples

SYNOPTIC ABSTRACTEfron's version of the “bootstrap” procedure was devised as a method for obtaining nonparametric estimates of Standard deviations and biases of estimators. Important applications include error rate of classifiers, non-linear regression, econometric modeling, discriminant analysis and principal components. Applications of the bootstrap procedure usually require the generation of “bootstrap samples.” This paper presents a connection between bootstrap samples and the classical occupancy problem. Small sample properties of a bootstrap sample are obtained based on the application of results on the occupancy problem. The implications of these results on the estimate of misclassification probability in discriminant analysis are addressed.