PAS205 Probability Modelling: Lecture 21 1 PAS205 PROBABILITY MODELLING: Lecture 21 21.1 Simulation The models we have studied <a href="/keyword/lend-themselves/" >lend themselves</a> to many mathematical techniques for obtaining explicit solutions, because of, for example, the nice properties of standard distributions and the generally strong independence assumptions. However, the more realistic a model becomes, the more difficult it tends to be to find such explicit solutions. One way of finding approximate solutions is simulation, which entails producing realisations or sample paths of a process by substituting for the random quantities in the model pseudo-random variates which are numbers produced by a deterministic algorithm but whose behaviour is intended to imitate the behaviour of numbers sampled independently from the appropriate distribution. Repeating this simulation a large number of times with different sets of pseudo-random variates may then enable us to estimate, for example, probabilities or means with reasonable accuracy. Neave Table 7.1 (random digits) is an example of a sequence of pseudo-random variates; each digit is meant to look as though it has been sampled from the uniform distribution on {0, 1, 2, . . . , 9}, independently of the rest. Such sequences may be tested in various ways to see whether they exhibit the desired behaviour e.g. do the ten digits appear approximately equally frequently in a long sequence? A good algorithm is one which passes all these tests; we shall not go into how these algorithms work. An important result in this area is that if U1 , U2 , . . . is a sequence of pseudo-random variates from the uniform distribution on the interval [0, 1], then for any univariate distribution function F , the transformation to F -1 (U1 ), F -1 (U2 ), . . . gives a sequence of pseudo-random variates from the distribution which has F as its distribution function, noting that if this distribution is discrete then F is a step function and we have to take care to define F -1 (u) := min{x : F (x) u} for 0 u 1. However, with a computer package we can specify the desired distribution out of a wide variety of standard distributions, and we may not need to do the above transformation ourselves. 21.2 Simulation of Poisson processes The properties which we have discussed of the basic Poisson process and its variations suggest ways of simulating these processes. Some examples follow. (i) Basic Poisson process via inter-occurrence times Since inter-occurrence times are independent exponential with parameter , we can simulate a Poisson process on a time interval by generating pseudo-random variates T1 , T2 , T3 , . . . from this exponential distribution, and then taking their successive partial sums T1 , T1 +T2 , T1 +T2 +T3 , . . . as the clock times of occurrences, carrying on until we have passed the end of the time interval. PAS205 Probability Modelling: Lecture 21 (ii) Basic Poisson process via conditioning property 2 To simulate the basic Poisson process in the time interval ]0, t] say, we may firstly generate a single variate from the Poisson distribution with parameter t, representing the total number of occurrences, and then generate this number of variates from the uniform distribution on the interval ]0, t] to represent the clock times of these occurrences. These will not in general come out in the right order, and will need to be sorted (possibly using another computer algorithm) into chronological order. It is easy to see from the property of conditioning on the number of occurrences in an interval that this construction gives a correct simulation of the basic Poisson process. (iii) Variable rate Poisson process via thinning Suppose we wish to simulate a variable rate process on an interval on which the maximum value of the rate on the interval is say. One possibility is firstly to simulate a Poisson process with rate on the interval using (i) say, and then to take each occurrence in this process independently and either retain it, with probability (t)/, or delete it, with probability 1 - (t)/, where t is its clock time of occurrence, and the decisions whether to retain or delete are made using further pseudo-random variates. (iv) Spatial Poisson process There is a method here analogous to that in (ii). Suppose, for instance, that we are in two dimensions and wish to simulate a Poisson process within a rectangle. We may simulate the total number of points of the process using a single variate from the appropriate Poisson distribution, and then simulate x and y co-ordinates of these points using independent variates from two uniform distributions, giving a two-dimensional uniform distribution over the rectangle. 21.3 <a href="/keyword/markov-chain-monte-carlo/" >markov <a href="/keyword/chain-monte-carlo/" ><a href="/keyword/chain-monte/" >chain monte</a> carlo</a> </a> (MCMC) methods These are methods used widely in Bayesian Statistics to simulate samples from an otherwise rather intractable posterior distribution. These samples are then used to get an idea of what the distribution looks like. Only a bare outline is given here. The idea is to construct a Markov chain (actually with continuous state space) which has the distribution of interest as its (unique) equilibrium distribution it may be shown that this is always possible via a relatively simple algorithm. The Markov chain is then simulated over a long period of time until it has effectively reached equilibrium, and its values are then sampled. Detailed coverage of this topic appears in the course PAS383. 21.4 Continuous time Markov chains In continuous time Markov chains, a state is occupied for an exponentially distributed length of time and then a jump takes place to another state according to some distribution, and so on. The role of the transition matrix is taken by the generator matrix which governs what happens over an infinitesimal period of time. Detailed coverage of this topic appears in the course PAS384.