Markov processes, lab 2

Transcription

1 Lunds Universitet Matematikcentrum Matematisk statistik FMSF15/MASC03 Markovprocesser Markov processes, lab 2 The first part of the lab is about simple Poisson processes. You will simulate and analyse Poisson processes for various intensities, on the line as well as in the plane. In addition, the intensity of a real process will be estimated. In the final part, you will do both simulation and estimation for a non-homogeneous processes. 1 Preparations Read through the instructions and answer the following questions. It is required that you bring your written answers with you to the computer lab. In the beginning of the computer session the lab supervisors will check your answers before you carry out the rest of the assignment. 1. What is meant by independent increments of a stochastic process? 2. What is the distribution of the number of events N(t) in the interval (0, t] for a Poisson process? What is the mean and standard deviation of N(t)? 3. What is the distribution of the time between two consecutive events in a Poisson process? 4. We have observed a process that may be modelled as a Poisson process in the interval (0, t]. (a) Derive the ML estimator of the intensity λ. (b) How can one compute a confidence interval for the estimator in (a)? 5. If one has observed two independent Poisson processes with different intensities λ 1 and λ 2, during different time intervals, how can one compute a confidence interval for the difference λ 1 λ 2? 6. How can one simulate a non-homogeneous Poisson process? 1

2 2 The Poisson process Log in at one of the PCs in the computer room MH:230 or MH:231, using the user name and password which you have obtained for this course. Click on the icon MClogin on the desktop and login again with the same user name and password. This will attach the hard drive L: where your working directory will be saved. Note that you need to do this before you start up the software package Matlab. Choose the latest version of Matlab from the Start menu. If you have problems either logging in or starting Matlab, ask for help. After starting Matlab write mh init( fmsf15 ). This will add the directory containing the m-files which you will use in this assignment. 2.1 Simulation of exponentially distributed holding times The distance between two events in a Poisson process is exponentially distributed with intensity λ, where λ is the intensity of the Poisson process. To simulate a Poisson process we need to simulate these distances, but standard MATLAB can only generate uniform and normal random variables (the Statistics Toolbox can do a lot more, however). We can generate exponentially distributed random variables through a transformation of uniform ones. Assume that we want to simulate random variables with the distribution function F (y), and that F (y) is invertible. Denote its inverse by F 1 (y) and let U be a random variable that has a uniform distribution on (0, 1). If we put Y = F 1 (U), then P (Y y) = P (F 1 (U) y) = P (F (F 1 (U)) F (y)) = P (U F (y)) = F (y). This shows that the transformed variable Y has distribution function F, and we can obtain exponentially distributed variables from a uniform U as Y = ln (1 U)/λ, where the intensity of the exponential distribution is λ. In MATLAB one can generate a vector of 100 exponentially distributed random variables with intensity λ through the commands: >> u=rand(1,100); >> y=-log(1-u)/lambda; 2.2 The counting process We now have a vector y of inter-event times, but our interest is often in the absolute time points of the events. We can obtain these by summing the interevent times. MATLAB provides a function cumsum, doing just that. Through 2

3 T=cumsum(y) we obtain a vector containing the time points of the simulated events. The value of the process is the number of events so far, hence 1 at the time point of the first event, 2 at the time point of the second one, and so on. Such a vector may be created as N=1:100, if we want 100 events. The process can be plotted by stairs(t,n). Choose an intensity λ and make a few simulations using different random sequences u. Plot them in the same figure using hold. 2.3 Estimation of a Poisson intensity The intensity λ of a Poisson process is the expected number of events per time unit. The parameter λ may be estimated by dividing the number of events during a time interval by the length of that time interval. Estimate the intensity for some realisations and compute confidence intervals. Do the cover they the true λ? Make a somewhat longer realisation and compute a confidence interval. How does the width of the confidence interval change with the number of events? 2.4 Estimation of the intensity of a real process The file coal.dat contains data about coal mine catastrophes in the United Kingdom from 1851 to The first column contains day of month, the second column contains the month, the third column contains the year, the fourth column contains the number of days elapsed thst year, the fifth column is the number of days since the last catastrophe and the sixth column contains the number of casualties in the catastrophe. The number of catastrophes since the start time may be viewed as a stochastic process that we may plot through summing the inter-event times. Load the data using the command load coal.dat and sum the inter-event times by T=cumsum(coal(:,5)). Put N=1:size(coal,1) and plot the process. Does is look like a Poisson process? 3

4 Estimate the intensity λ and compute a confidence interval. Give the answer in the unit catastrophes/year. The intensity does not appear constant throughout the whole period. Split the data into two time periods with two different intensities λ 1 and λ 2. Estimate the intensities and compute a confidence interval for the difference λ 1 λ 2. Plot the process with years on the x-axis by, for example, >> stairs(coal(:,3)+coal(:,4)/365,n) When do the changes occur? What can the underlying reasons be? Also look at the accumulative casualties process by >> stairs(coal(:,3)+coal(:,4)/365,cumsum(coal(:,6))) 2.5 A different method of simulation on the line and in the plane When we studied the Poisson process in one dimension, we used a method based on inter-event times. The lecture notes also describe a different method. This method is based on the result that, given n events until t, these n events are uniformly and independently distributed on [0, t]. Hence we can simulate a Poisson process on [0, t] by first drawing a number from Po(λt) and then drawing n random variables T i from a uniform distribution on the interval. Make a few simulations on [0, 100] for some different intensities. Use the function porand to simulate the Poisson variable. Use sort to sort the event times. Plot Poisson events as marks using e.g. plot(t,0*t, x ). The method can also be extended to simulate a Poisson process on the plane. The intensity is the the number of points per area unit. Simulate a Poisson process in the area 0 x 1, 0 y 1, and plot the result for some different intensities. Use, for example plot(x,y, x ). 4

5 Simulate a Poisson process on the triangle with corners in (0, 0), (0, 1) and (1, 0). Is there more that one way of doing that? 3 Non-homogeneous Poisson processes 3.1 Simulation Find a way to simulate a non-homogeneous process over the interval [0, t] with t = 10, having intensity λ(u) = 4(1 cos(u 2 /4)). Plot the Poisson points T i and the intensity function in the same graph, using e.g. >> u=0:0.01:10; >> plot(u,4*(1-cos(u.^2/4)),t,0*t, x ) Is the result as you expect it? 3.2 Simulation and estimation We will now consider an inhomogeneous Poisson process, N(t), on an interval t [0, S] with intensity function λ(t) = θ 1 + θ 2 cos(2πt/s) + θ 3 sin(2πt/s) where θ 1 θ θ 2 3 to ensure λ(t) 0 for all 0 t S. Given event times T 1,..., T N(S), the negative log-likelihood function is l(θ) = Λ(S) + ln(n(s)!) N(S) k=1 ln(λ(t k )) where Λ(S) = S 0 λ(t)dt = θ 1S. The following pre-written functions will be used in this part: inhom_poisson_simulate: Simulates a random sample from the model. inhom_poisson_est: Calculates estimates of the model parameters, the estimation covariances, and approximate 95% confidence intervals. 5

6 Use help inhom_poisson_simulate, etc for more information and also read the Matlab code. Start with parameter values S = 10 and θ = [20, 15, 0]. Use inhom_poisson_simulate and simulate a sample with these parameters. As before, plot the intensity function and simulated event points and convince yourself that the results are as expected. The pre-written function inhom_poisson_est mentioned above uses a robustified Newton optimisation on l(θ) to find the Maximum Likelihood estimate of the parameters for a given process sample. Find the estimates of the parameters using the sample you generated above. Do the confidence interval cover the true parameter values? The accuracy of the estimates depends on the true parameter values. Simulate new samples with different parameter values and run the estimation function for each sample. How low can you make the intensity before the estimates deteriorate significantly? Also increase the intensity; are the three parameters equally easy to estimate accurately? Why? 6