Predicting Precipitation in Darwin: An Experiment with Markov Chains (original) (raw)

2009, Australian Senior Mathematics Journal

Abstract

A s teachers of first-year college mathematics and science students, we are constantly on the lookout for simple classroom exercises that improve our students' analytical and computational skills. One such project, Predicting Precipitation in Darwin, is outlined below. In this project, students: • analyze and manipulate raw precipitation data; • build a prediction model using a Markov chain; • predict the long term distribution of precipitation-free and rainy days in Darwin, Northern Territory, Australia; • use a chi-square test to evaluate the effectiveness of the model they have constructed; • improve their prediction model. Beyond access to the Internet (to obtain the raw data) and a computer spreadsheet program or calculator, no special equipment is required. If the data is downloaded in advance, a well-prepared junior or senior high-school mathematics (or science) class should be able to perform this exercise in approximately 30-45 minutes of class time. Mathematical preliminaries: Markov chains A Markov chain is a sequence of identical trials, each of which can result in exactly one of a finite number of outcomes, called states. As the trials progress, the probability of moving from one state to another depends only on the state in which you are currently found. In most applications, Markov chains are represented by a state transition matrix, P. In this matrix, the entry in the (i,j)th position (row i, column j) is the probability that you will move to state j in the next trial if you are currently in state i. Properly constructed, the sum of each row in the matrix is one. Due to the properties of matrix multiplication, the (i,j)th entry in the matrix P 2 is the probability that you will move from state i to state j over the course of two trials; the (i,j)th entry in P 3 is the probability you will move from state i to state j in three trials, and so on.

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