Russ Hobbie and I like to use the homework problems in Intermediate Physics for Medicine and Biology to illustrate modeling. But rarely does a problem encompass the entire process of constructing and analyzing a mathematical model. Gene Surdutovich and I try to do better in the 6th edition of IPMB (due out in a few months). Here is a homework problem that is not in any edition of IPMB, but that requires the student to analyze a model in its entirety. At least, that’s the goal.
Section 11.1
Problem 4½. Consider a variable that changes discretely (in steps, not continuously). You measure it in consecutive steps and get
step data 1 3.2184 2 4.0680 3 2.2896 4 4.6490 5 0.3875 6 1.3951 Your hypothesis is that this data can be obtained from a logistic map. Develop a mathematical model. Quantify it, analyze it, determine any unknown parameters, and decide if the data support your hypothesis.
The central question to be answered by this problem is: can this data be explained by a logistic map? You can’t definitively answer this question, but you can assess if the data support this hypothesis.
First you have to quantify what a logistic map is. The first equation in Section 10.9 of IPMB states that the logistic map is represented by the equation
The counter j indicates which data point is which, and yj is the value of the jth data point. There are two parameters: a and y∞.
We will want to use least squares to determine these parameters. Least squares is simpler if the parameters enter the model linearly. Ours don’t, but we can define b = a/y∞ and then write the logistic map as
Now we are in a position to apply the method of linear least squares, as described in Section 11.1 of IPMB. Define the quantity Q as
Q represents the average of the squares of the differences between the data and the model. (Although we have six data points, the sum goes from one to five. We can’t use j = 6 because then we would need a seventh point for yj+1). Our goal is to minimize Q, thereby obtaining the best fit to the data. The variables we can vary to minimize Q are a and b. To find the minimum, we set ∂Q/∂a = ∂Q/∂b = 0. Setting ∂Q/∂a = 0 gives
which reduces to
Next, you need to calculate all these sums. I find it easiest to construct a table like that below
j yj yj2 yj3 yj4 yj yj+1 yj yj+12 1 3.2184 10.3581 33.3365 107.2902 13.0925 42.1367 2 4.0680 16.5486 67.3198 273.8570 9.3141 37.8897 3 2.2896 5.2423 12.0027 27.4814 10.6444 24.3713 4 4.6490 21.6132 100.4798 467.1305 1.8015 8.3751 5 0.3875 0.1502 0.0582 0.0226 0.5406 0.2095 6 1.3951 sum 53.9124 213.1970 875.7817 35.3931 112.9823
The two equations for a and b become
53.9124 a – 213.1970 b = 35.3931You can solve these two linear equations for the two unknowns. You will find
213.1970 a – 875.7817 b = 112.9823
a = 3.920 and b = 0.8253
which means that
a = 3.92 and y∞ = 4.75
These are exactly the parameters I used to construct the original data. If you calculate Q, you will get zero (expect, perhaps, for some round-off error) because I didn’t add any noise to the data. The model explains the data well.
The two parameters have different interpretations. The parameter y∞ merely scales the size of the data. Dividing yj by y∞ transforms the data so it lies in the range between zero and one. The parameter a, however, cannot be scaled away. It’s a fundamental parameter characteristic of the logistic map (Eq. 10.39 in IPMB). Moreover, a = 3.92 is well into the range for which the logistic map results are chaotic.
Other than for practice, why create this new problem? It requires the student to go through the entire modeling procedure. Translating the hypothesis (the logistic map) into quantitative form, identifying the unknown parameters (a and y∞), using least squares to evaluate the parameters from the data, and examining the quality of the fit to determine if the calculation supports the hypothesis. You are getting about as close to modeling as you can hope for with a simple homework problem.
The Five Step Method: Math Modelling, with Jason Bramburger
