A New Homework Problem on Modeling

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 y_j is the value of the j^th 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 y_j+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

Similarly, setting ∂Q/∂b = 0 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 = 35.3931

213.1970 a – 875.7817 b = 112.9823

You can solve these two linear equations for the two unknowns. You will find 

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 y_j 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  

https://www.youtube.com/watch?v=zw9Y4t-Nh3E 

Source: http://hobbieroth.blogspot.com/2026/06/russ-hobbie-and-i-like-to-use-homework.html