IS-609 Final Project Presentation

Final Project IS 609 - Mathematical Modeling Techniques for Data Analytics

Date: 05/25/2016

• Introduction
• Problem Description
• Assumptions
• Model Construction
• Methodology
• Least squares criterion Methodology details
• Least squares criterion Model results
• Chebyshev Criterion model
• Chebyshev LP model statement
• Chebyshev Criterion Geometric interpretation
• Conclusion

In this project, I will investigate the relationship of the body weights of birds and their pulse rates.

Heart rate of birds- Warm-blooded animals use large quantities of energy to maintain body temperature because of the heat loss through the body surface. In fact, biologists believe that the primary energy drain on a resting warm blooded animal is maintenance of body temperature.

(a) Construct a model relating blood flow through the heart to body weight.

(b) The following data relate weights of some birds to their heart rate measured in beats per minute. Construct a model that relates heart rate to body weight. Discuss the assumptions of your model. Use the data to check your model.

• It is well-known that small warm-blooded animals have much higher resting pulse rates than large ones. For examples, a canary, with a body mass of 20 grams, may have a heart rate of 1000 beats/minute, as compared to 70-80 beats/minute for a human being.

• Warm-blooded animals at rest use large quantities of energy to maintain body temperature, due to heat loss through the body surface. Let us make the following assumptions about energy balance over a fixed time interval (1 minute, say):

A. Energy lost is proportional to the surface area of the body.
B. Energy gained is proportional to the amount of blood flowing through the lungs, the source of oxygen for the body.
C. Energy lost = energy gained for a body in temperature equilibrium.

E = energy
S = surface area;
m = mass
W = weight of the animal
r = pulse rate
V = volume of the animal
b = volume of blood pumped by the heart in one stroke.

From assumption A, we can assume that the energy required to maintain the body temperature of a bird E is proportional to its body surface area S, i.e. \( E\propto S \)

Since blood flow through the lungs provides animals oxygen, larger animals have larger lungs and require more oxygen. Therefore, we assume that the amount of energy available is proportional to the blood flow b through the lungs, i.e \( E \propto b \)

Also, we assume that the least amount of blood needed to circulate, the amount of available energy will equal the amount of energy used to maintain the body temperature. This implies \( b \propto S \)

Since larger animals have larger hearts and larger lungs, we can assume the birds are geometrically similar. Thus, \( S\propto V^{\frac 2 3} \) where V is the volume of the animal.

Since \( V \propto m \propto W \) where m is the mass of the animal and W is the weight of the animal, we have \( S \propto W^{\frac 2 3} \) . This implies \( b \propto W^{\frac 2 3} \).

Also, since blood flow is determined by the total amount of blood pumped by the heart per unit time (minutes), and every time the heart beats the blood is pumped into the arteries, we have \( b \propto rV \propto rW \) where r is the pulse rate and Vh is heart volume. From \( b\propto W^{\frac 2 3} \) and \( b \propto rW \), we have \( W^{ \frac 2 3} \propto rW \), i.e. \( r \propto W^{\frac 1 3} \).

• We will be using Least squares criterion to fit our curve model of the form \( y= Ax^n \) where n is fixed to a given collection of data points. In our case, our curve is \( r = Aw^{\frac {-1} {3}} \). Therefore we will use the least squares criterion to estimate \( A \) with \( n = -1/3 \).

• In addition we will use the Chebyshev criterion to find the bounds

Recall model of form power curve \( f(x) = ax^n \) requires minimization of: \[ S = \sum_{i=1}^m[y_i -f(x_i)]^2 \] \[ s = \sum_{i=1}^m[y_i -ax_i^n]^2 \] solving the the equation for \( a \) yeilds: \[ a = \frac {\sum x_i^2y_i} {\sum x_i^{2n}} \]

     bird body_weight pulse_rate pulse_rate_fit
1  Canary          20       1000         949.21
2  Pigeon         300        185         384.89
3    Crow         341        378         368.80
4 Buzzard         658        300         296.23
5    Duck        1100        190         249.60
6     Hen        2000        312         204.50
7   Goose        2300        240         195.19
8  Turkey        8750        193         125.04
9 Ostrich       71000         60          62.22

Given N points (x_1, y_1),(x_2, y_2),(x_3, y_3), …., (x_n, y_n), to fit the model y=cx using Chebyshev criterion, we need to: \

\[ \epsilon = max_i\bigl| y_i - y(x_i) \bigr| \] \[ Goal: \ to \ minimize\ \epsilon \]

\[ r = max_i\bigl| y_i - cx_i \bigr| \] \[ Goal: to \ minimize\ r \] subject to: \[ r-(1000 -c*20) >=0 \] \[ r+(1000 -c*20) >=0 \] \[ r-(185 -c*300) >=0 \] \[ r+(185 -c*300) >=0 \] \[ r-(378 -c*341) >=0 \] \[ r+(378 -c*341) >=0 \] \[ r-(300 -c*658) >=0 \] \[ r+(300 -c*658) >=0 \] \[ r-(190 -c*1100) >=0 \] \[ r+(190 -c*1100) >=0 \] \[ r-(312 -c*2000) >=0 \] \[ r+(312 -c*2000) >=0 \] \[ r-(240 -c*2300) >=0 \] \[ r+(240 -c*2300) >=0 \] \[ r-(193 -c*8750) >=0 \] \[ r+(193 -c*8750) >=0 \] \[ r-(60 -c*71000) >=0 \] \[ r+(60 -c*71000) >=0 \]

For the majority of the birds, the model makes good predictions, and support the biology that larger animals have slower pulse rate. However, this is not always true for all birds. The pulse rate of a Pigeon is almost the same as the pulse rate of a Turkey, but the weight of a Turkey is 29.16 times of the weight of a pigeon. This model reveals that assuming there is no measurement error, there is something else affecting pigeon's pulse rate; thus further investigations and possible model refinement are needed for birds like pigeon.

However, a research by Florence Buchanan in his paper, The significance of the pulse rate in vertebrate animals University college, London 1910, explains the pigeon data discrepancy. The pigeon weights 300g and pulse rate of 185. However, if we examine its average dioxide per kilo per hour(in grams) which is 3.4 as compared to that of the goose 1.07 and yet body weight of 2300 and pulse rate of 240, we can derive that larger average dioxide per kilo per hour(in grams) leads to faster heart rates.. It is clearly demonstrated in by Florence Buchanan that larger average dioxide per kilo per hour (in grams) leads to faster heart rates. link: https://books.google.com/books?id=3jEsAAAAYAAJ&printsec=frontcover#v=onepage&q&f=false

In addition the Chebyshev resulted in optimal value of 67; however, large error of 340 which is not acceptable. In fact the feasible region could not be geometrically confirmed.