 Power Function Model
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Posted: 2017-07-09
Last Update: 2017-07-09

This is from a question in "Calculus Seventh Edition" by James Stewart. I have been using it to provide examples that I can work in python.

I've already lost the website that helped me sort this out by converting the data to log and then back again, but the scipy cookbook has a similar example (at the bottom of the page).  ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71``` ```#!/usr/bin/env python #Author: Mark Feineigle #Create Date: 2017/06/30 # Model a power function # Example: power_function_model.jpg # From: CALCULUS SEVENTH EDITION JAMES STEWART # Build xs and ys # Convert xs and ys to log10 # Do linear regression on the log10 data # Create sample data in log10 # m is the exponent and 10**b is the coefficient # to solve you must convert the target value to log10 # and then to raise 10**result to get out of log10 import matplotlib.pyplot as plt import numpy as np from scipy import stats # Build xs and ys xs = np.array([4, 40, 3459, 4411, 29418, 44218,]) ys = np.array([5, 9, 40, 39, 84, 76,]) # Convert xs and ys to log10 logxs = np.log10(xs) logys = np.log10(ys) # Linear regression m, b, r_value, p_value, std_err = stats.linregress(logxs, logys) print m, 10**b # m is the exponent and 10**b is the coefficient # m = 0.308044235477 10**b = 3.10462040171 answer = 10**(m*np.log10(291)+b) # prediction for 291 print answer # answer = 17.8236456399 # Test data, in log10 samps = np.log10(np.arange(1,50000,1)) # plot in log and linear scales fig1 = plt.figure() ax1 = fig1.add_subplot(111) ax1.scatter(logxs, logys) # in log scale ax1.loglog(samps, m*samps+b) plt.title("Logarithmic Scale") plt.xlabel("Island Area") plt.ylabel("Reptiles") fig2 = plt.figure() ax2 = fig2.add_subplot(111) ax2.scatter(xs, ys) ax2.plot(10**samps, 10**(m*samps+b)) # convert to linear ax2.scatter(291, answer, marker="x") plt.title("Linear Scale") plt.xlabel("Island Area") plt.ylabel("Reptiles") plt.annotate("(291,18)", xy=(291,18), xytext=(10000,17), arrowprops=dict(facecolor='black', shrink=0.05)) plt.show() ```  