Polynomial curve fitting

We have seen already how to a fit a given set of points minimizing an error function, now we will see how to find a fitting polynomial for the data using the function polyfit provided by numpy:

from numpy import *
import pylab

# data to fit
x = random.rand(6)
y = random.rand(6)

# fit the data with a 4th degree polynomial
z4 = polyfit(x, y, 4) 
p4 = poly1d(z4) # construct the polynomial 

z5 = polyfit(x, y, 5)
p5 = poly1d(z5)

xx = linspace(0, 1, 100)
pylab.plot(x, y, 'o', xx, p4(xx),'-g', xx, p5(xx),'-b')
pylab.legend(['data to fit', '4th degree poly', '5th degree poly'])
pylab.axis([0,1,0,1])
pylab.show()

Let's see the two polynomials:

Dice rolling experiment

If we roll a die a large number of times, and we compute the mean and variance, as exaplained here, we’d expect to obtain a mean = 3.5 and a variance = 2.916. Let's simulate that with a script:

import pylab
import math
 
# Rolling the die 1000 times
v = pylab.randint(1,7,size=(1000))

print 'mean',pylab.mean(v)
print 'variance',pylab.var(v)

pylab.hist(v, bins=6) # histogram of the outcoming
pylab.xlim(1,6)
pylab.show()

Here's the result:

mean 3.435
variance 2.781775

Prime factor decomposition of a number

The following function compute the prime factors (PFs) of an integer.

from math import floor

def factors(n):
 result = []
 for i in range(2,n+1): # test all integers between 2 and n
  s = 0;
  while n/i == floor(n/float(i)): # is n/i an integer?
   n = n/float(i)
   s += 1
  if s > 0:
   for k in range(s):
    result.append(i) # i is a pf s times
   if n == 1:
    return result

# test
print factors(90)

The result will be

[2, 3, 3, 5]

This means that 90 is equal to 2*3*3*5.