Least squares regression on a 2d array
The function numpy.linalg.lstsq(a,b)
takes an array of a
a size nx2 and 1-dimensional array b
, which is the dependent variable.
How can I do a least squares regression where the data points are represented as a 2d array generated from an image file? The array looks something like this:
[[0, 0, 0, 0, e]
[0, 0, c, d, 0]
[b, a, f, 0, 0]]
where a, b, c, d, e, f
are positive integer values.
I want to put a line at these points. Can I use np.linalg.lstsq
(and if so, how) or is there something that might make sense (and if so, how)?
Many thanks.
source to share
Use sklearn instead of numpy (sklearn is derived from numpy, but much better for this kind of calculation):
from sklearn import linear_model
clf = linear_model.LinearRegression()
clf.fit ([[0, 0], [1, 1], [2, 2]], [0, 1, 2])
LinearRegression (copy_X = True, fit_intercept = True, n_jobs = 1, normalize = False)
clf.coef_
array ([0.5, 0.5])
source to share
once in a while i saw a similar python program from
# Prac 2 for Monte Carlo methods in a nutshell
# Richard Chopping, ANU RSES and Geoscience Australia, October 2012
# Useage
# python prac_q2.py [number of bootstrap runs]
# e.g. python prac_q2.py 10000
# would execute this and perform 10 000 bootstrap runs.
# Default is 100 runs.
# sys cause I need to access the arguments the script was called with
import sys
# math cause it handy for scalar maths
import math
# time cause I want to benchmark how long things take
import time
# numpy cause it gives us awesome array / matrix manipulation stuff
import numpy
# scipy just in case
import scipy
# scipy.stats to make life simpler statistcally speaking
import scipy.stats as stats
def main():
print "Prac 2 solution: no graphs"
true_model = numpy.array([17.0, 10.0, 1.96])
# Here a nifty way to write out numpy arrays.
# Unlike the data table in the prac handouts, I've got time first
# and height second.
# You can mix up the order but you need to change a lot of calculations
# to deal with this change.
data = numpy.array([[1.0, 26.94],
[2.0, 33.45],
[3.0, 40.72],
[4.0, 42.32],
[5.0, 44.30],
[6.0, 47.19],
[7.0, 43.33],
[8.0, 40.13]])
# Perform the least squares regression to find the best fit solution
best_fit = regression(data)
# Nifty way to get out elements from an array
m1,m2,m3 = best_fit
print "Best fit solution:"
print "m1 is", m1, "and m2 is", m2, "and m3 is", m3
# Calculate residuals from the best fit solution
best_fit_resid = residuals(data, best_fit)
print "The residuals from the best fit solution are:"
print best_fit_resid
print ""
# Bootstrap part
# --------------
# Number of bootstraps to run. 100 is a minimum and our default number.
num_booties = 100
# If we have an argument to the python script, use this as the
# number of bootstrap runs
if len(sys.argv) > 1:
num_booties = int(sys.argv[1])
# preallocate an array to store the results.
ensemble = numpy.zeros((num_booties, 3))
print "Starting up the bootstrap routine"
# How to do timing within a Python script - here I start a stopwatch running
start_time = time.clock()
for index in range(num_booties):
# Print every 10 % so we know where we're up to in long runs
if print_progress(index, num_booties):
percent = (float(index) / float(num_booties)) * 100.0
print "Have completed", percent, "percent"
# For each iteration of the bootstrap algorithm,
# first calculate mixed up residuals...
resamp_resid = resamp_with_replace(best_fit_resid)
# ... then generate new data...
new_data = calc_new_data(data, best_fit, resamp_resid)
# ... then perform another regression to generate a new set of m1, m2, m3
bootstrap_model = regression(new_data)
ensemble[index] = (bootstrap_model[0], bootstrap_model[1], bootstrap_model[2])
# Done with the loop
# Calculate the time the run took - what the current time, minus when we started.
loop_time = time.clock() - start_time
print ""
print "Ensemble calculated based on", num_booties, "bootstrap runs."
print "Bootstrap runs took", loop_time, "seconds."
print ""
# Stats on the ensemble time
# --------------------------
B = num_booties
# Mean is pretty simple, 1.0/B to force it to use floating points
# This gives us an array of the means of the 3 model parameters
mean = 1.0/B * numpy.sum(ensemble, axis=0)
print "Mean is ([m1 m2 m3]):", mean
# Variance
var2 = 1.0/B * numpy.sum(((ensemble - mean)**2), axis=0)
print "Variance squared is ([m1 m2 m3]):", var2
# Bias
bias = mean - best_fit
print "Bias is ([m1 m2 m3]):", bias
bias_corr = best_fit - bias
print "Bias corrected solution is ([m1 m2 m3]):", bias_corr
print "The original solution was ([m1 m2 m3]):", best_fit
print "And the true solution is ([m1 m2 m3]):", true_model
print ""
# Confidence intervals
# ---------------------
# Sort column 1 to calculate confidence intervals
# Sorting in numpy sucks.
# Need to declare what the fields are (so it knows how to sort it)
# f8 => numpy floating point number
# Then need to delcare what we sort it on
# Here we sort on the first column, then the second, then the third.
# f0,f1,f2 field 0, then field 1, then field 2.
# Then we make sure we sort it by column (axis = 0)
# Then we take a view of that data as a float64 so it works properly
sorted_m1 = numpy.sort(ensemble.view('f8,f8,f8'), order=['f0','f1','f2'], axis=0).view(numpy.float64)
# stats is my name for scipy.stats
# This has a wonderful function that calculates percentiles, including performing interpolation
# (important for low numbers of bootstrap runs)
m1_perc0p5 = stats.scoreatpercentile(sorted_m1,0.5)[0]
m1_perc2p5 = stats.scoreatpercentile(sorted_m1,2.5)[0]
m1_perc16 = stats.scoreatpercentile(sorted_m1,16)[0]
m1_perc84 = stats.scoreatpercentile(sorted_m1,84)[0]
m1_perc97p5 = stats.scoreatpercentile(sorted_m1,97.5)[0]
m1_perc99p5 = stats.scoreatpercentile(sorted_m1,99.5)[0]
print "m1 68% confidence interval is from", m1_perc16, "to", m1_perc84
print "m1 95% confidence interval is from", m1_perc2p5, "to", m1_perc97p5
print "m1 99% confidence interval is from", m1_perc0p5, "to", m1_perc99p5
print ""
# Now column 2, sort it...
sorted_m2 = numpy.sort(ensemble.view('f8,f8,f8'), order=['f1','f0','f2'], axis=0).view(numpy.float64)
# ... and do stats.
m2_perc0p5 = stats.scoreatpercentile(sorted_m2,0.5)[1]
m2_perc2p5 = stats.scoreatpercentile(sorted_m2,2.5)[1]
m2_perc16 = stats.scoreatpercentile(sorted_m2,16)[1]
m2_perc84 = stats.scoreatpercentile(sorted_m2,84)[1]
m2_perc97p5 = stats.scoreatpercentile(sorted_m2,97.5)[1]
m2_perc99p5 = stats.scoreatpercentile(sorted_m2,99.5)[1]
print "m2 68% confidence interval is from", m2_perc16, "to", m2_perc84
print "m2 95% confidence interval is from", m2_perc2p5, "to", m2_perc97p5
print "m2 99% confidence interval is from", m2_perc0p5, "to", m2_perc99p5
print ""
# and finally column 3, again, sort it..
sorted_m3 = numpy.sort(ensemble.view('f8,f8,f8'), order=['f2','f1','f0'], axis=0).view(numpy.float64)
# ... and do stats.
m3_perc0p5 = stats.scoreatpercentile(sorted_m3,0.5)[1]
m3_perc2p5 = stats.scoreatpercentile(sorted_m3,2.5)[1]
m3_perc16 = stats.scoreatpercentile(sorted_m3,16)[1]
m3_perc84 = stats.scoreatpercentile(sorted_m3,84)[1]
m3_perc97p5 = stats.scoreatpercentile(sorted_m3,97.5)[1]
m3_perc99p5 = stats.scoreatpercentile(sorted_m3,99.5)[1]
print "m3 68% confidence interval is from", m3_perc16, "to", m3_perc84
print "m3 95% confidence interval is from", m3_perc2p5, "to", m3_perc97p5
print "m3 99% confidence interval is from", m3_perc0p5, "to", m3_perc99p5
print ""
# End of the main function
#
#
# Helper functions go down here
#
#
# regression
# This takes a 2D numpy array and performs a least-squares regression
# using the formula on the practical sheet, page 3
# Stored in the top are the real values
# Returns an array of m1, m2 and m3.
def regression(data):
# While testing, just return the real values
# real_values = numpy.array([17.0, 10.0, 1.96])
# Creating the G matrix
# ---------------------
# Because I'm using numpy arrays here, we need
# to learn some notation.
# data[:,0] is the FIRST column
# Length of this = number of time samples in data
N = len(data[:,0])
# numpy.sum adds up all data in a row or column.
# Axis = 0 implies add up each column. [0] at end
# returns the sum of the first column
# This is the sum of Ti for i = 1..N
sum_Ti = numpy.sum(data, axis=0)[0]
# numpy.power takes each element of an array and raises them to a given power
# In this one call we also take the sum of the columns (as above) after they have
# been squared, and then just take the t column
sum_Ti2 = numpy.sum(numpy.power(data, 2), axis=0)[0]
# Now we need to get the cube of Ti, then sum that result
sum_Ti3 = numpy.sum(numpy.power(data, 3), axis=0)[0]
# Finally we need the quartic of Ti, then sum that result
sum_Ti4 = numpy.sum(numpy.power(data, 4), axis=0)[0]
# Now we can construct the G matrix
G = numpy.array([[N, sum_Ti, -0.5 * sum_Ti2],
[sum_Ti, sum_Ti2, -0.5 * sum_Ti3],
[-0.5 * sum_Ti2, -0.5 * sum_Ti3, 0.25 * sum_Ti4]])
# We also need to take the inverse of the G matrix
G_inv = numpy.linalg.inv(G)
# Creating the d matrix
# ---------------------
# Hello numpy.sum, my old friend...
sum_Yi = numpy.sum(data, axis=0)[1]
# numpy.prod multiplies the values in an array.
# We need to do the products along axis 1 (i.e. row by row)
# Then sum all the elements
sum_TiYi = numpy.sum(numpy.prod(data, axis=1))
# The final element we need is a bit tricky.
# We need the product as above
TiYi = numpy.prod(data, axis=1)
# Then we get tricky. * works how we need it here,
# remember that the Ti column is referenced by data[:,0] as above
Ti2Yi = TiYi * data[:,0]
# Then we sum
sum_Ti2Yi = numpy.sum(Ti2Yi)
#With all the elements, we make the d matrix
d = numpy.array([sum_Yi,
sum_TiYi,
-0.5 * sum_Ti2Yi])
# Do the linear algebra stuff
# To multiple numpy arrays in a matrix style,
# we need to use numpy.dot()
# Not the most useful notation, but there you go.
# To help out the Matlab users: http://www.scipy.org/NumPy_for_Matlab_Users
result = G_inv.dot(d)
#Return this result
return result
# residuals:
# Takes in a data array, and an array of best fit paramers
# calculates the difference between the observed and predicted data
# and returns an array
def residuals(data, best_fit):
# Extract ti from the data array
ti = data[:,0]
# We also need an array of the square of ti
ti2 = numpy.power(ti, 2)
# Extract yi
yi = data[:,1]
# Calculate residual (data minus predicted)
result = yi - best_fit[0] - (best_fit[1] * ti) + (0.5 * best_fit[2] * ti2)
return result
# resamp_with_replace:
# Perform a dataset resampling with replacement on parameter set.
# Uses numpy.random to generate the random numbers to pick the indices to look up.
# So for item 0, ... N, we look up a random index from the set and put that in
# our resampled data.
def resamp_with_replace(set):
# How many things do we need to do this for?
N = len(set)
# Preallocate our result array
result = numpy.zeros(N)
# Generate N random integers between 0 and N-1
indices = numpy.random.randint(0, N - 1, N)
# For i from the set 0...N-1 (that what the range() command gives us),
# our result for that i is given by the index we randomly generated above
for i in range(N):
result[i] = set[indices[i]]
return result
# calc_new_data:
# Given a set of resampled residuals, use the model parameters to derive
# new data. This is used for bootstrapping the residuals.
# true_data is a numpy array of rows of ti, yi. We only need the ti column though.
# model is an array of three parameters, corresponding to m1, m2, m3.
# residuals are an array of our resudials
def calc_new_data(true_data, model, residuals):
# Extract the time information from the new data array
ti = true_data[:,0]
# Calculate new data using array maths
# This goes through and does the sums etc for each element of the array
# Nice and compact way to represent it.
y_new = residuals + model[0] + (model[1] * ti) - (0.5 * model[2] * ti**2)
# Our result needs to be an array of ti, y_new, so we need to combine them using
# the numpy.column_stack routine
result = numpy.column_stack((ti, y_new))
# Return this combined array
return result
# print_progress:
# Just a quick thing that returns true if we want to print for this index
# and false otherwise
def print_progress(index, total):
index = float(index)
total = float(total)
result = False
# Floating point maths is irritating
# We want to print at the start, every 10%, and at the end.
# This works up to index = 100,000
# Would also be lovely if Python had a switch statement
if (((index / total) * 100) <= 0.00001):
result = True
elif (((index / total) * 100) >= 9.99999) and (((index / total) * 100) <= 10.00001):
result = True
elif (((index / total) * 100) >= 19.99999) and (((index / total) * 100) <= 20.00001):
result = True
elif (((index / total) * 100) >= 29.99999) and (((index / total) * 100) <= 30.00001):
result = True
elif (((index / total) * 100) >= 39.99999) and (((index / total) * 100) <= 40.00001):
result = True
elif (((index / total) * 100) >= 49.99999) and (((index / total) * 100) <= 50.00001):
result = True
elif (((index / total) * 100) >= 59.99999) and (((index / total) * 100) <= 60.00001):
result = True
elif (((index / total) * 100) >= 69.99999) and (((index / total) * 100) <= 70.00001):
result = True
elif (((index / total) * 100) >= 79.99999) and (((index / total) * 100) <= 80.00001):
result = True
elif (((index / total) * 100) >= 89.99999) and (((index / total) * 100) <= 90.00001):
result = True
elif ((((index+1) / total) * 100) > 99.99999):
result = True
else:
result = False
return result
#
#
# End of helper functions
#
#
# So we can easily execute our script
if __name__ == "__main__":
main()
I think you can take a look, here is the link for complete information
source to share