Tutorial2PHY409 - University of British Columbiaphys409/PDFs/PythonTutorial2.pdf · Tutorial2PHY409 Anadi&Canepa [email protected]& Oﬃce,&TRIUMF&MOB&92 B (16042227330) & Alan&Manning&[email protected]

Tutorial 2 PHY409 Anadi Canepa

[email protected] Office, TRIUMF MOB 92B (1-‐604-‐222-‐7330)

Alan Manning [email protected]

Mohammad Samani [email protected]

During the 1st tutorial

We learnt •  What Python is •  How to import packages •  How to produce a 1D plots, scaUer plots, histograms We used •  matplotlib •  numpy

13/9/2013 Python TutoWal 2

HW #1 (due today)

1.  Install python 2.  Install matplotlib, numpy, scipy (the order maUers!) 3.  (Install Spyder -‐ opWonal) 4.  Write a script to plot sin(x):

1.  Range [0:2π] 2.  Add labels to the axis, Wtle, legend

5.  Save the plot in pdf format


The goal of today is to learn to fit

13/9/2013 4 Python TutoWal

1.   Example #1: Polynomial order 0 1.  Read in text file 2.  Store data 3.  Fit them and compute uncertainWes on the fit parameter

2.   Example #2: ExponenFal (hands-‐on) 1.  Modify the model 2.  Compute uncertainWes on the fit parameters

3.   Example #3: Gaussian (hands-‐on) 1.  Handling histograms

Example #1

13/9/2013 5 Python TutoWal

1.  Read in a text file containing data points 2.  Handle the data 3.  Fit the data

Module courtesy of Prof. C. Gay

Input file and script

hLp://trshare.triumf.ca/~canepa/PYTHON/Tutorial2/ •  input_p0.txt •  fiUest_p0.py


Important Concepts

1.  Objects 2.  Module 3.  Package


(1) Objects

Everything in Python is an object •  Strings, funcWons, classes •  The name is used to access the object •  The “dot” operator is used to access the object aUributes


str . index (‘e’)

Variable Delimiter AUribute Argument name

(2) Modules

•  A module is a file containing Python funcWons definiWons and executable statements –  The file name is the module name with the suffix .py appended

– Modules can be imported in scripts and in modules

•  Whenever you run a simple Python script, the interpreter treats it as a module called __main__, which gets its own namespace


(3) Packages

•  Packages are a way of structuring Python’s module namespace

•  For example, the module name A.B designates a sub-‐module named B in a package named A


Modules: import statement


import pylab as plb import scypy as sy import numpy as np

import matplotlib.pyplot as plt

from math import sqrt

from scipy.opWmize import leastsq

Import module (pylab) with a given alias

Access the funcWons using this alias

Import sub-‐module (pyplot)

from package (matplotlib)

Import name (sqrt)

from module (math)

How to read in a text file

FuncWon loadtxt of numpy to read in the text file Returns an ndarray

(mulWdimensional container of items of the same type)

13/9/2013 12

data = np.loadtxt(‘mydata.txt’, skiprows=2)

Working with the N-‐dimensional array

•  ndarray can be indexed: –  print data[1,2]

•  ndarray can be sliced:

13

x = data[:,0] y = data[:,1] sigma = data[,:2]

x y σ

X

x y σ

X X X X

13/9/2013 Python TutoWal

Least-‐square fi]ng

•  MinimizaWon of:

•  Where f(x) is the model we want to use to fit the data


Define the model funcFon

•  keyword def followed by the funcWon name and parentheses

•  input parameters or arguments should be placed within these parentheses

•  code block within every funcWon starts with a colon (:) and is indented

•  statement return [expression] exits a funcWon, opWonally passing back an expression to the caller


def func(x, a): return a*pow(x,0)

x0 x1 x2 … xn

1 1 1 … 1 a *

Fit the data (I)

•  Define the residual funcWon

•  Call the leastsq funcWon imported from scipy.opWmize


def residual (coeff, x, y, sigma): return (y-‐func(x, coeff[0]))/sigma

p = leastsq (residuals, coeff, args=(x, y, sigma))

Fit the data (II)

•  Leastsq returns the value of the fiUed parameter

•  It can also return the uncertainty if we request the full

output


print "Fit value: ", p[0]

p, cov, infodict, mesg, ier = leastsq (residuals, coeff, args=(x, y, sigma), full_output=True) print "Fit value: ", p[0] s_sq = (infodict['fvec']**2).sum()/(len(y)-‐1) cov_scaled = cov*s_sq print “Uncertainty", sqrt(cov_scaled[0])

Plo]ng the results

•  Some formarng …. (see script) •  Plot the data with error bars and overlay the fit result


plt.errorbar(x, y, sigma, linestyle='', marker='o’) plt.plot(x,func(x, p[0])) plt.show()

Example #1: Result


Example #2

•  Download the input file input_exp.txt from: hLp://trshare.triumf.ca/~canepa/PYTHON/Tutorial2/

•  Fit the data to the exponenWal:


Example #2 (I)

•  Replace the polynomial fit with the exponenWal fit •  Warning: number of free parameter changes from 1 to 2


coeff = [0,0] def func(x, a, b): return a * np.exp(-‐b * x) def residuals (coeff, x, y, sigma): return (y-‐func(x, coeff[0],coeff[1]))/sigma

Example #2 (II)

•  Replace the polynomial fit with a n exponenWal fit •  Warning: number of free parameter changes from 1 to 2


p = leastsq (residuals, coeff, args=(x,y,sigma), full_output=True) print "Fit values for a, b", p[0][0], p[0][1]

Example #3

•  Generate a sample of 1000 events distributed according to a Gaussian distribuWon (mean =0, variance = 1)

•  Fill a histogram with the data •  Fit the histogram to a Gaussian (take into account the

uncertainty on the bin content) •  Print the mean and the variance


Example #3 (I)

•  Extract the bin content using the return values of matplotlib.pyplot hist


tuple : (binContents, binEdges, patches)

Example #3 (II)


x = np.random.randn(10000) binContents, binEdges, patches = plt.hist(x,bins=60,range=(-‐3,3)) binWidth = binEdges[1]-‐binEdges[0] nBins = len(binContents) binCenters = range(nBins) binErrors = range(nBins) for i in range(len(binContents)): binCenters[i] = (binEdges[i+1]+binEdges[i])/2 binErrors[i] = sqrt(binContents[i])

In today’s tutorial

We learnt 1.  How to read in a text file 2.  Store its content in arrays 3.  Define and call funcWons 4.  Fit data 5.  Print fit results 6.  Plot fit results

Slides, scripts and input files can be found at: hLp://trshare.triumf.ca/~canepa/PYTHON/Tutorial2/


More material


Name

•  A name in Python is roughly analogous to a variable in just about any other language, but with a few extras.

•  Because of Python’s dynamic nature, you can apply a name to just about anything.


a = 5.0 a = ‘a’ a = [1,2,3] def fun():

print “Hello” a = fun

Name Space

•  A namespace is a space that holds names.

•  Each module gets it’s own global namespace •  Once you import that module into your script, you can

access the names by prefixing them with the module name

•  Each namespace is also completely isolated


Math package

•  FuncWons from the "math" package do not operate on numpy arrays (which you should always be using instead of lists)

•  Use funcWons from numpy to do this. Example:

import numpy import math x = numpy.asarray([0,2,3,4,5]) y = math.sqrt(x) #Error! y = numpy.sqrt(x) #No error


Documents

Tutorial2PHY409 - University of British Columbiaphys409/PDFs/PythonTutorial2.pdf · Tutorial2PHY409 Anadi&Canepa [email protected]& Oﬃce,&TRIUMF&MOB&92 B (16042227330) & Alan&Manning&[email protected]