Python Crash Course Data Fitting

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Python Crash Course Data Fitting. Sterrenkundig Practicum 2 V1.0 dd 08-01-2014 Hour 7. Data Fitting. We are given: observational data a model with a set of fit parameters Goal: Want to optimize some objective function. - PowerPoint PPT Presentation

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  • Python Crash Course Data FittingSterrenkundig Practicum 2V1.0dd 07-01-2015Hour 7

  • Data FittingWe are given:observational dataa model with a set of fit parameters

    Goal:Want to optimize some objective function.E.g.: Want to minimize squared difference between some model and some given data

    What we do not talk about:How to choose your objective function.How to choose your model.Least squares fitting

  • Data FittingFitting Lines to DataDefine function to fitDefine range and step size for all data pointsCalculate best fitting lineimport numpy as npimport pylab as pl

    # Data Fitting# First, we'll generate some fake data to usex = np.linspace(0,10,50) # 50 x points from 0 to 10

    # Remember, you can look at the help for linspace too:# help(np.linspace)

    # y = m x + by = 2.5 * x + 1.2

    # let's plot thatpl.clf()pl.plot(x,y)

  • Data FittingNoise

    # looks like a simple line. But we want to see the individual data pointspl.plot(x,y,marker='s')# We need to add noise firstnoise = pl.randn(y.size)# What's y.size?print y.sizeprint len(y)

    5050# looks like a simple line. But we want to see the individual data pointspl.plot(x,y,marker='s')# looks like a simple line. But we want to see the individual data pointspl.plot(x,y,marker='s')

  • Data Fitting# We can add arrays in python just like in IDLnoisy_flux = y + noise# We'll plot it too, but this time without any lines# between the points, and we'll use black dots# ('k' is a shortcut for 'black', '.' means 'point')pl.clf() # clear the figurepl.plot(x,noisy_flux,'k.')# We need labels, of coursepl.xlabel("Time")pl.ylabel("Flux")

  • Data Fitting# We'll use polyfit to find the values of the coefficients. The third# parameter is the "order"p = np.polyfit(x,noisy_flux,1)# help(polyfit) if you want to find out more

    # We'll use polyfit to find the values of the coefficients. The third# parameter is the "order"p = np.polyfit(x,noisy_flux,1)# help(polyfit) if you want to find out more[ 2.59289715 0.71443764]

    Now we're onto the fitting stage. We're going to fit a function of the formy=mx+bwhich is the same asf(x)=p[1]x+p[0]to the data. This is called "linear regression", but it is also a special case of a more general concept: this is a first-order polynomial. "First Order" means that the highest exponent of x in the equation is 1

  • Data Fittingnoisy_flux = y+noise*10p = polyfit(x,noisy_flux,1)print p[ 3.4289715 -3.65562359]

    # plot itpl.clf() # clear the figurepl.plot(x,noisy_flux,'k.') # repeated from abovepl.plot(x,np.polyval(p,x),'r-',label="Best fit") # A red solid linepl.plot(x,2.5*x+1.2,'b--',label="Input") # a blue dashed line showing the REAL linepl.legend(loc='best') # make a legend in the best locationpl.xlabel("Time") # labels againpl.ylabel("Flux")

    Let's do the same thing with a noisier data set. I'm going to leave out most of the comments this time.

  • Data Fittingpl.clf() # clear the figurepl.errorbar(x,noisy_flux,yerr=10,marker='.',color='k',linestyle='none') # errorbar requires some extras to look nicepl.plot(x,np.polyval(p,x),'r-',label="Best fit") # A red solid linepl.plot(x,2.5*x+1.2,'b--',label="Input") # a blue dashed line showing the REAL linepl.legend(loc='best') # make a legend in the best locationpl.xlabel("Time") # labels againpl.ylabel("Flux")

    Despite the noisy data, our fit is still pretty good! One last plotting trick.

  • Power Law Fittingt = np.linspace(0.1,10) a = 1.5b = 2.5z = a*t**b

    pl.clf()pl.plot(t,z)

    Power laws occur all the time in physis, so it's a good idea to learn how to use them.What's a power law? Any function of the form:f(t)=atbwhere t is your independent variable, a is a scale parameter, and b is the exponent (the power).When fitting power laws, it's very useful to take advantage of the fact that "a power law is linear in log-space". That means, if you take the log of both sides of the equation (which is allowed) and change variables, you get a linear equation!ln(f(t))=ln(atb)=ln(a)+bln(t)We'll use the substitutions y=ln(f(t)), A=ln(a), and x=ln(t), so thaty=a+bxwhich looks just like our linear equation from before (albeit with different letters for the fit parameters).

  • Power Law Fitting# Change the variables# np.log is the natural logy = np.log(z)x = np.log(t)pl.clf()pl.plot(x,y)pl.ylabel("log(z)")pl.xlabel("log(t)")

  • Power Law Fittingnoisy_z = z + pl.randn(z.size)*10pl.clf()pl.plot(t,z)pl.plot(t,noisy_z,'k.')

    noisy_y = np.log(noisy_z)pl.clf()pl.plot(x,y)pl.plot(x,noisy_y,'k.')pl.ylabel("log(z)")pl.xlabel("log(t)")

    It's a straight line. Now, for our "fake data", we'll add the noise before transforming from "linear" to "log" space

  • Power Law Fittingprint noisy_y[ 2.64567168 1.77226366 nan 1.49412348 nan 1.28855427 0.58257979 -0.50719229 1.84461177 2.72918637 nan -1.02747374 -0.65048639 3.4827123 2.60910913 3.48086587 3.84495668 3.90751514 4.10072678 3.76322421 4.06432005 4.23511474 4.26996586 4.25153639 4.4608377 4.5945862 4.66004888 4.8084364 4.81659305 4.97216304 5.05915226 5.02661678 5.05308181 5.26753675 5.23242563 5.36407125 5.41827503 5.486903 5.50263105 5.59005234 5.6588601 5.7546482 5.77382726 5.82299095 5.92653466 5.93264584 5.99879722 6.07711596 6.13466015 6.12248799]

    Note how different this looks from the "noisy line" we plotted earlier. Power laws are much more sensitive to noise! In fact, there are some data points that don't even show up on this plot because you can't take the log of a negative number. Any points where the random noise was negative enough that the curve dropped below zero ended up being "NAN", or "Not a Number". Luckily, our plotter knows to ignore those numbers, but polyfit doesnt.

  • Power Law Fitting# try to polyfit a linepars = np.polyfit(x,noisy_y,1)print pars[ nan nan]In order to get around this problem, we need to mask the data. That means we have to tell the code to ignore all the data points where noisy_y is nan.nan == nan is False

    OK = noisy_y == noisy_yprint OK[ True True False True False True True True True True False True True True True True True True True True True True True True True True True True True True True True True True True True True True True True True True True True True True True True True True]

  • Power Law Fittingprint "There are %i OK values" % (OK.sum())masked_noisy_y = noisy_y[OK]masked_x = x[OK]print "masked_noisy_y has length",len(masked_noisy_y)There are 47 OK valuesmasked_noisy_y has length 47

    # now polyfit againpars = np.polyfit(masked_x,masked_noisy_y,1)print pars[ 1.50239281 1.9907672 ]fitted_y = polyval(pars,x)pl.plot(x, fitted_y, 'r--')

    This OK array is a "boolean mask". We can use it as an "index array", which is pretty neat.

  • Power Law Fitting# Convert bag to linear-space to see what it "really" looks likefitted_z = np.exp(fitted_y)pl.clf()pl.plot(t,z)pl.plot(t,noisy_z,'k.')pl.plot(t,fitted_z,'r--')pl.xlabel('t')pl.ylabel('z')

    The noise seems to have affected our fit.

  • Power Law Fittingdef powerlaw(x,a,b): return a*(x**b)pars,covar = curve_fit(powerlaw,t,noisy_z)pl.clf()pl.plot(t,z)pl.plot(t,noisy_z,'k.')pl.plot(t,powerlaw(t,*pars),'r--')pl.xlabel('t')pl.ylabel('z')That's pretty bad. A "least-squares" approach, as with curve_fit, is probably going to be the better choice. However, in the absence of noise, this approach should work

  • Introduction to languageEnd

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