{
"cells": [
{
"cell_type": "markdown",
"metadata": {},
"source": [
"#Intro into IPython notebooks"
]
},
{
"cell_type": "code",
"execution_count": 1,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"Populating the interactive namespace from numpy and matplotlib\n"
]
}
],
"source": [
"%pylab inline"
]
},
{
"cell_type": "code",
"execution_count": 2,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/html": [
"\n",
" \n",
" "
],
"text/plain": [
""
]
},
"execution_count": 2,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"from IPython.display import YouTubeVideo\n",
"YouTubeVideo(\"qb7FT68tcA8\", width=600, height=400, theme=\"light\", color=\"blue\")"
]
},
{
"cell_type": "code",
"execution_count": 3,
"metadata": {
"collapsed": false
},
"outputs": [],
"source": [
"# You can ignore this, it's just for aesthetic purposes\n",
"matplotlib.rcParams['figure.figsize'] = (8,5)\n",
"rcParams['savefig.dpi'] = 100"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"#Fitting Lines to Data\n",
"\n",
"We'll cover very basic line fitting, largely ignoring the subtleties of the statistics in favor of showing you *how* to perform simple fits of models to data."
]
},
{
"cell_type": "code",
"execution_count": 4,
"metadata": {
"collapsed": false
},
"outputs": [],
"source": [
"# These import commands set up the environment so we have access to numpy and pylab functions\n",
"import numpy as np\n",
"import pylab as pl\n",
"\n",
"# Data Fitting\n",
"# First, we'll generate some fake data to use\n",
"x = np.linspace(0,10,50) # 50 x points from 0 to 10\n",
"\n",
"# Remember, you can look at the help for linspace too:\n",
"# help(np.linspace)"
]
},
{
"cell_type": "code",
"execution_count": 5,
"metadata": {
"collapsed": false
},
"outputs": [],
"source": [
"# y = m x + b\n",
"y = 2.5 * x + 1.2"
]
},
{
"cell_type": "code",
"execution_count": 6,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/plain": [
"[]"
]
},
"execution_count": 6,
"metadata": {},
"output_type": "execute_result"
},
{
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],
"text/plain": [
""
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"# let's plot that\n",
"pl.clf()\n",
"pl.plot(x,y)"
]
},
{
"cell_type": "code",
"execution_count": 7,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/plain": [
"[]"
]
},
"execution_count": 7,
"metadata": {},
"output_type": "execute_result"
},
{
"data": {
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],
"text/plain": [
""
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"# looks like a simple line. But we want to see the individual data points\n",
"pl.plot(x,y,marker='s')"
]
},
{
"cell_type": "code",
"execution_count": 8,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"50\n",
"50\n"
]
}
],
"source": [
"# We need to add noise first\n",
"noise = pl.randn(y.size)\n",
"# Like IDL, python has a 'randn' function that is centered at 0 with a standard deviation of 1. \n",
"# IDL's 'randomu' is 'pl.rand' instead\n",
"# What's y.size?\n",
"print y.size\n",
"print len(y)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"`y.size` is the number of elements in y, just like `len(y)` or, in IDL, `n_elements(y)`"
]
},
{
"cell_type": "code",
"execution_count": 9,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/plain": [
""
]
},
"execution_count": 9,
"metadata": {},
"output_type": "execute_result"
},
{
"data": {
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],
"text/plain": [
""
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"# We can add arrays in python just like in IDL\n",
"noisy_flux = y + noise\n",
"# We'll plot it too, but this time without any lines\n",
"# between the points, and we'll use black dots\n",
"# ('k' is a shortcut for 'black', '.' means 'point')\n",
"pl.clf() # clear the figure\n",
"pl.plot(x,noisy_flux,'k.')\n",
"# We need labels, of course\n",
"pl.xlabel(\"Time\")\n",
"pl.ylabel(\"Flux\")\n"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Now we're onto the fitting stage.\n",
"We're going to fit a function of the form\n",
"$$y = m*x + b$$\n",
"which is the same as\n",
"$$f(x) = p[1]*x + p[0]$$\n",
"to the data.\n",
"This is called \"linear regression\", but it is also a special case of a more\n",
"general concept: this is a first-order polynomial.\n",
"\"First Order\" means that the highest exponent of x in the equation is 1\n"
]
},
{
"cell_type": "code",
"execution_count": 10,
"metadata": {
"collapsed": false
},
"outputs": [],
"source": [
"# We'll use polyfit to find the values of the coefficients. The third\n",
"# parameter is the \"order\"\n",
"p = np.polyfit(x,noisy_flux,1)\n",
"# help(polyfit) if you want to find out more"
]
},
{
"cell_type": "code",
"execution_count": 11,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"[ 2.51109656 1.08260561]\n",
"\n"
]
}
],
"source": [
"# print our fit parameters. They are not exact because there's noise in the data!\n",
"# note that this is an array!\n",
"print p\n",
"print type(p) # you can ask python to tell you what type a variable is"
]
},
{
"cell_type": "code",
"execution_count": 12,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/plain": [
""
]
},
"execution_count": 12,
"metadata": {},
"output_type": "execute_result"
},
{
"data": {
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],
"text/plain": [
""
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"# Great! We've got our fit. Let's overplot the data and the fit now\n",
"pl.clf() # clear the figure\n",
"pl.plot(x,noisy_flux,'k.') # repeated from above\n",
"pl.plot(x,p[0]*x+p[1],'r-') # A red solid line\n",
"pl.xlabel(\"Time\") # labels again\n",
"pl.ylabel(\"Flux\")"
]
},
{
"cell_type": "code",
"execution_count": 13,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/plain": [
""
]
},
"execution_count": 13,
"metadata": {},
"output_type": "execute_result"
},
{
"data": {
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],
"text/plain": [
""
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"# Cool, but there's another (better) way to do this. We'll use the polyval\n",
"# function instead of writing out the m x + b equation ourselves\n",
"pl.clf() # clear the figure\n",
"pl.plot(x,noisy_flux,'k.') # repeated from above\n",
"pl.plot(x,np.polyval(p,x),'r-') # A red solid line\n",
"pl.xlabel(\"Time\") # labels again\n",
"pl.ylabel(\"Flux\")"
]
},
{
"cell_type": "code",
"execution_count": 14,
"metadata": {
"collapsed": false
},
"outputs": [],
"source": [
"# help(polyval) if you want to find out more"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Let's do the same thing with a noisier data set. I'm going to leave out most of the comments this time."
]
},
{
"cell_type": "code",
"execution_count": 15,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"[ 2.61096555 0.02605605]\n"
]
}
],
"source": [
"noisy_flux = y+noise*10\n",
"p = polyfit(x,noisy_flux,1)\n",
"print p"
]
},
{
"cell_type": "code",
"execution_count": 16,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/plain": [
""
]
},
"execution_count": 16,
"metadata": {},
"output_type": "execute_result"
},
{
"data": {
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],
"text/plain": [
""
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"# plot it\n",
"pl.clf() # clear the figure\n",
"pl.plot(x,noisy_flux,'k.') # repeated from above\n",
"pl.plot(x,np.polyval(p,x),'r-',label=\"Best fit\") # A red solid line\n",
"pl.plot(x,2.5*x+1.2,'b--',label=\"Input\") # a blue dashed line showing the REAL line\n",
"pl.legend(loc='best') # make a legend in the best location\n",
"pl.xlabel(\"Time\") # labels again\n",
"pl.ylabel(\"Flux\")"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Despite the noisy data, our fit is still pretty good! One last plotting trick, then we'll move on."
]
},
{
"cell_type": "code",
"execution_count": 17,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/plain": [
""
]
},
"execution_count": 17,
"metadata": {},
"output_type": "execute_result"
},
{
"data": {
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],
"text/plain": [
""
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"pl.clf() # clear the figure\n",
"pl.errorbar(x,noisy_flux,yerr=10,marker='.',color='k',linestyle='none') # errorbar requires some extras to look nice\n",
"pl.plot(x,np.polyval(p,x),'r-',label=\"Best fit\") # A red solid line\n",
"pl.plot(x,2.5*x+1.2,'b--',label=\"Input\") # a blue dashed line showing the REAL line\n",
"pl.legend(loc='best') # make a legend in the best location\n",
"pl.xlabel(\"Time\") # labels again\n",
"pl.ylabel(\"Flux\")"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"#Curve Fitting\n",
"\n",
"We'll now move on to more complicated curves. What if the data looks more like a sine curve? We'll create \"fake data\" in basically the same way as above."
]
},
{
"cell_type": "code",
"execution_count": 18,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/plain": [
"[]"
]
},
"execution_count": 18,
"metadata": {},
"output_type": "execute_result"
},
{
"data": {
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],
"text/plain": [
""
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"# this time we want our \"independent variable\" to be in radians\n",
"x = np.linspace(0,2*np.pi,50)\n",
"y = np.sin(x)\n",
"pl.clf()\n",
"pl.plot(x,y)"
]
},
{
"cell_type": "code",
"execution_count": 19,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/plain": [
"[]"
]
},
"execution_count": 19,
"metadata": {},
"output_type": "execute_result"
},
{
"data": {
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"text/plain": [
""
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"# We'll make it noisy again\n",
"noise = pl.randn(y.size)\n",
"noisy_flux = y + noise\n",
"pl.plot(x,noisy_flux,'k.') # no clear this time"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"That looks like kind of a mess. Let's see how well we can fit it.\n",
"The function we're trying to fit has the form:\n",
"$$f(x) = A * sin(x - B)$$\n",
"where $A$ is a \"scale\" parameter and $B$ is the side-to-side offset (or the \"delay\" if the x-axis is time). For our data, they are $A=1$ and $B=0$ respectively, because we made $y=sin(x)$"
]
},
{
"cell_type": "code",
"execution_count": 20,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"Help on function curve_fit in module scipy.optimize.minpack:\n",
"\n",
"curve_fit(f, xdata, ydata, p0=None, sigma=None, absolute_sigma=False, **kw)\n",
" Use non-linear least squares to fit a function, f, to data.\n",
" \n",
" Assumes ``ydata = f(xdata, *params) + eps``\n",
" \n",
" Parameters\n",
" ----------\n",
" f : callable\n",
" The model function, f(x, ...). It must take the independent\n",
" variable as the first argument and the parameters to fit as\n",
" separate remaining arguments.\n",
" xdata : An M-length sequence or an (k,M)-shaped array\n",
" for functions with k predictors.\n",
" The independent variable where the data is measured.\n",
" ydata : M-length sequence\n",
" The dependent data --- nominally f(xdata, ...)\n",
" p0 : None, scalar, or N-length sequence\n",
" Initial guess for the parameters. If None, then the initial\n",
" values will all be 1 (if the number of parameters for the function\n",
" can be determined using introspection, otherwise a ValueError\n",
" is raised).\n",
" sigma : None or M-length sequence, optional\n",
" If not None, these values are used as weights in the\n",
" least-squares problem.\n",
" absolute_sigma : bool, optional\n",
" If False, `sigma` denotes relative weights of the data points.\n",
" The returned covariance matrix `pcov` is based on *estimated*\n",
" errors in the data, and is not affected by the overall\n",
" magnitude of the values in `sigma`. Only the relative\n",
" magnitudes of the `sigma` values matter.\n",
" \n",
" If True, `sigma` describes one standard deviation errors of\n",
" the input data points. The estimated covariance in `pcov` is\n",
" based on these values.\n",
" \n",
" Returns\n",
" -------\n",
" popt : array\n",
" Optimal values for the parameters so that the sum of the squared error\n",
" of ``f(xdata, *popt) - ydata`` is minimized\n",
" pcov : 2d array\n",
" The estimated covariance of popt. The diagonals provide the variance\n",
" of the parameter estimate. To compute one standard deviation errors\n",
" on the parameters use ``perr = np.sqrt(np.diag(pcov))``.\n",
" \n",
" How the `sigma` parameter affects the estimated covariance\n",
" depends on `absolute_sigma` argument, as described above.\n",
" \n",
" See Also\n",
" --------\n",
" leastsq\n",
" \n",
" Notes\n",
" -----\n",
" The algorithm uses the Levenberg-Marquardt algorithm through `leastsq`.\n",
" Additional keyword arguments are passed directly to that algorithm.\n",
" \n",
" Examples\n",
" --------\n",
" >>> import numpy as np\n",
" >>> from scipy.optimize import curve_fit\n",
" >>> def func(x, a, b, c):\n",
" ... return a * np.exp(-b * x) + c\n",
" \n",
" >>> xdata = np.linspace(0, 4, 50)\n",
" >>> y = func(xdata, 2.5, 1.3, 0.5)\n",
" >>> ydata = y + 0.2 * np.random.normal(size=len(xdata))\n",
" \n",
" >>> popt, pcov = curve_fit(func, xdata, ydata)\n",
"\n"
]
}
],
"source": [
"# curve_fit is the function we need for this, but it's in another package called scipy\n",
"from scipy.optimize import curve_fit\n",
"# we need to know what it does:\n",
"help(curve_fit)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Look at the returns:\n",
"\n",
"\n",
" Returns\n",
" -------\n",
" popt : array\n",
" Optimal values for the parameters so that the sum of the squared error\n",
" of ``f(xdata, *popt) - ydata`` is minimized\n",
" pcov : 2d array\n",
" The estimated covariance of popt. The diagonals provide the variance\n",
" of the parameter estimate.\n",
" \n",
"\n",
"So the first set of returns is the \"best-fit parameters\", while the second set is the \"covariance matrix\""
]
},
{
"cell_type": "code",
"execution_count": 21,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"[ 1.02709234 0.02889065] [ 0.20008943 0.19095993]\n"
]
}
],
"source": [
"def sinfunc(x,a,b):\n",
" return a*np.sin(x-b)\n",
"fitpars, covmat = curve_fit(sinfunc,x,noisy_flux)\n",
"# The diagonals of the covariance matrix are variances\n",
"# variance = standard deviation squared, so we'll take the square roots to get the standard devations!\n",
"# You can get the diagonals of a 2D array easily:\n",
"variances = covmat.diagonal()\n",
"std_devs = np.sqrt(variances)\n",
"print fitpars,std_devs"
]
},
{
"cell_type": "code",
"execution_count": 22,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/plain": [
"[]"
]
},
"execution_count": 22,
"metadata": {},
"output_type": "execute_result"
},
{
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],
"text/plain": [
""
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"# Let's plot our best fit, see how well we did\n",
"# These two lines are equivalent:\n",
"pl.plot(x, sinfunc(x, fitpars[0], fitpars[1]), 'r-')\n",
"pl.plot(x, sinfunc(x, *fitpars), 'r-')"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Again, this is pretty good despite the noisiness."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"#Fitting a Power Law\n",
"\n",
"Power laws occur all the time in physis, so it's a good idea to learn how to use them.\n",
"\n",
"What's a power law? Any function of the form:\n",
"$$f(t) = a t^b$$\n",
"where $x$ is your independent variable, $a$ is a scale parameter, and $b$ is the exponent (the power).\n",
"\n",
"When fitting power laws, it's very useful to take advantage of the fact that \"a power law is linear in log-space\".\n",
"That means, if you take the log of both sides of the equation (which is allowed) and change variables, you get a \n",
"linear equation!\n",
"$$\\ln(f(t)) = \\ln(a t^b) = \\ln(a) + b \\ln(t)$$\n",
"We'll use the substitutions $y=\\ln(f(t))$, $A=\\ln(a)$, and $x=\\ln(t)$, so that\n",
"$$y=a+bx$$\n",
"which looks just like our linear equation from before (albeit with different letters for the fit parameters).\n",
"\n",
"We'll now go through the same fitting exercise as before, but using powerlaws instead of lines."
]
},
{
"cell_type": "code",
"execution_count": 23,
"metadata": {
"collapsed": false
},
"outputs": [],
"source": [
"t = np.linspace(0.1,10) \n",
"a = 1.5\n",
"b = 2.5\n",
"z = a*t**b "
]
},
{
"cell_type": "code",
"execution_count": 24,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/plain": [
"[]"
]
},
"execution_count": 24,
"metadata": {},
"output_type": "execute_result"
},
{
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],
"text/plain": [
""
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"pl.clf()\n",
"pl.plot(t,z)"
]
},
{
"cell_type": "code",
"execution_count": 25,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/plain": [
""
]
},
"execution_count": 25,
"metadata": {},
"output_type": "execute_result"
},
{
"data": {
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"+HqrrBeA64BVpV4zCbix1LFzCDsXbGf3rbIeBB7bj/klSZKkcvsT8FapY7vusLWvJWD1CXfYKr0U\n",
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],
"text/plain": [
""
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"# Change the variables\n",
"# np.log is the natural log\n",
"y = np.log(z)\n",
"x = np.log(t)\n",
"pl.clf()\n",
"pl.plot(x,y)\n",
"pl.ylabel(\"log(z)\")\n",
"pl.xlabel(\"log(t)\")"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"It's a straight line. Now, for our \"fake data\", we'll add the noise *before* transforming from \"linear\" to \"log\" space"
]
},
{
"cell_type": "code",
"execution_count": 26,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/plain": [
"[]"
]
},
"execution_count": 26,
"metadata": {},
"output_type": "execute_result"
},
{
"data": {
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"/99O5VxzOWHO8xfAKMpwIwNJkiRJkiRJkiRJkiRJkiRJkiRJkiRJkiRJkiRJkiRJkiRJkiRJkiRJ\n",
"kiRJkiRJkiRJkiSVof8PB6CFmR5ZujsAAAAASUVORK5CYII=\n"
],
"text/plain": [
""
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"noisy_z = z + pl.randn(z.size)*10\n",
"pl.clf()\n",
"pl.plot(t,z)\n",
"pl.plot(t,noisy_z,'k.')"
]
},
{
"cell_type": "code",
"execution_count": 27,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stderr",
"output_type": "stream",
"text": [
"-c:1: RuntimeWarning: invalid value encountered in log\n"
]
},
{
"data": {
"text/plain": [
""
]
},
"execution_count": 27,
"metadata": {},
"output_type": "execute_result"
},
{
"data": {
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],
"text/plain": [
""
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"noisy_y = np.log(noisy_z)\n",
"pl.clf()\n",
"pl.plot(x,y)\n",
"pl.plot(x,noisy_y,'k.')\n",
"pl.ylabel(\"log(z)\")\n",
"pl.xlabel(\"log(t)\")"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"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."
]
},
{
"cell_type": "code",
"execution_count": 28,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"[ nan 2.75771666 nan nan nan -0.47965175\n",
" 1.11478478 -1.26005945 2.80987409 1.960784 2.51234432 3.28748507\n",
" 0.87394124 2.04142372 3.38148253 3.07025932 3.7646366 3.39576211\n",
" 3.89415046 3.63523727 4.28055748 4.05990115 4.25603083 4.31912881\n",
" 4.37483666 4.36715417 4.43359341 4.6376513 4.86019646 4.83042249\n",
" 5.01487092 4.95886081 5.09708006 5.27513912 5.2566149 5.34589262\n",
" 5.43611065 5.47480662 5.58954413 5.56268956 5.66424772 5.72223943\n",
" 5.785335 5.85617183 5.90007055 5.94997552 6.02859912 6.03561701\n",
" 6.13080122 6.14716578]\n"
]
}
],
"source": [
"print noisy_y"
]
},
{
"cell_type": "code",
"execution_count": 29,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"[ nan nan]\n"
]
}
],
"source": [
"# try to polyfit a line\n",
"pars = np.polyfit(x,noisy_y,1)\n",
"print pars"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"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`.\n",
"\n",
"My favorite way to do this is to take advantage of a curious fact: $1=1$, but `nan`!=`nan`"
]
},
{
"cell_type": "code",
"execution_count": 30,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"True\n",
"False\n"
]
}
],
"source": [
"print 1 == 1\n",
"print np.nan == np.nan"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"So if we find all the places were `noisy_y` != `noisy_y`, we can get rid of them. Or we can just use the places where `noisy_y` equals itself."
]
},
{
"cell_type": "code",
"execution_count": 31,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"[False True False False False True True True True True True True\n",
" True True True True True True True True True True True True\n",
" True True True True True True True True True True True True\n",
" True True True True True True True True True True True True\n",
" True True]\n"
]
}
],
"source": [
"OK = noisy_y == noisy_y\n",
"print OK"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"This `OK` array is a \"boolean mask\". We can use it as an \"index array\", which is pretty neat. "
]
},
{
"cell_type": "code",
"execution_count": 32,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"There are 46 OK values\n",
"masked_noisy_y has length 46\n"
]
}
],
"source": [
"print \"There are %i OK values\" % (OK.sum())\n",
"masked_noisy_y = noisy_y[OK]\n",
"masked_x = x[OK]\n",
"print \"masked_noisy_y has length\",len(masked_noisy_y)"
]
},
{
"cell_type": "code",
"execution_count": 33,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"[ 2.05726448 1.10149221]\n"
]
}
],
"source": [
"# now polyfit again\n",
"pars = np.polyfit(masked_x,masked_noisy_y,1)\n",
"print pars"
]
},
{
"cell_type": "code",
"execution_count": 34,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/plain": [
"[]"
]
},
"execution_count": 34,
"metadata": {},
"output_type": "execute_result"
},
{
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],
"text/plain": [
""
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"# cool, it worked. But the fit looks a little weird!\n",
"fitted_y = polyval(pars,x)\n",
"pl.plot(x, fitted_y, 'r--')"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"The noise seems to have affected our fit. "
]
},
{
"cell_type": "code",
"execution_count": 35,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/plain": [
""
]
},
"execution_count": 35,
"metadata": {},
"output_type": "execute_result"
},
{
"data": {
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],
"text/plain": [
""
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"# Convert bag to linear-space to see what it \"really\" looks like\n",
"fitted_z = np.exp(fitted_y)\n",
"pl.clf()\n",
"pl.plot(t,z)\n",
"pl.plot(t,noisy_z,'k.')\n",
"pl.plot(t,fitted_z,'r--')\n",
"pl.xlabel('t')\n",
"pl.ylabel('z')"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"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 (i.e., on your homework), this approach *should* work"
]
},
{
"cell_type": "code",
"execution_count": 36,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/plain": [
""
]
},
"execution_count": 36,
"metadata": {},
"output_type": "execute_result"
},
{
"data": {
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""
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"def powerlaw(x,a,b):\n",
" return a*(x**b)\n",
"pars,covar = curve_fit(powerlaw,t,noisy_z)\n",
"pl.clf()\n",
"pl.plot(t,z)\n",
"pl.plot(t,noisy_z,'k.')\n",
"pl.plot(t,powerlaw(t,*pars),'r--')\n",
"pl.xlabel('t')\n",
"pl.ylabel('z')"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"#Tricks with Arrays\n",
"\n",
"We need to cover a few syntactic things comparing IDL and python.\n",
"\n",
"In IDL, if you wanted the maximum value in an array, you would do: \n",
"`maxval = max(array, location_of_max)`\n",
"\n",
"\n",
"In python, it's more straightforward: \n",
"`location_of_max = array.argmax()` \n",
"or \n",
"`location_of_max = np.argmax(array)` \n",
"\n",
"Now, say we want to determine the location of the maximum of a number of different functions. The functions we'll use are: \n",
"`sin(x)` \n",
"`sin`$^2$`(x)` \n",
"`sin`$^3$`(x)` \n",
"`sin(x)cos(x)` \n",
"\n",
"We'll define these functions, then loop over them. "
]
},
{
"cell_type": "code",
"execution_count": 37,
"metadata": {
"collapsed": false
},
"outputs": [],
"source": [
"# sin(x) is already defined\n",
"def sin2x(x):\n",
" \"\"\" sin^2 of x \"\"\"\n",
" return np.sin(x)**2\n",
"def sin3x(x):\n",
" \"\"\" sin^3 of x \"\"\"\n",
" return np.sin(x)**3\n",
"def sincos(x):\n",
" \"\"\" sin(x)*cos(x) \"\"\"\n",
" return np.sin(x)*np.cos(x)\n",
"list_of_functions = [np.sin, sin2x, sin3x, sincos]"
]
},
{
"cell_type": "code",
"execution_count": 38,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"The maximum of sin is 0.999486216201\n",
"The maximum of sin2x is 0.998972696375\n",
"The maximum of sin3x is 0.998459440388\n",
"The maximum of sincos is 0.4997431081\n"
]
}
],
"source": [
"# we want 0-2pi for these functions\n",
"t = np.linspace(0,2*np.pi)\n",
"# this is the cool part: we can make a variable function\n",
"for fun in list_of_functions:\n",
" # the functions know their own names (in a \"secret hidden variable\" called __name__)\n",
" print \"The maximum of \",fun.__name__,\" is \", fun(t).max()"
]
},
{
"cell_type": "code",
"execution_count": 39,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"The location of the maximum of sin is 12\n",
"The location of the maximum of sin2x is 12\n",
"The location of the maximum of sin3x is 12\n",
"The location of the maximum of sincos is 6\n"
]
}
],
"source": [
"# OK, but we wanted the location of the maximum....\n",
"for fun in list_of_functions:\n",
" print \"The location of the maximum of \",fun.__name__,\" is \", fun(t).argmax()"
]
},
{
"cell_type": "code",
"execution_count": 40,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"The location of the maximum of sin is 1.5387392589\n",
"The location of the maximum of sin2x is 1.5387392589\n",
"The location of the maximum of sin3x is 1.5387392589\n",
"The location of the maximum of sincos is 0.769369629451\n"
]
}
],
"source": [
"# well, that's not QUITE what we want, but it's close\n",
"# We want to know the value of t, not the index!\n",
"for fun in list_of_functions:\n",
" print \"The location of the maximum of \",fun.__name__,\" is \", t[fun(t).argmax()]"
]
},
{
"cell_type": "code",
"execution_count": 41,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"[1.5387392589011231, 1.5387392589011231, 1.5387392589011231, 0.76936962945056153]\n"
]
}
],
"source": [
"# Finally, what if we want to store all that in an array?\n",
"# Well, here's a cool trick: you can sort of invert the for loop\n",
"# This is called a \"list comprehension\":\n",
"maxlocs = [ t[fun(t).argmax()] for fun in list_of_functions ]\n",
"print maxlocs"
]
},
{
"cell_type": "code",
"execution_count": 42,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"[0, 1, 2, 3, 4, 5]\n",
"[0, 1, 4, 9, 16, 25]\n"
]
}
],
"source": [
"# Confused? OK. Try this one:\n",
"print range(6)\n",
"print [ii**2 for ii in range(6)]"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"#Further info on IPython Notebooks\n",
"\n",
"| Overview | link |\n",
"|--------------------------------------|------------------------------------------------------------------------------------|\n",
"| Blog of IPython creator | http://blog.fperez.org/2012/09/blogging-with-ipython-notebook.html |\n",
"| Blog of an avid IPython user | http://www.damian.oquanta.info/index.html |\n",
"| Turning notebook into a presentation | https://www.youtube.com/watch?v=rBS6hmiK-H8 |\n",
"| Tutorial on IPython & SciPy | https://github.com/esc/scipy2013-tutorial-numpy-ipython |\n",
"| IPython notebooks gallery | https://github.com/ipython/ipython/wiki/A-gallery-of-interesting-IPython-Notebooks |\n"
]
},
{
"cell_type": "code",
"execution_count": 43,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/html": [
"\n",
" \n",
" "
],
"text/plain": [
""
]
},
"execution_count": 43,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"from IPython.display import YouTubeVideo\n",
"YouTubeVideo(\"xe_ATRmw0KM\", width=600, height=400, theme=\"light\", color=\"blue\")"
]
},
{
"cell_type": "code",
"execution_count": 44,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/html": [
"\n",
" \n",
" "
],
"text/plain": [
""
]
},
"execution_count": 44,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"from IPython.display import YouTubeVideo\n",
"YouTubeVideo(\"zG8FYPFU9n4\", width=600, height=400, theme=\"light\", color=\"blue\")"
]
},
{
"cell_type": "code",
"execution_count": 44,
"metadata": {
"collapsed": false
},
"outputs": [],
"source": []
}
],
"metadata": {
"kernelspec": {
"display_name": "Python 2",
"language": "python",
"name": "python2"
},
"language_info": {
"codemirror_mode": {
"name": "ipython",
"version": 2
},
"file_extension": ".py",
"mimetype": "text/x-python",
"name": "python",
"nbconvert_exporter": "python",
"pygments_lexer": "ipython2",
"version": "2.7.8"
}
},
"nbformat": 4,
"nbformat_minor": 0
}