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@fonnesbeck
Created May 20, 2015 21:18
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{
"cells": [
{
"cell_type": "code",
"execution_count": 81,
"metadata": {
"collapsed": true,
"internals": {
"slide_helper": "subslide_end",
"slide_type": "subslide"
},
"slide_helper": "slide_end",
"slideshow": {
"slide_type": "skip"
}
},
"outputs": [],
"source": [
"%matplotlib inline\n",
"from pymc.gp import *\n",
"import numpy as np\n",
"import matplotlib.pylab as plt"
]
},
{
"cell_type": "markdown",
"metadata": {
"internals": {
"slide_type": "subslide"
},
"slideshow": {
"slide_type": "slide"
}
},
"source": [
"## Mean function\n",
"\n",
"The mean function of a GP can be interpreted as a \"prior guess\" at the form of the true function."
]
},
{
"cell_type": "code",
"execution_count": 82,
"metadata": {
"collapsed": false,
"internals": {},
"slideshow": {
"slide_type": "-"
}
},
"outputs": [],
"source": [
"# Generate mean\n",
"def quadfun(x, a, b, c):\n",
" return (a*x**2 + b*x + c)\n",
"\n",
"M = Mean(quadfun, a=1., b=0.5, c=2.)"
]
},
{
"cell_type": "code",
"execution_count": 83,
"metadata": {
"collapsed": false,
"internals": {},
"slideshow": {
"slide_type": "-"
}
},
"outputs": [
{
"data": {
"text/plain": [
"[<matplotlib.lines.Line2D at 0x10aa95940>]"
]
},
"execution_count": 83,
"metadata": {},
"output_type": "execute_result"
},
{
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],
"text/plain": [
"<matplotlib.figure.Figure at 0x10aaec630>"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"x = np.arange(-1,1,0.1)\n",
"plt.plot(x, M(x), 'k-')"
]
},
{
"cell_type": "markdown",
"metadata": {
"internals": {},
"slideshow": {
"slide_type": "-"
}
},
"source": [
"## Covariance function\n",
"\n",
"The behavior of individual realizations from the GP is governed by the covariance function. The Mat&#232;rn class of functions is a flexible choice."
]
},
{
"cell_type": "code",
"execution_count": 97,
"metadata": {
"collapsed": false,
"internals": {},
"slideshow": {
"slide_type": "-"
}
},
"outputs": [
{
"data": {
"text/plain": [
"<matplotlib.text.Text at 0x10c4d3ef0>"
]
},
"execution_count": 97,
"metadata": {},
"output_type": "execute_result"
},
{
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],
"text/plain": [
"<matplotlib.figure.Figure at 0x10c2a05f8>"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"from pymc.gp.cov_funs import matern\n",
"import numpy as np\n",
"C = Covariance(eval_fun=matern.euclidean, diff_degree=1.4, amp=0.4, scale=1, rank_limit=1000)\n",
"\n",
"plt.subplot(1,2,2)\n",
"plt.contourf(x, x, C(x,x).view(np.ndarray), origin='lower', extent=(-1,1,-1,1), cmap=plt.cm.bone)\n",
"plt.colorbar()\n",
"\n",
"plt.subplot(1,2,1)\n",
"plt.plot(x, C(x,0).view(np.ndarray), 'k-')\n",
"plt.ylabel('C(x,0)')"
]
},
{
"cell_type": "code",
"execution_count": 98,
"metadata": {
"collapsed": false,
"internals": {},
"slideshow": {
"slide_type": "-"
}
},
"outputs": [
{
"data": {
"text/plain": [
"array([ 0.16, 0.16, 0.16, 0.16, 0.16, 0.16, 0.16, 0.16, 0.16,\n",
" 0.16, 0.16, 0.16, 0.16, 0.16, 0.16, 0.16, 0.16, 0.16,\n",
" 0.16, 0.16, 0.16, 0.16, 0.16, 0.16, 0.16, 0.16, 0.16,\n",
" 0.16, 0.16, 0.16, 0.16, 0.16, 0.16, 0.16, 0.16, 0.16,\n",
" 0.16, 0.16, 0.16, 0.16, 0.16, 0.16, 0.16, 0.16, 0.16,\n",
" 0.16, 0.16, 0.16, 0.16, 0.16, 0.16, 0.16, 0.16, 0.16,\n",
" 0.16, 0.16, 0.16, 0.16, 0.16, 0.16, 0.16, 0.16, 0.16,\n",
" 0.16, 0.16, 0.16, 0.16, 0.16, 0.16, 0.16, 0.16, 0.16,\n",
" 0.16, 0.16, 0.16, 0.16, 0.16, 0.16, 0.16, 0.16, 0.16,\n",
" 0.16, 0.16, 0.16, 0.16, 0.16, 0.16, 0.16, 0.16, 0.16,\n",
" 0.16, 0.16, 0.16, 0.16, 0.16, 0.16, 0.16, 0.16, 0.16,\n",
" 0.16, 0.16, 0.16, 0.16, 0.16, 0.16, 0.16, 0.16, 0.16,\n",
" 0.16, 0.16, 0.16, 0.16, 0.16, 0.16, 0.16, 0.16, 0.16,\n",
" 0.16, 0.16, 0.16, 0.16, 0.16, 0.16, 0.16, 0.16, 0.16,\n",
" 0.16, 0.16, 0.16, 0.16, 0.16, 0.16, 0.16, 0.16, 0.16,\n",
" 0.16, 0.16, 0.16, 0.16, 0.16, 0.16, 0.16, 0.16, 0.16,\n",
" 0.16, 0.16, 0.16, 0.16, 0.16, 0.16, 0.16, 0.16, 0.16,\n",
" 0.16, 0.16, 0.16, 0.16, 0.16, 0.16, 0.16, 0.16, 0.16,\n",
" 0.16, 0.16, 0.16, 0.16, 0.16, 0.16, 0.16, 0.16, 0.16,\n",
" 0.16, 0.16, 0.16, 0.16, 0.16, 0.16, 0.16, 0.16, 0.16,\n",
" 0.16, 0.16, 0.16, 0.16, 0.16, 0.16, 0.16, 0.16, 0.16,\n",
" 0.16, 0.16, 0.16, 0.16, 0.16, 0.16, 0.16, 0.16, 0.16,\n",
" 0.16, 0.16])"
]
},
"execution_count": 98,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"# Returns the diagnonal\n",
"C(x)"
]
},
{
"cell_type": "code",
"execution_count": 99,
"metadata": {
"collapsed": false,
"internals": {},
"slideshow": {
"slide_type": "-"
}
},
"outputs": [
{
"data": {
"text/plain": [
"matrix([[ 0.16, 0.16, 0.16, 0.16, 0.16, 0.16, 0.16, 0.16, 0.16,\n",
" 0.16, 0.16, 0.16, 0.16, 0.16, 0.16, 0.16, 0.16, 0.16,\n",
" 0.16, 0.16, 0.16, 0.16, 0.16, 0.16, 0.16, 0.16, 0.16,\n",
" 0.16, 0.16, 0.16, 0.16, 0.16, 0.16, 0.16, 0.16, 0.16,\n",
" 0.16, 0.16, 0.16, 0.16, 0.16, 0.16, 0.16, 0.16, 0.16,\n",
" 0.16, 0.16, 0.16, 0.16, 0.16, 0.16, 0.16, 0.16, 0.16,\n",
" 0.16, 0.16, 0.16, 0.16, 0.16, 0.16, 0.16, 0.16, 0.16,\n",
" 0.16, 0.16, 0.16, 0.16, 0.16, 0.16, 0.16, 0.16, 0.16,\n",
" 0.16, 0.16, 0.16, 0.16, 0.16, 0.16, 0.16, 0.16, 0.16,\n",
" 0.16, 0.16, 0.16, 0.16, 0.16, 0.16, 0.16, 0.16, 0.16,\n",
" 0.16, 0.16, 0.16, 0.16, 0.16, 0.16, 0.16, 0.16, 0.16,\n",
" 0.16, 0.16, 0.16, 0.16, 0.16, 0.16, 0.16, 0.16, 0.16,\n",
" 0.16, 0.16, 0.16, 0.16, 0.16, 0.16, 0.16, 0.16, 0.16,\n",
" 0.16, 0.16, 0.16, 0.16, 0.16, 0.16, 0.16, 0.16, 0.16,\n",
" 0.16, 0.16, 0.16, 0.16, 0.16, 0.16, 0.16, 0.16, 0.16,\n",
" 0.16, 0.16, 0.16, 0.16, 0.16, 0.16, 0.16, 0.16, 0.16,\n",
" 0.16, 0.16, 0.16, 0.16, 0.16, 0.16, 0.16, 0.16, 0.16,\n",
" 0.16, 0.16, 0.16, 0.16, 0.16, 0.16, 0.16, 0.16, 0.16,\n",
" 0.16, 0.16, 0.16, 0.16, 0.16, 0.16, 0.16, 0.16, 0.16,\n",
" 0.16, 0.16, 0.16, 0.16, 0.16, 0.16, 0.16, 0.16, 0.16,\n",
" 0.16, 0.16, 0.16, 0.16, 0.16, 0.16, 0.16, 0.16, 0.16,\n",
" 0.16, 0.16, 0.16, 0.16, 0.16, 0.16, 0.16, 0.16, 0.16,\n",
" 0.16, 0.16]])"
]
},
"execution_count": 99,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"np.diag(C(x,x))"
]
},
{
"cell_type": "markdown",
"metadata": {
"internals": {},
"slideshow": {
"slide_type": "-"
}
},
"source": [
"## Drawing realizations from a GP"
]
},
{
"cell_type": "code",
"execution_count": 103,
"metadata": {
"collapsed": false,
"internals": {},
"slideshow": {
"slide_type": "-"
}
},
"outputs": [
{
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],
"text/plain": [
"<matplotlib.figure.Figure at 0x10c13add8>"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"# Generate realizations\n",
"f_list = [Realization(M, C) for i in range(3)]\n",
"\n",
"# Plot mean and covariance\n",
"x = np.arange(-1,1,0.01)\n",
"plot_envelope(M, C, x)\n",
"\n",
"# Add realizations\n",
"for f in f_list:\n",
" plt.plot(x, f(x))"
]
},
{
"cell_type": "markdown",
"metadata": {
"internals": {},
"slideshow": {
"slide_type": "-"
}
},
"source": [
"## Non-parametric Regression"
]
},
{
"cell_type": "markdown",
"metadata": {
"internals": {},
"slideshow": {
"slide_type": "-"
}
},
"source": [
"Under a GP prior for an unknown function `f`, when the observation error is normally distributed, the posterior us another GP with new mean and covariance functions."
]
},
{
"cell_type": "code",
"execution_count": 110,
"metadata": {
"collapsed": false,
"internals": {},
"slideshow": {
"slide_type": "-"
}
},
"outputs": [],
"source": [
"M = Mean(quadfun, a=1., b=0.5, c=2.)\n",
"C = Covariance(eval_fun=matern.euclidean, diff_degree=1.4, amp=0.4, scale=1, rank_limit=1000)\n",
"\n",
"obs_x = np.array([-.5, .5])\n",
"V = np.array([0.002, 0.002])\n",
"data = np.array([3.1, 2.9])\n",
"\n",
"observe(M=M, C=C, obs_mesh=obs_x, obs_V=V, obs_vals=data)\n",
"\n",
"# Generate realizations from posterior\n",
"f_list = [Realization(M,C) for i in range(3)]"
]
},
{
"cell_type": "markdown",
"metadata": {
"internals": {},
"slideshow": {
"slide_type": "-"
}
},
"source": [
"The function `observe` informs the mean and covariance functions that values on `obs_mesh` with observation variance `V`. Making observations with no error is called `conditioning`. This is useful when, for example, forcing a rate function to be zero when a population's size is zero."
]
},
{
"cell_type": "code",
"execution_count": 111,
"metadata": {
"collapsed": false,
"internals": {},
"slideshow": {
"slide_type": "-"
}
},
"outputs": [
{
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],
"text/plain": [
"<matplotlib.figure.Figure at 0x10300c390>"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"plot_envelope(M, C, mesh=x)\n",
"for f in f_list:\n",
" plt.plot(x, f(x))"
]
},
{
"cell_type": "markdown",
"metadata": {
"internals": {},
"slideshow": {
"slide_type": "-"
}
},
"source": [
"## Salmon Example"
]
},
{
"cell_type": "code",
"execution_count": 112,
"metadata": {
"collapsed": true,
"internals": {},
"slideshow": {
"slide_type": "-"
}
},
"outputs": [],
"source": [
"class salmon:\n",
" \"\"\"\n",
" Reads and organizes data from csv files,\n",
" acts as a container for mean and covariance objects,\n",
" makes plots.\n",
" \"\"\"\n",
" def __init__(self, name, data):\n",
"\n",
" # Read in data\n",
"\n",
" self.name = name\n",
"\n",
" self.abundance = data[:,0].ravel()\n",
" self.frye = data[:,1].ravel()\n",
"\n",
" # Specify priors\n",
"\n",
" # Function for prior mean\n",
" def line(x, slope):\n",
" return slope * x\n",
"\n",
" self.M = Mean(line, slope = np.mean(self.frye / self.abundance))\n",
"\n",
" self.C = Covariance( matern.euclidean,\n",
" diff_degree = 1.4,\n",
" scale = 100. * self.abundance.max(),\n",
" amp = 200. * self.frye.max())\n",
"\n",
" observe(self.M,self.C,obs_mesh = 0, obs_vals = 0, obs_V = 0)\n",
"\n",
" self.xplot = np.linspace(0,1.25 * self.abundance.max(),100)\n",
"\n",
"\n",
"\n",
" def plot(self):\n",
" \"\"\"\n",
" Plot posterior from simple nonstochetric regression.\n",
" \"\"\"\n",
" plt.figure()\n",
" plot_envelope(self.M, self.C, self.xplot)\n",
" for i in range(3):\n",
" f = Realization(self.M, self.C)\n",
" plt.plot(self.xplot,f(self.xplot))\n",
"\n",
" plt.plot(self.abundance, self.frye, 'k.', markersize=4)\n",
" plt.xlabel('Female abundance')\n",
" plt.ylabel('Frye density')\n",
" plt.title(self.name)\n",
" plt.axis('tight')"
]
},
{
"cell_type": "code",
"execution_count": 113,
"metadata": {
"collapsed": false,
"internals": {},
"slideshow": {
"slide_type": "-"
}
},
"outputs": [],
"source": [
"sockeye_data = np.reshape([2986,9,\n",
"3424,12.39,\n",
"1631,4.5,\n",
"784,2.56,\n",
"9671,32.62,\n",
"2519,8.19,\n",
"1520,4.51,\n",
"6418,15.21,\n",
"10857,35.05,\n",
"15044,36.85,\n",
"10287,25.68,\n",
"16525,52.75,\n",
"19172,19.52,\n",
"17527,40.98,\n",
"11424,26.67,\n",
"24043,52.6,\n",
"10244,21.62,\n",
"30983,56.05,\n",
"12037,29.31,\n",
"25098,45.4,\n",
"11362,18.88,\n",
"24375,19.14,\n",
"18281,33.77,\n",
"14192,20.44,\n",
"7527,21.66,\n",
"6061,18.22,\n",
"15536,42.9,\n",
"18080,46.09,\n",
"17354,38.82,\n",
"17301,42.22,\n",
"11486,21.96,\n",
"20120,45.05,\n",
"10700,13.7,\n",
"12867,27.71,], (34,2))"
]
},
{
"cell_type": "code",
"execution_count": 114,
"metadata": {
"collapsed": false,
"internals": {},
"slideshow": {
"slide_type": "-"
}
},
"outputs": [],
"source": [
"# Instantiate salmon object with sockeye abundance\n",
"sockeye = salmon('sockeye', sockeye_data)\n",
"\n",
"# Observe some data\n",
"observe(sockeye.M, sockeye.C, obs_mesh = sockeye.abundance, obs_vals = sockeye.frye, obs_V = .25*sockeye.frye)"
]
},
{
"cell_type": "code",
"execution_count": 115,
"metadata": {
"collapsed": false,
"internals": {},
"slideshow": {
"slide_type": "-"
}
},
"outputs": [
{
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],
"text/plain": [
"<matplotlib.figure.Figure at 0x10eddd828>"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"sockeye.plot()"
]
},
{
"cell_type": "markdown",
"metadata": {
"collapsed": false,
"internals": {},
"slideshow": {
"slide_type": "-"
}
},
"source": [
"## Multidimensional Function Approximation\n",
"\n",
"A non-linear function to approximate:"
]
},
{
"cell_type": "code",
"execution_count": 116,
"metadata": {
"collapsed": false,
"internals": {
"slide_helper": "subslide_end"
},
"slide_helper": "slide_end",
"slideshow": {
"slide_type": "-"
}
},
"outputs": [],
"source": [
"def sinxfun(x=0,y=0):\n",
" ''' for each (x,y) point, it returns d*sin(d), where d is the distance from the origin.'''\n",
" d = np.sqrt(x**2+y**2)\n",
" return d*np.sin(d)"
]
},
{
"cell_type": "code",
"execution_count": 117,
"metadata": {
"collapsed": false
},
"outputs": [],
"source": [
"x_mesh = np.arange(-10, 10, 0.1)\n",
"y_mesh = np.arange(-10, 10, 0.1)\n",
"xx, yy = np.meshgrid(x_mesh, y_mesh)\n",
"z_mesh = sinxfun(xx,yy)"
]
},
{
"cell_type": "code",
"execution_count": 118,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/plain": [
"<mpl_toolkits.mplot3d.art3d.Poly3DCollection at 0x10c7215f8>"
]
},
"execution_count": 118,
"metadata": {},
"output_type": "execute_result"
},
{
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"SUVORK5CYII=\n"
],
"text/plain": [
"<matplotlib.figure.Figure at 0x10c720dd8>"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"from mpl_toolkits.mplot3d import Axes3D\n",
"fig = plt.figure()\n",
"ax = fig.gca(projection='3d')\n",
"ax.plot_surface(xx, yy, z_mesh, cmap=plt.cm.Blues, vmin=-15, vmax=15)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"We define our GP using a mean function and a covariance function."
]
},
{
"cell_type": "code",
"execution_count": 119,
"metadata": {
"collapsed": true
},
"outputs": [],
"source": [
"#assumed mean of gassian process = 0\n",
"def meanfun(xy):\n",
" return np.zeros((len(xy),1))\n",
"\n",
"M = Mean(meanfun)\n",
"\n",
"\n",
"#covariance is what was used in the example above\n",
"C = Covariance(eval_fun = matern.euclidean, \n",
" diff_degree = 10.4, \n",
" amp = 10.4, \n",
" scale = 10.)\n"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Let's draw sample functions from the prior"
]
},
{
"cell_type": "code",
"execution_count": 120,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/plain": [
"array([[-10. , -10. ],\n",
" [ -9.9, -10. ],\n",
" [ -9.8, -10. ],\n",
" ..., \n",
" [ 9.7, 9.9],\n",
" [ 9.8, 9.9],\n",
" [ 9.9, 9.9]])"
]
},
"execution_count": 120,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"xy = np.c_[xx.ravel(), yy.ravel()]\n",
"\n",
"xy"
]
},
{
"cell_type": "code",
"execution_count": 121,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/plain": [
"(-15, 15)"
]
},
"execution_count": 121,
"metadata": {},
"output_type": "execute_result"
},
{
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],
"text/plain": [
"<matplotlib.figure.Figure at 0x10c724400>"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"r = Realization(M, C)\n",
"\n",
"fig = plt.figure()\n",
"ax = fig.gca(projection='3d')\n",
"ax.plot_surface(xx, yy, np.reshape(r(xy), xx.shape),\n",
" alpha=.3,\n",
" cmap=plt.cm.Blues,\n",
" vmin=-15,vmax=15)\n",
"ax.set_zlim(-15,15)"
]
},
{
"cell_type": "code",
"execution_count": 122,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/plain": [
"(-15, 15)"
]
},
"execution_count": 122,
"metadata": {},
"output_type": "execute_result"
},
{
"data": {
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],
"text/plain": [
"<matplotlib.figure.Figure at 0x10cc7ad30>"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"r = Realization(M, C)\n",
"\n",
"fig = plt.figure()\n",
"ax = fig.gca(projection='3d')\n",
"ax.plot_surface(xx, yy, np.reshape(r(xy), xx.shape),\n",
" alpha=.3,\n",
" cmap=plt.cm.Blues,\n",
" vmin=-15,vmax=15)\n",
"ax.set_zlim(-15,15)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Now suppose we observe some data (5 points)"
]
},
{
"cell_type": "code",
"execution_count": 123,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/plain": [
"array([-5.30449219, -0.51763162, -0.90444381, 1.54728181, 1.56982855])"
]
},
"execution_count": 123,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"# Observational variances\n",
"V = np.array([.002,.002]) \n",
"\n",
"xobs = np.random.uniform(-10,10, (2,5))\n",
"yobs = sinxfun(*xobs)\n",
"yobs"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Update prior mean and covariance functions"
]
},
{
"cell_type": "code",
"execution_count": 124,
"metadata": {
"collapsed": false
},
"outputs": [],
"source": [
"observe(M, C, \n",
" obs_mesh=xobs.T, \n",
" obs_V=V, \n",
" obs_vals=yobs)\n",
"\n",
"r = Realization(M,C)"
]
},
{
"cell_type": "code",
"execution_count": 125,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/plain": [
"[<mpl_toolkits.mplot3d.art3d.Line3D at 0x1030227f0>]"
]
},
"execution_count": 125,
"metadata": {},
"output_type": "execute_result"
},
{
"data": {
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"SURAugECBAgwiQhIN0CAAAEmEf8He0E2LIlEAwcAAAAASUVORK5CYII=\n"
],
"text/plain": [
"<matplotlib.figure.Figure at 0x1099caba8>"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"fig = plt.figure()\n",
"ax = fig.gca(projection='3d')\n",
"ax.plot_surface(xx, yy, np.reshape(r(xy),xx.shape),\n",
" alpha=.3,\n",
" cmap=plt.cm.Blues,\n",
" vmin=-15,vmax=15)\n",
"ax.set_zlim(-15,15)\n",
"ax.plot(xobs[0], xobs[1], yobs, '.r', markersize=10)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Now observe 500 points."
]
},
{
"cell_type": "code",
"execution_count": 126,
"metadata": {
"collapsed": false
},
"outputs": [],
"source": [
"xobs = np.random.uniform(-10,10, (2,500))\n",
"yobs = sinxfun(*xobs)"
]
},
{
"cell_type": "code",
"execution_count": 127,
"metadata": {
"collapsed": true
},
"outputs": [],
"source": [
"observe(M, C, \n",
" obs_mesh=xobs.T, \n",
" obs_V=V, \n",
" obs_vals=yobs)\n",
"\n",
"r = Realization(M,C)"
]
},
{
"cell_type": "code",
"execution_count": 128,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/plain": [
"(-15, 15)"
]
},
"execution_count": 128,
"metadata": {},
"output_type": "execute_result"
},
{
"data": {
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],
"text/plain": [
"<matplotlib.figure.Figure at 0x10cc821d0>"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"fig = plt.figure()\n",
"ax = fig.gca(projection='3d')\n",
"ax.plot_surface(xx, yy, np.reshape(r(xy),xx.shape),\n",
" alpha=.3,\n",
" cmap=plt.cm.Blues,\n",
" vmin=-15,vmax=15)\n",
"ax.set_zlim(-15,15)"
]
},
{
"cell_type": "code",
"execution_count": 131,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/plain": [
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]
},
"execution_count": 131,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"M.reg_mat"
]
},
{
"cell_type": "code",
"execution_count": null,
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