{ "metadata": { "name": "Projection_Ex" }, "nbformat": 3, "nbformat_minor": 0, "worksheets": [ { "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "As before, we can easily extend this projection operator to cases where the measure of distance between $\\mathbf{y}$ and the subspace $\\mathbf{v}$ is weighted (i.e. non-uniform). We can accomodate these weighted distances by re-writing the projection operator as\n", "\n", "$$\\mathbf{P}_{V} = \\mathbf{V} ( \\mathbf{V}^T \\mathbf{Q V})^{-1} \\mathbf{V}^T \\mathbf{Q}$$\n", "\n", "where $\\mathbf{Q}$ is positive definite matrix. Earlier, we started with a point $\\mathbf{y}$ and inflated a sphere centered at $\\mathbf{y}$ until it just touched the plane defined by $\\mathbf{v}_i$ and this point was closest point on the subspace to $\\mathbf{y}$. In the general case with a weighted distance except now we inflate an ellipsoid, not a sphere, until the ellipsoid touches the line.\n", "\n", "The code and figure below illustrate what happens using the weighted $\\mathbf{P}_v$. It is basically the same code we used above. You can download the IPython notebook corresponding to this post and try different values on the diagonal of $\\mathbf{Q}$." ] }, { "cell_type": "code", "collapsed": false, "input": [ "x = np.linspace(0,2,50)\n", "y = x + x**2 + np.random.randn(len(x))\n", "\n", "V = matrix(np.vstack([ones(x.shape),x,x**2]).T)\n", "Q = np.matrix(np.eye(V.shape[0]))\n", "i,j =np.diag_indices_from(Q)\n", "Q[i[:20],j[:20]]=100\n", "R = np.matrix(np.diag([1,1,13]))*0\n", "\n", "Pv = V*inv(V.T*Q*V+R)*V.T*Q\n", "\n", "p=np.polyfit(x,y,2)\n", "\n", "fig, ax = subplots()\n", "fig.set_size_inches(5,5)\n", "\n", "ax.plot(x,y,'o',label='data',alpha=0.3)\n", "ax.plot(x,np.dot(Pv,y).flat,label='projection')\n", "ax.plot(x,np.polyval(p,x),'-',label='polyfit',alpha=0.3)\n", "ax.grid()\n", "ax.legend(loc=0)" ], "language": "python", "metadata": {}, "outputs": [ { "output_type": "pyout", "prompt_number": 69, "text": [ "" ] }, { "output_type": "display_data", "png": 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MZvV6G6LGBrhkJ6PBqQfO3i1CQMAElX1oOy+dseJ17G1CQgLWrVuHI0eOcFYw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\nIMaPcnrmrrYWyMxUrFMxcCD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"text": [ "" ] } ], "prompt_number": 69 }, { "cell_type": "code", "collapsed": false, "input": [ "x = np.random.rand(50)*2\n", "x.sort() # sort for plotting\n", " \n", "y = x + x**2 + np.random.randn(len(x))\n", "\n", "V = matrix(np.vstack([ones(x.shape),x,x**2]).T)\n", "Pv = V*inv(V.T*V+eye(3)*4)*V.T\n", "\n", "p=np.polyfit(x,y,2)\n", "\n", "fig, ax = subplots()\n", "fig.set_size_inches(5,5)\n", "\n", "ax.plot(x,y,'o',label='data',alpha=0.3)\n", "ax.plot(x,np.dot(Pv,y).flat,'-s',label='projection')\n", "#ax.plot(x,np.polyval(p,x),'s',label='polyfit')\n", "ax.grid()\n", "ax.legend(loc=0);" ], "language": "python", "metadata": {}, "outputs": [ { "output_type": "display_data", "png": 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"text": [ "" ] } ], "prompt_number": 2 }, { "cell_type": "code", "collapsed": false, "input": [ "import numpy as np\n", "import matplotlib.pyplot as plt\n", "from matplotlib.widgets import Slider \n", "\n", "fig, ax = plt.subplots()\n", "fig.set_size_inches(5,5)\n", "\n", "plt.subplots_adjust(left=0.25, bottom=0.30,right=1)\n", "\n", "x = np.linspace(0,2,50)\n", "y = x + x**2 + np.sin(2*np.pi*x) + np.random.randn(len(x))\n", "\n", "V = np.matrix(np.vstack([np.ones(x.shape),x,x**2]).T)\n", "Q = np.matrix(np.eye(V.shape[0]))\n", "i,j =np.diag_indices_from(Q)\n", "#Q[i[:20],j[:20]]=100\n", "R = np.matrix(np.diag([1,1,1]))*0\n", "\n", "Pv = V*np.linalg.inv(V.T*Q*V+R)*V.T*Q\n", "p=np.polyfit(x,y,2)\n", "\n", "ax.plot(x,y,'o',label='data',alpha=0.3)\n", "ax.set_title('err^2=%3.2f'%(np.linalg.norm(y)**2 - np.linalg.norm(np.dot(Pv,y))**2 ))\n", "ls_line,=ax.plot(x,np.dot(Pv,y).flat,label='projection')\n", "ax.plot(x,np.polyval(p,x),'-',label='polyfit',alpha=0.3)\n", "ax.legend(loc=0)\n", "ax.grid()\n", "\n", "axw0 = plt.axes([0.25, 0.2, 0.65, 0.03] )\n", "sw0 = Slider(axw0, 'w0', 0.1, 30.0, valinit = 1)\n", "axw1 = plt.axes([0.25, 0.15, 0.65, 0.03] )\n", "sw1 = Slider(axw1, 'w1', 0.1, 30.0, valinit = 1)\n", "axw2 = plt.axes([0.25, 0.10, 0.65, 0.03] )\n", "sw2 = Slider(axw2, 'w2', 0.1, 30.0, valinit = 1)\n", "\n", "def update_w0(val):\n", " w = sw0.val\n", " R[0,0]=w\n", " Pv = V*np.linalg.inv(V.T*Q*V+R)*V.T*Q\n", " ls_line.set_ydata(np.dot(Pv,y).flat)\n", " ax.set_title('err=%3.2f'%(np.linalg.norm(y)**2 - np.linalg.norm(np.dot(Pv,y))**2 ))\n", " plt.draw()\n", "sw0.on_changed(update_w0)\n", "\n", "def update_w1(val):\n", " w = sw1.val\n", " R[1,1]=w\n", " Pv = V*np.linalg.inv(V.T*Q*V+R)*V.T*Q\n", " ls_line.set_ydata(np.dot(Pv,y).flat)\n", " ax.set_title('err=%3.2f'%(np.linalg.norm(y)**2 - np.linalg.norm(np.dot(Pv,y))**2 ))\n", " plt.draw()\n", "sw1.on_changed(update_w1)\n", "\n", "def update_w2(val):\n", " w = sw2.val\n", " R[2,2]=w\n", " Pv = V*np.linalg.inv(V.T*Q*V+R)*V.T*Q\n", " ls_line.set_ydata(np.dot(Pv,y).flat)\n", " ax.set_title('err=%3.2f'%(np.linalg.norm(y)**2 - np.linalg.norm(np.dot(Pv,y))**2 ))\n", " plt.draw()\n", "sw2.on_changed(update_w2)\n", "\n", "plt.show()\n" ], "language": "python", "metadata": {}, "outputs": [ { "output_type": "display_data", "png": 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"text": [ "" ] } ], "prompt_number": 77 }, { "cell_type": "heading", "level": 3, "metadata": {}, "source": [ "References" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "This post was created using the [nbconvert](https://github.com/ipython/nbconvert) utility from the source [IPython Notebook](www.ipython.org) which is available for [download](https://github.com/unpingco/Python-for-Signal-Processing/blob/master/Projection_mdim.ipynb) from the main github [site](https://github.com/unpingco/Python-for-Signal-Processing) for this blog. The projection concept is masterfully discussed in the classic Strang, G. (2003). *Introduction to linear algebra*. Wellesley Cambridge Pr. Also, some of Dr. Strang's excellent lectures are available on [MIT Courseware](http://ocw.mit.edu/courses/mathematics/18-06-linear-algebra-spring-2010/). I highly recommend these as well as the book." ] } ], "metadata": {} } ] }