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+{
+ "cells": [
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "# Chapter 10 - Curve fitting regression"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "## Example No. 10_01 Pg No. 326"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 13,
+ "metadata": {
+ "collapsed": false
+ },
+ "outputs": [
+ {
+ "name": "stdout",
+ "output_type": "stream",
+ "text": [
+ "b = [-1 -1 0 0 1]\n",
+ "a = [ 3 3 1 1 -2]\n",
+ "y= [-x + 3 -x + 3 1 1 x - 2]\n"
+ ]
+ }
+ ],
+ "source": [
+ "from numpy import array\n",
+ "from sympy.abc import x\n",
+ "#Fitting a Straight Line\n",
+ "\n",
+ "X = range(1,6)\n",
+ "Y = array([[ 3, 4, 5 ,6 ,8 ]])\n",
+ "n = len(X)#\n",
+ "X=array(X)\n",
+ "b = ( n*sum(X*Y) - sum(X)*sum(Y) )/( n*sum(X*X) - (sum(X))**2 )\n",
+ "a = sum(Y)/n - b*sum(X)/n\n",
+ "print 'b = ',b\n",
+ "print 'a = ',a\n",
+ "y = a + b*x\n",
+ "print 'y=',y"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "## Example No. 10_02 Pg No. 331"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 1,
+ "metadata": {
+ "collapsed": false
+ },
+ "outputs": [
+ {
+ "name": "stdout",
+ "output_type": "stream",
+ "text": [
+ "b = 2.0\n",
+ "lna = -0.69314718056\n",
+ "a = 0.5\n",
+ "\n",
+ " The power function equation obtained is \n",
+ " y = 0.5x**2\n"
+ ]
+ }
+ ],
+ "source": [
+ "from numpy import array,log,exp\n",
+ "from sympy.abc import x\n",
+ "\n",
+ "#Fitting a Power-Function model to given data\n",
+ "\n",
+ "X = array(range(1,6))\n",
+ "Y = [ 0.5, 2 ,4.5 ,8 ,12.5 ]\n",
+ "Xnew = log(X)\n",
+ "Ynew = log(Y)\n",
+ "n = len(Xnew)\n",
+ "b = ( n*sum(Xnew*Ynew) - sum(Xnew)*sum(Ynew) )/( n*sum(Xnew*Xnew) - ( sum(Xnew) )**2 )\n",
+ "lna = sum(Ynew)/n - b*sum(Xnew)/n\n",
+ "a = exp(lna)\n",
+ "print 'b = ',b\n",
+ "print 'lna = ',lna\n",
+ "print 'a = ',a\n",
+ "print '\\n The power function equation obtained is \\n y = %.4Gx**%.4G'%(a,b)#"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "## Example No. 10_03 Pg No. 332"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 1,
+ "metadata": {
+ "collapsed": false
+ },
+ "outputs": [
+ {
+ "name": "stdout",
+ "output_type": "stream",
+ "text": [
+ "b = 37.6294062985\n",
+ "a = 20.9234245534\n",
+ "The relationship between T and t is \n",
+ "T = 37.63*e**(t/4) + 20.92 \n",
+ "\n",
+ "The temperature at t = 6 is 189.566723485\n"
+ ]
+ }
+ ],
+ "source": [
+ "from numpy import array,log,exp\n",
+ "time = array(range(1,5))\n",
+ "T = [ 70 ,83 ,100 ,124 ]\n",
+ "t = 6\n",
+ "Fx = exp(time/4.0)\n",
+ "n = len(Fx)\n",
+ "Y = T #\n",
+ "b = ( n*sum(Fx*Y) - sum(Fx)*sum(Y) )/( n*sum(Fx*Fx) - (sum(Fx))**2 )\n",
+ "a = sum(Y)/n - b*sum(Fx)/n\n",
+ "print 'b = ',b\n",
+ "print 'a = ',a\n",
+ "print 'The relationship between T and t is \\nT = %.4G*e**(t/4) + %.4G \\n'%(b,a)\n",
+ "#deff('T = T(t)'%('T = b*exp(t/4) + a '\n",
+ "def T(t):\n",
+ " tt=b*exp(t/4.0)+a\n",
+ " return tt\n",
+ " \n",
+ "T_6 = T(6)\n",
+ "print 'The temperature at t = 6 is',T_6"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "## Example No. 10_04 Pg NO. 335"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 1,
+ "metadata": {
+ "collapsed": false
+ },
+ "outputs": [
+ {
+ "name": "stdout",
+ "output_type": "stream",
+ "text": [
+ "Using CA = B form , we get\n",
+ "B [ 6. 62. 190.]\n",
+ "C [[ 1. 1. 4.]\n",
+ " [ 1. 4. 10.]\n",
+ " [ 4. 10. 30.]]\n",
+ "A = [[ 20. 103.33333333 -190. ]\n",
+ " [ 10. 144.66666667 -190. ]\n",
+ " [ -6. -62. 95. ]]\n",
+ "Therefore the least sqaures polynomial is\n",
+ " y = 1J + 1J*x + 1J*x**2 \n",
+ "[ 20. 103.33333333 -190. ]\n",
+ "[ 10. 144.66666667 -190. ]\n",
+ "[ -6. -62. 95.]\n"
+ ]
+ }
+ ],
+ "source": [
+ "from numpy import array,ones,identity\n",
+ "from numpy.linalg import inv\n",
+ "\n",
+ "#Curve Fitting\n",
+ "\n",
+ "x = array(range(1,5))\n",
+ "y = [6, 11, 18, 27 ]\n",
+ "n = len(x) #Number of data points\n",
+ "m = 2+1 #Number of unknowns\n",
+ "print 'Using CA = B form , we get'\n",
+ "C=identity(m)\n",
+ "B=ones(m)\n",
+ "for j in range(0,m):\n",
+ " for k in range(0,m):\n",
+ " C[j,k] = sum(x**(j+k-2))\n",
+ " \n",
+ " B[j] = sum( y*( x**( j-1 ) ) )\n",
+ "\n",
+ "print 'B',B\n",
+ "print 'C',C\n",
+ "A = inv(C)*B\n",
+ "print 'A = ',A\n",
+ "print 'Therefore the least sqaures polynomial is\\n y = 1J + 1J*x + 1J*x**2 \\n',(A[0])\n",
+ "print A[1]\n",
+ "print A[2]"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "## Example No. 10_05 Pg No. 342"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 1,
+ "metadata": {
+ "collapsed": false
+ },
+ "outputs": [
+ {
+ "name": "stdout",
+ "output_type": "stream",
+ "text": [
+ "C=\n",
+ "[[ 4. 10. 6.]\n",
+ " [ 10. 30. 20.]\n",
+ " [ 6. 20. 14.]]\n",
+ "B=\n",
+ "[ 84. 240. 156.]\n",
+ "\n",
+ " The regression plane is \n",
+ " y = 5 + 6*x + 0*z \n",
+ "\n"
+ ]
+ }
+ ],
+ "source": [
+ "from numpy import array,ones,identity,arange,vstack,transpose\n",
+ "from scipy.sparse.linalg import lsqr\n",
+ "#Plane Fitting\n",
+ "\n",
+ "x = range(1,5)\n",
+ "z = range(0,4)\n",
+ "y = arange(12,31,6)\n",
+ "n = len(x) #Number of data points\n",
+ "m = 3 #Number of unknowns\n",
+ "G = vstack([ones(n),x,z])\n",
+ "H = transpose(G)\n",
+ "C = G.dot(H)\n",
+ "B = y.dot(H)\n",
+ "D = lsqr(C,B)\n",
+ "print 'C=\\n',C\n",
+ "print 'B=\\n',B\n",
+ "print '\\n The regression plane is \\n y = %d + %.f*x + %d*z \\n'%(D[0][0],D[0][1],D[0][2])\n"
+ ]
+ }
+ ],
+ "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": "2.7.9"
+ }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 0
+}
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