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"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"
]
}
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