[2526] | 1 | #!/usr/bin/env python |
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| 2 | """Auxiliary numerical tools |
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| 3 | |
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| 4 | """ |
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| 5 | |
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| 6 | |
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[2531] | 7 | #Establish which Numeric package to use |
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| 8 | #(this should move to somewhere central) |
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| 9 | try: |
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[2972] | 10 | from scipy import ArrayType, array, sum, innerproduct, ravel, sqrt, searchsorted, sort, concatenate, Float, arange |
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[2531] | 11 | except: |
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[2633] | 12 | #print 'Could not find scipy - using Numeric' |
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[2972] | 13 | from Numeric import ArrayType, array, sum, innerproduct, ravel, sqrt, searchsorted, sort, concatenate, Float, arange |
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[2526] | 14 | |
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[2704] | 15 | # Getting an infinite number to use when using Numeric |
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[3452] | 16 | #INF = (array([1])/0.)[0] |
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[2573] | 17 | |
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[3452] | 18 | NAN = (array([1])/0.)[0] |
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| 19 | # Note, INF is used instead of NAN (Not a number), since Numeric has no NAN |
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| 20 | # if we use a package that has NAN, this should be updated to use NAN. |
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[2526] | 21 | |
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| 22 | |
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[2704] | 23 | def angle(v1, v2=None): |
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| 24 | """Compute angle between 2D vectors v1 and v2. |
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| 25 | |
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| 26 | If v2 is not specified it will default |
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| 27 | to e1 (the unit vector in the x-direction) |
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[2526] | 28 | |
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[2704] | 29 | The angle is measured as a number in [0, 2pi] from v2 to v1. |
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| 30 | """ |
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| 31 | from math import acos, pi, sqrt |
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| 32 | |
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| 33 | # Prepare two Numeric vectors |
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| 34 | if v2 is None: |
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| 35 | v2 = [1.0, 0.0] # Unit vector along the x-axis |
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| 36 | |
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| 37 | v1 = ensure_numeric(v1, Float) |
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| 38 | v2 = ensure_numeric(v2, Float) |
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[2526] | 39 | |
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[2704] | 40 | # Normalise |
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| 41 | v1 = v1/sqrt(sum(v1**2)) |
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| 42 | v2 = v2/sqrt(sum(v2**2)) |
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| 43 | |
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| 44 | # Compute angle |
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| 45 | p = innerproduct(v1, v2) |
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[2710] | 46 | c = innerproduct(v1, normal_vector(v2)) # Projection onto normal |
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| 47 | # (negative cross product) |
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[3676] | 48 | #print "p",p |
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| 49 | #print "v1", v1 |
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| 50 | #print "v2", v2 |
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| 51 | |
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[2704] | 52 | |
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[3676] | 53 | # Warning, this is a hack. It could cause code to go in loop forever |
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| 54 | if False: |
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| 55 | try: |
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| 56 | theta = acos(p) |
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| 57 | #print "theta",theta |
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| 58 | except ValueError: |
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| 59 | print "Doing a hack in numerical tools." |
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| 60 | print "p",p |
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| 61 | print "v1", v1 |
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| 62 | print "v2", v2 |
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| 63 | if p > (1.0 - 1e-12): #sus, checking a float |
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| 64 | # Throw a warning |
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| 65 | theta = 0.0 |
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| 66 | else: |
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| 67 | raise |
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| 68 | else: |
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| 69 | theta = acos(p) |
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| 70 | |
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| 71 | # print "problem with p",p |
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| 72 | # as p goes to 1 theta goes to 0 |
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[2704] | 73 | |
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| 74 | # Correct if v1 is in quadrant 3 or 4 with respect to v2 (as the x-axis) |
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| 75 | # If v2 was the unit vector [1,0] this would correspond to the test |
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| 76 | # if v1[1] < 0: theta = 2*pi-theta |
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[2710] | 77 | # In general we use the sign of the projection onto the normal. |
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| 78 | if c < 0: |
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[2704] | 79 | #Quadrant 3 or 4 |
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| 80 | theta = 2*pi-theta |
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| 81 | |
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[2526] | 82 | return theta |
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| 83 | |
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[2704] | 84 | |
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[2526] | 85 | def anglediff(v0, v1): |
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[2709] | 86 | """Compute difference between angle of vector v0 (x0, y0) and v1 (x1, y1). |
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[2526] | 87 | This is used for determining the ordering of vertices, |
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| 88 | e.g. for checking if they are counter clockwise. |
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| 89 | |
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| 90 | Always return a positive value |
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| 91 | """ |
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| 92 | |
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| 93 | from math import pi |
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| 94 | |
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| 95 | a0 = angle(v0) |
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| 96 | a1 = angle(v1) |
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| 97 | |
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| 98 | #Ensure that difference will be positive |
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| 99 | if a0 < a1: |
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| 100 | a0 += 2*pi |
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| 101 | |
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| 102 | return a0-a1 |
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| 103 | |
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[2704] | 104 | def normal_vector(v): |
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[3103] | 105 | """Normal vector to v. |
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| 106 | |
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| 107 | Returns vector 90 degrees counter clockwise to and of same length as v |
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[2704] | 108 | """ |
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| 109 | |
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| 110 | return array([-v[1], v[0]], Float) |
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[2526] | 111 | |
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[2704] | 112 | |
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[2710] | 113 | #def crossproduct_length(v1, v2): |
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| 114 | # return v1[0]*v2[1]-v2[0]*v1[1] |
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[2704] | 115 | |
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| 116 | |
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[2526] | 117 | def mean(x): |
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| 118 | """Mean value of a vector |
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| 119 | """ |
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[2531] | 120 | return(float(sum(x))/len(x)) |
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[2526] | 121 | |
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| 122 | |
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| 123 | def cov(x, y=None): |
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| 124 | """Covariance of vectors x and y. |
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| 125 | |
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| 126 | If y is None: return cov(x, x) |
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| 127 | """ |
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| 128 | |
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| 129 | if y is None: |
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| 130 | y = x |
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| 131 | |
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| 132 | assert(len(x)==len(y)) |
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| 133 | N = len(x) |
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| 134 | |
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| 135 | cx = x - mean(x) |
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| 136 | cy = y - mean(y) |
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| 137 | |
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[2531] | 138 | p = innerproduct(cx,cy)/N |
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[2526] | 139 | return(p) |
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| 140 | |
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| 141 | |
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| 142 | def err(x, y=0, n=2, relative=True): |
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| 143 | """Relative error of ||x-y|| to ||y|| |
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| 144 | n = 2: Two norm |
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| 145 | n = None: Max norm |
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| 146 | |
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[3103] | 147 | If denominator evaluates to zero or |
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| 148 | if y is omitted or |
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| 149 | if keyword relative is False, |
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[2526] | 150 | absolute error is returned |
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| 151 | """ |
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| 152 | |
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| 153 | x = ensure_numeric(x) |
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| 154 | if y: |
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| 155 | y = ensure_numeric(y) |
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| 156 | |
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| 157 | if n == 2: |
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| 158 | err = norm(x-y) |
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| 159 | if relative is True: |
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| 160 | try: |
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| 161 | err = err/norm(y) |
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| 162 | except: |
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| 163 | pass |
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| 164 | |
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| 165 | else: |
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| 166 | err = max(abs(x-y)) |
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| 167 | if relative is True: |
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| 168 | try: |
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| 169 | err = err/max(abs(y)) |
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| 170 | except: |
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| 171 | pass |
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| 172 | |
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| 173 | return err |
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| 174 | |
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| 175 | |
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| 176 | def norm(x): |
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| 177 | """2-norm of x |
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| 178 | """ |
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| 179 | |
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[2531] | 180 | y = ravel(x) |
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| 181 | p = sqrt(innerproduct(y,y)) |
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[2526] | 182 | return p |
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| 183 | |
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| 184 | |
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| 185 | def corr(x, y=None): |
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| 186 | """Correlation of x and y |
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| 187 | If y is None return autocorrelation of x |
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| 188 | """ |
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| 189 | |
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| 190 | from math import sqrt |
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| 191 | if y is None: |
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| 192 | y = x |
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| 193 | |
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| 194 | varx = cov(x) |
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| 195 | vary = cov(y) |
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| 196 | |
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| 197 | if varx == 0 or vary == 0: |
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| 198 | C = 0 |
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| 199 | else: |
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| 200 | C = cov(x,y)/sqrt(varx * vary) |
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| 201 | |
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| 202 | return(C) |
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| 203 | |
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| 204 | |
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| 205 | |
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| 206 | def ensure_numeric(A, typecode = None): |
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| 207 | """Ensure that sequence is a Numeric array. |
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| 208 | Inputs: |
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| 209 | A: Sequence. If A is already a Numeric array it will be returned |
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| 210 | unaltered |
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| 211 | If not, an attempt is made to convert it to a Numeric |
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| 212 | array |
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| 213 | typecode: Numeric type. If specified, use this in the conversion. |
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| 214 | If not, let Numeric decide |
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| 215 | |
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| 216 | This function is necessary as array(A) can cause memory overflow. |
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| 217 | """ |
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| 218 | |
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| 219 | if typecode is None: |
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| 220 | if type(A) == ArrayType: |
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| 221 | return A |
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| 222 | else: |
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| 223 | return array(A) |
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| 224 | else: |
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| 225 | if type(A) == ArrayType: |
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| 226 | if A.typecode == typecode: |
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| 227 | return array(A) #FIXME: Shouldn't this just return A? |
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| 228 | else: |
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[2778] | 229 | return array(A,typecode) |
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[2526] | 230 | else: |
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[2778] | 231 | return array(A,typecode) |
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[2526] | 232 | |
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| 233 | |
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| 234 | |
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[2533] | 235 | |
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[2976] | 236 | def histogram(a, bins, relative=False): |
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[2533] | 237 | """Standard histogram straight from the Numeric manual |
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[2976] | 238 | |
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| 239 | If relative is True, values will be normalised againts the total and |
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| 240 | thus represent frequencies rather than counts. |
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[2533] | 241 | """ |
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| 242 | |
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| 243 | n = searchsorted(sort(a), bins) |
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| 244 | n = concatenate( [n, [len(a)]] ) |
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| 245 | |
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[2976] | 246 | hist = n[1:]-n[:-1] |
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| 247 | |
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| 248 | if relative is True: |
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| 249 | hist = hist/float(sum(hist)) |
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| 250 | |
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| 251 | return hist |
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| 252 | |
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[2972] | 253 | def create_bins(data, number_of_bins = None): |
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| 254 | """Safely create bins for use with histogram |
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| 255 | If data contains only one point or is constant, one bin will be created. |
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| 256 | If number_of_bins in omitted 10 bins will be created |
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| 257 | """ |
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[2533] | 258 | |
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[2972] | 259 | mx = max(data) |
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| 260 | mn = min(data) |
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[2533] | 261 | |
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[2972] | 262 | if mx == mn: |
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| 263 | bins = array([mn]) |
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| 264 | else: |
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| 265 | if number_of_bins is None: |
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| 266 | number_of_bins = 10 |
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| 267 | |
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| 268 | bins = arange(mn, mx, (mx-mn)/number_of_bins) |
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| 269 | |
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| 270 | return bins |
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| 271 | |
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| 272 | |
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| 273 | |
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[2526] | 274 | #################################################################### |
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| 275 | #Python versions of function that are also implemented in numerical_tools_ext.c |
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| 276 | # |
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| 277 | |
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| 278 | def gradient_python(x0, y0, x1, y1, x2, y2, q0, q1, q2): |
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| 279 | """ |
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| 280 | """ |
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| 281 | |
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| 282 | det = (y2-y0)*(x1-x0) - (y1-y0)*(x2-x0) |
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| 283 | a = (y2-y0)*(q1-q0) - (y1-y0)*(q2-q0) |
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| 284 | a /= det |
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| 285 | |
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| 286 | b = (x1-x0)*(q2-q0) - (x2-x0)*(q1-q0) |
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| 287 | b /= det |
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| 288 | |
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| 289 | return a, b |
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| 290 | |
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| 291 | |
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| 292 | def gradient2_python(x0, y0, x1, y1, q0, q1): |
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| 293 | """Compute radient based on two points and enforce zero gradient |
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| 294 | in the direction orthogonal to (x1-x0), (y1-y0) |
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| 295 | """ |
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| 296 | |
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| 297 | #Old code |
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| 298 | #det = x0*y1 - x1*y0 |
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| 299 | #if det != 0.0: |
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| 300 | # a = (y1*q0 - y0*q1)/det |
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| 301 | # b = (x0*q1 - x1*q0)/det |
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| 302 | |
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| 303 | #Correct code (ON) |
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| 304 | det = (x1-x0)**2 + (y1-y0)**2 |
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| 305 | if det != 0.0: |
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| 306 | a = (x1-x0)*(q1-q0)/det |
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| 307 | b = (y1-y0)*(q1-q0)/det |
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| 308 | |
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| 309 | return a, b |
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| 310 | |
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| 311 | |
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| 312 | ############################################## |
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| 313 | #Initialise module |
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| 314 | |
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[3514] | 315 | from anuga.utilities import compile |
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[2526] | 316 | if compile.can_use_C_extension('util_ext.c'): |
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| 317 | from util_ext import gradient, gradient2 |
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| 318 | else: |
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| 319 | gradient = gradient_python |
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| 320 | gradient2 = gradient2_python |
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| 321 | |
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| 322 | |
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| 323 | if __name__ == "__main__": |
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| 324 | pass |
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| 325 | |
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[2704] | 326 | |
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| 327 | |
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| 328 | def angle_obsolete(v): |
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| 329 | """Compute angle between e1 (the unit vector in the x-direction) |
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| 330 | and the specified vector v. |
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| 331 | |
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| 332 | Return a number in [0, 2pi] |
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| 333 | """ |
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| 334 | from math import acos, pi, sqrt |
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| 335 | |
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| 336 | # Normalise v |
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| 337 | v = ensure_numeric(v, Float) |
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| 338 | v = v/sqrt(sum(v**2)) |
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| 339 | |
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| 340 | # Compute angle |
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| 341 | theta = acos(v[0]) |
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| 342 | |
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| 343 | if v[1] < 0: |
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| 344 | #Quadrant 3 or 4 |
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| 345 | theta = 2*pi-theta |
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| 346 | |
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| 347 | return theta |
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| 348 | |
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