massive transfer of risk-reliability book

figures/data_axle.csv

@@ -0,0 +1,992 @@
+41.19,69.75
+55.84,65.1
+31.76,43.73
+36.24,61.63
+47.79,45.38
+35.2,43.71
+44.38,76.57
+39.67,60.09
+60.28,100.97
+35.69,63.03
+41.28,41.09
+52.11,99.61
+35.06,96.74
+28.87,56.66
+31.45,26.19
+31.98,63.83
+17.46,32.42
+45.79,54.33
+31.65,67.07
+73.71,85.48
+43.62,103.52
+41.74,97.76
+33.64,64.44
+43.4,74.62
+55.17,88.91
+45.03,54.11
+36.84,67.6
+31.9,30.56
+36.75,71.89
+50.78,66.16
+32.28,44.52
+36.83,36.5
+49.42,44
+37.31,33.38
+28.61,33.47
+21.76,32.07
+48.28,43.22
+49.02,66.21
+42.3,57.11
+36.73,45.33
+36.94,38.49
+37.33,93.22
+31.83,56.14
+38.56,62.72
+30.55,53.31
+45.7,46.1
+33.03,29.26
+23.33,46.33
+31.18,23.39
+22.79,51.07
+35.16,34.86
+21.73,33.9
+41.21,73.51
+21.32,39.55
+32.33,41.2
+51.38,48.64
+45.05,78.96
+37.49,41.22
+52.95,65.65
+50.52,50.57
+53.12,46.78
+45.42,104.07
+28.27,38.58
+33.86,51.8
+46.44,60.72
+37.15,84.29
+24.1,30.99
+54.36,100.85
+48.75,48.19
+40.59,52.18
+53.96,135.17
+52.32,79.16
+17.32,35.93
+37.73,66.25
+38.21,55.26
+16.94,39.99
+29.85,38.7
+31.55,80.72
+47.39,44.49
+30.04,75.43
+36.73,45.98
+46.74,75.45
+36.63,58.68
+36.4,65.96
+32.86,50.45
+35.99,47.93
+33.82,61.05
+49.55,82.05
+43.67,59.7
+30.09,30.99
+40.28,61.28
+34.16,54.66
+44.89,72.86
+18.72,30.77
+50.07,59.27
+21.9,36.14
+28.59,36.53
+35.7,45.61
+53.24,88.22
+31.92,45.12
+47.89,68.85
+31.69,32.75
+60.76,78.1
+28.44,34.68
+29.78,29.44
+33.21,61.18
+26.62,63.66
+26.96,33.58
+55.37,83.64
+26.57,32.54
+53.69,64.75
+32.05,59.94
+36.14,49.43
+48.74,78.89
+30.7,28.08
+36.84,44.78
+53.85,63.85
+29.51,47.65
+30.14,37.13
+25.54,62.9
+30.13,67.55
+31.34,69.29
+39.19,62.61
+26.65,36.15
+40.32,55.47
+31.61,25.75
+46.04,59.3
+48.87,55.39
+43.3,43.42
+16.22,31.51
+47.37,76.99
+57.21,71.58
+56.49,77.52
+22.79,35.8
+57.95,67.38
+43.59,66.05
+33.96,61.22
+32.08,73.03
+45.41,75.97
+44.82,46.98
+26.78,52.05
+47.25,61.51
+27.06,56.87
+51.99,45.83
+39.04,60.16
+54.09,78.71
+19.76,41.26
+32.06,54.54
+25.14,44.79
+31.04,75.62
+39.99,41.25
+30.52,37.28
+59.7,64.95
+32.29,40.96
+31.24,36.07
+29.78,40.52
+33.25,41.6
+40.32,63.45
+49.42,43.61
+31.36,43.9
+33.18,55.93
+39.16,60.16
+45.11,49.68
+37.29,61
+30.32,69.67
+41.32,57.25
+31.55,81.22
+52.99,79.57
+43.23,59.32
+38.58,38.91
+40.17,112.23
+51.24,27.06
+46.16,80.08
+58.95,79.15
+38.82,48.57
+36.76,58.04
+31.08,36.24
+37.81,58.4
+55.35,102.93
+32.46,32.17
+51.54,82.24
+38.01,56.99
+27.67,36.81
+22.83,30.18
+31.9,57.46
+37.97,45.6
+23.47,32.75
+34.26,70.78
+38.27,60.5
+47.64,42.59
+36.47,63.14
+44.68,72.4
+37.47,56.1
+39.95,72.6
+40.05,40.78
+56.51,92.1
+50.71,62.54
+48.17,63.31
+46.23,73.78
+44.69,43.74
+49.62,91.48
+44.93,70.42
+32.37,27.98
+48.49,81.86
+20.35,34.58
+25.83,45.19
+16.16,45.02
+31.13,38.84
+52.76,41.82
+36,73.76
+25.49,30.29
+28.46,61.92
+26.58,29.87
+32.58,61.77
+39.67,54.28
+35.3,59.7
+32.41,50.83
+57.52,57.81
+36.15,37.32
+38.98,65.29
+38.29,71.45
+39.58,37.4
+30.18,85.63
+31.88,67.67
+45.36,52.55
+28.39,57.32
+34.96,71.02
+30.75,41.8
+43.3,67.82
+48.59,62.63
+38.12,52.65
+40.88,64.49
+29.72,32.05
+51.65,89.01
+56.75,54.44
+43.69,46.72
+30.73,58.58
+32.78,78.48
+27.53,35.02
+41.75,57.37
+27.02,37.38
+48.96,78.4
+36.35,56.66
+37.06,42.47
+55.29,127.55
+39.37,80.78
+31.8,38.03
+43.51,94.15
+36.08,52.42
+24.73,71.52
+24.66,68.5
+35.72,35.22
+43.59,65.6
+35.53,57.06
+32.88,65
+18.03,32
+54.17,63.05
+38.9,56.24
+51.66,56.57
+47.91,71.9
+45.06,50.79
+25.53,39.3
+44.81,71.61
+36.91,57.3
+25.42,31.82
+33.96,34.52
+28.03,32.53
+38.31,104.16
+50.74,50.69
+29.61,31.29
+54.02,61.51
+35.71,50.83
+42.6,41.47
+39.96,62.6
+30.1,68.41
+33.33,42.35
+33.64,35.68
+47.44,49.45
+31.86,54.14
+44.52,54.58
+47.1,85.78
+49.9,64.69
+38.41,40.67
+42.3,60.69
+37.72,55.65
+31.15,69.65
+37.96,47.58
+50.72,54.7
+34.28,35.09
+36.85,52.53
+33.36,42.01
+45.37,48.64
+45.64,72.53
+32.34,29.05
+42.72,36.87
+41.34,54.3
+45.85,50.91
+49.7,51.51
+41.68,40.35
+43.93,38.44
+50.5,67.76
+37.87,56.92
+35.28,30.2
+36.29,38.29
+41.41,73.29
+49.99,53.5
+50.43,71.87
+28.24,43.9
+34.61,53.97
+35.68,30.57
+48.56,59.58
+30.82,65.43
+45.97,68.4
+30.41,31.3
+29.67,31.61
+17.79,30.19
+60.03,53.98
+49.87,71.54
+36.04,45.05
+30.14,67.63
+49.63,42.12
+37.63,77.9
+43.76,46.76
+34.82,66.99
+41.37,47.64
+50.02,77.77
+42.64,50.11
+51.99,64.67
+38,62.13
+29.98,32.1
+51.52,52.8
+30.88,33.13
+36.94,93.45
+37.97,59.91
+29.4,53.14
+34.47,80.16
+57.39,75.45
+54.13,62
+26.59,32.27
+34.48,48.11
+36.77,49.78
+40.84,51.1
+41.73,42.98
+39.93,67.02
+53.99,75.58
+39.57,64.03
+52.43,43.74
+31.19,73.46
+43.9,47.64
+41.35,35.87
+51.08,46.02
+40.2,58.92
+36.37,53.56
+34.21,49.56
+35.15,47.51
+37.89,48.52
+38.03,43.23
+21.11,55.85
+34.5,29.56
+37.08,47.58
+46.62,60.53
+32.14,75.61
+41.86,60.51
+44.75,74.45
+48.43,69.28
+46.31,62.36
+52.01,74.55
+51.48,52.5
+51.67,67.45
+31.97,39.05
+15.81,30.61
+47.39,47.39
+26.64,34.81
+58.38,80.23
+30.81,31.11
+26.07,33.1
+47.6,69.85
+37.82,70.15
+25.66,34.01
+43.2,51.71
+32.22,45.24
+46.09,75.02
+48.02,74.93
+37.04,37.65
+41.72,74.52
+52.87,92.15
+36.92,63.99
+31.11,38.71
+29.78,66.84
+41.85,58.07
+45.2,69.4
+37.51,42
+22.82,40.02
+55.44,74.28
+49.99,88.59
+54.64,79.12
+15.48,31.85
+47.97,64.54
+27.19,29.83
+62.1,90.03
+48.74,71.84
+28.94,30.7
+31.51,62.9
+51.72,62.77
+43.14,42.85
+30.33,31.16
+27.47,53.51
+21.28,34.61
+38.85,62.92
+50.79,70.25
+30.3,35.66
+37.58,66.48
+28.23,53.97
+36.12,58.19
+46.3,74.19
+30.41,59.43
+48.98,68.8
+42.3,46.47
+28.59,30.74
+40.05,69.63
+41.41,67.14
+49.84,71.54
+43.12,45.85
+38.31,88.17
+51.61,98.95
+33.78,53.95
+22.21,34.72
+38.05,74.17
+43.97,71.1
+34.51,70.63
+24.6,43.17
+43.46,39.02
+30.86,29.39
+42.91,75.84
+32.04,29.75
+28.48,43.9
+18.91,33.69
+24.53,44.32
+47.15,90.03
+46.09,58.22
+31.52,51.99
+29.79,35.18
+29.77,37.78
+26.88,51.19
+37.21,72.74
+27.29,39.88
+27.26,41.07
+39.37,88.9
+51.4,81.82
+52.07,56.78
+69.39,139.79
+40.91,78.92
+35.06,29.26
+20.63,44.17
+63.75,105.64
+34.59,36.73
+25.5,43.5
+42.29,43.4
+47.11,104.64
+33.28,48.33
+28.18,29.9
+27.34,35
+26.4,51.57
+28.92,40.43
+41.87,63.45
+30.56,42.55
+24.6,33.62
+37.34,85.16
+52.82,89.94
+55.49,62.83
+32.32,61.52
+47.75,55.55
+30.69,49.75
+35.08,57.23
+31.59,30.4
+57.08,75.32
+47.42,54.33
+31.3,70.86
+43.05,49.06
+30.26,42.61
+55.64,131.67
+27.3,32.1
+44.88,66.39
+46.2,54.35
+31.2,34.22
+24.02,39.03
+23.88,45.41
+70.01,78.45
+50.68,75.11
+43.49,76.57
+30.76,35.47
+24.93,35.92
+21.33,36.11
+28.43,38.68
+51.31,70.33
+62.92,101.62
+37.1,55.58
+34.81,49.12
+61.18,80.98
+26.97,32.19
+52.17,104.76
+40.25,68.32
+39.24,37.5
+31.82,96.1
+40.1,58.23
+24.83,30.43
+42.63,63.51
+21.84,40.89
+33.91,36.91
+33.67,38.55
+36.15,56.27
+45.85,91.27
+43.37,56.63
+54,42.56
+46.82,72.68
+37.76,74.79
+53.95,68.17
+56.37,53.55
+30.28,72.18
+35.92,66.25
+35.7,55.77
+24.04,39.91
+33.75,75.78
+41.35,63.63
+42.28,72.45
+47.92,68.94
+23.78,55.79
+48.76,72.69
+26.49,63.07
+30.63,25.25
+34.67,61.24
+49.33,46.97
+32.12,32.3
+39.35,45.69
+32.13,40.27
+43.84,23.75
+26.17,72.78
+41.91,69.96
+27.52,31.65
+52.49,91.11
+17.69,31.38
+42.68,74.33
+54.64,22.44
+29.1,30.96
+30.45,27.5
+19.47,32.56
+32.01,79.27
+46.71,73.38
+43.06,53.24
+61.78,75.23
+19.26,39.8
+43.32,50.98
+26.14,30.1
+40.64,95.11
+26.52,62.56
+49.93,69.5
+45.81,47.44
+53.48,72.88
+38.81,70.03
+38.96,68.82
+31.68,34.57
+37.92,39.23
+26.75,30.77
+35.57,51.43
+44.06,69.52
+44.94,51.63
+35.25,43.97
+57.64,74.01
+33.46,65.76
+49.94,85.61
+39.2,54.09
+26.82,43.08
+29.99,69.9
+43.59,65.6
+46.21,83.03
+37.23,59.95
+30.58,63.39
+35.44,62.56
+28.12,47.27
+29.6,54.31
+33.14,53.35
+31.09,74.28
+40.99,76.24
+32.69,21.81
+51.29,79.77
+31.18,48.81
+48.11,44.23
+63.02,85.38
+52.77,64.15
+20.12,29.44
+23.24,35.11
+29.08,32.27
+50.41,81.01
+40.94,39.75
+26.12,30.73
+52.11,55.08
+48.68,75.82
+32.16,55.05
+29.89,29.31
+22.33,29.73
+36.33,57.04
+39.42,64.34
+32.7,56.96
+43.98,77.69
+30.55,43.29
+34.84,39.19
+37.1,67.98
+57.88,89.04
+41.82,66.48
+32.26,72.13
+35.24,38.73
+40.84,66.46
+41.87,51.17
+31.15,78.63
+43.53,56.93
+33.15,59.09
+49.44,63.45
+30.41,34.37
+31.26,45.29
+28.3,29.57
+36.27,54.64
+38.31,38.75
+26.7,49.07
+32.45,41.94
+44.46,42.84
+39.18,59.53
+38.2,43.01
+27.7,36.6
+55.28,80.56
+32.58,38.18
+44,64.73
+26.46,36.75
+26.59,39.41
+34.69,50.88
+26.07,31.79
+29.88,40.88
+48.46,60.32
+43.71,88.08
+19.74,35.82
+26.01,57.81
+45.35,46.92
+24.88,58.35
+24.98,53.66
+29.46,28.25
+41.23,46.64
+29.98,38.98
+47.94,78.34
+18.05,31.91
+33.18,71.75
+42.3,117.99
+33.24,26.44
+45.77,35.01
+23.5,56.1
+50.99,87.01
+35.19,44.6
+31.2,60.11
+32.96,21.05
+37.3,73.24
+29.97,33.53
+41.2,48.47
+27.77,33.4
+22.67,68.98
+39.4,59.88
+27.69,63.19
+36.71,44.88
+39.3,69.33
+38.8,47.42
+23.93,30.61
+39.86,66.37
+52.32,67.19
+47.21,58.78
+54.48,70.63
+48.26,41.58
+30.91,26.35
+57.36,65.78
+34.01,29.79
+23.84,62.86
+50.09,52.55
+40.18,52.62
+48.48,65.8
+36.58,66.28
+44.62,68.85
+53.3,59.64
+31.96,67.71
+29.49,67.86
+36.82,64.1
+39.39,62.85
+52.4,60.62
+30.06,41.69
+38.04,48.08
+39.47,67.08
+31.43,57.67
+38.54,64.55
+36.91,75.43
+43.29,67.44
+36.95,50.71
+30.73,68.56
+26.29,40.61
+33.82,21.36
+47.73,58.35
+33.52,74.55
+34.32,36.64
+32.61,50.68
+36.95,60.18
+41.57,67.85
+38.75,53.7
+24.52,30.59
+45.55,39.17
+30.58,43.44
+34.4,42.69
+31.54,44.71
+57.68,72.15
+27.08,43.08
+51.49,66.05
+32.95,60.37
+37.37,56.93
+53.74,60.01
+32.75,48
+32.7,46.13
+46.87,93.33
+38.19,56.12
+27.64,33.74
+28.65,54.99
+34.54,44.82
+41.75,54.66
+44.97,44.43
+30.26,32.7
+31.04,33.92
+25.92,32.85
+68.14,118.12
+34.29,83.85
+42.17,45.86
+30.37,31.27
+36.43,38.57
+31.06,41.85
+43.63,49.58
+31.06,54.71
+36.03,60.75
+41.75,84.41
+29.78,27.42
+52.73,43.1
+47.08,41.84
+29.28,66.52
+36.48,72.25
+31.26,63.31
+29.58,68.12
+22.8,47.79
+26.63,30.47
+30.04,31.21
+44.71,39.69
+31.8,29.71
+64.83,72.21
+51.54,42.06
+35.96,36.96
+51.83,84.84
+30.89,46.24
+30.95,31.46
+34.99,90.31
+54.05,60.95
+39.39,44.87
+37.02,57.81
+27.28,38.44
+32.01,46.73
+46.09,54.62
+44.01,55.64
+57.29,79.06
+23.6,40.82
+46.12,65.27
+49.32,69.87
+33.75,35.21
+27.21,40.16
+35.08,47.58
+31.23,25.62
+31.83,29.26
+49.08,75.59
+27.14,57.46
+46.21,82.42
+32.11,36.91
+54.09,27.45
+38.18,54.57
+30.84,52.69
+30.53,24.43
+40.46,60.2
+55.38,65.09
+32.98,46.05
+27.56,54.44
+39.86,38.8
+29.84,37.75
+45.77,43.3
+49.45,49.62
+50.38,53.56
+24.94,64.63
+26.88,52.1
+39.39,65.87
+19.88,34.9
+47.19,57.74
+33.35,42.74
+29.92,34.9
+43.42,47.26
+54.02,51.62
+50.72,42.01
+27.21,41.2
+28.28,30.7
+39.26,148.09
+35.04,32.04
+26.43,48.95
+47.45,64.3
+37.14,78.1
+18.87,36.66
+41.98,62.13
+24.28,30.06
+38.41,45.16
+29.77,41.39
+29.16,47.54
+31.43,72
+39.78,79.23
+27.87,34.2
+32.09,51.97
+29.95,31.06
+45.87,61.33
+42.02,69.32
+30.07,32.57
+49.27,68.26
+46.52,82.86
+48.84,81.3
+31.56,46.57
+46.08,54.03
+38.39,37.2
+41.42,77.34
+30.66,32.64
+28.34,57.08
+50.82,50.24
+51.34,28.93
+46.56,57.49
+33.64,44.84
+48.45,59.31
+67.82,117.15
+33.28,29.76
+16.58,30.23
+30.09,65.38
+51.65,57.12
+39.54,50.1
+49.74,46.08
+51.52,85.69
+23.38,34.34
+36.01,39.11
+47.9,50.9
+37.67,21.67
+29.56,30.8
+37.33,54.54
+31.85,40.8
+46.53,73.19
+32.8,52.67
+42.36,45.1
+39.85,51.13
+38.7,67.41
+34.81,69.39
+29.88,63.84
+30.53,58.86
+35.33,40.95
+44.11,56.93
+48.19,61.28
+47.01,77.67
+42.58,48.4
+54.41,88.94
+46.13,59.09
+17.67,30.85
+31.55,48.89
+37.75,43.49
+45.15,47.87
+53.79,66.8
+29.57,50.68
+34.89,52
+36.55,36.47
+58.62,110.2
+39.39,39.14
+33.61,53.24
+35.82,39.88
+27.88,34.26
+33.24,28.49
+46.78,80.49
+49.99,26.56
+40.15,45.81
+31.99,51.06
+51.4,85.48
+45.29,54.98
+34.15,76.57
+39.97,66.4
+58.5,80.01
+31,78.74
+64.17,106.28
+39.1,55.44
+39.79,37.47
+36.49,66.74
+27.27,32.21
+33.77,40.51
+24.21,68.85
+26.38,48.36
+33.78,75.07
+30.61,33.76
+28.87,71.18
+22.98,29.72
+20.93,31.2
+23.31,54.07
+39.59,75.42
+51.79,50.66
+28.58,34.73
+38.74,73.32
+55.25,83.64
+25.3,35.36
+33.99,71.85
+35.32,55.34
+24.48,59.19
+46.23,101.86
+31.72,81.32
+39.08,58.23
+39.49,88.26
+26.78,35.01
+26.73,33.79
+31.19,79.46
+51.56,56.46
+27.91,39.84
+36.08,37.96
+43.74,49.52
+23.9,77.07
+44.48,52.03
+26.34,59.22
+35.6,66.55
+32.06,39.73
+44.5,76
+40.07,65.99
+29.24,43.74
+45.05,44.53
+34.74,46.82
+40.87,76.86
+32.43,44.38
+32.03,43.4
+49.23,55.55
+44.92,58.8
+33.52,69.83
+35.71,75.7
+39.17,55.58
+32.12,65.54
+41.34,43.95
+22.74,72.82
+40.49,65.72
+30.79,63.25
+44.97,56.67
+34.59,35.89
+36.8,66.96
+29.64,61.99
+49.58,68.88
+64.49,113.28
+49.93,61.48
+23.08,56.45
+50.47,58.7
+49.97,58.01
+41.67,42.61
+44.73,74.03
+34.3,83.7
+52.41,59.94
+36.14,75.78
+70.07,70.25
+30,53.42
+41.47,43.4
+34.24,71.84
+46.46,38.58
+48.94,66.15
+59.37,62.79
+28.84,33.23
+34.33,51.11
+59.6,91.75
+24.7,85.84
+29.52,46.73
+39.29,57.91
+52.86,61.59
+40.4,71.06
+22.96,37.46
+41.42,45.78
+26.4,30.66
+19.79,40.65
+18.25,33.15
+37.28,50.53
+38.82,46.68
+31.5,53.62
+44.46,79.83
+44.62,79.11
+39.72,46.07
+42,62.93
+27.32,42.47
+43.87,70.57
+33.96,50.75

figures/helper.py

@@ -0,0 +1,109 @@
+import numpy as np
+import matplotlib.pyplot as plt
+import matplotlib.cm as cm
+import pyvinecopulib as cop
+
+class Bivariate:
+    """Bivariate dependent distribution, design to function in the 
+    same way as an instance of the scipy.stats.rv_continuous class,
+    specifically, with methods: cdf, pdf, rvs.
+    """
+    
+    def __init__(self, X1, X2, rho):
+        """Constructor to initialize Bivariate object.
+        
+        Arguments:
+        ---
+        X1: scipy.stats.rv_continuous object. Marginal distribution
+        X2: scipy.stats.rv_continuous object. Marginal distribution
+        rho: float. Pearson correlation coefficient.
+        """
+        self.X1 = X1
+        self.X2 = X2
+        self.rho = rho
+        self.copula_uv = cop.Bicop(family=cop.BicopFamily.gaussian, parameters = [rho])
+        
+    def cdf(self, x):
+        """Compute cdf with array or list as argument: [x1, x2]."""
+        x1 = x[0]
+        x2 = x[1]
+        u = self.X1.cdf(x1)
+        v = self.X2.cdf(x2)
+        return self.copula_uv.cdf(np.array([[u, v]]))[0]
+
+    def pdf(self, x):
+        """Compute pdf with array or list as argument: [x1, x2]."""
+        x1 = x[0]
+        x2 = x[1]
+        u = self.X1.cdf(x1)
+        v = self.X2.cdf(x2)
+        return self.copula_uv.pdf(np.array([[u, v]]))[0]*self.X1.pdf(x1)*self.X2.pdf(x2)
+
+    def rvs(self, size=1):
+        """Create random sample with keyword argument size."""
+        sample = self.copula_uv.simulate(size)
+        sample_X1 = self.X1.ppf(sample[:,0])
+        sample_X2 = self.X2.ppf(sample[:,1])
+        return np.vstack([sample_X1.T, sample_X2.T]).T
+
+def plot_contour(bivar, xy_lim, data=None,
+                 design_case=None, design_region=None,
+                 nb_points=100):
+    """Contour plot of PDF in the bivariate plane (X,Y).
+    
+    bivar: bivariate distribution, defined by
+        scipy.stats.multivariate_normal.
+    xy_lim: list. 1x4 list of xlim and ylim (in order) for plot limits.
+        
+    Optional arguments:
+    
+    data: array. A 2xN array of points.
+    design_case: list. Typically a 1x2 list defining a key
+        calculation point; plots as a red dot(s)
+    design_region: array. A 2xN array of points that define the
+        boundary of a region that will be shaded in gray between
+        the boundary and the max y-axis value (ylim[1]).
+    nb_points: int. Size of the grid (default: 200).
+
+    returns: matplotlib.pyplot Figure and Axis objects. 
+    """
+    
+    f, ax = plt.subplots(1)
+
+    xlim = [xy_lim[0], xy_lim[1]]
+    ylim = [xy_lim[2], xy_lim[3]]
+
+    x = np.linspace(xlim[0], xlim[1], nb_points)
+    y = np.linspace(ylim[0], ylim[1], nb_points)
+    X,Y = np.meshgrid(x,y)
+    pdf = np.zeros(X.shape)
+    for i in range(X.shape[0]):
+            for j in range(X.shape[1]):
+                # if X[i,j]>0 and Y[i,j]>0:
+                pdf[i,j] = bivar.pdf([X[i,j], Y[i,j]])
+    
+    ax.contour(X, Y, pdf, levels=8, cmap=cm.Blues)
+    
+    if isinstance(data, np.ndarray):
+        ax.scatter(data[0,:], data[1,:], s=10.0,
+                   facecolors='none', edgecolors='darkgray',
+                   label='Data')
+    
+    if isinstance(design_region, np.ndarray):
+        ax.plot(design_region[0,:], design_region[1,:],
+                label='Design Boundary', color='k')
+        ax.fill_between(design_region[0,:], design_region[1,:],
+                        ylim[1], label='Design Region', color='grey')
+    
+    if design_case:
+        ax.plot(design_case[0], design_case[1], 'ro',
+                label='Design Case')
+    
+    ax.set_aspect("equal")
+    ax.set_xlim(xlim)
+    ax.set_ylim(ylim)
+    ax.set_xlabel(r"$X_1$", fontsize=14)
+    ax.set_ylabel(r"$X_2$", fontsize=14)
+    if isinstance(design_region, np.ndarray) or design_case:
+        ax.legend()
+    return f, ax 

figures/solution.py

@@ -0,0 +1,264 @@
+"""Set of helper functions for Week 8 part of Project 4, 2023.
+
+Primary purpose is:
+    1) define dictionaries to track information about bivariate
+       and univariate probability models, as well as a few
+       important characteristics,
+    2) print and plot results,
+    3) perform simple calculations (e.g., covariance, ecdf etc).
+       
+Organized in order used in solution notebooks.
+"""
+
+import numpy as np
+import matplotlib.pyplot as plt
+import scipy.stats as st
+
+from helper import plot_contour
+from helper import Bivariate
+
+def univariate_RV_dict(name, symbol, unit):
+    """Create a dictionary to track a random variable."""
+    d = {"name":name, "symbol":symbol, "unit":unit}
+    d = create_plot_label(d)
+    return d
+      
+def create_plot_label(univariate_dict):
+    """Combine RV name, symbol, unit into key xlabel."""
+    univariate_dict["label"] = (univariate_dict["name"] + ", "
+                                + univariate_dict["symbol"] + " ["
+                                + univariate_dict["unit"] + "]")
+    return univariate_dict
+      
+def univariate_model_dict(dist_type, parameters,
+                          distribution, empirical=False, note=None):
+    """Create a dictionary to track a distribution model.
+    
+    dist_type: string,
+    parameters: tuple, from scipy.stats rv_continuous
+    distribution: frozen instance of scipy.stats.rv_continuous
+    empirical: boolean. Set to True if distribution is a sample
+      (array). This is just a placeholder to support other functions
+      that will be used for printing results
+      (e.g., compare_univar_prob).
+    note: string (optional)
+    """
+    model = {"type":dist_type}
+    model["parameters"] = parameters
+    model["distribution"] = distribution
+    model["empirical"] = empirical
+    model["note"] = note
+    
+    if empirical:
+        model["N"] = len(distribution)
+        model["sample"] = distribution
+    
+    return model
+
+def print_dist_model_summary(RV, model_key):
+    """Prints info about a model."""
+    print(f'Random variable {RV["name"]}, model \"{model_key}\" is:\n'
+          f'    distribution: {RV[model_key]["type"]}\n'
+          f'      parameters: {[f"{i:.3f}" for i in RV[model_key]["parameters"]]}\n'
+          f'      (μ,σ) dist: '
+          f'({RV[model_key]["distribution"].mean():.3f}, '
+          f'{RV[model_key]["distribution"].std():.3f})\n'
+          f'      (μ,σ) data: '
+          f'({RV["data"].mean():.3f}, {RV["data"].std():.3f})\n')
+    
+def calculate_covariance(X1, X2):
+    """Covariance of X1 and X2 (numpy arrays)."""
+    mean_x1 = X1.mean()
+    mean_x2 = X2.mean()
+    diff_x1 = X1 - mean_x1
+    diff_x2 = X2 - mean_x2
+    product = diff_x1 * diff_x2
+    covariance = product.mean()
+    return covariance
+
+def pearson_correlation(X1, X2):
+    """Pearson correlation of X1 and X2 (numpy arrays)."""
+    covariance = calculate_covariance(X1, X2)
+    correl_coeff = covariance/(X1.std()*X2.std())
+    return correl_coeff
+
+def bivariate_model_dict(model_name,
+                         X1_key, X1, X2_key, X2, rho):
+    """Create dictionary to track the bivariate model.
+    
+    model_name: string. Arg is root name, which has
+      marginal info added; used for plot title.
+    X1, X2: dict.from univariate_RV_dict()
+    X1_key, X2_key: string. Name of model of X1, X2
+    rho: correlation coefficient
+    
+    returns: dictionary
+    """
+    
+    d = {"X1_key":X1_key, "X1":X1[X1_key],
+         "X2_key":X2_key, "X2":X2[X2_key],
+         "rho":rho}
+    
+    d["title"] = (model_name + ": "
+                  + r"$X_1 \sim$" + X1_key + ", "
+                  + r"$X_2 \sim$" + X2_key + ", "
+                  + r"$\rho$ = " + f"{rho:.2f}")
+    
+    d["xlabel"], d["ylabel"] = X1["label"], X2["label"]
+    d["X1X2"] = Bivariate(d["X1"]["distribution"],
+                          d["X2"]["distribution"],
+                          d["rho"])
+    d["data"] = np.array([X1["data"], X2["data"]])
+    d["data_X1"], d["data_X2"] = X1["data"], X2["data"]
+    
+    return d
+
+def compare_bivar_prob(point, model):
+    """Compare probabilities at a given point (bivariate).
+    
+    point: array or list; size = 2.
+    model: dictionary defining bivariate distribution.
+    
+    returns: none, except printed output.
+    """
+    
+    data_X1 = model["data_X1"]
+    data_X2 = model["data_X2"]
+    N = data_X1.size
+    
+    left_emp = sum(data_X1 < point[0])/N
+    bottom_emp = sum(data_X2 < point[1])/N
+    lower_left_emp = sum((data_X1 < point[0])*(data_X2 < point[1]))/N
+
+    union_emp = 1 - lower_left_emp
+    intersection_emp = 1 - (left_emp + bottom_emp - lower_left_emp)
+    
+    lower_left = model["X1X2"].cdf(point)
+    union = 1 - lower_left
+
+    left = model["X1"]["distribution"].cdf(point[0])
+    bottom = model["X2"]["distribution"].cdf(point[1])
+    intersection = 1 - (left + bottom - lower_left)
+    
+    print(f'-------------------------------------------------')
+    print(f'      Comparison, Empirical Values in ( )')
+    print(f'-------------------------------------------------')
+    print(f'Lower left, P[X1<x1, X2<x2] = {lower_left:.5f}',
+          f'({lower_left_emp:.5f})')
+    print(f'Left side,         P[X1<x1] = {left:.5f}',
+          f'({left_emp:.5f})')
+    print(f'Bottom half,       P[X2<x2] = {bottom:.5f}',
+          f'({bottom_emp:.5f})')
+    print(f'-------------------------------------------------')
+    print(f'Case 1: P[X1>x1 or X2>x2]   = {union:.5f}',
+          f'({union_emp:.5f})')
+    print(f'Case 2: P[X1>x1 and X2>x2]  = {intersection:.5f}',
+          f'({intersection_emp:.5f})')
+    print(f'-------------------------------------------------')
+    
+def compare_univar_prob(point, model, dist_key):
+    """Compare probabilities at a given point (univariate).
+    
+    point: float
+    model: dictionary defining univariate distribution
+    dist_key: specify which distribution model to use
+    
+    returns: none, except printed output.
+    """
+    
+    data = model["data"]
+    N = data.size
+    p_data = sum(data < point)/N
+    if model[dist_key]["empirical"]:
+        N_model = model[dist_key]["N"]
+        n_eval = sum(model[dist_key]["sample"]<point)
+        p_model = n_eval/N_model
+    else:
+        X = model[dist_key]["distribution"]
+        p_model = X.cdf(point)
+    
+    print(f'---------------------------------------')
+    print(f'  Results for {model["name"]}, case {dist_key}')
+    print(f'  Comparison, Empirical Values in ( )')
+    print(f'---------------------------------------')
+    print(f'P[X<x] = {p_model:.5f} ({p_data:.5f})')
+    print(f'P[X>x] = {1 - p_model:.5f} ({1 - p_data:.5f})')
+    print(f'There are {N} points in the data set.')
+    if model[dist_key]["empirical"]:
+        print(f'This distribution is from MCS:')
+        print(f'    number of smaples = {N_model}')
+    print(f'---------------------------------------')
+    
+def ecdf(var):
+    """Empirical cdf from list or array. Returns [F(x), x]."""
+    x = np.sort(var)
+    n = x.size
+    y = np.arange(1, n+1) / (n + 1)
+    return [y, x]
+
+def ecdf_inverse(p, data, exceedance=False):
+    """Compute x = F^-1(p) for an empirical distribution."""
+    N = len(data)
+    if exceedance:
+        p = 1 - p
+    index = int(np.floor(p*(N + 1)))
+    x = np.sort(data)[index - 1]
+    return x
+
+def semi_log_plots(dist_1, dist_2, exceedance=True):
+    """Helper function for plot_univariate_comparison.
+    
+    dist_1, dist_2: ndarray, list. Empirical distributions.
+    exceedance: bool. Default (True) plots exceedance, 1 - F(x)
+    
+    returns: matplotlib objects fig, ax
+    """
+    fig, ax = plt.subplots(1, 1, figsize=(8, 5))
+    
+    e_dist_1 = ecdf(dist_1)
+    e_dist_2 = ecdf(dist_2)
+    if exceedance:
+        e_dist_1[0] = 1 - e_dist_1[0]
+        e_dist_2[0] = 1 - e_dist_2[0]
+    
+    ax.step(e_dist_1[1], e_dist_1[0], 
+              color='k', label='Data')
+    ax.step(e_dist_2[1], e_dist_2[0],
+              color='r', label='MCS Sample')
+    ax.set_xlabel('$Z(X_1,X_2)$')
+    ax.set_yscale('log')
+    ax.legend()
+    ax.grid()
+    return fig, ax
+
+def plot_univariate_comparison(rv_dictionary, dist_key, plot_values):
+    """Compare univariate plots.
+    
+    Assumes model is an empirical distribution (MCS
+    
+    rv_dictionary: dict. Created from univariate_RV_dict()
+    dist_key: str. Name of probability model
+    plot_values: array, list (size 4). Limits of X and Y, respectively.
+    returns: none, except printed output.
+    """
+    
+    data = rv_dictionary["data"]
+    model = rv_dictionary[dist_key]["sample"]
+    
+    fig, ax = plt.subplots(1)
+    ax.hist([data, model],
+             bins=plot_values,
+             density=True,
+             edgecolor='black');
+    ax.legend(['Data', 'MCS Sample'])
+    ax.set_xlabel('$Z(X_1,X_2)$')
+    ax.set_ylabel('Empirical Density [--]')
+    ax.set_title('Comparison of Distributions of $Z$');
+    
+    fig, ax = semi_log_plots(data, model, exceedance=False)
+    ax.set_ylabel('CDF, $\mathrm{P}[Z < z]$')
+    ax.set_title('Comparison: CDF (log scale expands lower tail)')
+    
+    fig, ax = semi_log_plots(data, model, exceedance=True)
+    ax.set_ylabel('CDF, $\mathrm{P}[Z < z]$')
+    ax.set_title('Comparison: CDF (log scale expands lower tail)')

figures/template_blank.svg

@@ -0,0 +1,534 @@
+<?xml version="1.0" encoding="utf-8" standalone="no"?>
+<!DOCTYPE svg PUBLIC "-//W3C//DTD SVG 1.1//EN"
+  "http://www.w3.org/Graphics/SVG/1.1/DTD/svg11.dtd">
+<svg xmlns:xlink="http://www.w3.org/1999/xlink" width="576pt" height="432pt" viewBox="0 0 576 432" xmlns="http://www.w3.org/2000/svg" version="1.1">
+ <metadata>
+  <rdf:RDF xmlns:dc="http://purl.org/dc/elements/1.1/" xmlns:cc="http://creativecommons.org/ns#" xmlns:rdf="http://www.w3.org/1999/02/22-rdf-syntax-ns#">
+   <cc:Work>
+    <dc:type rdf:resource="http://purl.org/dc/dcmitype/StillImage"/>
+    <dc:date>2024-02-18T07:45:46.054013</dc:date>
+    <dc:format>image/svg+xml</dc:format>
+    <dc:creator>
+     <cc:Agent>
+      <dc:title>Matplotlib v3.8.2, https://matplotlib.org/</dc:title>
+     </cc:Agent>
+    </dc:creator>
+   </cc:Work>
+  </rdf:RDF>
+ </metadata>
+ <defs>
+  <style type="text/css">*{stroke-linejoin: round; stroke-linecap: butt}</style>
+ </defs>
+ <g id="figure_1">
+  <g id="patch_1">
+   <path d="M 0 432 
+L 576 432 
+L 576 0 
+L 0 0 
+z
+" style="fill: #ffffff"/>
+  </g>
+  <g id="axes_1">
+   <g id="patch_2">
+    <path d="M 72 384.48 
+L 518.4 384.48 
+L 518.4 51.84 
+L 72 51.84 
+z
+" style="fill: #ffffff"/>
+   </g>
+   <g id="matplotlib.axis_1">
+    <g id="text_1">
+     <!-- xlabel [$x-units$] -->
+     <g transform="translate(233.36 400.64) scale(0.16 -0.16)">
+      <defs>
+       <path id="ArialMT-78" d="M 47 0 
+L 1259 1725 
+L 138 3319 
+L 841 3319 
+L 1350 2541 
+Q 1494 2319 1581 2169 
+Q 1719 2375 1834 2534 
+L 2394 3319 
+L 3066 3319 
+L 1919 1756 
+L 3153 0 
+L 2463 0 
+L 1781 1031 
+L 1600 1309 
+L 728 0 
+L 47 0 
+z
+" transform="scale(0.015625)"/>
+       <path id="ArialMT-6c" d="M 409 0 
+L 409 4581 
+L 972 4581 
+L 972 0 
+L 409 0 
+z
+" transform="scale(0.015625)"/>
+       <path id="ArialMT-61" d="M 2588 409 
+Q 2275 144 1986 34 
+Q 1697 -75 1366 -75 
+Q 819 -75 525 192 
+Q 231 459 231 875 
+Q 231 1119 342 1320 
+Q 453 1522 633 1644 
+Q 813 1766 1038 1828 
+Q 1203 1872 1538 1913 
+Q 2219 1994 2541 2106 
+Q 2544 2222 2544 2253 
+Q 2544 2597 2384 2738 
+Q 2169 2928 1744 2928 
+Q 1347 2928 1158 2789 
+Q 969 2650 878 2297 
+L 328 2372 
+Q 403 2725 575 2942 
+Q 747 3159 1072 3276 
+Q 1397 3394 1825 3394 
+Q 2250 3394 2515 3294 
+Q 2781 3194 2906 3042 
+Q 3031 2891 3081 2659 
+Q 3109 2516 3109 2141 
+L 3109 1391 
+Q 3109 606 3145 398 
+Q 3181 191 3288 0 
+L 2700 0 
+Q 2613 175 2588 409 
+z
+M 2541 1666 
+Q 2234 1541 1622 1453 
+Q 1275 1403 1131 1340 
+Q 988 1278 909 1158 
+Q 831 1038 831 891 
+Q 831 666 1001 516 
+Q 1172 366 1500 366 
+Q 1825 366 2078 508 
+Q 2331 650 2450 897 
+Q 2541 1088 2541 1459 
+L 2541 1666 
+z
+" transform="scale(0.015625)"/>
+       <path id="ArialMT-62" d="M 941 0 
+L 419 0 
+L 419 4581 
+L 981 4581 
+L 981 2947 
+Q 1338 3394 1891 3394 
+Q 2197 3394 2470 3270 
+Q 2744 3147 2920 2923 
+Q 3097 2700 3197 2384 
+Q 3297 2069 3297 1709 
+Q 3297 856 2875 390 
+Q 2453 -75 1863 -75 
+Q 1275 -75 941 416 
+L 941 0 
+z
+M 934 1684 
+Q 934 1088 1097 822 
+Q 1363 388 1816 388 
+Q 2184 388 2453 708 
+Q 2722 1028 2722 1663 
+Q 2722 2313 2464 2622 
+Q 2206 2931 1841 2931 
+Q 1472 2931 1203 2611 
+Q 934 2291 934 1684 
+z
+" transform="scale(0.015625)"/>
+       <path id="ArialMT-65" d="M 2694 1069 
+L 3275 997 
+Q 3138 488 2766 206 
+Q 2394 -75 1816 -75 
+Q 1088 -75 661 373 
+Q 234 822 234 1631 
+Q 234 2469 665 2931 
+Q 1097 3394 1784 3394 
+Q 2450 3394 2872 2941 
+Q 3294 2488 3294 1666 
+Q 3294 1616 3291 1516 
+L 816 1516 
+Q 847 969 1125 678 
+Q 1403 388 1819 388 
+Q 2128 388 2347 550 
+Q 2566 713 2694 1069 
+z
+M 847 1978 
+L 2700 1978 
+Q 2663 2397 2488 2606 
+Q 2219 2931 1791 2931 
+Q 1403 2931 1139 2672 
+Q 875 2413 847 1978 
+z
+" transform="scale(0.015625)"/>
+       <path id="ArialMT-20" transform="scale(0.015625)"/>
+       <path id="ArialMT-5b" d="M 434 -1272 
+L 434 4581 
+L 1675 4581 
+L 1675 4116 
+L 997 4116 
+L 997 -806 
+L 1675 -806 
+L 1675 -1272 
+L 434 -1272 
+z
+" transform="scale(0.015625)"/>
+       <path id="DejaVuSans-Oblique-78" d="M 3841 3500 
+L 2234 1784 
+L 3219 0 
+L 2559 0 
+L 1819 1388 
+L 531 0 
+L -166 0 
+L 1556 1844 
+L 641 3500 
+L 1300 3500 
+L 1972 2234 
+L 3144 3500 
+L 3841 3500 
+z
+" transform="scale(0.015625)"/>
+       <path id="DejaVuSans-2212" d="M 678 2272 
+L 4684 2272 
+L 4684 1741 
+L 678 1741 
+L 678 2272 
+z
+" transform="scale(0.015625)"/>
+       <path id="DejaVuSans-Oblique-75" d="M 428 1388 
+L 838 3500 
+L 1416 3500 
+L 1006 1409 
+Q 975 1256 961 1147 
+Q 947 1038 947 966 
+Q 947 700 1109 554 
+Q 1272 409 1569 409 
+Q 2031 409 2368 721 
+Q 2706 1034 2809 1563 
+L 3194 3500 
+L 3769 3500 
+L 3091 0 
+L 2516 0 
+L 2631 550 
+Q 2388 244 2052 76 
+Q 1716 -91 1338 -91 
+Q 878 -91 622 161 
+Q 366 413 366 863 
+Q 366 956 381 1097 
+Q 397 1238 428 1388 
+z
+" transform="scale(0.015625)"/>
+       <path id="DejaVuSans-Oblique-6e" d="M 3566 2113 
+L 3156 0 
+L 2578 0 
+L 2988 2091 
+Q 3016 2238 3031 2350 
+Q 3047 2463 3047 2528 
+Q 3047 2791 2881 2937 
+Q 2716 3084 2419 3084 
+Q 1956 3084 1622 2776 
+Q 1288 2469 1184 1941 
+L 800 0 
+L 225 0 
+L 903 3500 
+L 1478 3500 
+L 1363 2950 
+Q 1603 3253 1940 3418 
+Q 2278 3584 2650 3584 
+Q 3113 3584 3367 3334 
+Q 3622 3084 3622 2631 
+Q 3622 2519 3608 2391 
+Q 3594 2263 3566 2113 
+z
+" transform="scale(0.015625)"/>
+       <path id="DejaVuSans-Oblique-69" d="M 1172 4863 
+L 1747 4863 
+L 1606 4134 
+L 1031 4134 
+L 1172 4863 
+z
+M 909 3500 
+L 1484 3500 
+L 800 0 
+L 225 0 
+L 909 3500 
+z
+" transform="scale(0.015625)"/>
+       <path id="DejaVuSans-Oblique-74" d="M 2706 3500 
+L 2619 3053 
+L 1472 3053 
+L 1100 1153 
+Q 1081 1047 1072 975 
+Q 1063 903 1063 863 
+Q 1063 663 1183 572 
+Q 1303 481 1569 481 
+L 2150 481 
+L 2053 0 
+L 1503 0 
+Q 991 0 739 200 
+Q 488 400 488 806 
+Q 488 878 497 964 
+Q 506 1050 525 1153 
+L 897 3053 
+L 409 3053 
+L 500 3500 
+L 978 3500 
+L 1172 4494 
+L 1747 4494 
+L 1556 3500 
+L 2706 3500 
+z
+" transform="scale(0.015625)"/>
+       <path id="DejaVuSans-Oblique-73" d="M 3200 3397 
+L 3091 2853 
+Q 2863 2978 2609 3040 
+Q 2356 3103 2088 3103 
+Q 1634 3103 1373 2948 
+Q 1113 2794 1113 2528 
+Q 1113 2219 1719 2053 
+Q 1766 2041 1788 2034 
+L 1972 1978 
+Q 2547 1819 2739 1644 
+Q 2931 1469 2931 1166 
+Q 2931 609 2489 259 
+Q 2047 -91 1331 -91 
+Q 1053 -91 747 -37 
+Q 441 16 72 128 
+L 184 722 
+Q 500 559 806 475 
+Q 1113 391 1394 391 
+Q 1816 391 2080 572 
+Q 2344 753 2344 1031 
+Q 2344 1331 1650 1516 
+L 1591 1531 
+L 1394 1581 
+Q 956 1697 753 1886 
+Q 550 2075 550 2369 
+Q 550 2928 970 3256 
+Q 1391 3584 2113 3584 
+Q 2397 3584 2667 3537 
+Q 2938 3491 3200 3397 
+z
+" transform="scale(0.015625)"/>
+       <path id="ArialMT-5d" d="M 1363 -1272 
+L 122 -1272 
+L 122 -806 
+L 800 -806 
+L 800 4116 
+L 122 4116 
+L 122 4581 
+L 1363 4581 
+L 1363 -1272 
+z
+" transform="scale(0.015625)"/>
+      </defs>
+      <use xlink:href="#ArialMT-78" transform="translate(0 0.015625)"/>
+      <use xlink:href="#ArialMT-6c" transform="translate(50 0.015625)"/>
+      <use xlink:href="#ArialMT-61" transform="translate(72.216797 0.015625)"/>
+      <use xlink:href="#ArialMT-62" transform="translate(127.832031 0.015625)"/>
+      <use xlink:href="#ArialMT-65" transform="translate(183.447266 0.015625)"/>
+      <use xlink:href="#ArialMT-6c" transform="translate(239.0625 0.015625)"/>
+      <use xlink:href="#ArialMT-20" transform="translate(261.279297 0.015625)"/>
+      <use xlink:href="#ArialMT-5b" transform="translate(289.0625 0.015625)"/>
+      <use xlink:href="#DejaVuSans-Oblique-78" transform="translate(316.845703 0.015625)"/>
+      <use xlink:href="#DejaVuSans-2212" transform="translate(395.507812 0.015625)"/>
+      <use xlink:href="#DejaVuSans-Oblique-75" transform="translate(498.779297 0.015625)"/>
+      <use xlink:href="#DejaVuSans-Oblique-6e" transform="translate(562.158203 0.015625)"/>
+      <use xlink:href="#DejaVuSans-Oblique-69" transform="translate(625.537109 0.015625)"/>
+      <use xlink:href="#DejaVuSans-Oblique-74" transform="translate(653.320312 0.015625)"/>
+      <use xlink:href="#DejaVuSans-Oblique-73" transform="translate(692.529297 0.015625)"/>
+      <use xlink:href="#ArialMT-5d" transform="translate(744.628906 0.015625)"/>
+     </g>
+    </g>
+   </g>
+   <g id="matplotlib.axis_2">
+    <g id="text_2">
+     <!-- ylabel [$y-units$] -->
+     <g transform="translate(64.48 280) rotate(-90) scale(0.16 -0.16)">
+      <defs>
+       <path id="ArialMT-79" d="M 397 -1278 
+L 334 -750 
+Q 519 -800 656 -800 
+Q 844 -800 956 -737 
+Q 1069 -675 1141 -563 
+Q 1194 -478 1313 -144 
+Q 1328 -97 1363 -6 
+L 103 3319 
+L 709 3319 
+L 1400 1397 
+Q 1534 1031 1641 628 
+Q 1738 1016 1872 1384 
+L 2581 3319 
+L 3144 3319 
+L 1881 -56 
+Q 1678 -603 1566 -809 
+Q 1416 -1088 1222 -1217 
+Q 1028 -1347 759 -1347 
+Q 597 -1347 397 -1278 
+z
+" transform="scale(0.015625)"/>
+       <path id="DejaVuSans-Oblique-79" d="M 1588 -325 
+Q 1188 -997 936 -1164 
+Q 684 -1331 294 -1331 
+L -159 -1331 
+L -63 -850 
+L 269 -850 
+Q 509 -850 678 -719 
+Q 847 -588 1056 -206 
+L 1234 128 
+L 459 3500 
+L 1069 3500 
+L 1650 819 
+L 3256 3500 
+L 3859 3500 
+L 1588 -325 
+z
+" transform="scale(0.015625)"/>
+      </defs>
+      <use xlink:href="#ArialMT-79" transform="translate(0 0.015625)"/>
+      <use xlink:href="#ArialMT-6c" transform="translate(50 0.015625)"/>
+      <use xlink:href="#ArialMT-61" transform="translate(72.216797 0.015625)"/>
+      <use xlink:href="#ArialMT-62" transform="translate(127.832031 0.015625)"/>
+      <use xlink:href="#ArialMT-65" transform="translate(183.447266 0.015625)"/>
+      <use xlink:href="#ArialMT-6c" transform="translate(239.0625 0.015625)"/>
+      <use xlink:href="#ArialMT-20" transform="translate(261.279297 0.015625)"/>
+      <use xlink:href="#ArialMT-5b" transform="translate(289.0625 0.015625)"/>
+      <use xlink:href="#DejaVuSans-Oblique-79" transform="translate(316.845703 0.015625)"/>
+      <use xlink:href="#DejaVuSans-2212" transform="translate(395.507812 0.015625)"/>
+      <use xlink:href="#DejaVuSans-Oblique-75" transform="translate(498.779297 0.015625)"/>
+      <use xlink:href="#DejaVuSans-Oblique-6e" transform="translate(562.158203 0.015625)"/>
+      <use xlink:href="#DejaVuSans-Oblique-69" transform="translate(625.537109 0.015625)"/>
+      <use xlink:href="#DejaVuSans-Oblique-74" transform="translate(653.320312 0.015625)"/>
+      <use xlink:href="#DejaVuSans-Oblique-73" transform="translate(692.529297 0.015625)"/>
+      <use xlink:href="#ArialMT-5d" transform="translate(744.628906 0.015625)"/>
+     </g>
+    </g>
+   </g>
+   <g id="patch_3">
+    <path d="M 72 384.48 
+L 72 51.84 
+" style="fill: none; stroke: #000000; stroke-width: 2; stroke-linejoin: miter; stroke-linecap: square"/>
+   </g>
+   <g id="patch_4">
+    <path d="M 72 384.48 
+L 518.4 384.48 
+" style="fill: none; stroke: #000000; stroke-width: 2; stroke-linejoin: miter; stroke-linecap: square"/>
+   </g>
+   <g id="text_3">
+    <!-- this is the title -->
+    <g transform="translate(237.045 45.84) scale(0.192 -0.192)">
+     <defs>
+      <path id="ArialMT-74" d="M 1650 503 
+L 1731 6 
+Q 1494 -44 1306 -44 
+Q 1000 -44 831 53 
+Q 663 150 594 308 
+Q 525 466 525 972 
+L 525 2881 
+L 113 2881 
+L 113 3319 
+L 525 3319 
+L 525 4141 
+L 1084 4478 
+L 1084 3319 
+L 1650 3319 
+L 1650 2881 
+L 1084 2881 
+L 1084 941 
+Q 1084 700 1114 631 
+Q 1144 563 1211 522 
+Q 1278 481 1403 481 
+Q 1497 481 1650 503 
+z
+" transform="scale(0.015625)"/>
+      <path id="ArialMT-68" d="M 422 0 
+L 422 4581 
+L 984 4581 
+L 984 2938 
+Q 1378 3394 1978 3394 
+Q 2347 3394 2619 3248 
+Q 2891 3103 3008 2847 
+Q 3125 2591 3125 2103 
+L 3125 0 
+L 2563 0 
+L 2563 2103 
+Q 2563 2525 2380 2717 
+Q 2197 2909 1863 2909 
+Q 1613 2909 1392 2779 
+Q 1172 2650 1078 2428 
+Q 984 2206 984 1816 
+L 984 0 
+L 422 0 
+z
+" transform="scale(0.015625)"/>
+      <path id="ArialMT-69" d="M 425 3934 
+L 425 4581 
+L 988 4581 
+L 988 3934 
+L 425 3934 
+z
+M 425 0 
+L 425 3319 
+L 988 3319 
+L 988 0 
+L 425 0 
+z
+" transform="scale(0.015625)"/>
+      <path id="ArialMT-73" d="M 197 991 
+L 753 1078 
+Q 800 744 1014 566 
+Q 1228 388 1613 388 
+Q 2000 388 2187 545 
+Q 2375 703 2375 916 
+Q 2375 1106 2209 1216 
+Q 2094 1291 1634 1406 
+Q 1016 1563 777 1677 
+Q 538 1791 414 1992 
+Q 291 2194 291 2438 
+Q 291 2659 392 2848 
+Q 494 3038 669 3163 
+Q 800 3259 1026 3326 
+Q 1253 3394 1513 3394 
+Q 1903 3394 2198 3281 
+Q 2494 3169 2634 2976 
+Q 2775 2784 2828 2463 
+L 2278 2388 
+Q 2241 2644 2061 2787 
+Q 1881 2931 1553 2931 
+Q 1166 2931 1000 2803 
+Q 834 2675 834 2503 
+Q 834 2394 903 2306 
+Q 972 2216 1119 2156 
+Q 1203 2125 1616 2013 
+Q 2213 1853 2448 1751 
+Q 2684 1650 2818 1456 
+Q 2953 1263 2953 975 
+Q 2953 694 2789 445 
+Q 2625 197 2315 61 
+Q 2006 -75 1616 -75 
+Q 969 -75 630 194 
+Q 291 463 197 991 
+z
+" transform="scale(0.015625)"/>
+     </defs>
+     <use xlink:href="#ArialMT-74"/>
+     <use xlink:href="#ArialMT-68" x="27.783203"/>
+     <use xlink:href="#ArialMT-69" x="83.398438"/>
+     <use xlink:href="#ArialMT-73" x="105.615234"/>
+     <use xlink:href="#ArialMT-20" x="155.615234"/>
+     <use xlink:href="#ArialMT-69" x="183.398438"/>
+     <use xlink:href="#ArialMT-73" x="205.615234"/>
+     <use xlink:href="#ArialMT-20" x="255.615234"/>
+     <use xlink:href="#ArialMT-74" x="283.398438"/>
+     <use xlink:href="#ArialMT-68" x="311.181641"/>
+     <use xlink:href="#ArialMT-65" x="366.796875"/>
+     <use xlink:href="#ArialMT-20" x="422.412109"/>
+     <use xlink:href="#ArialMT-74" x="450.195312"/>
+     <use xlink:href="#ArialMT-69" x="477.978516"/>
+     <use xlink:href="#ArialMT-74" x="500.195312"/>
+     <use xlink:href="#ArialMT-6c" x="527.978516"/>
+     <use xlink:href="#ArialMT-65" x="550.195312"/>
+    </g>
+   </g>
+  </g>
+ </g>
+</svg>