update on logreg slides
This commit is contained in:
Morten Hjorth-Jensen
@@ -1,101 +1,100 @@
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ID Age Agegroup CHD
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1 21 1 0
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2 23 1 0
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3 25 1 1
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4 29 1 0
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5 21 1 0
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6 24 1 0
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7 27 1 0
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8 29 1 0
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9 28 1 0
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10 26 1 0
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11 30 2 0
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12 31 2 0
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13 31 2 0
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14 31 2 1
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15 32 2 0
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16 34 2 0
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17 34 2 0
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18 31 2 0
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19 32 2 0
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20 32 2 0
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21 33 2 0
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22 34 2 0
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23 31 2 1
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24 30 2 0
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25 33 2 0
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26 36 3 1
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27 35 3 0
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28 35 3 0
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29 38 3 0
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30 37 3 1
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31 36 3 0
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32 35 3 0
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33 39 3 0
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34 39 3 0
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35 38 3 1
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36 37 3 0
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37 37 3 0
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38 40 4 0
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39 41 4 1
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40 44 4 0
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41 44 4 0
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42 43 4 1
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43 42 4 0
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44 41 4 0
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45 40 4 1
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46 42 4 0
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47 42 4 0
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48 43 4 0
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49 44 4 1
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50 44 4 0
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51 42 4 0
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52 41 4 1
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53 45 5 0
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54 45 5 1
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55 49 5 0
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56 48 5 1
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57 47 5 0
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58 49 5 1
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59 46 5 1
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60 45 5 0
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61 49 5 1
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62 48 5 0
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63 47 5 1
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64 46 5 0
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65 47 5 0
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66 50 6 1
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67 51 6 1
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68 51 6 0
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69 54 6 1
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70 53 6 1
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71 51 6 0
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72 52 6 1
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73 54 6 0
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74 55 7 1
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75 56 7 1
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76 58 7 0
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77 59 7 1
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78 59 7 1
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79 58 7 0
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80 55 7 1
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81 56 7 1
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82 57 7 1
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83 58 7 1
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84 59 7 0
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85 55 7 1
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86 56 7 1
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87 57 7 1
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88 58 7 0
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89 59 7 1
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90 56 7 1
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91 60 8 1
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92 65 8 1
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93 67 8 1
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94 66 8 0
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95 63 8 1
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96 61 8 1
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97 69 8 1
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98 65 8 1
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99 64 8 1
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100 63 8 0
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1, 21, 1, 0
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2, 23, 1, 0
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3, 25, 1, 1
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4, 29, 1, 0
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5, 21, 1, 0
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6, 24, 1, 0
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7, 27, 1, 0
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8, 29, 1, 0
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9, 28, 1, 0
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10, 26, 1, 0
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11, 30, 2, 0
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12, 31, 2, 0
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13, 31, 2, 0
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14, 31, 2, 1
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15, 32, 2, 0
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16, 34, 2, 0
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17, 34, 2, 0
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18, 31, 2, 0
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19, 32, 2, 0
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20, 32, 2, 0
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21, 33, 2, 0
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22, 34, 2, 0
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23, 31, 2, 1
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24, 30, 2, 0
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25, 33, 2, 0
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26, 36, 3, 1
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27, 35, 3, 0
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28, 35, 3, 0
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29, 38, 3, 0
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30, 37, 3, 1
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31, 36, 3, 0
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32, 35, 3, 0
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33, 39, 3, 0
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34, 39, 3, 0
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35, 38, 3, 1
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36, 37, 3, 0
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37, 37, 3, 0
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38, 40, 4, 0
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39, 41, 4, 1
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40, 44, 4, 0
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41, 44, 4, 0
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42, 43, 4, 1
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43, 42, 4, 0
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44, 41, 4, 0
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45, 40, 4, 1
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46, 42, 4, 0
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47, 42, 4, 0
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48, 43, 4, 0
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49, 44, 4, 1
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50, 44, 4, 0
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51, 42, 4, 0
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52, 41, 4, 1
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53, 45, 5, 0
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54, 45, 5, 1
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55, 49, 5, 0
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56, 48, 5, 1
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57, 47, 5, 0
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58, 49, 5, 1
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59, 46, 5, 1
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60, 45, 5, 0
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61, 49, 5, 1
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62, 48, 5, 0
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63, 47, 5, 1
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64, 46, 5, 0
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65, 47, 5, 0
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66, 50, 6, 1
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67, 51, 6, 1
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68, 51, 6, 0
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69, 54, 6, 1
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70, 53, 6, 1
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71, 51, 6, 0
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72, 52, 6, 1
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73, 54, 6, 0
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74, 55, 7, 1
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75, 56, 7, 1
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76, 58, 7, 0
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77, 59, 7, 1
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78, 59, 7, 1
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79, 58, 7, 0
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80, 55, 7, 1
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81, 56, 7, 1
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82, 57, 7, 1
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83, 58, 7, 1
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84, 59, 7, 0
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85, 55, 7, 1
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86, 56, 7, 1
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87, 57, 7, 1
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88, 58, 7, 0
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89, 59, 7, 1
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90, 56, 7, 1
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91, 60, 8, 1
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92, 65, 8, 1
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93, 67, 8, 1
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94, 66, 8, 0
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95, 63, 8, 1
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96, 61, 8, 1
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97, 69, 8, 1
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98, 65, 8, 1
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99, 64, 8, 1
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100, 63, 8, 0
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@@ -8,6 +8,10 @@ from sklearn.model_selection import train_test_split
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from sklearn.utils import resample
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from sklearn.metrics import mean_squared_error
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from IPython.display import display
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from pylab import plt, mpl
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plt.style.use('seaborn')
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mpl.rcParams['font.family'] = 'serif'
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# Where to save the figures and data files
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PROJECT_ROOT_DIR = "Results"
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FIGURE_ID = "Results/FigureFiles"
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@@ -33,11 +37,27 @@ def save_fig(fig_id):
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infile = open(data_path("chddata.csv"),'r')
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# Read the chd data as csv file and organize the data into two arrays with density and energies
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# Read the chd data as csv file and organize the data into arrays with age group, age, and chd
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chd = pd.read_csv(infile, names=('ID', 'Age', 'Agegroup', 'CHD'))
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display(chd)
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chd.columns = ['ID', 'Age', 'Agegroup', 'CHD']
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output = chd['CHD']
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age = chd['Age']
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agegroup = chd['Agegroup']
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numberID = chd['ID']
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display(output)
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display(chd)
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plt.scatter(age, output, marker='o')
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plt.axis([18,70.0,-0.1, 1.2])
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plt.xlabel(r'Age')
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plt.ylabel(r'CHD')
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plt.title(r'Age distribution and Coronary heart disease')
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plt.show()
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agegroupmean = np.array([0.1, 0.133, 0.250, 0.333, 0.462, 0.625, 0.765, 0.800])
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group = np.array([1, 2, 3, 4, 5, 6, 7, 8])
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plt.plot(group, agegroupmean, "r-")
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plt.axis([0,9,0, 1.0])
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plt.xlabel(r'Age group')
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plt.ylabel(r'CHD mean values')
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plt.title(r'Mean values for each age group')
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plt.show()
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@@ -1040,6 +1040,93 @@ literature. This model is extremely simple. However, in many cases it is more
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favorable to use a ``soft" classifier that outputs
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the probability of a given category. This leads us to the logistic function.
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!split
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===== Simple example =====
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The following example on data for coronary heart disease (CHD) as function of age may serve as an illustration. In the code here we read and plot for whether a person has had CHD (output = 1) or not (output = 0) is plotted against age. Clearly, the figure shows that attempting to make a standard lineae regression fit may not be very meaningful.
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!bc pycod
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# Common imports
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import os
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import numpy as np
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import pandas as pd
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import matplotlib.pyplot as plt
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from sklearn.linear_model import LinearRegression, Ridge, Lasso
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from sklearn.model_selection import train_test_split
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from sklearn.utils import resample
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from sklearn.metrics import mean_squared_error
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from IPython.display import display
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from pylab import plt, mpl
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plt.style.use('seaborn')
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mpl.rcParams['font.family'] = 'serif'
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# Where to save the figures and data files
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PROJECT_ROOT_DIR = "Results"
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FIGURE_ID = "Results/FigureFiles"
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DATA_ID = "DataFiles/"
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if not os.path.exists(PROJECT_ROOT_DIR):
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os.mkdir(PROJECT_ROOT_DIR)
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if not os.path.exists(FIGURE_ID):
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os.makedirs(FIGURE_ID)
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if not os.path.exists(DATA_ID):
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os.makedirs(DATA_ID)
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def image_path(fig_id):
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return os.path.join(FIGURE_ID, fig_id)
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def data_path(dat_id):
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return os.path.join(DATA_ID, dat_id)
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def save_fig(fig_id):
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plt.savefig(image_path(fig_id) + ".png", format='png')
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infile = open(data_path("chddata.csv"),'r')
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# Read the chd data as csv file and organize the data into arrays with age group, age, and chd
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chd = pd.read_csv(infile, names=('ID', 'Age', 'Agegroup', 'CHD'))
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chd.columns = ['ID', 'Age', 'Agegroup', 'CHD']
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output = chd['CHD']
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age = chd['Age']
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agegroup = chd['Agegroup']
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numberID = chd['ID']
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display(chd)
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plt.scatter(age, output, marker='o')
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plt.axis([18,70.0,-0.1, 1.2])
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plt.xlabel(r'Age')
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plt.ylabel(r'CHD')
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plt.title(r'Age distribution and Coronary heart disease')
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plt.show()
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!ec
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!split
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===== Plotting the mean value for each group =====
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What we could attempt however is to plot the mean value for each group.
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!bc pycod
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agegroupmean = np.array([0.1, 0.133, 0.250, 0.333, 0.462, 0.625, 0.765, 0.800])
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group = np.array([1, 2, 3, 4, 5, 6, 7, 8])
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plt.plot(group, agegroupmean, "r-")
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plt.axis([0,9,0, 1.0])
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plt.xlabel(r'Age group')
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plt.ylabel(r'CHD mean values')
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plt.title(r'Mean values for each age group')
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plt.show()
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!ec
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We are now trying to find a function $f(y\vert x)$, that is a function which gives us an expected value for the output $y$ with a given input $x$.
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In standard linear regression with a linear dependence on $x$, we would write this in terms of our model
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!bt
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\[
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f(y_i\vert x_i)=\beta_0+\beta_1 x_i.
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\]
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!et
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This expression implies however that $f(y_i\vert x_i)$ could take any value from minus infinity to plus infinity. If we however let $f(y\vert y)$ be represented by the mean value, the above example shows us that we can constain to be between zero and one, that is we have $0 \le f(y_i\vert x_i) \le 1$. Looking at our last curve we see also that it has an S-shaped form. This leads us to a very popular model for the function $f$, namely the so-called Sigmoid function or logistic model. We will consider this function as representing the probability for finding a value of $y_i$ with a given $x_i$.
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!split
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===== The logistic function =====
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@@ -1318,58 +1405,10 @@ descent method. Newton's method and gradient descent methods are
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discussed in the material on "optimization
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methods":"https://compphysics.github.io/MachineLearning/doc/pub/Splines/html/Splines-bs.html".
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This will be discussed next week.
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!split
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===== A simple classification problem =====
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!bc pycod
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import numpy as np
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from sklearn import datasets, linear_model
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import matplotlib.pyplot as plt
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def generate_data():
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np.random.seed(0)
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X, y = datasets.make_moons(200, noise=0.20)
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return X, y
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def visualize(X, y, clf):
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plot_decision_boundary(lambda x: clf.predict(x), X, y)
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def plot_decision_boundary(pred_func, X, y):
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# Set min and max values and give it some padding
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x_min, x_max = X[:, 0].min() - .5, X[:, 0].max() + .5
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y_min, y_max = X[:, 1].min() - .5, X[:, 1].max() + .5
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h = 0.01
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# Generate a grid of points with distance h between them
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xx, yy = np.meshgrid(np.arange(x_min, x_max, h), np.arange(y_min, y_max, h))
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# Predict the function value for the whole gid
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Z = pred_func(np.c_[xx.ravel(), yy.ravel()])
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Z = Z.reshape(xx.shape)
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# Plot the contour and training examples
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plt.contourf(xx, yy, Z, cmap=plt.cm.Spectral)
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plt.scatter(X[:, 0], X[:, 1], c=y, cmap=plt.cm.Spectral)
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plt.show()
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def classify(X, y):
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clf = linear_model.LogisticRegressionCV()
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clf.fit(X, y)
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return clf
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def main():
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X, y = generate_data()
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# visualize(X, y)
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clf = classify(X, y)
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visualize(X, y, clf)
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if __name__ == "__main__":
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main()
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!ec
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!split
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