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@zsh-o 2018-07-09T16:51:38.000000Z 字数 6670 阅读 936

5 - 决策树

《统计学习方法》


  1. %matplotlib inline
  2. import numpy as np
  3. from matplotlib import pyplot as plt
  4. epsilon = 1e-5
  5. np.seterr(divide='ignore',invalid='ignore')
{'divide': 'warn', 'over': 'warn', 'under': 'ignore', 'invalid': 'warn'}
  1. import pydot
  2. from IPython.display import Image, display
  3. def viewPydot(pdot):
  4. plt = Image(pdot.create_png())
  5. display(plt)
  1. ## 给定离散的概率,求熵,e为底
  2. def H(P):
  3. P = P + epsilon
  4. return -(np.sum(P * np.log2(P)))
  1. def information_gain(X, Y):
  2. N, M = X.shape
  3. K = np.max(Y) + 1
  4. SX = np.max(X, axis=0) + 1
  5. NY = np.zeros(K)
  6. for i in range(K):
  7. NY[i] = np.count_nonzero(Y==i)
  8. NX = []
  9. for m in range(M):
  10. NX_i = np.zeros((SX[m],1))
  11. for i in range(SX[m]):
  12. NX_i[i,0] = np.count_nonzero(X[:, m]==i)
  13. NX.append(NX_i)
  14. HX = H(NY / N)
  15. NY_A = []
  16. for i in range(M):
  17. NY_Ai = np.zeros((SX[i], K))
  18. for j in range(SX[i]):
  19. t = Y[X[:, i]==j]
  20. for k in range(K):
  21. NY_Ai[j, k] = np.count_nonzero(t==k)
  22. NY_A.append(NY_Ai)
  23. PY_A = []
  24. PX = []
  25. for m in range(M):
  26. PY_A.append(NY_A[m] / NX[m]) ## NY_A[m][i, k] / NX[m][i] ## p(Y = y_k | X^m = i)
  27. PX.append(NX[m] / N)
  28. HY_X = []
  29. for m in range(M):
  30. HY_Xi = np.zeros(SX[m])
  31. for i in range(SX[m]):
  32. HY_Xi[i] = H(PY_A[m][i, :])
  33. HY_X.append(HY_Xi)
  34. HY_A = np.zeros(M)
  35. for m in range(M):
  36. HY_A[m] = np.sum(HY_X[m] * PX[m].reshape(-1)) ## numpy由于有Broadcasting存在,所以写程序的时候一定要注意维度要匹配 https://docs.scipy.org/doc/numpy-1.13.0/user/basics.broadcasting.html
  37. HA_X = np.zeros(M)
  38. for m in range(M):
  39. HA_X[m] = H(NX[m] / N)
  40. gain = HX - HY_A
  41. return gain, gain / HA_X
  1. X = np.array([
  2. [0, 0, 0, 0, 0],
  3. [0, 0, 0, 1, 0],
  4. [0, 1, 0, 1, 1],
  5. [0, 1, 1, 0, 1],
  6. [0, 0, 0, 0, 0],
  7. [1, 0, 0, 0, 0],
  8. [1, 0, 0, 1, 0],
  9. [1, 1, 1, 1, 1],
  10. [1, 0, 1, 2, 1],
  11. [1, 0, 1, 2, 1],
  12. [2, 0, 1, 2, 1],
  13. [2, 0, 1, 1, 1],
  14. [2, 1, 0, 1, 1],
  15. [2, 1, 0, 2, 1],
  16. [2, 0, 0, 1, 0],
  17. ])
  18. Y = X[:, -1]
  19. Y = Y.reshape((-1,1))
  20. X = X[:, :-1]
  1. information_gain(X, Y)
(array([0.08300555, 0.32359689, 0.41990845, 0.29479317]),
 array([0.05237053, 0.35239124, 0.43247518, 0.1926588 ]))
  1. class TreeNode(object):
  2. def __init__(self, prop=None, Nprop=None, label=None):
  3. self.children = dict()
  4. self.prop = prop
  5. self.Nprop = Nprop
  6. self.label = label
  7. self.leaf = False
  1. def ID3(X, Y, threshold):
  2. N, M = X.shape
  3. K = np.max(Y) + 1
  4. SX = np.max(X, axis=0) + 1
  5. def build_tree(cX, cY):
  6. NcY = np.zeros(K)
  7. unique_Y = np.unique(cY)
  8. if len(unique_Y)==1: ## 属于同一类
  9. t = TreeNode()
  10. t.label = unique_Y[0]
  11. t.leaf = True
  12. return t
  13. for i in range(K):
  14. NcY[i] = np.count_nonzero(cY==i)
  15. gain, ratio = information_gain(cX, cY)
  16. iP = np.argmax(gain)
  17. max_gain = gain[iP]
  18. t = TreeNode(prop=iP, Nprop=SX[iP], label = np.argmax(NcY))
  19. if max_gain < threshold:
  20. t.leaf = True
  21. return t
  22. for i in range(SX[iP]):
  23. cindex = (cX[:,iP] == i)
  24. t.children[i] = build_tree(cX[cindex], cY[cindex])
  25. return t
  26. return build_tree(X, Y)
  1. root = ID3(X, Y, 0.)
  1. dot = pydot.Dot()
  2. global level
  3. level = 1
  4. def create_dot(p):
  5. global level
  6. p_name = "%d # %s, %s, %s" % (level, str(p.prop), str(p.Nprop), str(p.label))
  7. dot.add_node(pydot.Node(name=p_name))
  8. if p.prop == None:
  9. return
  10. for i in range(p.Nprop):
  11. level = level + 1
  12. c = p.children[i]
  13. c_name = "%d # %s, %s, %s" % (level, str(c.prop), str(c.Nprop), str(c.label))
  14. dot.add_edge(pydot.Edge(dst=c_name, src=p_name))
  15. create_dot(c)
  16. level = level -1
  1. create_dot(root)
  2. viewPydot(dot)

output_11_0.png-21kB

剪枝

前序遍历计算每个节点的H,后续遍历对有叶节点进行剪枝

  1. class TreeNode(object):
  2. def __init__(self, prop=None, Nprop=None, label=None):
  3. self.children = dict()
  4. self.prop = prop
  5. self.Nprop = Nprop
  6. self.label = label
  7. self.leaf = False
  8. self.H = None
  9. def ID3(X, Y, threshold):
  10. N, M = X.shape
  11. K = np.max(Y) + 1
  12. SX = np.max(X, axis=0) + 1
  13. global visited
  14. visited = np.zeros(M, dtype=np.bool)
  15. def build_tree(cX, cY):
  16. global visited
  17. NcY = np.zeros(K)
  18. unique_Y = np.unique(cY)
  19. if len(unique_Y)==1: ## 属于同一类
  20. t = TreeNode()
  21. t.label = unique_Y[0]
  22. t.leaf = True
  23. t.H = 0.
  24. return t
  25. for i in range(K):
  26. NcY[i] = np.count_nonzero(cY==i)
  27. gain, ratio = information_gain(cX, cY)
  28. gain[visited] = - np.inf
  29. iP = np.argmax(gain)
  30. if len(np.unique(cX[:,iP]))==1:
  31. t1 = TreeNode(prop=None, Nprop=None, label=np.argmax(NcY))
  32. t1.H = H(NcY / len(cY))
  33. t1.leaf = True
  34. return t1
  35. max_gain = gain[iP]
  36. t = TreeNode(prop=iP, Nprop=SX[iP], label = np.argmax(NcY))
  37. t.H = H(NcY / len(cY))
  38. if max_gain < threshold:
  39. t.leaf = True
  40. return t
  41. visited[iP] = True
  42. for i in range(SX[iP]):
  43. cindex = (cX[:,iP] == i)
  44. t.children[i] = build_tree(cX[cindex], cY[cindex])
  45. visited[iP] = False
  46. return t
  47. return build_tree(X, Y)
  1. root = ID3(X, Y, 0.)
  1. dot = pydot.Dot()
  2. global level
  3. level = 1
  4. def create_dot(p):
  5. global level
  6. p_name = "%d # %s, %s, %s, %.3f" % (level, str(p.prop), str(p.Nprop), str(p.label), p.H)
  7. dot.add_node(pydot.Node(name=p_name))
  8. if p.leaf is True:
  9. return
  10. for i in range(p.Nprop):
  11. level = level + 1
  12. c = p.children[i]
  13. c_name = "%d # %s, %s, %s, %.3f" % (level, str(c.prop), str(c.Nprop), str(c.label), c.H)
  14. dot.add_edge(pydot.Edge(dst=c_name, src=p_name))
  15. create_dot(c)
  16. level = level -1
  17. create_dot(root)
  18. viewPydot(dot)

output_15_0.png-26.6kB

  1. Watermelon = np.array([
  2. [0, 0, 0, 0, 0, 0, 0],
  3. [1, 0, 1, 0, 0, 0, 0],
  4. [1, 0, 0, 0, 0, 0, 0],
  5. [0, 0, 1, 0, 0, 0, 0],
  6. [2, 0, 0, 0, 0, 0, 0],
  7. [0, 1, 0, 0, 1, 1, 0],
  8. [1, 1, 0, 1, 1, 1, 0],
  9. [1, 1, 0, 0, 1, 0, 0],
  10. [1, 1, 1, 1, 1, 0, 1],
  11. [0, 2, 2, 0, 2, 1, 1],
  12. [2, 2, 2, 2, 2, 0, 1],
  13. [2, 0, 0, 2, 2, 1, 1],
  14. [0, 1, 0, 1, 0, 0, 1],
  15. [2, 1, 1, 1, 0, 0, 1],
  16. [1, 1, 0, 0, 1, 1, 1],
  17. [2, 0, 0, 2, 2, 0, 1],
  18. [0, 0, 1, 1, 1, 0, 1],
  19. ])
  20. Y = Watermelon[:, -1]
  21. Y = Y.reshape((-1,1))
  22. X = Watermelon[:, :-1]
  1. root = ID3(X, Y, 0.)
  1. dot = pydot.Dot()
  2. create_dot(root)
  3. viewPydot(dot)

output_18_0.png-38.5kB

不是leaf并且子节点的leaf==true

  1. def pruning(root, threshold):
  2. if root.leaf is True:
  3. return;
  4. print(root.H, root.leaf)
  5. for p in root.children.values():
  6. pruning(p, threshold)
  7. if root.children[0].leaf is True:
  8. t = 0.
  9. l = len(root.children)
  10. for p in root.children.values():
  11. t = t + p.H
  12. if t - root.H < (l-1) * threshold:
  13. for i in range(l):
  14. root.children[i] = None
  15. root.leaf = True
  1. pruning(root, .0)
0.9974937421855691 False
0.764200977141265 False
0.7219256790976065 False
  1. dot = pydot.Dot()
  2. create_dot(root)
  3. viewPydot(dot)

output_22_0.png-37kB


CART

需要在当前节点对应的数据集下计算所有属性和属性值对应的Gini指数,对最大Gini指数的进行划分,小于阈值跳出

  1. class TreeNode(object):
  2. def __init__(self, prop, value, label):
  3. self.prop = prop
  4. self.value = value
  5. self.label = label
  6. self.lc = None
  7. self.rc = None
  8. self.leaf = False
  9. self.N_leaf = 0
  10. self.H = 0.
  11. self.g = 0.
  1. def CART_C(X, Y, threshold, purning_thre):
  2. N, M = X.shape
  3. SX = np.max(X, axis=0) + 1
  4. K = np.max(Y) + 1
  5. global visited
  6. visited = []
  7. for m in range(M):
  8. visited.append(np.zeros(SX[m], dtype=np.bool))
  9. def build_tree(cX, cY):
  10. global visited
  11. PX_Y = []
  12. for m in range(M):
  13. PXi = np.zeros((SX[m], K))
  14. for i in range(SX[m]):
  15. index = (cX[:, m] == i)
  16. iY = cY[index]
  17. l = len(iY)
  18. for k in range(K):
  19. PXi[i, k] = len(np.where(iY==k)[0]) / (l+epsilon)
  20. if l == 0:
  21. PXi[i, k] = 1
  22. PX_Y.append(PXi)
  23. ginis = []
  24. mginis = np.zeros(M)
  25. mindex = np.zeros(M, dtype=np.int)
  26. for m in range(M):
  27. PXi = PX_Y[m]
  28. gs = 1 - np.sum(np.power(PXi, 2), axis=1)
  29. gs[visited[m]] = -np.inf
  30. mm = np.argmax(gs)
  31. mindex[m] = mm
  32. mginis[m] = gs[mm]
  33. mpi = np.argmax(mginis)
  34. mpii = mindex[mpi]
  35. mmg = mginis[mpi]
  36. t = TreeNode(prop=mpi, value=mpii, label=np.argmax(PX_Y[mpi][mpii,:]))
  37. t.H = H(PX_Y[mpi][mpii,:])
  38. if mmg <= (threshold+epsilon):
  39. t.leaf = True
  40. t.N_leaf = 1
  41. t.g = 0.
  42. return t
  43. t.leaf = False
  44. visited[mpi][mpii] = True
  45. index = (cX[:, mpi] == mpii)
  46. t.lc = build_tree(cX[index], cY[index])
  47. t.rc = build_tree(cX[np.logical_not(index)], cY[np.logical_not(index)])
  48. t.N_leaf = t.lc.N_leaf + t.rc.N_leaf
  49. visited[mpi][mpii] = False
  50. return t
  51. root = build_tree(X, Y)
  52. return root
  1. root = CART_C(X, Y, 0.45, 0.1)
  1. dot = pydot.Dot()
  2. global level
  3. level = 1
  4. def create_dot(p):
  5. global level
  6. p_name = "%s # %s, %s, %s, %.3f" % (p.__str__(), str(p.prop), str(p.value), str(p.label), p.H)
  7. dot.add_node(pydot.Node(name=p_name))
  8. if p.leaf is True:
  9. return
  10. for c in [p.lc, p.rc]:
  11. level = level + 1
  12. c_name = "%s # %s, %s, %s, %.3f" % (c.__str__(), str(c.prop), str(c.value), str(c.label), c.H)
  13. dot.add_edge(pydot.Edge(dst=c_name, src=p_name))
  14. create_dot(c)
  15. level = level -1
  16. create_dot(root)
  17. viewPydot(dot)

output_28_0.png-301kB

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