在現實生活中，很多問題并非“非黑即白”的問題，而是可以分為多個不同的類別，這些問題可以視為多分類學習任務，
多分類學習任務可以基于二分類演算法進行推廣后解決，有些二分類演算法可以直接推廣用于解決多分類問題，但是在更多情形下需要基于一些基本策略來對二分類演算法進行處理從而更有效的解決多分類問題，
最經典和最基礎的拆分策略包括兩種：“一對一”（One vs. One，簡稱OvO）和“一對其余”（“One vs. Rest”，簡稱OvR），本實訓專案主要介紹這兩類多分類處理策
OvO多分類策略
import numpy as np
# 邏輯回歸
class tiny_logistic_regression(object):
def __init__(self):
#W
self.coef_ = None
#b
self.intercept_ = None
#所有的W和b
self._theta = None
#01到標簽的映射
self.label_map = {}
def _sigmoid(self, x):
return 1. / (1. + np.exp(-x))
#訓練
def fit(self, train_datas, train_labels, learning_rate=1e-4, n_iters=1e3):
#loss
def J(theta, X_b, y):
y_hat = self._sigmoid(X_b.dot(theta))
try:
return -np.sum(y*np.log(y_hat)+(1-y)*np.log(1-y_hat)) / len(y)
except:
return float('inf')
# 算theta對loss的偏導
def dJ(theta, X_b, y):
return X_b.T.dot(self._sigmoid(X_b.dot(theta)) - y) / len(y)
# 批量梯度下降
def gradient_descent(X_b, y, initial_theta, leraning_rate, n_iters=1e2, epsilon=1e-6):
theta = initial_theta
cur_iter = 0
while cur_iter < n_iters:
gradient = dJ(theta, X_b, y)
last_theta = theta
theta = theta - leraning_rate * gradient
if (abs(J(theta, X_b, y) - J(last_theta, X_b, y)) < epsilon):
break
cur_iter += 1
return theta
unique_labels = list(set(train_labels))
labels = train_labels.copy()
self.label_map[0] = unique_labels[0]
labels[train_labels == unique_labels[0]] = 0
self.label_map[1] = unique_labels[1]
labels[train_labels == unique_labels[1]] = 1
X_b = np.hstack([np.ones((len(train_datas), 1)), train_datas])
initial_theta = np.zeros(X_b.shape[1])
self._theta = gradient_descent(X_b, labels, initial_theta, learning_rate, n_iters)
self.intercept_ = self._theta[0]
self.coef_ = self._theta[1:]
return self
#預測概率分布
def predict_proba(self, X):
X_b = np.hstack([np.ones((len(X), 1)), X])
return self._sigmoid(X_b.dot(self._theta))
#預測
def predict(self, X):
proba = self.predict_proba(X)
result = np.array(proba >= 0.5, dtype='int')
for i in range(len(result)):
if result[i] == 0:
result[i] = self.label_map[0]
else:
result[i] = self.label_map[1]
return result
class OvO(object):
def __init__(self):
# 用于保存訓練時各種模型的list
self.models = []
def fit(self, train_datas, train_labels):
'''
OvO的訓練階段，將模型保存到self.models中
:param train_datas: 訓練集資料，型別為ndarray
:param train_labels: 訓練集標簽，型別為ndarray，shape為(-1,)
:return:None
'''
#********* Begin *********#
unique_labels = list(set(train_labels))
for i in range(len(unique_labels)):
for j in range(i+1, len(unique_labels)):
datas = train_datas[(train_labels == unique_labels[i]) | (train_labels == unique_labels[j])]
labels = train_labels[(train_labels == unique_labels[i]) | (train_labels == unique_labels[j])]
lr = tiny_logistic_regression()
lr.fit(datas, labels)
self.models.append(lr)
#********* End *********#
def predict(self, test_datas):
'''
OvO的預測階段
:param test_datas:測驗集資料，型別為ndarray
:return:預測結果，型別為ndarray
'''
#********* Begin *********#
def _predict(models, test_data):
# 變形
test_data = np.reshape(test_data, (1, -1))
vote = {}
# 計票
for model in models:
pred = model.predict(test_data)[0]
if pred not in vote:
vote[pred] = 1
else:
vote[pred] += 1
vote = sorted(vote.items(), key=lambda x: x[1], reverse=True)
return vote[0][0]
predict = []
for data in test_datas:
predict.append(_predict(self.models, data))
return np.array(predict)
#********* End *********#
OvR多分類策略
import numpy as np
# 邏輯回歸
class tiny_logistic_regression(object):
def __init__(self):
#W
self.coef_ = None
#b
self.intercept_ = None
#所有的W和b
self._theta = None
#01到標簽的映射
self.label_map = {}
def _sigmoid(self, x):
return 1. / (1. + np.exp(-x))
#訓練
def fit(self, train_datas, train_labels, learning_rate=1e-4, n_iters=1e3):
#loss
def J(theta, X_b, y):
y_hat = self._sigmoid(X_b.dot(theta))
try:
return -np.sum(y*np.log(y_hat)+(1-y)*np.log(1-y_hat)) / len(y)
except:
return float('inf')
# 算theta對loss的偏導
def dJ(theta, X_b, y):
return X_b.T.dot(self._sigmoid(X_b.dot(theta)) - y) / len(y)
# 批量梯度下降
def gradient_descent(X_b, y, initial_theta, leraning_rate, n_iters=1e2, epsilon=1e-6):
theta = initial_theta
cur_iter = 0
while cur_iter < n_iters:
gradient = dJ(theta, X_b, y)
last_theta = theta
theta = theta - leraning_rate * gradient
if (abs(J(theta, X_b, y) - J(last_theta, X_b, y)) < epsilon):
break
cur_iter += 1
return theta
X_b = np.hstack([np.ones((len(train_datas), 1)), train_datas])
initial_theta = np.zeros(X_b.shape[1])
self._theta = gradient_descent(X_b, train_labels, initial_theta, learning_rate, n_iters)
self.intercept_ = self._theta[0]
self.coef_ = self._theta[1:]
return self
#預測X中每個樣本label為1的概率
def predict_proba(self, X):
X_b = np.hstack([np.ones((len(X), 1)), X])
return self._sigmoid(X_b.dot(self._theta))
#預測
def predict(self, X):
proba = self.predict_proba(X)
result = np.array(proba >= 0.5, dtype='int')
return result
class OvR(object):
def __init__(self):
# 用于保存訓練時各種模型的list
self.models = []
# 用于保存models中對應的正例的真實標簽
# 例如第1個模型的正例是2，則real_label[0]=2
self.real_label = []
def fit(self, train_datas, train_labels):
'''
OvO的訓練階段，將模型保存到self.models中
:param train_datas: 訓練集資料，型別為ndarray
:param train_labels: 訓練集標簽，型別為ndarray，shape為(-1,)
:return:None
'''
#********* Begin *********#
unique_labels = list(set(train_labels))
self.real_label = unique_labels
# 劃分并訓練
for i in range(len(unique_labels)):
# label變為0，1
label_1 = list(np.ones(len(train_labels[train_labels == unique_labels[i]])))
label_0 = list(np.zeros(len(train_labels[train_labels != unique_labels[i]])))
label_1.extend(label_0)
labels = np.array(label_1)
data_1 = list(train_datas[train_labels == unique_labels[i]])
data_0 = list(train_datas[train_labels != unique_labels[i]])
data_1.extend(data_0)
datas = np.array(data_1)
lr = tiny_logistic_regression()
lr.fit(datas, labels)
self.models.append(lr)
#********* End *********#
def predict(self, test_datas):
'''
OvO的預測階段
:param test_datas:測驗集資料，型別為ndarray
:return:預測結果，型別為ndarray
'''
#********* Begin *********#
def _predict(models, real_labels, test_data):
# 變形
test_data = np.reshape(test_data, (1, -1))
max_prob = 0
max_prob_idx = 0
for i, model in enumerate(models):
pred_prob = model.predict_proba(test_data)[0]
if pred_prob > max_prob:
max_prob = pred_prob
max_prob_idx = i
return max_prob_idx
predict = []
for data in test_datas:
predict.append(self.real_label[_predict(self.models, self.real_label, data)])
return np.array(predict)
#********* End *********#
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