6.5 KiB
6.5 KiB
| title | localeTitle |
|---|---|
| Support Vector Machine | 支持向量机 |
支持向量机
支持向量机（SVM）是由分离超平面正式定义的判别分类器。换句话说，给定标记的训练数据（监督学习），算法输出最佳超平面，其对新示例进行分类。它通过最小化超平面附近的数据点之间的边距来实现这一点。
SVM成本函数试图用分段线性逼近逻辑函数。该ML算法用于分类问题，并且是监督学习算法子集的一部分。
成本函数
成本函数用于训练SVM。通过最小化J（theta）的值，我们可以确保SVM尽可能准确。在等式中，函数cost1和cost0指的是y = 1的示例的成本和y = 0的示例的成本。 SVM的成本由内核（相似性）函数决定。
仁
多项式特征可能在计算上很昂贵，并且可能会减慢大型数据集的运行时间。 不要添加更多的多项式特征，而是添加“地标”，用它来测试其他数据点的接近程度。 训练集的每个成员都是一个里程碑。 内核是“相似度函数”，用于衡量输入与特定标记的接近程度。
大边距分类器
SVM将找到以最大边距分割数据的线（或更一般情况下的超平面）。 虽然异常值可能会使线条向一个方向摆动，但足够小的C值将强制执行正则化。 这个新的正则化与1 / \ lambda的作用相同，如线性和逻辑回归中所见，但在这里我们修改成本组件。
更多信息：
Andrew Ng的ML课程 独立视频讲座 维基百科上的SVM
以下是为python中的SVM训练，预测和查找准确性而编写的代码。这是使用Numpy完成的，但是，我们也可以在函数调用中使用scikit-learn编写。
import numpy as np
class Svm (object):
"""" Svm classifier """
def __init__ (self, inputDim, outputDim):
self.W = None
# - Generate a random svm weight matrix to compute loss #
# with standard normal distribution and Standard deviation = 0.01. #
sigma =0.01
self.W = sigma * np.random.randn(inputDim,outputDim)
def calLoss (self, x, y, reg):
"""
Svm loss function
D: Input dimension.
C: Number of Classes.
N: Number of example.
Inputs:
- x: A numpy array of shape (batchSize, D).
- y: A numpy array of shape (N,) where value < C.
- reg: (float) regularization strength.
Returns a tuple of:
- loss as single float.
- gradient with respect to weights self.W (dW) with the same shape of self.W.
"""
loss = 0.0
dW = np.zeros_like(self.W)
# - Compute the svm loss and store to loss variable. #
# - Compute gradient and store to dW variable. #
# - Use L2 regularization #
#Calculating score matrix
s = x.dot(self.W)
#Score with yi
s_yi = s[np.arange(x.shape[0]),y]
#finding the delta
delta = s- s_yi[:,np.newaxis]+1
#loss for samples
loss_i = np.maximum(0,delta)
loss_i[np.arange(x.shape[0]),y]=0
loss = np.sum(loss_i)/x.shape[0]
#Loss with regularization
loss += reg*np.sum(self.W*self.W)
#Calculating ds
ds = np.zeros_like(delta)
ds[delta > 0] = 1
ds[np.arange(x.shape[0]),y] = 0
ds[np.arange(x.shape[0]),y] = -np.sum(ds, axis=1)
dW = (1/x.shape[0]) * (xT).dot(ds)
dW = dW + (2* reg* self.W)
return loss, dW
def train (self, x, y, lr=1e-3, reg=1e-5, iter=100, batchSize=200, verbose=False):
"""
Train this Svm classifier using stochastic gradient descent.
D: Input dimension.
C: Number of Classes.
N: Number of example.
Inputs:
- x: training data of shape (N, D)
- y: output data of shape (N, ) where value < C
- lr: (float) learning rate for optimization.
- reg: (float) regularization strength.
- iter: (integer) total number of iterations.
- batchSize: (integer) number of example in each batch running.
- verbose: (boolean) Print log of loss and training accuracy.
Outputs:
A list containing the value of the loss at each training iteration.
"""
# Run stochastic gradient descent to optimize W.
lossHistory = []
for i in range(iter):
xBatch = None
yBatch = None
# - Sample batchSize from training data and save to xBatch and yBatch #
# - After sampling xBatch should have shape (batchSize, D) #
# yBatch (batchSize, ) #
# - Use that sample for gradient decent optimization. #
# - Update the weights using the gradient and the learning rate. #
#creating batch
num_train = np.random.choice(x.shape[0], batchSize)
xBatch = x[num_train]
yBatch = y[num_train]
loss, dW = self.calLoss(xBatch,yBatch,reg)
self.W= self.W - lr * dW
lossHistory.append(loss)
# Print loss for every 100 iterations
if verbose and i % 100 == 0 and len(lossHistory) is not 0:
print ('Loop {0} loss {1}'.format(i, lossHistory[i]))
return lossHistory
def predict (self, x,):
"""
Predict the y output.
Inputs:
- x: training data of shape (N, D)
Returns:
- yPred: output data of shape (N, ) where value < C
"""
yPred = np.zeros(x.shape[0])
# - Store the predict output in yPred #
s = x.dot(self.W)
yPred = np.argmax(s, axis=1)
return yPred
def calAccuracy (self, x, y):
acc = 0
# - Calculate accuracy of the predict value and store to acc variable
yPred = self.predict(x)
acc = np.mean(y == yPred)*100
return acc