<jupyter_start><jupyter_text>Logistic Regression不是真正用来Regression，而是用来classification和Linear Regression既有区别也有联系 [笔记](https://blog.csdn.net/qq_29598161/article/details/107477883)---Logistic Regression的模型f需要经过sigmoid，以此符合概率在0~1的分布(其w和b，实际上是由概率公式推导得到的)$w = \left(\mu^{1}-\mu^{2}\right)^{T} \Sigma^{-1}$$b = -\frac{1}{2}\left(\mu^{1}\right)^{T} \Sigma^{-1} \mu^{1}+\frac{1}{2}\left(\mu^{2}\right)^{T} \Sigma^{-1} \mu^{2}+\ln \frac{N_{1}}{N_{2}}$和Linear Regression中w和b代表的含义不一样，---损失不一样，Logistic Regression是交叉熵，Linear Regression一般使用Square Error---Logistic Regression是Discriminative model，和Porbabilistic generative model区分是Discriminative model 通过梯度下降求w和bPorbabilistic generative model通过求分布得到w和b  Logistic Regression(Discriminative Model)<jupyter_code># import pandas as pd
import numpy as np
np.random.seed(0)
X_train_fpath = './data/X_train.csv'
Y_train_fpath = './data/Y_train.csv'
X_test_fpath = './data/X_test.csv'
output_fpath = './output_{}.csv'<jupyter_output><empty_output><jupyter_text>数据预处理 读取数据转成numpy<jupyter_code>with open(X_train_fpath) as f:
    next(f)
    X_train = np.array([line.strip('\n').split(',')[1:] for line in f], dtype = float)
with open(Y_train_fpath) as f:
    next(f)
    Y_train = np.array([line.strip('\n').split(',')[1] for line in f], dtype = float)
with open(X_test_fpath) as f:
    next(f)
    X_test = np.array([line.strip('\n').split(',')[1:] for line in f], dtype = float)<jupyter_output><empty_output><jupyter_text>归一化<jupyter_code>def _normalize(X, train = True, specified_column = None, X_mean = None, X_std = None):
    # This function normalizes specific columns of X.
    # The mean and standard variance of training data will be reused when processing testing data.
    #
    # Arguments:
    #     X: data to be processed
    #     train: 'True' when processing training data, 'False' for testing data
    #     specific_column: indexes of the columns that will be normalized. If 'None', all columns
    #         will be normalized.
    #     X_mean: mean value of training data, used when train = 'False'
    #     X_std: standard deviation of training data, used when train = 'False'
    # Outputs:
    #     X: normalized data
    #     X_mean: computed mean value of training data
    #     X_std: computed standard deviation of training data

    if specified_column == None:
        specified_column = np.arange(X.shape[1])
    if train:
        X_mean = np.mean(X[:, specified_column] ,0).reshape(1, -1)
        X_std  = np.std(X[:, specified_column], 0).reshape(1, -1)

    X[:,specified_column] = (X[:, specified_column] - X_mean) / (X_std + 1e-8)
     
    return X, X_mean, X_std

def _train_dev_split(X, Y, dev_ratio = 0.25):
    # This function spilts data into training set and development set.
    train_size = int(len(X) * (1 - dev_ratio))
    return X[:train_size], Y[:train_size], X[train_size:], Y[train_size:]

# Normalize training and testing data
X_train, X_mean, X_std = _normalize(X_train, train = True)
X_test, _, _= _normalize(X_test, train = False, specified_column = None, X_mean = X_mean, X_std = X_std)
    
# Split data into training set and development set
dev_ratio = 0.1
X_train, Y_train, X_dev, Y_dev = _train_dev_split(X_train, Y_train, dev_ratio = dev_ratio)

train_size = X_train.shape[0]
dev_size = X_dev.shape[0]
test_size = X_test.shape[0]
data_dim = X_train.shape[1]
print('Size of training set: {}'.format(train_size))
print('Size of development set: {}'.format(dev_size))
print('Size of testing set: {}'.format(test_size))
print('Dimension of data: {}'.format(data_dim))<jupyter_output>Size of training set: 48830
Size of development set: 5426
Size of testing set: 27622
Dimension of data: 510<jupyter_text>共用的函数<jupyter_code>def _shuffle(X, Y):
    # This function shuffles two equal-length list/array, X and Y, together.
    randomize = np.arange(len(X))
    np.random.shuffle(randomize)
    return (X[randomize], Y[randomize])

def _sigmoid(z):
    # Sigmoid function can be used to calculate probability.
    # To avoid overflow, minimum/maximum output value is set.
    return np.clip(1 / (1.0 + np.exp(-z)), 1e-8, 1 - (1e-8))

def _predict(X, w, b):
    # This function returns a truth value prediction for each row of X 
    # by rounding the result of logistic regression function.
    return np.round(_f(X, w, b)).astype(np.int)
    
def _accuracy(Y_pred, Y_label):
    # This function calculates prediction accuracy
    acc = 1 - np.mean(np.abs(Y_pred - Y_label))
    return acc<jupyter_output><empty_output><jupyter_text>回归模型$f_{w, b}(x)=\sum_{i} w_{i} x_{i}+b$<jupyter_code>def _f(X, w, b):
    # This is the logistic regression function, parameterized by w and b
    #
    # Arguements:
    #     X: input data, shape = [batch_size, data_dimension]
    #     w: weight vector, shape = [data_dimension, ]
    #     b: bias, scalar
    # Output:
    #     predicted probability of each row of X being positively labeled, shape = [batch_size, ]
    return _sigmoid(np.matmul(X, w) + b)<jupyter_output><empty_output><jupyter_text>交叉熵$C\left(f\left(x^{n}\right), \hat{y}^{n}\right)=-\left[\hat{y}^{n} \ln f\left(x^{n}\right)+\left(1-\hat{y}^{n}\right) \ln \left(1-f\left(x^{n}\right)\right)\right]$<jupyter_code>def _cross_entropy_loss(y_pred, Y_label):
    # This function computes the cross entropy.
    #
    # Arguements:
    #     y_pred: probabilistic predictions, float vector
    #     Y_label: ground truth labels, bool vector
    # Output:
    #     cross entropy, scalar
    cross_entropy = -np.dot(Y_label, np.log(y_pred)) - np.dot((1 - Y_label), np.log(1 - y_pred))
    return cross_entropy<jupyter_output><empty_output><jupyter_text>梯度更新$w_{i} \leftarrow w_{i}-\eta \sum_{n}\left(\hat{y}^{n}-f_{w, b}\left(x^{n}\right)\right) x_{i}^{n}$<jupyter_code>def _gradient(X, Y_label, w, b):
    # This function computes the gradient of cross entropy loss with respect to weight w and bias b.
    y_pred = _f(X, w, b)
    pred_error = Y_label - y_pred
    w_grad = -np.sum(pred_error * X.T, 1)
    b_grad = -np.sum(pred_error)
    return w_grad, b_grad<jupyter_output><empty_output><jupyter_text>Training<jupyter_code># Zero initialization for weights ans bias
w = np.zeros((data_dim,)) 
b = np.zeros((1,))

# Some parameters for training    
max_iter = 100
batch_size = 8
learning_rate = 0.2

# Keep the loss and accuracy at every iteration for plotting
train_loss = []
dev_loss = []
train_acc = []
dev_acc = []

# Calcuate the number of parameter updates
step = 1

# Iterative training
for epoch in range(max_iter):
    # Random shuffle at the begging of each epoch
    X_train, Y_train = _shuffle(X_train, Y_train)
        
    # Mini-batch training
    for idx in range(int(np.floor(train_size / batch_size))):
        X = X_train[idx*batch_size:(idx+1)*batch_size]
        Y = Y_train[idx*batch_size:(idx+1)*batch_size]

        # Compute the gradient
        w_grad, b_grad = _gradient(X, Y, w, b)
            
        # gradient descent update
        # learning rate decay with time
        w = w - learning_rate/np.sqrt(step) * w_grad
        b = b - learning_rate/np.sqrt(step) * b_grad

        step = step + 1
            
    # Compute loss and accuracy of training set and development set
    y_train_pred = _f(X_train, w, b)
    Y_train_pred = np.round(y_train_pred)
    train_acc.append(_accuracy(Y_train_pred, Y_train))
    train_loss.append(_cross_entropy_loss(y_train_pred, Y_train) / train_size)

    y_dev_pred = _f(X_dev, w, b)
    Y_dev_pred = np.round(y_dev_pred)
    dev_acc.append(_accuracy(Y_dev_pred, Y_dev))
    dev_loss.append(_cross_entropy_loss(y_dev_pred, Y_dev) / dev_size)
    
#     print('Training loss: {}'.format(train_loss[-1]))
#     print('Development loss: {}'.format(dev_loss[-1]))
#     print('Training accuracy: {}'.format(train_acc[-1]))
#     print('Development accuracy: {}'.format(dev_acc[-1]))
#     print('------')

print('Training loss: {}'.format(train_loss[-1]))
print('Development loss: {}'.format(dev_loss[-1]))
print('Training accuracy: {}'.format(train_acc[-1]))
print('Development accuracy: {}'.format(dev_acc[-1]))
import matplotlib.pyplot as plt

# Loss curve
plt.plot(train_loss)
plt.plot(dev_loss)
plt.title('Loss')
plt.legend(['train', 'dev'])
plt.savefig('loss.png')
plt.show()

# Accuracy curve
plt.plot(train_acc)
plt.plot(dev_acc)
plt.title('Accuracy')
plt.legend(['train', 'dev'])
plt.savefig('acc.png')
plt.show()<jupyter_output><empty_output><jupyter_text>Porbabilistic generative model  数据预处理 和Logistic Regression一样，直接调用  计算均值和方差 求分布，这里没有用最大似然，因为假设分布是正态分布，只需求均值和方差就知道分布了下面分别求2类的分布<jupyter_code># Compute in-class mean
X_train_0 = np.array([x for x, y in zip(X_train, Y_train) if y == 0])
X_train_1 = np.array([x for x, y in zip(X_train, Y_train) if y == 1])

mean_0 = np.mean(X_train_0, axis = 0)
mean_1 = np.mean(X_train_1, axis = 0)  

# Compute in-class covariance
cov_0 = np.zeros((data_dim, data_dim))
cov_1 = np.zeros((data_dim, data_dim))

for x in X_train_0:
    cov_0 += np.dot(np.transpose([x - mean_0]), [x - mean_0]) / X_train_0.shape[0]
for x in X_train_1:
    cov_1 += np.dot(np.transpose([x - mean_1]), [x - mean_1]) / X_train_1.shape[0]<jupyter_output><empty_output><jupyter_text>有一个可以提高准确率和减少参数的方法就是方差Σ，是两个分布方差的均值<jupyter_code># Shared covariance is taken as a weighted average of individual in-class covariance.
cov = (cov_0 * X_train_0.shape[0] + cov_1 * X_train_1.shape[0]) / (X_train_0.shape[0] + X_train_1.shape[0])<jupyter_output><empty_output><jupyter_text>使用SVD求逆<jupyter_code># Compute inverse of covariance matrix.
# Since covariance matrix may be nearly singular, np.linalg.inv() may give a large numerical error.
# Via SVD decomposition, one can get matrix inverse efficiently and accurately.
u, s, v = np.linalg.svd(cov, full_matrices=False)
inv = np.matmul(v.T * 1 / s, u.T)<jupyter_output><empty_output><jupyter_text>$w = \left(\mu^{1}-\mu^{2}\right)^{T} \Sigma^{-1}$$b = -\frac{1}{2}\left(\mu^{1}\right)^{T} \Sigma^{-1} \mu^{1}+\frac{1}{2}\left(\mu^{2}\right)^{T} \Sigma^{-1} \mu^{2}+\ln \frac{N_{1}}{N_{2}}$<jupyter_code># Directly compute weights and bias
w = np.dot(inv, mean_0 - mean_1)
b =  (-0.5) * np.dot(mean_0, np.dot(inv, mean_0)) + 0.5 * np.dot(mean_1, np.dot(inv, mean_1))\
    + np.log(float(X_train_0.shape[0]) / X_train_1.shape[0]) 

# Compute accuracy on training set
Y_train_pred = 1 - _predict(X_train, w, b)
print('Training accuracy: {}'.format(_accuracy(Y_train_pred, Y_train)))
# Predict testing labels
predictions = 1 - _predict(X_test, w, b)
with open(output_fpath.format('generative'), 'w') as f:
    f.write('id,label\n')
    for i, label in  enumerate(predictions):
        f.write('{},{}\n'.format(i, label))

# Print out the most significant weights
ind = np.argsort(np.abs(w))[::-1]
with open(X_test_fpath) as f:
    content = f.readline().strip('\n').split(',')
features = np.array(content)
for i in ind[0:10]:
    print(features[i], w[i])<jupyter_output>5th or 6th grade -3.2322260765358806
 21 -3.0626443377695978
 12 -2.511015271767974
 Bachelors degree(BA AB BS) -2.460119580850005
 Some college but no degree -2.0836510490626097
 Panama -1.9747587339952588
 38 -1.6180884456261992
 Other Rel <18 never marr child of subfamily RP 1.5947637981444132
 Holand-Netherlands -1.5784552804543637
 Local government 1.5688253603875637
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