【作业4】林轩田机器学习基石

作业四的代码题目主要是基于ridge regression来做的，并加上了各种cross-validation的情况。

由于ridge regression是有analytic solution，所以直接求逆矩阵就OK了，过程并不复杂。只有在做cross-validation的时候遇上了些问题。

#encoding=utf8
import sys
import numpy as np
import math
from random import *

# read input data ( train or test )
def read_input_data(path):
    x = []
    y = []
    for line in open(path).readlines():
        items = line.strip().split(‘ ‘)
        tmp_x = []
        for i in range(0,len(items)-1): tmp_x.append(float(items[i]))
        x.append(tmp_x)
        y.append(float(items[-1]))
    return np.array(x),np.array(y)

def calculate_W_rigde_regression(x,y,LAMBDA):
    Z_v = np.linalg.inv(np.dot(x.transpose(),x)+LAMBDA*np.eye(x.shape[1]))
    return  np.dot(np.dot(Z_v,x.transpose()),y)

# test result
def calculate_E(w, x, y):
    scores = np.dot(w, x.transpose())
    predicts = np.where(scores>=0,1.0,-1.0)
    Eout = sum(predicts!=y)
    return (Eout*1.0) / predicts.shape[0]

if __name__ == ‘__main__‘:

    # prepare train and test data
    x,y = read_input_data("train.dat")
    x = np.hstack((np.ones(x.shape[0]).reshape(-1,1),x))
    test_x,test_y = read_input_data("test.dat")
    test_x = np.hstack((np.ones(test_x.shape[0]).reshape(-1,1),test_x))
    # lambda
    LAMBDA_set = [ i for i in range(2,-11,-1) ]

    ## Q13~Q15
    min_Ein = 1
    min_Eout = 1
    target_lambda = 2
    for LAMBDA in LAMBDA_set:
        # calculate ridge regression W
        W = calculate_W_rigde_regression(x,y,pow(10, LAMBDA))
        Ein = calculate_E(W, x, y)
        Eout = calculate_E(W, test_x, test_y)
        # update Ein Eout lambda
        if Eout<min_Eout:
            target_lambda = LAMBDA
            min_Ein = Ein
            min_Eout = Eout
    #print min_Ein
    #print min_Eout
    #print target_lambda

    ## Q16~Q18
    min_Etrain = 1
    min_Eval = 1
    min_Eout = 1
    target_lambda = 2
    split = 120
    for LAMBDA in LAMBDA_set:
        # calculate ridge regression W
        W = calculate_W_rigde_regression(x[:split], y[:split], pow(10, LAMBDA))
        Etrain = calculate_E(W, x[:split], y[:split])
        Eval = calculate_E(W, x[split:], y[split:])
        Eout = calculate_E(W, test_x, test_y)
        # update Ein Eout lambda
        if Eval<min_Eval:
            target_lambda = LAMBDA
            min_Etrain = Etrain
            min_Eval = Eval
            min_Eout = Eout
    #print min_Etrain
    #print min_Eval
    #print min_Eout
    #print target_lambda

    W = calculate_W_rigde_regression(x,y,pow(10,target_lambda))
    optimal_Ein = calculate_E(W,x,y)
    optimal_Eout = calculate_E(W,test_x,test_y)
    #print optimal_Ein
    #print optimal_Eout

    ## Q19~Q20
    min_Ecv = 1
    target_lambda = 2
    V = 5
    V_range = []
    for i in range(0,V): V_range.append([i*(x.shape[0]/V),(i+1)*(x.shape[0]/V)])

    for LAMBDA in LAMBDA_set:
        total_Ecv = 0
        for i in range(0,V):
            # train x, y
            train_x = []
            train_y = []
            for j in range(0,V):
                if j!=i :
                    train_x.extend( x[range(V_range[j][0],V_range[j][1])].tolist() )
                    train_y.extend( y[range(V_range[j][0],V_range[j][1])].tolist() )
            train_x = np.array(train_x)
            train_y = np.array(train_y)
            # test x, y
            test_x = x[range(V_range[i][0],V_range[i][1])]
            test_y = y[range(V_range[i][0],V_range[i][1])]
            W = calculate_W_rigde_regression(train_x, train_y, pow(10,LAMBDA))
            Ecv = calculate_E(W, test_x, test_y)
            total_Ecv = total_Ecv + Ecv
        print "total Ecv:" + str(total_Ecv)
        if min_Ecv>(total_Ecv*1.0)/V:
            min_Ecv = (total_Ecv*1.0)/V
            target_lambda = LAMBDA
    print min_Ecv
    print target_lambda

    W = calculate_W_rigde_regression(x, y, pow(10,target_lambda))
    Ein = calculate_E(W, x, y)
    test_x,test_y = read_input_data("test.dat")
    test_x = np.hstack((np.ones(test_x.shape[0]).reshape(-1,1),test_x))
    Eout = calculate_E(W, test_x, test_y)
    print Ein
    print Eout 

这里还留有一个疑问：

在讲Linear Regression的时候有：

X‘X这个矩阵当时说，可能是可逆的，也可能不是？但是肯定是实对称阵，跟正定有什么关系？

这里又说到这个矩阵了Z‘Z是半正定的矩阵，配上单位阵就一定是正定的了？这个是怎么得来的。