Question

Você pode me mostrar como o OLS pode ser usado para classificação e os problemas causados por outliers?

Answer 1

O código abaixo desenvolvido em Python explora a classificação usando OLS e os problemas causados por outliers. Note particularmente que com a presença de outliers a superfície discriminante começa a errar (como mostra a figura abaixo). Mais detalhes sobre o problema podem ser encontrados na seção 4.1.3 do livro Pattern Recognition and Machine Learning - Christopher Bishop

import matplotlib.pyplot as plt
import numpy as np
from sklearn import linear_model
np.random.seed(50)

fig=plt.figure(num=None, figsize=(8, 4), dpi=80, facecolor='w', edgecolor='k')


def plot_classificator(beta0,beta1,beta2,x1):
    minx1=np.min(x1)
    maxx1=np.max(x1)    

    minx2=(beta1*minx1)/(-beta2)
    maxx2=(beta1*maxx1)/(-beta2)

    plt.plot([minx1,maxx1],[minx2,maxx2],'k-')  


def data_generation1(n,beta0,beta1,beta2):
    x1=np.random.normal(0,0.64,n) # Primeiro regressor
    x2=np.random.normal(0,0.36,n) # Segundo regressor
    z=beta0*np.ones(n)+beta1*x1+beta2*x2 # Variavel latente
    y=np.empty([n])
    for i in range(n):
        if(z[i]>beta0): # In order to ensure that in average each class has the same number of points
            y[i]=1
        else:
            y[i]=0
    return y,x1,x2

def data_generation2(n,beta0,beta1,beta2,x1,x2):
    x1[95]=2.29
    x2[95]=-0.4   
    x1[96]=2.3
    x2[96]=-0.5   
    x1[97]=2.35
    x2[97]=-0.45   
    x1[98]=2.38
    x2[98]=-0.55   
    x1[99]=2.4
    x2[99]=-0.6   

    z=beta0*np.ones(n)+beta1*x1+beta2*x2 # Variavel latente
    y=np.empty([n])
    for i in range(n):
        if(z[i]>beta0): # In order to ensure that in average each class has the same number of points
            y[i]=1
        else:
            y[i]=0
    return y,x1,x2

def ols_estimator(x,y):
    linModel= linear_model.LinearRegression(fit_intercept=False)
    linModel.fit(x,y)
    return linModel.coef_

if __name__ == '__main__':
    n=100 # numero de observacoes
    beta0=2
    beta1=-1.5
    beta2=-1

    # Case 1

    ax=plt.subplot(121)

    [y,x1,x2]=data_generation1(n,beta0,beta1,beta2)        

    for i in range(n):
        if(y[i]==1):
            plt.plot(x1[i],x2[i],u'ro',markersize=5, markeredgewidth=0)
        else:
            plt.plot(x1[i],x2[i],u'bo',markersize=5, markeredgewidth=0)   
    ax.set_xlabel('$x_1$')
    ax.set_ylabel('$x_2$')          
    ax.set_title('No outliers')

    x=np.stack((np.ones(len(x1)).T, x1.T,x2.T),axis=1)
    coefficients=ols_estimator(x,y)
    print coefficients

    hatBeta0=coefficients[0]
    hatBeta1=coefficients[1]
    hatBeta2=coefficients[2]
    plot_classificator(hatBeta0,hatBeta1,hatBeta2,x1)

    # Case 2

    ax=plt.subplot(122)

    [y,x1,x2]=data_generation2(n,beta0,beta1,beta2,x1,x2)        

    for i in range(n):
        if(y[i]==1):
            plt.plot(x1[i],x2[i],u'ro',markersize=5, markeredgewidth=0)
        else:
            plt.plot(x1[i],x2[i],u'bo',markersize=5, markeredgewidth=0)   
    ax.set_xlabel('$x_1$')
    ax.set_ylabel('$x_2$')          
    ax.set_title('With outliers')

    x=np.stack((np.ones(len(x1)).T, x1.T,x2.T),axis=1)
    coefficients=ols_estimator(x,y)
    print coefficients

    hatBeta0=coefficients[0]
    hatBeta1=coefficients[1]
    hatBeta2=coefficients[2]
    plot_classificator(hatBeta0,hatBeta1,hatBeta2,x1)    

    fig.savefig("ols.eps")