p141_linear_descriminant_analsys.py

''' linear_descriminant_analsys.py 
Linear Descriminant Analysis seeks to find a reduced dimensionality of 
the features by maximizing class separability A hyperplane between features from 
each class is calculated that separates the classes.  A new sample can be classified 
depending on which side of the hyperplane it is on.

In a non linearly separable set of features, then the separation is not 
perfect, i.e. some samples from class A might end up on the class B side
of the hyperplane.



    Note:  
        1) the technique assumes that the data is normally distributed.
        This is a red flag.  Normally distributed data usually does not exist.
        It is a figment of the imagination.
        2) LDA is supervised, meaning the class labels are known for each
        input sample, whereas Principle Component Analysis is unsupervised, 
        needing no labels.
        
    Procedure:
        1) Standardize the training set, because LDA is sensitive to scaling
        2) For each class, compute the d -dimensional mean vector.
        3) Construct the between-class scatter matrix S B and the within-class 
        scatter matrix S w
        4) apply the steps of Principal Component Anaysis - eigenpairs,  etc
        
        
Created on Jul 2, 2016

from Python Machine Learning by Sebastian Raschka under the following license

The MIT License (MIT)

Copyright (c) 2015, 2016 SEBASTIAN RASCHKA (mail@sebastianraschka.com)

Permission is hereby granted, free of charge, to any person obtaining a copy
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in the Software without restriction, including without limitation the rights
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SOFTWARE.

@author: richard lyman
'''
import numpy as np
import ocr_utils
import matplotlib.pyplot as plt
#from sklearn.lda import LDA
from sklearn.discriminant_analysis import  LinearDiscriminantAnalysis as LDA   
print_limit = 20
chars_to_train = range(48,58)
n_classes = len(chars_to_train)
columnsXY=range(0,20)
column_str = 'column_sum{}'.format(list(columnsXY))

input_filters_dict = {'m_label': chars_to_train, 'font': 'E13B'}

# output  the character label and the image and column sums
output_feature_list = ['m_label','image',column_str] 

# read the complete image (20x20) = 400 pixels for each character
ds = ocr_utils.read_data(input_filters_dict=input_filters_dict, 
                            output_feature_list=output_feature_list, 
                            test_size=.2,
                            random_state=0)
   
y_train = ds.train.features[0]
X_train_image = ds.train.features[1]
X_train = ds.train.features[2]

y_test = ds.test.features[0]
X_test_image = ds.test.features[1]
X_test = ds.test.features[2]


from sklearn.preprocessing import StandardScaler
# 
sc = StandardScaler()
X_train_std = sc.fit_transform(X_train_image)
X_test_std = sc.fit_transform(X_test_image)
# X_train_std = X_train_image
# X_test_std = X_test_image
unique_labels=np.unique(y_train)
num_unique_labels = len(unique_labels)
np.set_printoptions(precision=4)

mean_vecs = []

for label in unique_labels:
#for label in range(1,4):
    mean_vecs.append(np.mean(X_train_std[y_train==label], axis=0))
#     print('Mean Vector {}: {}\n'.format(label, mean_vecs[i]))

d = mean_vecs[0].shape[0] # number of features

S_W = np.zeros((d, d))
for label, mv in zip(unique_labels, mean_vecs):
    class_scatter = np.zeros((d, d))
    for row in X_train_std[y_train == label]:
        row, mv = row.reshape(d, 1), mv.reshape(d, 1)
        class_scatter += (row-mv).dot((row-mv).T)
    S_W += class_scatter
   
print('Within-class scatter matrix: {}x{}'.format(S_W.shape[0], S_W.shape[1]))

print('Class label distribution: %s'
    % np.bincount(np.array(y_train,dtype='int32'))[int(min(y_train)):])

d = S_W.shape[1] # number of features
S_W = np.zeros((d, d))
for label,mv in zip(unique_labels, mean_vecs):
    class_scatter = np.cov(X_train_std[y_train==label].T)
    S_W += class_scatter
print('Scaled within-class scatter matrix: %sx%s' % (S_W.shape[0], S_W.shape[1]))

mean_overall = np.mean(X_train_std, axis=0)
#d = 13 # number of features
S_B = np.zeros((d, d))
for i,mean_vec in enumerate(mean_vecs):
    #n = X[y==i+1, :].shape[0]    
    n = X_train_std[y_train==np.unique(y_train)[i], :].shape[0]
    mean_vec = mean_vec.reshape(d, 1) # make column vector
    mean_overall = mean_overall.reshape(d, 1) # make column vector
    S_B += n * (mean_vec - mean_overall).dot((mean_vec - mean_overall).T)

print('Between-class scatter matrix: %sx%s' % (S_B.shape[0], S_B.shape[1]))
eigen_vals, eigen_vecs = np.linalg.eig(np.linalg.inv(S_W).dot(S_B))


# Make a list of (eigenvalue, eigenvector) tuples
eigen_pairs = [(np.abs(eigen_vals[i]), eigen_vecs[:,i]) for i in range(len(eigen_vals))]

# Sort the (eigenvalue, eigenvector) tuples from high to low
eigen_pairs = sorted(eigen_pairs, key=lambda k: k[0], reverse=True)

# Visually confirm that the list is correctly sorted by decreasing eigenvalues

print('Eigenvalues in decreasing order:\n')
for eigen_val in eigen_pairs[:print_limit]:
    print(eigen_val[0])

tot = sum(eigen_vals.real)
discr = [(i / tot) for i in sorted(eigen_vals.real, reverse=True)][:20]
cum_discr = np.cumsum(discr)

plt.bar(range(1, len(discr)+1), discr, alpha=0.5, align='center',
        label='individual "discriminability"')
plt.step(range(1, len(cum_discr)+1), cum_discr, where='mid',
         label='cumulative "discriminability"')
plt.ylabel('"discriminability" ratio')
plt.xlabel('Linear Discriminants')
plt.ylim([-0.1, 1.1])
plt.legend(loc='best')
title='Discriminability'
plt.title(title)
plt.tight_layout()
ocr_utils.show_figures(plt,title)

w = np.hstack((eigen_pairs[0][1][:, np.newaxis].real,
                      eigen_pairs[1][1][:, np.newaxis].real))
print('Matrix W:\n', w[:print_limit])

X_train_lda = X_train_std.dot(w)

markers = ('o', 'v', '^', '<', '>', '8', 's', 'p', '*', 'h', 'H', 'D', 'd')
colors = ('red', 'blue', 'lightgreen', 'gray', 'cyan','orange','green','brown','lightblue','pink')

for l, c, m in zip(np.unique(y_train), colors, markers):
    plt.scatter(X_train_lda[y_train==l, 0], 
                X_train_lda[y_train==l, 1], 
                c=c, label=l, marker=m)

plt.xlabel('LD 1')
plt.ylabel('LD 2')
plt.legend(loc='upper right')
title='Projecting Feature Set onto New Feature Space'
plt.title(title)
plt.tight_layout()
ocr_utils.show_figures(plt,title)

###############################################################################3
lda = LDA(n_components=2)
X_train_lda = lda.fit_transform(X_train_std, y_train)
X_test_lda =  lda.transform(X_test_std)

from sklearn.linear_model import LogisticRegression
lr = LogisticRegression(solver='liblinear', multi_class='auto')
lr = lr.fit(X_train_lda, y_train)

title = 'Linear Descriminant Analysis Training Set'
ocr_utils.plot_decision_regions(X_train_lda, y_train, classifier=lr, labels=['LD 1','LD 2'], title=title)

title = 'Linear Descriminant Analysis Test Set'

ocr_utils.plot_decision_regions(X_test_lda, y_test, classifier=lr, labels=['LD 1','LD 2'], title=title)


###############################################################################
n_components = 10
lda = LDA(n_components=min(n_components,n_classes-1))
X_train_lda = lda.fit_transform(X_train_std, y_train)
X_test_lda = lda.transform(X_test_std)

print ('n_components={}'.format(lda.n_components))

lr = LogisticRegression(solver='liblinear', multi_class='auto')
logistic_fitted = lr.fit(X_train_lda, y_train)

from sklearn.metrics import accuracy_score

y_pred_train = logistic_fitted.predict(X_train_lda)
y_pred_test = logistic_fitted.predict(X_test_lda)
print('\nLDA Train Accuracy: {:4.6f}, n_components={}'.format(accuracy_score(y_train, y_pred_train), lda.n_components))
print('LDA Test Accuracy: {:4.6f}, n_components={}'.format(accuracy_score(y_test, y_pred_test), lda.n_components))

X_errors_image = X_test_image[y_test!=y_pred_test]
y_errors = y_test[y_test!=y_pred_test]

# change to a 2D shape 
X2D=np.reshape(X_errors_image, (X_errors_image.shape[0], ds.train.num_rows, ds.train.num_columns))
ocr_utils.montage(X2D,title='LDA E13B Error Character,components={}'.format(n_components))

###############################################################################
n_components = 10
lda = LDA(n_components=n_components-1, solver='eigen')
X_train_lda = lda.fit_transform(X_train_std, y_train)
X_test_lda = lda.transform(X_test_std)

print ('n_components={}'.format(lda.n_components))

lr = LogisticRegression(solver='liblinear', multi_class='auto')
logistic_fitted = lr.fit(X_train_lda, y_train)

from sklearn.metrics import accuracy_score

y_pred_train = logistic_fitted.predict(X_train_lda)
y_pred_test = logistic_fitted.predict(X_test_lda)
print('\nLDA eigen Train Accuracy: {:4.6f}, n_components={}'.format(accuracy_score(y_train, y_pred_train), lda.n_components))
print('LDA eigen Test Accuracy: {:4.6f}, n_components={}'.format(accuracy_score(y_test, y_pred_test), lda.n_components))

X_errors_image = X_test_image[y_test!=y_pred_test]
y_errors = y_test[y_test!=y_pred_test]

# change to a 2D shape 
X2D=np.reshape(X_errors_image, (X_errors_image.shape[0], ds.train.num_rows, ds.train.num_columns))
ocr_utils.montage(X2D,title='LDA eigen E13B Error Character,components={}'.format(n_components))

print ('\n########################### No Errors ####################################')