.. _sec_naive_bayes:
Naive Bayes
===========
Ao longo das seções anteriores, aprendemos sobre a teoria da
probabilidade e variáveis aleatórias. Para colocar essa teoria em
prática, vamos apresentar o classificador *Naive Bayes*. Isso usa apenas
fundamentos probabilísticos para nos permitir realizar a classificação
dos dígitos.
Aprender é fazer suposições. Se quisermos classificar um novo exemplo de
dados que nunca vimos antes, temos que fazer algumas suposições sobre
quais exemplos de dados são semelhantes entre si. O classificador *Naive
Bayes*, um algoritmo popular e notavelmente claro, assume que todos os
recursos são independentes uns dos outros para simplificar o cálculo.
Nesta seção, vamos aplicar este modelo para reconhecer personagens em
imagens.
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%matplotlib inline
import math
from mxnet import gluon, np, npx
from d2l import mxnet as d2l
npx.set_np()
d2l.use_svg_display()
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%matplotlib inline
import math
import torch
import torchvision
from d2l import torch as d2l
d2l.use_svg_display()
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%matplotlib inline
import math
import tensorflow as tf
from d2l import tensorflow as d2l
d2l.use_svg_display()
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def transform(data, label):
return np.floor(data.astype('float32') / 128).squeeze(axis=-1), label
mnist_train = gluon.data.vision.MNIST(train=True, transform=transform)
mnist_test = gluon.data.vision.MNIST(train=False, transform=transform)
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data_transform = torchvision.transforms.Compose(
[torchvision.transforms.ToTensor()])
mnist_train = torchvision.datasets.MNIST(
root='./temp', train=True, transform=data_transform, download=True)
mnist_test = torchvision.datasets.MNIST(
root='./temp', train=False, transform=data_transform, download=True)
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0.4%Downloading http://yann.lecun.com/exdb/mnist/train-images-idx3-ubyte.gz
Downloading http://yann.lecun.com/exdb/mnist/train-images-idx3-ubyte.gz to ./temp/MNIST/raw/train-images-idx3-ubyte.gz
77.4%
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((train_images, train_labels), (
test_images, test_labels)) = tf.keras.datasets.mnist.load_data()
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image, label = mnist_train[2]
image.shape, label
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((28, 28), array(4, dtype=int32))
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image, label = mnist_train[2]
image.shape, label
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(torch.Size([1, 28, 28]), 4)
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image, label = train_images[2], train_labels[2]
image.shape, label
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((28, 28), 4)
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image.shape, image.dtype
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((28, 28), dtype('float32'))
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image.shape, image.dtype
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(torch.Size([1, 28, 28]), torch.float32)
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image.shape, image.dtype
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((28, 28), dtype('uint8'))
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label, type(label), label.dtype
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(array(4, dtype=int32), mxnet.numpy.ndarray, dtype('int32'))
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label, type(label)
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(4, int)
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label, type(label)
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(4, numpy.uint8)
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images, labels = mnist_train[10:38]
images.shape, labels.shape
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((28, 28, 28), (28,))
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images = torch.stack([mnist_train[i][0] for i in range(10,38)],
dim=1).squeeze(0)
labels = torch.tensor([mnist_train[i][1] for i in range(10,38)])
images.shape, labels.shape
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(torch.Size([28, 28, 28]), torch.Size([28]))
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images = tf.stack([train_images[i] for i in range(10, 38)], axis=0)
labels = tf.constant([train_labels[i] for i in range(10, 38)])
images.shape, labels.shape
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(TensorShape([28, 28, 28]), TensorShape([28]))
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d2l.show_images(images, 2, 9);
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d2l.show_images(images, 2, 9);
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d2l.show_images(images, 2, 9);
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X, Y = mnist_train[:] # All training examples
n_y = np.zeros((10))
for y in range(10):
n_y[y] = (Y == y).sum()
P_y = n_y / n_y.sum()
P_y
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array([0.09871667, 0.11236667, 0.0993 , 0.10218333, 0.09736667,
0.09035 , 0.09863333, 0.10441667, 0.09751666, 0.09915 ])
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X = torch.stack([mnist_train[i][0] for i in range(len(mnist_train))],
dim=1).squeeze(0)
Y = torch.tensor([mnist_train[i][1] for i in range(len(mnist_train))])
n_y = torch.zeros(10)
for y in range(10):
n_y[y] = (Y == y).sum()
P_y = n_y / n_y.sum()
P_y
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tensor([0.0987, 0.1124, 0.0993, 0.1022, 0.0974, 0.0904, 0.0986, 0.1044, 0.0975,
0.0992])
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X = tf.stack([train_images[i] for i in range(len(train_images))], axis=0)
Y = tf.constant([train_labels[i] for i in range(len(train_labels))])
n_y = tf.Variable(tf.zeros(10))
for y in range(10):
n_y[y].assign(tf.reduce_sum(tf.cast(Y == y, tf.float32)))
P_y = n_y / tf.reduce_sum(n_y)
P_y
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n_x = np.zeros((10, 28, 28))
for y in range(10):
n_x[y] = np.array(X.asnumpy()[Y.asnumpy() == y].sum(axis=0))
P_xy = (n_x + 1) / (n_y + 1).reshape(10, 1, 1)
d2l.show_images(P_xy, 2, 5);
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n_x = torch.zeros((10, 28, 28))
for y in range(10):
n_x[y] = torch.tensor(X.numpy()[Y.numpy() == y].sum(axis=0))
P_xy = (n_x + 1) / (n_y + 1).reshape(10, 1, 1)
d2l.show_images(P_xy, 2, 5);
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n_x = tf.Variable(tf.zeros((10, 28, 28)))
for y in range(10):
n_x[y].assign(tf.cast(tf.reduce_sum(
X.numpy()[Y.numpy() == y], axis=0), tf.float32))
P_xy = (n_x + 1) / tf.reshape((n_y + 1), (10, 1, 1))
d2l.show_images(P_xy, 2, 5);
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def bayes_pred(x):
x = np.expand_dims(x, axis=0) # (28, 28) -> (1, 28, 28)
p_xy = P_xy * x + (1 - P_xy)*(1 - x)
p_xy = p_xy.reshape(10, -1).prod(axis=1) # p(x|y)
return np.array(p_xy) * P_y
image, label = mnist_test[0]
bayes_pred(image)
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array([0., 0., 0., 0., 0., 0., 0., 0., 0., 0.])
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def bayes_pred(x):
x = x.unsqueeze(0) # (28, 28) -> (1, 28, 28)
p_xy = P_xy * x + (1 - P_xy)*(1 - x)
p_xy = p_xy.reshape(10, -1).prod(dim=1) # p(x|y)
return p_xy * P_y
image, label = mnist_test[0]
bayes_pred(image)
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tensor([0., 0., 0., 0., 0., 0., 0., 0., 0., 0.])
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def bayes_pred(x):
x = tf.expand_dims(x, axis=0) # (28, 28) -> (1, 28, 28)
p_xy = P_xy * x + (1 - P_xy)*(1 - x)
p_xy = tf.math.reduce_prod(tf.reshape(p_xy, (10, -1)), axis=1) # p(x|y)
return p_xy * P_y
image, label = tf.cast(train_images[0], tf.float32), train_labels[0]
bayes_pred(image)
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a = 0.1
print('underflow:', a**784)
print('logarithm is normal:', 784*math.log(a))
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underflow: 0.0
logarithm is normal: -1805.2267129073316
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a = 0.1
print('underflow:', a**784)
print('logarithm is normal:', 784*math.log(a))
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underflow: 0.0
logarithm is normal: -1805.2267129073316
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a = 0.1
print('underflow:', a**784)
print('logarithm is normal:', 784*tf.math.log(a).numpy())
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underflow: 0.0
logarithm is normal: -1805.2267379760742
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log_P_xy = np.log(P_xy)
log_P_xy_neg = np.log(1 - P_xy)
log_P_y = np.log(P_y)
def bayes_pred_stable(x):
x = np.expand_dims(x, axis=0) # (28, 28) -> (1, 28, 28)
p_xy = log_P_xy * x + log_P_xy_neg * (1 - x)
p_xy = p_xy.reshape(10, -1).sum(axis=1) # p(x|y)
return p_xy + log_P_y
py = bayes_pred_stable(image)
py
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array([-269.00424, -301.73447, -245.21458, -218.8941 , -193.46907,
-206.10315, -292.54315, -114.62834, -220.35619, -163.18881])
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log_P_xy = torch.log(P_xy)
log_P_xy_neg = torch.log(1 - P_xy)
log_P_y = torch.log(P_y)
def bayes_pred_stable(x):
x = x.unsqueeze(0) # (28, 28) -> (1, 28, 28)
p_xy = log_P_xy * x + log_P_xy_neg * (1 - x)
p_xy = p_xy.reshape(10, -1).sum(axis=1) # p(x|y)
return p_xy + log_P_y
py = bayes_pred_stable(image)
py
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tensor([-274.1814, -302.0445, -254.1889, -223.6053, -199.4294, -212.9662,
-298.5672, -119.8299, -223.9705, -169.0555])
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log_P_xy = tf.math.log(P_xy)
# TODO: Look into why this returns infs
log_P_xy_neg = tf.math.log(1 - P_xy)
log_P_y = tf.math.log(P_y)
def bayes_pred_stable(x):
x = tf.expand_dims(x, axis=0) # (28, 28) -> (1, 28, 28)
p_xy = log_P_xy * x + log_P_xy_neg * (1 - x)
p_xy = tf.math.reduce_sum(tf.reshape(p_xy, (10, -1)), axis=1) # p(x|y)
return p_xy + log_P_y
py = bayes_pred_stable(image)
py
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# Convert label which is a scalar tensor of int32 dtype
# to a Python scalar integer for comparison
py.argmax(axis=0) == int(label)
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array(True)
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py.argmax(dim=0) == label
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tensor(True)
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tf.argmax(py, axis=0) == label
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def predict(X):
return [bayes_pred_stable(x).argmax(axis=0).astype(np.int32) for x in X]
X, y = mnist_test[:18]
preds = predict(X)
d2l.show_images(X, 2, 9, titles=[str(d) for d in preds]);
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def predict(X):
return [bayes_pred_stable(x).argmax(dim=0).type(torch.int32).item()
for x in X]
X = torch.stack([mnist_train[i][0] for i in range(10,38)], dim=1).squeeze(0)
y = torch.tensor([mnist_train[i][1] for i in range(10,38)])
preds = predict(X)
d2l.show_images(X, 2, 9, titles=[str(d) for d in preds]);
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def predict(X):
return [tf.cast(tf.argmax(bayes_pred_stable(x), axis=0), tf.int32).numpy()
for x in X]
X = tf.stack(
[tf.cast(train_images[i], tf.float32) for i in range(10, 38)], axis=0)
y = tf.constant([train_labels[i] for i in range(10, 38)])
preds = predict(X)
# TODO: The preds are not correct due to issues with bayes_pred_stable()
d2l.show_images(X, 2, 9, titles=[str(d) for d in preds]);
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X, y = mnist_test[:]
preds = np.array(predict(X), dtype=np.int32)
float((preds == y).sum()) / len(y) # Validation accuracy
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0.8426
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X = torch.stack([mnist_train[i][0] for i in range(len(mnist_test))],
dim=1).squeeze(0)
y = torch.tensor([mnist_train[i][1] for i in range(len(mnist_test))])
preds = torch.tensor(predict(X), dtype=torch.int32)
float((preds == y).sum()) / len(y) # Validation accuracy
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0.8364
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X = tf.stack([tf.cast(train_images[i], tf.float32) for i in range(
len(test_images))], axis=0)
y = tf.constant([train_labels[i] for i in range(len(test_images))])
preds = tf.constant(predict(X), dtype=tf.int32)
# TODO: The accuracy is not correct due to issues with bayes_pred_stable()
tf.reduce_sum(tf.cast(preds == y, tf.float32)) / len(y) # Validation accuracy
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`Discussões `__
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`Discussões `__
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`Discussões `__
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