This blog’s aim is demo the neural network, including both the algorithm and implementation
import the necessary modules
#show the figures buildin the notebook
%matplotlib inline
import numpy as np
import matplotlib.pyplot as plt
import scipy.io #Used to load the OCTAVE *.mat files
import scipy.misc #Used to show matrix as an image
from scipy import optimize
import matplotlib.cm as cm #Used to display images in a specific colormap
import random
load the data from source file
fileName = 'ex4/data/ex4data1.mat'
mat = scipy.io.loadmat(fileName)
X = mat['X']
Y = mat['y']
X = np.insert(X,0,1,axis=1)
Y = Y - 1#Y[Y==10]=0 # convert 10 to 0
print "'X' shape:%s.X[0] shape:%s" %(X.shape,X[0].shape)
print "'y' shape:%s. Unique elements in y:%s" %(Y.shape,np.unique(Y))
The output is:
'X' shape:(5000, 401).X[0] shape:(401,)
'y' shape:(5000, 1). Unique elements in y:[0 1 2 3 4 5 6 7 8 9]
visualization
3 steps to get done
- reshape a sample to a 20*20 array
- fill the returned array to a big array
- visulize the big array
problems
- from the result image, the vetical axis is not correct
def getDatumImg(row):
"""
function that is handed a single np array with shape 1*400,
create an image from object from it, and return.
"""
width, height = (20,20)
return row[1:].reshape(width, height)
def displayData(arr=None):
"""
function that picks 100 samples from X to display as images.
"""
width, height = (20,20)
nrows, ncols = (10,10)
if not arr:
arr = random.sample(xrange(X.shape[0]),nrows*ncols)
big_picture = np.zeros((height * nrows, width * ncols))
irow, icols = (0,0)
for idx in arr:
if icols==ncols:
irow = irow+1
icols = 0
sample = getDatumImg(X[idx])
big_picture[irow * height:irow*height + sample.shape[0],icols*width:icols*width + sample.shape[1]] = sample.T
icols = icols + 1
fig = plt.figure(figsize=(8,8))
img = scipy.misc.toimage(big_picture)
plt.imshow(img,cmap=cm.Greys_r)
plt.show()
return
displayData()
print 'visualise the digit has done'
visualise the digit has done
loss function
- implement a loss function without penalty
- implement a loss function with penalty
def sigmoid(arr, theta):
"""
function that sigmoid both the input samples and the parameters
"""
z = np.dot(arr, theta)
return 1.0 / (1 + np.exp(-z))
def randInitializeWeights(input_layer_size, hidden_layer_size):
"""
episilon = np.sqrt(6)/np.sqrt(Lin + Lout)
Lin = the number of input layer unit
Lout = the number of the adjacent layer unit
"""
episilon = 0.12
return np.random.rand(input_layer_size,hidden_layer_size+1) * 2.0 * episilon - episilon
def sigmoidGradient(arr, theta):
sig = sigmoid(arr, theta)
return sig * ( 1 - sig)
def reshapeParams(nn_params, input_layer_size=400, hidden_layer_size=25, num_labels=10):
"""
function is used to reshape the input parameter:theta with type:list as 2 arrays, return it
"""
print "the type of nn_params in reshapeParams is:%s" % type(nn_params)
theta1 = np.array(nn_params[:(input_layer_size+1) * hidden_layer_size]).reshape((hidden_layer_size,input_layer_size + 1))
theta2 = np.array(nn_params[-num_labels * (hidden_layer_size+1):]).reshape((num_labels, hidden_layer_size+1))
return (theta1, theta2)
def formatY(Y,num_labels=10):
result = np.zeros((Y.shape[0],num_labels))
for idx in xrange(Y.shape[0]):
result[idx,Y[idx,0]] = 1
return result
def nnCostFunction(nn_params, X, Y, lamda=0.0,input_layer_size=400, hidden_layer_size=25,
num_labels=10):
"""
function to calculate the loss error of the samples
"""
print "the type of nn_params in nnCostFunction is:%s" % type(nn_params)
theta1, theta2 = reshapeParams(nn_params, input_layer_size, hidden_layer_size, num_labels)
a1 = sigmoid(X, theta1.T) # m * hidden_layer_size
a1 = np.insert(a1,0, 1, axis=1) # m * (hidden_layer_size + 1)
#print a1[:10]
#print "a1's shape:(%d,%d)" % a1.shape
a2 = sigmoid(a1, theta2.T) # m * num_labels
#print a2[:10]
#print "a2's shape:(%d,%d)" % a2.shape
# format Y from m * 1 to a m*num_labels array
fY = formatY(Y,num_labels)
#print "Y's shape:(%d,%d)" % fY.shape
J = -(np.sum(np.log(a2[fY==1])) + np.sum(np.log(1.0 - a2[fY==0])))
m = len(X)
J = J/m + lamda * (np.sum(theta1**2) + np.sum(theta2**2)) /(2*m)
print "cost value:%f" % J
return J
paramFile = 'ex4/data/ex4weights.mat'
params = scipy.io.loadmat(paramFile)
Theta1 = params['Theta1']
Theta2 = params['Theta2']
input_layer_size=400 # NO of features of samples
hidden_layer_size=25 # NO of Hidden Units
num_labels = 10 # NO of Output Units
#print "Theta1's shape:%s, Theta2's shape:%s" % (Theta1.shape, Theta2.shape)
# flatten both Theta1 and Theta2 into one list
theta = np.append(Theta1.flatten(),Theta2.flatten())
#print type(theta)
#print "theta size:%d" % theta.size
#print X.dtype,Y.dtype,theta.dtype
#theta1, theta2 = reshapeParams(theta)
# if theta1=Theta1, theta2=Theta2, then the sum would be zero
#print np.sum(theta1!=Theta1)
#print np.sum(theta2!=Theta2)
print nnCostFunction(theta,X,Y,0.0)
print nnCostFunction(theta,X,Y,1.0)
the output is:
the type of nn_params in nnCostFunction is:<type 'numpy.ndarray'>
the type of nn_params in reshapeParams is:<type 'numpy.ndarray'>
a1's shape:(5000,26)
a2's shape:(5000,10)
Y's shape:(5000,10)
cost value:0.287629
0.287629165161
the type of nn_params in nnCostFunction is:<type 'numpy.ndarray'>
the type of nn_params in reshapeParams is:<type 'numpy.ndarray'>
a1's shape:(5000,26)
a2's shape:(5000,10)
Y's shape:(5000,10)
cost value:0.384488
0.384487796243
back propagation
def backpropagation(nn_params, X, Y, lamda=0.0,input_layer_size=400, hidden_layer_size=25,
num_labels=10):
theta1, theta2 = reshapeParams(nn_params, input_layer_size, hidden_layer_size, num_labels)
a2 = sigmoid(X, theta1.T) # m * hidden_layer_size
a2 = np.insert(a2,0, 1, axis=1) # m * (hidden_layer_size + 1)
a3 = sigmoid(a2, theta2.T) # m * num_labels
# format Y from m * 1 to a m*num_labels array
fY = formatY(Y,num_labels)
delta3 = a3 - fY # m * num_labels
delta2 = np.dot(delta3, theta2[:,1:]) * sigmoidGradient(X, theta1.T) # m * (hidden_layer_size)
grad2 = np.dot(delta3.T, a2) / X.shape[0] # num_labels * (hidden_layer_size+1)
grad2[:,1:] = grad2[:,1:] + (lamda * theta2[:,1:]/X.shape[0])
grad1 = np.dot(delta2.T, X) / X.shape[0] # (hidden_layer_size) * (input_layer_size+1)
grad1[:,1:] = grad1[:,1:] + (lamda * theta1[:,1:]/X.shape[0])
return np.append(grad1.flatten(),grad2.flatten())
def computeNumericalGradient(mytheta, X, Y, mylambda=0.0,input_layer_size=400, hidden_layer_size=25,
num_labels=10):
"""
mytheta is a flatten array
"""
print input_layer_size,hidden_layer_size,num_labels
print mytheta.shape
ngrad = np.zeros((len(mytheta),1))
episode = 0.0001
for i in xrange(len(mytheta)):
theta_plus = mytheta.copy()
theta_plus[i]=theta_plus[i] + episode
theta_minus = mytheta.copy()
theta_minus[i] = theta_minus[i] - episode
ngrad[i]=(nnCostFunction(theta_plus, X, Y,mylambda,input_layer_size,hidden_layer_size,num_labels) - nnCostFunction(theta_minus, X, Y,mylambda,input_layer_size,hidden_layer_size,num_labels))/ (2 * episode)
return ngrad
def checkNNGradient(mylambda=0.0):
input_layer_size = 3;
hidden_layer_size = 5;
num_labels = 3;
m = 5;
theta1 = randInitializeWeights(hidden_layer_size,input_layer_size);
theta2 = randInitializeWeights(num_labels,hidden_layer_size);
X = randInitializeWeights(m, input_layer_size - 1)
X = np.insert(X,0,1,axis=1)
Y = (np.arange(m) % 3).reshape(m,1)
ngrad = computeNumericalGradient(np.append(theta1.flatten(),theta2.flatten()),X,Y,mylambda,input_layer_size,hidden_layer_size,num_labels)
print ngrad.shape
grad = backpropagation(np.append(theta1.flatten(),theta2.flatten()), X, Y, mylambda,input_layer_size, hidden_layer_size,num_labels)
print grad.shape
#print (ngrad.flatten(),grad.flatten())
print "%.15f" % (norm(ngrad.flatten() - grad) / norm(ngrad.flatten() + grad))
def norm(arr):
return np.sqrt(np.dot(arr,arr.T))
checkNNGradient()
then, the output is:
3 5 3
(38,)
the type of nn_params in nnCostFunction is:<type 'numpy.ndarray'>
the type of nn_params in reshapeParams is:<type 'numpy.ndarray'>
a1's shape:(5,6)
a2's shape:(5,3)
Y's shape:(5,3)
cost value:2.088275
the type of nn_params in nnCostFunction is:<type 'numpy.ndarray'>
the type of nn_params in reshapeParams is:<type 'numpy.ndarray'>
a1's shape:(5,6)
a2's shape:(5,3)
Y's shape:(5,3)
cost value:2.088260
(38, 1)
the type of nn_params in reshapeParams is:<type 'numpy.ndarray'>
(38,)
0.000000000040026
checkNNGradient(3)
3 5 3
(38,)
the type of nn_params in nnCostFunction is:<type 'numpy.ndarray'>
the type of nn_params in reshapeParams is:<type 'numpy.ndarray'>
a1's shape:(5,6)
a2's shape:(5,3)
Y's shape:(5,3)
cost value:2.121677
the type of nn_params in nnCostFunction is:<type 'numpy.ndarray'>
the type of nn_params in reshapeParams is:<type 'numpy.ndarray'>
a1's shape:(5,6)
a2's shape:(5,3)
Y's shape:(5,3)
cost value:2.121665
(38, 1)
the type of nn_params in reshapeParams is:<type 'numpy.ndarray'>
(38,)
0.106736304870728
nnCostFunction(theta, X, Y, lamda=3.0,input_layer_size=400, hidden_layer_size=25,num_labels=10)
then the output is:
the type of nn_params in nnCostFunction is:<type 'numpy.ndarray'>
the type of nn_params in reshapeParams is:<type 'numpy.ndarray'>
a1's shape:(5000,26)
a2's shape:(5000,10)
Y's shape:(5000,10)
cost value:0.578205
0.57820505840604419
build the model
init_theta1 = randInitializeWeights(25,400)
init_theta2 = randInitializeWeights(10,25)
init_theta = np.append(init_theta1.flatten(),init_theta2.flatten())
print len(init_theta)==len(theta)
res = optimize.minimize(nnCostFunction,init_theta,args=(X,Y,1.0,400,25,10),method='BFGS',
jac=backpropagation,options={'maxiter':50,'disp':True})
Predict
ret_theta = res.x
ret_theta1 = np.array(ret_theta)[:25*401].reshape(25,401)
ret_theta2 = np.array(ret_theta)[25*401:].reshape(10,26)
def predict(X,Y,ret_theta1, ret_theta2):
a2 = sigmoid(X, ret_theta1.T)
a2 = np.insert(a2,0,1,axis=1)
a3 = sigmoid(a2, ret_theta2.T)
ret = np.argmax(a3,axis=1).reshape(-1,1)
ret = ret
return np.mean(Y==ret) * 100
print "the precision is:%f" % predict(X,Y,ret_theta1, ret_theta2)
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