-Since the dataset is not linearly separable, the logistic regression algorithm is not suggested for the data. Here we implemented an normal neural network(NN) method.The general methodology is referenced by the tutorial codes.
-picture of NN presentation and fomulas here
Notation:
After this segment you will be able to:
In [46]:
""" Import_packages"""
from rdkit import Chem
from rdkit.Chem import AllChem
from rdkit.Chem.Draw import IPythonConsole
from IPython.display import SVG
from sklearn.cross_validation import train_test_split
import numpy as np
from scipy.stats import norm
import time
from sklearn.neighbors.kde import KernelDensity
import matplotlib.pyplot as plt
import pandas as pd
import csv
from synbioTools import tensorChem
In [47]:
""" Read csv file and return it as a panda dataframe(dictionary) """
def read_csv(path):
""" Read csv, return a panda dataframe(dictionary) """
df = pd.read_csv(path)
return df
In [48]:
"""Load chemicals information and convert the chemical info to SMILES format """
def load_chemicals(path):
SolubilityData = read_csv(path) # read csv
chems=[] # variable to store the
# print("List loaded:\n") # view the list
# print(SolubilityData)
# change column names of
SolubilityData.rename(columns={ SolubilityData.columns[1]: "Solubility" }, inplace=True)
SolubilityData.rename(columns={ SolubilityData.columns[0]: "Compound" }, inplace=True)
SolubilityData.rename(columns={ SolubilityData.columns[2]: "SMILES" }, inplace=True)
for row in range(0,len(SolubilityData['SMILES'])):
chems.append( Chem.MolFromSmiles(SolubilityData['SMILES'][row] ) )
SolubilityData['SMILES'] = chems
return SolubilityData # return the data list which contains the three input
# chems = SolubilityData['SMILES'] # read columns
# compounds = SolubilityData['Compound ID']
# solubilities = SolubilityData['measured log(solubility:mol/L)']
# data = {"chems": chems,
# "compounds": compounds,
# "solubilities": solubilities,
data = load_chemicals("C:/Users/DR/Desktop/P2/Latent-master/data/solubility/delaney.csv")
In [49]:
"""Visualize chemical in SMILES format"""
data['SMILES'][1]
Out[49]:
After this segment you will be able to:
Imported RDKFingerprint function from Chem:
Function: Returns an RDKit topological fingerprint for a molecule
ARGUMENTS for RDKFingerprint:
- mol: the molecule to use
- minPath: (optional) minimum number of bonds to include in the subgraphs
Defaults to 1.
- maxPath: (optional) maximum number of bonds to include in the subgraphs
Defaults to 7.
- fpSize: (optional) number of bits in the fingerprint
Defaults to 2048.
In [50]:
"""Fingerprints in different depth: see how max and min depth affect the SMILES binary vector"""
pix=AllChem.RDKFingerprint(data['SMILES'][11], 1, 1, fpSize=20)
fpix = [int(x) for x in list(pix.ToBitString())]
print(fpix)
fpix=AllChem.RDKFingerprint(data['SMILES'][11], 5, 5, fpSize=20)
fpix = [int(x) for x in list(pix.ToBitString())]
print(fpix)
In [51]:
"""Convert SMILES into fingerprint
def chemFP(chem, FINGERPRINT_SIZE, MIN_PATH, MAX_PATH):
tmp=[]
length=[]
fpix=AllChem.RDKFingerprint(chem[0], minPath=MIN_PATH, maxPath=MAX_PATH, fpSize=FINGERPRINT_SIZE)
fpix = [int(x) for x in list(fpix.ToBitString())]
for i in range(1,len(chem)):
tmp = AllChem.RDKFingerprint(chem[i], minPath=MIN_PATH, maxPath=MAX_PATH, fpSize=FINGERPRINT_SIZE) # convert SMILE to fingerprint
tmp = [int(x) for x in list(tmp.ToBitString())] # convert footprint object to binary vector
length.append(len(tmp)) # append length of each fingerprint
fpix=np.vstack((fpix,tmp)) # stack each fingerprint
return length,fpix
# Test the function
minPath = 1
maxPath = 5
fpSize = 1024
length,px = chemFP(data['SMILES'], fpSize, minPath, maxPath)
# plot the fingerprint length distribution
plt.plot(np.squeeze(length))
plt.ylabel('length')
plt.xlabel('fingerprints')
plt.title("fingerprint length distribution" )
plt.show()"""
Out[51]:
In [52]:
"""Convert SMILES into fingerprint"""
def chemFP(chem, FINGERPRINT_SIZE, MIN_PATH=1, MAX_PATH=5):
fpix = AllChem.RDKFingerprint(chem, minPath=MIN_PATH, maxPath=MAX_PATH, fpSize=FINGERPRINT_SIZE)
fpix = [int(x) for x in list(fpix.ToBitString())]
return fpix
In [53]:
""" Encode a chemical as a tensor by concatenating fingerprints up to desired depth """
def tensorChem(chems, FINGERPRINT_SIZE, CHEMDEPTH):
TRAIN_BATCH_SIZE = len(chems)
Xs = np.zeros( (TRAIN_BATCH_SIZE, FINGERPRINT_SIZE, CHEMDEPTH) )
# print(Xs.shape)
for i in range(0, len(chems)-1):
for k in range(0, CHEMDEPTH):
fpix = chemFP(chems[i],FINGERPRINT_SIZE, k+1, k+1)
Xs[i, :, k] = fpix
return Xs
In [54]:
""" Flatten the tensor into a two dimentional vector(feature mapping) """
# The original vector shape
depth = 4
train_y_b4=np.zeros((1,len(data['Solubility'])))
fpSize = 1024
tc = tensorChem(data['SMILES'],fpSize, depth)
print('The original vector shape:\n'+str(tc.shape))
# The flattened vector shape
train_x_flatten = tc.reshape(tc.shape[0], -1).T
print('The flattened vector shape:\n '+str(train_x_flatten.shape))
# The shape of label vector
train_y_b4[0]=np.squeeze(data['Solubility'])
print('The solubility vector shape:\n '+str(train_y_b4.shape))
In [58]:
# # Visualize the data:
# plt.scatter(train_x_flatten[0],train_x_flatten[1], c=train_y[0], s=40, cmap=plt.cm.Spectral);
# # Here we should add PCA#####################################################
# """This dataset is a little noisy, but it looks like a diagonal line separating the
# upper left half (blue) from the lower right half (red) would work well."""
In [57]:
# """ Visualize tensor vector """
# %matplotlib inline
# import matplotlib.pyplot as plt
# plt.imshow(tc[0,:,:])
# plt.set_cmap('hot')
# plt.xlabel('depth')
# plt.ylabel('fingerprint')
After this segment you will be able to:
In [59]:
""" Visualize solubility """
# plot the solubility distribution
plt.plot(np.squeeze(train_y_b4))
plt.ylabel('solubility')
plt.xlabel('fingerprints')
plt.title("fingerprint and solubility distribution" )
plt.show()
In [60]:
# plot the histogram of solubility
import seaborn as sns
train_y_plot = pd.Series( np.squeeze(train_y_b4), name="Solubility")
mean = train_y_b4.mean()
std = train_y_b4.std()
print("The mean of the solubility is: " + str(mean))
print("The S.D. of the solubility is: " + str(std))
sns.distplot(train_y_plot, kde=True, rug=True, hist=True)
"""In statistics, kernel density estimation (KDE) is a non-parametric way to estimate the probability density function
of a random variable. Kernel density estimation is a fundamental data smoothing problem where inferences about the population
are made, based on a finite data sample."""
Out[60]:
In [61]:
"""Hardmax the labels"""
# convert train_y into a vector range from 0 to 1
train_y=np.zeros((1,len(data['Solubility'])))
for i in range(0,len(train_y_b4[0])):
if (train_y_b4[0][i] >=mean):
train_y[0][i]=1
else:
train_y[0][i]=0
print('There are '+ str(list(np.squeeze(train_y)).count(1)) + ' soluble chemicals (positive samples) and ' + str(list(np.squeeze(train_y)).count(0)) + ' insoluble chemicals (negative samples).')
# plot the input fingerprint length distribution plot
plt.plot(np.squeeze(train_y))
plt.ylabel('solubility')
plt.xlabel('fingerprints')
plt.title("fingerprint and solubility distribution in binary classification" )
plt.show()
After this segment you will be able to:
Notation:
Codes based on Andrew N.g's model code
Implementing the $L$-layer Neural Net, we design a function that replicates(linear_activation_forward with RELU) $L-1$ times, then follows that with one linear_activation_forward with SIGMOID.
Sigmoid: $\sigma(Z) = \sigma(W A + b) = \frac{1}{ 1 + e^{-(W A + b)}}$. We have provided you with the sigmoid function. This function returns two items: the activation value "a" and a "cache" that contains "Z" (it's what we will feed in to the corresponding backward function). To use it you could just call:
A, activation_cache = sigmoid(Z)
ReLU: The mathematical formula for ReLu is $A = RELU(Z) = max(0, Z)$. We have provided you with the relu function. This function returns two items: the activation value "A" and a "cache" that contains "Z" (it's what we will feed in to the corresponding backward function). To use it you could just call:
``` python
A, activation_cache = relu(Z)
Steps:
c to a list, you can use list.append(c).
In [62]:
"""Define activation functions"""
def sigmoid(Z):
"""
Implements the sigmoid activation in numpy
Arguments:
Z -- numpy array of any shape
Returns:
A -- output of sigmoid(z), same shape as Z
cache -- returns Z as well, useful during backpropagation
"""
A = 1/(1+np.exp(-Z))
cache = Z
return A, cache
def relu(Z):
"""
Implement the RELU function.
Arguments:
Z -- Output of the linear layer, of any shape
Returns:
A -- Post-activation parameter, of the same shape as Z
cache -- a python dictionary containing "A" ; stored for computing the backward pass efficiently
"""
A = np.maximum(0,Z)
assert(A.shape == Z.shape)
cache = Z
return A, cache
def relu_backward(dA, cache):
"""
Implement the backward propagation for a single RELU unit.
Arguments:
dA -- post-activation gradient, of any shape
cache -- 'Z' where we store for computing backward propagation efficiently
Returns:
dZ -- Gradient of the cost with respect to Z
"""
Z = cache
dZ = np.array(dA, copy=True) # just converting dz to a correct object.
# When z <= 0, you should set dz to 0 as well.
dZ[Z <= 0] = 0
assert (dZ.shape == Z.shape)
return dZ
def sigmoid_backward(dA, cache):
"""
Implement the backward propagation for a single SIGMOID unit.
Arguments:
dA -- post-activation gradient, of any shape
cache -- 'Z' where we store for computing backward propagation efficiently
Returns:
dZ -- Gradient of the cost with respect to Z
"""
Z = cache
s = 1/(1+np.exp(-Z))
dZ = dA * s * (1-s)
assert (dZ.shape == Z.shape)
return dZ
def dictionary_to_vector(parameters):
"""
Roll all our parameters dictionary into a single vector satisfying our specific required shape.
"""
keys = []
count = 0
for key in ["W1", "b1", "W2", "b2", "W3", "b3"]:
# flatten parameter
new_vector = np.reshape(parameters[key], (-1,1))
keys = keys + [key]*new_vector.shape[0]
if count == 0:
theta = new_vector
else:
theta = np.concatenate((theta, new_vector), axis=0)
count = count + 1
return theta, keys
def vector_to_dictionary(theta):
"""
Unroll all our parameters dictionary from a single vector satisfying our specific required shape.
"""
parameters = {}
parameters["W1"] = theta[:20].reshape((5,4))
parameters["b1"] = theta[20:25].reshape((5,1))
parameters["W2"] = theta[25:40].reshape((3,5))
parameters["b2"] = theta[40:43].reshape((3,1))
parameters["W3"] = theta[43:46].reshape((1,3))
parameters["b3"] = theta[46:47].reshape((1,1))
return parameters
def gradients_to_vector(gradients):
"""
Roll all our gradients dictionary into a single vector satisfying our specific required shape.
"""
count = 0
for key in ["dW1", "db1", "dW2", "db2", "dW3", "db3"]:
# flatten parameter
new_vector = np.reshape(gradients[key], (-1,1))
if count == 0:
theta = new_vector
else:
theta = np.concatenate((theta, new_vector), axis=0)
count = count + 1
return theta
"He Initialization" is named for the first author of He et al., 2015. (If you have heard of "Xavier initialization", this is similar except Xavier initialization uses a scaling factor for the weights $W^{[l]}$ of sqrt(1./layers_dims[l-1]) where He initialization would use sqrt(2./layers_dims[l-1]).)
Hint: This function is similar to the previous initialize_parameters_random(...). The only difference is that instead of multiplying np.random.randn(..,..) by 10, you will multiply it by $\sqrt{\frac{2}{\text{dimension of the previous layer}}}$, which is what He initialization recommends for layers with a ReLU activation.
In [63]:
"""Init_parameters_he"""
def initialize_parameters_he(layers_dims):
"""
Arguments:
layer_dims -- python array (list) containing the size of each layer.
Returns:
parameters -- python dictionary containing your parameters "W1", "b1", ..., "WL", "bL":
W1 -- weight matrix of shape (layers_dims[1], layers_dims[0])
b1 -- bias vector of shape (layers_dims[1], 1)
...
WL -- weight matrix of shape (layers_dims[L], layers_dims[L-1])
bL -- bias vector of shape (layers_dims[L], 1)
"""
np.random.seed(3)
parameters = {}
L = len(layers_dims) - 1 # integer representing the number of layers
for l in range(1, L + 1):
### START CODE HERE ### (≈ 2 lines of code)
parameters['W' + str(l)] = np.random.randn(layers_dims[l], layers_dims[l - 1]) * np.sqrt(2 / layers_dims[l - 1])
parameters['b' + str(l)] = np.zeros((layers_dims[l], 1))
### END CODE HERE ###
return parameters
Now that you have initialized your parameters, you will do the forward propagation module. You will start by implementing some basic functions that you will use later when implementing the model. You will complete three functions in this order:
The linear forward module (vectorized over all the examples) computes the following equations:
$$Z^{[l]} = W^{[l]}A^{[l-1]} +b^{[l]}\tag{4}$$where $A^{[0]} = X$.
Exercise: Build the linear part of forward propagation.
Reminder:
The mathematical representation of this unit is $Z^{[l]} = W^{[l]}A^{[l-1]} +b^{[l]}$. You may also find np.dot() useful. If your dimensions don't match, printing W.shape may help.
In [64]:
def linear_forward(A, W, b):
"""
Implement the linear part of a layer's forward propagation.
Arguments:
A -- activations from previous layer (or input data): (size of previous layer, number of examples)
W -- weights matrix: numpy array of shape (size of current layer, size of previous layer)
b -- bias vector, numpy array of shape (size of the current layer, 1)
Returns:
Z -- the input of the activation function, also called pre-activation parameter
cache -- a python dictionary containing "A", "W" and "b" ; stored for computing the backward pass efficiently
"""
Z = np.dot(W,A)+b
assert(Z.shape == (W.shape[0], A.shape[1]))
cache = (A, W, b)
return Z, cache
In this notebook, you will use two activation functions:
Sigmoid: $\sigma(Z) = \sigma(W A + b) = \frac{1}{ 1 + e^{-(W A + b)}}$. We have provided you with the sigmoid function. This function returns two items: the activation value "a" and a "cache" that contains "Z" (it's what we will feed in to the corresponding backward function). To use it you could just call:
A, activation_cache = sigmoid(Z)
ReLU: The mathematical formula for ReLu is $A = RELU(Z) = max(0, Z)$. We have provided you with the relu function. This function returns two items: the activation value "A" and a "cache" that contains "Z" (it's what we will feed in to the corresponding backward function). To use it you could just call:
A, activation_cache = relu(Z)
In [65]:
def linear_activation_forward(A_prev, W, b, activation,keep_prob=1):
"""
Implement the forward propagation for the LINEAR->ACTIVATION layer
Arguments:
A_prev -- activations from previous layer (or input data): (size of previous layer, number of examples)
W -- weights matrix: numpy array of shape (size of current layer, size of previous layer)
b -- bias vector, numpy array of shape (size of the current layer, 1)
activation -- the activation to be used in this layer, stored as a text string: "sigmoid" or "relu"
Returns:
A -- the output of the activation function, also called the post-activation value
cache -- a python dictionary containing "linear_cache" and "activation_cache";
stored for computing the backward pass efficiently
"""
if activation == "sigmoid":
# Inputs: "A_prev, W, b". Outputs: "A, activation_cache".
### START CODE HERE ### (≈ 2 lines of code)
Z, linear_cache = linear_forward(A_prev,W,b)
A, activation_cache = sigmoid(Z)
### END CODE HERE ###
Dt = np.random.rand(A.shape[0], A.shape[1])
elif activation == "relu":
# Inputs: "A_prev, W, b". Outputs: "A, activation_cache".
### START CODE HERE ### (≈ 2 lines of code)
Z, linear_cache = linear_forward(A_prev,W,b)
A, activation_cache = relu(Z)
# Dropout
Dt = np.random.rand(A.shape[0], A.shape[1]) # Step 1: initialize matrix D2 = np.random.rand(..., ...)
Dt = Dt < keep_prob # Step 2: convert entries of D2 to 0 or 1 (using keep_prob as the threshold)
A = A * Dt # Step 3: shut down some neurons of A2
A = A / keep_prob
### END CODE HERE ###
assert (A.shape == (W.shape[0], A_prev.shape[1]))
cache = (linear_cache, activation_cache,Dt)
return A, cache
For even more convenience when implementing the $L$-layer Neural Net, you will need a function that replicates the previous one (linear_activation_forward with RELU) $L-1$ times, then follows that with one linear_activation_forward with SIGMOID.
Instruction: In the code below, the variable AL will denote $A^{[L]} = \sigma(Z^{[L]}) = \sigma(W^{[L]} A^{[L-1]} + b^{[L]})$. (This is sometimes also called Yhat, i.e., this is $\hat{Y}$.)
Tips:
c to a list, you can use list.append(c).
In [66]:
def L_model_forward(X, parameters, keep_prob):
"""
Implement forward propagation for the [LINEAR->RELU]*(L-1)->LINEAR->SIGMOID computation
Arguments:
X -- data, numpy array of shape (input size, number of examples)
parameters -- output of initialize_parameters_he()
Returns:
AL -- last post-activation value
caches -- list of caches containing:
every cache of linear_activation_forward() (there are L-1 of them, indexed from 0 to L-1)
"""
caches = []
A = X
L = len(parameters) // 2 # number of layers in the neural network
keep_probtmp=keep_prob
# Implement [LINEAR -> RELU]*(L-1). Add "cache" to the "caches" list.
for l in range(1, L):
A_prev = A
### START CODE HERE ### (≈ 2 lines of code)
A, cache = linear_activation_forward(A_prev,parameters['W'+str(l)], parameters['b'+str(l)], activation='relu',keep_prob=keep_probtmp)
caches.append(cache)
### END CODE HERE ###
# Implement LINEAR -> SIGMOID. Add "cache" to the "caches" list.
### START CODE HERE ### (≈ 2 lines of code)
AL, cache = linear_activation_forward(A, parameters['W'+str(L)], parameters['b'+str(L)], activation='sigmoid')
caches.append(cache)
### END CODE HERE ###
assert(AL.shape == (1,X.shape[1]))
return AL, caches
In [67]:
def compute_cost(AL, Y):
"""
Implement the cost function defined by equation (7).
Arguments:
AL -- probability vector corresponding to your label predictions, shape (1, number of examples)
Y -- true "label" vector (for example: containing 0 if non-cat, 1 if cat), shape (1, number of examples)
Returns:
cost -- cross-entropy cost
"""
m = Y.shape[1]
# Compute loss from aL and y.
### START CODE HERE ### (≈ 1 lines of code)
cost = (-1/m)*np.sum(np.multiply(Y,np.log(AL))+np.multiply(1-Y,np.log(1-AL)))
### END CODE HERE ###
cost = np.squeeze(cost) # To make sure your cost's shape is what we expect (e.g. this turns [[17]] into 17).
assert(cost.shape == ())
return cost
def compute_cost_with_regularization(AL, Y, parameters, lambd):
"""
Implement the cost function with L2 regularization. See formula (2) above.
Arguments:
A3 -- post-activation, output of forward propagation, of shape (output size, number of examples)
Y -- "true" labels vector, of shape (output size, number of examples)
parameters -- python dictionary containing parameters of the model
Returns:
cost - value of the regularized loss function (formula (2))
"""
m = Y.shape[1]
L2 =0
for i in range(1,len(AL)):
L2 = L2 + lambd * (np.sum( parameters['W'+str(i)]) ) / (2 * m)
cross_entropy_cost = compute_cost(AL, Y) # This gives you the cross-entropy part of the cost
### START CODE HERE ### (approx. 1 line)
L2_regularization_cost = (-1/m)*np.sum(np.multiply(Y,np.log(AL))+np.multiply(1-Y,np.log(1-AL))) + L2
### END CODER HERE ###
cost = cross_entropy_cost + L2_regularization_cost
return cost
Just like with forward propagation, you will implement helper functions for backpropagation. Remember that back propagation is used to calculate the gradient of the loss function with respect to the parameters.
Reminder:
Now, similar to forward propagation, you are going to build the backward propagation in three steps:
For layer $l$, the linear part is: $Z^{[l]} = W^{[l]} A^{[l-1]} + b^{[l]}$ (followed by an activation).
Suppose you have already calculated the derivative $dZ^{[l]} = \frac{\partial \mathcal{L} }{\partial Z^{[l]}}$. You want to get $(dW^{[l]}, db^{[l]} dA^{[l-1]})$.
The three outputs $(dW^{[l]}, db^{[l]}, dA^{[l]})$ are computed using the input $dZ^{[l]}$.Here are the formulas you need: $$ dW^{[l]} = \frac{\partial \mathcal{L} }{\partial W^{[l]}} = \frac{1}{m} dZ^{[l]} A^{[l-1] T} \tag{8}$$ $$ db^{[l]} = \frac{\partial \mathcal{L} }{\partial b^{[l]}} = \frac{1}{m} \sum_{i = 1}^{m} dZ^{[l](i)}\tag{9}$$ $$ dA^{[l-1]} = \frac{\partial \mathcal{L} }{\partial A^{[l-1]}} = W^{[l] T} dZ^{[l]} \tag{10}$$
In [68]:
def linear_backward(dZ, cache,keep_prob,D):
"""
Implement the linear portion of backward propagation for a single layer (layer l)
Arguments:
dZ -- Gradient of the cost with respect to the linear output (of current layer l)
cache -- tuple of values (A_prev, W, b) coming from the forward propagation in the current layer
Returns:
dA_prev -- Gradient of the cost with respect to the activation (of the previous layer l-1), same shape as A_prev
dW -- Gradient of the cost with respect to W (current layer l), same shape as W
db -- Gradient of the cost with respect to b (current layer l), same shape as b
"""
A_prev, W, b = cache
m = A_prev.shape[1]
### START CODE HERE ### (≈ 3 lines of code)
dW = (1/m)*np.dot(dZ,cache[0].T)
db = (1/m)*np.sum(dZ,axis=1,keepdims=True)
dA_prev = np.dot(cache[1].T,dZ)
if keep_prob<1:
dA_prev = dA_prev * D # Step 1: Apply mask D2 to shut down the same neurons as during the forward propagation
dA_prev = dA_prev / keep_prob
### END CODE HERE ###
assert (dA_prev.shape == A_prev.shape)
assert (dW.shape == W.shape)
assert (db.shape == b.shape)
return dA_prev, dW, db
# FUNCTION: backward_propagation_with_regularization
def backward_propagation_with_regularization(dZ, cache, lambd,keep_prob,D):
"""
Implement the linear portion of backward propagation for a single layer (layer l)
Arguments:
dZ -- Gradient of the cost with respect to the linear output (of current layer l)
cache -- tuple of values (A_prev, W, b) coming from the forward propagation in the current layer
Returns:
dA_prev -- Gradient of the cost with respect to the activation (of the previous layer l-1), same shape as A_prev
dW -- Gradient of the cost with respect to W (current layer l), same shape as W
db -- Gradient of the cost with respect to b (current layer l), same shape as b
"""
A_prev, W, b = cache
m = A_prev.shape[1]
### START CODE HERE ### (≈ 3 lines of code)
dW = (1/m)*np.dot(dZ,cache[0].T) + (lambd * W) / m
db = (1/m)*np.sum(dZ,axis=1,keepdims=True)
dA_prev = np.dot(cache[1].T,dZ)
if keep_prob<1:
dA_prev = dA_prev * D # Step 1: Apply mask D2 to shut down the same neurons as during the forward propagation
dA_prev = dA_prev / keep_prob # Step 2: Scale the value of neurons that haven't been shut down
### END CODE HERE ###
assert (dA_prev.shape == A_prev.shape)
assert (dW.shape == W.shape)
assert (db.shape == b.shape)
return dA_prev, dW, db
### START CODE HERE ### (approx. 1 line)
Next, you will create a function that merges the two helper functions: linear_backward and the backward step for the activation linear_activation_backward.
To help you implement linear_activation_backward, we provided two backward functions:
sigmoid_backward: Implements the backward propagation for SIGMOID unit. You can call it as follows:dZ = sigmoid_backward(dA, activation_cache)
relu_backward: Implements the backward propagation for RELU unit. You can call it as follows:dZ = relu_backward(dA, activation_cache)
If $g(.)$ is the activation function,
sigmoid_backward and relu_backward compute $$dZ^{[l]} = dA^{[l]} * g'(Z^{[l]}) \tag{11}$$.
Exercise: Implement the backpropagation for the LINEAR->ACTIVATION layer.
In [69]:
def linear_activation_backward(dA, cache, activation, regu,lambd,keep_prob):
"""
Implement the backward propagation for the LINEAR->ACTIVATION layer.
Arguments:
dA -- post-activation gradient for current layer l
cache -- tuple of values (linear_cache, activation_cache) we store for computing backward propagation efficiently
activation -- the activation to be used in this layer, stored as a text string: "sigmoid" or "relu"
Returns:
dA_prev -- Gradient of the cost with respect to the activation (of the previous layer l-1), same shape as A_prev
dW -- Gradient of the cost with respect to W (current layer l), same shape as W
db -- Gradient of the cost with respect to b (current layer l), same shape as b
"""
linear_cache, activation_cache, D = cache
if activation == "relu":
### START CODE HERE ### (≈ 2 lines of code)
if regu == True:
dZ = relu_backward(dA,activation_cache)
dA_prev, dW, db = backward_propagation_with_regularization(dZ,linear_cache,lambd,keep_prob,D)
else:
dZ = relu_backward(dA,activation_cache)
dA_prev, dW, db = linear_backward(dZ,linear_cache,keep_prob,D)
### END CODE HERE ###
elif activation == "sigmoid":
### START CODE HERE ### (≈ 2 lines of code)
dZ = sigmoid_backward(dA,activation_cache)
if regu == True:
dA_prev, dW, db = backward_propagation_with_regularization(dZ,linear_cache,lambd,keep_prob,D)
else:
dA_prev, dW, db = linear_backward(dZ,linear_cache,keep_prob,D)
### END CODE HERE ###
return dA_prev, dW, db
Now you will implement the backward function for the whole network. Recall that when you implemented the L_model_forward function, at each iteration, you stored a cache which contains (X,W,b, and z). In the back propagation module, you will use those variables to compute the gradients. Therefore, in the L_model_backward function, you will iterate through all the hidden layers backward, starting from layer $L$. On each step, you will use the cached values for layer $l$ to backpropagate through layer $l$. Figure 5 below shows the backward pass.
Initializing backpropagation:
To backpropagate through this network, we know that the output is,
$A^{[L]} = \sigma(Z^{[L]})$. Your code thus needs to compute dAL $= \frac{\partial \mathcal{L}}{\partial A^{[L]}}$.
To do so, use this formula (derived using calculus which you don't need in-depth knowledge of):
dAL = - (np.divide(Y, AL) - np.divide(1 - Y, 1 - AL)) # derivative of cost with respect to AL
You can then use this post-activation gradient dAL to keep going backward. As seen in Figure 5, you can now feed in dAL into the LINEAR->SIGMOID backward function you implemented (which will use the cached values stored by the L_model_forward function). After that, you will have to use a for loop to iterate through all the other layers using the LINEAR->RELU backward function. You should store each dA, dW, and db in the grads dictionary. To do so, use this formula :
For example, for $l=3$ this would store $dW^{[l]}$ in grads["dW3"].
Exercise: Implement backpropagation for the [LINEAR->RELU] $\times$ (L-1) -> LINEAR -> SIGMOID model.
In [70]:
def L_model_backward(AL, Y, caches, regu,lambd,keep_prob):
"""
Implement the backward propagation for the [LINEAR->RELU] * (L-1) -> LINEAR -> SIGMOID group
Arguments:
AL -- probability vector, output of the forward propagation (L_model_forward())
Y -- true "label" vector (containing 0 if right, 1 if opposive)
caches -- list of caches containing:
every cache of linear_activation_forward() with "relu" (it's caches[l], for l in range(L-1) i.e l = 0...L-2)
the cache of linear_activation_forward() with "sigmoid" (it's caches[L-1])
Returns:
grads -- A dictionary with the gradients
grads["dA" + str(l)] = ...
grads["dW" + str(l)] = ...
grads["db" + str(l)] = ...
"""
grads = {}
L = len(caches) # the number of layers
m = AL.shape[1]
Y = Y.reshape(AL.shape) # after this line, Y is the same shape as AL
# Initializing the backpropagation
### START CODE HERE ### (1 line of code)
dAL = - (np.divide(Y, AL) - np.divide(1 - Y, 1 - AL))
### END CODE HERE ###
# Lth layer (SIGMOID -> LINEAR) gradients. Inputs: "dAL, current_cache". Outputs: "grads["dAL-1"], grads["dWL"], grads["dbL"]
### START CODE HERE ### (approx. 2 lines)
current_cache = caches[L-1]
if regu == True:
grads["dA" + str(L-1)], grads["dW" + str(L)], grads["db" + str(L)] = linear_activation_backward(dAL,current_cache,'sigmoid',True,lambd,keep_prob)
else:
grads["dA" + str(L-1)], grads["dW" + str(L)], grads["db" + str(L)] = linear_activation_backward(dAL,current_cache,'sigmoid',False,lambd,keep_prob)
### END CODE HERE ###
# Loop from l=L-2 to l=0
for l in reversed(range(L-1)):
# lth layer: (RELU -> LINEAR) gradients.
# Inputs: "grads["dA" + str(l + 1)], current_cache". Outputs: "grads["dA" + str(l)] , grads["dW" + str(l + 1)] , grads["db" + str(l + 1)]
### START CODE HERE ### (approx. 5 lines)
current_cache = caches[l]
if regu == True:
dA_prev_temp, dW_temp, db_temp = linear_activation_backward(grads["dA"+str(l+1)],current_cache,"relu",True,lambd,keep_prob)
else:
dA_prev_temp, dW_temp, db_temp = linear_activation_backward(grads["dA"+str(l+1)],current_cache,"relu",False,lambd,keep_prob)
grads["dA" + str(l)] = dA_prev_temp
grads["dW" + str(l + 1)] = dW_temp
grads["db" + str(l + 1)] = db_temp
### END CODE HERE ###
return grads
In this section you will update the parameters of the model, using gradient descent:
$$ W^{[l]} = W^{[l]} - \alpha \text{ } dW^{[l]} \tag{16}$$$$ b^{[l]} = b^{[l]} - \alpha \text{ } db^{[l]} \tag{17}$$where $\alpha$ is the learning rate. After computing the updated parameters, store them in the parameters dictionary.
In [71]:
"""update vectors W,b for gradient descent"""
def update_parameters(parameters, grads, learning_rate):
"""
Update parameters using gradient descent
Arguments:
parameters -- python dictionary containing your parameters
grads -- python dictionary containing your gradients, output of L_model_backward
Returns:
parameters -- python dictionary containing your updated parameters
parameters["W" + str(l)] = ...
parameters["b" + str(l)] = ...
"""
L = len(parameters) // 2 # number of layers in the neural network
# Update rule for each parameter. Use a for loop.
### START CODE HERE ### (≈ 3 lines of code)
for l in range(L):
parameters["W" + str(l+1)] = parameters["W" + str(l+1)]-learning_rate*grads["dW"+str(l+1)]
parameters["b" + str(l+1)] = parameters["b" + str(l+1)]-learning_rate*grads["db"+str(l+1)]
### END CODE HERE ###
return parameters
In [72]:
def L_layer_model(X, Y, layers_dims, learning_rate = 0.0075, num_iterations = 3000, print_cost = True, lambd = 0, keep_prob = 1,grad_check=False):#lr was 0.009
"""
Implements a L-layer neural network: [LINEAR->RELU]*(L-1)->LINEAR->SIGMOID.
Arguments:
X -- data, numpy array of shape (number of examples, num_px * num_px * 3)
Y -- true "label" vector (containing 0 if cat, 1 if non-cat), of shape (1, number of examples)
layers_dims -- list containing the input size and each layer size, of length (number of layers + 1).
learning_rate -- learning rate of the gradient descent update rule
num_iterations -- number of iterations of the optimization loop
print_cost -- if True, it prints the cost every 100 steps
Returns:
parameters -- parameters learnt by the model. They can then be used to predict.
"""
costs = [] # keep track of cost
lambdtmp = lambd
# Parameters initialization. (≈ 1 line of code)
### START CODE HERE ###
parameters = initialize_parameters_he(layers_dims)
### END CODE HERE ###
# Loop (gradient descent)
for i in range(0, num_iterations):
# Forward propagation: [LINEAR -> RELU]*(L-1) -> LINEAR -> SIGMOID.
### START CODE HERE ### (≈ 1 line of code)
if keep_prob == 1:
AL, caches = L_model_forward(X,parameters,keep_prob)
elif keep_prob < 1:
AL, caches = L_model_forward(X,parameters,keep_prob)
### END CODE HERE ###
# Compute cost.
### START CODE HERE ### (≈ 1 line of code)
if lambd == 0:
cost = compute_cost(AL,Y)
else:
cost = compute_cost_with_regularization(AL, Y, parameters, lambd)
### END CODE HERE ###
# Backward propagation.
### START CODE HERE ### (≈ 1 line of code)
if lambd == 0 and keep_prob == 1:
grads = L_model_backward(AL,Y,caches,False,lambd,keep_prob)
elif lambd != 0:
grads = L_model_backward(AL,Y,caches,True,lambd,keep_prob)
elif keep_prob < 1:
grads = L_model_backward(AL,Y,caches,False,lambd,keep_prob)
### END CODE HERE ###
# difference = gradient_check_n(parameters, grads, X, Y)
# Update parameters.
### START CODE HERE ### (≈ 1 line of code)
parameters = update_parameters(parameters,grads,learning_rate)
### END CODE HERE ###
# Print the cost every 100 training example
if print_cost and i % 100 == 0:
print ("Cost after iteration %i: %f" %(i, cost))
if grad_check==True:
print ("Difference after iteration %i: %f" %(i, difference))
if i % 100 == 0:
costs.append(cost)
# plot the cost
plt.plot(np.squeeze(costs))
plt.ylabel('cost')
plt.xlabel('iterations (per tens)')
plt.title("Learning rate =" + str(learning_rate))
plt.show()
return parameters,costs
After this assignment you will be able to:
Notation:
train_x_flatten: the binary vector of a chemicals list
train_y: the solubility feature of the vector
Holdout method
Split training set/dev set/test set = 90/10/10 (dev=test distribution)
Split training set/dev set/test set = 99/0.5/0.5 (dev=test distribution)
k-fold cross-validation
smaller data
to maximize our ability to evaluate the NN performance.
Leave-p-out cross-validation
Leave-p-out cross-validation (LpO CV) involves using p observations as the validation set and the remaining observations as the training set. This is repeated on all ways to cut the original sample on a validation set of p observations and a training set.
In [73]:
"""Transpose the matricesin shape (samples,features) for sklearn"""
train_yy=train_y.T
train_xx=train_x_flatten.T
""" Define_NN_structures """
# 4-layers model of neurons with layers 20,7,5,1
layers_dims = [train_x_flatten.shape[0], 20, 7, 5, 1]
In [82]:
"""Hold-out method"""
# Split the data in train/test = 0.9/0.1
X_train, X_test, y_train, y_test = train_test_split(train_xx, train_yy, test_size=0.10)
In [83]:
X_train=X_train.T
X_test=X_test.T
y_train=y_train.T
y_test=y_test.T
In [97]:
"""StratifiedKFold"""
from sklearn.model_selection import StratifiedKFold
# Split the dataset in 3 folds
sfolder = StratifiedKFold(n_splits=3,random_state=0,shuffle=False)
sfolder.get_n_splits(train_xx,train_yy)
for train, test in sfolder.split(train_xx,train_yy):
X_train, X_test = train_xx[train].T, train_xx[test].T
y_train, y_test = train_yy[train].T, train_yy[test].T
# Train the model with each combination of folds
parameters,costs = L_layer_model(X_train, y_train, layers_dims, learning_rate = 0.0075, num_iterations = 5000, print_cost = True,lambd = 0)
# Predict the model
predictions = predict(parameters, X_train)
print ('Accuracy %d' % float((np.dot(y_train,predictions.T) + np.dot(1-y_train,1-predictions.T))/float(y_train.size)*100) + '%'+" on the training set.")
predictions = predict(parameters, X_test)
print ('Accuracy %d' % float((np.dot(y_test,predictions.T) + np.dot(1-y_test,1-predictions.T))/float(y_test.size)*100) + '%'+" on the test set.")
In [96]:
"""Prediction based on training set"""
#Should be testing set########
def predict(parameters, X):
"""
Using the learned parameters, predicts a class for each example in X
Arguments:
parameters -- python dictionary containing your parameters
X -- input data of size (n_x, m)
Returns
predictions -- vector of predictions of our model (non-significant: 0 / significant: 1)
"""
# Computes probabilities using forward propagation, and classifies to 0/1 using 0.5 as the threshold.
### START CODE HERE ### (≈ 2 lines of code)
A2, cache = L_model_forward(X,parameters,keep_prob=1)
predictions = np.round(A2)
### END CODE HERE ###
return predictions
In [158]:
"""Train the model"""
# Train the 4-layer model; layers 20,7,5,1; 5000 iterations; no regularization and dropout; 0.75 alpha; gradient descent
parameters,costs = L_layer_model(X_train, y_train, layers_dims, learning_rate = 0.0075, num_iterations = 5000, print_cost = True,lambd = 0)
In [68]:
# Print accuracy
predictions = predict(parameters, X_train)
print ('Accuracy %d' % float((np.dot(y_train,predictions.T) + np.dot(1-y_train,1-predictions.T))/float(y_train.size)*100) + '%'+" on the training set.")
predictions = predict(parameters, X_test)
print ('Accuracy %d' % float((np.dot(y_test,predictions.T) + np.dot(1-y_test,1-predictions.T))/float(y_test.size)*100) + '%'+" on the test set.")
The standard way to avoid overfitting is called L2 regularization. It consists of appropriately modifying your cost function, from: $$J = -\frac{1}{m} \sum\limits_{i = 1}^{m} \large{(}\small y^{(i)}\log\left(a^{[L](i)}\right) + (1-y^{(i)})\log\left(1- a^{[L](i)}\right) \large{)} \tag{1}$$ To: $$J_{regularized} = \small \underbrace{-\frac{1}{m} \sum\limits_{i = 1}^{m} \large{(}\small y^{(i)}\log\left(a^{[L](i)}\right) + (1-y^{(i)})\log\left(1- a^{[L](i)}\right) \large{)} }_\text{cross-entropy cost} + \underbrace{\frac{1}{m} \frac{\lambda}{2} \sum\limits_l\sum\limits_k\sum\limits_j W_{k,j}^{[l]2} }_\text{L2 regularization cost} \tag{2}$$
``` Note that you have to do this for $W^{[1]}$, $W^{[2]}$ and $W^{[3]}$, then sum the three terms and multiply by $ \frac{1}{m} \frac{\lambda}{2} $.
In [94]:
"""Train the model after regularization"""
parameters,costs = L_layer_model(X_train, y_train, layers_dims, learning_rate = 0.3, num_iterations = 3000, print_cost = True,lambd = 0.5)
# Print accuracy
predictions = predict(parameters, X_train)
print ('Accuracy: %d' % float((np.dot(y_train,predictions.T) + np.dot(1-y_train,1-predictions.T))/float(y_train.size)*100) + '%'+" on the training set:")
predictions = predict(parameters, X_test)
print ('Accuracy: %d' % float((np.dot(y_test,predictions.T) + np.dot(1-y_test,1-predictions.T))/float(y_test.size)*100) + '%'+" on the test set:")
In [ ]:
# """ plot the decision boundary of your trained model to see if there is over-fitting"""
# plt.title("Model without regularization")
# axes = plt.gca()
# axes.set_xlim([-0.75,0.40])
# axes.set_ylim([-0.75,0.65])
# plot_decision_boundary(lambda x: predict_dec(parameters, x.T), train_X, train_Y)
Inverted dropout
When you shut some neurons down, you actually modify your model. The idea behind drop-out is that at each iteration, you train a different model that uses only a subset of your neurons. With dropout, your neurons thus become less sensitive to the activation of one other specific neuron, because that other neuron might be shut down at any time.
In [98]:
"""Train the model after regularization"""
parameters,costs = L_layer_model(X_train, y_train, layers_dims, learning_rate = 0.075, num_iterations = 1400, print_cost = True,lambd = 0,keep_prob=0.7)
In [99]:
# Print accuracy
predictions = predict(parameters, X_train)
print ('Accuracy: %d' % float((np.dot(y_train,predictions.T) + np.dot(1-y_train,1-predictions.T))/float(y_train.size)*100) + '%'+" on the training set:")
predictions = predict(parameters, X_test)
print ('Accuracy: %d' % float((np.dot(y_test,predictions.T) + np.dot(1-y_test,1-predictions.T))/float(y_test.size)*100) + '%'+" on the test set:")
In [107]:
# Tunning hidden layer size in hidden layer 2nd
hidden_layer_sizes = [1, 2, 3, 4, 5, 20, 50]
costs=[]
for i, n_h in enumerate(hidden_layer_sizes):
layers_dims = [n_x, 20, n_h, 5, n_y] # 4-layers model with n_h hidden units
parameters,costs = L_layer_model(train_x_flatten, train_y, layers_dims, num_iterations = 2500, print_cost = False)
# Here we should add PCA#####################################################
#plot_decision_boundary(lambda x: predict(parameters, x.T), X, Y)
predictions = predict(parameters,train_x_flatten)
accuracy = float((np.dot(train_y,predictions.T) + np.dot(1-train_y,1-predictions.T))/float(train_y.size)*100)
print ("Accuracy for {} hidden units: {} %".format(n_h, accuracy))
In [77]:
"""Tuning learning rate"""
#Try PyTOUCH########################################
learning_rates = [0.01, 0.001, 0.0001]
layers_dims = [n_x, 20, 1, 1, n_y]
costs_list=[]
for i in range(0,len(learning_rates)-1):
print ("learning rate is: " + str(learning_rates[i]))
start = time.time()
parameters,costs = L_layer_model(train_x_flatten, train_y, layers_dims,learning_rate=learning_rates[i], num_iterations = 2500, print_cost = False)
elapsed = (time.time() - start)
print ("Time for learning rate{} : {} %".format(str(learning_rates[i]), elapsed))
costs_list.append(costs)
predictions = predict(parameters,train_x_flatten)
accuracy = float((np.dot(train_y,predictions.T) + np.dot(1-train_y,1-predictions.T))/float(train_y.size)*100)
print ("Accuracy for learning rate{} : {} %".format(str(learning_rates[i]), accuracy))
print ('\n' + "-------------------------------------------------------" + '\n')
for i in range(0,len(learning_rates)-1):
plt.plot(np.squeeze(costs_list[i]), label= str(costs_list[i]))
plt.ylabel('cost')
plt.xlabel('iterations (hundreds)')
legend = plt.legend(loc='upper center', shadow=True)
frame = legend.get_frame()
frame.set_facecolor('0.90')
plt.show()
Backpropagation computes the gradients $\frac{\partial J}{\partial \theta}$, where $\theta$ denotes the parameters of the model. $J$ is computed using forward propagation and your loss function.
Because forward propagation is relatively easy to implement, you're confident you got that right, and so you're almost 100% sure that you're computing the cost $J$ correctly. Thus, you can use your code for computing $J$ to verify the code for computing $\frac{\partial J}{\partial \theta}$.
Let's look back at the definition of a derivative (or gradient): $$ \frac{\partial J}{\partial \theta} = \lim_{\varepsilon \to 0} \frac{J(\theta + \varepsilon) - J(\theta - \varepsilon)}{2 \varepsilon} \tag{1}$$
If you're not familiar with the "$\displaystyle \lim_{\varepsilon \to 0}$" notation, it's just a way of saying "when $\varepsilon$ is really really small."
We know the following:
How does gradient checking work?.
As in 1) and 2), you want to compare "gradapprox" to the gradient computed by backpropagation. The formula is still:
$$ \frac{\partial J}{\partial \theta} = \lim_{\varepsilon \to 0} \frac{J(\theta + \varepsilon) - J(\theta - \varepsilon)}{2 \varepsilon} \tag{1}$$However, $\theta$ is not a scalar anymore. It is a dictionary called "parameters". We implemented a function "dictionary_to_vector()" for you. It converts the "parameters" dictionary into a vector called "values", obtained by reshaping all parameters (W1, b1, W2, b2, W3, b3) into vectors and concatenating them.
The inverse function is "vector_to_dictionary" which outputs back the "parameters" dictionary.
We have also converted the "gradients" dictionary into a vector "grad" using gradients_to_vector(). You don't need to worry about that.
Exercise: Implement gradient_check_n().
Instructions: Here is pseudo-code that will help you implement the gradient check.
For each i in num_parameters:
J_plus[i]:np.copy(parameters_values)forward_propagation_n(x, y, vector_to_dictionary($\theta^{+}$ )). J_minus[i]: do the same thing with $\theta^{-}$Thus, you get a vector gradapprox, where gradapprox[i] is an approximation of the gradient with respect to parameter_values[i]. You can now compare this gradapprox vector to the gradients vector from backpropagation. Just like for the 1D case (Steps 1', 2', 3'), compute:
$$ difference = \frac {\| grad - gradapprox \|_2}{\| grad \|_2 + \| gradapprox \|_2 } \tag{3}$$
In [102]:
# GRADED FUNCTION: gradient_check_n
def gradient_check_n(parameters, gradients, X, Y, epsilon = 1e-7):
"""
Checks if backward_propagation_n computes correctly the gradient of the cost output by forward_propagation_n
Arguments:
parameters -- python dictionary containing your parameters "W1", "b1", "W2", "b2", "W3", "b3":
grad -- output of backward_propagation_n, contains gradients of the cost with respect to the parameters.
x -- input datapoint, of shape (input size, 1)
y -- true "label"
epsilon -- tiny shift to the input to compute approximated gradient with formula(1)
Returns:
difference -- difference (2) between the approximated gradient and the backward propagation gradient
"""
# Set-up variables
parameters_values, _ = dictionary_to_vector(parameters)
grad = gradients_to_vector(gradients)
num_parameters = parameters_values.shape[0]
J_plus = np.zeros((num_parameters, 1))
J_minus = np.zeros((num_parameters, 1))
gradapprox = np.zeros((num_parameters, 1))
# Compute gradapprox
for i in range(num_parameters):
# Compute J_plus[i]. Inputs: "parameters_values, epsilon". Output = "J_plus[i]".
# "_" is used because the function you have to outputs two parameters but we only care about the first one
### START CODE HERE ### (approx. 3 lines)
thetaplus =np.copy(parameters_values) # Step 1
thetaplus[i][0] = thetaplus[i][0] + epsilon # Step 2
J_plus[i], _ = forward_propagation_n(X, Y, vector_to_dictionary(thetaplus)) # Step 3
### END CODE HERE ###
# Compute J_minus[i]. Inputs: "parameters_values, epsilon". Output = "J_minus[i]".
### START CODE HERE ### (approx. 3 lines)
thetaminus = np.copy(parameters_values) # Step 1
thetaminus[i][0] = thetaminus[i][0] - epsilon # Step 2
J_minus[i], _ = forward_propagation_n(X, Y, vector_to_dictionary(thetaminus)) # Step 3
### END CODE HERE ###
# Compute gradapprox[i]
### START CODE HERE ### (approx. 1 line)
gradapprox[i] = (J_plus[i] - J_minus[i]) / (2 * epsilon)
### END CODE HERE ###
# Compare gradapprox to backward propagation gradients by computing difference.
### START CODE HERE ### (approx. 1 line)
numerator = np.linalg.norm(grad - gradapprox) # Step 1'
denominator = np.linalg.norm(grad) + np.linalg.norm(gradapprox) # Step 2'
difference = numerator / denominator # Step 3'
### END CODE HERE ###
if difference > 2e-7:
print ("\033[93m" + "There is a mistake in the backward propagation! difference = " + str(difference) + "\033[0m")
else:
print ("\033[92m" + "Your backward propagation works perfectly fine! difference = " + str(difference) + "\033[0m")
return difference
In [178]:
"""Train the model after regularization"""
parameters,costs = L_layer_model(X_train, y_train, layers_dims, learning_rate = 0.075, num_iterations = 1400, print_cost = True,lambd = 0,keep_prob=1,grad_check=True)
Keras is a high-level neural networks API, written in Python and capable of running on top of TensorFlow, CNTK, or Theano. It was developed with a focus on enabling fast experimentation. Being able to go from idea to result with the least possible delay is key to doing good research.This is possible in Keras because we can “wrap” any neural network such that it can use the evaluation features available in scikit-learn, including k-fold cross-validation.
Steps:
In [90]:
# Load libraries
import numpy as np
from keras import models
from keras import optimizers
from keras import layers
from keras.wrappers.scikit_learn import KerasClassifier
from sklearn.model_selection import cross_val_score
from sklearn.datasets import make_classification
# Set random seed
np.random.seed(0)
In [91]:
# Number of features
number_of_features = 4096
# Generate features matrix and target vector
features, target = make_classification(n_samples = 1144,
n_features = number_of_features,
n_informative = 3,
n_redundant = 0,
n_classes = 2,
weights = [.5, .5],
random_state = 0)
In [26]:
# Create function returning a compiled network
def create_network():
# Start neural network
network = models.Sequential()
# Add fully connected layer with a ReLU activation function
network.add(layers.Dense(units=20, activation='relu', input_shape=(number_of_features,)))
# Add fully connected layer with a ReLU activation function
network.add(layers.Dense(units=7, activation='relu'))
# Add fully connected layer with a ReLU activation function
network.add(layers.Dense(units=5, activation='relu'))
# Add fully connected layer with a sigmoid activation function
network.add(layers.Dense(units=1, activation='sigmoid'))
# Compile neural network
network.compile(loss='binary_crossentropy', # Cross-entropy
optimizer='rmsprop', # Root Mean Square Propagation
metrics=['accuracy']) # Accuracy performance metric
network.compile(loss='categorical_crossentropy', optimizer=sgd,metrics=['accuracy'])
# Return compiled network
return network
In [85]:
# Wrap Keras model so it can be used by scikit-learn
neural_network = KerasClassifier(build_fn=create_network,
epochs=5000,
batch_size=100,
verbose=0)
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# Prediction of accuracy in folds of 3
cross_val_score(neural_network,train_x_flatten.T, train_y.T, cv=3)
Out[86]:
In [92]:
#Try Adam as optimizer and implement time-based learning rate decay lr *= (1. / (1. + self.decay * self.iterations))
def create_network():
# Start neural network
network = models.Sequential()
# Add fully connected layer with a ReLU activation function
network.add(layers.Dense(units=20, activation='relu', input_shape=(number_of_features,)))
# Add fully connected layer with a ReLU activation function
network.add(layers.Dense(units=7, activation='relu'))
# Add fully connected layer with a ReLU activation function
network.add(layers.Dense(units=5, activation='relu'))
# Add fully connected layer with a sigmoid activation function
network.add(layers.Dense(units=1, activation='sigmoid'))
# Setup hyperparameters for Adam optimizer
Adam = optimizers.Adam(lr=0.0075, beta_1=0.9, beta_2=0.999, epsilon=None, decay=1e-6, amsgrad=False)
# Compile neural network
network.compile(loss='binary_crossentropy', # Cross-entropy
optimizer= Adam, # Root Mean Square Propagation
metrics=['accuracy']) # Accuracy performance metric
# Return compiled network
return network
In [93]:
# Wrap Keras model so it can be used by scikit-learn
neural_network_Adam = KerasClassifier(build_fn=create_network,
epochs=5000,
batch_size=100,
verbose=0)
In [94]:
# Prediction of accuracy in folds of 3
cross_val_score(neural_network_Adam,train_x_flatten.T, train_y.T, cv=3)
Out[94]:
After this segment you will be able to:
Notation:
Codes based on Andrew N.g's model code
Implementing the $L$-layer Neural Net, we design a function that replicates(linear_activation_forward with RELU) $L-1$ times, then follows that with one linear_activation_forward with SIGMOID.
Sigmoid: $\sigma(Z) = \sigma(W A + b) = \frac{1}{ 1 + e^{-(W A + b)}}$. We have provided you with the sigmoid function. This function returns two items: the activation value "a" and a "cache" that contains "Z" (it's what we will feed in to the corresponding backward function). To use it you could just call:
A, activation_cache = sigmoid(Z)
ReLU: The mathematical formula for ReLu is $A = RELU(Z) = max(0, Z)$. We have provided you with the relu function. This function returns two items: the activation value "A" and a "cache" that contains "Z" (it's what we will feed in to the corresponding backward function). To use it you could just call:
``` python
A, activation_cache = relu(Z)
Steps:
c to a list, you can use list.append(c).
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if not data_augmentation:
print('Not using data augmentation.')
model.fit(X_train, Y_train,
batch_size=batch_size,
nb_epoch=nb_epoch,
validation_data=(X_test, Y_test),
shuffle=True)
else:
print('Using real-time data augmentation.')
datagen = ImageDataGenerator( #1
featurewise_center=False,
samplewise_center=False,
featurewise_std_normalization=False,
samplewise_std_normalization=False,
zca_whitening=False,
rotation_range=0,
width_shift_range=0.1,
height_shift_range=0.1,
horizontal_flip=True,
vertical_flip=False)
datagen.fit(X_train) #2
model.fit_generator(datagen.flow(X_train, Y_train, #3
batch_size=batch_size),
samples_per_epoch=X_train.shape[0],
nb_epoch=nb_epoch,
validation_data=(X_test, Y_test))
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