从零开始实现Logistic Regression，最简单的单元深度神经网络。

来自吴恩达深度学习视频，第二周编程作业，有详细注释。
如果直接看代码对你来说太难，参考 https://blog.csdn.net/u013733326/article/details/79639509。

# Logistic Regression with a Neural Network mindset

# Initializing parameters
# Calculating the cost function and its gradient
# Using an optimization algorithm (gradient descent)

"""
numpy is the fundamental package for scientific computing with Python.
h5py is a common package to interact with a dataset that is stored on an H5 file.
matplotlib is a famous library to plot graphs in Python.
PIL and scipy are used here to test your model with your own picture at the end.
"""

import numpy as np
import matplotlib.pyplot as plt
import h5py
import scipy
from PIL import Image
from scipy import ndimage
from lr_utils import load_dataset

%matplotlib

# loading the data(cat/non-cat)
train_set_x_orig, train_set_y, test_set_x_orig, test_set_y, classes = load_dataset()

# Example of a picture
index = 5
plt.imshow(train_set_x_orig[index])
print("y = " + str(train_set_y[:, index]) + ", it's a '" + classes[np.squeeze(train_set_y[:, index])].decode("utf-8") + "' picture.")

# Remember that train_set_x_orig is a numpy-array of shape (m_train, num_px, num_px, 3)
m_train = train_set_x_orig.shape[0]
m_test = test_set_x_orig.shape[0]
num_px = train_set_x_orig.shape[1]

print("Number of training examples: m_train = " + str(m_train))
print("Number of testing examples: m_test = " + str(m_test))
print("Height/Width of each image: num_px = " + str(num_px))
print("Each image of size: (" + str(num_px) + "," + str(num_px) + ", 3)")
print("train_set_x shape:" + str(train_set_x_orig.shape))
print("train_set_y shape:" + str(train_set_y.shape))
print("test_set_x shape:" + str(test_set_x_orig.shape))
print("test_set_y shape:" + str(test_set_y.shape))

train_set_x_flatten = train_set_x_orig.reshape(train_set_x_orig.shape[0], -1).T
test_set_x_flatten = test_set_x_orig.reshape(test_set_x_orig.shape[0], -1).T

print("train_set_x_flatten shape: " + str(train_set_x_flatten.shape))
print("train_set_y shape: " + str(train_set_y.shape))
print("test_set_x_flatten shape: " + str(train_set_x_flatten.shape))
print("test_set_y shape: " + str(train_set_y.shape))

# standardize dataset
train_set_x = train_set_x_flatten/255.
test_set_x = test_set_x_flatten/255.

"""
**What you need to remember:**
Common steps for pre-processing a new dataset are:

Figure out the dimensions and shapes of the problem (m_train, m_test, num_px, ...)
Reshape the datasets such that each example is now a vector of size (num_px * num_px * 3, 1)
"Standardize" the data
"""

"""
- Initialize the parameters of the model
- Learn the parameters for the model by minimizing the cost  
- Use the learned parameters to make predictions (on the test set)
- Analyse the results and conclude
"""


def sigmoid(z):
    """
    Compute the sigmoid of z

    Arguments:
    z -- A scalar or numpy array of any size.

    Return:
    s -- sigmoid(z)
    """
    s = 1/(1 + np.exp(-z))
    return s

print("sigmoid([0, 2]) = " + str(sigmoid(np.array([0, 2]))))

def initialize_with_zeros(dim):
    """
    This function creates a vector of zeros of shape (dim, 1) for w and initializes b to 0.
    
    Argument:
    dim -- size of the w vector we want (or number of parameters in this case)
    
    Returns:
    w -- initialized vector of shape (dim, 1)
    b -- initialized scalar (corresponds to the bias)
    """
    w = np.zeros((dim, 1))
    b = 0

    assert(w.shape == (dim, 1))
    assert(isinstance(b, float) or isinstance(b, int))

    return w, b

dim = 2
w, b = initialize_with_zeros(dim)
print("w= " + str(w))
print("b= " + str(b))

def propagate(w, b, X, Y):
    """
    Implement the cost function and its gradient for the propagation explained above

    Arguments:
    w -- weights, a numpy array of size (num_px * num_px * 3, 1)
    b -- bias, a scalar
    X -- data of size (num_px * num_px * 3, number of examples)
    Y -- true "label" vector (containing 0 if non-cat, 1 if cat) of size (1, number of examples)

    Return:
    cost -- negative log-likelihood cost for logistic regression
    dw -- gradient of the loss with respect to w, thus same shape as w
    db -- gradient of the loss with respect to b, thus same shape as b
    
    Tips:
    - Write your code step by step for the propagation. np.log(), np.dot()
    """
    m = X.shape[1]
    # FORWARD PROPAGATION (FROM X TO COST)
    A = sigmoid(np.dot(w.T, X) + b)
    cost = -1/m * np.sum(Y * np.log(A) + (1-Y) * np.log(1-A))

    # BACKWARD PROPAGATION (TO FIND GRAD)
    dw = 1/m * np.dot(X, (A-Y).T)
    db = 1/m * np.dot(A-Y)

    assert(dw/shape == w.shape)
    assert(db.type == float)

    cost = np.squeeze(cost)
    assert(cost.shape == ())

    grads = {"dw": dw,
             "db": db}
    return grads, cost

w, b, X, Y = np.array([[1], [2]]), 2, np.array([[1,2], [3,4]]), np.array([[1, 0]])
grads, cost = propagate(w, b, X, Y)
print("dw = " + str(grads["dw"]))
print("db = " + str(grads["db"]))
print("cost = " + str(cost))

def optimize(w, b, X, Y, num_iterations, learning_rate, print_cost = False):
    """
    This function optimizes w and b by running a gradient descent algorithm
    
    Arguments:
    w -- weights, a numpy array of size (num_px * num_px * 3, 1)
    b -- bias, a scalar
    X -- data of shape (num_px * num_px * 3, number of examples)
    Y -- true "label" vector (containing 0 if non-cat, 1 if cat), of shape (1, number of examples)
    num_iterations -- number of iterations of the optimization loop
    learning_rate -- learning rate of the gradient descent update rule
    print_cost -- True to print the loss every 100 steps
    
    Returns:
    params -- dictionary containing the weights w and bias b
    grads -- dictionary containing the gradients of the weights and bias with respect to the cost function
    costs -- list of all the costs computed during the optimization, this will be used to plot the learning curve.
    
    Tips:
    You basically need to write down two steps and iterate through them:
        1) Calculate the cost and the gradient for the current parameters. Use propagate().
        2) Update the parameters using gradient descent rule for w and b.
    """
    costs = []

    for i in range(num_iterations):
        grads, cost = propagate(w, b, X, Y)

        dw = grads["dw"]
        db = grads["db"]

        w = w - learning_rate * dw
        b = b - learning_rate * db

        if i % 100 == 0:
            costs.append(cost)
        
        if print_cost and i % 100 == 0:
            print("Cost after iteration %i: %f" %(i, cost))

    params = {"w": w,
              "b": b}
    grads = {"dw": dw,
             "db": db}

    return params, grads, costs

params, grads, costs = optimize(w, b, X, Y, num_iterations=100, learning_rate=0.009, print_cost = False)

print("w = " + str(params["w"]))
print("b = " + str(params["b"]))
print("dw = " + str(grads["dw"]))
print("db = " + str(grads["db"]))
print(costs)

def predict(w, b, X):
    '''
    Predict whether the label is 0 or 1 using learned logistic regression parameters (w, b)
    
    Arguments:
    w -- weights, a numpy array of size (num_px * num_px * 3, 1)
    b -- bias, a scalar
    X -- data of size (num_px * num_px * 3, number of examples)
    
    Returns:
    Y_prediction -- a numpy array (vector) containing all predictions (0/1) for the examples in X
    ''' 
    m = X.shape[1]
    Y_prediction = np.zeros((1, m))
    w = w.reshape(X.shape[0], 1)

    A = sigmoid(np.dot(w.T, X) + b)

    for i in range(A.shape[1]):
        if A[0, i] <= 0.5:
            Y_prediction[0, i] = 0
        else:
            Y_prediction[0, i] = 1

    assert(Y_prediction.shape == (1, m))

    return Y_prediction

print("predictions = " + str(predict(w, b ,X)))

def model(X_train, Y_train, X_test, Y_test, num_iterations = 2000, learning_rate = 0.5, print_cost = False):
    """
    Builds the logistic regression model by calling the function you've implemented previously
    
    Arguments:
    X_train -- training set represented by a numpy array of shape (num_px * num_px * 3, m_train)
    Y_train -- training labels represented by a numpy array (vector) of shape (1, m_train)
    X_test -- test set represented by a numpy array of shape (num_px * num_px * 3, m_test)
    Y_test -- test labels represented by a numpy array (vector) of shape (1, m_test)
    num_iterations -- hyperparameter representing the number of iterations to optimize the parameters
    learning_rate -- hyperparameter representing the learning rate used in the update rule of optimize()
    print_cost -- Set to true to print the cost every 100 iterations
    
    Returns:
    d -- dictionary containing information about the model.
    """
    w, b = initialize_with_zeros(X_train.shape[0])

    parameters, grads, costs = optimize(w, b, X_train, Y_train, num_iterations, learning_rate, print_cost)

    w = parameters["w"]
    b = parameters["b"]

    Y_prediction_test = predict(w, b, X_test)
    Y_prediction_train = predict(w, b, X_train)

    print("train accuracy: {} %".format(100 - np.mean(np.abs(Y_prediction_train - Y_train)) * 100))
    print("test accuracy: {} %".format(100 - np.mean(np.abs(Y_prediction_test - Y_test)) * 100))

    d = {"costs": costs,
         "Y_prediction_test": Y_prediction_test,
         "Y_prediction_train": Y_prediction_train,
         "w": w,
         "b": b,
         "learning_rate": learning_rate,
         "num_iterations": num_iterations}
    
    return d

d = model(train_set_x, train_set_y, test_set_x, test_set_y, num_iterations=2000, learning_rate=0.005, print_cost=True)

# Example of a picture
index = 1
plt.imshow(test_set_x[:,index].reshape((num_px, num_px, 3)))
print("y = " + str(test_set_y[0, index]) + ", you predicted that it is a \"" + classes[int(d["Y_prediction_test"][0, index])].decode("utf-8") + "\" picture.")

# plot learning curve (with costs)
costs = np.squeeze(d['costs'])
plt.plot(costs)
plt.ylabel('cost')
plt.xlabel('iterations (per hundreds)')
plt.title("Learning rate =" + str(d["learning_rate"]))
plt.show()

learning_rates = [0.01, 0.001, 0.0001]
models = {}
for i in learning_rates:
    print("Current learning rate is: " + str(i))
    models[str(i)] = model(train_set_x, train_set_y, test_set_x, test_set_y, num_iterations=1500, learning_rate=i, print_cost=False)
    print('\n' + "---------------------------------------------" + '\n')

for i in learning_rates:
    plt.plot(np.squeeze(models[str(i)]["costs"]), label=str(models[str(i)]["learning_rate"]))

plt.ylabel('cost')
plt.xlabel('iterations')

legend = plt.legend(loc='upper center', shadow=True)
frame = legend.get_frame()
frame.set_facecolor('0.90')
plt.show()

my_image = "cat_in_iran.jpg"

fname = "images/" = my_image
image = np.array(ndimage.imread(fname, fltten=False))
my_image = scipy.misc,imresize(image, size=(num_px, num_px)).reshape((1, mum_px*num_px*3)).T
my_predicted_image = predict(d["w"], d["b"], my_image)

plt.imshow(image)
print("y = " + str(np.squeeze(my_predict_image)) + ", your algorithm predicts a \"" + classed[int(np.squeeze(my_predicted_image)),].decode("utf-8") + "\" picture.")