I am trying to implement gradient descent after transforming some random data using sklearns polynomial transformer. My code works when not using polynomial features, but gives really high coefficients when transforming.
Is there an issue with my code (below)?
l= 20
np.random.seed(0)
X = 2 - 3 * np.random.normal(0, 1, l)
y = X - 2 * (X ** 2) + 0.5 * (X ** 3) + np.random.normal(-3, 3, l)
plt.scatter(X,y, s=10)
plt.show()
X = X.reshape(-1,1)
m, n = X.shape
p = PolynomialFeatures(degree=2)
xbp = p.fit_transform(X)
# initiate coefs
theta = np.ones((xbp.shape[1], 1))
for _ in range(500):
err = xbp.dot(theta) - y.reshape(-1, 1)
gradients = 2 / m * xbp.T.dot(err)
theta -= 0.01 * gradients
print(theta)
print()
print(LinearRegression().fit(xbp, y).coef_)
output
[[ 802.60118234]
[ 360.65857329]
[12234.00939771]]
[ 0. 8.48492679 -1.62853134]
code snippet
Thanks
While implementing Gradient Descent Algorithm in linear regression, the prediction that my algorithm is making and the resulting regression line are coming as a wrong output. Could anyone please have a look at my implementation and help me out? Also, please guide me that how can I know what value of "learning rate" and "number of iterations" to choose in specific regression problem?
theta0 = 0 #first parameter
theta1 = 0 #second parameter
alpha = 0.001 #learning rate (denoted by alpha)
num_of_iterations = 100 #total number of iterations performed by Gradient Descent
m = float(len(X)) #total number of training examples
for i in range(num_of_iterations):
y_predicted = theta0 + theta1 * X
derivative_theta0 = (1/m) * sum(y_predicted - Y)
derivative_theta1 = (1/m) * sum(X * (y_predicted - Y))
temp0 = theta0 - alpha * derivative_theta0
temp1 = theta1 - alpha * derivative_theta1
theta0 = temp0
theta1 = temp1
print(theta0, theta1)
y_predicted = theta0 + theta1 * X
plt.scatter(X,Y)
plt.plot(X, y_predicted, color = 'red')
plt.show()
Resulting regression line about which I need some help
Your learning rate is to high, I got it working by reducing the learning rate to alpha = 0.0001.
I wrote some code that performs gradient descent on a couple of data points.
For some reason the curve is not converging correctly, but I have no idea why that is. I always end up with an exploding tail.
Am I doing one of the computations wrong? Am I actually getting stuck in a local minimum or is it something else?
Here is my code:
import numpy as np
import matplotlib.pyplot as plt
def estimate(weights, x, order):
est = 0
for i in range(order):
est += weights[i] * x ** i
return est
def cost_function(x, y, weights, m):
cost = 0
for i in range(m-1):
cost += (((weights[i] * x ** i) - y) ** 2)
return (np.sum(cost ** 2) / ( 2 * m ))
def descent(A, b, iterations, descent_rate, order):
x = A.T[0]
y = b.reshape(4)
# features
ones = np.vstack(np.ones(len(A)))
x = np.vstack(A.T[0])
x2 = np.vstack(A.T[0] ** 2)
# Our feature matrix
features = np.concatenate((ones,x,x2), axis = 1).T
# Initialize our coefficients to zero
weights = np.zeros(order + 1)
m = len(y)
# gradient descent
for i in range(iterations):
est = estimate(weights, x, order).T
difference = est - y
weights = weights + (-descent_rate * (1/m) * np.matmul(difference, features.T)[0])
cost = cost_function(x, y, weights, m)
print(cost)
plt.scatter(x,y)
u = np.linspace(0,3,100)
plt.plot(u, (u ** 2) * weights[2] + u * weights[1] + weights[0], '-')
plt.show()
A = np.array(((0,1),
(1,1),
(2,1),
(3,1)))
b = np.array((1,2,0,3), ndmin = 2 ).T
iterations = 150
descent_rate = 0.01
order = 2
descent(A, b, iterations, descent_rate, order)
I would like to avoid getting stuck in such a minimum. I have attempted setting the initial weights to random values but to no avail, sometimes it dips a bit more but then gives me the same behaviour again.
Here is the one of the plots that I am getting:
And here is the expected result obtained by a least squares solution:
Your estimate function should be
def estimate(weights, x, order):
est = 0
for i in range(order+1):
est += weights[i] * x ** i
return est
Better yet, since the order information is already present in the size of the weights vector, remove the redundancy with:
def estimate(weights, x):
est = 0
for i in range(len(weights)):
est += weights[i] * x ** i
return est
This is what I got when using your code and running 2000 iterations:
I'm starting the ML journey and I'm having troubles with this coding exercise
here is my code
import numpy as np
import pandas as pd
import scipy.optimize as op
# Read the data and give it labels
data = pd.read_csv('ex2data2.txt', header=None, name['Test1', 'Test2', 'Accepted'])
# Separate the features to make it fit into the mapFeature function
X1 = data['Test1'].values.T
X2 = data['Test2'].values.T
# This function makes more features (degree)
def mapFeature(x1, x2):
degree = 6
out = np.ones((x1.shape[0], sum(range(degree + 2))))
curr_column = 1
for i in range(1, degree + 1):
for j in range(i+1):
out[:,curr_column] = np.power(x1, i-j) * np.power(x2, j)
curr_column += 1
return out
# Separate the data into training and target, also initialize theta
X = mapFeature(X1, X2)
y = np.matrix(data['Accepted'].values).T
m, n = X.shape
cols = X.shape[1]
theta = np.matrix(np.zeros(cols))
#Initialize the learningRate(sigma)
learningRate = 1
# Define the Sigmoid Function (Output between 0 and 1)
def sigmoid(z):
return 1 / (1 + np.exp(-z))
def cost(theta, X, y, learningRate):
# This is require to make the optimize function work
theta = theta.reshape(-1, 1)
error = sigmoid(X # theta)
first = np.multiply(-y, np.log(error))
second = np.multiply(1 - y, np.log(1 - error))
j = np.sum((first - second)) / m + (learningRate * np.sum(np.power(theta, 2)) / 2 * m)
return j
# Define the gradient of the cost function
def gradient(theta, X, y, learningRate):
# This is require to make the optimize function work
theta = theta.reshape(-1, 1)
error = sigmoid(X # theta)
grad = (X.T # (error - y)) / m + ((learningRate * theta) / m)
grad_no = (X.T # (error - y)) / m
grad[0] = grad_no[0]
return grad
Result = op.minimize(fun=cost, x0=theta, args=(X, y, learningRate), method='TNC', jac=gradient)
opt_theta = np.matrix(Result.x)
def predict(theta, X):
sigValue = sigmoid(X # theta.T)
p = sigValue >= 0.5
return p
p = predict(opt_theta, X)
print('Train Accuracy: {:f}'.format(np.mean(p == y) * 100))
So, when the learningRate = 1, the accuracy should be around 83,05% but I'm getting 80.5% and when the learningRate = 0, the accuracy should be 91.52% but I'm getting 87.28%
So the question is What am I doing wrong? Why my accuracy is below the problem default answer?
Hope someone can guide me in the right direction. Thanks!
P.D: Here is the dataset, maybe it can help
https://raw.githubusercontent.com/TheGirlWhiteWithBandages/Machine-Learning-Algorithms/master/Logistic%20Regression/ex2data2.txt
Hey guys I found a way to make it even better!
Here is the code
import numpy as np
import pandas as pd
import scipy.optimize as op
from sklearn.preprocessing import PolynomialFeatures
# Read the data and give it labels
data = pd.read_csv('ex2data2.txt', header=None, names=['Test1', 'Test2', 'Accepted'])
# Separate the data into training and target
X = (data.iloc[:, 0:2]).values
y = (data.iloc[:, 2:3]).values
# Modify the features to a certain degree (Polynomial)
poly = PolynomialFeatures(6)
m = y.size
XX = poly.fit_transform(data.iloc[:, 0:2].values)
# Initialize Theta
theta = np.zeros(XX.shape[1])
# Define the Sigmoid Function (Output between 0 and 1)
def sigmoid(z):
return(1 / (1 + np.exp(-z)))
# Define the Regularized cost function
def costFunctionReg(theta, reg, *args):
# This is require to make the optimize function work
h = sigmoid(XX # theta)
first = np.log(h).T # - y
second = np.log(1 - h).T # (1 - y)
J = (1 / m) * (first - second) + (reg / (2 * m)) * np.sum(np.square(theta[1:]))
return J
# Define the Regularized gradient function
def gradientReg(theta, reg, *args):
theta = theta.reshape(-1, 1)
h = sigmoid(XX # theta)
grad = (1 / m) * (XX.T # (h - y)) + (reg / m) * np.r_[[[0]], theta[1:]]
return grad.flatten()
# Define the predict Function
def predict(theta, X):
sigValue = sigmoid(X # theta.T)
p = sigValue >= 0.5
return p
# A loop to test between different values for sigma (reg parameter)
for i, Sigma in enumerate([0, 1, 100]):
# Optimize costFunctionReg
res2 = op.minimize(costFunctionReg, theta, args=(Sigma, XX, y), method=None, jac=gradientReg)
# Get the accuracy of the model
accuracy = 100 * sum(predict(res2.x, XX) == y.ravel()) / y.size
# Get the Error between different weights
error1 = costFunctionReg(res2.x, Sigma, XX, y)
# print the accuracy and error
print('Train accuracy {}% with Lambda = {}'.format(np.round(accuracy, decimals=4), Sigma))
print(error1)
Thanks for all your help!
try out this:
# import library
import pandas as pd
import numpy as np
dataset = pd.read_csv('ex2data2.csv',names = ['Test #1','Test #2','Accepted'])
# splitting to x and y variables for features and target variable
x = dataset.iloc[:,:-1].values
y = dataset.iloc[:,-1].values
print('x[0] ={}, y[0] ={}'.format(x[0],y[0]))
m, n = x.shape
print('#{} Number of training samples, #{} features per sample'.format(m,n))
# import library FeatureMapping
from sklearn.preprocessing import PolynomialFeatures
# We also add one column of ones to interpret theta 0 (x with power of 0 = 1) by
include_bias as True
pf = PolynomialFeatures(degree = 6, include_bias = True)
x_poly = pf.fit_transform(x)
pd.DataFrame(x_poly).head(5)
m,n = x_poly.shape
# define theta as zero
theta = np.zeros(n)
# define hyperparameter λ
lambda_ = 1
# reshape (-1,1) because we just have one feature in y column
y = y.reshape(-1,1)
def sigmoid(z):
return 1/(1+np.exp(-z))
def lr_hypothesis(x,theta):
return np.dot(x,theta)
def compute_cost(theta,x,y,lambda_):
theta = theta.reshape(n,1)
infunc1 = -y*(np.log(sigmoid(lr_hypothesis(x,theta)))) - ((1-y)*(np.log(1 - sigmoid(lr_hypothesis(x,theta)))))
infunc2 = (lambda_*np.sum(theta[1:]**2))/(2*m)
j = np.sum(infunc1)/m+ infunc2
return j
# gradient[0] correspond to gradient for theta(0)
# gradient[1:] correspond to gradient for theta(j) j>0
def compute_gradient(theta,x,y,lambda_):
gradient = np.zeros(n).reshape(n,)
theta = theta.reshape(n,1)
infunc1 = sigmoid(lr_hypothesis(x,theta))-y
gradient_in = np.dot(x.transpose(),infunc1)/m
gradient[0] = gradient_in[0,0] # theta(0)
gradient[1:] = gradient_in[1:,0]+(lambda_*theta[1:,]/m).reshape(n-1,) # theta(j) ; j>0
gradient = gradient.flatten()
return gradient
You can now test your cost and gradient without optimization. Th below code will optimize the model:
# hyperparameters
m,n = x_poly.shape
# define theta as zero
theta = np.zeros(n)
# define hyperparameter λ
lambda_array = [0, 1, 10, 100]
import scipy.optimize as opt
for i in range(0,len(lambda_array)):
# Train
print('======================================== Iteration {} ===================================='.format(i))
optimized = opt.minimize(fun = compute_cost, x0 = theta, args = (x_poly, y,lambda_array[i]),
method = 'TNC', jac = compute_gradient)
new_theta = optimized.x
# Prediction
y_pred_train = predictor(x_poly,new_theta)
cm_train = confusion_matrix(y,y_pred_train)
t_train,f_train,acc_train = acc(cm_train)
print('With lambda = {}, {} correct, {} wrong ==========> accuracy = {}%'
.format(lambda_array[i],t_train,f_train,acc_train*100))
Now you should see output like this :
=== Iteration 0 === With lambda = 0, 104 correct, 14 wrong ==========> accuracy = 88.13559322033898%
=== Iteration 1 === With lambda = 1, 98 correct, 20 wrong ==========> accuracy = 83.05084745762711%
=== Iteration 2 === With lambda = 10, 88 correct, 30 wrong ==========> accuracy = 74.57627118644068%
=== Iteration 3 === With lambda = 100, 72 correct, 46 wrong ==========> accuracy = 61.016949152542374%
I am trying to mimic the gradient descent algorithm for linear regression from Andrew NG's Machine learning course to Python, but for some reason my implementation is not working correctly.
Here's my implementation in Octave, it works correctly:
function [theta, J_history] = gradientDescent(X, y, theta, alpha, num_iters)
J_history = zeros(num_iters, 1);
for iter = 1:num_iters
prediction = X*theta;
margin_error = prediction - y;
gradient = 1/m * (alpha * (X' * margin_error));
theta = theta - gradient;
J_history(iter) = computeCost(X, y, theta);
end
end
However, when I translate this to Python for some reason it is not giving me accurate results. The cost seems to be going up rather than descending.
Here's my implementation in Python:
def gradientDescent(x, y, theta, alpha, iters):
m = len(y)
J_history = np.matrix(np.zeros((iters,1)))
for i in range(iters):
prediction = x*theta.T
margin_error = prediction - y
gradient = 1/m * (alpha * (x.T * margin_error))
theta = theta - gradient
J_history[i] = computeCost(x,y,theta)
return theta,J_history
My code is compiling and there isn't anything wrong. Please note this is theta:
theta = np.matrix(np.array([0,0]))
Alpha and iters is set to this:
alpha = 0.01
iters = 1000
When I run it, opt_theta, cost = gradientDescent(x, y, theta, alpha, iters) and print out opt_theta, I get this:
matrix([[ 2.36890383e+16, -1.40798902e+16],
[ 2.47503758e+17, -2.36890383e+16]])
when I should get this:
matrix([[-3.24140214, 1.1272942 ]])
What am I doing wrong?
Edit:
Cost function
def computeCost(x, y, theta):
# Get length of data set
m = len(y)
# We get theta transpose because we are working with a numpy array [0,0] for example
prediction = x * theta.T
J = 1/(2*m) * np.sum(np.power((prediction - y), 2))
return J
Look there:
>>> A = np.matrix([3,3,3])
>>> B = np.matrix([[1,1,1], [2,2,2]])
>>> A-B
matrix([[2, 2, 2],
[1, 1, 1]])
Matrices are broadcasted together.
"it's because np.matrix inherits from np.array. np.matrix overrides multiplication, but not addition and subtraction"
In yours situation theta(1x2) subtract gradient(2x1) and in result you have got 2x2. Try to transpose gradient before subtracting.
theta = theta - gradient.T