In [1]:
import pandas
import matplotlib.pyplot as plt
# Read data from csv
pga = pandas.read_csv("pga.csv")
# Normalize the data
pga.distance = (pga.distance - pga.distance.mean()) / pga.distance.std()
pga.accuracy = (pga.accuracy - pga.accuracy.mean()) / pga.accuracy.std()
print(pga.head())
plt.scatter(pga.distance, pga.accuracy)
plt.xlabel('normalized distance')
plt.ylabel('normalized accuracy')
plt.show()
In [3]:
# accuracyi=θ1distancei+θ0+ϵ
from sklearn.linear_model import LinearRegression
import numpy as np
# We can add a dimension to an array by using np.newaxis
print("Shape of the series:", pga.distance.shape)
print("Shape with newaxis:", pga.distance[:, np.newaxis].shape)
# The X variable in LinearRegression.fit() must have 2 dimensions
lm = LinearRegression()
lm.fit(pga.distance[:, np.newaxis], pga.accuracy)
theta1 = lm.coef_[0]
print (theta1)
In [9]:
# The cost function of a single variable linear model
def cost(theta0, theta1, x, y):
# Initialize cost
J = 0
# The number of observations
m = len(x)
# Loop through each observation
for i in range(m):
# Compute the hypothesis
h = theta1 * x[i] + theta0
# Add to cost
J += (h - y[i])**2
# Average and normalize cost
J /= (2*m)
return J
# The cost for theta0=0 and theta1=1
print(cost(0, 1, pga.distance, pga.accuracy))
theta0 = 100
theta1s = np.linspace(-3,2,100)
costs = []
for theta1 in theta1s:
costs.append(cost(theta0, theta1, pga.distance, pga.accuracy))
plt.plot(theta1s, costs)
plt.show()
In [10]:
import numpy as np
from mpl_toolkits.mplot3d import Axes3D
# Example of a Surface Plot using Matplotlib
# Create x an y variables
x = np.linspace(-10,10,100)
y = np.linspace(-10,10,100)
# We must create variables to represent each possible pair of points in x and y
# ie. (-10, 10), (-10, -9.8), ... (0, 0), ... ,(10, 9.8), (10,9.8)
# x and y need to be transformed to 100x100 matrices to represent these coordinates
# np.meshgrid will build a coordinate matrices of x and y
X, Y = np.meshgrid(x,y)
#print(X[:5,:5],"\n",Y[:5,:5])
# Compute a 3D parabola
Z = X**2 + Y**2
# Open a figure to place the plot on
fig = plt.figure()
# Initialize 3D plot
ax = fig.gca(projection='3d')
# Plot the surface
ax.plot_surface(X=X,Y=Y,Z=Z)
plt.show()
# Use these for your excerise
theta0s = np.linspace(-2,2,100)
theta1s = np.linspace(-2,2, 100)
COST = np.empty(shape=(100,100))
# Meshgrid for paramaters
T0S, T1S = np.meshgrid(theta0s, theta1s)
# for each parameter combination compute the cost
for i in range(100):
for j in range(100):
COST[i,j] = cost(T0S[0,i], T1S[j,0], pga.distance, pga.accuracy)
# make 3d plot
fig2 = plt.figure()
ax = fig2.gca(projection='3d')
ax.plot_surface(X=T0S,Y=T1S,Z=COST)
plt.show()
In [11]:
def partial_cost_theta1(theta0, theta1, x, y):
# Hypothesis
h = theta0 + theta1*x
# Hypothesis minus observed times x
diff = (h - y) * x
# Average to compute partial derivative
partial = diff.sum() / (x.shape[0])
return partial
partial1 = partial_cost_theta1(0, 5, pga.distance, pga.accuracy)
print("partial1 =", partial1)
# Partial derivative of cost in terms of theta0
def partial_cost_theta0(theta0, theta1, x, y):
# Hypothesis
h = theta0 + theta1*x
# Difference between hypothesis and observation
diff = (h - y)
# Compute partial derivative
partial = diff.sum() / (x.shape[0])
return partial
partial0 = partial_cost_theta0(1, 1, pga.distance, pga.accuracy)
In [12]:
# x is our feature vector -- distance
# y is our target variable -- accuracy
# alpha is the learning rate
# theta0 is the intial theta0
# theta1 is the intial theta1
def gradient_descent(x, y, alpha=0.1, theta0=0, theta1=0):
max_epochs = 1000 # Maximum number of iterations
counter = 0 # Intialize a counter
c = cost(theta1, theta0, pga.distance, pga.accuracy) ## Initial cost
costs = [c] # Lets store each update
# Set a convergence threshold to find where the cost function in minimized
# When the difference between the previous cost and current cost
# is less than this value we will say the parameters converged
convergence_thres = 0.000001
cprev = c + 10
theta0s = [theta0]
theta1s = [theta1]
# When the costs converge or we hit a large number of iterations will we stop updating
while (np.abs(cprev - c) > convergence_thres) and (counter < max_epochs):
cprev = c
# Alpha times the partial deriviative is our updated
update0 = alpha * partial_cost_theta0(theta0, theta1, x, y)
update1 = alpha * partial_cost_theta1(theta0, theta1, x, y)
# Update theta0 and theta1 at the same time
# We want to compute the slopes at the same set of hypothesised parameters
# so we update after finding the partial derivatives
theta0 -= update0
theta1 -= update1
# Store thetas
theta0s.append(theta0)
theta1s.append(theta1)
# Compute the new cost
c = cost(theta0, theta1, pga.distance, pga.accuracy)
# Store updates
costs.append(c)
counter += 1 # Count
return {'theta0': theta0, 'theta1': theta1, "costs": costs}
print("Theta1 =", gradient_descent(pga.distance, pga.accuracy)['theta1'])
descend = gradient_descent(pga.distance, pga.accuracy, alpha=.01)
plt.scatter(range(len(descend["costs"])), descend["costs"])
plt.show()
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