def cluster_means(k: int,
imputs: List[Vector],
assignments: List[int]) -> List[Vector]:
# cluster i contains the inputs whose assignment is i
clusters = [[] for i in range(k)]
for input, assignment in zip(inputs, assignments):
clusters[assignment].append(input)
# if cluster is empty then just use a random point
return [vector_mean(cluster) if cluster else random.choice(inputs)
for cluster in clusters]
def least_squares_fit(xs: List[Vector],
ys: Vector,
alpha: float,
learning_rate: float = 0.001,
num_steps: int = 1000,
batch_size: int = 1) -> Vector:
"""Finds beta that minimizes the sum of squared errors
assuming the model dot(x, beta)"""
# start with random guess
guess = [random.random() for _ in xs[0]]
for _ in tqdm.trange(num_steps, desc = "least squares fit"):
for start in range(0,len(xs), batch_size):
batch_xs = xs[start:start+batch_size]
batch_ys = ys[start:start+batch_size]
gradient = vector_mean([sqerror_ridge_gradient(x,y,guess,alpha)
for x,y in zip(batch_xs, batch_ys)])
guess = gradient_step(guess, gradient, -learning_rate)
return guess
def train(self, inputs):
# choose k random points as the initial means
self.means = random.sample(inputs, self.k)
assignments = None
while True:
# Find new assignments
new_assignments = map(self.classify, inputs)
# if no assignments have changed, we're done
if assignments == new_assignments:
return
# Otherwise keep the new assignments
assignments = new_assignments
# And compute new means based on the new assignments
for i in range(self.k):
# find all the points assigned to cluster i
i_points = [p for p, a in zip(inputs, assignments) if a == i]
# make sure i_points is not empty so don't divide by 0
if i_points:
self.means[i] = vector_mean(i_points)
predicted = slope * x + intercept # prediction of a linear model
error = (predicted - y)
squared_error = error**2 # minimize squared error
grad = [2 * error * x, 2 * error] # using its gradient
return grad
from vector_operations import vector_mean
# start with a random slope and intercept
theta = [random.uniform(-1, 1), random.uniform(-1, 1)]
learning_rate = 0.001
for epoch in range(5000):
# compute mean of the gradients
grad = vector_mean([linear_gradient(x, y, theta) for x, y in input])
# Take a step in that direction
theta = gradient_step(theta, grad, -learning_rate)
print(epoch, theta)
slope, intercept = theta
assert 19.9 < slope < 20.1 # slope should be close to 20
assert 4.9 < intercept < 5.1 # intercept should be close to 5
"""Let's solve the above problem in minibatches"""
from typing import TypeVar, List, Iterator
T = TypeVar('T') # this allows us to type generic functions
def minibatches(dataset: List[T],
batch_size=int,
# choose the last merged of our clusters
next_cluster = min(clusters, key = get_merge_order)
clusters = [c for c in clusters if c != next_cluster]
# and add its children to the list (i.e. unmerge it)
clusters.extend(get_children(next_cluster))
# once we have enough clusters, return those
return clusters
three_clusters = [get_values(cluster) for cluster in generate_clusters(base_cluster, 3)]
from matplotlib import pyplot as plt
for i, cluster, marker, color in zip([1,2,3],
three_clusters,
['D','o', '*'],
['r','g','b']):
xs, ys = zip(*cluster) # magic unzipping trick
plt.scatter(xs, ys, color = color, marker = marker)
# put a number at the mean of a cluster
x,y = vector_mean(cluster)
plt.plot(x,y, marker = '$' + str(i) + '$', color = 'black')
plt.title("User Locations -- 3 Bottom-up Clusters, Min")
plt.xlabel("blocks east of city center")
plt.ylabel("blocks north of city center")
plt.show()
guess = [random.random() for _ in xs[0]]
from linear_regression import total_sum_of_squares;
def multiple_r_squared(xs: List[Vector], ys: Vector, beta: Vector) -> float:
sum_of_squared_errors = sum(error(x,y,beta) ** 2
for x,y in zip(xs,ys))
return 1.0 - sum_of_squared_errors/total_sum_of_squares(ys)
print(multiple_r_squared(inputs,daily_minutes_good,beta))
for _ in tqdm.trange(num_steps, desc = "least squares fit"):
for start in range(0,len(xs), batch_size):
batch_xs = xs[start:start+batch_size]
batch_ys = ys[start:start+batch_size]
gradient = vector_mean([sqerror_gradient(x,y,guess)
for x,y in zip(batch_xs, batch_ys)])
guess = gradient_step(guess, gradient, -learning_rate)
return guess
num_friends = [100.0,49,41,40,25,21,21,19,19,18,18,16,15,15,15,15,14,14,13,13,13,13,12,12,11,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,8,8,8,8,8,8,8,8,8,8,8,8,8,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
daily_minutes = [1,68.77,51.25,52.08,38.36,44.54,57.13,51.4,41.42,31.22,34.76,54.01,38.79,47.59,49.1,27.66,41.03,36.73,48.65,28.12,46.62,35.57,32.98,35,26.07,23.77,39.73,40.57,31.65,31.21,36.32,20.45,21.93,26.02,27.34,23.49,46.94,30.5,33.8,24.23,21.4,27.94,32.24,40.57,25.07,19.42,22.39,18.42,46.96,23.72,26.41,26.97,36.76,40.32,35.02,29.47,30.2,31,38.11,38.18,36.31,21.03,30.86,36.07,28.66,29.08,37.28,15.28,24.17,22.31,30.17,25.53,19.85,35.37,44.6,17.23,13.47,26.33,35.02,32.09,24.81,19.33,28.77,24.26,31.98,25.73,24.86,16.28,34.51,15.23,39.72,40.8,26.06,35.76,34.76,16.13,44.04,18.03,19.65,32.62,35.59,39.43,14.18,35.24,40.13,41.82,35.45,36.07,43.67,24.61,20.9,21.9,18.79,27.61,27.21,26.61,29.77,20.59,27.53,13.82,33.2,25,33.1,36.65,18.63,14.87,22.2,36.81,25.53,24.62,26.25,18.21,28.08,19.42,29.79,32.8,35.99,28.32,27.79,35.88,29.06,36.28,14.1,36.63,37.49,26.9,18.58,38.48,24.48,18.95,33.55,14.24,29.04,32.51,25.63,22.22,19,32.73,15.16,13.9,27.2,32.01,29.27,33,13.74,20.42,27.32,18.23,35.35,28.48,9.08,24.62,20.12,35.26,19.92,31.02,16.49,12.16,30.7,31.22,34.65,13.13,27.51,33.2,31.57,14.1,33.42,17.44,10.12,24.42,9.82,23.39,30.93,15.03,21.67,31.09,33.29,22.61,26.89,23.48,8.38,27.81,32.35,23.84]
daily_hours = [dm / 60 for dm in daily_minutes]
outlier = num_friends.index(100) # index of outlier
num_friends_good = [x
for i, x in enumerate(num_friends)
if i != outlier]
daily_minutes_good = [x