In [101]:
import math
import numpy as np
import matplotlib.pyplot as plt
import plotly
import plotly.plotly as py
import plotly.graph_objs as go
%matplotlib inline
plotly.offline.init_notebook_mode(connected=True)
from IPython.core.display import display, HTML, Markdown
# The polling here is to ensure that plotly.js has already been loaded before
# setting display alignment in order to avoid a race condition.
display(HTML(
'<script>'
'var waitForPlotly = setInterval( function() {'
'if( typeof(window.Plotly) !== "undefined" ){'
'MathJax.Hub.Config({ SVG: { font: "STIX-Web" }, displayAlign: "center" });'
'MathJax.Hub.Queue(["setRenderer", MathJax.Hub, "SVG"]);'
'clearInterval(waitForPlotly);'
'}}, 250 );'
'</script>'
))
import colorlover as cl
def formatted(f):
return format(f, '.2f').rstrip('0').rstrip('.')
In [102]:
seed_train = 1
seed_test = 2
sample_sizes = np.arange(100, 10100, 100)
error = {}
In [103]:
numFeatures = 10
def generateDataSample(x):
"""
Generate a noisy data sample from a given data point in the range [0,1]
:param x: A scalar dependent variable for which to calculate the output y_noisy
:returns: The output with gaussian noise added
"""
y = 1 - x - math.sin(-2 * math.pi * x ** 3) * math.cos(-2 * math.pi * x ** 3) * math.exp(-x ** 4)
sigma = 0.1
noise = sigma * np.random.random()
y_noisy = y + noise
return y_noisy
centers = np.linspace(0.0, 1.0, numFeatures)
widthConstant = (1.0 - 0.0) / numFeatures / 10
widths = np.ones(numFeatures,) * widthConstant
alpha = 0.9
def phiOutput(input):
"""
Get the output of the Gaussian features for a given input variable(s)
:param input: A single or vector of dependent variables with size [Ns] for which to calculate the FA features
:returns: A vector of feature outputs with size [NumFeats x Ns]
"""
global numFeatures, centers, widths
if np.size(input) == 1:
phi = np.exp(-np.divide(np.square(input - centers), widths))
elif np.size(input) > 1:
numEvals = np.shape(input)[0]
# Repeat vectors to vectorize output calculation
inputMat = np.array([input, ] * numFeatures)
centersMat = np.array([centers, ] * numEvals).transpose()
widthsMat = np.array([widths, ] * numEvals).transpose()
phi = np.exp(-np.divide(np.square(inputMat - centersMat), widthsMat))
return phi
def f(input, *user_theta):
"""
Get the FA output for a given input variable(s)
:param input: A single or vector of dependent variables with size [Ns] for which to calculate the FA features
:param user_theta: (Variable argument) A vector of theta variables to apply to the FA. If left blank the method will default to using the trained thetas in self.theta. This is only used for visualization purposes.
:returns: A vector of function approximator outputs with size [Ns]
"""
global theta
phi = phiOutput(input)
if not user_theta:
Theta = theta
else:
Theta = np.array(user_theta)
out = np.dot(phi, Theta.transpose())
return out
generateDataSample = np.vectorize(generateDataSample)
f = np.vectorize(f)
model = 'GD'
error[model] = {}
error[model]['train'] = []
error[model]['test'] = []
for sample_size in sample_sizes:
theta = np.random.random(numFeatures)
xHistory = []
yHistory = []
error_evolution = []
np.random.seed(seed_train)
def train_GD(maxIter):
global theta, xHistory, yHistory, alpha, E
iterationCount = 0
while iterationCount < maxIter:
#----------------------#
# Training Algorithm #
#----------------------#
# Draw a random sample on the interval [0,1]
x = np.random.random()
y = generateDataSample(x)
xHistory.append(x)
yHistory.append(y)
#-----------------------------#
# Modifications #
#-----------------------------#
fval = f(x)
e = y - fval
delta = phiOutput(x)
theta += alpha*e*delta
error_evolution.append(e)
#-----------------------------#
# End of Training Algorithm #
#-----------------------------#
iterationCount += 1
train_GD(sample_size)
error[model]['train'].append(np.mean(np.abs(np.array(error_evolution))))
np.random.seed(seed_test)
xs_test = np.random.random(sample_size)
error[model]['test'].append(np.mean(np.abs(generateDataSample(xs_test) - f(xs_test))))
#----------- Plotting ---------------#
xs = np.linspace(0.0,1.0,1000)
z = map(f, xs)
idx = 0
def feat(input):
global theta, idx
phi = phiOutput(input)
return phi[idx] * theta[idx]
for i in range(numFeatures):
idx = i
z2 = map(feat, xs)
plt.plot(xs, z2)
plt.plot(xHistory, yHistory, 'o')
plt.plot(xs, z, lw = 3, color = 'red')
plt.show()
In [104]:
data1 = [
go.Scatter(
x=sample_sizes,
y=error['GD']['train'],
name='Train error'
)
]
data2 = [
go.Scatter(
x=sample_sizes,
y=error['GD']['test'],
name='Test error'
)
]
layout = go.Layout(
title='Comparison of train and test error for gradient descent',
xaxis=dict(
title='Number of input data points'
),
yaxis=dict(
title='Average Error',
ticklen= 5,
gridwidth= 2
)
)
plotly.offline.iplot(go.Figure(data=data1+data2, layout=layout))
In [105]:
numFeatures = 10
def generateDataSample(x):
"""
Generate a noisy data sample from a given data point in the range [0,1]
:param x: A scalar dependent variable for which to calculate the output y_noisy
:returns: The output with gaussian noise added
"""
y = 1 - x - math.sin(-2 * math.pi * x ** 3) * math.cos(-2 * math.pi * x ** 3) * math.exp(-x ** 4)
sigma = 0.1
noise = sigma * np.random.random()
y_noisy = y + noise
return y_noisy
centers = np.linspace(0.0, 1.0, numFeatures)
widthConstant = (1.0 - 0.0) / numFeatures / 10
widths = np.ones(numFeatures,) * widthConstant
def phiOutput(input):
"""
Get the output of the Gaussian features for a given input variable(s)
:param input: A single or vector of dependent variables with size [Ns] for which to calculate the FA features
:returns: A vector of feature outputs with size [NumFeats x Ns]
"""
global numFeatures, centers, widths
if np.size(input) == 1:
phi = np.exp(-np.divide(np.square(input - centers), widths))
elif np.size(input) > 1:
numEvals = np.shape(input)[0]
# Repeat vectors to vectorize output calculation
inputMat = np.array([input, ] * numFeatures)
centersMat = np.array([centers, ] * numEvals).transpose()
widthsMat = np.array([widths, ] * numEvals).transpose()
phi = np.exp(-np.divide(np.square(inputMat - centersMat), widthsMat))
return phi
def f(input, *user_theta):
"""
Get the FA output for a given input variable(s)
:param input: A single or vector of dependent variables with size [Ns] for which to calculate the FA features
:param user_theta: (Variable argument) A vector of theta variables to apply to the FA. If left blank the method will default to using the trained thetas in self.theta. This is only used for visualization purposes.
:returns: A vector of function approximator outputs with size [Ns]
"""
global theta
phi = phiOutput(input)
if not user_theta:
Theta = theta
else:
Theta = np.array(user_theta)
out = np.dot(phi, Theta.transpose())
return out
generateDataSample = np.vectorize(generateDataSample)
f = np.vectorize(f)
model = 'LS'
error[model] = {}
error[model]['train'] = []
error[model]['test'] = []
for sample_size in sample_sizes:
theta = np.random.random(numFeatures)
np.random.seed(seed_train)
x = np.random.random(sample_size)
y = map(generateDataSample, x)
def train_LS():
global x, y, numfeatures, theta
#----------------------#
# # Training Algorithm ##
#----------------------#
Phi = phiOutput(x)
A = Phi.dot(Phi.T)
b = Phi.dot(y)
theta = np.dot(np.linalg.pinv(A),b)
#-----------------------------#
# # End of Training Algorithm ##
#-----------------------------#
train_LS()
error[model]['train'].append(np.mean(np.abs(y - f(x))))
np.random.seed(seed_test)
xs_test = np.random.random(sample_size)
error[model]['test'].append(np.mean(np.abs(generateDataSample(xs_test) - f(xs_test))))
#----------- Plotting ---------------#
xs = np.linspace(0.0, 1.0, 1000)
z = map(f, xs)
idx = 0
def feat(input):
global theta, idx
phi = phiOutput(input)
return phi[idx] * theta[idx]
for i in range(numFeatures):
idx = i
z2 = map(feat, xs)
plt.plot(xs, z2)
plt.plot(x, y, 'o')
plt.plot(xs, z, lw=3, color='red')
plt.show()
In [106]:
data1 = [
go.Scatter(
x=sample_sizes,
y=error['LS']['train'],
name='Train error'
)
]
data2 = [
go.Scatter(
x=sample_sizes,
y=error['LS']['test'],
name='Test error'
)
]
layout = go.Layout(
title='Comparison of train and test error for the Least Squares method',
xaxis=dict(
title='Number of input data points'
),
yaxis=dict(
title='Average Error',
ticklen= 5,
gridwidth= 2
)
)
plotly.offline.iplot(go.Figure(data=data1+data2, layout=layout))
In [107]:
numFeatures = 10
def generateDataSample(x):
"""
Generate a noisy data sample from a given data point in the range [0,1]
:param x: A scalar dependent variable for which to calculate the output y_noisy
:returns: The output with gaussian noise added
"""
y = 1 - x - math.sin(-2 * math.pi * x ** 3) * math.cos(-2 * math.pi * x ** 3) * math.exp(-x ** 4)
sigma = 0.1
noise = sigma * np.random.random()
y_noisy = y + noise
return y_noisy
centers = np.linspace(0.0, 1.0, numFeatures)
widthConstant = (1.0 - 0.0) / numFeatures / 10
widths = np.ones(numFeatures,) * widthConstant
def phiOutput(input):
"""
Get the output of the Gaussian features for a given input variable(s)
:param input: A single or vector of dependent variables with size [Ns] for which to calculate the FA features
:returns: A vector of feature outputs with size [NumFeats x Ns]
"""
global numFeatures, centers, widths
if np.size(input) == 1:
phi = np.exp(-np.divide(np.square(input - centers), widths))
elif np.size(input) > 1:
numEvals = np.shape(input)[0]
# Repeat vectors to vectorize output calculation
inputMat = np.array([input, ] * numFeatures)
centersMat = np.array([centers, ] * numEvals).transpose()
widthsMat = np.array([widths, ] * numEvals).transpose()
phi = np.exp(-np.divide(np.square(inputMat - centersMat), widthsMat))
return phi
def f(input, *user_theta):
"""
Get the FA output for a given input variable(s)
:param input: A single or vector of dependent variables with size [Ns] for which to calculate the FA features
:param user_theta: (Variable argument) A vector of theta variables to apply to the FA. If left blank the method will default to using the trained thetas in self.theta. This is only used for visualization purposes.
:returns: A vector of function approximator outputs with size [Ns]
"""
global theta
phi = phiOutput(input)
if not user_theta:
Theta = theta
else:
Theta = np.array(user_theta)
out = np.dot(phi, Theta.transpose())
return out
generateDataSample = np.vectorize(generateDataSample)
f = np.vectorize(f)
model = 'RLS'
error[model] = {}
error[model]['train'] = []
error[model]['test'] = []
for sample_size in sample_sizes:
xHistory = []
yHistory = []
error_evolution = []
np.random.seed(seed_train)
def train_RLS(maxIter):
global theta, xHistory, yHistory, E
## Initialize A and b ##
A = np.zeros(shape=(numFeatures,numFeatures))
b = np.zeros( numFeatures )
iterationCount = 0
# Begin training
while iterationCount < maxIter:
# Draw a random sample on the interval [0,1]
x = np.random.random()
y = generateDataSample(x)
xHistory.append(x)
yHistory.append(y)
e=0
#----------------------#
## Training Algorithm ##
#----------------------#
theta = np.zeros((numFeatures,))
A += np.outer(phiOutput(x),phiOutput(x))
b += phiOutput(x)*y
theta = np.dot(np.linalg.pinv(A),b)
error_evolution.append(y - f(x))
#-----------------------------#
## End of Training Algorithm ##
#-----------------------------#
iterationCount+=1
theta = np.dot(np.linalg.pinv(A),b)
train_RLS(sample_size)
error[model]['train'].append(np.mean(np.abs(np.array(error_evolution))))
np.random.seed(seed_test)
xs_test = np.random.random(sample_size)
error[model]['test'].append(np.mean(np.abs(generateDataSample(xs_test) - f(xs_test))))
#----------- Plotting ---------------#
xs = np.linspace(0.0,1.0,1000)
z = map(f, xs)
idx = 0
def feat(input):
global theta, idx
phi = phiOutput(input)
return phi[idx] * theta[idx]
for i in range(numFeatures):
idx = i
z2 = map(feat, xs)
plt.plot(xs, z2)
plt.plot(xHistory, yHistory, 'o')
plt.plot(xs, z, lw = 3, color = 'red')
plt.show()
In [108]:
data1 = [
go.Scatter(
x=sample_sizes,
y=error['RLS']['train'],
name='Train error'
)
]
data2 = [
go.Scatter(
x=sample_sizes,
y=error['RLS']['test'],
name='Test error'
)
]
layout = go.Layout(
title='Comparison of train and test error for the Recursive Least Squares method',
xaxis=dict(
title='Number of input data points'
),
yaxis=dict(
title='Average Error',
ticklen= 5,
gridwidth= 2
)
)
plotly.offline.iplot(go.Figure(data=data1+data2, layout=layout))
In [109]:
numFeatures = 10
def generateDataSample(x):
"""
Generate a noisy data sample from a given data point in the range [0,1]
:param x: A scalar dependent variable for which to calculate the output y_noisy
:returns: The output with gaussian noise added
"""
y = 1 - x - math.sin(-2 * math.pi * x ** 3) * math.cos(-2 * math.pi * x ** 3) * math.exp(-x ** 4)
sigma = 0.1
noise = sigma * np.random.random()
y_noisy = y + noise
return y_noisy
centers = np.linspace(0.0, 1.0, numFeatures)
widthConstant = (1.0 - 0.0) / numFeatures / 10
widths = np.ones(numFeatures,) * widthConstant
def phiOutput(input):
"""
Get the output of the Gaussian features for a given input variable(s)
:param input: A single or vector of dependent variables with size [Ns] for which to calculate the FA features
:returns: A vector of feature outputs with size [NumFeats x Ns]
"""
global numFeatures, centers, widths
if np.size(input) == 1:
phi = np.exp(-np.divide(np.square(input - centers), widths))
elif np.size(input) > 1:
numEvals = np.shape(input)[0]
# Repeat vectors to vectorize output calculation
inputMat = np.array([input, ] * numFeatures)
centersMat = np.array([centers, ] * numEvals).transpose()
widthsMat = np.array([widths, ] * numEvals).transpose()
phi = np.exp(-np.divide(np.square(inputMat - centersMat), widthsMat))
return phi
def f(input, *user_theta):
"""
Get the FA output for a given input variable(s)
:param input: A single or vector of dependent variables with size [Ns] for which to calculate the FA features
:param user_theta: (Variable argument) A vector of theta variables to apply to the FA. If left blank the method will default to using the trained thetas in self.theta. This is only used for visualization purposes.
:returns: A vector of function approximator outputs with size [Ns]
"""
global theta
phi = phiOutput(input)
if not user_theta:
Theta = theta
else:
Theta = np.array(user_theta)
out = np.dot(phi, Theta.transpose())
return out
generateDataSample = np.vectorize(generateDataSample)
f = np.vectorize(f)
model = 'RLS2'
error[model] = {}
error[model]['train'] = []
error[model]['test'] = []
for sample_size in sample_sizes:
xHistory = []
yHistory = []
error_evolution = []
np.random.seed(seed_train)
def train_RLS(maxIter):
global theta, xHistory, yHistory, E
## Initialize b and A_sharp ##
b = np.zeros( numFeatures )
A_sharp = np.eye(numFeatures)
iterationCount = 0
# Begin training
while iterationCount < maxIter:
# Draw a random sample on the interval [0,1]
x = np.random.random()
y = generateDataSample(x)
xHistory.append(x)
yHistory.append(y)
#----------------------#
## Training Algorithm ##
#----------------------#
phi = phiOutput(x)
A_sharp -= A_sharp.dot(np.outer(phi, phi).dot(A_sharp))/(1+phi.dot(A_sharp.dot(phi)))
b += phi.dot(y)
error_evolution.append(y - f(x))
#-----------------------------#
## End of Training Algorithm ##
#-----------------------------#
iterationCount+=1
theta = np.dot(A_sharp,b)
train_RLS(sample_size)
error[model]['train'].append(np.mean(np.abs(np.array(error_evolution))))
np.random.seed(seed_test)
xs_test = np.random.random(sample_size)
error[model]['test'].append(np.mean(np.abs(generateDataSample(xs_test) - f(xs_test))))
#----------- Plotting ---------------#
xs = np.linspace(0.0,1.0,1000)
z = map(f, xs)
idx = 0
def feat(input):
global theta, idx
phi = phiOutput(input)
return phi[idx] * theta[idx]
for i in range(numFeatures):
idx = i
z2 = map(feat, xs)
plt.plot(xs, z2)
plt.plot(xHistory, yHistory, 'o')
plt.plot(xs, z, lw = 3, color = 'red')
plt.show()
In [110]:
data1 = [
go.Scatter(
x=sample_sizes,
y=error['RLS2']['train'],
name='Train error'
)
]
data2 = [
go.Scatter(
x=sample_sizes,
y=error['RLS2']['test'],
name='Test error'
)
]
layout = go.Layout(
title='Comparison of train and test error for the Recursive Least Squares method with Sherman-Morrison',
xaxis=dict(
title='Number of input data points'
),
yaxis=dict(
title='Average Error',
ticklen= 5,
gridwidth= 2
)
)
plotly.offline.iplot(go.Figure(data=data1+data2, layout=layout))
In [111]:
numFeatures = 10
def generateDataSample(x):
"""
Generate a noisy data sample from a given data point in the range [0,1]
:param x: A scalar dependent variable for which to calculate the output y_noisy
:returns: The output with gaussian noise added
"""
y = 1 - x - math.sin(-2 * math.pi * x ** 3) * math.cos(-2 * math.pi * x ** 3) * math.exp(-x ** 4)
sigma = 0.1
noise = sigma * np.random.random()
y_noisy = y + noise
return y_noisy
centers = np.linspace(0.0, 1.0, numFeatures)
widthConstant = (1.0 - 0.0) / numFeatures / 10
widths = np.ones(numFeatures,) * widthConstant
def phiOutput(input):
"""
Get the output of the Gaussian features for a given input variable(s)
:param input: A single or vector of dependent variables with size [Ns] for which to calculate the FA features
:returns: A vector of feature outputs with size [NumFeats x Ns]
"""
global numFeatures, centers, widths
if np.size(input) == 1:
phi = np.exp(-np.divide(np.square(input - centers), widths))
elif np.size(input) > 1:
numEvals = np.shape(input)[0]
# Repeat vectors to vectorize output calculation
inputMat = np.array([input, ] * numFeatures)
centersMat = np.array([centers, ] * numEvals).transpose()
widthsMat = np.array([widths, ] * numEvals).transpose()
phi = np.exp(-np.divide(np.square(inputMat - centersMat), widthsMat))
return phi
def w(input):
"""
Get the output of the features for a given input variable(s)
:param input: A single or vector of dependent variables with size [Ns] for which to calculate the FA features
:returns: A vector of feature outputs with size [NumFeats x Ns]
"""
if np.size(input) == 1:
w = np.vstack(([input], [1]))
elif np.size(input) > 1:
w = np.vstack((input, np.ones((1,np.size(input)))))
return w
def f(input):
"""
Get the FA output for a given input variable(s)
:param input: A single or vector of dependent variables with size [Ns] for which to calculate the FA features
:returns: A vector of function approximator outputs with size [Ns]
"""
global theta
wval = w(input)
Phi = phiOutput(input)
g = (np.dot(wval.transpose(), theta)).transpose() #[numFeats x Ns]
out = np.sum(Phi*g, axis=0) / np.sum(Phi, axis=0)
return out
generateDataSample = np.vectorize(generateDataSample)
model = 'LWLS'
error[model] = {}
error[model]['train'] = []
error[model]['test'] = []
for sample_size in sample_sizes:
theta = np.zeros((2, numFeatures))
np.random.seed(seed_train)
x = np.random.random(sample_size)
y = map(generateDataSample, x)
def train_LWLS():
global x, y, numFeatures, theta
#----------------------#
## Training Algorithm ##
#----------------------#
Phi = phiOutput(x)
W = w(x)
for k in range(numFeatures):
Wphi = W.dot(np.diag(Phi[k]))
A_i = Wphi.dot(W.T)
b_i = Wphi.dot(y)
theta[:,k] = np.dot(np.linalg.pinv(A_i),b_i)
#-----------------------------#
## End of Training Algorithm ##
#-----------------------------#
train_LWLS()
error[model]['train'].append(np.mean(np.abs(np.array(y - f(x)))))
np.random.seed(seed_test)
xs_test = np.random.random(sample_size)
error[model]['test'].append(np.mean(np.abs(generateDataSample(xs_test) - f(xs_test))))
#----------- Plotting ---------------#
xs = np.linspace(0.0, 1.0, 1000)
z = f(xs)
plt.plot(x, y, 'o')
plt.plot(xs, z, lw=3, color='red')
plt.show()
In [113]:
data1 = [
go.Scatter(
x=sample_sizes,
y=error['LWLS']['train'],
name='Train error'
)
]
data2 = [
go.Scatter(
x=sample_sizes,
y=error['LWLS']['test'],
name='Test error'
)
]
layout = go.Layout(
title='Comparison of train and test error for the Locally Weighted Least Squares method',
xaxis=dict(
title='Number of input data points'
),
yaxis=dict(
title='Average Error',
ticklen= 5,
gridwidth= 2
)
)
plotly.offline.iplot(go.Figure(data=data1+data2, layout=layout))
In [114]:
layout_fin = go.Layout(
title= 'Comparison of train and test errors for all the models',
hovermode= 'closest',
xaxis= dict(
title= 'Number of input data points',
ticklen= 5,
zeroline= False,
gridwidth= 2,
),
yaxis=dict(
title= 'Average Error',
ticklen= 5,
gridwidth= 2,
)
)
legend_every = 1
models = ['GD', 'LS', 'RLS', 'RLS2', 'LWLS']
colors = cl.scales['{}'.format(2*len(models))]['qual']['Paired']
# Plotting the evolution
traces = []
for i, m in enumerate(models):
for j, test_train in enumerate(['test', 'train']):
traces.append(
go.Scatter(
x = sample_sizes,
y = error[m][test_train],
mode = 'lines',
name = '{}: {} error'.format(m, test_train),
line = dict(
width = 2,
color = colors[2*i+j],
dash = 'solid'
),
hoverlabel = dict(
namelength = -1
),
showlegend = (i % legend_every == 0)
)
)
plotly.offline.iplot(go.Figure(data=traces, layout=layout_fin))
In [115]:
trace_bar = {}
def bar_chart_error(list_classifiers, trace_bar = trace_bar):
learning_sets = np.arange(1, 10)
colors = cl.scales['{}'.format(2*len(list_classifiers))]['qual']['Paired']
for i, classifier in enumerate(list_classifiers):
trace_bar[classifier] = {}
for j, test_train in enumerate(['test', 'train']):
trace_bar[classifier][test_train] = go.Bar(
x = ['{} k'.format(k) for k in learning_sets],
y = [error[classifier][test_train][10*k] for k in learning_sets],
name = '{}: {} error'.format(classifier, test_train),
marker=dict(
color=colors[2*i+j],
),
opacity=0.75
)
return trace_bar
list_classifiers = ['GD', 'LS', 'RLS', 'RLS2', 'LWLS']
trace_bar = bar_chart_error(list_classifiers)
layout_bar = go.Layout(
barmode='group',
title = 'Average Errors: Comparison',
xaxis = dict(
title='Number of data points'
),
yaxis = dict(
title = 'Error'
),
)
fig = go.Figure(data=[trace_bar[key][tt] for key in trace_bar for tt in trace_bar[key]], layout=layout_bar)
plotly.offline.iplot(fig)