TF/03_Linear_Regression/05Deming06Lasso Ridge 07Elasticnet
05 Implementing Deming Regression
Deming regression, also known as total regression, is regular regression that minimizes the shortest distance to the line. Contrast this to regular regression, in which we aim to minimize the vertical distance between the model output and the y-target values.
Model
The model will be the same as regular linear regression:
y = A * x + b
Instead of measuring the vertical L2 distance, we will measure the shortest distance between the line and the predicted point in the loss function.
loss = |y_target - (A * x_input + b)| / sqrt(A^2 + 1)
Graph of Linear Fit
05_deming_regression.py
# Deming Regression
#----------------------------------
#
# This function shows how to use TensorFlow to
# solve linear Deming regression.
# y = Ax + b
#
# We will use the iris data, specifically:
# y = Sepal Length
# x = Petal Width
import matplotlib.pyplot as plt
import numpy as np
import tensorflow as tf
from sklearn import datasets
from tensorflow.python.framework import ops
ops.reset_default_graph()
# Create graph
sess = tf.Session()
# Load the data
# iris.data = [(Sepal Length, Sepal Width, Petal Length, Petal Width)]
iris = datasets.load_iris()
x_vals = np.array([x[3] for x in iris.data])
y_vals = np.array([y[0] for y in iris.data])
# Declare batch size
batch_size = 50
# Initialize placeholders
x_data = tf.placeholder(shape=[None, 1], dtype=tf.float32)
y_target = tf.placeholder(shape=[None, 1], dtype=tf.float32)
# Create variables for linear regression
A = tf.Variable(tf.random_normal(shape=[1,1]))
b = tf.Variable(tf.random_normal(shape=[1,1]))
# Declare model operations
model_output = tf.add(tf.matmul(x_data, A), b)
# Declare Demming loss function
demming_numerator = tf.abs(tf.subtract(y_target, tf.add(tf.matmul(x_data, A), b)))
demming_denominator = tf.sqrt(tf.add(tf.square(A),1))
loss = tf.reduce_mean(tf.truediv(demming_numerator, demming_denominator))
# Declare optimizer
my_opt = tf.train.GradientDescentOptimizer(0.1)
train_step = my_opt.minimize(loss)
# Initialize variables
init = tf.global_variables_initializer()
sess.run(init)
# Training loop
loss_vec = []
for i in range(250):
rand_index = np.random.choice(len(x_vals), size=batch_size)
rand_x = np.transpose([x_vals[rand_index]])
rand_y = np.transpose([y_vals[rand_index]])
sess.run(train_step, feed_dict={x_data: rand_x, y_target: rand_y})
temp_loss = sess.run(loss, feed_dict={x_data: rand_x, y_target: rand_y})
loss_vec.append(temp_loss)
if (i+1)%50==0:
print('Step #' + str(i+1) + ' A = ' + str(sess.run(A)) + ' b = ' + str(sess.run(b)))
print('Loss = ' + str(temp_loss))
# Get the optimal coefficients
[slope] = sess.run(A)
[y_intercept] = sess.run(b)
# Get best fit line
best_fit = []
for i in x_vals:
best_fit.append(slope*i+y_intercept)
# Plot the result
plt.plot(x_vals, y_vals, 'o', label='Data Points')
plt.plot(x_vals, best_fit, 'r-', label='Best fit line', linewidth=3)
plt.legend(loc='upper left')
plt.title('Sepal Length vs Pedal Width')
plt.xlabel('Pedal Width')
plt.ylabel('Sepal Length')
plt.show()
# Plot loss over time
plt.plot(loss_vec, 'k-')
plt.title('L2 Loss per Generation')
plt.xlabel('Generation')
plt.ylabel('L2 Loss')
plt.show()Step #50 A = [[ 2.91271663]] b = [[ 1.92914867]]
Loss = 0.467175
Step #100 A = [[ 2.81875157]] b = [[ 2.08339]]
Loss = 0.410863
Step #150 A = [[ 2.72617555]] b = [[ 2.26403856]]
Loss = 0.385974
Step #200 A = [[ 2.61317611]] b = [[ 2.42952514]]
Loss = 0.363812
Step #250 A = [[ 2.49734712]] b = [[ 2.59445238]]
Loss = 0.38427406 Implementing Lasso and Ridge Regression
# LASSO and Ridge Regression
#
# This function shows how to use TensorFlow to solve LASSO or
# Ridge regression for
# y = Ax + b
#
# We will use the iris data, specifically:
# y = Sepal Length
# x = Petal Width
# import required libraries
import matplotlib.pyplot as plt
import sys
import numpy as np
import tensorflow as tf
from sklearn import datasets
from tensorflow.python.framework import ops
# Specify 'Ridge' or 'LASSO'
regression_type = 'LASSO'
# clear out old graph
ops.reset_default_graph()
# Create graph
sess = tf.Session()
###
# Load iris data
###
# iris.data = [(Sepal Length, Sepal Width, Petal Length, Petal Width)]
iris = datasets.load_iris()
x_vals = np.array([x[3] for x in iris.data])
y_vals = np.array([y[0] for y in iris.data])
###
# Model Parameters
###
# Declare batch size
batch_size = 50
# Initialize placeholders
x_data = tf.placeholder(shape=[None, 1], dtype=tf.float32)
y_target = tf.placeholder(shape=[None, 1], dtype=tf.float32)
# make results reproducible
seed = 13
np.random.seed(seed)
tf.set_random_seed(seed)
# Create variables for linear regression
A = tf.Variable(tf.random_normal(shape=[1,1]))
b = tf.Variable(tf.random_normal(shape=[1,1]))
# Declare model operations
model_output = tf.add(tf.matmul(x_data, A), b)
###
# Loss Functions
###
# Select appropriate loss function based on regression type
if regression_type == 'LASSO':
# Declare Lasso loss function
# Lasso Loss = L2_Loss + heavyside_step,
# Where heavyside_step ~ 0 if A < constant, otherwise ~ 99
lasso_param = tf.constant(0.9)
heavyside_step = tf.truediv(1., tf.add(1., tf.exp(tf.multiply(-50., tf.subtract(A, lasso_param)))))
regularization_param = tf.multiply(heavyside_step, 99.)
loss = tf.add(tf.reduce_mean(tf.square(y_target - model_output)), regularization_param)
elif regression_type == 'Ridge':
# Declare the Ridge loss function
# Ridge loss = L2_loss + L2 norm of slope
ridge_param = tf.constant(1.)
ridge_loss = tf.reduce_mean(tf.square(A))
loss = tf.expand_dims(tf.add(tf.reduce_mean(tf.square(y_target - model_output)), tf.multiply(ridge_param, ridge_loss)), 0)
else:
print('Invalid regression_type parameter value',file=sys.stderr)
###
# Optimizer
###
# Declare optimizer
my_opt = tf.train.GradientDescentOptimizer(0.001)
train_step = my_opt.minimize(loss)
###
# Run regression
###
# Initialize variables
init = tf.global_variables_initializer()
sess.run(init)
# Training loop
loss_vec = []
for i in range(1500):
rand_index = np.random.choice(len(x_vals), size=batch_size)
rand_x = np.transpose([x_vals[rand_index]])
rand_y = np.transpose([y_vals[rand_index]])
sess.run(train_step, feed_dict={x_data: rand_x, y_target: rand_y})
temp_loss = sess.run(loss, feed_dict={x_data: rand_x, y_target: rand_y})
loss_vec.append(temp_loss[0])
if (i+1)%300==0:
print('Step #' + str(i+1) + ' A = ' + str(sess.run(A)) + ' b = ' + str(sess.run(b)))
print('Loss = ' + str(temp_loss))
print('\n')
###
# Extract regression results
###
# Get the optimal coefficients
[slope] = sess.run(A)
[y_intercept] = sess.run(b)
# Get best fit line
best_fit = []
for i in x_vals:
best_fit.append(slope*i+y_intercept)
###
# Plot results
###
# Plot regression line against data points
plt.plot(x_vals, y_vals, 'o', label='Data Points')
plt.plot(x_vals, best_fit, 'r-', label='Best fit line', linewidth=3)
plt.legend(loc='upper left')
plt.title('Sepal Length vs Pedal Width')
plt.xlabel('Pedal Width')
plt.ylabel('Sepal Length')
plt.show()
# Plot loss over time
plt.plot(loss_vec, 'k-')
plt.title(regression_type + ' Loss per Generation')
plt.xlabel('Generation')
plt.ylabel('Loss')
plt.show()Step #300 A = [[ 0.77170753]] b = [[ 1.82499862]]
Loss = [[ 10.2647295]]
Step #600 A = [[ 0.75908542]] b = [[ 3.22206354]]
Loss = [[ 3.06291938]]
Step #900 A = [[ 0.74843585]] b = [[ 3.99758244]]
Loss = [[ 1.23220396]]
Step #1200 A = [[ 0.73752165]] b = [[ 4.42974091]]
Loss = [[ 0.57872051]]
Step #1500 A = [[ 0.72942668]] b = [[ 4.67253113]]
Loss = [[ 0.40874988]]
Step #300 A = [[ 1.70595658]] b = [[ 1.55541837]]
Loss = [ 8.26485348]
Step #600 A = [[ 1.61488771]] b = [[ 2.56744456]]
Loss = [ 4.79188538]
Step #900 A = [[ 1.34542716]] b = [[ 3.24855161]]
Loss = [ 3.07603455]
Step #1200 A = [[ 1.11086905]] b = [[ 3.76259446]]
Loss = [ 2.06162453]
Step #1500 A = [[ 0.93269861]] b = [[ 4.15556765]]
Loss = [ 1.48733997]
07 Implementing Elasticnet Regression
# Elastic Net Regression
#----------------------------------
#
# This function shows how to use TensorFlow to
# solve elastic net regression.
# y = Ax + b
#
# We will use the iris data, specifically:
# y = Sepal Length
# x = Pedal Length, Petal Width, Sepal Width
import matplotlib.pyplot as plt
import numpy as np
import tensorflow as tf
from sklearn import datasets
from tensorflow.python.framework import ops
###
# Set up for TensorFlow
###
ops.reset_default_graph()
# Create graph
sess = tf.Session()
###
# Obtain data
###
# Load the data
# iris.data = [(Sepal Length, Sepal Width, Petal Length, Petal Width)]
iris = datasets.load_iris()
x_vals = np.array([[x[1], x[2], x[3]] for x in iris.data])
y_vals = np.array([y[0] for y in iris.data])
###
# Setup model
###
# make results reproducible
seed = 13
np.random.seed(seed)
tf.set_random_seed(seed)
# Declare batch size
batch_size = 50
# Initialize placeholders
x_data = tf.placeholder(shape=[None, 3], dtype=tf.float32)
y_target = tf.placeholder(shape=[None, 1], dtype=tf.float32)
# Create variables for linear regression
A = tf.Variable(tf.random_normal(shape=[3,1]))
b = tf.Variable(tf.random_normal(shape=[1,1]))
# Declare model operations
model_output = tf.add(tf.matmul(x_data, A), b)
# Declare the elastic net loss function
elastic_param1 = tf.constant(1.)
elastic_param2 = tf.constant(1.)
l1_a_loss = tf.reduce_mean(tf.abs(A))
l2_a_loss = tf.reduce_mean(tf.square(A))
e1_term = tf.multiply(elastic_param1, l1_a_loss)
e2_term = tf.multiply(elastic_param2, l2_a_loss)
loss = tf.expand_dims(tf.add(tf.add(tf.reduce_mean(tf.square(y_target - model_output)), e1_term), e2_term), 0)
# Declare optimizer
my_opt = tf.train.GradientDescentOptimizer(0.001)
train_step = my_opt.minimize(loss)
###
# Train model
###
# Initialize variables
init = tf.global_variables_initializer()
sess.run(init)
# Training loop
loss_vec = []
for i in range(1000):
rand_index = np.random.choice(len(x_vals), size=batch_size)
rand_x = x_vals[rand_index]
rand_y = np.transpose([y_vals[rand_index]])
sess.run(train_step, feed_dict={x_data: rand_x, y_target: rand_y})
temp_loss = sess.run(loss, feed_dict={x_data: rand_x, y_target: rand_y})
loss_vec.append(temp_loss[0])
if (i+1)%250==0:
print('Step #' + str(i+1) + ' A = ' + str(sess.run(A)) + ' b = ' + str(sess.run(b)))
print('Loss = ' + str(temp_loss))
###
# Extract model results
###
# Get the optimal coefficients
[[sw_coef], [pl_coef], [pw_ceof]] = sess.run(A)
[y_intercept] = sess.run(b)
###
# Plot results
###
# Plot loss over time
plt.plot(loss_vec, 'k-')
plt.title('Loss per Generation')
plt.xlabel('Generation')
plt.ylabel('Loss')
plt.show()
Step #250 A = [[ 1.26014626]
[ 0.4016138 ]
[ 0.40159121]] b = [[-0.14889474]]
Loss = [ 1.59188592]
Step #500 A = [[ 1.17897248]
[ 0.46715766]
[ 0.29896322]] b = [[-0.0677181]]
Loss = [ 1.46314836]
Step #750 A = [[ 1.13416576]
[ 0.51899701]
[ 0.21090427]] b = [[ 0.01904623]]
Loss = [ 1.37157845]
Step #1000 A = [[ 1.09745109]
[ 0.54604095]
[ 0.13102382]] b = [[ 0.10402214]]
Loss = [ 1.27774763]