ML-Fundamentals - Lineare Regression - Exercise: Multivariate Linear Regression

Introduction

In this exercise you will implement the multiivariate linear regression, a model with two or more predictors and one response variable (opposed to one predictor using univariate linear regression). The whole exercise consists of the following steps:

  1. Generate values for two predictors/features (x1,x2)(x_1, x_2)
  2. Implement a linear function as hypothesis (model)
  3. Generate values for the response (Y / target values)
  4. Plot the ((x1,x2),y)((x_1, x_2), y) values in a 3D plot.
  5. Write a function to quantify your model (cost function)
  6. Implement the gradient descent algorithm to train your model (optimizer)
  7. Visualize your training process and results
  8. Apply feature scaling (pen & paper)

Requirements

Knowledge

You should have a basic knowledge of:

  • Univariate lineare regression
  • Multivariate linear regression
  • Squared error
  • Gradient descent
  • numpy
  • matplotlib

Suitable sources for acquiring this knowledge are:

Python Modules

By deep.TEACHING convention, all python modules needed to run the notebook are loaded centrally at the beginning.

# External Modules
import numpy as np
import matplotlib.pyplot as plt
from mpl_toolkits.mplot3d.axes3d import Axes3D
from matplotlib import cm

%matplotlib notebook

Exercise - Multivariate Linear Regression

We will only use two features in this notebook, so we are still able to plot them together with the target in a 3D plot. But your implementation should also be capable of handling more (except the plots).

Create Features

First we will create some features. The features should be in a 2D numpy array, the rows seperating the different feature vectors, the columns containing the features. Each feature should be uniformly distributed in a specifiable range.

Task:

Implement the function to generate a feature matrix (numpy array).

def create_feature_matrix(sample_size, n_features, x_min, x_max):
    '''creates random feature vectors based on a lienar function in a given interval
    
    Args:
        sample_size: number feature vectors
        n_features: number of features for each vector
        x_min: lower bound value ranges
        x_max: upper bound value ranges
    
    Returns:
        x: 2D array containing feature vecotrs with shape (sample_size, n_features)
    '''
    raise NotImplementedError("You should implement this!")
sample_size = 100
n_features = 2
x_min = [1.5,-0.5]
x_max = [11.,5.0]

X = create_feature_matrix(sample_size, n_features, x_min, x_max)
assert len(X[:,0]) == sample_size
assert len(X[0,:]) == n_features
for i in range(n_features):
    assert np.max(X[:,i]) <= x_max[i]
    assert np.min(X[:,i]) >= x_min[i]

Linear Hypothesis

A short recap, a hypothesis hθ(x)h_\theta(x) is a certain function that we believe is similar to a target function that we like to model. A hypothesis hθ(x)h_\theta(x) is a function of xx with fixed parameters θ\theta.

Here we have nn features x=[x1,,xn]x = [x_1, \ldots, x_n ] and n+1n+1 θ\thetas:

hθ(x)=θ0+θ1x1+θnxnh_\theta(x) = \theta_{0} + \theta_{1} x_1 + \ldots \theta_n x_n

adding an extra element to xx for convenience, this could also be rewritten as:

hθ(x)=θ0x0+θ1x1+θnxnh_\theta(x) = \theta_{0} x_0 + \theta_{1} x_1 + \ldots \theta_n x_n

with x0=1x_0 = 1 for all feature vectors.

Or treating xx and θ\theta as vectors:

h(x)=θTxh(\vec x) = \vec \theta^T \vec x'

with:

x=(x1x2xn) and x=(1x1x2xn)\vec x = \begin{pmatrix} x_1 & x_2 & \ldots & x_n \\ \end{pmatrix} \text{ and } \vec x' = \begin{pmatrix} 1 & x_1 & x_2 & \ldots & x_n \\ \end{pmatrix}

Or for the whole data set at once:

h(x)=θTXh(\vec x) = \vec \theta^T X'

with:

X=(x11xn1x12xn2x1mxnm) and X=(1x11xn11x12xn21x1mxnm)X = \begin{pmatrix} x_1^1 & \ldots & x_n^1 \\ x_1^2 & \ldots & x_n^2 \\ \vdots &\vdots &\vdots \\ x_1^m & \ldots & x_n^m \\ \end{pmatrix} \text{ and } X' = \begin{pmatrix} 1 & x_1^1 & \ldots & x_n^1 \\ 1 & x_1^2 & \ldots & x_n^2 \\ \vdots &\vdots &\vdots &\vdots \\ 1 & x_1^m & \ldots & x_n^m \\ \end{pmatrix}

Task:

Implement hypothesis hθ(x)h_\theta(x) in the method linear_hypothesis and return it as a function. Implement it the computationally efficient (pythonic) way by not using any loops and handling all data at once (use XX respectively XX').

Hint:

Of course you are free to implement as many helper functions as you like, e.g. for transforming XX to XX', though you do not have to. Up to you.

def linear_hypothesis(thetas):
    ''' Combines given list argument in a linear equation and returns it as a function
    
    Args:
        thetas: list of coefficients
        
    Returns:
        lambda that models a linear function based on thetas and x
    '''
    raise NotImplementedError("You should implement this!")
(len(X[:,0]), 1)
assert len(linear_hypothesis([.1,.2,.3])(X)) == sample_size

Generate Target Values

Task:

Use your implemented linear_hypothesis inside the next function to generate some target values YY. Additionally add some Gaussian noise.

def generate_targets(X, theta, sigma):
    ''' Combines given arguments in a linear equation with X, 
    adds some Gaussian noise and returns the result
    
    Args:
        X: 2D numpy feature matrix
        theta: list of coefficients
        sigma: standard deviation of the gaussian noise
        
    Returns:
        target values for X
    '''
    raise NotImplementedError("You should implement this!")
theta = (2., 3., -4.)
sigma = 3.
y = generate_targets(X, theta, sigma)
assert len(y) == sample_size

Plot The Data

Task:

Plot the data D={(x1(1),x2(1),y(1)),,(x1(n),x2(n),y(n))}D = \{(x^{(1)}_1,x^{(1)}_2,y^{(1)}), \ldots, (x^{(n)}_1,x^{(n)}_2,y^{(n)})\} in a 3D scatter plot. The plot should look like the following:

Sidenote:

The command %matplotlib notebook (instead of %matplotlib inline) creates an interactive (e.g. rotatable) plot.

%matplotlib notebook

def plot_data_scatter(features, targets):
    """ Plots the features and the targets in a 3D scatter plot
    
    Args:
        features: 2D numpy-array features
        targets: ltargets
    """
    raise NotImplementedError("You should implement this!")
plot_data_scatter(X, y)

Cost Function

A cost function JJ depends on the given training data DD and hypothesis hθ(x)h_\theta(x). In the context of the linear regression, the cost function measures how wrong a model is regarding its ability to estimate the relationship between xx and yy for specific θ\theta values. Later we will treat this as an optimization problem and try to minimize the cost function JD(θ)J_D(\theta) to find optimal θ\theta values for our hypothesis hθ(x)h_\theta(x). The cost function we use in this exercise is the Mean-Squared-Error cost function:

JD(θ)=12mi=1m(hθ(xi)yi)2J_D(\theta)=\frac{1}{2m}\sum_{i=1}^{m}{(h_\theta(x_i)-y_i)^2}

Implement the cost function JD(θ)J_D(\theta) in the method mse_cost_function. The method should return a function that takes the values of θ0\theta_0 and θ1\theta_1 as an argument.

Sidenote, the terms loss function or error function are often used interchangeably in the field of Machine Learning.

def mse_cost_function(x, y):
    ''' Implements MSE cost function as a function J(theta) on given traning data 
    
    Args:
        x: vector of x values 
        y: vector of ground truth values y 
        
    Returns:
        lambda J(theta) that models the cost function
    '''
    raise NotImplementedError("You should implement this!")
J = mse_cost_function(X, y)
print(J(theta))

Gradient Descent

A short recap, the gradient descent algorithm is a first-order iterative optimization for finding a minimum of a function. From the current position in a (cost) function, the algorithm steps proportional to the negative of the gradient and repeats this until it reaches a local or global minimum and determines. Stepping proportional means that it does not go entirely in the direction of the negative gradient, but scaled by a fixed value α\alpha also called the learning rate. Implementing the following formalized update rule is the core of the optimization process:

θjnewθjoldαδδθjoldJ(θold)\theta_{j_{new}} \leftarrow \theta_{j_{old}} - \alpha * \frac{\delta}{\delta\theta_{j_{old}}} J(\theta_{old})

Task:

Implement the function to update all theta values.

def update_theta(x, y, theta, learning_rate):
    ''' Updates learnable parameters theta 
    
    The update is done by calculating the partial derivities of 
    the cost function including the linear hypothesis. The 
    gradients scaled by a scalar are subtracted from the given 
    theta values.
    
    Args:
        x: 2D numpy array of x values
        y: array of y values corresponding to x
        theta: current theta values
        learning_rate: value to scale the negative gradient 
        
    Returns:
        t0: Updated theta_0
        t1: Updated theta_1
    '''
    raise NotImplementedError("You should implement this!")

Using the update_theta method, you can now implement the gradient descent algorithm. Iterate over the update rule to find the values for θ\theta that minimize our cost function JD(θ)J_D(\theta). This process is often called training of a machine learning model.

Task:

  • Implement the function for the gradient descent.
  • Create a history of all theta and cost values and return them.
def gradient_descent(learning_rate, theta, iterations, x, y):
    ''' Minimize theta values of a linear model based on MSE cost function
    
    Args:
        learning_rate: scalar, scales the negative gradient 
        theta: initial theta values
        x: vector, x values from the data set
        y: vector, y values from the data set
        iterations: scalar, number of theta updates
        
    Returns:
        history_cost: cost after each iteration
        history_theta: Updated theta values after each iteration
    '''
    raise NotImplementedError("You should implement this!")

Training and Evaluation

Task:

Choose an appropriate learning rate, number of iterations and initial theta values and start the training

# Your implementation:

alpha = 42.42 # assign an appropriate value
nb_iterations = 1337 # assign an appropriate value
start_values_theta = [42., 42., 42.] # assign appropriate values
history_cost, history_theta = gradient_descent(alpha, start_values_theta, nb_iterations, X, y)

Now that the training has finished we can visualize our results.

Task:

Plot the costs over the iterations. If you have used fig = plt.figure() and ax = fig.add_subplot(111) in the last plot, use it again here, else the plot will be added to the last plot instead of a new one.

Your plot should look similar to this one:

def plot_progress(costs):
    """ Plots the costs over the iterations
    
    Args:
        costs: history of costs
    """
    raise NotImplementedError("You should implement this!")
plot_progress(history_cost)
print("costs before the training:\t ", history_cost[0])
print("costs after the training:\t ", history_cost[-1])

Task:

Finally plot the decision hyperplane (just a plain plane here though) together with the data in a 3D plot.

Your plot should look similar to this one:

def evaluation_plt(x, y, final_theta):
    ''' Plots a cost curve and the decision boundary
    
    The Method plots a cost curve from a given training process (cost_hist). 
    It also plots the data set (x,y) and draws a linear decision boundary 
    with the parameters theta_0 and theta_1 into the plotted data set.
    
    Args:
        cost_hist: vector, history of all cost values from a opitmization
        theta_0: scalar, model parameter for boundary
        theta_1: scalar, model parameter for boundary
        x: vector, x values from the data set
        y: vector, y values from the data set
    '''
    raise NotImplementedError("You should implement this!")
evaluation_plt(X, y, history_theta[-1])
print("thetas before the training:\t", history_theta[0])
print("thetas after the training:\t", history_theta[-1])

Feature Scaling

Now suppose the following features XX:

X = np.array([[0.0001, 2000],
       [0.0002, 1800],
       [0.0003, 1600]], dtype=np.float32)

sample_size = len(X[:,0])
X.std(axis=0)
v = ((0.0001 - 0.0002)**2 + (0.0003 - 0.0002)**2)/3
s = np.sqrt(v)
s

Optional:

You can even execute the cell above and start running your notebook again from top (all except executing the cell to generate your features, which would overwrite these new features).

When you start training you should notice that your costs do not decrease, maybe even increase, if you have not adjusted your learning rate (training might also throw an overflow warning).

Task:

This task can be done via pen & paper or by inserting some code below. Either way, you should be able to solve both tasks below on paper only using a calculator.

  1. Apply feature scaling onto X using the mean and the standard deviation. What values do the scaled features have?

  2. After the training with scaled features your new θ\theta' values will be very high, something like: θ=[7197,326,326]\theta'=[-7197, 326, -326] (you can try it but you do not have to). Suppose θ=[7197,326,326]\theta'=[-7197, 326, -326]. What are the corresponding θ\theta values for the unscaled data?

  3. Answer the question: What would happen if we did not apply feature scaling and we would use polynomial regression for X?

X_scaled = (X - mu) / std
X_scaled

Summary and Outlook

During this exercise, the linear regression was extended to multidimensional feature space and feature scaling was practiced. You should be able to answer the following questions:

  • How does the implementation of the multivariate regression differ from the univariate one?
  • Why do we apply feature scaling?
  • Why does feature scaling help?

Licenses

Notebook License (CC-BY-SA 4.0)

The following license applies to the complete notebook, including code cells. It does however not apply to any referenced external media (e.g., images).

Exercise: Multivariate Linear Regression
by Christian Herta, Klaus Strohmenger
is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License.
Based on a work at https://gitlab.com/deep.TEACHING.

Code License (MIT)

The following license only applies to code cells of the notebook.

Copyright 2018 Christian Herta, Klaus Strohmenger

Permission is hereby granted, free of charge, to any person obtaining a copy of this software and associated documentation files (the "Software"), to deal in the Software without restriction, including without limitation the rights to use, copy, modify, merge, publish, distribute, sublicense, and/or sell copies of the Software, and to permit persons to whom the Software is furnished to do so, subject to the following conditions:

The above copyright notice and this permission notice shall be included in all copies or substantial portions of the Software.

THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE SOFTWARE.