In the previous lecture, we learned what is a supervised machine learning problem.
Before we turn our attention to Linear Regression, we will first dive deeper into the question of optimization.
At a high level, a supervised machine learning problem has the following structure:
$$ \text{Dataset} + \underbrace{\text{Learning Algorithm}}_\text{Model Class + Objective + Optimizer } \to \text{Predictive Model} $$The predictive model is chosen to model the relationship between inputs and targets. For instance, it can predict future targets.
At a high-level an optimizer takes
Intuitively, this is the function that bests "fits" the data on the training dataset $\mathcal{D} = \{(x^{(i)}, y^{(i)}) \mid i = 1,2,...,n\}$.
We will use the a quadratic function as our running example for an objective $J$.
import numpy as np
import matplotlib.pyplot as plt
plt.rcParams['figure.figsize'] = [8, 4]
def quadratic_function(theta):
"""The cost function, J(theta)."""
return 0.5*(2*theta-1)**2
We can visualize it.
# First construct a grid of theta1 parameter pairs and their corresponding
# cost function values.
thetas = np.linspace(-0.2,1,10)
f_vals = quadratic_function(thetas[:,np.newaxis])
plt.plot(thetas, f_vals)
plt.xlabel('Theta')
plt.ylabel('Objective value')
plt.title('Simple quadratic function')
Text(0.5, 1.0, 'Simple quadratic function')
Recall that the derivative $$\frac{d f(\theta_0)}{d \theta}$$ of a univariate function $f : \mathbb{R} \to \mathbb{R}$ is the instantaneous rate of change of the function $f(\theta)$ with respect to its parameter $\theta$ at the point $\theta_0$.
def quadratic_derivative(theta):
return (2*theta-1)*2
df0 = quadratic_derivative(np.array([[0]])) # derivative at zero
f0 = quadratic_function(np.array([[0]]))
line_length = 0.2
plt.plot(thetas, f_vals)
plt.annotate('', xytext=(0-line_length, f0-line_length*df0), xy=(0+line_length, f0+line_length*df0),
arrowprops={'arrowstyle': '-', 'lw': 1.5}, va='center', ha='center')
plt.xlabel('Theta')
plt.ylabel('Objective value')
plt.title('Simple quadratic function')
Text(0.5, 1.0, 'Simple quadratic function')
pts = np.array([[0, 0.5, 0.8]]).reshape((3,1))
df0s = quadratic_derivative(pts)
f0s = quadratic_function(pts)
plt.plot(thetas, f_vals)
for pt, f0, df0 in zip(pts.flatten(), f0s.flatten(), df0s.flatten()):
plt.annotate('', xytext=(pt-line_length, f0-line_length*df0), xy=(pt+line_length, f0+line_length*df0),
arrowprops={'arrowstyle': '-', 'lw': 1}, va='center', ha='center')
plt.xlabel('Theta')
plt.ylabel('Objective value')
plt.title('Simple quadratic function')
Text(0.5, 1.0, 'Simple quadratic function')
The partial derivative $$\frac{\partial f(\theta_0)}{\partial \theta_j}$$ of a multivariate function $f : \mathbb{R}^d \to \mathbb{R}$ is the derivative of $f$ with respect to $\theta_j$ while all othe other inputs $\theta_k$ for $k\neq j$ are fixed.
The gradient $\nabla_\theta f$ further extends the derivative to multivariate functions $f : \mathbb{R}^d \to \mathbb{R}$, and is defined at a point $\theta_0$ as
$$ \nabla_\theta f (\theta_0) = \begin{bmatrix} \frac{\partial f(\theta_0)}{\partial \theta_1} \\ \frac{\partial f(\theta_0)}{\partial \theta_2} \\ \vdots \\ \frac{\partial f(\theta_0)}{\partial \theta_d} \end{bmatrix}.$$The $j$-th entry of the vector $\nabla_\theta f (\theta_0)$ is the partial derivative $\frac{\partial f(\theta_0)}{\partial \theta_j}$ of $f$ with respect to the $j$-th component of $\theta$.
We will use a quadratic function as a running example.
def quadratic_function2d(theta0, theta1):
"""Quadratic objective function, J(theta0, theta1).
The inputs theta0, theta1 are 2d arrays and we evaluate
the objective at each value theta0[i,j], theta1[i,j].
We implement it this way so it's easier to plot the
level curves of the function in 2d.
Parameters:
theta0 (np.array): 2d array of first parameter theta0
theta1 (np.array): 2d array of second parameter theta1
Returns:
fvals (np.array): 2d array of objective function values
fvals is the same dimension as theta0 and theta1.
fvals[i,j] is the value at theta0[i,j] and theta1[i,j].
"""
theta0 = np.atleast_2d(np.asarray(theta0))
theta1 = np.atleast_2d(np.asarray(theta1))
return 0.5*((2*theta1-2)**2 + (theta0-3)**2)
Let's visualize this function.
theta0_grid = np.linspace(-4,7,101)
theta1_grid = np.linspace(-1,4,101)
theta_grid = theta0_grid[np.newaxis,:], theta1_grid[:,np.newaxis]
J_grid = quadratic_function2d(theta0_grid[np.newaxis,:], theta1_grid[:,np.newaxis])
X, Y = np.meshgrid(theta0_grid, theta1_grid)
contours = plt.contour(X, Y, J_grid, 10)
plt.clabel(contours)
plt.axis('equal')
(-4.0, 7.0, -1.0, 4.0)
Let's write down the derivative of the quadratic function.
def quadratic_derivative2d(theta0, theta1):
"""Derivative of quadratic objective function.
The inputs theta0, theta1 are 1d arrays and we evaluate
the derivative at each value theta0[i], theta1[i].
Parameters:
theta0 (np.array): 1d array of first parameter theta0
theta1 (np.array): 1d array of second parameter theta1
Returns:
grads (np.array): 2d array of partial derivatives
grads is of the same size as theta0 and theta1
along first dimension and of size
two along the second dimension.
grads[i,j] is the j-th partial derivative
at input theta0[i], theta1[i].
"""
# this is the gradient of 0.5*((2*theta1-2)**2 + (theta0-3)**2)
grads = np.stack([theta0-3, (2*theta1-2)*2], axis=1)
grads = grads.reshape([len(theta0), 2])
return grads
We can visualize the derivative.
theta0_pts, theta1_pts = np.array([2.3, -1.35, -2.3]), np.array([2.4, -0.15, 2.75])
dfs = quadratic_derivative2d(theta0_pts, theta1_pts)
line_length = 0.2
contours = plt.contour(X, Y, J_grid, 10)
for theta0_pt, theta1_pt, df0 in zip(theta0_pts, theta1_pts, dfs):
plt.annotate('', xytext=(theta0_pt, theta1_pt),
xy=(theta0_pt-line_length*df0[0], theta1_pt-line_length*df0[1]),
arrowprops={'arrowstyle': '->', 'lw': 2}, va='center', ha='center')
plt.scatter(theta0_pts, theta1_pts)
plt.clabel(contours)
plt.xlabel('Theta0')
plt.ylabel('Theta1')
plt.title('Gradients of the quadratic function')
plt.axis('equal')
(-4.0, 7.0, -1.0, 4.0)
Next, we will use gradients to define an important algorithm called gradient descent.
The gradient $\nabla_\theta f$ further extends the derivative to multivariate functions $f : \mathbb{R}^d \to \mathbb{R}$, and is defined at a point $\theta_0$ as
$$ \nabla_\theta f (\theta_0) = \begin{bmatrix} \frac{\partial f(\theta_0)}{\partial \theta_1} \\ \frac{\partial f(\theta_0)}{\partial \theta_2} \\ \vdots \\ \frac{\partial f(\theta_0)}{\partial \theta_d} \end{bmatrix}.$$The $j$-th entry of the vector $\nabla_\theta f (\theta_0)$ is the partial derivative $\frac{\partial f(\theta_0)}{\partial \theta_j}$ of $f$ with respect to the $j$-th component of $\theta$.
theta0_pts, theta1_pts = np.array([2.3, -1.35, -2.3]), np.array([2.4, -0.15, 2.75])
dfs = quadratic_derivative2d(theta0_pts, theta1_pts)
line_length = 0.2
contours = plt.contour(X, Y, J_grid, 10)
for theta0_pt, theta1_pt, df0 in zip(theta0_pts, theta1_pts, dfs):
plt.annotate('', xytext=(theta0_pt, theta1_pt),
xy=(theta0_pt-line_length*df0[0], theta1_pt-line_length*df0[1]),
arrowprops={'arrowstyle': '->', 'lw': 2}, va='center', ha='center')
plt.scatter(theta0_pts, theta1_pts)
plt.clabel(contours)
plt.xlabel('Theta0')
plt.ylabel('Theta1')
plt.title('Gradients of the quadratic function')
plt.axis('equal')
(-4.0, 7.0, -1.0, 4.0)
Gradient descent is a very common optimization algorithm used in machine learning.
The intuition behind gradient descent is to repeatedly obtain the gradient to determine the direction in which the function decreases most steeply and take a step in that direction.
More formally, if we want to optimize $J(\theta)$, we start with an initial guess $\theta_0$ for the parameters and repeat the following update until $\theta$ is no longer changing: $$ \theta_i := \theta_{i-1} - \alpha \cdot \nabla_\theta J(\theta_{i-1}). $$
As code, this method may look as follows:
theta, theta_prev = random_initialization()
while norm(theta - theta_prev) > convergence_threshold:
theta_prev = theta
theta = theta_prev - step_size * gradient(theta_prev)
In the above algorithm, we stop when $||\theta_i - \theta_{i-1}||$ is small.
It's easy to implement this function in numpy.
convergence_threshold = 2e-1
step_size = 2e-1
theta, theta_prev = np.array([[-2], [3]]), np.array([[0], [0]])
opt_pts = [theta.flatten()]
opt_grads = []
while np.linalg.norm(theta - theta_prev) > convergence_threshold:
# we repeat this while the value of the function is decreasing
theta_prev = theta
gradient = quadratic_derivative2d(*theta).reshape([2,1])
theta = theta_prev - step_size * gradient
opt_pts += [theta.flatten()]
opt_grads += [gradient.flatten()]
We can now visualize gradient descent.
opt_pts = np.array(opt_pts)
opt_grads = np.array(opt_grads)
contours = plt.contour(X, Y, J_grid, 10)
plt.clabel(contours)
plt.scatter(opt_pts[:,0], opt_pts[:,1])
for opt_pt, opt_grad in zip(opt_pts, opt_grads):
plt.annotate('', xytext=(opt_pt[0], opt_pt[1]),
xy=(opt_pt[0]-0.8*step_size*opt_grad[0], opt_pt[1]-0.8*step_size*opt_grad[1]),
arrowprops={'arrowstyle': '->', 'lw': 2}, va='center', ha='center')
plt.axis('equal')
(-4.0, 7.0, -1.0, 4.0)
Let's now use gradient descent to derive a supervised learning algorithm for linear models.
If we want to optimize $J(\theta)$, we start with an initial guess $\theta_0$ for the parameters and repeat the following update: $$ \theta_i := \theta_{i-1} - \alpha \cdot \nabla_\theta J(\theta_{i-1}). $$
As code, this method may look as follows:
theta, theta_prev = random_initialization()
while norm(theta - theta_prev) > convergence_threshold:
theta_prev = theta
theta = theta_prev - step_size * gradient(theta_prev)
Recall that a linear model has the form \begin{align*} y & = \theta_0 + \theta_1 \cdot x_1 + \theta_2 \cdot x_2 + ... + \theta_d \cdot x_d \end{align*} where $x \in \mathbb{R}^d$ is a vector of features and $y$ is the target. The $\theta_j$ are the parameters of the model.
By using the notation $x_0 = 1$, we can represent the model in a vectorized form $$ f_\theta(x) = \sum_{j=0}^d \theta_j \cdot x_j = \theta^\top x. $$
Let's define our model in Python.
def f(X, theta):
"""The linear model we are trying to fit.
Parameters:
theta (np.array): d-dimensional vector of parameters
X (np.array): (n,d)-dimensional data matrix
Returns:
y_pred (np.array): n-dimensional vector of predicted targets
"""
return X.dot(theta)
We pick $\theta$ to minimize the mean squared error (MSE). Slight variants of this objective are also known as the residual sum of squares (RSS) or the sum of squared residuals (SSR). $$J(\theta)= \frac{1}{2n} \sum_{i=1}^n(y^{(i)}-\theta^\top x^{(i)})^2$$ In other words, we are looking for the best compromise in $\theta$ over all the data points.
Let's implement mean squared error.
def mean_squared_error(theta, X, y):
"""The cost function, J, describing the goodness of fit.
Parameters:
theta (np.array): d-dimensional vector of parameters
X (np.array): (n,d)-dimensional design matrix
y (np.array): n-dimensional vector of targets
"""
return 0.5*np.mean((y-f(X, theta))**2)
Let's work out what a partial derivative is for the MSE error loss for a linear model.
\begin{align*} \frac{\partial J(\theta)}{\partial \theta_j} & = \frac{\partial}{\partial \theta_j} \frac{1}{2} \left( f_\theta(x) - y \right)^2 \\ & = \left( f_\theta(x) - y \right) \cdot \frac{\partial}{\partial \theta_j} \left( f_\theta(x) - y \right) \\ & = \left( f_\theta(x) - y \right) \cdot \frac{\partial}{\partial \theta_j} \left( \sum_{k=0}^d \theta_k \cdot x_k - y \right) \\ & = \left( f_\theta(x) - y \right) \cdot x_j \end{align*}We can use this derivation to obtain an expression for the gradient of the MSE for a linear model \begin{align*} \nabla_\theta J (\theta) = \begin{bmatrix} \frac{\partial f(\theta)}{\partial \theta_1} \ \frac{\partial f(\theta)}{\partial \theta_2} \ \vdots \ \frac{\partial f(\theta)}{\partial \theta_d}
= \left( f_\theta(x) - y \right) \cdot \bf{x} . \end{align*}
Let's implement the gradient.
def mse_gradient(theta, X, y):
"""The gradient of the cost function.
Parameters:
theta (np.array): d-dimensional vector of parameters
X (np.array): (n,d)-dimensional design matrix
y (np.array): n-dimensional vector of targets
Returns:
grad (np.array): d-dimensional gradient of the MSE
"""
return np.mean((f(X, theta) - y) * X.T, axis=1)
In this section, we are going to again use the UCI Diabetes Dataset.
%matplotlib inline
import matplotlib.pyplot as plt
plt.rcParams['figure.figsize'] = [8, 4]
import numpy as np
import pandas as pd
from sklearn import datasets
# Load the diabetes dataset
X, y = datasets.load_diabetes(return_X_y=True, as_frame=True)
# add an extra column of onens
X['one'] = 1
# Collect 20 data points and only use bmi dimension
X_train = X.iloc[-20:].loc[:, ['bmi', 'one']]
y_train = y.iloc[-20:] / 300
plt.scatter(X_train.loc[:,['bmi']], y_train, color='black')
plt.xlabel('Body Mass Index (BMI)')
plt.ylabel('Diabetes Risk')
Text(0, 0.5, 'Diabetes Risk')
Putting this together with the gradient descent algorithm, we obtain a learning method for training linear models.
theta, theta_prev = random_initialization()
while abs(J(theta) - J(theta_prev)) > conv_threshold:
theta_prev = theta
theta = theta_prev - step_size * (f(x, theta)-y) * x
This update rule is also known as the Least Mean Squares (LMS) or Widrow-Hoff learning rule.
threshold = 1e-3
step_size = 4e-1
theta, theta_prev = np.array([2,1]), np.ones(2,)
opt_pts = [theta]
opt_grads = []
iter = 0
while np.linalg.norm(theta - theta_prev) > threshold:
if iter % 100 == 0:
print('Iteration %d. MSE: %.6f' % (iter, mean_squared_error(theta, X_train, y_train)))
theta_prev = theta
gradient = mse_gradient(theta, X_train, y_train)
theta = theta_prev - step_size * gradient
opt_pts += [theta]
opt_grads += [gradient]
iter += 1
Iteration 0. MSE: 0.171729 Iteration 100. MSE: 0.014765 Iteration 200. MSE: 0.014349 Iteration 300. MSE: 0.013997 Iteration 400. MSE: 0.013701
x_line = np.stack([np.linspace(-0.1, 0.1, 10), np.ones(10,)])
y_line = opt_pts[-1].dot(x_line)
plt.scatter(X_train.loc[:,['bmi']], y_train, color='black')
plt.plot(x_line[0], y_line)
plt.xlabel('Body Mass Index (BMI)')
plt.ylabel('Diabetes Risk')
Text(0, 0.5, 'Diabetes Risk')
In practice, there is a more effective way than gradient descent to find linear model parameters.
We will see this method here, which will lead to our first non-toy algorithm: Ordinary Least Squares.
The gradient $\nabla_\theta f$ further extends the derivative to multivariate functions $f : \mathbb{R}^d \to \mathbb{R}$, and is defined at a point $\theta_0$ as
$$ \nabla_\theta f (\theta_0) = \begin{bmatrix} \frac{\partial f(\theta_0)}{\partial \theta_1} \\ \frac{\partial f(\theta_0)}{\partial \theta_2} \\ \vdots \\ \frac{\partial f(\theta_0)}{\partial \theta_d} \end{bmatrix}.$$In other words, the $j$-th entry of the vector $\nabla_\theta f (\theta_0)$ is the partial derivative $\frac{\partial f(\theta_0)}{\partial \theta_j}$ of $f$ with respect to the $j$-th component of $\theta$.
In this section, we are going to again use the UCI Diabetes Dataset.
%matplotlib inline
import matplotlib.pyplot as plt
plt.rcParams['figure.figsize'] = [8, 4]
import numpy as np
import pandas as pd
from sklearn import datasets
# Load the diabetes dataset
X, y = datasets.load_diabetes(return_X_y=True, as_frame=True)
# add an extra column of onens
X['one'] = 1
# Collect 20 data points
X_train = X.iloc[-20:]
y_train = y.iloc[-20:]
plt.scatter(X_train.loc[:,['bmi']], y_train, color='black')
plt.xlabel('Body Mass Index (BMI)')
plt.ylabel('Diabetes Risk')
Text(0, 0.5, 'Diabetes Risk')
Machine learning algorithms are most easily defined in the language of linear algebra. Therefore, it will be useful to represent the entire dataset as one matrix $X \in \mathbb{R}^{n \times d}$, of the form: $$ X = \begin{bmatrix} x^{(1)}_1 & x^{(1)}_2 & \ldots & x^{(1)}_d \ x^{(2)}_1 & x^{(2)}_2 & \ldots & x^{(2)}_d \ \vdots \ x^{(n)}_1 & x^{(n)}_2 & \ldots & x^{(n)}_d
.$$
We can view the design matrix for the diabetes dataset.
X_train.head()
| age | sex | bmi | bp | s1 | s2 | s3 | s4 | s5 | s6 | one | |
|---|---|---|---|---|---|---|---|---|---|---|---|
| 422 | -0.078165 | 0.050680 | 0.077863 | 0.052858 | 0.078236 | 0.064447 | 0.026550 | -0.002592 | 0.040672 | -0.009362 | 1 |
| 423 | 0.009016 | 0.050680 | -0.039618 | 0.028758 | 0.038334 | 0.073529 | -0.072854 | 0.108111 | 0.015567 | -0.046641 | 1 |
| 424 | 0.001751 | 0.050680 | 0.011039 | -0.019442 | -0.016704 | -0.003819 | -0.047082 | 0.034309 | 0.024053 | 0.023775 | 1 |
| 425 | -0.078165 | -0.044642 | -0.040696 | -0.081414 | -0.100638 | -0.112795 | 0.022869 | -0.076395 | -0.020289 | -0.050783 | 1 |
| 426 | 0.030811 | 0.050680 | -0.034229 | 0.043677 | 0.057597 | 0.068831 | -0.032356 | 0.057557 | 0.035462 | 0.085907 | 1 |
Similarly, we can vectorize the target variables into a vector $y \in \mathbb{R}^n$ of the form $$ y = \begin{bmatrix} y^{(1)} \\ y^{(2)} \\ \vdots \\ y^{(n)} \end{bmatrix}.$$
Recall that we may fit a linear model by choosing $\theta$ that minimizes the squared error: $$J(\theta)=\frac{1}{2}\sum_{i=1}^n(y^{(i)}-\theta^\top x^{(i)})^2$$ In other words, we are looking for the best compromise in $\beta$ over all the data points.
We can write this sum in matrix-vector form as: $$J(\theta) = \frac{1}{2} (y-X\theta)^\top(y-X\theta) = \frac{1}{2} \|y-X\theta\|^2,$$ where $X$ is the design matrix and $\|\cdot\|$ denotes the Euclidean norm.
We can a gradient for the mean squared error as follows.
\begin{align*} \nabla_\theta J(\theta) & = \nabla_\theta \frac{1}{2} (X \theta - y)^\top (X \theta - y) \\ & = \frac{1}{2} \nabla_\theta \left( (X \theta)^\top (X \theta) - (X \theta)^\top y - y^\top (X \theta) + y^\top y \right) \\ & = \frac{1}{2} \nabla_\theta \left( \theta^\top (X^\top X) \theta - 2(X \theta)^\top y \right) \\ & = \frac{1}{2} \left( 2(X^\top X) \theta - 2X^\top y \right) \\ & = (X^\top X) \theta - X^\top y \end{align*}We used the facts that $a^\top b = b^\top a$ (line 3), that $\nabla_x b^\top x = b$ (line 4), and that $\nabla_x x^\top A x = 2 A x$ for a symmetric matrix $A$ (line 4).
Setting the above derivative to zero, we obtain the normal equations: $$ (X^\top X) \theta = X^\top y.$$
Hence, the value $\theta^*$ that minimizes this objective is given by: $$ \theta^* = (X^\top X)^{-1} X^\top y.$$
Note that we assumed that the matrix $(X^\top X)$ is invertible; if this is not the case, there are easy ways of addressing this issue.
Let's apply the normal equations.
import numpy as np
theta_best = np.linalg.inv(X_train.T.dot(X_train)).dot(X_train.T).dot(y_train)
theta_best_df = pd.DataFrame(data=theta_best[np.newaxis, :], columns=X.columns)
theta_best_df
| age | sex | bmi | bp | s1 | s2 | s3 | s4 | s5 | s6 | one | |
|---|---|---|---|---|---|---|---|---|---|---|---|
| 0 | -3.888868 | 204.648785 | -64.289163 | -262.796691 | 14003.726808 | -11798.307781 | -5892.15807 | -1136.947646 | -2736.597108 | -393.879743 | 155.698998 |
We can now use our estimate of theta to compute predictions for 3 new data points.
# Collect 3 data points for testing
X_test = X.iloc[:3]
y_test = y.iloc[:3]
# generate predictions on the new patients
y_test_pred = X_test.dot(theta_best)
Let's visualize these predictions.
# visualize the results
plt.xlabel('Body Mass Index (BMI)')
plt.ylabel('Diabetes Risk')
plt.scatter(X_train.loc[:, ['bmi']], y_train)
plt.scatter(X_test.loc[:, ['bmi']], y_test, color='red', marker='o')
plt.plot(X_test.loc[:, ['bmi']], y_test_pred, 'x', color='red', mew=3, markersize=8)
plt.legend(['Model', 'Prediction', 'Initial patients', 'New patients'])
<matplotlib.legend.Legend at 0x128d89668>
So far, we have learned about a very simple linear model. These can capture only simple linear relationships in the data. How can we use what we learned so far to model more complex relationships?
We will now see a simple approach to model complex non-linear relationships called least squares.
Recall that a polynomial of degree $p$ is a function of the form $$ a_p x^p + a_{p-1} x^{p-1} + ... + a_{1} x + a_0. $$
Below are some examples of polynomial functions.
import warnings
warnings.filterwarnings("ignore")
plt.figure(figsize=(16,4))
x_vars = np.linspace(-2, 2)
plt.subplot('131')
plt.title('Quadratic Function')
plt.plot(x_vars, x_vars**2)
plt.legend(["$x^2$"])
plt.subplot('132')
plt.title('Cubic Function')
plt.plot(x_vars, x_vars**3)
plt.legend(["$x^3$"])
plt.subplot('133')
plt.title('Third Degree Polynomial')
plt.plot(x_vars, x_vars**3 + 2*x_vars**2 + x_vars + 1)
plt.legend(["$x^3 + 2 x^2 + x + 1$"])
<matplotlib.legend.Legend at 0x128ed2ac8>
Specifically, given a one-dimensional continuous variable $x$, we can defining a feature function $\phi : \mathbb{R} \to \mathbb{R}^{p+1}$ as $$ \phi(x) = \begin{bmatrix} 1 \\ x \\ x^2 \\ \vdots \\ x^p \end{bmatrix}. $$
The class of models of the form $$ f_\theta(x) := \sum_{j=0}^p \theta_p x^p = \theta^\top \phi(x) $$ with parameters $\theta$ and polynomial features $\phi$ is the set of $p$-degree polynomials.
In this section, we are going to again use the UCI Diabetes Dataset.
%matplotlib inline
import matplotlib.pyplot as plt
plt.rcParams['figure.figsize'] = [8, 4]
import numpy as np
import pandas as pd
from sklearn import datasets
# Load the diabetes dataset
X, y = datasets.load_diabetes(return_X_y=True, as_frame=True)
# add an extra column of onens
X['one'] = 1
# Collect 20 data points
X_train = X.iloc[-20:]
y_train = y.iloc[-20:]
plt.scatter(X_train.loc[:,['bmi']], y_train, color='black')
plt.xlabel('Body Mass Index (BMI)')
plt.ylabel('Diabetes Risk')
Text(0, 0.5, 'Diabetes Risk')
Let's now obtain linear features for this dataset.
X_bmi = X_train.loc[:, ['bmi']]
X_bmi_p3 = pd.concat([X_bmi, X_bmi**2, X_bmi**3], axis=1)
X_bmi_p3.columns = ['bmi', 'bmi2', 'bmi3']
X_bmi_p3['one'] = 1
X_bmi_p3.head()
| bmi | bmi2 | bmi3 | one | |
|---|---|---|---|---|
| 422 | 0.077863 | 0.006063 | 0.000472 | 1 |
| 423 | -0.039618 | 0.001570 | -0.000062 | 1 |
| 424 | 0.011039 | 0.000122 | 0.000001 | 1 |
| 425 | -0.040696 | 0.001656 | -0.000067 | 1 |
| 426 | -0.034229 | 0.001172 | -0.000040 | 1 |
By training a linear model on this featurization of the diabetes set, we can obtain a polynomial model of diabetest risk as a function of BMI.
# Fit a linear regression
theta = np.linalg.inv(X_bmi_p3.T.dot(X_bmi_p3)).dot(X_bmi_p3.T).dot(y_train)
# Show the learned polynomial curve
x_line = np.linspace(-0.1, 0.1, 10)
x_line_p3 = np.stack([x_line, x_line**2, x_line**3, np.ones(10,)], axis=1)
y_train_pred = x_line_p3.dot(theta)
plt.xlabel('Body Mass Index (BMI)')
plt.ylabel('Diabetes Risk')
plt.scatter(X_bmi, y_train)
plt.plot(x_line, y_train_pred)
[<matplotlib.lines.Line2D at 0x1292c99e8>]
We can also take this approach to construct non-linear function of multiples variable by using multivariate polynomials.
For example, a polynomial of degree $2$ over two variables $x_1, x_2$ is a function of the form $$ a_{20} x_1^2 + a_{10} x_1 + a_{02} x_2^2 + a_{01} x_2 + a_{11} x_1 x_2 + a_{00}. $$
In general, a polynomial of degree $p$ over two variables $x_1, x_2$ is a function of the form $$ f(x_1, x_2) = \sum_{i,j \geq 0 : i+j \leq p} a_{ij} x_1^i x_2^j. $$
In our two-dimensional example, this corresponds to a feature function $\phi : \mathbb{R}^2 \to \mathbb{R}^6$ of the form $$ \phi(x) = \begin{bmatrix} 1 \\ x_1 \\ x_1^2 \\ x_2 \\ x_2^2 \\ x_1 x_2 \end{bmatrix}. $$
The same approach holds for polynomials of an degree and any number of variables.
Any non-linear feature map $\phi(x) : \mathbb{R}^d \to \mathbb{R}^p$ can be used in this way to obtain general models of the form $$ f_\theta(x) := \theta^\top \phi(x) $$ that are highly non-linear in $x$ but linear in $\theta$.
For example, here is a way of modeling complex periodic functions via a sum of sines and cosines.
import warnings
warnings.filterwarnings("ignore")
plt.figure(figsize=(16,4))
x_vars = np.linspace(-5, 5)
plt.subplot('131')
plt.title('Cosine Function')
plt.plot(x_vars, np.cos(x_vars))
plt.legend(["$cos(x)$"])
plt.subplot('132')
plt.title('Sine Function')
plt.plot(x_vars, np.sin(2*x_vars))
plt.legend(["$x^3$"])
plt.subplot('133')
plt.title('Combination of Sines and Cosines')
plt.plot(x_vars, np.cos(x_vars) + np.sin(2*x_vars) + np.cos(4*x_vars))
plt.legend(["$cos(x) + sin(2x) + cos(4x)$"])
<matplotlib.legend.Legend at 0x129571160>
