Abstract

In the last session we explored least squares for univariate and multivariate regression. We introduced matrices, linear algebra and derivatives.

In this session we will introduce basis functions which allow us to implement non-linear regression models.

%matplotlib inline

Nonlinear Regression

We’ve now seen how we may perform linear regression. Now, we are going to consider how we can perform non-linear regression. However, before we get into the details of how to do that we first need to consider in what ways the regression can be non-linear. Multivariate linear regression allows us to build models that take many features into account when making our prediction. In this session we are going to introduce basis functions. The term seems complicted, but they are actually based on rather a simple idea. If we are doing a multivariate linear regression, we get extra features that might help us predict our required response varible (or target value), y. But what if we only have one input value? We can actually artificially generate more input values with basis functions.

Non-linear in the Inputs

When we refer to non-linear regression, we are normally referring to whether the regression is non-linear in the input space, or non-linear in the covariates. The covariates are the observations that move with the target (or response) variable. In our notation we have been using $\inputVector_i$ to represent a vector of the covariates associated with the ith observation. The coresponding response variable is $\dataScalar_i$. If a model is non-linear in the inputs, it means that there is a non-linear function between the inputs and the response variable. Linear functions are functions that only involve multiplication and addition, in other words they can be represented through linear algebra. Linear regression involves assuming that a function takes the form
$$ \mappingFunction(\inputVector) = \mappingVector^\top \inputVector $$
where $\mappingVector$ are our regression weights. A very easy way to make the linear regression non-linear is to introduce non-linear functions. When we are introducing non-linear regression these functions are known as basis functions.

Basis Functions

Basis Functions [edit]

Here’s the idea, instead of working directly on the original input space, $\inputVector$, we build models in a new space, $\basisVector(\inputVector)$ where $\basisVector(\cdot)$ is a vector-valued function that is defined on the space $\inputVector$.

Quadratic Basis

Remember, that a vector-valued function is just a vector that contains functions instead of values. Here’s an example for a one dimensional input space, x, being projected to a quadratic basis. First we consider each basis function in turn, we can think of the elements of our vector as being indexed so that we have
$$ \begin{align*} \basisFunc_1(\inputScalar) = 1, \\ \basisFunc_2(\inputScalar) = x, \\ \basisFunc_3(\inputScalar) = \inputScalar^2. \end{align*} $$
Now we can consider them together by placing them in a vector,
$$ \basisVector(\inputScalar) = \begin{bmatrix} 1\\ x \\ \inputScalar^2\end{bmatrix}. $$
For the vector-valued function, we have simply collected the different functions together in the same vector making them notationally easier to deal with in our mathematics.

When we consider the vector-valued function for each data point, then we place all the data into a matrix. The result is a matrix valued function,
$$ \basisMatrix(\inputVector) = \begin{bmatrix} 1 & \inputScalar_1 & \inputScalar_1^2 \\ 1 & \inputScalar_2 & \inputScalar_2^2\\ \vdots & \vdots & \vdots \\ 1 & \inputScalar_n & \inputScalar_n^2 \end{bmatrix} $$
where we are still in the one dimensional input setting so $\inputVector$ here represents a vector of our inputs with $\numData$ elements.

Let’s try constructing such a matrix for a set of inputs. First of all, we create a function that returns the matrix valued function

import numpy as np

def quadratic(x, **kwargs):
    """Take in a vector of input values and return the design matrix associated 
    with the basis functions."""
    return np.hstack([np.ones((x.shape[0], 1)), x, x**2])

Functions Derived from Quadratic Basis

$$ \mappingFunction(\inputScalar) = {\color{cyan}\mappingScalar_0} + {\color{green}\mappingScalar_1 \inputScalar} + {\color{yellow}\mappingScalar_2 \inputScalar^2} $$

{

{

import pods
from ipywidgets import IntSlider

pods.notebook.display_plots('\basisfunction{num_basis:0>3}.svg', 
                            directory='../slides/diagrams/ml', 
                            num_basis=IntSlider(0,0,2,1))

This function takes in an $\numData \times 1$ dimensional vector and returns an $\numData \times 3$ dimensional design matrix containing the basis functions. We can plot those basis functions against there input as follows.

The actual function we observe is then made up of a sum of these functions. This is the reason for the name basis. The term basis means ‘the underlying support or foundation for an idea, argument, or process’, and in this context they form the underlying support for our prediction function. Our prediction function can only be composed of a weighted linear sum of our basis functions.

Quadratic Functions

import pods
from ipywidgets import IntSlider

pods.notebook.display_plots('quadratic_function{num_function:0>3}.svg', 
                            directory='../slides/diagrams/ml', 
                            num_basis=IntSlider(0,0,2,1))

Different Bases [edit]

Our choice of basis can be made based on what our beliefs about what is appropriate for the data. For example, the polynomial basis extends the quadratic basis to aribrary degree, so we might define the jth basis function associated with the model as
$$ \basisFunc_j(\inputScalar_i) = \inputScalar_i^j $$
which is known as the polynomial basis.

Polynomial Basis [edit]

import matplotlib.pyplot as plt
import mlai
import teaching_plots as plot

%load -s polynomial mlai.py

Functions Derived from Polynomial Basis

$$ \mappingFunction(\inputScalar) = {\color{cyan}\mappingScalar_0} + {\color{green}\mappingScalar_1 \inputScalar} + {\color{yellow}\mappingScalar_2 \inputScalar^2} $$

import pods
from ipywidgets import IntSlider

pods.notebook.display_plots('polynomial_basis{num_basis:0>3}.svg', 
                            directory='../slides/diagrams/ml', 
                num_basis=IntSlider(1,1,4,1))

To aid in understanding how a basis works, we’ve provided you with a small interactive tool for exploring this polynomial basis. The tool can be summoned with the following command.

import pods

pods.notebook.display_prediction(basis=mlai.polynomial, num_basis=4)

Try moving the sliders around to change the weight of each basis function. Click the control box display_basis to show the underlying basis functions (in red). The prediction function is shown in a thick blue line. Warning the sliders aren’t presented quite in the correct order. w_0 is associated with the bias, w_1 is the linear term, w_2 the quadratic and here (because we have four basis functions) we have w_3 for the cubic term. So the subscript of the weight parameter is always associated with the corresponding polynomial’s degree.

Different Basis

The polynomial basis is widely used in Engineering and graphics, but it has some drawbacks in machine learning: outside the input region between -1 and 1, the values of the polynomial basis rise very quickly.

Now we look at basis functions that have been used as the activation functions in neural network model.

Radial Basis Functions [edit]

Another type of basis is sometimes known as a ‘radial basis’ because the effect basis functions are constructed on ‘centres’ and the effect of each basis function decreases as the radial distance from each centre increases.

$$\basisFunc_j(\inputScalar) = \exp\left(-\frac{(\inputScalar-\mu_j)^2}{\lengthScale^2}\right)$$

import matplotlib.pyplot as plt
import mlai
import teaching_plots as plot

%load -s radial mlai.py

pods.notebook.display_prediction(basis=mlai.radial, num_basis=4)

from ipywidgets import IntSlider
import pods

pods.notebook.display_plots('radial_basis{num_basis:0>3}.svg', 
                            directory='../slides/diagrams/ml', 
                            num_basis=IntSlider(0,0,2,1))

Functions Derived from Radial Basis

$$ \mappingFunction(\inputScalar) = {\color{cyan}\mappingScalar_1 e^{-2(\inputScalar+1)^2}} + {\color{green}\mappingScalar_2e^{-2\inputScalar^2}} + {\color{yellow}\mappingScalar_3 e^{-2(\inputScalar-1)^2}} $$

from ipywidgets import IntSlider
import pods

pods.notebook.display_plots('radial_function{func_num:0>3}.svg', directory='../slides/diagrams/ml', func_num=IntSlider(0,0,2,1))

Rectified Linear Units [edit]

import numpy as np

%load -s relu mlai.py

import pods

pods.notebook.display_prediction(basis=mlai.relu, num_basis=4)

Rectified linear units are popular in the current generation of multilayer perceptron models, or deep networks. These basis functions start flat, and then become linear functions at a certain threshold.

import matplotlib.pyplot as plt
import teaching_plots as plot
import mlai

Functions Derived from Relu Basis

$$ \mappingFunction(\inputScalar) = {\color{cyan}\mappingScalar_0} + {\color{green}\mappingScalar_1 xH(x+1.0) } + {\color{yellow}\mappingScalar_2 xH(x+0.33) } + {\color{magenta}\mappingScalar_3 xH(x-0.33)} + {\color{red}\mappingScalar_4 xH(x-1.0)} $$

import pods
from ipywidgets import IntSlider

pods.notebook.display_plots('relu_basis{num_basis:0>3}.svg', 
                            directory='../slides/diagrams/ml', 
                num_basis=IntSlider(0,0,4,1))

Hyperbolic Tangent Basis [edit]

%load -s tanh mlai.py

import pods

pods.notebook.display_prediction(basis=mlai.tanh, num_basis=4)

Sigmoid or hyperbolic tangent basis was popular in the original generation of multilayer perceptron models, or deep networks. These basis functions start flat, rise and then saturate.

Functions Derived from Tanh Basis

$$ \mappingFunction(\inputScalar) = {\color{cyan}\mappingScalar_0} + {\color{green}\mappingScalar_1 } + {\color{yellow}\mappingScalar_3 } $$

import pods
from ipywidgets import IntSlider

pods.notebook.display_plots('tanh_basis{num_basis:0>3}.svg', 
                            directory='../slides/diagrams/ml', 
                            num_basis=IntSlider(0,0,4,1))

Fourier Basis [edit]

Joseph Fourier suggested that functions could be converted to a sum of sines and cosines. A Fourier basis is a linear weighted sum of these functions.
$$\basisFunc_j(\inputScalar) = \mappingScalar_0 + \mappingScalar_1 \sin(\inputScalar) + \mappingScalar_2 \cos(\inputScalar) + \mappingScalar_3 \sin(2\inputScalar) + \mappingScalar_4 \cos(2\inputScalar)$$

import numpy as np

%load -s fourier mlai.py

import matplotlib.pyplot as plt
import mlai
import teaching_plots as plot

import pods
from ipywidgets import IntSlider

pods.notebook.display_plots('fourier_basis{num_basis:0>3}.svg', 
                            directory='../slides/diagrams/ml', 
                            num_basis=IntSlider(0,0,4,1))

In this code, basis functions with an odd index are sine and basis functions with an even index are cosine. The first basis function (index 0, so cosine) has a frequency of 0 and then frequencies increase every time a sine and cosine are included.

pods.notebook.display_prediction(basis=mlai.fourier, num_basis=5)

Functions Derived from Fourier Basis

$$ \mappingFunction(\inputScalar) = {\color{cyan}\mappingScalar_0} + {\color{green}\mappingScalar_1 \sin(\inputScalar)} + {\color{yellow}\mappingScalar_2 \cos(\inputScalar)} + {\color{magenta}\mappingScalar_3 \sin(2\inputScalar)} + {\color{red}\mappingScalar_4 \cos(2\inputScalar)} $$

import pods
from ipywidgets import IntSlider

pods.notebook.display_plots('fourier_function{func_num:0>3}.svg', directory='../slides/diagrams/ml', func_num=IntSlider(0,0,2,1))

Fitting to Data [edit]

Now we are going to consider how these basis functions can be adjusted to fit to a particular data set. We will return to the olympic marathon data from last time. First we will scale the output of the data to be zero mean and variance 1.

import pods
import numpy as np
import matplotlib.pyplot as plt

data = pods.datasets.olympic_marathon_men()
y = data['Y']
x = data['X']
y -= y.mean()
y /= y.std()

%load -s polynomial mlai.py

pods

pods.notebook.display_prediction(basis=dict(radial=mlai.radial, 
                                            polynomial=mlai.polynomial, 
                                            fourier=mlai.fourier, 
                                            relu=mlai.relu), 
                                 data_limits=(1888, 2020),
                                 fig=fig, ax=ax,
                                 offset=0.,
                                 wlim = (-4., 4.),
                                 num_basis=4)

np.asarray([[1, 2, 3, 4]]).shape

Basis Function Models

$$ \mappingFunction(\inputVector_i) = \sum_{j=1}^\numBasisFunc \mappingScalar_j \basisFunc_{i, j} $$

$$ \mappingFunction(\inputVector_i) = \mappingVector^\top \basisVector_i $$

Log Likelihood for Basis Function Model

$$ p\left(\dataScalar_i|\inputScalar_i\right)=\frac{1}{\sqrt{2\pi\dataStd^2}}\exp\left(-\frac{\left(\dataScalar_i-\mappingVector^{\top}\basisVector_i\right)^{2}}{2\dataStd^2}\right). $$

$$ L(\mappingVector,\dataStd^2)= -\frac{\numData}{2}\log \dataStd^2-\frac{\numData}{2}\log 2\pi -\frac{\sum_{i=1}^{\numData}\left(\dataScalar_i-\mappingVector^{\top}\basisVector_i\right)^{2}}{2\dataStd^2}. $$

Objective Function

$$ \errorFunction(\mappingVector,\dataStd^2)= \frac{\numData}{2}\log\dataStd^2 + \frac{\sum_{i=1}^{\numData}\left(\dataScalar_i-\mappingVector^{\top}\basisVector_i\right)^{2}}{2\dataStd^2}. $$

Expand the Brackets

$$ \begin{align} \errorFunction(\mappingVector,\dataStd^2) = &\frac{\numData}{2}\log \dataStd^2 + \frac{1}{2\dataStd^2}\sum_{i=1}^{\numData}\dataScalar_i^{2}-\frac{1}{\dataStd^2}\sum_{i=1}^{\numData}\dataScalar_i\mappingVector^{\top}\basisVector_i\\ &+\frac{1}{2\dataStd^2}\sum_{i=1}^{\numData}\mappingVector^{\top}\basisVector_i\basisVector_i^{\top}\mappingVector+\text{const}. \end{align} $$

Expand the Brackets

$$\begin{align} \errorFunction(\mappingVector, \dataStd^2) = & \frac{\numData}{2}\log \dataStd^2 + \frac{1}{2\dataStd^2}\sum_{i=1}^{\numData}\dataScalar_i^{2}-\frac{1}{\dataStd^2} \mappingVector^\top\sum_{i=1}^{\numData}\basisVector_i \dataScalar_i\\ & +\frac{1}{2\dataStd^2}\mappingVector^{\top}\left[\sum_{i=1}^{\numData}\basisVector_i\basisVector_i^{\top}\right]\mappingVector+\text{const}.\end{align}$$

Design Matrices

We like to make use of design matrices for our data. Design matrices, as you will recall, involve placing the data points into rows of the matrix and data features into the columns of the matrix. By convention, we are referincing a vector with a bold lower case letter, and a matrix with a bold upper case letter. The design matrix is therefore given by
$$ \basisMatrix = \begin{bmatrix} \mathbf{1} & \inputVector & \inputVector^2\end{bmatrix} $$
so that
$$ \basisMatrix \in \Re^{\numData \times \dataDim}. $$

Multivariate Derivatives Reminder

$$\frac{\text{d}\mathbf{a}^{\top}\mappingVector}{\text{d}\mappingVector}=\mathbf{a}$$
and
$$\frac{\text{d}\mappingVector^{\top}\mathbf{A}\mappingVector}{\text{d}\mappingVector}=\left(\mathbf{A}+\mathbf{A}^{\top}\right)\mappingVector$$
or if A is symmetric (i.e. A = A^⊤)
$$\frac{\text{d}\mappingVector^{\top}\mathbf{A}\mappingVector}{\text{d}\mappingVector}=2\mathbf{A}\mappingVector.$$

Differentiate

Differentiating with respect to the vector $\mappingVector$ we obtain
$$\frac{\text{d} E\left(\mappingVector,\dataStd^2 \right)}{\text{d}\mappingVector}=-\frac{1}{\dataStd^2} \sum_{i=1}^{\numData}\basisVector_i\dataScalar_i+\frac{1}{\dataStd^2} \left[\sum_{i=1}^{\numData}\basisVector_i\basisVector_i^{\top}\right]\mappingVector$$
Leading to
$$\mappingVector^{*}=\left[\sum_{i=1}^{\numData}\basisVector_i\basisVector_i^{\top}\right]^{-1}\sum_{i=1}^{\numData}\basisVector_i\dataScalar_i,$$

Matrix Notation

Rewrite in matrix notation:
$$ \sum_{i=1}^{\numData}\basisVector_i\basisVector_i^\top = \basisMatrix^\top \basisMatrix$$

$$\sum _{i=1}^{\numData}\basisVector_i\dataScalar_i = \basisMatrix^\top \dataVector $$

Update Equations

• Update for $\mappingVector^{*}$
  $$ \mappingVector^{*} = \left(\basisMatrix^\top \basisMatrix\right)^{-1} \basisMatrix^\top \dataVector $$
  
• The equation for $\left.\dataStd^2\right.^{*}$ may also be found
  $$ \left.\dataStd^2\right.^{*}=\frac{\sum_{i=1}^{\numData}\left(\dataScalar_i-\left.\mappingVector^{*}\right.^{\top}\basisVector_i\right)^{2}}{\numData}. $$
  

Avoid Direct Inverse

• E.g. Solve for $\mappingVector$
  $$ \left(\basisMatrix^\top \basisMatrix\right)\mappingVector = \basisMatrix^\top \dataVector $$
  
• See np.linalg.solve
• In practice use QR decomposition (see lab class notes).

Polynomial Fits to Olympic Data

import numpy as np
from matplotlib import pyplot as plt
import teaching_plots as plot
import mlai
import pods

basis = mlai.polynomial

data = pods.datasets.olympic_marathon_men()

x = data['X']
y = data['Y']

xlim = [1892, 2020]
max_basis = 27

ll = np.array([np.nan]*(max_basis))
sum_squares = np.array([np.nan]*(max_basis))
basis=mlai.Basis(mlai.polynomial, number=1, data_limits=xlim)

from ipywidgets import IntSlider

pods.notebook.display_plots('olympic_LM_polynomial_number{num_basis:0>3}.svg',
                            directory='../slides/diagrams/ml', 
                            num_basis=IntSlider(1,1,28,1))

Non-linear but Linear in the Parameters

One rather nice aspect of our model is that whilst it is non-linear in the inputs, it is still linear in the parameters $\mappingVector$. This means that our derivations from before continue to operate to allow us to work with this model. In fact, although this is a non-linear regression it is still known as a linear model because it is linear in the parameters,

$$ \mappingFunction(\inputVector) = \mappingVector^\top \basisVector(\inputVector) $$
where the vector $\inputVector$ appears inside the basis functions, making our result, $\mappingFunction(\inputVector)$ non-linear in the inputs, but $\mappingVector$ appears outside our basis function, making our result linear in the parameters. In practice, our basis function itself may contain its own set of parameters,
$$ \mappingFunction(\inputVector) = \mappingVector^\top \basisVector(\inputVector; \boldsymbol{\theta}), $$
that we’ve denoted here as θ. If these parameters appear inside the basis function then our model is non-linear in these parameters.

Fitting the Model Yourself

You now have everything you need to fit a non- linear (in the inputs) basis function model to the marathon data.

Lecture on Basis Functions from GPRS Uganda

Use of QR Decomposition for Numerical Stability

In the last session we showed how rather than computing $\inputMatrix^\top\inputMatrix$ as an intermediate step to our solution, we could compute the solution to the regressiond directly through QR-decomposition. Now we will consider an example with non linear basis functions where such computation is critical for forming the right answer.

TODO example with polynomials.

import numpy as np

x = np.random.normal(size=(10, 1))

Phi = fourier(x, 5)

(np.dot(Phi.T,Phi))

Phi*Phi

References