Linear regression

Machine Learning
Linear Regression

Agenda
• Single Dimension Linear Regression

• Multi Dimension Linear Regression

• Gradient Descent

• Generalisation, Over-ﬁtting & Regularisation

• Categorical Inputs

What is Linear Regression?
• Learning

• A supervised algorithm that learns from a set of training samples.

• Each training sample has one or more input values and a single output value.

• The algorithm learns the line, plane or hyper-plane that best ﬁts the training
samples.

• Prediction

• Use the learned line, plane or hyper-plane to predict the output value for any
input sample.

Single Dimension Linear
Regression

Single Dimension Linear Regression
• Single dimension linear regression
has pairs of x and y values as input
training samples.

• It uses these training sample to
derive a line that predicts values of y.

• The training samples are used to
derive the values of a and b that
minimise the error between actual
and predicated values of y.

• We want a line that minimises the
error between the Y values in
training samples and the Y values
that the line passes through.

• Or put another way, we want the
line that “best ﬁts’ the training
samples.

• So we deﬁne the error function for
our algorithm so we can minimise
that error.

• To determine the value of a that
minimises the error E, we look for
where the partial diﬀerential of E
with respect to a is zero.

• To determine the value of b that
where the partial diﬀerential of E
with respect to b is zero.

• By substituting the ﬁnal equations
from the previous two slides we
derive equations for a and b that
minimise the error

• We also define a function which we can
use to score how well derived line fits.

• A value of 1 indicates a perfect fit.

• A value of 0 indicates a fit that is no
better than simply predicting the mean
of the input y values.

• A negative value indicates a fit that is
even worse than just predicting the
mean of the input y values.

Multi Dimension Linear
Regression

Multi Dimension Linear Regression
• Each training sample has an x made
up of multiple input values and a
corresponding y with a single value.

• The inputs can be represented as
an X matrix in which each row is
sample and each column is a
dimension.

• The outputs can be represented as
y matrix in which each row is a
sample.

• Our predicated y values are
calculated by multiple the X matrix
by a matrix of weights, w.

• If there are 2 dimension, then this
equation deﬁnes plane. If there are
more dimensions then it deﬁnes a
hyper-plane.

• We want a plane or hyper-plane
that minimises the error between
the y values in training samples
and the y values that the plane or
hyper-plane passes through.

• Or put another way, we want the
plane/hyper-plane that “best ﬁts’
the training samples.

• So we deﬁne the error function for
our algorithm so we can minimise
that error.

• To determine the value of w that
where the diﬀerential of E with
respect to w is zero.

• We use the Matrix Cookbook to
help with the diﬀerentiation!

• We also define a function which we can
use to score how well derived line fits.

• A value of 1 indicates a perfect fit.

• A value of 0 indicates a fit that is no
better than simply predicting the mean
of the input y values.

• A negative value indicates a fit that is
even worse than just predicting the
mean of the input y values.

• In addition to using the X matrix to represent basic features our training
data, we can can also introduce additional dimensions (i.e. columns in
our X matrix) that are derived from those basic feature values.

• If we introduce derived features whose values are powers of basic
features, our multi-dimensional linear regression can then derive
polynomial curves, planes and hyper-planes.

• For example, if we have just one
basic feature in each sample of X, we
can include a range of powers of that
value into our X matrix like this:

• In non-matrix form our multi-
dimensional linear equation is:

• Inserting the powers of the basic
feature that we have introduced this
becomes a polynomial:

Singular Matrices
• As we have seen, we can use
numpy’s linalg.solve() function to
determine the value of the weights
that result in the lowest possible error.

• But this doesn’t work if np.dot(X.T, X)
is a singular matrix.

• It results in the matrix equivalent of a
divide by zero.

• Gradient descent is an alternative
approach to determining the optimal
weights that in works for all cases,
including this singular matrix case.

Gradient Descent
• Gradient descent is a technique we can use to ﬁnd the minimum of
arbitrarily complex error functions.

• In gradient descent we pick a random set of weights for our algorithm and
iteratively adjust those weights in the direction of the gradient of the error
with respect to each weight.

• As we iterate, the gradient approaches zero and we approach the
minimum error.

• In machine learning we often use gradient descent with our error function
to ﬁnd the weights that give the lowest errors.

Gradient Descent
• Here is an example with a very
simple function:

• The gradient of this function is
given by:

• We choose an random initial
value for x and a learning rate of
0.1 and then start descent.

• On each iteration our x value is
decreasing and the gradient (2x)
is converging towards 0.

Gradient Descent
• The learning rate is a what is know as a hyper-parameter.

• If the learning rate is too small then convergence may take a very long
time.

• If the learning rate is too large then convergence may never happen
because our iterations bounce from one side of the minima to the other.

• Choosing a suitable value for hyper-parameters is an art so try diﬀerent
values and plot the results until you ﬁnd suitable values.

with Gradient Descent
• For multi dimension linear
regression our error function
is:

• Diﬀerentiating this with
respect to the weights vector
gives:

• We can iteratively reduce the
error by adjusting the weights
in the direction of these
gradients.

with Gradient Descent

Generalisation, Over-ﬁtting &
Regularisation

Generalisation & Over-fitting
• As we train our model with more and more data the it may start to fit the training data more and
more accurately, but become worse at handling test data that we feed to it later.

• This is know as “over-fitting” and results in an increased generalisation error.

• To minimise the generalisation error we should

• Collect as much sample data as possible.

• Use a random subset of our sample data for training.

• Use the remaining sample data to test how well our model copes with data it was not trained
with.

• Also, experiment with adding higher degrees of polynomials (X2, X3, etc) as this can reduce
overfitting.

L1 Regularisation (Lasso)
• Having a large number of samples (n) with respect to the number of
dimensionality (d) increases the quality of our model.

• One way to reduce the eﬀective number of dimensions is to use those that
most contribute to the signal and ignore those that mostly act as noise.

• L1 regularisation achieves this by adding a penalty that results in the
weight for the dimensions that act as noise becoming 0.

• L1 regularisation encourages a sparse vector of weights in which few are
non-zero and many are zero.

L1 Regularisation (Lasso)
• In L1 regularisation we add a penalty to
the error function:

• Expanding this we get:

• Take the derivative with respect to w to
ﬁnd our gradient:

• Where sign(w) is -1 if w < 0, 0 if w = 0
and +1 if w > 0

• Note that because sign(w) has no
inverse function we cannot solve for w
and so must use gradient descent.

L2 Regularisation (Ridge)
• Another way to reduce the complexity of our model and prevent overﬁtting
to outliers is L2 regression, which is also known as ridge regression.

• In L2 Regularisation we introduce an additional term to the cost function
that has the eﬀect of penalising large weights and thereby minimising this
skew.

• In L2 regularisation we the sum of
the squares of the weights to the
error function.

• Expanding this we get:

• Take the derivative with respect to
w to ﬁnd our gradient:

• Solving for the values of w that give
minimal error:

L1 & L2 Regularisation (Elastic Net)
• L1 Regularisation minimises the impact of dimensions that have low
weights and are thus largely “noise”.

• L2 Regularisation minimise the impacts of outliers in our training data.

• L1 & L2 Regularisation can be used together and the combination is
referred to as Elastic Net regularisation.

• Because the diﬀerential of the error function contains the sigmoid which
has no inverse, we cannot solve for w and must use gradient descent.

One-hot Encoding
• When some inputs are categories (e.g. gender) rather than numbers (e.g.
age) we need to represent the category values as numbers so they can be
used in our linear regression equations.

• In one-hot encoding we allocate each category value it's own dimension in
the inputs. So, for example, we allocate X1 to Audi, X2 to BMW & X3 to
Mercedes.

• For Audi X = [1,0,0]

• For BMW X = [0,1,0])

• For Mercedes X = [0,0,1]

Summary
• Single Dimension Linear Regression

• Multi Dimension Linear Regression

• Gradient Descent

• Generalisation, Over-ﬁtting & Regularisation

• Categorical Inputs

Linear regression

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