4.2 KiB
Exercise 7
Generating points according to gaussian distributions
The firts task of esercise 7 is to generate two sets of 2D points $(x, y)$ according to two bivariate gaussian distributions with parameters:
\text{signal} \quad
\begin{cases}
\mu = (0, 0) \\
\sigma_x = \sigma_y = 0.3 \\
\rho = 0.5
\end{cases}
\et
\text{noise} \quad
\begin{cases}
\mu = (4, 4) \\
\sigma_x = \sigma_y = 1 \\
\rho = 0.4
\end{cases}
where \mu stands for the mean, \sigma_x and \sigma_y stand for the
standard deviations in x and y directions respectively and \rho is the
correlation.
In the code, default settings are N_s = 800 points for the signal and $n_n =
1000$ points for the noise but can be changed from the command-line. Both
samples were handled as matrices of dimension n x 2, where n is the number
of points in the sample. The library gsl_matrix provided by GSL was employed
for this purpose and the function gsl_ran_bivariate_gaussian() was used for
generating the points.
Then, a model of classification must be implemented in order to assign each point to the right class (signal or noise) to which it 'most probably' belongs to. The point is how 'most probably' can be interpreted and implemented.
Fisher linear discriminant
The Fisher linear discriminant (FLD) is a linear classification model based on dimensionality reduction. It allows to reduce this 2D classification problem into a one-dimensional decision surface.
Consider the case of two classes, (in this case the signal and the noise): the
simplest representation of a linear discriminant is obtained by taking a linear
function of a sampled point 2D x so that:
\hat{x} = w x + w_0
where w is called 'weight vector' and w_0 is a bias. The negative of the
bias is called 'threshold'. An input point x is assigned to the first class
if \hat{x} \geqslant 0 and to the second one otherwise.
In general, the projection onto one dimension leads to a considerable loss of
information and classes that are well separated in the original 2D space may
become strongly overlapping in one dimension. However, by adjusting the
components of the weight vector, a projection that maximizes the classes
separation can be selected.
To begin with, consider a two-classes problem in which there are N_1 points of
class C_1 and N_2 points of class C_2, so that the means n_1 and $m_2$
of the two classes are given by:
m_1 = \frac{1}{N_1} \sum_{n \in C_1} x_n
\et
m_2 = \frac{1}{N_2} \sum_{n \in C_2} x_n
The simplest measure of the separation of the classes is the separation of the
projected class means. This suggests that to choose w so as to maximize:
\hat{m}_2 − \hat{m}_1 = w (m_2 − m_1)
{#fig:overlap}
This expression can be made arbitrarily large simply by increasing the magnitude
of w. To solve this problem, w can be costrained to have unit length, so
that | w^2 | = 1. Using a Lagrange multiplier to perform the constrained
maximization, it can be find that w \propto (m_2 − m_1).
There is still a problem with this approach, however, as illustrated in
@fig:overlap: the two classes are well separated in the original 2D space but
have considerable overlap when projected onto the line joining their means.
The idea to solve it is to maximize a function that will give a large separation
between the projected classes means while also giving a small variance within
each class, thereby minimizing the class overlap.
The within-classes variance of the transformed data of each k class is given
by:
s_k^2 = \sum_{n \in C_k} (\hat{x}_n - \hat{m}_k)^2
The total within-classes variance for the whole data set can be simply defined
as s^2 = s_1^2 + s_2^2. The Fisher criterion is derefore defined to be the
ratio of the between-classes distance to the within-class variance and is given
by:
J(w) = \frac{(\hat{m}_2 - \hat{m}_1)^2}{s^2}