Introduction to LDA (Linear Discrimination Analysis)

Linear Discrimination Analysis

锻炼一下ability of english writing : ) 光看不写感觉不行哇～

First of all, we try to solve a problem and then guide the LDA out here ：）

Here is the question that there are two different points in this picture, what‘s the evidence in mathmatic that  you classify there two different points.

After classification, try to input some generic points and classify these inputed points by that mathmatic evidence that you have found.

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Method One:  the distance between projected means

Just compute the different mean value of different class points. Compare the distance between the inputed new points and the mean value‘s location. If the inputed points is close to mean value location of RED class,
we treat its as red class, verse via.

The green point in this picture is the mean location of blue class.

The yellow point in this picture is the mean location of red  class.

Which points close to the mean point of which class, it belongs to that class.

However, the distance between projected means is not a good measure  since it does not account for the standard deviation within classes

Method two : Fisher’s linear discriminant

This is a fantastic discriminant method  \(^o^)/~

What is our target if we want to descrininate different class datas ?

Fisher suggested
maximizing the difference between the means,  normalized by a measure of the within-class scatter

Attention! y is a vector but not a normal single dimention varible !

So one of our target is to make the within-class scatter as min as possible.

On the another hand, we should consider about the relationship between the two different class of datas.

If could find a matrix W_t which multiple vector x could translate vectorx
into scale y , our will finish half of our work.

Function J(w) is very helpful. It describe the target of our discriminant. The more bigger of J(w), the better of our discriminant

Within-class scatter

Matrix Si describe the level of scatter inner of class-i

Beblow this is a description about within-class scatter in scale y

To get matrix Sw, we could sum all matrix-Si.

Between-class scatter

This Matrix describe the level of scatter between different class.

Everything is more and more clearly...

At this moment, we may memory back the operation of differential on matrix.

After this, you will have the ability to understand proof beblow here.

Attention！ SB  is  a diagonal
matrix.  A_t == A

Let‘s have a exercise ：）

Go back and consider about there data points in below picture.

First of all we should set the data.

Class_1 = [ 4,1;
            2,4;
            2,3;
            3,6;
            4,4];

Class_2 = [9,10;
    6,8;
    9,5;
    8,7;
    10,8];

And the compute the mean location of each class data collection.

mean_Class_1 = [mean(Class_1(:,1)),mean(Class_1(:,2))];
mean_Class_2 = [mean(Class_2(:,1)),mean(Class_2(:,2))];

mean_Class_1 =

3.0000    3.6000

mean_Class_2 =

8.4000    7.6000

And then compute S1 and S2

S_w =

13.2000   -2.2000

-2.2000   26.4000

S_b =

29.1600   21.6000

21.6000   16.0000

S_w = S_1 + S_2;

S_b = (mean_Class_1 - mean_Class_2)' * (mean_Class_1 - mean_Class_2);

Look！ The red line is what we want! Just project every points onto the red line in the picture.

(y - original_y)/(x - original_x) = -1/slope;

new_location_x = ((1/slope)*x_original_point + y_original_point)/(slope + 1/slope);

new_location_y = slope*new_location_x;

Now，I will give my code in matlab which draw that picture out here.

clear all
close all
clc

Class_1 = [ 4,1;
            2,4;
            2,3;
            3,6;
            4,4];

Class_2 = [9,10;
    6,8;
    9,5;
    8,7;
    10,8];

figure(1);
hold on;
scatter(Class_1(:,1),Class_1(:,2),'fill','r');
scatter(Class_2(:,1),Class_2(:,2),'fill','b');

mean_Class_1 = [mean(Class_1(:,1)),mean(Class_1(:,2))];
mean_Class_2 = [mean(Class_2(:,1)),mean(Class_2(:,2))];

scatter(mean_Class_1(1,1),mean_Class_1(1,2),'fill','y');
scatter(mean_Class_2(1,1),mean_Class_2(1,2),'fill','g');

value = Class_1;

S_1 = [0 0;0 0];

for temp = 1:size(Class_1,1)
    value(temp,1) = Class_1(temp,1) - mean_Class_1(1,1);
    value(temp,2) = Class_1(temp,2) - mean_Class_1(1,2);

    S_1 = S_1 + value(temp,:)'*value(temp,:);
end

S_2 = [0 0;0 0];

for temp = 1:size(Class_1,1)
    value(temp,1) = Class_2(temp,1) - mean_Class_2(1,1);
    value(temp,2) = Class_2(temp,2) - mean_Class_2(1,2);

    S_2 = S_2 + value(temp,:)'*value(temp,:);
end

S_w = S_1 + S_2;

S_b = (mean_Class_1 - mean_Class_2)' * (mean_Class_1 - mean_Class_2);

Temp_matrix = inv(S_w)*S_b;

%% compute the eig dialog matrix by function eig()
[V,D] = eig(Temp_matrix);

eig_value = max(D(:));

Temp_matrix(1,1) = Temp_matrix(1,1) - eig_value;
Temp_matrix(2,2) = Temp_matrix(2,2) - eig_value;

slope = -Temp_matrix(1,1)./Temp_matrix(1,2);

x = [1:16];
y = slope*x;

plot(x,y);

Projection_Class_1(:,1) = ...
(Class_1(:,1).*(1/slope) + Class_1(:,2))./(slope + (1/slope));

Projection_Class_1(:,2) = Projection_Class_1(:,1).*slope;

scatter(Projection_Class_1(:,1),Projection_Class_1(:,2),'r');

Projection_Class_2(:,1) = ...
(Class_2(:,1).*(1/slope) + Class_2(:,2))./(slope + (1/slope));

Projection_Class_2(:,2) = Projection_Class_2(:,1).*slope;

scatter(Projection_Class_2(:,1),Projection_Class_2(:,2),'b');