1. Introduction to Statistical Methods for Measuring “Omics” and Field Data
PCA, PcoA, distance measure, AMOVA

2. Outline Covariance matrix and correlation matrix Matrix determinant
Identity matrix
Eigenvalues
Eigenvectors
Principal Component Analysis (PCA)
Distance measure
Principal Coordinate Analysis (PcoA)
Analysis of Molecular Variance (AMOVA)

3. Covariance matrix and correlation matrix
X Y X Y
Variable Number X Y 1
53,047
62,490 2
49,958
58,850 3
41,974
49,445 4
44,366
52,263 5
40,470
47,674 6
36,963
43,542 7
31,474
75,113 8
54,376
72,265 9
60,880
98,675 10
66,774
104,543
Correlation matrix
X y X Y

4. Matrix determinant The determinant of a matrix A represented by |A|
Determinant of 2 x 2 matrix
a11 a12
a21 a22 determinant = (a11 X a22) – (a12 x a21)
Covariance matrix
X Y X Y
The determinant of the covariance matrix
determinant = ( X ) – ( X )
determinant = x 1016

5. Matrix determinant The determinant of a matrix A represented by |A|
Determinant of 2 x 2 matrix
a11 a12
a21 a22 determinant = (a11 X a22) – (a12 x a21)
Correlation matrix
X y X Y
The determinant of the correlation matrix
determinant = (1.00 X 1.00) – ( X )
determinant =

6. Identity matrix
Identity matrix is denoted by I is a square matrix with ones on the main (NW - SE) diagonal and zeros elsewhere.
I =
(2 x 2)
I =
(3 x 3)
I =
(4 x 4)
I =
(5 x 5)

7. Eigenvalues
Let A be a k x k square matrix and I be the k x k identity matrix. Then eigenvalues 𝜆1, 𝜆2, 𝜆k satisfy the polynomial equation given by
A - 𝝀I = 0 Characteristics equation
For example: A = 1
- 𝝀
- 𝝀 =
(1 - 𝝀) =
(3 - 𝝀)=𝟎
3 - 𝝀
Eigenvalues are: 𝝀=1, 𝝀 = 3

8. Eigenvectors
Let A be a matrix of dimension k x k and 𝜆 be an eigenvalue of A
Eigenvector x of a matrix A associated with eigenvalue 𝜆: Ax = 𝜆x
For example: A = ,
Eigenvectors associated with the eigenvalues are determined by solving the following equations
= for 𝝀=1; = for
Eigenvalues are: 𝝀=1, 𝝀 = 3 x1 x2 x1 x2 x1 x2 x1 x2
𝝀 = 3

9. Eigenvectors
Eigenvectors associated with the eigenvalues of 1 and 3 are given below
For 𝝀 = 1; eigenvector is -2 1 1
For 𝝀 = 3; eigenvector is

10. Principal Component Analysis

11. PCA: Principal Component Analysis
PCA is a mathematical procedure that transforms a set of variables into a smaller set of uncorrelated variables called principal components (PCs). These PCs are linear combinations of the original variables and can be thought of as “new” variables.
Uses of PCA:
a) Data screening (identify outlier)
b) Clustering
c) Dimension reduction

12. PCA From k original variables: x1,x2,...,xk:
Produce k new variables: y1,y2,...,yk:
y1 = a11x1 + a12x a1kxk
y2 = a21x1 + a22x a2kxk
...
yk = ak1x1 + ak2x akkxk
yk's are
Principal Components
Where:
a11 is an eigenvector

13. PCA Raw data: Original variables Covariance/Correlation Matrix
Determinant matrix
Eigenvalues
Eigenvectors
Principal component scores = Original variables x Eigenvectors

14. PCA for clustering using PC scores
Variables
Protein M1 M2 M3
Protein_X1
124 99
4.3
Protein_X2
106 67
7.5
Protein_X3
111 90
9.2
Protein_X4
109
Protein_X5
113
112
9.6
Protein_X6 89 72
10.1
Protein_X7 78
7.7
Protein_X8
190 87
6.8
Protein_X9
123 68
7.6
Protein_X10
116
7.8
Protein_X11
Protein_X12 M2 M3 M1
plot
PC2
PCA
PC1 14

15. Distance measures

16. Similarity and Dissimilarity distance
Similarity distance measure:
Euclidean distance
Manhattan distance
Dissimilarity distance measure:
Jaccard
Dice

17. Euclidean Distance
Euclidean Distance is the most common use of distance. In most cases when people said about distance , they will refer to Euclidean distance. Euclidean distance or simply 'distance' examines the root of square differences between coordinates of a pair of objects.
Euclidean distance
(4,5)
(1,1)

18. Euclidean Distance Euclidean distance Features cost time weight
Example:
Point A has coordinate (0, 3, 4, 5) and point B has coordinate (7, 6, 3, -1).
The Euclidean Distance between point A and B is
Euclidean distance
Features
cost
time
weight
incentive
Plant A 3 4 5
Plant B 7 6 -1

19. Manhattan
It is also known as Manhattan distance, boxcar distance, absolute value distance. It examines the absolute differences between coordinates of a pair of objects.
Features
cost
time
weight
incentive
Plant A 3 4 5
Plant B 7 6 -1

20. Dissimilarity distance
Marker1
Marker2
Marker3
Marker4
Marker5
Marker6
Marker7 i 1 j

21. Genetic distance Marker1 Marker2 Marker3 Marker4 Marker5 Marker6
Sample 1 1
Sample 2
Sample 1 1
Sample 2
Fa=3
Fb=1
Fc=2
Fd=1
N= Fa+Fb+Fc+Fd
Simple Match distance = Fa/N= 3/7= 0.43
Genetic distance (Jaccard) = Fa/(Fa+Fb+Fc) = 3/6= 0.5

22. Distance Matrix
Gives the matrix of distances between each pair of elements A B C D E 63 94
111 67 79 96 16 47 83
100 A B C D E

23. Principal Coordinate Analysis

24. Principal Coordinate Analysis (PcoA)
Raw data: Original variables
Distance matrix
Determinant matrix
Eigenvalues
Eigenvectors
Principal coordinate scores = Original variables x Eigenvectors

25. Examples of PcoA plot

26. Analysis of molecular variance (AMOVA)

27. Analysis of molecular variance (AMOVA)
AMOVA used to detect population differentiation utilizing molecular markers.
It uses distance matrix
A P-value is calculated by measuring the fraction of 1000 randomizations of the rows and columns in a distance matrix.
It is not ANOVA :
No requirement of assumption of a normality.

28. AMOVA
An example of AMOVA model This AMOVA model measures gene diversity among populations with specific reference to areas of a region in a continent
We have: i = individuals, j = alleles, k = populations

29. RStudio