UNIT_-V_MA(1).ppt (1).pdf multivariate analysis

multivariate analysis

1. MULTIVARIATE
ANALYSIS
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UNIT - V

2. Random Vectors and Matrices - Mean
vectors and Covariance matrices –
Multivariate Normal density and its
properties - Principal components
Population principal components -
Principal components from standardized
variables.
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SYLLABUS

3. A random vector is a vector whose elements are random variables.
Similarly, a random matrix whose elements are random variables.
Random Vectors & Matrices
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4. Expected Value of a Random Matrix
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The expected value of a random matrix (or vector) is the matrix
(vector) consisting of the expected values of each of the elements.

5. Mean Vectors
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6. Covariance Matrices
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7. Covariance Matrix
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⚫Covariance matrix captures the variance and linear
correlation in multivariate/ multidimensional data.
⚫If data is an n x p matrix, the Covariance Matrix is a p x p
square matrix
⚫.Think of n as the number of data instances (rows) and p
the number of attributes (columns).

8. Covariance
⚫The covariance of the return is



⚫It is always true that

⚫i.

⚫ii.

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9. Mean Matrix
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10. Covariance Matrix
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11. Covariance Matrix
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12. Example
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Find the mean & covariance matrix for the 2 r.v. X
1
& X
2
for the given
joint probability function P
12
(x
1
,x
2
) is






Soln:
Marginal Distribution of X
X
1
-1 0 1
P(X
1
) 0.3 0.3 0.4

13. X
2
0 1
P(X
2
) 0.8 0.2
Marginal Distribution of Y
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Example

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Example

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Example

16. Sample Covariance
⚫Example. The table provides the returns on three assets
over three years






⚫Mean returns

Year 1Year 2Year 3
A 10 12 11
B 10 14 12
C 12 6 9
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17. Sample Covariance
⚫Covariance between A and B is



⚫Covariance between A and C is



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18. Variance-Covariance Matrix

⚫Covariance between B and C is



⚫The matrix is symmetric




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19. Variance-Covariance Matrix
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⚫For the example the variance-covariance matrix is

20. Correlation Coefficient
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Let the population correlation coefficient matrix be the p x p symmetric
matrix

21. Standard Deviation
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Let the p x p standard deviation be
Then it is verified that

22. Example
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23. IFETCE/H&S- II/MATHS/MATHIVADHANA/IYEAR/
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24. Linear Combination of Random
Variables
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Prove that the linear combination cʹX = aX
1
+ bX
2
has
Mean = E(cʹX) = cʹμ
Var = Var(cʹX) = cʹΣc
Where μ = E(X) & Σ = cov(X)
Soln:

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26. IFETCE/H&S- II/MATHS/MATHIVADHANA/IYEAR/
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The previous result can be extended to a linear combination of
p random varaibles:
The linear combination cʹX = c
1
X
1
+ c
2
X
2
+… + c
p
X
p
has
Mean = E(cʹX) = cʹμ
Var = Var(cʹX) = cʹΣc


In general, consider for q linear combinations Z=CX of the p
random varaibles X
1
, X
2
, …, X
p


μ
Z
= E(Z) = E(CX) = C μ
X

Σ
Z
= cov(Z) = cov(CX) = CΣ
X
Cʹ

27. Example
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