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MH4501 Multivariate analysis

创建日期：2024-06-11 15:38:09

所属分类：数学mathematics

标签： MH4501

Multivariate analysis

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MH4501 Multivariate analysis
Outline
Sample statistics
Definition of the multivariate normal distribution
1
Sample statistics
Sample mean
• Sample mean x¯j ∈ R of variable j is the average of {x1j , . . . , xnj} :
x¯j =
1
n
n∑
i=1
xij , j = 1, . . . , p
• Sample mean vector x¯ = 1n
∑n
i=1 xi ∈ Rp
x¯ = (x¯1, . . . , x¯p) =
1
n
n∑
i=1
xi
• Matrix form of the sample mean vector:
x¯ =

1
n
∑n
i=1 xi1
...
1
n
∑n
i=1 xip
 = 1n
x11 x21 · · · xn1... ... . . . ...
x1p x2p · · · xnp

1...
1
 = 1nX⊤1n,
where 1n = (1, . . . , 1)⊤ ∈ Rn.
2
Sample mean
Define an estimate x¯ of the population mean vector µ: Suppose the sample data matrix
X = (x1, . . . , xn)⊤ is collected from a multivariate population of dimension p, i.e.
xi = (xi1, . . . , xip)⊤ ∼ f
where f denotes the population pdf. Let µp×1 be the population mean vector and Σp×p be the
population covariance matrix.
Theorem
x¯ is an unbiased estimator for µ, i.e. E[x¯ ] = µ.
Proof
Note that x¯ = 1n
∑n
i=1 xi and for each xi ,E [xi ] = µ, thus
E[x¯ ] = 1n
n∑
i=1
µ = µ
3
Sample Covariance Matrix
• Sample variance of variable j is sjj = 1n−1
∑n
i=1 (xij − x¯j)2;
• Sample covariance between variables j and k is sjk = 1n−1
∑n
i=1 (xij − x¯j) (xik − x¯k);
• These sjk ’s are put together to form a (symmetric) sample covariance matrix S :
S =
s11 s12 · · · s1p... ... . . . ...
sp1 sp2 · · · spp

• Compute S by matrices. Define the sum of squares and cross-products matrix:
S = 1n − 1

∑n
i=1 (xi1 − x¯1)2 · · ·
∑n
i=1 (xi1 − x¯1) (xip − x¯p)
... . . .
...∑n
i=1 (xip − x¯p) (xi1 − x¯1) · · ·
∑n
i=1 (xip − x¯p)2

= 1n − 1
(x11 − x¯1) · · · (xn1 − x¯1)... . . . ...
(x1p − x¯p) · · · (xnp − x¯p)

(x11 − x¯1) · · · (x1p − x¯p)... . . . ...
(xn1 − x¯1) · · · (xnp − x¯p)
 .
4
Sample Covariance Matrix
Notice that(x11 − x¯1) · · · (x1p − x¯p)... . . . ...
(xn1 − x¯1) · · · (xnp − x¯p)
 =
x11 · · · x1p... . . . ...
xn1 · · · xnp
−
x¯1 · · · x¯p... . . . ...
x¯1 · · · x¯p

= X − 1x¯⊤ = X − 1n11
⊤X =
(
I − 1n11
⊤
)
X ,
then we have two expressions of S :
S = 1n − 1
(
X − 1x¯⊤)⊤ (X − 1x¯⊤)
= 1n − 1
n∑
i=1
(xi − x¯) (xi − x¯)⊤
5
Sample Covariance Matrix
Properties:
• S is symmetric as S⊤ = S.
• S is positive-semi-definite as S = B⊤B with
B = 1√
n − 1
(
X − 1x¯⊤)
Theorem
The sample covariance matrix S provides an unbiased estimator of the population covariance
matrix Σ, i.e. E[S] = Σ.
Proof
Hint:
S = 1n − 1
n∑
i=1
(xi − x¯) (xi − x¯)⊤ = 1n − 1
[ n∑
i=1
xix⊤i − nx¯ x¯⊤
]
6
Sample Correlation Matrix
Sample correlation coefficient between variables j and k is rjk :
rjk =
sjk√sjj√skk =
∑n
i=1 (xij − x¯j) (xik − x¯k)√∑n
i=1 (xij − x¯j)2
√∑n
i=1 (xik − x¯k)2
where sjk is sample covariance between variables j and k; sjj is sample variance of variable j .
• Clearly, rjj = 1, i.e. the sample correlation of variable j with itself is 1.
• When collecting all rjk ’s together, we have the sample correlation matrix R:
R =

1 r12 · · · r1p
r21 1 · · · r2p
...
... . . .
...
rp1 rp2 · · · 1

It is easy to check that rjk = rkj ⇒ R is symmetric.
7
Sample Correlation Matrix
Compute R using the sample covariance matrix S: let D = diag (s11, . . . , spp), then
R = D−1/2SD−1/2, S = D1/2RD1/2
Comments:
• R is positive semi-definite.
• The quantity det(S) is called the generalised sample variance. det(S) and det(R) are
related via
det(S) = s11 . . . spp det(R)
8
Sample Correlation Matrix
• Define the standardisation of xij by
zij =
xij − x¯j√sjj
Now define vectors zi = (zi1, . . . , zip)⊤, i = 1, . . . , n. We want to investigate the
covariance matrix of the standardised values, i.e. Sz .
As z¯j = 0, the (j , k)-th element of Sz is
Sz(j , k) =
1
n − 1
n∑
i=1
zijzik =
1
n − 1
n∑
i=1
xij − x¯j√sjj
xik − x¯k√skk
= 1n − 1
1√sjj√skk
n∑
i=1
(xij − x¯j) (xik − x¯k) = sjk√sjj√skk = Rx (j , k),
i.e. the correlation matrix of X is equal to the covariance matrix of the standardised X .
9
Sample Correlation Matrix
• Many multivariate methods use the concept of generalised distance:
d2 = (x − µ)⊤Σ−1(x − µ) distance between x and µ
or d2 = (x − x¯)⊤S−1(x − x¯) distance between x and x¯
Notes: From the graph we see that variances along x1 or x2 directions are different. When
considering a point (x∗1 , x∗2 ) and its distance to (x¯1, x¯2), the matrix S accounts for the
difference in variation.
10
Linear Combinations
Suppose X is a p × 1 random vector with mean µp×1 and covariance matrix Σp×p. Let Ak1×p
and Ck2×p be matrices, bk1×1 and dk2×1 be vectors. Define Y = AX + b and Z = CX + d .
Then we have sample means, variances, and covariances of Y and Z that are related to x¯ and
S by
Sample mean vector of Y is Ax¯ + b,
Sample covariance matrix of Y is ASA⊤,
Sample mean vector of Z is Cx¯ + d ,
Sample covariance matrix of Z is CSC⊤,
Sample covariance matrix between Y and Z is ASC⊤.
It is easy to see that all these sample estimates are unbiased estimates.
11
Definition of the multivariate
normal distribution
Definition
The univariate normal distribution N
(
µ, σ2
)
has the density function:
f (x) = 1√
2πσ
exp
(
− (x − µ)
2
2σ2
)
Graph of f (x)
Note that (x − µ)2/σ2 in the exponent is a general distance measurement:
(x − µ)2/σ2 = (x − µ)⊤ (σ2)−1 (x − µ).
12
Definition
Thus we can extend it to multivariate situation by using
(x − µ)⊤1×pΣ−1p×p(x − µ)p×1
fx (xp×1) = c exp
(
−12 (x − µ)
⊤Σ−1(x − µ)
)
,
where µ = E [Xp×1] ,Σ = Var (Xp×1), and c is a constant. From the requirement that∫
f (x)dx = 1, we have
c = 1(2π)p/2|Σ|1/2 , |Σ| denotes the determinant.
Definition
A random vector xp×1 is said to follow a Np(µ,Σ) (p-dimensional multivariate normal)
distribution if its density function is
fx (x) =
1
(2π)p/2|Σ|1/2 exp
(
−12 (x − µ)
⊤Σ−1(x − µ)
)
.
13
Examples
Example 2.1: When p = 1, the p-dimensional multivariate normal pdf reduces to that of
univariate normal distribution.
Example 2.2: When p = 2, we have x = (x1, x2)⊤ , µ = (µ1, µ2)⊤, and
Σ =
(
σ11 σ12
σ21 σ22
)
=
(
σ21 ρσ1σ2
ρσ1σ2 σ22
)
, where ρ = Corr (x1, x2) and σ2j = Var (xj). As
|Σ| = σ21σ22 − ρ2σ21σ22 =
(
1− ρ2)σ21σ22
and
Σ−1 = 1(1− ρ2)σ21σ22
(
σ22 −ρσ1σ2
−ρσ1σ2 σ21
)
= 1(1− ρ2)
(
σ−21 −ρσ−11 σ−12
−ρσ−11 σ−12 σ−22
)
We have
(x − µ)⊤Σ−1(x − µ) = 1(1− ρ2)
[(
x1 − µ1
σ1
)2
− 2ρ
(
x1 − µ1
σ1
)(
x2 − µ2
σ2
)
+
(
x2 − µ2
σ2
)2]
14
Examples
Hence the bivariate normal has the density function
f (x1, x2) =
1
2πσ1σ2
√
1− ρ2 ·
exp
{
− 12 (1− ρ2)
[(
x1 − µ1
σ1
)2
− 2ρ
(
x1 − µ1
σ1
)(
x2 − µ2
σ2
)
+
(
x2 − µ2
σ2
)2]}
.
Graph of bivariate normal density f (x1, x2) 15
Contours of the multivariate normal pdf
The contour at a fixed height h is determined by f (x) = h, or
1
(2π)p/2|Σ|1/2 exp
(
−12 (x − µ)
⊤Σ−1(x − µ)
)
= h ⇔ (x − µ)⊤Σ−1(x − µ) = k
for some constant k > 0.
Question: What does (x − µ)⊤Σ−1(x − µ) = k look like?
From the eigenvalue decomposition of Σ, we have
Σ−1 = UΛ−1U⊤,
where Λp×p = diagonal matrix of eigenvalues = diag (λ1, . . . , λp) and Up×p = eigenvector
matrix = (u1, . . . , up).
Since the eigenvalues are positive, (x − µ)⊤Σ−1(x − µ) = k is an ellipsoid.
16
Contours of the multivariate normal pdf
Example 2.3: When p = 2, (x−µ)⊤Σ−1(x−µ) = k gives
an ellipse as shown in the graph.
If observations (x1, x2)⊤ are from a bivariate normal distri-
bution, then the plot of x1 vs. x2 (i.e. scatter plot) usually
results in an ellipse shape.

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EMATM0051 COMP220 BLAW1002 E20 TCP8001 ECON61001 EART40005 ECONM1022 MATH 365 MATH 367 ELEC6218 COMP 5812M EDUC 5061M ELEC ENG 1101 equation MATH6174 ST334 C4 EC9570 A33648 EBUS603 ECON47815 stata WFMS0001 PEM102 INF6060 ionically ELECTRICAL MATH 375 MMW 12 MKTG860 ECON60401 ALY6010 SOST20131 Introduction COM3524 MME2 GGR305 JNB 538 EECS101 COMM5030 IEOR E4601 2B CSC336 EN2315 LAW 432 CSC263 MATH2001 OLET1143 CSCB63 SIT772 COMM1170 STAT 441 MS930 INFT3050 GLBH0030 Stat 120B MAST10007 MGEC61 Machine GGR 252H1S ECON7030 Financial Marketing MGMT30019 ECON5120 SDSC 8009 CSC165H1S DSCI 551 3081W ACM41000 ECA 5304 MATH 1064 ECSE444 DATA 604 COMP7105 MIE1615 ECON4004 ENV200H1S EC481 MKT 306 EC1B1 MATH08051 ALY6140 CS260C CSE30 MSIN0226 CS4103 CSC3064 AM16 COMP11212 CSE4101 PSTAT 174 Math 380W CNT 5106C STAT 310 FIN2028 CS5014 STAT 425 COMP 4102A ELEC2140 MANF9544 QRM 9 QRM 8 ECON 457 COMPSCI 4091 MATH 523 DNSC-4211 PEAS 6000 MSIN0180 COMP2119 A2A HT 2022 IB93F0 MAS316/414 ACCT2522 MSIN0181 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