aarjaneiro/planetmath huggingface dataset:
trimmed mean

trimmed mean

Let

x

1

,

x

2

,

…

,

x

n

be a set of real-valued data

observations

. Let

x

(

1

)

≤

x

(

2

)

≤

…

≤

x

(

n

)

be the

order statistics

of the observations. The

k

th

trimmed mean

x

¯

k

is defined as:

x

¯

k

=

x

(

k

+

1

)

+

x

(

k

+

2

)

+

…

+

x

(

n

-

k

)

n

-

2

⁢

k

=

1

n

-

2

⁢

k

⁢

∑

i

=

k

+

1

n

-

k

x

(

i

)

.

By ordering the original observations, and taking away the first

k

smallest observations and the first

k

largest obserations, the trimmed mean takes the arithmetic

average

of the resulting data. The idea of a trimmed mean is to eliminate

outliers

, or extreme observations that do not seem to have any logical explanations in calculating the overall mean of a

population

.

For example, suppose 10 new lightbulbs are drawn from a population of 100 to find the average lifetime of a typical lightbulb, measured in number of hours. The measurements are 802, 854, 823, 428, 815, 840, 833, 809, 843, 821. The (arithmetic) mean of the measurement is

802

+

854

+

823

+

428

+

815

+

840

+

833

+

809

+

843

+

821

10

=

786.8

,

with sample

standard deviation

= 127.1, whereas the 1st trimmed mean gives:

802

+

823

+

815

+

840

+

833

+

809

+

843

+

821

8

=

823.25

,

with sample standard deviation = 14.6, greatly reducing the sample deviation.

The trimmed mean gives a much more

robust estimation

(an estimation not greatly affected by outliers) of the average than the

arithmetic mean

.

Another robust estimtor of a mean is the

winsorized mean

. Like the trimmed mean, the winsorized mean eliminates the outliers at both ends of an ordered set of observations. Unlike the trimmed mean, the winsorized mean replaces the outliers with observed values, rather than discarding them. The formal definition of the

k

th winsorized mean

w

k

is:

w

k

=

(

k

+

1

)

⁢

x

(

k

+

1

)

+

x

(

k

+

2

)

+

…

+

x

(

n

-

k

-

1

)

+

(

k

+

1

)

⁢

x

(

n

-

k

)

n

=

k

⁢

x

(

k

+

1

)

+

(

n

-

2

⁢

k

)

⁢

x

¯

k

+

k

⁢

x

(

n

-

k

)

n

.

From the definition, we see that the winsorized mean is the average of the observations where the first

k

smallest values are replaced by the

k

+

1

th smallest value,

x

(

k

+

1

)

, and the first

k

largest values are replaced by the

k

+

1

th largest value,

x

(

n

-

k

)

.

From the above example, the 1st winsorized mean is

802

+

843

+

823

+

802

+

815

+

840

+

833

+

809

+

843

+

821

10

=

823.1

,

with sample standard deviation = 16.1, fairly close to the answer given by the trimmed mean.

Title

trimmed mean

Canonical name

TrimmedMean

Date of creation

2013-03-22 14:42:02

Last modified on

2013-03-22 14:42:02

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

5

Author

CWoo (3771)

Entry type

Definition

Classification

msc 62F10

Classification

msc 62F35

Defines

winsorized mean

Defines

outlier

Defines

robust estimation

Generated on Sat Feb 10 12:04:45 2018 by

LaTeXML
consistent estimator

consistent estimator

Given a set of samples

X

1

,

…

,

X

n

from a given probability
distribution

f

with an unknown parameter

θ

∈

Θ

, where

Θ

is the

parameter space

that is a subset of

ℝ

m

.
Let

U

(

=

U

⁢

(

X

1

,

…

,

X

n

)

)

be an

estimator

of

θ

. Allowing
the sample size

n

to vary, we get a

sequence

of estimators for

θ

:

U

1

=

U

⁢

(

X

1

)

,

⋮

U

n

=

U

⁢

(

X

1

,

…

,

X

n

)

,

⋮

We say that the sequence of estimators

{

U

n

}

consistent

(or that

U

is a

consistent estimator

of

θ

), if

U

i

converges in probability

to

θ

for

every

θ

∈

Θ

. That is, for every

ε

>

0

,

lim

n

→

∞

P

(

|

h

n

-

θ

|

≥

ε

)

=

0

for all

θ

∈

Θ

.

Remark

. Suppose

U

is an estimator of

θ

such that
the sequence

{

U

n

}

is consistent. If

α

n

→

α

∈

ℝ

and

β

n

→

β

∈

ℝ

m

are two convergent sequences of

constants

with

0

<

|

α

|

<

∞

and

|

β

|

<

∞

, then the sequence

{

V

n

}

, defined by

V

n

:=

α

n

⁢

U

n

+

β

n

, is consistent,

V

is an estimator of

α

⁢

θ

+

β

.

Proof.

First, observe that

|

V

n

-

(

α

⁢

θ

+

β

)

|

=

|

α

n

⁢

U

n

+

β

n

-

α

⁢

θ

-

β

|

≤

|

α

n

⁢

U

n

-

α

⁢

θ

|

+

|

β

n

-

β

|

=

|

α

n

⁢

U

n

-

α

n

⁢

θ

+

α

n

⁢

θ

-

α

⁢

θ

|

+

|

β

n

-

β

|

≤

|

α

n

⁢

U

n

-

α

n

⁢

θ

|

+

|

α

n

⁢

θ

-

α

⁢

θ

|

+

|

β

n

-

β

|

=

|

α

n

|

⁢

|

U

n

-

θ

|

+

|

α

n

-

α

|

⁢

|

θ

|

+

|

β

n

-

β

|

.

This implies

P

(

|

V

n

-

(

α

θ

+

β

)

|

≥

ε

)

≤

P

(

|

α

n

|

|

U

n

-

θ

|

+

|

α

n

-

α

|

|

θ

|

+

|

β

n

-

β

|

≥

ε

)

=

P

(

|

U

n

-

θ

|

≥

ε

-

|

β

n

-

β

|

-

|

α

n

-

α

|

⁢

|

θ

|

|

α

n

|

)

.

As

n

→

∞

,

|

β

n

-

β

|

→

0

,

|

α

n

-

α

|

⁢

|

θ

|

→

0

, and

|

α

n

|

→

|

α

|

≠

0

.
So the last expression goes to

0

as

n

→

∞

. Therefore,

lim

n

→

∞

P

(

|

V

n

-

(

α

θ

+

β

)

|

≥

ε

)

=

0

,

and thus

{

V

n

}

is a consistent sequence of estimators
of

α

⁢

θ

+

β

.
∎

Title

consistent estimator

Canonical name

ConsistentEstimator

Date of creation

2013-03-22 15:26:34

Last modified on

2013-03-22 15:26:34

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

5

Author

CWoo (3771)

Entry type

Definition

Classification

msc 62F12

Defines

consistent sequence of estimators

Generated on Sat Feb 10 12:04:48 2018 by

LaTeXML
Glivenko-Cantelli lemma

Glivenko-Cantelli lemma

Let

X

1

,

…

,

X

n

be iid as

X

with (unknown)

distribution function

F

. Let

ω

be the outcome and

F

n

⁢

(

x

,

ω

)

be the

empirical distribution function

based on

observations

X

1

⁢

(

ω

)

,

…

,

X

n

⁢

(

ω

)

. Then, as

n

→

∞

,

sup

-

∞

<

x

<

∞

⁡

|

F

n

⁢

(

x

,

ω

)

-

F

⁢

(

x

)

|

→

0

⁢

a.s.

,

where a.s. means almost surely.

Title

Glivenko-Cantelli lemma

Canonical name

GlivenkoCantelliLemma

Date of creation

2013-03-22 14:33:24

Last modified on

2013-03-22 14:33:24

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

4

Author

CWoo (3771)

Entry type

Theorem

Classification

msc 62G20

Generated on Sat Feb 10 12:04:50 2018 by

LaTeXML
empirical distribution function

empirical distribution function

Let

X

1

,

…

,

X

n

be

random variables

with realizations

x

i

=

X

i

⁢

(

ω

)

∈

ℝ

,

i

=

1

,

…

,

n

. The

empirical distribution function

F

n

⁢

(

x

,

ω

)

based on

x

1

,

…

,

x

n

is

F

n

⁢

(

x

,

ω

)

=

1

n

⁢

∑

i

=

1

n

χ

A

i

⁢

(

x

,

ω

)

,

where

χ

A

i

is the

indicator function

(or

characteristic function

) and

A

i

=

{

(

x

,

ω

)

∣

x

i

≤

x

}

. Note that each indicator function is itself a random variable.

The empirical

function

can be alternatively and equivalently defined by using the

order statistics

X

(

i

)

of

X

i

as:

F

n

⁢

(

x

,

ω

)

=

{

0

if

x

<

x

(

1

)

;

1

n

if

x

(

1

)

≤

x

<

x

(

2

)

,

1

≤

k

<

2

;

2

n

if

x

(

2

)

≤

x

<

x

(

3

)

,

2

≤

k

<

3

;

⋮

i

n

if

x

(

i

)

≤

x

<

x

(

i

+

1

)

,

i

≤

k

<

i

+

1

;

⋮

1

if

x

≥

x

(

n

)

;

where

x

(

i

)

is the realization of the random variable

X

(

i

)

with outcome

ω

.

Title

empirical distribution function

Canonical name

EmpiricalDistributionFunction

Date of creation

2013-03-22 14:33:27

Last modified on

2013-03-22 14:33:27

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

7

Author

CWoo (3771)

Entry type

Definition

Classification

msc 62G30

Generated on Sat Feb 10 12:04:52 2018 by

LaTeXML
order statistics

order statistics

Let

X

1

,

…

,

X

n

be

random variables

with realizations in

ℝ

. Given an outcome

ω

, order

x

i

=

X

i

⁢

(

ω

)

in non-decreasing order so that

x

(

1

)

≤

x

(

2

)

≤

⋯

≤

x

(

n

)

.

Note that

x

(

1

)

=

min

⁡

(

x

1

,

…

,

x

n

)

and

x

(

n

)

=

max

⁡

(

x

1

,

…

,

x

n

)

. Then each

X

(

i

)

, such that

X

(

i

)

⁢

(

ω

)

=

x

(

i

)

, is a random variable.

Statistics

defined by

X

(

1

)

,

…

,

X

(

n

)

are called

order statistics

of

X

1

,

…

,

X

n

. If all the

orderings

are strict, then

X

(

1

)

,

…

,

X

(

n

)

are

the

order statistics of

X

1

,

…

,

X

n

. Furthermore, each

X

(

i

)

is called the

i

th order statistic of

X

1

,

…

,

X

n

.

Remark.

If

X

1

,

…

,

X

n

are iid as

X

with

probability density function

f

X

(assuming

X

is a continuous random variable), Let

Z

be the vector of the order statistics

(

X

(

1

)

,

…

,

X

(

n

)

)

(with strict orderings), then one can show that the joint probability density function

f

𝐙

of the order statistics is:

f

𝐙

⁢

(

𝒛

)

=

n

!

⁢

∏

i

=

1

n

f

X

⁢

(

z

i

)

,

where

𝒛

=

(

z

1

,

…

,

z

n

)

and

z

1

<

⋯

<

z

n

.

Title

order statistics

Canonical name

OrderStatistics

Date of creation

2013-03-22 14:33:30

Last modified on

2013-03-22 14:33:30

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

8

Author

CWoo (3771)

Entry type

Definition

Classification

msc 62G30

Generated on Sat Feb 10 12:04:54 2018 by

LaTeXML
Jacobi determinant

Jacobi determinant

Let

f

=

f

⁢

(

x

)

=

f

⁢

(

x

1

,

…

,

x

n

)

be a function of

n

variables

, and let

u

=

u

⁢

(

x

)

=

(

u

1

⁢

(

x

)

,

…

,

u

n

⁢

(

x

)

)

be a function of

x

, where inversely

x

can be expressed as a function of

u

,

x

=

x

⁢

(

u

)

=

(

x

1

⁢

(

u

)

,

…

,

x

n

⁢

(

u

)

)

The formula for a change of variable in an

n

-dimensional integral is then

∫

Ω

f

⁢

(

x

)

⁢

d

n

⁢

x

=

∫

u

⁢

(

Ω

)

f

⁢

(

x

⁢

(

u

)

)

⁢

|

det

⁡

(

d

⁢

x

/

d

⁢

u

)

|

⁢

d

n

⁢

u

Ω

is an integration region, and one

integrates

over all

x

∈

Ω

, or equivalently, all

u

∈

u

⁢

(

Ω

)

.

d

⁢

x

/

d

⁢

u

=

(

d

⁢

u

/

d

⁢

x

)

-

1

is the Jacobi matrix and

|

det

⁡

(

d

⁢

x

/

d

⁢

u

)

|

=

|

det

⁡

(

d

⁢

u

/

d

⁢

x

)

|

-

1

is the

absolute value

of the

Jacobi determinant

or

Jacobian

.

As an example, take

n

=

2

and

Ω

=

{

(

x

1

,

x

2

)

|

0

<

x

1

≤

1

,

0

<

x

2

≤

1

}

Define

ρ

=

-

2

⁢

log

⁡

(

x

1

)

φ

=

2

⁢

π

⁢

x

2

u

1

=

ρ

⁢

cos

⁡

φ

u

2

=

ρ

⁢

sin

⁡

φ

Then by the chain rule and

definition

of the Jacobi matrix,

d

⁢

u

/

d

⁢

x

=

∂

⁡

(

u

1

,

u

2

)

/

∂

⁡

(

x

1

,

x

2

)

=

(

∂

⁡

(

u

1

,

u

2

)

/

∂

⁡

(

ρ

,

φ

)

)

⁢

(

∂

⁡

(

ρ

,

φ

)

/

∂

⁡

(

x

1

,

x

2

)

)

=

(

cos

⁡

φ

-

ρ

⁢

sin

⁡

φ

sin

⁡

φ

ρ

⁢

cos

⁡

φ

)

⁢

(

-

1

/

ρ

⁢

x

1

0

0

2

⁢

ϕ

)

The Jacobi determinant is

det

⁡

(

d

⁢

u

/

d

⁢

x

)

=

det

⁡

{

∂

⁡

(

u

1

,

u

2

)

/

∂

⁡

(

ρ

,

φ

)

}

⁢

det

⁡

{

∂

⁡

(

ρ

,

φ

)

/

∂

⁡

(

x

1

,

x

2

)

}

=

ρ

⁢

(

-

2

⁢

π

/

ρ

⁢

x

1

)

=

-

2

⁢

π

/

x

i

and

d

2

⁢

x

=

|

det

⁡

(

d

⁢

x

/

d

⁢

u

)

|

⁢

d

2

⁢

u

=

|

det

⁡

(

d

⁢

u

/

d

⁢

x

)

|

-

1

⁢

d

2

⁢

u

=

(

x

1

/

2

⁢

π

)

=

(

1

/

2

⁢

π

)

⁢

exp

⁡

(

-

(

u

1

2

+

u

2

2

/

2

)

)

⁢

d

2

⁢

u

This shows that if

x

1

and

x

2

are

independent

random variables

with

uniform distributions

between 0 and 1, then

u

1

and

u

2

as defined above are independent random variables with

standard normal distributions

.

References

•

Originally from The Data Analysis Briefbook
(

http://rkb.home.cern.ch/rkb/titleA.html

http://rkb.home.cern.ch/rkb/titleA.html)

Title

Jacobi determinant

Canonical

name

JacobiDeterminant

Date of creation

2013-03-22 12:07:08

Last modified on

2013-03-22 12:07:08

Owner

akrowne (2)

Last modified by

akrowne (2)

Numerical id

8

Author

akrowne (2)

Entry type

Definition

Classification

msc 62H05

Classification

msc 15-00

Synonym

Jacobian

Related topic

ChainRuleSeveralVariables

Related topic

MultidimensionalGaussianIntegral

Related topic

ChangeOfVariablesInIntegralOnMathbbRn

Generated on Sat Feb 10 12:04:58 2018 by

LaTeXML
joint normal distribution

joint normal distribution

A

finite set

of

random variables

X

1

,

…

,

X

n

are said to have a

joint normal distribution

or

multivariate normal
distribution

if all real

linear combinations

λ

1

⁢

X

1

+

λ

2

⁢

X

2

+

⋯

+

λ

n

⁢

X

n

are normal (

http://planetmath.org/NormalRandomVariable

). This implies, in particular, that the

individual

random variables

X

i

are each normally distributed. However, the

converse

is not not true and sets of normally distributed random variables need not, in general, be jointly normal.

If

𝑿

=

(

X

1

,

X

2

,

…

,

X

n

)

is joint normal, then its probability distribution is uniquely determined by the means

𝝁

∈

ℝ

n

and the

n

×

n

positive semidefinite

covariance matrix

𝚺

,

μ

i

=

𝔼

⁢

[

X

i

]

,

Σ

i

⁢

j

=

Cov

⁡

(

X

i

,

X

j

)

=

𝔼

⁢

[

X

i

⁢

X

j

]

-

𝔼

⁢

[

X

i

]

⁢

𝔼

⁢

[

X

j

]

.

Then, the joint normal distribution is commonly denoted as

N

⁡

(

𝝁

,

𝚺

)

. Conversely, this

distribution

exists for any such

𝝁

and

𝚺

.

Figure 1:

Density

of joint normal variables

X

,

Y

with

Var

⁡

(

X

)

=

2

,

Var

⁡

(

Y

)

=

1

and

Cov

⁡

(

X

,

Y

)

=

-

1

.

The joint normal distribution has the following properties:

1.

If

𝑿

has the

N

⁡

(

𝝁

,

𝚺

)

distribution for nonsigular

𝚺

then it has the multidimensional

Gaussian

probability density function

f

𝑿

⁢

(

𝒙

)

=

1

(

2

⁢

π

)

n

⁢

det

⁡

(

𝚺

)

⁢

exp

⁡

(

-

1

2

⁢

(

𝒙

-

𝝁

)

T

⁢

𝚺

-

1

⁢

(

𝒙

-

𝝁

)

)

.

2.

If

𝑿

has the

N

⁡

(

𝝁

,

𝚺

)

distribution and

𝝀

∈

ℝ

n

then

𝝀

⋅

𝑿

=

λ

1

⁢

X

1

+

⋯

+

λ

n

⁢

X

n

∼

N

⁡

(

𝝀

⋅

𝝁

,

𝝀

T

⁢

𝚺

⁢

𝝀

)

.

3.

Sets of linear combinations of joint normals are themselves joint normal. In particular, if

𝑿

∼

N

⁡

(

𝝁

,

𝚺

)

and

A

is an

m

×

n

matrix, then

A

⁢

𝑿

has the joint normal distribution

N

⁡

(

A

⁢

𝝁

,

A

⁢

𝚺

⁢

A

T

)

.

4.

The

characteristic function

is given by

φ

𝑿

⁢

(

𝒂

)

≡

𝔼

⁢

[

exp

⁡

(

i

⁢

𝒂

⋅

𝑿

)

]

=

exp

⁡

(

i

⁢

𝒂

⋅

𝝁

-

1

2

⁢

𝒂

T

⁢

𝚺

⁢

𝒂

)

,

for

𝑿

∼

N

⁡

(

𝝁

,

𝚺

)

and any

𝒂

∈

ℂ

n

.

5.

A pair

X

,

Y

of jointly normal random variables are

independent

if and only if they have zero

covariance

.

6.

Let

𝑿

be a

random vector

whose distribution is jointly
normal. Suppose the

coordinates

of

𝑿

are partitioned
into two groups, forming random vectors

𝑿

𝟏

and

𝑿

𝟐

, then the conditional distribution of

𝑿

𝟏

given

𝑿

𝟐

=

𝒄

is
jointly normal.

Title

joint normal distribution

Canonical name

JointNormalDistribution

Date of creation

2013-03-22 15:22:34

Last modified on

2013-03-22 15:22:34

Owner

gel (22282)

Last modified by

gel (22282)

Numerical id

14

Author

gel (22282)

Entry type

Definition

Classification

msc 62H05

Classification

msc 60E05

Synonym

multivariate Gaussian distribution

Related topic

NormalRandomVariable

Defines

jointly normal

Defines

multivariate normal distribution

Generated on Sat Feb 10 12:05:00 2018 by

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Wishart distribution

Wishart distribution

Let

U

i

∼

N

p

⁢

(

μ

i

,

Σ

)

,

i

=

1

,

…

,

k

be

independent

p

-dimensional

random variables

, which are
multivariate normally distributed (

http://planetmath.org/jointnormaldistribution

).
Let

S

=

∑

i

=

1

k

U

i

⁢

U

i

T

. Let

M

be the

k

×

p

matrix
with

μ

1

,

…

,

μ

k

as rows.
Then the

joint distribution

of the
elements of

S

is said to be a

Wishart distribution

on

k

of freedom

, and is
denoted by

W

p

⁢

(

k

,

Σ

,

M

)

. If

M

=

0

, the

distribution

is said to be

central

and is denoted by

W

p

⁢

(

k

,

Σ

)

.
The Wishart distribution is a multivariate

generalization

of the

χ

2

distribution.

W

p

has a

density function

when

k

≥

p

.

Title

Wishart distribution

Canonical

name

WishartDistribution

Date of creation

2013-03-22 16:12:31

Last modified on

2013-03-22 16:12:31

Owner

Mathprof (13753)

Last modified by

Mathprof (13753)

Numerical id

9

Author

Mathprof (13753)

Entry type

Definition

Classification

msc 62H05

Defines

central Wishart distribution

Generated on Sat Feb 10 12:05:02 2018 by

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multivariate gamma function (complex-valued)

multivariate gamma function (complex-valued)

The

complex

multivariate gamma function

is defined as

Γ

~

m

⁢

(

a

)

=

∫

𝔄

e

-

Tr

⁡

A

⁢

|

A

|

a

-

m

⁢

d

A

,

(1)

where

𝔄

is the set of all

m

×

m

positive, complex-valued

Hermitian matrices

, i.e.

𝔄

=

{

A

∈

ℂ

m

×

m

|

A

=

A

H

,

A

>

0

}

.

(2)

It can also be expressed in terms of the

gamma function

as follows

Γ

~

m

⁢

(

a

)

=

π

1

2

⁢

m

⁢

(

m

-

1

)

⁢

∏

i

=

1

m

Γ

⁢

(

a

-

i

+

1

)

.

(3)

Reference

A. T. James,“

Distributions

of

matrix variates

and latent roots derived from normal samples,”

Ann. Math. Statist.

, vol. 35, pp. 475-501, 1964.

Title

multivariate gamma function (complex-valued)

Canonical name

MultivariateGammaFunctioncomplexvalued

Date of creation

2013-03-22 14:22:10

Last modified on

2013-03-22 14:22:10

Owner

mathpeter (5480)

Last modified by

mathpeter (5480)

Numerical id

14

Author

mathpeter (5480)

Entry type

Definition

Classification

msc 62H10

Defines

gamma function (

multivariate

complex)

Generated on Sat Feb 10 12:05:04 2018 by

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multivariate gamma function (real-valued)

multivariate gamma function (real-valued)

The real-valued

multivariate gamma function

is defined by

Γ

m

⁢

(

a

)

=

∫

𝔖

e

-

Tr

⁡

S

⁢

|

S

|

a

-

1

2

⁢

(

m

+

1

)

⁢

d

S

,

(1)

where

𝔖

is the set of all

m

×

m

real,

positive definite

symmetric matrices

, i.e.

𝔖

=

{

S

∈

ℝ

m

×

m

∣

S

>

0

,

x

T

⁢

S

⁢

x

>

0

⁢

∀

x

∈

ℝ

m

×

1

∖

{

0

}

}

.

(2)

The real-valued multivariate gamma function can also be expressed in terms of the

gamma function

as follows

Γ

m

⁢

(

a

)

=

π

1

4

⁢

m

⁢

(

m

-

1

)

⁢

∏

i

=

1

m

Γ

⁢

(

a

-

1

2

⁢

(

i

-

1

)

)

.

(3)

Reference

A. T. James, “

Distributions

of

matrix variates

and latent roots derived from normal samples,”

Ann. Math. Statist.

, vol. 35, pp. 475-501, 1964.

Title

multivariate gamma function (real-valued)

Canonical name

MultivariateGammaFunctionrealvalued

Date of creation

2013-03-22 14:22:06

Last modified on

2013-03-22 14:22:06

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

15

Author

rspuzio (6075)

Entry type

Definition

Classification

msc 62H10

Defines

gamma function (

multivariate

real)

Generated on Sat Feb 10 12:05:05 2018 by

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contingency table

contingency table

Given a random sample of

N

observations

𝐙

i

=

(

Y

i

,

X

i

⁢

1

,

…

,

X

i

⁢

k

)

where

1.

the response variables

Y

i

are

identically distributed

as

Y

2.

Y

is categorical in nature (coming from a

multinomial distribution

)

3.

each of the explanatory variables

X

i

⁢

j

is categorical in nature

Then we can analyze the data by forming a

contingency table

. The table is customarily formed by

labeling

the categories for the response across the top, and then the

combinations

of the levels for each explanatory variable down the left-most columns. Then the cells are filled with counts or frequencies of

occurrences

corresponding to the specific explanatory variable level combination to the left and the response to the top.

The simplest example of a contingency table is where the response variable

Y

comes from a

binomial distribution

(with two possible responses

r

1

and

r

2

) and there is only one explanatory variable

X

, which has only two levels,

A

1

and

A

2

. This is an

instance

of a

2 way contingency table

:

Title

contingency table

Canonical name

ContingencyTable

Date of creation

2013-03-22 14:32:53

Last modified on

2013-03-22 14:32:53

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

11

Author

CWoo (3771)

Entry type

Definition

Classification

msc 62H17

\@unrecurse

Generated on Sat Feb 10 12:05:08 2018 by

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Simpson’s paradox

Simpson’s paradox

Before describing what a

Simpson’s paradox

is, let’s start with a hypothetical example. During a particular summer, an experiment was conducted to find out the preference between two types of beverages: soda and lemonade. The data was drawn from two locations: city and rural. In each location, the gender and the choice of drinks were collected. The results are summarized as follows:

Title

Simpson’s paradox

Canonical

name

SimpsonsParadox

Date of creation

2013-03-22 14:42:05

Last modified on

2013-03-22 14:42:05

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

8

Author

CWoo (3771)

Entry type

Definition

Classification

msc 62H17

\@unrecurse

Generated on Sat Feb 10 12:05:12 2018 by

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canonical correlation

canonical correlation

Let

X

be the

(

T

,

n

)

matrix corresponding to the

n

signals and

Y

be a

(

T

,

p

)

matrix corresponding to one set of

p

signals. Time indexes each row of the matrix (

T

time samples). Let

Σ

11

and

Σ

22

be the sample covariance matrices of

X

and

Y

, respectively, and let

Σ

12

=

Σ

21

′

be the sample covariance matrix between

X

and

Y

. For simplicity, we suppose that all signals have zero mean.

Canonical correlation

analysis (CCA) finds the

linear combinations

of the column of

X

and

Y

that has the largest

correlation

; i.e., it finds the weight vectors (loadings)

a

and

b

that maximize:

ρ

=

a

′

⁢

Σ

12

⁢

b

a

′

⁢

Σ

11

⁢

a

⁢

b

′

⁢

Σ

22

⁢

b

.

(1)

We follow the derivations of Johnson and we do a

change of basis

:

c

=

Σ

11

1

/

2

⁢

a

and

d

=

Σ

22

1

/

2

⁢

b

.

ρ

=

c

′

⁢

Σ

11

-

1

/

2

⁢

Σ

12

⁢

Σ

22

-

1

/

2

⁢

d

c

′

⁢

c

⁢

d

′

⁢

d

(2)

By the

Cauchy-Schwartz inequality

:

ρ

≤

c

′

⁢

Σ

11

-

1

/

2

⁢

Σ

12

⁢

Σ

22

-

1

/

2

⁢

Σ

22

-

1

/

2

⁢

Σ

21

⁢

Σ

11

-

1

/

2

⁢

c

⁢

d

′

⁢

d

c

′

⁢

c

⁢

d

′

⁢

d

=

c

′

⁢

Σ

11

-

1

/

2

⁢

Σ

12

⁢

Σ

22

-

1

⁢

Σ

21

⁢

Σ

11

-

1

/

2

⁢

c

c

′

⁢

c

.

(3)

The inequality above is an equality when

Σ

22

-

1

/

2

⁢

Σ

21

⁢

Σ

11

-

1

/

2

⁢

c

and

d

are collinear. The right hand side of the

expression

above is a Rayleigh quotient and it is maximum when

c

is the

eigenvector

corresponding to the largest eingenvalue of

Σ

11

-

1

/

2

⁢

Σ

12

⁢

Σ

22

-

1

⁢

Σ

21

⁢

Σ

11

-

1

/

2

(we obtain the other rows by using the other

eigenvalues

in decreasing magnitude). This results if the basis of the CCA. We can compute the two canonical variables:

U

1

=

X

⁢

a

and

V

1

=

Y

⁢

b

.

We can continue this way to find the subsequent vectors

Title

canonical correlation

Canonical

name

CanonicalCorrelation

Date of creation

2013-03-22 19:16:11

Last modified on

2013-03-22 19:16:11

Owner

tony_bruguier (26297)

Last modified by

tony_bruguier (26297)

Numerical id

4

Author

tony_bruguier (26297)

Entry type

Definition

Classification

msc 62H20

Generated on Sat Feb 10 12:05:14 2018 by

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