AbsoluteCorrelationFunction

✖
`AbsoluteCorrelationFunction`

✖

AbsoluteCorrelationFunction[data,hspec]

estimates the absolute correlation function at lags hspec from data.

✖

AbsoluteCorrelationFunction[proc,hspec]

represents the absolute correlation function at lags hspec for the random process proc.

✖

AbsoluteCorrelationFunction[proc,s,t]

represents the absolute correlation function at times s and t for the random process proc.

Details

AbsoluteCorrelationFunction is also known as the autocorrelation function.
The following specifications can be given for hspec:

	τ	at time or lag τ
	{τ_max}	unit spaced from 0 to τ_max
	{τ_min,τ_max}	unit spaced from τ_min to τ_max
	{τ_min,τ_max,dτ}	from τ_min to τ_max in steps of dτ
	{{τ₁,τ₂,…}}	use explicit {τ₁,τ₂,…}

AbsoluteCorrelationFunction[{x₁,…,x_n},h] is equivalent to .
When data is TemporalData containing an ensemble of paths, the output represents the average across all paths.
AbsoluteCorrelationFunction for a process proc with value x[t] at time t is given by:
Expectation[x[s] x[t]] for a scalar-valued process

Expectation[x[s]⊗x[t]] for a vector-valued process
The symbol ⊗ represents KroneckerProduct.
AbsoluteCorrelationFunction[proc,h] is defined only if proc is a weakly stationary process and is equivalent to AbsoluteCorrelationFunction[proc,h,0].
The process proc can be any random process such as ARMAProcess and WienerProcess.

Examples

open allclose all

Basic Examples (4)Summary of the most common use cases

Estimate the absolute correlation function at lag 2:

In[1]:=1

✖

https://wolfram.com/xid/0tqflznxbkgi9u-7xwnvr

Out[1]=1

Sample the absolute correlation function for a random sample from an autoregressive time series:

In[1]:=1

✖

https://wolfram.com/xid/0tqflznxbkgi9u-mzk4

In[2]:=2

✖

https://wolfram.com/xid/0tqflznxbkgi9u-4ic4n4

Out[2]=2

The absolute correlation function for a discrete-time process:

In[1]:=1

✖

https://wolfram.com/xid/0tqflznxbkgi9u-64o9n

Out[1]=1

In[2]:=2

✖

https://wolfram.com/xid/0tqflznxbkgi9u-du47fs

Out[2]=2

The absolute correlation function for a continuous-time process:

In[1]:=1

✖

https://wolfram.com/xid/0tqflznxbkgi9u-jlgu4j

Out[1]=1

In[2]:=2

✖

https://wolfram.com/xid/0tqflznxbkgi9u-cb5pyd

Out[2]=2

Scope (13)Survey of the scope of standard use cases

Empirical Estimates (7)

Estimate the absolute correlation function for some data at lag 5:

In[1]:=1

✖

https://wolfram.com/xid/0tqflznxbkgi9u-e2gx6r

Out[1]=1

Obtain empirical estimates of the correlation function up to lag 9:

In[1]:=1

✖

https://wolfram.com/xid/0tqflznxbkgi9u-lpyp8t

Out[1]=1

Compute the absolute correlation function for lags 1 to 9 in steps of 2:

In[1]:=1

✖

https://wolfram.com/xid/0tqflznxbkgi9u-bdsgu1

Out[1]=1

Compute the absolute correlation function for a time series:

In[1]:=1

✖

https://wolfram.com/xid/0tqflznxbkgi9u-fk4573

The absolute correlation function of a time series for multiple lags is given as a time series:

In[2]:=2

✖

https://wolfram.com/xid/0tqflznxbkgi9u-83mv4

Out[2]=2

In[3]:=3

✖

https://wolfram.com/xid/0tqflznxbkgi9u-5ep7bs

Out[3]=3

Estimate the absolute correlation function for an ensemble of paths:

In[1]:=1

✖

https://wolfram.com/xid/0tqflznxbkgi9u-f8uvhg

Out[1]=1

In[2]:=2

✖

https://wolfram.com/xid/0tqflznxbkgi9u-myqy3

In[3]:=3

✖

https://wolfram.com/xid/0tqflznxbkgi9u-c1pvcs

Out[3]=3

Compare empirical and theoretical absolute correlation functions:

In[1]:=1

✖

https://wolfram.com/xid/0tqflznxbkgi9u-jmctrn

In[2]:=2

✖

https://wolfram.com/xid/0tqflznxbkgi9u-ffbta1

In[3]:=3

✖

https://wolfram.com/xid/0tqflznxbkgi9u-nfsjnw

Out[3]=3

Plot the absolute cross-correlation for vector data:

In[1]:=1

✖

https://wolfram.com/xid/0tqflznxbkgi9u-5ejkpt

In[2]:=2

✖

https://wolfram.com/xid/0tqflznxbkgi9u-emzkrl

Out[2]=2

Random Processeses (6)

The absolute correlation function for a weakly stationary discrete-time process:

In[1]:=1

✖

https://wolfram.com/xid/0tqflznxbkgi9u-jlvt9

Out[1]=1

In[2]:=2

✖

https://wolfram.com/xid/0tqflznxbkgi9u-c08pf3

Out[2]=2

The absolute correlation function only depends on the antidiagonal :

In[3]:=3

✖

https://wolfram.com/xid/0tqflznxbkgi9u-drb7n6

Out[3]=3

In[4]:=4

✖

https://wolfram.com/xid/0tqflznxbkgi9u-9gw5a

Out[4]=4

The absolute correlation function for a weakly stationary continuous-time process:

In[1]:=1

✖

https://wolfram.com/xid/0tqflznxbkgi9u-bsxrpt

Out[1]=1

In[2]:=2

✖

https://wolfram.com/xid/0tqflznxbkgi9u-6xr1r

Out[2]=2

The absolute correlation function only depends on the antidiagonal :

In[3]:=3

✖

https://wolfram.com/xid/0tqflznxbkgi9u-ct9g81

Out[3]=3

In[4]:=4

✖

https://wolfram.com/xid/0tqflznxbkgi9u-drlf7

Out[4]=4

The absolute correlation function for a non-weakly stationary discrete-time process:

In[1]:=1

✖

https://wolfram.com/xid/0tqflznxbkgi9u-rlsomi

Out[1]=1

In[2]:=2

✖

https://wolfram.com/xid/0tqflznxbkgi9u-spx2b3

Out[2]=2

The absolute correlation function depends on both time arguments:

In[3]:=3

✖

https://wolfram.com/xid/0tqflznxbkgi9u-8vday0

Out[3]=3

In[4]:=4

✖

https://wolfram.com/xid/0tqflznxbkgi9u-iykr60

Out[4]=4

The absolute correlation function for a non-weakly stationary continuous-time process:

In[1]:=1

✖

https://wolfram.com/xid/0tqflznxbkgi9u-5eak2t

Out[1]=1

In[2]:=2

✖

https://wolfram.com/xid/0tqflznxbkgi9u-sl444x

Out[2]=2

The absolute correlation function depends on both time arguments:

In[3]:=3

✖

https://wolfram.com/xid/0tqflznxbkgi9u-sv9v1p

Out[3]=3

In[4]:=4

✖

https://wolfram.com/xid/0tqflznxbkgi9u-cf7247

Out[4]=4

The correlation function for some time series processes:

In[1]:=1

✖

https://wolfram.com/xid/0tqflznxbkgi9u-89jvi

In[2]:=2

✖

https://wolfram.com/xid/0tqflznxbkgi9u-3bfn4x

Out[2]=2

Absolute cross-correlation plots for a vector ARProcess:

In[1]:=1

✖

https://wolfram.com/xid/0tqflznxbkgi9u-soinyv

In[2]:=2

✖

https://wolfram.com/xid/0tqflznxbkgi9u-izvng2

Out[2]=2

Applications (2)Sample problems that can be solved with this function

Determine whether the following data is best modeled with an MAProcess or an ARProcess:

In[2]:=2

✖

https://wolfram.com/xid/0tqflznxbkgi9u-c3c54

Out[2]=2

It is difficult to determine the underlying process from sample paths:

In[3]:=3

✖

https://wolfram.com/xid/0tqflznxbkgi9u-xt365

In[4]:=4

✖

https://wolfram.com/xid/0tqflznxbkgi9u-gd19j1

Out[5]=5

The absolute correlation function of the data decays slowly:

In[6]:=6

✖

https://wolfram.com/xid/0tqflznxbkgi9u-ka6ks

Out[6]=6

ARProcess is clearly a better candidate model than MAProcess:

In[7]:=7

✖

https://wolfram.com/xid/0tqflznxbkgi9u-9y8np

Out[7]=7

Use the absolute correlation function to determine if a process is mean ergodic:

In[1]:=1

✖

https://wolfram.com/xid/0tqflznxbkgi9u-rmoejj

The process is weakly stationary:

In[2]:=2

✖

https://wolfram.com/xid/0tqflznxbkgi9u-uv261e

Out[2]=2

Calculate the absolute correlation function:

In[3]:=3

✖

https://wolfram.com/xid/0tqflznxbkgi9u-1drnsx

Out[3]=3

Find the value of the strip integral:

In[4]:=4

✖

https://wolfram.com/xid/0tqflznxbkgi9u-p270ef

Out[4]=4

Check if the limit of the integral is 0 to conclude mean ergodicity:

In[5]:=5

✖

https://wolfram.com/xid/0tqflznxbkgi9u-77w6az

Out[5]=5

Properties & Relations (13)Properties of the function, and connections to other functions

Sample absolute correlation function is a biased estimator for the process absolute correlation function:

In[1]:=1

✖

https://wolfram.com/xid/0tqflznxbkgi9u-2r99uh

Calculate the sample absolute correlation function:

In[36]:=36

✖

https://wolfram.com/xid/0tqflznxbkgi9u-bmgyvh

Out[36]=36

In[37]:=37

✖

https://wolfram.com/xid/0tqflznxbkgi9u-ehmlan

Out[37]=37

Absolute correlation function for the process:

In[38]:=38

✖

https://wolfram.com/xid/0tqflznxbkgi9u-oz6gf8

Out[38]=38

Plot both functions:

In[39]:=39

✖

https://wolfram.com/xid/0tqflznxbkgi9u-bfv26f

Out[39]=39

Absolute correlation function for a list can be calculated using AbsoluteCorrelation:

In[1]:=1

✖

https://wolfram.com/xid/0tqflznxbkgi9u-jo60yq

Calculate absolute correlation function for the data:

In[2]:=2

✖

https://wolfram.com/xid/0tqflznxbkgi9u-fcpmch

Out[2]=2

Use absolute correlation:

In[3]:=3

✖

https://wolfram.com/xid/0tqflznxbkgi9u-do8q1x

Out[3]=3

In[4]:=4

✖

https://wolfram.com/xid/0tqflznxbkgi9u-jixg72

Out[4]=4

AbsoluteCorrelationFunction is the off-diagonal entry in the absolute correlation matrix:

In[1]:=1

✖

https://wolfram.com/xid/0tqflznxbkgi9u-c8x3e0

In[2]:=2

✖

https://wolfram.com/xid/0tqflznxbkgi9u-cs2zy3

Out[2]=2

In[3]:=3

✖

https://wolfram.com/xid/0tqflznxbkgi9u-hpmt

Out[3]=3

In[4]:=4

✖

https://wolfram.com/xid/0tqflznxbkgi9u-dmok3u

Out[4]=4

Sample absolute correlation function at lag 0 estimates the second Moment:

In[1]:=1

✖

https://wolfram.com/xid/0tqflznxbkgi9u-0zuug9

Out[1]=1

In[2]:=2

✖

https://wolfram.com/xid/0tqflznxbkgi9u-qrqcji

Out[2]=2

In[3]:=3

✖

https://wolfram.com/xid/0tqflznxbkgi9u-sxji9t

Out[3]=3

Sample absolute correlation function is related to CovarianceFunction:

In[1]:=1

✖

https://wolfram.com/xid/0tqflznxbkgi9u-zusqzf

In[2]:=2

✖

https://wolfram.com/xid/0tqflznxbkgi9u-00kicv

Out[2]=2

In[3]:=3

✖

https://wolfram.com/xid/0tqflznxbkgi9u-q4nm4x

Out[3]=3

In[4]:=4

✖

https://wolfram.com/xid/0tqflznxbkgi9u-b4n3g4

Out[4]=4

Sample absolute correlation function is related to CorrelationFunction:

In[1]:=1

✖

https://wolfram.com/xid/0tqflznxbkgi9u-qr7h7n

In[2]:=2

✖

https://wolfram.com/xid/0tqflznxbkgi9u-klbxo4

Out[2]=2

Scale by the first element:

In[3]:=3

✖

https://wolfram.com/xid/0tqflznxbkgi9u-or7b5

Out[3]=3

Compare to the sample correlation function:

In[4]:=4

✖

https://wolfram.com/xid/0tqflznxbkgi9u-mebes9

Out[4]=4

In[5]:=5

✖

https://wolfram.com/xid/0tqflznxbkgi9u-innlhe

Out[5]=5

Use Expectation to calculate the absolute correlation function:

In[1]:=1

✖

https://wolfram.com/xid/0tqflznxbkgi9u-d76wac

In[2]:=2

✖

https://wolfram.com/xid/0tqflznxbkgi9u-kf4qex

Out[2]=2

In[3]:=3

✖

https://wolfram.com/xid/0tqflznxbkgi9u-4zi09y

Out[3]=3

In[4]:=4

✖

https://wolfram.com/xid/0tqflznxbkgi9u-hhorvz

Out[4]=4

The absolute correlation function is related to the Moment function:

In[1]:=1

✖

https://wolfram.com/xid/0tqflznxbkgi9u-o1uye

In[2]:=2

✖

https://wolfram.com/xid/0tqflznxbkgi9u-4mmjhb

Out[2]=2

Verify equality , where is the moment function:

In[3]:=3

✖

https://wolfram.com/xid/0tqflznxbkgi9u-9q3x99

Out[3]=3

In[4]:=4

✖

https://wolfram.com/xid/0tqflznxbkgi9u-j7bfw8

Out[4]=4

The absolute correlation function is related to the CovarianceFunction :

In[1]:=1

✖

https://wolfram.com/xid/0tqflznxbkgi9u-esdh0p

In[2]:=2

✖

https://wolfram.com/xid/0tqflznxbkgi9u-boa2zq

Out[2]=2

Verify equality , where is the mean function:

In[3]:=3

✖

https://wolfram.com/xid/0tqflznxbkgi9u-3iqn1

Out[3]=3

In[4]:=4

✖

https://wolfram.com/xid/0tqflznxbkgi9u-bh0hcn

Out[4]=4

The absolute correlation function equals CovarianceFunction when the mean of the process is zero:

In[1]:=1

✖

https://wolfram.com/xid/0tqflznxbkgi9u-iemvfy

Out[1]=1

In[2]:=2

✖

https://wolfram.com/xid/0tqflznxbkgi9u-ni991p

Out[2]=2

In[3]:=3

✖

https://wolfram.com/xid/0tqflznxbkgi9u-0t69ag

Out[3]=3

The absolute correlation function is invariant for ToInvertibleTimeSeries:

In[1]:=1

✖

https://wolfram.com/xid/0tqflznxbkgi9u-lzmm3o

In[2]:=2

✖

https://wolfram.com/xid/0tqflznxbkgi9u-sc3ri6

Out[2]=2

In[3]:=3

✖

https://wolfram.com/xid/0tqflznxbkgi9u-0gzrsk

Out[3]=3

In[4]:=4

✖

https://wolfram.com/xid/0tqflznxbkgi9u-ng1jmj

Out[4]=4

The absolute correlation function is not invariant to centralizing:

In[1]:=1

✖

https://wolfram.com/xid/0tqflznxbkgi9u-d9qtiq

The data has nonzero mean:

In[2]:=2

✖

https://wolfram.com/xid/0tqflznxbkgi9u-3y2jx0

Out[2]=2

Centralize data:

In[3]:=3

✖

https://wolfram.com/xid/0tqflznxbkgi9u-qwlcfc

Compare absolute correlation functions:

In[4]:=4

✖

https://wolfram.com/xid/0tqflznxbkgi9u-n5g69

Out[4]=4

PowerSpectralDensity is a transform of the absolute correlation function for mean zero processes:

In[1]:=1

✖

https://wolfram.com/xid/0tqflznxbkgi9u-hmwxsw

Out[1]=1

Use FourierSequenceTransform with appropriate parameters:

In[2]:=2

✖

https://wolfram.com/xid/0tqflznxbkgi9u-czau7j

Out[2]=2

Compare to the power spectrum:

In[3]:=3

✖

https://wolfram.com/xid/0tqflznxbkgi9u-jr5e2r

Out[3]=3

In[4]:=4

✖

https://wolfram.com/xid/0tqflznxbkgi9u-tlkjuj

Out[4]=4

Possible Issues (1)Common pitfalls and unexpected behavior

AbsoluteCorrelationFunction output may contain DifferenceRoot:

In[1]:=1

✖

https://wolfram.com/xid/0tqflznxbkgi9u-zoh06c

Out[1]=1

Use FunctionExpand to recover explicit powers:

In[2]:=2

✖

https://wolfram.com/xid/0tqflznxbkgi9u-eoxost

Out[2]=2

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Read

AbsoluteCorrelationFunction

✖
`AbsoluteCorrelationFunction`

Details

Examples

Basic Examples (4)Summary of the most common use cases

Scope (13)Survey of the scope of standard use cases

Empirical Estimates (7)

Random Processeses (6)

Applications (2)Sample problems that can be solved with this function

Properties & Relations (13)Properties of the function, and connections to other functions

Possible Issues (1)Common pitfalls and unexpected behavior

Text

CMS

APA

BibTeX

BibLaTeX

	Expectation[x[s] x[t]]	for a scalar-valued process
	Expectation[x[s]⊗x[t]]	for a vector-valued process

AbsoluteCorrelationFunction ✖ AbsoluteCorrelationFunction

Details

Examples

Basic Examples (4)Summary of the most common use cases

Scope (13)Survey of the scope of standard use cases

Empirical Estimates (7)

Random Processeses (6)

Applications (2)Sample problems that can be solved with this function

Properties & Relations (13)Properties of the function, and connections to other functions

Possible Issues (1)Common pitfalls and unexpected behavior

See Also

Related Guides

History

Text

CMS

APA

BibTeX

BibLaTeX

AbsoluteCorrelationFunction

✖
`AbsoluteCorrelationFunction`