Namespaces
Variants
Actions

Difference between revisions of "Periodogram"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
m (tex encoded by computer)
Line 1: Line 1:
A function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072230/p0722301.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072230/p0722302.png" />, with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072230/p0722303.png" /> a positive integer, defined on a sample <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072230/p0722304.png" /> of a [[Stationary stochastic process|stationary stochastic process]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072230/p0722305.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072230/p0722306.png" /> as follows:
+
<!--
 +
p0722301.png
 +
$#A+1 = 28 n = 0
 +
$#C+1 = 28 : ~/encyclopedia/old_files/data/P072/P.0702230 Periodogram
 +
Automatically converted into TeX, above some diagnostics.
 +
Please remove this comment and the {{TEX|auto}} line below,
 +
if TeX found to be correct.
 +
-->
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072230/p0722307.png" /></td> </tr></table>
+
{{TEX|auto}}
 +
{{TEX|done}}
 +
 
 +
A function  $  I _ {N} ( \lambda ) $,
 +
$  - \infty < \lambda < \infty $,
 +
with  $  N $
 +
a positive integer, defined on a sample  $  X( 1) \dots X( N) $
 +
of a [[Stationary stochastic process|stationary stochastic process]]  $  X( t) $,
 +
$  t = 0, \pm  1 \dots $
 +
as follows:
 +
 
 +
$$
 +
I _ {N} ( \lambda )  =
 +
\frac{1}{2 \pi N }
 +
| d _ {N}  ^ {(} X) ( \lambda ) |  ^ {2} ,
 +
$$
  
 
where
 
where
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072230/p0722308.png" /></td> </tr></table>
+
$$
 +
d _ {N}  ^ {(} X) ( \lambda )  = \sum _ { t= } 1 ^ { N }  \mathop{\rm exp} \{ - it \lambda \} X( t).
 +
$$
 +
 
 +
A periodogram is a function that is periodic in  $  \lambda $
 +
with period  $  2 \pi $.
 +
The differentiable [[Spectral density|spectral density]]  $  f( \lambda ) $
 +
of the stationary process  $  X( t) $
 +
with mean  $  c = {\mathsf E} X( t) $
 +
can be estimated by means of the periodogram for  $  \lambda \neq 0 $
 +
$  (  \mathop{\rm mod}  2 \pi ) $:
  
A periodogram is a function that is periodic in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072230/p0722309.png" /> with period <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072230/p07223010.png" />. The differentiable [[Spectral density|spectral density]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072230/p07223011.png" /> of the stationary process <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072230/p07223012.png" /> with mean <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072230/p07223013.png" /> can be estimated by means of the periodogram for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072230/p07223014.png" /> <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072230/p07223015.png" />:
+
$$
 +
{\mathsf E} I _ {N} ( \lambda )  = f( \lambda ) +
 +
\frac{1}{2 \pi N }
 +
 +
\frac{\sin  ^ {2}  N
 +
\lambda /2 }{\sin  ^ {2}  \lambda /2 }
 +
c  ^ {2} + O( N  ^ {-} 1 ).
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072230/p07223016.png" /></td> </tr></table>
+
At the same time, the periodogram is not a [[Consistent estimator|consistent estimator]] for  $  f( \lambda ) $(
 +
cf. [[#References|[1]]]). Consistent estimators of the spectral density (cf. [[Spectral density, estimator of the|Spectral density, estimator of the]]) can be obtained by some further constructions that employ the asymptotic lack of correlation for the periodograms for different frequencies  $  \lambda _ {1} \neq \lambda _ {2} $,
 +
with the result that averaging  $  I _ {N} ( x) $
 +
with respect to frequencies close to  $  \lambda $
 +
may lead to an asymptotically-consistent estimator. In the case of an  $  n $-
 +
dimensional stochastic process  $  X( t) = \{ X _ {k} ( t) \} _ {k=} 1  ^ {n} $,
 +
the matrix periodogram  $  I _ {N} ( \lambda ) $
 +
is determined by its elements
  
At the same time, the periodogram is not a [[Consistent estimator|consistent estimator]] for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072230/p07223017.png" /> (cf. [[#References|[1]]]). Consistent estimators of the spectral density (cf. [[Spectral density, estimator of the|Spectral density, estimator of the]]) can be obtained by some further constructions that employ the asymptotic lack of correlation for the periodograms for different frequencies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072230/p07223018.png" />, with the result that averaging <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072230/p07223019.png" /> with respect to frequencies close to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072230/p07223020.png" /> may lead to an asymptotically-consistent estimator. In the case of an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072230/p07223021.png" />-dimensional stochastic process <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072230/p07223022.png" />, the matrix periodogram <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072230/p07223023.png" /> is determined by its elements
+
$$
 +
I _ {N} ^ {( i, j) } ( \lambda ) = \
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072230/p07223024.png" /></td> </tr></table>
+
\frac{1}{2 \pi N }
 +
d _ {N} ^ {X _ {i} } (
 +
\lambda ) {d _ {N} ^ {X _ {j} } ( \lambda ) } bar .
 +
$$
  
Along with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072230/p07223025.png" />, which is also called a second-order periodogram, one sometimes also considers the periodogram of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072230/p07223027.png" />:
+
Along with $  I _ {N} ( \lambda ) $,  
 +
which is also called a second-order periodogram, one sometimes also considers the periodogram of order $  m $:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072230/p07223028.png" /></td> </tr></table>
+
$$
 +
I _ {N} ^ {( k _ {1} \dots k _ {m} ) }
 +
( \lambda _ {1} \dots \lambda _ {m} )  =
 +
\frac{1}{( 2 \pi )  ^ {m-} 1 N }
 +
\prod _ { j= } 1 ^ { m }
 +
d _ {N} ^ {X _ {k _ {j}  } } ( \lambda _ {j} ),
 +
$$
  
which is used in constructing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072230/p07223029.png" />-th order estimators of the spectral density (see [[Spectral semi-invariant|Spectral semi-invariant]]).
+
which is used in constructing $  m $-
 +
th order estimators of the spectral density (see [[Spectral semi-invariant|Spectral semi-invariant]]).
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  D.R. Brillinger,  "Time series. Data analysis and theory" , Holt, Rinehart &amp; Winston  (1974)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  E.J. Hannan,  "Multiple time series" , Wiley  (1970)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  D.R. Brillinger,  "Time series. Data analysis and theory" , Holt, Rinehart &amp; Winston  (1974)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  E.J. Hannan,  "Multiple time series" , Wiley  (1970)</TD></TR></table>

Revision as of 08:05, 6 June 2020


A function $ I _ {N} ( \lambda ) $, $ - \infty < \lambda < \infty $, with $ N $ a positive integer, defined on a sample $ X( 1) \dots X( N) $ of a stationary stochastic process $ X( t) $, $ t = 0, \pm 1 \dots $ as follows:

$$ I _ {N} ( \lambda ) = \frac{1}{2 \pi N } | d _ {N} ^ {(} X) ( \lambda ) | ^ {2} , $$

where

$$ d _ {N} ^ {(} X) ( \lambda ) = \sum _ { t= } 1 ^ { N } \mathop{\rm exp} \{ - it \lambda \} X( t). $$

A periodogram is a function that is periodic in $ \lambda $ with period $ 2 \pi $. The differentiable spectral density $ f( \lambda ) $ of the stationary process $ X( t) $ with mean $ c = {\mathsf E} X( t) $ can be estimated by means of the periodogram for $ \lambda \neq 0 $ $ ( \mathop{\rm mod} 2 \pi ) $:

$$ {\mathsf E} I _ {N} ( \lambda ) = f( \lambda ) + \frac{1}{2 \pi N } \frac{\sin ^ {2} N \lambda /2 }{\sin ^ {2} \lambda /2 } c ^ {2} + O( N ^ {-} 1 ). $$

At the same time, the periodogram is not a consistent estimator for $ f( \lambda ) $( cf. [1]). Consistent estimators of the spectral density (cf. Spectral density, estimator of the) can be obtained by some further constructions that employ the asymptotic lack of correlation for the periodograms for different frequencies $ \lambda _ {1} \neq \lambda _ {2} $, with the result that averaging $ I _ {N} ( x) $ with respect to frequencies close to $ \lambda $ may lead to an asymptotically-consistent estimator. In the case of an $ n $- dimensional stochastic process $ X( t) = \{ X _ {k} ( t) \} _ {k=} 1 ^ {n} $, the matrix periodogram $ I _ {N} ( \lambda ) $ is determined by its elements

$$ I _ {N} ^ {( i, j) } ( \lambda ) = \ \frac{1}{2 \pi N } d _ {N} ^ {X _ {i} } ( \lambda ) {d _ {N} ^ {X _ {j} } ( \lambda ) } bar . $$

Along with $ I _ {N} ( \lambda ) $, which is also called a second-order periodogram, one sometimes also considers the periodogram of order $ m $:

$$ I _ {N} ^ {( k _ {1} \dots k _ {m} ) } ( \lambda _ {1} \dots \lambda _ {m} ) = \frac{1}{( 2 \pi ) ^ {m-} 1 N } \prod _ { j= } 1 ^ { m } d _ {N} ^ {X _ {k _ {j} } } ( \lambda _ {j} ), $$

which is used in constructing $ m $- th order estimators of the spectral density (see Spectral semi-invariant).

References

[1] D.R. Brillinger, "Time series. Data analysis and theory" , Holt, Rinehart & Winston (1974)
[2] E.J. Hannan, "Multiple time series" , Wiley (1970)
How to Cite This Entry:
Periodogram. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Periodogram&oldid=48159
This article was adapted from an original article by I.G. Zhurbenko (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article