ALLAN VARIANCE PLOT - NIST

Graphics Commands

ALLAN VARIANCE PLOT

ALLAN VARIANCE PLOT

PURPOSE Generates an Allan variance plot in order to examine the low-frequency component of a spectrum of an equi-spaced time series, and to estimate the exponent in a low-frequency power-law spectral model.

DESCRIPTION The Allan variance plot is a graphical data analysis technique for examining the low-frequency component of a time series. The horizontal axis is the subsample size (up to N/2). The vertical axis is the Allan variance (AV(K)), which is the variance of the squared deltas as defined below. For subsample size 1: delta1 = x(1)-x(2) delta2 = x(3)-x(4) delta3 = x(5)-x(6) ... deltan = x(n-1)-x(n) For subsample size 2: delta1 = (x(1)+x(2))-(x(3)+x(4)) delta2 = (x(5)+x(6))-(x(7)+x(8)) ... For subsample size 3: delta1 = (x(1)+x(2)+x(3))-(x(4)+x(5)+x(6)) delta2 = (x(7)+x(8)+x(9))-(x(10)+x(11)+x(12)) ... ...

The Allan variance plot is usually viewed on a loglog scale. A common frequency domain model for the spectrum S(w) of a lowfrequency time series is the power-law:

S(w) = w

There is a one-to-one correspondence between the slope of the loglog spectrum (the ) and the slope of the loglog Allan variance plot:

Time Series Model Slope of Loglog Spectrum Slope of Loglog AV Plot

(--1)/2

----------------------------------------------------------------------------------------

Random Walk

-2

0.5

Flicker

-1

0

White Noise

0

-0.5

Super Flicker

1

-1

Super White

2

-1.5

If one has a time series with a dominant low-frequency component, then the Allan variance plot is a useful tool for assessing the nature of the low-frequency component and for estimating the power () of the power-law spectral power-law model. The slope of the Allan variance plot indicates the nature of the underlying time series model.

The response variable must have at least 3 elements.

SYNTAX

ALLAN VARIANCE PLOT

where is a response variable;

and where the is optional.

EXAMPLES ALLAN VARIANCE PLOT Y AV PLOT Y

NOTE 1 The Allan variance plot and the Allan standard deviation plot have equivalent information content (and differ only by a factor of 2). The Allan variance plot is more heavily used than the Allan standard deviation plot.

DATAPLOT Reference Manual

February 28, 1997

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ALLAN VARIANCE PLOT

Graphics Commands

DEFAULT None

SYNONYMS AV PLOT

RELATED COMMANDS

SPECTRAL PLOT

=

ALLAN STAND DEVIATION PLOT =

Generates a spectral plot. Generates an Allan standard deviation plot.

REFERENCE Dave Allan, NIST in Boulder

APPLICATIONS Frequency Time Series Analysis

IMPLEMENTATION DATE 87/1

PROGRAM 1 . THIS IS AN EXAMPLE OF AN ALLAN VARAIANCE PLOT . FOR WHITE NOISE DATA S(W) = W**0 . (THUS THE LOGLOG SPECTRUM HAS SLOPE 0 AND . AND THE ALLAN SD PLOT HAS SLOPE (-(0)-1)/2 = -1/2 LET Y = NORMAL RANDOM NUMBERS FOR I = 1 1 500 TITLE WHITE NOISE MULTIPLOT 2 2; MULTIPLOT CORNER COORDINATES 0 0 100 100 GRID ON; X3LABEL AUTOMATIC PLOT Y; SPECTRUM Y LOGLOG; SPECTRUM Y; ALLAN VARIANCE PLOT Y END OF MULTIPLOT

3 2 1 0 -1 -2 -3 -4

0

WHITE NOISE

100

200

300

400

500

PLOT Y

WHITE NOISE

1

0

-1

-3

-2

-1

0

SPECTRUM Y

WHITE NOISE

6

5

4

3

2

1

0 0

0.1

0.2

0.3

0.4

0.5

SPECTRUM Y

WHITE NOISE

0

-1

-2

-3

-4

-5

-6

-7

0

1

2

3

ALLAN VARIANCE PLOT Y

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February 28, 1997

DATAPLOT Reference Manual

Graphics Commands

ALLAN VARIANCE PLOT

PROGRAM 2 . THIS IS AN EXAMPLE OF AN ALLAN VARIANCE PLOT . FOR RANDOM WALK DATA S(W) = W**(-2) . (THUS THE LOGLOG SPECTRUM HAS SLOPE -2 AND . AND THE ALLAN VARIANCE PLOT HAS SLOPE (-(-2)-1)/2 = 1/2 SKIP 25; READ RANDWALK.DAT Y TITLE RANDOM WALK MULTIPLOT 2 2; MULTIPLOT CORNER COORDINATES 0 0 100 100 GRID ON; X3LABEL AUTOMATIC PLOT Y; SPECTRUM Y LOGLOG; SPECTRUM Y; ALLAN VARIANCE PLOT Y END OF MULTIPLOT

8 7 6 5 4 3 2 1 0 -1 -2

0

3

2

1

0

-1

-2 -3

RANDOM WALK

100

200

300

400

500

PLOT Y

RANDOM WALK

-2

-1

0

SPECTRUM Y

700 600 500 400 300 200 100

0 0

2

1

0

-1

-2 0

RANDOM WALK

0.1

0.2

0.3

0.4

0.5

SPECTRUM Y

RANDOM WALK

1

2

3

ALLAN VARIANCE PLOT Y

DATAPLOT Reference Manual

February 28, 1997

2-11

ALLAN VARIANCE PLOT

Graphics Commands

PROGRAM 3 . THIS IS AN EXAMPLE OF AN ALLAN VARIANCE PLOT . FOR FLICKER NOISE DATA S(W) = W**(-1) . (THUS THE LOGLOG SPECTRUM HAS SLOPE -1 AND . AND THE ALLAN VARIANCE PLOT HAS SLOPE (-(-1)-1)/2 = 0 SKIP 25; READ FLICKER.DAT Y TITLE FLICKER DATA MULTIPLOT 2 2; MULTIPLOT CORNER COORDINATES 0 0 100 100 GRID ON; X3LABEL AUTOMATIC PLOT Y; SPECTRUM Y LOGLOG; SPECTRUM Y; ALLAN VARIANCE PLOT Y END OF MULTIPLOT

FLICKER DATA

0.02

0.01

0

-0.01

-0.02 0

100

200

300

400

500

600

700

800

900

1000

PLOT Y

FLICKER DATA

-2

-3

-4

-5

-6

-4

-3

-2

-1

0

SPECTRUM Y

0.006 0.005 0.004 0.003 0.002 0.001

0 0

-4 -5 -6 -7 -8 -9 -10

0

FLICKER DATA

0.1

0.2

0.3

0.4

0.5

SPECTRUM Y

FLICKER DATA

1

2

3

ALLAN VARIANCE PLOT Y

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February 28, 1997

DATAPLOT Reference Manual

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