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discrepancy between periodogram implementations ? per and spec.pgram

3 messages · Lieven Desmet, Brian Ripley

Lieven Desmet
# 
hello,

I have been using the per function in package longmemo to obtain a 
simple raw periodogram.
I am considering to switch to the function spec.pgram since I want to be 
able to do tapering.
To compare both I used spec.pgram with the options as suggested in the 
documentation of per {longmemo} to make them correspond.

Now I have found on a variety of examples that there is a shift between 
the log of the periodogram with per and that with spec.pgram. This 
vertical shift amounts to  approx. 1.8  on the log scale  (the graph of 
spec.pgram being above the one from per).

Is there some explanation for this ? Is the one from spec.pgram the 
better one as suggested in the documentation of per {longmemo}? Finally 
how are these related to an estimate of the spectral density obtained 
from spec.arima ?

Many thanks for help and clarification.

Lieven Desmet


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Brian Ripley
# 
There are several definitions of a periodgram.  Note that
[1] 1.837877

See the comments in ?spectrum about scalings.

I think the comments in ?per incorrectly ignore the scaling issues: per() 
does not take the base frequency into account and has an extra divisor of 
2*pi.  E.g.
[1] 6.283185 6.283185 6.283185 6.283185 6.283185 6.283185 6.283185 6.283185
  [9] 6.283185 6.283185 6.283185 6.283185 6.283185 6.283185 6.283185 6.283185
[17] 6.283185 6.283185 6.283185 6.283185 6.283185 6.283185 6.283185 6.283185
[25] 6.283185 6.283185 6.283185 6.283185 6.283185 6.283185 6.283185 6.283185
On Wed, 12 Dec 2007, Lieven Desmet wrote:

            

      
      
      
    

        
What is spec.arima?  If you meant spec.ar, that is on the same scale as 
spec.pgram for series with base frequency 1 (and for all series for R >= 
2.7.0).

            

      
      
      
    

        

  
    

      
      
    

  


  


      

    

    
1 day later

    

      
      

        


  

        

      Lieven Desmet
    

    

      
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Prof Brian Ripley wrote:

        

            

      
      
      
    

        
Dear Prof. Ripley,

thanks very much for a quick and helpful response. In the last question 
I wanted to hint at
specARIMA which I am using to get the theoretical spectral density of an 
ARMA process.

This works very well in general, however, in a simple example

X_t=0.7*X_{t-1}+epsilon_t

I obtain a value 1.768253 for funscaled[1] ( the first Fourier frequency 
0.003141593)

using

str(f<-specARIMA(eta=c(H=0.5,phi=c(0.7),psi=c()),p=1,q=0,m=2000))
funscaled<-numeric(length(f$freq))
funscaled<-f$spec*f$theta1

where the theoretical value should be 0.901878 with

b<-0.7
omega<-0.003141593
1/(2*pi)*(1-b^2)/(1+b^2-2*b*cos(omega))
[1] 0.9018088


using the formula (2.40) in Fan and Yao, Nonlinear Time Series ( 
Springer 2003 ), page 54-55

Is there also a simple explanation for this ? am I overlooking something ?

Thanks and best regards,

Lieven Desmet,

maths dept - KULeuven - Belgium

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