ATMO 632: Week 4

Time Series Analysis: Spectral Peaks, Resampling, and Monte Carlo methods

                    Course Home Page: http://www.cgd.ucar.edu/~svn/atmo632

 

Reading Material (overlaps with week 3)

 

  1. Wilks: Chapter 8.4-8.5 (Harmonic and Spectral Analysis), 5.2.3 (Serial correlations), Chapter 5.3.2 (Resampling methods)
  2. North: Chapter 6 (Spectral Analysis of Time Series)
  3. Hartmann: Chapter 6.2 (Harmonic Analysis)

 

 

Concepts

 

            Power spectra

                        How to smooth spectra

                        Different ways to plot spectra

 

            Sums of uncorrelated time series

 

            Choosing the right null hypothesis

                        AR(0)

                        AR(1)

                        AR(2)

 

            Sampling errors in power spectra

                        a priori vs. a posteriori confidence intervals (see Hartmann’s notes: 6.2.6)

 

            Serial correlations and degrees of freedom (Wilks 5.2.3; North 4.10)

 

            Resampling/Bootstrapping (Wilks: 5.3.2)

 

            Monte Carlo methods

                        Generating synthetic data

                        Lag-1 autocorrelation

                        Variance of AR(1) process

                        Spatial correlations

                        EOFs

 

 

 

           

           


ATMO 632: Homework for week 4 (due Sep. 28)

           

NOTE: You can obtain the datasets used below by visiting the course home page

                        http://www.cgd.ucar.edu/~svn/atmo632 and clicking the appropriate links.

You do not need to type in the URLs given below, which are provided for reference.

 

  1. Reference red-noise spectrum: Consider 100 years (1904-2003) of the NAO time series analyzed in Week 2 homework.
    1. Compute the standard deviation sx of the time series and the lag-1 autocorrelation r1, i.e., the correlation between NAO index values for successive years.

 

    1. Use the formula below to plot the reference red-noise power spectrum, for the frequency range f=0,..,0.5 yr-1, with increments Df = 0.01 (=1/N), where N=100:

             (Wilks: Sec. 8.5.5)

Note that S(fj) satisfies the normalization condition, with . This ensures that for uncorrelated (r1=0) data, we have.

 

    1. Compute 95% confidence limits associated with the above red-noise spectrum, assuming a distribution with degrees of freedom . Recall that the  statistic is used to compare the sample variance to the population variance. For each , assume the red-noise spectrum S(fj) represents the population variance . The sample variance can therefore be expressed as

Look up thevalues defining the 95% confidence limit, and use it to compute  using the above formula to obtain the confidence limits on the power spectrum. Overlay the confidence limits on the red-noise spectrum plot (using a different line style).

 

  1. Synthetic red noise and “spectral peaks”:
    1. Generate a synthetic red-noise time series, with 100 data points starting from , using the formula

,

where  is a gaussian random variable with zero mean and unit variance. Compute the frequency power spectrum S(fj) by applying a discrete fourier transform on the time series. Use the same normalization condition for S(fj) as in Problem 1. Overlay power spectrum on the previous plot of the reference red noise spectrum. Note how many “spectral peaks” lie above the 95% limit.

 

    1. Generate two more synthetic time series, using different realizations of random variable . Plot their power spectrum, along with the reference red noise spectrum and the 95% confidence limits. Note how many “spectral peaks” lie above the 95% limit for each of the two power spectra.

 

    1. What do you learn from the power spectra of the three synthetic red-noise time series?

 

  1. NAO power spectrum: Consider 100 years (1904-2003) of the NAO time series analyzed in Week 2 homework.
    1. Compute the frequency power spectrum S(fj) by applying a discrete fourier transform on the time series. Remember to detrend the data before computing the transform. Use the same normalization condition for S(fj) as in Problem 1. Plot the power spectrum and overlay the reference red noise spectrum and the 95% confidence limits. Are there any spectral peaks above the 95% limit?

 

    1. Average together neighboring power spectral estimates in groups of five (f=0.01-0.05, 0.06-0.10, ..., 0.46-0.50). Plot the 10 binned power spectral estimates as a function of the middle frequency value for each bin. Again plot the reference red-noise spectrum and 95% confidence limits using the appropriate value of . Are there any significant spectral peaks?

 

    1. Based upon the above two plots, would you accept the null hypothesis that NAO is just red noise? How do these results relate to what you learn from Problem 2?

 

 

“Extra credit”

 

Spectral confidence limits using Monte Carlo approach: Generate 1000 time synthetic red-noise series, each with 100 data points, as in Problem 2. Compute the frequency-power spectrum for each of the time series, binning frequencies in groups of five as in Problem 3(b), resulting in 10 frequency values fj. Compute the PDF of the power spectrum and the 2.5 percentile and 97.5 percentile power for each of the 10 frequency values fj, and plot them as a function of fj. Compare to the 95% confidence limits in Problem 3(b). Discuss the advantages and disadvantages of using the Monte Carlo approach to computing confidence limits, as opposed to using the traditional chi-square test.