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Name: Fan Zhang EAS6490-HW4 October 22, 2012

1 y(t) = 2 + 3t + 10sin(1t) + 5cos(2t) + 2cos(3t) + n(t)

Original and detrended time series are shown in Figure 1.

Fig. 1. Original (black) and detrended (red) time series. Note that y-axis limits for twotime series are different.

1.1 Power Spectrum of the Detrended Time Series

t = 1 month, N= 1000.

Angular frequencies (rad/s) are

1 = 60Nt

, 2 = 100Nt

, 3 = 20Nt

.

Corresponding frequencies (s1) are

f1 =30

Nt, f2 =

50

Nt, f3 =

10

Nt,

and periods ( 1f

, month) are 33.3 months, 20 months, 100 months.

Figure 2 shows the power spectrum (black) and normalized power spectrum (red).

Without normalization, magnitudes of the power spectrum correspond to well the equa-tion, with three dominant frequencies and have magnitudes of 2, 10 and 5.

After normalization, the magnitude (y-axis) changes, but variations remain the same:the black and red overlap with each other; the peaks are the same: 0.01, 0.03 and 0.05. Thisdemonstrates that normalization will not affect the shape of the power spectrum.

If we change the x-axis from frequency (month1) to period (month), dominant periods(20 months, 33.3 months and 100 months) become more clear.

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Fig. 2. Power spectrum vs. frequency for the detrended time series. Normalized (red) andwithout normalization (black, ).

Fig. 3. Normalized power spectrum vs. period for the detrended time series.

1.2 Power Spectrum of the Original Time Series

If the time series is not detrended prior computing power spectrum. The power spectrum(black line in Figure 4) is really not satisfactory, because the trend magnitude (approximately3000) is several orders higher than the sin and cos components (Figure 1). Thus, we couldnot tell the dominant frequencies (black line in Figure 4).

However, after normalization, dominant frequencies stand out.

1.3 Comparison between Detrend and Without Detrend

Finally, normalized power spectra with and without detrending are compared (Figure 5).Both captured the dominant frequencies well, though magnitudes differ.

From above comparisons, it is inferred that detrending and normalization are

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Fig. 4. Power spectrum vs. frequency for the detrended time series. Normalized (red) andwithout normalization (black). Due to the large range of power spectrum prior normalization,the logarithmic scale is used.

Fig. 5. Normalized power spectrum vs. frequency for the detrended () and original (red)time series. Since interested frequencies are all lower than 0.2, limits of x-axis is set to [0,0.2] for better visual.

important before further analysis is carried out in examining and interpreting

geophysical time series. This becomes particularly important when the trend of

the time series is so strong that it may dwarf components with cycles.

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2 HW Timeseries1.txt

Original and detrended time series are shown in Figure 6. The overlapping of two timeseries indicates that the data provided has been detrended. And that is why the black lineare almost invisible in the figure.

Fig. 6. Original (black) and detrended (red) time series. Note that y-axis limits for twotime series are different. The overlapping of two time series indicates that the data providedhas been detrended. And that is why the black line are almost invisible in the figure.

2.1 Case a. no window, no smoothing.

In this case, since there is no window and no smoothing. The degree of freedom (DoF)for each spectral estimate is 2.

In F-statistics,

F =s21

s22

,

where s21

and s22

are variances of two time series.

In this case, DoF1 = 2 (the time series provided) and DoF2 = (the red noise). TheF-statistics value for 95% significance is determined from the table (from http://home.comcast.net/~sharov/PopEcol/tables/f005.html ) with 1 = 2 and 2 > 1000. Thus, forthe 95% significance, the F-statistics value is 3.00.

The result is shown in Figure 7. It reveals that the time series has a dominant frequency0.1 month1 (a period of 10 month) that is above the 95% significance curve. If this timeseries is taken as a geophysical time series, this period may be associated with seasonal cycle.

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http://home.comcast.net/~sharov/PopEcol/tables/f005.htmlhttp://home.comcast.net/~sharov/PopEcol/tables/f005.htmlhttp://home.comcast.net/~sharov/PopEcol/tables/f005.htmlhttp://home.comcast.net/~sharov/PopEcol/tables/f005.htmlhttp://home.comcast.net/~sharov/PopEcol/tables/f005.htmlhttp://home.comcast.net/~sharov/PopEcol/tables/f005.html

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Fig. 7. Power spectrum of the data (black), power spectrum of the red-noise (red) and 95%

significance curve of red noise spectrum (red, dashed). DoF = 2, F0.95 = 3.00.

2.2 Case b. No window, 5-point running mean.

The 5-point running mean is shown in Figure 8. As expected, the power spectrumbecomes more smooth.

The running mean procedure increases the DoF to 10. Consequently, the F-statistic valueF0.95 = 1.83.

The dominant frequency appears more clear compared to that before the running-mean.However, running-mean decreases the resolution of the power spectrum, since the running-mean was also run on the frequency.

Fig. 8. Five-point running mean of power spectrum (black), power spectrum of the red-noise(red) and 95% significance curve of red noise spectrum (red, dashed). DoF = 10, F0.95 = 1.83.

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2.3 Case c. No window, no smoothing. Ten subsets.

Without decreasing the frequency resolution, DoF is increased to 20 by dividing the timeseries into ten subsets. Correspondingly, F0.95 = 1.57.

Power spectra of ten subsets are shown in Figure 9 and the averaged spectra in Figure10. Autocorrelation at 1 lat of each subset was calculated.

Similarly, a peak occurred at f 0.1month1.

Fig. 9. Power spectrum of ten subsets.

Fig. 10. Averaged power spectrum of ten subsets (black), power spectrum of the red-noise

(red) and 95% significance curve of red noise spectrum (red, dashed). DoF = 20, F0.95 = 1.57.

2.4 Case d. Hanning window, no smoothing. Nine subsets.

A Hanning window with a length of 200 was applied when computing the power spectrafor nine subsets (each has a length of 200, the same as the window). Practically, if thewindow length is shorter than the time series, ZEROs will be padded.

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The Hanning window is generated by:

w(t) =1

2(1 cos(

2t

T))(0 t T).

Since nine subsets were averaged, DoF = 18, F0.95 = 1.61.Power spectra of nine subsets are shown in Figure 11 and the averaged in Figure 12.

Though not below the red-noise power spectrum, two lower frequency peaks (f = 0.05and f= 0.01) arise in this case (Figure 12), compared to those in Figure 10.

Fig. 11. Power spectrum of nine subsets.

Fig. 12. Averaged power spectrum of nine subsets (black), power spectrum of the red-noise(red) and 95% significance curve of red noise spectrum (red, dashed). DoF = 18, F0.95 = 1.61

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3 ENSO

Original and detrended ENSO index are shown in Figure 13. The overlapping of twotime series indicates that the data provided has been detrended. And that is why the blackline are almost invisible in the figure.

Fig. 13. Original (black, square) and detrended (red) ENSO index. The overlapping of twotime series indicates that the ENSO index provided has been detrended. And that is whythe black line are almost invisible in the figure

Figure 14 illustrates the power spectrum and five-point running mean power spectrum.Two peaks above the 95% significance curve were found:

f1 = 0.02344 month1, f2 = 0.03906 month

1.

Correspondingly, periods are:

T1 = 1/f1 = 42.7 months = 3.5 years,T2 = 1/f2 = 25.6 months = 2.1 years.

However, we noted that a uniform autocorrelation ( = 0.72) for Nino 3 SST was usedby Torrence and Compo (1998). They also employed the five-point running mean. When thesame procedures were applied to our ENSO index, we obtained different results and there isno periods that goes over the 95% significance curve (Figure 15 compared to their Figure 6).

One possible reason for this discrepancy is that probably our ENSO index data is different

from Nino 3 SST.

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Fig. 14. Power spectrum of the ENSO index (black, thin), the 5-point running mean (black,bold), the red noise spectrum (red) and 95% confidence curve for red-noise spectrum (red,dashed). DoF = 10, F0.95 = 1.83, = 0.9258.

Fig. 15. Power spectrum of the ENSO index (black, thin), the 5-point running mean (black,bold), the red noise spectrum (red) and 95% confidence curve for red-noise spectrum (red,dashed). DoF = 10, F0.95 = 1.83, = 0.72.

References

Harrison, D., and N. Larkin (1998), El nino-southern oscillation sea surface temperature and

wind anomalies, 19461993, Reviews of Geophysics, 36(3), 353399.Kao, H., and J. Yu (2009), Contrasting eastern-pacific and central-pacific types of enso,

Journal of Climate, 22(3), 615632.

Torrence, C., and G. Compo (1998), A practical guide to wavelet analysis, Bulletin of theAmerican Meteorological Society, 79(1), 6178.

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