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This page explains what the EOF is and how the customer can use them. This page describs a part of the data analysis services we offer at CRI. Please click "Data Analysis" button above to see other types of data analysis we offer.

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2. Complex (time domain) EOF
2-1 What is it?
The complex (time domain) EOF was introduced to analyze a set of time series data that have phase lag among them by adding components that are the original time series data rotated by 90 degrees on a complex plane using a mathematical method called Hilbert transform. This EOF is close to frequency domain EOF, which will be described later, but this method does not require converting data in the time domain into the frequency domain explicitly in the process. Thus, we call this method "Complex (time domain) EOF" in this page. We would not get into any further detail of mathematics but show an example of this type of EOF right away.

2-2 Example
We use same data set as before (Figure 1a). Figure 6a shows time series plots of mode 1 and mode 2 components at 40m. We multiplied mode 2 component by three in this figure.

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Since phase lag among time series data are now allowed, not only the amplitude but the shape of these components changes also at different depths. Figure 6b shows mode 1 component (blue line) and input data (red line, same as blue line in figure 1a) at several depths. Those at depths below 150m are multiplied by 2 because their amplitude is too small to see.

For the comparison, we made a similar figure, figure 6c, for non-complex time domain EOF using a result of section 1-2. In this case only the amplitude and sign ("hill" becomes "basin" or the other way around) of time series data generated by EOF change at different depths but the positions of "hills" and "basins" do not change. Comparing Figure 6b with 6c, you probably get an impression that the time series data generated by complex (time domain) EOF achieve better agreement with input data, especially at 250m.
Figure 6d shows how much variance in the original time series data these components explain just like Figure 2b. The mode 1 explains 60.2% of variance in the original time series data. This is 6.7% higher than the case of previous EOF. Figure 6e and 6f shows vertical how the amplitude and phase difference relative to those at 40m of mode 1 vary at different depths. These are eigenvectors that are complex now. The amplitude in figure 6e corresponds to the absolute value (if a value is negative, make it a positive value by changing the sigh) of the amplitude shown in Figure 2c. The phase difference at 100m relative to that at 40m is about 180 degrees. This means that the variations at 100m are opposite to those at 40m. The relative phase difference between 100m and 140m is about -180 degrees but -180 degrees is equal to 180 degrees. Any value which is larger than 180 degrees is shown as -180+value in this figure.
The non-complex time domain EOF (Figure 2) allows only 0 and 180 degrees phase differences. We will omit further analysis regarding to these figures. The important point here is that these figures tell us how the amplitude of mode1 varies at different depths and how it delays. Information such as that would be very useful to understand how the ocean responds to wind forcing. Figure 6g shows time series plots of mode 1 (blue line) and east-west wind speed (red line). This figure corresponds to Figure 3a.
2-3 Some cautions of using complex (time domain) EOF
We still need some cautions when we use complex (time domain) EOF.


(1) Eigenvectors (phase and amplitude) and eigenvalues are supposed to be reasonably constant.
This is basically the same as (2) of section 1-3, so we would not repeat explanation.

(2) Complex (time domain) EOF allows phase lag but phase lag is not the same as time lag.
As described previously, phase difference is not the same as time difference (lag). If there are certain reasons to believe that the time series data have variations, which have a constant time lag rather than a constant phase lag among them, then using a complex (time domain) EOF might not be a good idea. Applying a band-pass filter before computing complex (time domain) EOF might reduce the risk of this problem. Alternatively we might try using frequency domain EOF, which will be described later, in the case like that.

3 EOF for vector data(Complex, time domain)
We described a family of EOF applied to scalar data that have only the amplitude up to now. We picked up east-west component of oceanic velocity data in our previous examples but velocity data also have north-south component. Since the equator is geophysically rather a special place, picking up only the east-west component of ocean current makes sense. However, we might want to analyze east-west and north-south components of simultaneously without analyzing these components separately. Can we do that? Yes. We could apply EOF to vector data that have both amplitude and direction. We generate complex input data using east-west component as a real part and north-south component as a imaginary part of them. After that we could apply EOF to these complex data just like in our previous example of complex (time domain) EOF.

We omit examples here because we do not have any good data to demonstrate usefulness of this method.

4 Frequency domain EOF (complex)
4-1 What is it?
In short, the frequency domain EOF is a principal component analysis applied to a matrix of cross-spectral density function. Roughly speaking the cross-spectral density function shows relationships between two time series at different frequencies. The frequency domain EOF is computed separately at each of these frequencies. Thus, this type of EOF allows different phase lag at different frequencies among input time series data unlike complex time domain EOF. This particular feature is quite suited for certain cases.

4-2 Example
We use same data as before except that we did not apply a band-pass filter this time because that is not necessary for this type of EOF. We computed frequency domain EOF at 12 frequency-bands separately and show how the square of amplitude of mode 1 variations distributes in the depth-period space in Figure 7a. The vertical black solid lines indicate center frequency of these frequency-bands.
We converted the result in this figure so that we can compare it directly with power spectral density function (Figure 7b) which is proportional to the square of amplitude of variations of original data at each frequency. We have to apply a window function to cross-spectral density function in order to improve accuracy when we compute frequency domain EOF. This procedure is basically averaging raw cross-spectral density function at each frequency band with those at nearby frequency bands with or without weighting (it resembles to a running mean but should be done in the frequency domain). The window width in this example is three frequency-bands, which is very narrow, to show the fine features. We made a contour of log10 of the result. What it means is that the largest value in this figure (dark brown) is actually 100000 times larger than the smallest value in this figure (dark blue; 10 to (6-1)=5th power=100000). Figure 7b shows that the variations of original time series data are large (red area) at periods longer than 40-day near the surface and at periods between 40 and 80-day with maxima at about 110m, 180m and 230m. The amplitude of the maximum near 110m at about 50-day period (red) is more than 100 times larger than the amplitude at periods longer than 80-day or at periods shorter than 45-day (green).


Comparing figure 7b with 7a we noticed that the pattern of these two figures are quite similar although the amplitudes of mode1 (Figure 7a) are slightly smaller than those of psd (Figure 7b). Adding all the EOF modes (Figure not shown), however, result of EOF becomes almost the same as psd as it should be within the computational error. Figure 7c shows phase of mode 1. Please note that -180 degree (dark blue) is equal to +180 degree (dark brown). Phases at depths from about 70 to 100m and at periods from about 80 to 30-day do not jump from one extreme to another. This figure shows that the contour lines at periods longer than about 30-day are mostly horizontal. This means that the vertical variation of phase is relatively independent on period, and it suggests that the applying complex (time domain) EOF instead of frequency domain EOF to this data set might not be a bad idea.

Figure 7d and 7e show coherency function and phase between mode 1 and east-west component of wind speed, respectively. Figure 7d indicates that the mode 1 of ocean current speed is strongly related with wind speed at periods between 40 and 80-day and at periods shorter than 30-day. Figure 7e indicates that the variations at 40 to 80-day periods between about 100 and 150m are 180 degrees different from (mirror image to) those above and below as suggested by previous examples of complex (time domain) EOF.

Like this we can learn more detailed information about variations regarding to their dominant periods (or frequencies) from frequency domain EOF. We will stop further analysis of these figures because this web page is not aim to publish our research activities.


4-3 The relation between complex (time domain) EOF and frequency domain EOF
The result of complex (time domain) EOF becomes practically the same as the result of frequency domain EOF if we apply a band-pass filter, which allows very narrow frequency range to pass through, to data before computing complex (time domain) EOF. We would like to mention that if we apply a band-pass filter which passes vary narrow frequency range to input data, resultant time series data look like simple sinusoidal variations with varying amplitude (We could play music by applying a band-pass filter to a white noise.). Data such as those are not that much suited to compare with other data visually because they tend to lose "signatures" to compare with.

Here, we computed frequency domain EOF with very wide window width and compared the result of it with that of complex (time domain) EOF instead of computing complex (time domain) EOF at several frequencies and sum them up. We used data at depths only between 40 and 80m. The blue line of Figure 8a shows phase of mode 1 of frequency domain EOF at 34.6-day period.
The window at this period covers from 121-day to 20.2-day in this computation. The red line shows phase of mode 1 of complex (time domain) EOF. The band pass filter we applied to data before computing complex (time domain) EOF allows 121-day to 20.2-day period variations to path through.
Figure 8b shows amplitude of these results. We multiplied amplitude of mode 1 of frequency domain EOF by the number of bands (12) between 121-day and 20.2-day period. We added vertical profile of the variations of original time series data for a comparison (black line). We would say that the results of these two methods are reasonably close.

4-4 Some cautions of using frequency domain EOF
(1) How to compare the results of frequency domain EOF with other time series data that are not included in the computation of frequency domain EOF
If you want to compare results of frequency domain EOF with other time series data, we recommend computing a coherency function as we did in our example. If the frequency range we are interested in is wide, computational time of frequency domain EOF might become painfully long because it should be computed at many frequencies separately. However, if we compute frequency domain EOF only at a few frequencies and generate time series data from them, then these time series data will be vary monotonous and it will be difficult to identify any signatures on them to compare with.

We would think that it would be more straightforward to apply time domain EOFs if what we want to have is a set of new time series data.


(2) The nature of variations might be quite different at different frequencies even if they are the same mode.
This is because we compute frequency domain EOF separately at different frequencies. The vertical black lines in figure 7a shows these frequencies (periods). The result at any frequency is independent on the results at other frequencies if they are outside of "window", the width of which is three in our example. This means that the result of EOF at any frequency band is related to those only at immediate left and right in figure 7a. So, for example, there is no guarantee that the mode 1 at about 100-day period has any relation with the mode 1 at about 40-day period. There is a possibility that the responses caused by the same source might become mode 1 at some frequencies but falls into mode 2 or other modes at other frequencies. Therefore, it needs some caution to interpret figures like figure 7a and 7c.

As a matter of fact, you might have noticed that the coherency function in figure 7d sharply decreases at period slightly shorter than 50 days (blue area centered at about 40-day period). Figure 9a shows coherency function between east-west component of wind speed and original ocean current time series data without applying EOF.
Figure 9b is the coherency function between east-west component of wind speed and mode1 of frequency domain EOF applied to the original ocean current time series data. This is the copy of figure 7d. Figure 9c is the coherency function between east-west component of wind speed and mode 2. The high coherency function area between mode 1 and wind speed is wider than the high coherency function area between original ocean current and wind speed except at about 40-day periods and the area of period shorter than 30 days and of depth deeper than 200m. Figure 9a shows that the ocean current at about 40-day period is still highly coherent with wind speed. Figure 9c indicates that the variations of ocean current highly coherent with wind variations at this period become mode 2 instead of mode 1.

5 Rotary EOF (complex; frequency domain )
In the ocean there are motions that rotate clock-wise or anti-clock-wise. The rotary EOF was used to detect these motions. It uses complex time series, real part of which is east-west component of velocity and the imaginary part of which is north-south component of velocity just like EOF applied to vector data. The next step is generating cross spectral density function matrix from these data analogous to the computation of rotary cross spectral density function. Then you can compute EOF just like ordinary frequency domain EOF. As such, this method is basically for the vector data and rotation is not necessarily required. We omit examples here because we do not have any good data to demonstrate usefulness of this method.
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