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This page explains what the coherency, phase and gain functions are 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. We prepared explanatory pages with some examples for underlined words in blue. If you want to see those pages, please click underlined words in blue below. What are coherency function, phase function and gain function? What do coherency function, phase function and gain function of actual data look like? 

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There are 9620 data and we shift the zero level of time series data at 165E (red) vertically for clarity. It is fairly apparent that these two time series data are quite similar each other. Our attentions tend to focus on variations of periods of a month (720 on horizontal axis) to a few months because they are so obvious in this example but we are going to show how much similar these data are at other periods.  
Figure 2a shows coherency function. The coherency is over 0.8 at long periods (low frequency) but it starts dropping rather fast as period decreases. It becomes 0.5 at about 40day period (dotted black vertical line) and it hits floor at about 20day period. From there it changes rapidly but stays low as period further decreases. Thus, coherency function shows what appears to be highly correlated these two sets of time series are actually highly coherent only at period longer than about one month. Figure 2b shows phase function and horizontal black line indicates 0 degree (no phase/time lag between these data). The phase changes the sign within a range where the coherency is high. This means that the variations of current at 156E lagged behind those at 165E at very long periods but the order changes at about 60day period. However, the phase at period longer than about 25day is not statistically significantly different from zero. The gain factor, figure 2c shows that the amplitude of variations at 165E is smaller than that at 156E at almost entire period range (values are less than 1.0). There is a slight tendency that amplitude of variations at 165E relative to that at 156E increases as period decreases within a range where coherency is high. Finally, figure 2d shows power spectral density function of the current at 156E. It is noted that there are relatively strong energy at periods shorter than 40day period. This suggests that those variations at periods where the coherency was low are not negligible. 

Figure 3a shows time series plots of these current data after applying a lowpass filter, cut off period of which is 40day. As you would have expected, these two sets of time series are quite resemble each other (the maximum lagged correlation coefficient is 0.84) except that shorter period variations of 165E (red) is slightly exaggerated.  
This might be expected from figure 2b. Figure 3b shows time series plots of these current data after applying a bandpass filter, which allows 40 to 2day period variations. Except for first 3000 data, they don't look resemble each other (the maximum lagged correlation coefficient is 0.26) . Even about those first 3000 data, only the long period variations are similar and it might not be a surprise because the coherency is about 0.5 at 40day period. The important point here is that the amplitudes of variations shown in figure 3b are not very small compared with those shown in figure 3a as suggested by figure 2d and, thus, they are not negligible despite of our initial impression after seeing figure 1.  
We hope, this example shows how useful coherency and other related functions are. We would like to mention that the coherence between the current at 156E and at 165E at long periods is unusually high considering the distance between these two position is about 1000km(about 540 nautical miles and 625 land miles). This is a characteristic of near surface current on the equator. If you are interested in this topic, please click here or see the site (NOAA/PMEL) where we downloaded these data. Is coherency function always useful? Yes and no. Since coherency function is computed from cross spectral density function and power spectral density functions, same cautions described in the explanation of power spectral density function are necessary. What can I do if I have more than three sets of time series data and want to analyze the relationships among them? 
