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Phase differences and their interpretation 

Phase differences and their interpretation 

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WaveletComp is an R package for continuous wavelet-based analysis of univariate and bivariate time series. Wavelet functions are implemented in WaveletComp such that a wide range of intermediate and final results are easily accessible. The null hypothesis that there is no (joint) periodicity in the series is tested via p-values obtained from simula...

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Context 1
... absolute value less (larger) than π/2 indicates that the two series move in phase (anti-phase, respectively) referring to the instantaneous time as time origin and at the frequency (or period) in question, while the sign of the phase difference shows which series is the leading one in this relationship. Figure 2 (in the style of a diagram by Aguiar-Conraria and Soares [2]) illustrates the range of possible phase differences and their interpretation. In line with this style, phase differences are displayed as arrows in the image plot of cross-wavelet power. ...
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... 250000 (the latter was not found to have an adverse effect on file sizes). A grayscale can be convenient for black-and-white printout; it is easy to realize in WaveletComp (see Figure 20): ...
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... following code constructs a data frame with all weekdays of 2018 and a time series with weekly (low on Mondays, high on Fridays) and monthly (for simplicity, we let a month have 22 workdays) seasonality: The result is shown in Figure 23. So far, so good. ...
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... far, so good. Now suppose the company is closed for vacation from the beginning of July to mid-August, so that the time series is interrupted, retaining only what is outside of the shaded interval in Figure 22: The code above, with my.data.part substituted for my.data, then leads to the result shown in Figure 24. ...
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... suppose the company is closed for vacation from the beginning of July to mid-August, so that the time series is interrupted, retaining only what is outside of the shaded interval in Figure 22: The code above, with my.data.part substituted for my.data, then leads to the result shown in Figure 24. There are two problems here, due to the data interruption: ...
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... The plot looks as if there were a structural break -although the underlying data-generating model (and even the very realization to be transformed) is the same in Figures 23 and 24. ...
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... The time axis in Figure 24 pretends that data were available throughout July and August. To be sure, fixing the time axis won't make the plot more meaningful. ...
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... is, of course, true for all the tools of statistical analysis.) The idea here is that x and y (see Figure 25) represent hourly observations from a 96-day interval. The time axis labels of Figure 25 simply count through the observations; for certain applications, it may be more intuitive to display days (see also Section 2.7). ...
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... idea here is that x and y (see Figure 25) represent hourly observations from a 96-day interval. The time axis labels of Figure 25 simply count through the observations; for certain applications, it may be more intuitive to display days (see also Section 2.7). Series x and y have joint constant periods 1, 2, 4, 8, 16, but different amplitudes at period 16. ...
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... produces the plot in Figure 26. Horizontal arrows pointing to the right indicate that the two series x and y are in phase at the respective period with vanishing phase differences. ...
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... is also possible to define siglvl as a vector.) Figure 27 confirms that the rectified version of cross-wavelet power gives sound results for all periods. (Omitting rectification would severely underestimate the lower periods' power.) ...
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... time series x shares period 128 with series y within a certain range of time, and y is shifted: The result, which is somewhat counterintuitive, is displayed in Figure 29. Period 128 shows significance across the entire time interval, while one expects period 128 to be jointly important only in the middle, according to the construction of x and y. ...
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... 128 shows significance across the entire time interval, while one expects period 128 to be jointly important only in the middle, according to the construction of x and y. The arrows indicate that x and y are in phase in the middle, with x leading (see also Figure 2), and they tilt away off the middle. This example illustrates the dilemma of the cross-wavelet power (see also the comment in Section 1.2), which corresponds to the covariance - it can be large even if only one component swings widely. ...
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... Limit the area where arrows are drawn to the region where both individual wavelet transforms of x and y show significance (set which.arrow.sig = "wt"), and avoid the artifacts of the image in Figure 29 resulting from the steep power gradient by choosing color.key = "interval": ...
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... computation time, but no arrows indicating phase differences will be plotted.) Figure 32 results again from smoothing with Bartlett windows, but window.size.s = 1 now defines a window of length 101 (an even number will be increased by 1) and produces more blurring for low periods (high frequencies): This leads to less granularity in high frequency areas. ...
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... resulting wavelet power spectrum is shown in Figure 42. A cluttered plot will appear with option plot.waves ...
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... plots in Figure 53 are not quite satisfactory: The three series' wavelet transforms have different maximum powers (as might have been guessed from Figure 52). The plots in Figure 53 thus cannot be directly compared. ...
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... Plot time axes similar (with respect to ticks and labels) to those in Figure 52. ...
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... on period 365 days, the arrows in the cross-wavelet power spectra in Figure 56 can then be interpreted with the help of Figure 2 (and remembering that a full circle corresponds to one year): ...
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... arrows (see also Figure 2) now reveal that both pairs under scrutiny are more or less in sync at the 24-hour period, but there is a very important difference between the pairs at the 12-hour period: Trump pos/neutral is leading over Clinton pos/neutral, while Clinton neg is leading over Trump neg most of the time. It looks like Trump supporters and Clinton opponents were eager to post media, while Trump opponents and Clinton supporters were sluggish. ...
Context 21
... relevant angle series can now be extracted, and we can compute the lead time in minutes: A plot of the lead time (or phase difference) is shown in Figure 62. (Remember that we also need to compute my.wc with my.pair = c("trump.neg", ...

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