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32 FIGURE 3.14 The kernel wf(f) for the windowed Fourier operator, for n = 8 and = 6. Three aspects of the complex kernel are shown: (I) the envelope (= absolute value), (2) the real part (starting on top), and (3) the imaginary part (starting at zero). Similar to Equation 3.51, we obtain an iterative computational formula for the complex EMA: EMA[f;Z](f„) = ; z](r„ i) + 7£ Zn\ + 77 Zn 1 + ik 1 + ik with a  f (tn  r„ i) fJL = (3.80) where v depends on the chosen interpolation scheme as given by Equation 3.52. We define the (complex) kernel wf(f) of the windowed Fourier transform WF as wf[t,M](f) = ma[r, n](t) ~ / 1 A 1 {t\]~[ n 7 = 1 (71)! \t Y]ema[f, j](t) 7 = 1 (3.81) The kernel is shown in Figure 3.14. Another appropriate name for this operator might be CMA for "complex moving average." The normalization is such that,
for a constant function c(t) = c, /Vwf = WF[£,«; c] =  (1 + ik)J To provide a more convenient real quantity, with the mean of the signal subtracted, we can define a (nonlinear) normed windowed Fourier transform as In Equation 3.82, we are only interested in the amplitude of the measured frequency; by taking the absolute value we have lost information on the phase of the oscillations. Windowed Fourier transforms can be computed for a set of different values to obtain a full spectrum. However, there is an upper limit in the range of computable frequencies. Results are reliable if clearly exceeds the average time interval between ticks. For values smaller than the average tick interval, results become biased and noisy; this sentence applies not only to windowed Fourier transforms but also to most other time series operators. Figure 3.15 shows an example of the normed windowed Fourier transform for the example week. The stock market crash is again nicely spotted as the peak on Tuesday, October 28. Using our computational toolbox of operators, other quantities of interest can be easily derived. For example, we can compute the relative share of a certain frequency in the total volatility. This would mean a volatility correction of the normed windowed Fourier transform. A way to achieve this is to divide NormedWF by a suitable volatility, or to replace z by the standardized time series z in Equation 3.82. 3.4 MICROSCOPIC OPERATORS As discussed in Section 3.1, it is in general better to use macroscopic operators because they are well behaved with respect to the sampling frequency. Some microscopic operators allow the extraction of tickrelated information at the highest possible frequency. An example of such an operator is the microscopic volatility defined later. The computation of the tick frequency requires (by definition) microscopic operators. We also want to extend to inhomogeneous time series the usual operators applied to homogeneous time series, such as the shift operator. NormedWF[£, n; z] = WF[£, n\z]  A/WF [ , n; ]I (3.82) The normalization is chosen so that NorrnedWFl£,n;cl =0
3.4.1 Backward Shift and Time Translation Operators The backward shift operator shifts the value of the time series by one event backward B[z]i = (?;, z/i), but the time associated to each event is not changed. Some authors use the equivalent lag operator L instead. It shifts the time series values but leaves the time part untouched. The inverse operator will shift the series forward. It is well defined for regular and irregular time series. Only for a homogeneous time series spaced by St, this operator is equivalent to a time translation by St (followed by a shift of the time and value series by one event with respect to the irrelevant index i). The operator T translates the time series by St forward T[St; z]i = (tt+St, zi); namely it shifts the time part but leaves the time series values untouched. Note that for an inhomogeneous time series, this operator defines a series with another set of time points. 3.4.2 Regular Time Series Operator From the time series z, irregularly spaced in time, the operator RTS[to, St] constructs an artificial homogeneous time series at times to + kS t, regularly spaced by St, rooted at ?o This involves an interpolation scheme as discussed in Section 3.2.1. Depending on this scheme, the RTS operator can be causal or not. The regular time series can also be constructed as being regular on a given business time scale rather than in physical time. The RTS operator allows us to move from inhomogeneous to the homogeneous time series as presented in Section 3.2.2. For many computations, it is mandatory to
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