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28 This family of operators can be extended by "peeling off" some EMAs with small k: j wsup MAr,ninf,nsup] =  V* EMA[r,fc] "sup  ninf + 1 /1 V k=nM with , 2r r = "sup ""H "inf and with 1 < nmf < nsup. By choosing such a modified MA with > 1, we can generate a lagged operator with a kernel whose rectangularlike form starts after a lag rather than immediately. At the same time, the kernel loses its abrupt behavior at / = 0 and becomes fully continuous, thus reducing noise in the results even further. However, the time delay implied by the lag makes such kernels less attractive for realtime applications. Almost everywhere, a moving average operator can be used instead of a sample average. The sample average of z(t) is defined by E[z] =  / dtz(t) (3.59) where the dependency on starttime ts and endtime te is implicit on the lefthand side. This dependency can be made explicit, for example with the notation E [te  t„; z\Ue), thus demonstrating the parallelism between the sample average and a moving average MA[2r; z](t). The conceptual difference is that when using a sample average, ts and te are fixed, and the sample average is a number (the sample average is a functional from the space of time series to K), whereas the MA operator produces another time series. Keeping this difference in mind, we can replace the sample average E [•] by a moving average MAL] For example, we can construct a standardized time series z (as defined in Section 3.3.1), a moving skewness, or a moving correlation (see the various definitions below). Yet be aware that sample averages and MAs can behave differently, for example E [(z  E [zj)2] = E [z2]  E z]2, whereas MA[(z  MA[z])2] MA[z2]  MA[z2. 3.3.8 Moving Norm, Variance, and Standard Deviation With the efficient moving average operator, we can define the moving norm, moving variance, and moving standard deviation operators: MNormfr, p;z] = MA[r; zp1/p MVarfr, p;z\ = MA[r; z  MA[r; z]\p] (3.60) MSDLr, p\ z.\ = MA[r; z  MA[r; z]\p]Vp The norm and standard deviation are homogeneous of degree 1 with respect to . The pmoment is related to the norm by \xp = MA[zp] = MNorm[z]/. Usually, p = 2 is taken. Lower values for p provide a more robust estimate (see
FIGURE 3.7 A schematic differential kernel. Section 3.3.4), and p = I is another common choice. Yet even lower values can be used, for example, p = 1 /2. In the formulae for MVar and MSD, there are two MA operators with the same range r and the same kernel. This choice is in line with common practice for the calculation of empirical means and variances in the same sample. Yet other choices can be interesting, for example the sample mean can be estimated with a longer time range. 3.3.9 Differential As argued in the introduction, a lownoise differential operator suitable to stochastic processes should compute an "average differential", namely the difference between an average around time "now" over a time interval x\ and an average around time "now  r" on a time interval xi The kernel may look like that in Figure 3.7. Kernels of a similar kind are used for wavelet transforms. This analogy also applies to other kernel forms and is further discussed in Section 3.3.14. Usually, r, x\ and t2 are related and only the r parameter appears, with x\ ~ t2 ~ r/2. The normalization of the differential is chosen so that ; ] 0 for a constant function = c(t) = constant, and [ ; t\ = . Note that our point of view is different from that used in continuoustime stochastic analysis. In continuous time, the limit r > 0 is taken, leading to the Ito derivative with its subtleties. In our case, we keep the range r finite in order to be able to analyze
IGURE 3.8 An example of a differential operator kernel (full line) for r = 1. The lotted curve corresponds to the first two terms of the operator y(EMA[ar, 1] + 3MA[ar, 21), the dashed curve to the last term 2y EMA[ai, 4]. process at different time scales (i.e., for different orders of magnitudes of r). floreover, for financial data, the limit r > 0 cannot be taken because a process s known only on a discrete set of time points (and probably does not exist in ontinuous time). The following operator can be selected as a suitable differential operator: [ ] = (EMA[ar, 1] + EMA[ar, 2]  2 EMA[a/3r, 4]) (3.61) vith = 1.22208, f3 = 0.65 and a"1 = (8/3  3). This operator has a wellshaving kernel that is plotted in Figure 3.8. The value of is fixed so that the integral of the kernel from the origin to the nrst zero is one. The value of a is fixed by the normalization condition and the alue of /3 is chosen in order to get a short tail. The tail can be seen in Figure 3.9. ""his shows that after t = 3.25r, the kernel is smaller than 10~3, which translates nto a small required buildup time of about 4 . In finance, the main purpose of operator is computing returns of a time ,eries of (logarithmic) prices x with a given time interval . Returns are normally lefined as changes of x over r; we prefer the alternative return definition r[z \ = [ ; ]. This computation requires the evaluation of six EMAs and is therefore efficient, timewise and memorywise. An example using our standard week is >iotted in Figure 3.10, demonstrating the low noise level of the differential. The •onventionally computed return r[z](t)  x(t)  x(t  r) is very inefficient to evaluate for inhomogeneous time series. The computation of xt  r) requires high, unbounded number of old f,, x, values to be kept in memory, and the
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