Real-Time Filtering and Turning-Point Detection: Application of ...

Real-Time Filtering and Turning-Point Detection: Application of Customized Criteria (DFA) to the US Business Cycle Marc Wildi∗and Simone Elmer†

Abstract Signal-extraction concerns the estimation of interesting components in time series. We use band-pass filters to extract the business cycle - the signal - in US-GDP. Ideally, signal extraction filters are symmetric moving-average filters. Unfortunately, only asymmetric filters can be used in real-time i.e. towards the most recent observation. We apply a new real-time filter technique which is based on a set of ‘customized’ optimization criteria. The customization allows the filter to be adjusted to reflect particular user preferences. To illustrate the usefulness of the customization approach, we examine the problem of real-time measurement of the business cycle (minimum meansquare criterion) as well as the real-time detection of business cycle turning-points (TP-detection criteria). In the latter case, symmetric and asymmetric penalty-functions are introduced which emphasize either synchronicity (coincident indicator) or a prospective perspective (leading indicator). We show how users with different preferences for precision and timeliness can adjust the performance of the real-time filter through suitable parameter-settings.

Keywords. Customized optimization criteria, real-time concurrent filter, time-shift, phase-shift, business cycle, turning point, amplitude, modelmisspecification JEL classification: C32, C61, E32, E37

1

Introduction

While business cycle estimates are of central importance to policy makers and other economic agents, the practical difficulty of their estimation is ∗ Institute of Data Analysis and Process Design, University of Technical Sciences, Rosentrasse 3, 8401 Winterthur, Switzerland, email: [email protected] † KOF Swiss Economic Institute, ETH Zurich, WEH D4, Weinbergstrasse 35, 8092 Zurich, Switzerland, email: [email protected]

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sometimes understated. Orphanides and van Norden (2002) illustrate difficulties encountered in the context of real-time output-gap measures and Wildi (2008b) analyzes real-time trend estimation problems in the context of leading indicators.Widely-used filter-designs such as HP- (Hodrick-Prescott (1997)) or CF- filters (Christiano-Fitzgerald (2003)) rely on simple ‘fixed’ model assumptions when computing real-time signal estimates. In order to extend the scope of their applicability, model-based methods such as X-12ARIMA, X-13, TRAMO-SEATS or STAMP rely on ARIMA- or state-space model-representations of the data generating process (DGP). In these modelbased settings, efficiency of signal-estimates is critically related to the identification of the ‘true’ data generating process. Wildi (2008b), chapters 4, 5 and 6, and Wildi-Schips (2005) have shown that typical unit-root misspecifications can severely affect the efficiency of real-time model-based signal estimates. Moreover, the above approaches target the level of the interesting signal, whereas in practice users are often interested in detecting turningpoints. Therefore, alternative methods have been proposed such as, Markovswitching models1 or Logit-models2 . Besides similar model-misspecification issues - as in the ‘traditional’ model-based approach - one can identify two additional model-specific shortcomings. First, the criteria do not distinguish leads and lags - in practice leads are often beneficial -. Second, user preferences (risk-profile, -aversion) cannot be accounted for explicitly. We here propose a set of optimization criteria - the Direct Filter Approach (DFA) - designed explicitly for real-time signal extraction applications, see Wildi (2008b). The DFA outperformed established linear and nonlinear methods in the context of real-time business-cycle analysis, outputgap measures, recession dating and multi-step ahead forecasting3 , see Wildi (2008b), Sturm/Wildi (2008) (multivariate filter) and Hoenecke/Wildi (2008)(nonlinear real-time filters). Whereas all these methods emphasize traditional mean-square performances we here extend the scope of the method by considering turning-points also. A novelty of our approach is that various ‘riskprofiles’ - user preferences - can be accounted for explicitly by the filter design. Moreover, symmetric and asymmetric penalty functions are proposed which emphasize either synchronicity (coincident indicator) or a prospective perspective (leading indicator). In the latter case, ‘leads’ (false antic1 See for example Hamilton (1989), (1994), Smith and Summers (2002), Mc Culloch and Tsay (1994), Harding and Pagan (2002), Koskinen and Oeller (2004), Anas et al. (2004), Krolzig (1997), (2003), Diebold and Rudebusch (2005), Chauvet and Piger (2003), (2005), (2007). 2 See for example Birchenhall et al. (1999), Chin et al. (2000), Sensier et al. (2004), Estrella and Mishkin (1997), (1998), Lamy (1997) and Filardo (2004). 3 The DFA won the recent NN3- (2007) and NN5- (2008) forecasting competitions and outperformed, among others, winner and runner-up of the prestigious M3 competition, see http://www.neural-forecasting-competition.com/NN3/results.htm and http://www.neural-forecasting-competition.com/results.htm.

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ipations of turning-points) are ignored. Both criteria account for one- and multi-step ahead performances simultaneously. Therefore, typical modelmisspecification issues (false unit-roots) and overfitting are alleviated, see Wildi (2008b), chapters 7 and 8. Section 2 introduces the DFA by distinguishing mean-square and turningpoint criteria. In the latter case, we propose criteria for coincident as well as leading indicators. Section 3 presents empirical results based on US- as well as Euro-zone data. Our main findings are summarized in section 4.

2

Direct Filter Approach

Due to space consideration, we here present the DFA, first introduced in Wildi (1998), in a concise, informal and intuitive way. We refer the reader interested in formal technical results to Wildi (2004), (2008a) and (2008b).

2.1

Level Criteria

Real-time signal extraction is about an optimal approximation of a predefined symmetric filter - an interesting signal - by an asymmetric filter that allows to estimate the signal towards the last available data point. Assume we wish to estimate a particular signal Yt Yt =

∞ X

γk Xt−k

(1)

k=−∞

based on a (doubly-infinite) time series Xt . In practice a finite sample X1 , ..., XT of Xt is available only. Moreover, one is often interested in estimating the signal towards the last data point t = T (real-time estimation problem). Therefore, filter coefficients γˆk , k = 0, ..., T − 1 must be computed such that the real-time estimate YˆT =

TX −1

γˆk XT −k

(2)

k=0

minimizes the mean-square filter error E[(YT − YˆT )2 ] → min

(3)

The latter expression is sometimes called revision error variance. In a traditional time-domain perspective, the estimation problem may be solved as follows YˆT =

−1 X

k=−∞

ˆ T −k + γk X

TX −1

γk XT −k +

0

3

∞ X T

ˆ T −k γk X

(4)

where missing observations are replaced by back- and forecasts of the data, see Cleveland (1972) and Bell (1984). X-12-ARIMA and TRAMO/SEATS for example rely on (Reg-)ARIMA-forecasting models whereas STAMP relies on state-space models. The correspondence with (2) is established by noting that back- and forecasts are typically linear functions of the data. The main problem of this ‘traditional’ approach is that model-parameters are optimized with respect to short-term (one-step ahead) forecasting performances - whereas a whole bunch of multi-step ahead forecasts enters into (4) - and that important model-misspecifications cannot be detected (see Wildi (2008b), chapters 3 and 4, for a comprehensive account on these topics and for comparisons of the above approaches with the DFA). An alternative approach is based on minimizing (3) ‘directly’ as a function of the filter coefficients (DFA). By doing so, one- and multi-step ahead forecasts are addressed simultaneously. Unfortunately, Yt in (3) as well as the theoretical expectation are generally unknown. Therefore, the expectation has to be replaced by an estimate. The main idea, first introduced in Wildi (1998), is to find an efficient estimate of the expected squared filter error. Assume, first, that Xt is a stationary process and consider the following sequence of approximations: E[(Yt − Yˆt )2 ] = =

T 1X (Yt − Yˆt )2 + eT T t=1

2π T

(5)

T /2

X

ˆ k )|2 IT X (ωk ) + eT + e′ |Γ(ωk ) − Γ(ω T

(6)

k=−T /2

ˆ k ) is where IT X (ωk ) is the periodogram of Xt and ωk = k2π/T and Γ(ω the transfer function of the (asymmetric) real-time filter. Wildi (2008b) (univariate case) and (2008a) (multivariate case) establish that eT , in the first stage, is asymptotically smallest possible (efficiency) and that e′T , in the second stage, is of smaller magnitude than eT (superconsistency) under suitable regularity assumptions. Note that whereas both expressions in (5) are generally unobservable (Yt is the output of a bi-infinite filter) the frequency-domain statistic in (6) is observable. Moreover, one can show see the above references - that its advantageous statistical properties apply uniformly under the same regularity assumptions (which can be very easily implemented in practice). Therefore, ‘best’ mean-square level filters can be derived from the DFA-criterion 2π T

T /2

X

ˆ k )|2 IT X (ωk ) → min |Γ(ωk ) − Γ(ω

(7)

k=−T /2

Efficiency here means that the solution of (7) minimizes a superconsistent estimate of an efficient estimate of the mean-square filter error (3). It seems 4

reasonable to attribute part of the observed performance gains in applications to the immanent efficiency concept of the DFA. Generalizations to non-stationary processes are straightforward, see Wildi (2008b) chapter 6 (univariate filtering and unit-roots) and Wildi (2008a) (multivariate filtering and cointegration). ˆ is the transfer function of a particular ARMAIn our applications, Γ(·) filter Yˆt =

Q X

ak Yˆt−k +

k=1

q X

bk Xt−k

(8)

k=−r

The real-time property is obtained by setting r = 0. Filter parameters obtained for a fast filter turning-point filter are reported in table 3 in the appendix. It is to be noted that we do not explicitly optimize the ‘timedomain’ AR- and MA-parameters but that we use a set of parsimonious frequency-domain parameters (so-called ZPC-filters) that match the salient features of the data in the frequency-domain, namely location, height and width of the dominant spectral peaks, see the ZPC-filter design in the appendix. We conclude our presentation of the level criterion by picking-up a ‘frequently asked question’ addressing the inconsistency of the spectral estimate, the periodogram, in criterion (7). Besides formal technical proofs - which are out of the scope of this article - we may provide two informal explanations justifying its use in this context. First, under fairly general assumptions, the periodogram is a sufficient statistic for inferring the frequency of a cycle, see Bretthorst (1988) chap. 6. Therefore, the DGP enters into the optimization via a sufficient statistic in the frequency-domain. Second, the periodogram ˆ k )|2 is implicitly smoothed in the above expression (7) because |Γ(ωk ) − Γ(ω is a smooth - often infinitely differentiable - function. Consider that we do not want to estimate the spectral density at a particular frequency - in which case the periodogram would be unsuitable - but instead we are interested in estimating the mean-square filter error which is a weighted integral of the spectral density. Now the central limit theorem ensures that the idiosyncratic error component in the periodogram is smoothed out in the above sum (discrete integral). Based on extensive experience, we clearly disadvise the use of non-parametric (smoothed periodogram) estimates or the use of ‘exotic’ spectral estimates (for example Hilbert spectrum) in conjunction with the DFA-criterion (7).

2.2

Turning-Point Criterion

We now leave the mean-square ‘scenery’ and propose a generalization of (7) which emphasizes turning-points by pointing towards speed and reliabil-

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ity of the real-time filter.For this purpose, consider the following geometric decomposition of (7): 2π T

T /2

X

ˆ k )|2 IT X (ωk ) |Γ(ωk ) − Γ(ω

k=−T /2

2π = T

T /2

X

ˆ k ))2 W (ωk )IT X (ωk ) (Γ(ωk ) − A(ω

k=−T /2 T /2

+

h  i X 2π ˆ k ) 1 − cos Φ(ω ˆ k ) W (ωk )IT X (ωk ) 2Γ(ωk )A(ω λ T k=−T /2

(9)

→ min ˆ k ) and Φ(ω ˆ k ) are amplitude and phase functions of the real-time where A(ω filter and where λ = 1 and W (·) ≡ 1. It is shown in Wildi (2008b), section 3.1.3, that the first summand corresponds to that portion of the total mean-square filter error which is attributable to the (necessarily) imperfect amplitude function of the real time filter. The second summand then summarizes effects due to the non-vanishing phase of the real-time filter. The particular setting λ = 1 and W (·) ≡ 1 reproduces the previous mean-square optimization criterion. However, the additional degrees of freedom provided by λ and W (·) ≥ 0 enable a powerful extension of the DFA: increasing λ emphasizes the error due to the time shift and W (·) can be used to emphasize stop-band properties of the filter. Therefore, we are able to control for ‘speed’ - trough λ - and ‘reliability’ - through W (·) - of the real-time filter simultaneously. As always, there is a price to be paid for that additional flexibility: strongly emphasizing stopband properties and heavily weighting time shifts in the passband induce distortions in the amplitude function in the passband (because the latter is not emphasized by artificial weights). As a consequence, the solutions of the above turning-point criterion are no more optimal in a mean-square sense: mean-square level performances and the ability to detect turning-points fast and reliably are to some extent incongruent criteria. This obvious fact is generally ignored by users relying on traditional methods that emphasize mean-square performances. Note that the penalty-function in (9) is symmetric i.e. delays and lags are penalized equally. Such a design would match the framework of coincident indicators. In the prospective perspective conveyed by leading indicators, anticipations would be welcome. We therefore modify the above criterion

6

such that delays only are penalized 2π T

T /2 X

ˆ k ))2 W (ωk )IT X (ωk ) (Γ(ωk ) − A(ω

k=−T /2 T /2

+

h  i X 2π ˆ k ) 1 − cos Φ(ω ˆ k )I ˆ λ 2Γ(ωk )A(ω W (ωk )IT X (ωk ) {Φ(ωk )>0} T k=−T /2

→ min

(10)

where I{Φ(ω ˆ k )>0} is the ordinary indicator function. Criterion (10) emphasizes explicitly a purely prospective perspective (which justified an extensive application in the framework of recent forecasting competitions).

3

Empirical Results

The following experimental framework relies on a design in Wildi (2008a) and Wildi/Sturm (2008). The authors analyze business-cycles in US-GDP based on a CF-filter with a band-pass between 5.8 and 38 quarters4 . The filter is applied to the (de-trended and log-transformed) US-GDP available from the first quarter of 1967 to the second quarter of 2005.

3.1

Level Filters

The real-time DFA-level filter is based on criterion (7) and the real-time CF-filter is based on (4) using the common random-walk hypothesis, see Christiano and Fitzgerald (2003) and Valle e Azevedo (2008). Real-time filter outputs and signal are plotted in fig.2. As can be seen, the real-time CF-filter performs comparatively poorly between 1987 and 1992 as well as from 1995 to 2000 where the dynamics of the signal are missed. Meansquare performances based on the whole sample are reported in table 1. For sake of comparison we included a new efficient multivariate MDFA design proposed in Wildi/Sturm (2008): this design performs better with respect the MSE-measure (because it relies on additional explanatory variables) but it cannot account explicitly for time-shift issues, see the following section. The two time spans (72-02) and (88-02) are taken from Valle e Azevedo. Out of sample results are reported in table 2. In the latter case, uni- and multivariate DFA-filters - periodograms - are computed on the time span from Q1-1967 to Q4-1987 (21 years). 4 This setting is very close to the ‘standard’ definition 6-32 quarters and it allows the boundaries of the bandpass in the frequency-domain to coincide with the discrete frequency ordinates ω4 and ω26 in 7 which favors a tighter approximation. We refer the reader interested in robustness issues (variations of the band-pass definition) to Wildi/Sturm (2008), section 5.

7

HP CF DFA MDFA

MSE (72-02) 0.320 0.298 0.254 0.186

MSE(88-02) 0.141 0.124 0.057 0.034

Table 1: Mean-square error in-sample HP CF DFA MDFA

MSE(88-02) 0.141 0.124 0.071 0.041

Table 2: Mean-square error out-of-sample Amplitude and time shift functions can be seen in figs. 3 and 4. Note that the negative time shift - the apparent anticipation-ability - of the DFA is obtained by a MA-unit-root (difference) in the real-time bandpass-filter that generates a vanishing amplitude function in frequency zero.

3.2

Turning-Point Filters

The following ‘fast’ turning-point filter is based on data from Q1-1967 to Q41987 and it relies on criterion (10) with λ := 15 and W (ωk ) := exp(2 ∗ k). Smoothing in the stop-band and small time delays in the pass-band are strongly emphasized (at costs of distortions of the amplitude function in the pass-band), see figs. 9 and 10. Accordingly, mean-square performances are poor. Note that the negative time-shift of the ‘fast’ TP-DFA filter suggests the ability to foresee turning-points. This statement is confirmed by the filter outputs in figs. 5 (TP-DFA and signal). Figs. 6 (TP-DFA vs. levelDFA), 7 (TP-DFA vs. CF) and 8 (TP-DFA vs. new multivariate DFA) - for ease of visual inspection we standardized all series -. As expected, the noiseattenuation obtained by the TP-filter in the stop-band (as emphasized by W (·)) is more pronounced than for the optimal mean-square level filter, see fig. 9. Also, the time-shift in the passband is smaller since it is everywhere negative: this is a direct consequence of the asymmetry introduced in (10). Next, we would like to account for different ‘user-preferences’, i.e. a higher risk-aversion. For that purpose we weaken the strong speed-constraint entailed by λ = 15 in the previous example. The output of the resulting filter will be smoother and more reliable but slightly delayed. To be more specific we set λ = 3 and W (ωk ) = exp(3k) in (10). Filter outputs of ‘fast’ 8

and ‘reliable’ DFA-TP designs are plotted in fig. 11. As can be seen, the output of the new filter is smoother but slightly delayed, as confirmed by amplitude and time shift-functions in figs. 12 and 13.

4

Conclusion

The novelty of the proposed DFA lies in the ability to solve important realtime estimation problems efficiently in various user-relevant frameworks. This user-specific perspective is obtained through a customization of the underlying optimization criteria. The DFA optimization criteria (7), (9) and (10) address mean-square level approximation, synchronous turningpoint detection (coincident indicator) and prospective turning-point detection (leading indicator). They allow for general time series dynamics including stationary and non-stationary integrated processes (see Wildi (2008b), chap.6). Empirical examples based on real-time band-pass filtering of USGDP data illustrate efficiency gains of the DFA when compared to the wellknown CF- and HP-filters. The dedicated DFA-criteria outperform the competing (real-time) filters both in terms of mean-square performances and in the ability to date turning-points fast and reliably. The new multivariate filter (MDFA) outperforms the DFA with regards to mean-square performances but the univariate TP-filter is faster than the multivariate filter. Therefore, we would like to combine the univariate TPcriteria (9) (coincident) and (10) (leading) with the generalized multivariate (mean-square) criterion in Wildi (2008a) and Wildi/Sturm (2008) in future work.

Appendix Filter Design: ZPC-Filters We now briefly present the particular ARMA-ZPC filter design used in our method and the relevant filter constraints. Technical details are to be found in Wildi (2008b), section 3.2. Consider the following recursive ARMA-filter equation ˆt = X

Q X

ˆ t−k + ak X

k=1

q X

bk Xt−k

(11)

k=−r

For real-time filters r = 0. Stability means that the roots of the characterisP k tic AR-polynomial 1 − Q k=1 ak z lie outside the unit circle. Invertibility or, equivalently, the minimum-phase property is achieved by requiring all zeroes of the characteristic MA-polynomial to lie outside the unit circle. The 9

transfer function of the ARMA-filter 11 is given by ˆ Γ(ω) = =

Pq

k=−r bk exp(−ikω) P 1− Q k=1 ak exp(−ikω) Qn j=1 (Z2j−1 − exp(−iω))(Z2j C exp(irω) Qn′ k=1 (P2k−1 − exp(−iω))(P2k Qq+r j=2n+1 (Zj − exp(−iω)) · QQ k=2n′ +1 (Pk − exp(−iω))

− exp(−iω)) − exp(−iω))

where Z2j := Z¯2j−1 , j = 1, ..., n are complex conjugate zeroes, P2k := P¯2k−1 are complex conjugate poles, Zj , j = 2n + 1, ..., q + r are real zeroes, Pk , k = 2n′ + 1, ..., Q are real poles and QQ

Pk j=1 Zj

C := b−r Qk=1 q+r

We now consider a particular restriction linking zeroes and poles of the above filter. The resulting design is called Zero-Pole-Combination (ZPC) filter. The following ARMA(1,1)-transfer function Z − exp(−iω) ˆ Γ(ω) =C P − exp(−iω) is called an elementary ZPC-filter, if arg(Z) = arg(P ). An ARMA(p, p)-filter ˆ is called ZPC-filter if poles Pk , k = 1, ..., p and zeroes Zj , j = 1, ..., p can Γ(·) be grouped into pairs (Zk , Pj(k) ) defining elementary ZPC-filters, where j(k) is a suitable (bijective) renumbering of the poles. In order to illustrate the purpose of the proposed restriction (ZPC design) amplitude and time shift functions of three real ARMA(2,2)-ZPC filters (Z − exp(−iω))(Z − exp(−iω)) (P − exp(−iω))(P − exp(−iω)) (where thus arg(Z) = arg(P )) are plotted in fig. 1. Proposition 3.9 and theorem 3.10 in Wildi (2004) show that the amplitude function of an elementary ZPC-filter has a unique extremum in the common argument −λ of zero and pole (π/4 and π/2 in the above examples5 ). The extremum is a maximum if the pole is closer to the unit circle (shaded line in fig. 1, left 5

The location of the extrema of the composed real filter can slightly deviate because conjugate and non-conjugate pairs interact. However, this potential ‘interference’ is negligible in most applications.

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panel), otherwise it is a minimum (solid and dotted lines). The amplitude function in λ is |Z| exp(−iλ) − exp(−iλ) |Z| − 1 |P | exp(−iλ) − exp(−iλ) = |P | − 1

Therefore, the ratio (|Z| − 1)/(|P | − 1) controls the height of the extremum: in the above examples 0.5 (solid line), 0.05 (dotted line) and 1.2 (shaded line)6 . Finally, given the location λ and the extremal value (|Z|−1)/(|P |−1), the width of the trough (peak) can be controlled by the ratio |Z|/|P |: for |Z|/|P | → 1 the trough (peak) almost disappears due to canceling zeroes and poles. In the above example, we measure 0.9166 (solid line), 0.9814 (dotted line) and 1.0666 for the corresponding ratios. Due to their straightforward interpretability, we have adopted the transformed parameter set p1 := λ, p2 := (|Z| − 1)/(|P | − 1) and p3 := |Z|/|P |. ‘Traditional’ moduli of zeroes and poles are derived from the latter two parameters. The transformed parameter space (p1 , p2 , p3 ) of real ARMA(2,2)ZPC-filters is designed specifically for matching the location, the height and the width of particular (dominant) spectral peaks. Therefore, the parsimony concept inherent to ARMA-designs becomes strengthened by a constraint (the common argument of ZPC-filters) which emphasizes interpretability of parameters.

Filter-parameters We here report AR- and MA-parameters of the fast turning-point filter. The design of the level filter is slightly different, see Wildi/Sturm (2008).

6 These values slightly differ from the plotted extrema because the latter are computed for the real ARMA(2,2)-filter: the complex conjugate ZPC-filters disturb marginally the results.

11

0 1 2 3 4 5 6 7 8

AR 1.000000e+00 -3.178092e+00 4.593060e+00 -3.735145e+00 1.624444e+00 -2.685037e-01 1.756150e-02 -4.109931e-04 7.024016e-06

MA 1.146403e-01 -1.778080e-01 9.199430e-02 -1.595269e-02 1.073855e-03 -2.635328e-05 4.653549e-07 -7.627756e-10 9.800258e-13

Table 3: AR- and MA-parameters of the fast real-time DFA-TP filter

Graphs

Time Shifts ZPC filters

0.0

0.2

−0.5

0.4

0.6

0.0

0.8

1.0

0.5

1.2

Amplitude ZPC filters

0

pi/6

2pi/6

3pi/6

4pi/6

5pi/6

pi

0

pi/6

2pi/6

3pi/6

4pi/6

5pi/6

pi

Figure 1: Amplitude and time shift functions of ZPC-filters: Z = 1.1 exp(−iπ/4), P = 1.2 exp(−iπ/4) (solid line) and Z = 1.001 exp(−iπ/4), P = 1.02 exp(−iπ/4) (dotted line) and Z = 1.6 exp(−iπ/2), P = 1.5 exp(−iπ/2) (shaded line)

12

0 −1 −2 1971

1975

1979

1983

1987

1991

1995

1999

Time−Span Signal

DFA

CF

Figure 2: Signal and Real-Time Filter Outputs

0

0.5

1

Amplitude Function Univariate

Levels

Levels

1

DFA vs. CF Real−Time Filters

0

pi/2

Band−Pass

pi

Spectrum

DFA

Figure 3: Amplitude Function Univariate Level-Filter

13

−4

Levels

−2

0

Time−Shift Function Univariate

0

pi/2

pi

Band−Pass

DFA

Figure 4: Time-Shift Function Univariate Level-Filter

14

−1

0

1

2

Signal (green) and DFA−TP (black)

1972

1974

1976

1977

1979

1981

1983

−2 −1

0

1

2

Signal (green) and DFA−TP (black)

1983

1987

1990

1994

1998

Figure 5: Signal and Real-Time TP-DFA

15

2001

2005

−1

0

1

2

Signal (green) and TP (black) vs. DFA−level (red)

1972

1974

1976

1977

1979

1981

1983

−2

−1

0

1

2

Signal (green) and TP (black) vs. DFA−level (red)

1983

1987

1990

1994

1998

2001

Figure 6: Real Time Filters: DFA-TP vs. DFA-Level

16

2005

−1

0

1

2

Signal (green) and TP (black) vs. CF (red)

1972

1974

1976

1977

1979

1981

1983

−2

0

1

2

Signal (green) and TP (black) vs. CF (red)

1983

1987

1990

1994

1998

Figure 7: Real Time Filters: DFA-TP vs. CF

17

2001

2005

−2

−1

0

1

2

Signal (green), DFA−TP (black) and Multivariate DFA (red)

1972

1974

1976

1977

1979

1981

1983

−2

0

1

2

Signal (green), DFA−TP (black) and Multivariate DFA (red)

1983

1987

1990

1994

1998

2001

Figure 8: Real Time Filters: DFA-TP vs. Multivariate DFA

18

2005

Levels

0

0.5

1

Amplitude Function Univariate

0

pi/2

Band−Pass

pi

Spectrum

Level−DFA

TP−DFA

Figure 9: Amplitude Function: TP vs. Level

−4

Levels

−2

0

Time−Shift Function Univariate

0

pi/2

Band−Pass

pi

Level−DFA

TP−DFA

Figure 10: Time-Shift Function: TP vs. Level

19

−1

0

1

2

Signal (green), TP Fast (black) vs. TP−Reliable (red)

1972

1974

1976

1977

1979

1981

1983

−2

0

1

2

3

Signal (green), TP Fast (black) vs. TP−Reliable (red)

1983

1987

1990

1994

1998

Figure 11: Reliable and Fast TP-DFA

20

2001

2005

0

0.5

Levels

1

1.5

Amplitude Functions

0

pi/2

Band−Pass

pi

Spectrum

TP−Reliable

TP−Fast

Figure 12: Amplitude Function: Reliable and Fast TP-DFA

0 −2

Levels

2

Time−Shift Functions

0

pi/2

Band−Pass

pi

TP−Reliable

TP−Fast

Figure 13: Time-Shift Function: Reliable and Fast TP-DFA

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