Wavelet Functional Data Analysis for FANOVA Models under Dependent Errors
WWavelet Functional Data Analysis for FANOVA Models underDependent Errors
Airton Kist (1) , Alu´ısio Pinheiro (2) ∗ (1) University of Ponta Grossa, Brazil, (2)
University of Campinas, Brazil
Abstract
We extend the wavelet tests for fixed effects FANOVA models with iid errors, proposed inAbramovich et al, 2004 to FANOVA models with dependent errors and provide an iterativeCochrane-Orcutt type procedure to estimate the parameters and the functional. The functionis estimated through a nonlinear wavelet estimator. Nonparametric tests based on the optimalperformance of nonlinear wavelet estimators are also proposed. The method is illustrated on realdata sets and in simulated studies. The simulation also addresses the test performance underrealistic sample sizes.Keywords: Cochrane-Orcut; Functional Data Analysis; Nonparametric Test; Nonparametric In-ference.MSC2000: primary-62G10; secondary-62G20;
Consider the Ornstein-Uhlenbeck diffusion process as follows: dy ( t ) = ρ c y ( t ) dt + σdW ( t ); y = b, t > , (1)where ρ c and σ > { W ( t ) : t ≥ } is a standard Brownianmotion. The unique solution for { y ( t ) : t ≥ } in the mean square sense (Arnold, 1974) is givenby y t = e ρ c t b + σ (cid:90) t e ρ c ( t − s ) dW ( s ) = e ρ c t b + σJ ρ c ( t ) , (2)where J ρ c ( t ) = (cid:82) t e ρ c ( t − s ) dW ( s ). Note that J ρ c ( t ) ∼ N(0 , ( e ρ c t − / ρ c ).Suppose one is interested in estimating ρ c based on a single observed path { y ( t ) } ≤ t ≤ T . Themean squared estimator of ρ is given byˆ ρ c = (cid:90) T y ( t ) dy ( t ) (cid:46) (cid:90) T y ( t ) dt. (3)which is also the unrestricted maximum likelihood estimator if b = 0.The discrete representation of such process is given by y t,h = e ρ c h y ( t − ,h + u t,h , t > , y = b, (4)where u t,h ∼ N(0 , σ ( e ρ c h − / ρ c ) and h is the sampling interval.In this discretized version of the problem, one aims to estimate ρ h = e ρ c h given the observations { y t,h } nt =0 , where n = T /h . The minimum squared error estimator, which is also the conditional(on y ) maximum likelihood solution, is given byˆ ρ h = n (cid:88) t =1 y t,h y ( t − ,h (cid:46) n (cid:88) t =1 y t − ,h . ∗ corresponding author: [email protected] a r X i v : . [ s t a t . M E ] O c t erron (1991) studies the limit distribution of n (ˆ ρ h − ρ h ) under (4), when h → T ,and proves that it is identical to the limiting distribution of T (ˆ ρ c − ρ c ) under (1). Suppose one isinterested in estimating and testing functions f in the model defined by: dy i ( t ) = f i ( t ) dt + ε i ( t ) dt, t ∈ [0 , , i = 1 , . . . , r, (5)where r is the number of curves being compared and { ε ( t ) : t ≥ } is a CAR(1) model. Theaforementioned CAR(1) is also called the Ornstein-Uhlenbeck process and is the only stationarysolution of the PDE dε ( t ) + αε ( t ) dt = σdW ( t ) , (6)where α and σ are unknown positive parameters. Note that E [ ε ( t ) | ε (0)] = e − αt ε (0) (7)Var[ ε ( t ) | ε (0)] = σ α (1 − e − αt ) . (8)Moreover, { ε ( t ) } can be written as ε ( t ) = e − αt ε (0) + σ (cid:90) t e − α ( t − s ) dW ( s ) = e − αt ε (0) + σJ α ( t )where J α ( t ) = (cid:90) t e − α ( t − s ) dW ( s ), which is distributed as N(0 , (1 − e − αt ) / α ). This solution isstationary, so that α > ε ( t ) ∼ N(0 , σ / α ) ∀ t ≥
0. Consequently, E [ y i ( t ) | f i ( t ) , σ , α ] = f i ( t ) , Var[ y i ( t ) | f i ( t ) , σ , α ] = σ / (2 α ) andCorr[ y i ( s ) , y i ( t ) | f i ( t ) , σ , α ] = e − α ( t − s ) , for s < t. Suppose the model defined by (5). If the sample on the CAR(1) is equally spaced, i.e. such that h = 1 /n one can write ρ = ρ ( α, h ) = e − αh . Given the discrete process variance σ p , one can write σ p = σ / (2 α ), and σ p (1 − e − αh ) = σ p (1 − ρ ).Following (4), we discretize the model as y i,t = f i,t + ε i,t , i = 1 , . . . r, (9)where ε i,t = ρε i,t − + u i,t are independent CAR(1) processes, i = 1 , . . . , r , and the u i,t ∼ (0 , σ u )uncorrelated with ε i,s , s < t for each i . The Fisher information matrix for this model is given by: I ( f t , ρ, σ u ) = σ u (cid:0) − ρ + ( n − − ρ ) (cid:1) n − − n ) ρ (1 − ρ ) σ u (1 − ρ )0 1 σ u (1 − ρ ) n σ u , which, being block-diagonal, justifies applying diverse methods for f t and ( ρ, σ u ). For instance,we estimate the former by wavelets, and the latter by ML.The procedure can be resumed as follows:(E1) Initial solution for ρ : ˆ ρ ∈ ( − , y t − ˆ ρy t − = f t − ˆ ρf t − + u t , i.e., z t = g t + u t , and estimate g ( g t ) by ˆ g (ˆ g t );(E3) Estimate f by ˆ f t = ˆ g t + ˆ ρ ˆ f t − , with ˆ f = y ;(E4) Estimate ρ by ˆ ρ = (cid:80) nt =2 e t e t − / (cid:80) nt =2 e t , where e t = y t − ˆ f t ;(E5) Testing convergence by subsequent estimated values of ρ ; E6) Repete steps (2)-(5) until convergence is attained, or the maximum number of iterations isreached.
Remarks:
1. In step (2) one may estimate g linearly on nonlinearly.2. The variance σ u must be estimated in each iteration if non-linear wavelet estimators areused in (2). Otherwise, one only needs it at the end of the process. In the simulation studiesand in the application, we compared MAD and STD estimates for σ u (Vidakovic, 1999 pp.196-7).It is known that high-dimensional models, and functional models as well, pose a problem forthe classical criteria such as Neyman optimality. For that reason some shrinkage must be appliedin order to get statistically sound solutions. In the HANOVA setup, Fan (1996) and Fan and Lin(1998) present adaptive Neyman tests for high-dimensional parameters and curves, respectively.Abramovich et al. (2004) proposes a FANOVA model for dy i ( t ) = f i ( t ) dt + ε i ( t ) dt, t ∈ [0 , , i = 1 , . . . , r, (10)where r is the number of curves being compared and { ε ( t ) : t ≥ } is a Brownian motion. (10)and (5) may be seen as equivalent models except for the error structure.The FANOVA decomposition is given by f i ( t ) = m + µ ( t ) + a i + γ i ( t ) , i = 1 , . . . , r ; t ∈ [0 ,
1] (11)where: m is the overall mean; µ ( t ) is the main effect in t ; a i is the main effect in i ; γ i ( t ) is theinteraction between i e t , with the following identificability conditions (cid:90) µ ( t ) dt = 0; r (cid:88) i =1 a i = 0; r (cid:88) i =1 γ i ( t ) = 0; (cid:90) γ i ( t ) dt = 0 , ∀ i = 1 . . . . , r ; t ∈ [0 , . (12)One would be interested in hypotheses such as:H : µ ( t ) ≡ , t ∈ [0 , : a i = 0 , i = 1 , . . . , r ; and (14)H : γ i ( t ) ≡ , i = 1 , . . . , r t ∈ [0 , . (15)While (14) can be treated as the usual parametric hypotheses, (13) (15) are intrinsically func-tional. Donoho and Johnstone (1995, 1998) and Spokoiny (1996) present optimal minimax ratesin Besov spaces which are attained by wavelet procedures. For a significance level α ∈ (0 ,
1) let φ ∗ be the test defined by φ ∗ = (cid:40) { T ( j ( s )) > v ( j ( s )) z − α } if p ≥
2; or (cid:110) T ( j ( s )) + Q ( j ( s )) > (cid:112) v ( j ( s )) + w ( j ( s )) z − α (cid:111) if 1 ≤ p < , (16)where p , q , s and C are considered known. We refer the readers to Abramovich et al. (2004) fordetails. This test is proven to be optimal in the minimax sense. The non-adaptative test alsoproposed by Abramovich et al. (2004), which is optimal as well, is given by φ ∗ η = 1 if: φ ∗ η = max j min ≤ j ( s ) ≤ j η − (cid:40) T ( j ( s )) + Q ( j ( s )) (cid:112) v ( j ( s )) + w ( j ( s )) (cid:41) > (cid:112) η − . If one knows that p ≥
2, then φ ∗ η = 1 if φ ∗ η = max j min ≤ j ( s ) ≤ j η − (cid:40) T ( j ( s )) (cid:112) v ( j ( s )) (cid:41) > (cid:112) η − . ne assumes above that the f i ( t ) belong to a Besov ball of radius C >
0, in [0 , B p,qs ( C ),where s > /p and 1 ≤ p, q ≤ ∞ .For the dependent error model (5), a two-step iterative procedure is employed, where in eachiteration ρ and f are successively estimated by a Cochrane-Orkutt type procedure, specified in(E1)-(E6), employing the estimation (and testing) procedures proposed by Abramovich et al.(2004) for f .One can then prove that reasonable error norms are minimized by this procedure. In particular,it can be proven that the L error norm is improved in each iteration. We present simulation studies to evaluate the testing procedure. Twelve classical test functions(Figure 1), five sample sizes ( n = 512; 1024; 2048; 4096; 8192), three signal-to-noise ratios (SNR =1; 3; 7), two values of ρ (0 .
99 and 0 . Figure 1: Test functions
Since α = − n log ρ one has for ρ = 0 .
99 and n = 512; 1024; 2048; 4096; 8192, α = 5 .
14; 10 .
29; 20 .
58; 41 .
16; 82 . ρ = 0 . α = 0 . . . . . V for n = 512, V for n = 1024; 2048 and V for n = 4096; 8192 are employed. For the latter, thresholding in the levels4 − n = 512; 1024; 2048 and 5 − n = 4096; 8192 are performed. In the last iteration, after ρ is estimated, the function is estimated by thresholding either term-by-term (Spokoiny, 1996;Abramovich et al., 2004) or by blocks (Cai, 1999, 2002).Data was generated by the model y t = f t + ε t , t = 1 , . . . , n, (17)where ε t = ρε t − + u t is a discretized CAR(1), ρ = e − α/n , u t ∼ N(0 , (1 − ρ ) σ / α ), ε ∼ N(0 , σ / α ) and f t a test function.For each combination of ρ and n , one uses σ = 1 and rescales the test function to get thedesired SNR. In each estimation procedure, 50 randomly selected values in ( − ,
1) are used asinitial values of ˆ ρ . MAD estimates of σ u are employed for the non-linear estimates. The stoppingcriterium was a difference on the subsequent ρ estimates smaller than < − or 250 iterations. smaller preliminary simulation was performed with the single aim of assessing the real needof estimating ρ . Table 1 shows the performance of the estimators of f as a function of the em-ployed value of ρ . 1000 replications onf each combination of f ( t ) = sin(2 πt ), n = 1025, SNR = 7and ρ = 0 .
99; 0 . . f estimation. The iterationswere made with thresholding from levels 4 to 7. Moreover, to show the effects of ignoring thedependence on ε , twelve estimators are compared. Eleven fixed values for ρ are considered: − . , − . , − . , − . , − . , , . , . , . , . , .
9. The Integrated Mean Squared Error (IMSE)is used to compare the results. We summarize the results in Table 1 through the IMSE ranks foreach case ( ρ = 0 .
99; 0 . . α = 10 . . . ρ . Table 1: Average and Median Rank on the proposed IMSE (out of 1000 replications) f ( t ) = sin(2 πt ),SNR = 7, ‘db6’ and n = 1025. Rank \ ρ We present in Table 2 an overview for all test functions. The lines within each column representthe number of times (out of 15) in which the titled technique or basis achieved the best performance.The Appendix has a detailed presentation for the
Doppler test function. The details for the othertest functions can be made available as Supplementary Material.Some of the simulation results were expected. For instance, for either value of ρ , the biasof the linear estimator of ρ generally decreases in the SNR and in the sample size. Some orderdiscrepancies on n are observed, but the fact that the functional projection is made in a fixed V j can be taken as cause for them. A similar behavior is seen when the non-linear estimatoris employed, albeit in much milder terms. However, one should pay special attention to thethresholding procedure. When comparing the linear and non-linear procedures, the latter is thebest performer. For ρ = 0 .
99, it attains precisions of two or more decimal places against onefor the former whenever n ≥ ρ = 0 . ρ isthat it is heavily influenced by any bias on the estimation of f , be it by mistaken projection orover- and under-shrinkage. No significant effect of the initial value of ρ has been found.The results regarding linear vs non-linear functional estimation are somewhat similar to theones for ρ . There are cases in which the linear estimator outperforms the non-linear estimator.However, there are no instance of really poor performance by the non-linear estimator while thelinear estimator fails severely for some test functions. Moreover,in the case where the linearestimators have a better performance, the differences are small. The numerical differences due tothe basies are small but favor in general the ’db6’ and ’db3’ bases.The overall recommendation is to use the term-by-term thresholding wavelet estimator for allbut the last iteration. Then after the final estimation of ρ , a blocking-thresholded estimator of f yields the best performance. The data was obtained from SONDA (National System of Environmental Data, INPE-Brazil, http://sonda.ccst.inpe.br ). The SONDA network has minute-by-minute environmental data. Weanalyze the variables air temperateure at surface, air relative humidity, and air pressure. The fourinitial months for each season, March, June, September and December, were selected.The Weather stations in Bras´ılia, Ourinhos and S˜ao Luiz were chosen, given their latitude,longitude and altitude characteristics. The integrity and the nature and consistency of the datawere also important factors in the choice of cities and variables analyzed. We consider 22.75 days a b l e : O v e r v i e w o f t h e S i m u l a t i o nS t ud i e s f o r E s t i m a t i o n o f f , ρ . T y p e ( ) B i a s a ndS m a ll e s t B i a s S m a ll e s t M S E I M S E S m a ll e s t I M S E M S E N L ‘ db ’‘ db ’‘ s y m ’‘ db ’‘ db ’‘ s y m ’ N L b l o c k s ‘ db ’‘ db ’‘ s y m ’ D o11129421128258 D o21096015915159 H e H e B u B u B k B k Sp Sp B p B p C o11563637541302 C o2148437536294 W a1152112212253120 W a211258456123120 A n A n P a1156633111110141 P a21443844792112 T s T s C u C u ( ) D o* - D opp l e r ; H e * - H e a v i S i n e ; B u * - B u m p s ; B k * - B l o ck s ; Sp * - S p i ke s ; B p * - B l i p ; C o* - C o r n e r ; W a* - W a ve ; A n * - A n g l e s ; P a* - P a r a b o l a s ; T s * - T i m e S h if t e d S i n e ; C u * - C u s p . F o r a ll c a s e s ,i f * = , ρ = . . I f * = , ρ = . . N L m e a n s ( n o n li n e a r )t h r e s h o l d i n g . ρ based on ‘db6’. ˆ ρ , ˆ ρ ,ˆ ρ e ˆ ρ are the ρ estimates for March, June, September and December, respectively. The analysisis based upon n = 2 data points, which corresponds to 22.75 days for each month. The iterativeprocedure employs term-by-term thresholding from levels 5 to 8.Station Lat.(S) Long.(W) Alt.(m) Estimator Temperature Humidity Pressureˆ ρ ◦
36’ 47 ◦
42’ 1023 ˆ ρ ρ ρ ρ ◦
56’ 49 ◦
53’ 446 ˆ ρ ρ ρ ρ ◦
35’ 44 ◦
12’ 40 ˆ ρ ρ ρ for each month, which provides a total of 2 = 32 , ρ are employed, randomly generated from a U ( − , ρ is estimated. After that, in the last iteration, only levels 8 and 9 werethresholded, as proposed by Abramovich et al. (2004).Table 3 presents the estimates for ρ for the SONDA’s stations. First, we’d like to emphasizethat the final estimate does not depend on its initial value, i.e., in all cases the five initial valuesyield the same ρ estimates. The general conclusion is that a reasonable variation is observed inthe estimated values of ρ which corroborates the necessity of adjusting the data set to their effects.Moreover, high values of ρ are observed.Some specific results should be discussed. The three cities, which have very different weatherconditions, do present different behaviors in the ρ values as well. For instance, Bras´ılia presents verystable values of ρ for temperature and pressure, but humidity’s ρ vary during the year. Ourinhoshas similar behavior for the values of ρ over time and among the environmental variables. S˜ao Luizpresents reasonably stable values of ρ for humidity and pressure, but temperature in Septemberhas quite a different value of ρ from the rest of the year.In the remaining text we analyze the data of Bras´ılia. Figures 2, 3 and 4 show the temperature,humidity and atmospheric pressure observed and estimated the first 22 days of the month of March2009 for the city of Bras´ılia, respectively. A noisy curve of interpolated observed values and asmoothed estimated curve is presented for each day. Figures 5, 6 and 7 present the estimatesfor the Bras´ılia’s daily curves of temperature, humidity and pressure, respectively. One sees somesimilar behavior on the daily temperature and atmospheric pressure curves for each variable withineach month, with some exceptions. In general, these daily curves are very regular with one localminimum and maximum per day for the temperature and two local maxima and minima for thepressure. The humidity curves present much wider daily amplitude and much less regular behaviorwithin each month, specially for the rainy season months, i.e. March and December.Average Bras´ılia daily estimates curves for the first 22 days of March, June, September andDecember 2009 are shown in Figure 8. One sees that June presents the coldest days, and Septem-ber, the hottest days. The month of September is usually the driest, and December the wettest.Finally the months of March and December present lower atmospheric pressures compared to Juneand September.To verify that the curves of climatic variables have identical behavior from one year to anotherapplied the test described in the proposed model to the observed data in 2009 and 2010 in Bras´ılia.Has taken the observed curves for June and September 2010 from Bras´ılia to perform this test.
10 20202530 time (h) t e m p . ( ° C ) t e m p . ( ° C ) t e m p . ( ° C ) t e m p . ( ° C ) t e m p . ( ° C ) t e m p . ( ° C ) t e m p . ( ° C ) t e m p . ( ° C ) t e m p . ( ° C ) t e m p . ( ° C ) t e m p . ( ° C ) t e m p . ( ° C ) t e m p . ( ° C ) t e m p . ( ° C ) t e m p . ( ° C ) t e m p . ( ° C ) t e m p . ( ° C ) t e m p . ( ° C ) t e m p . ( ° C ) t e m p . ( ° C ) t e m p . ( ° C ) t e m p . ( ° C ) Figure 2: Bras´ılia’s estimated and observed temperature curves for the month of March 2009. hu m i d . ( % ) hu m i d . ( % ) hu m i d . ( % ) hu m i d . ( % ) hu m i d . ( % ) hu m i d . ( % ) hu m i d . ( % ) hu m i d . ( % ) hu m i d . ( % ) hu m i d . ( % ) hu m i d . ( % ) hu m i d . ( % ) hu m i d . ( % ) hu m i d . ( % ) hu m i d . ( % ) hu m i d . ( % ) hu m i d . ( % ) hu m i d . ( % ) hu m i d . ( % ) hu m i d . ( % ) hu m i d . ( % ) hu m i d . ( % ) Figure 3: Bras´ılia’s estimated and observed humidity curves for the month of March 2009.
The data for March are incomplete and of December were lost. Since the data are correlated,one can not perform the test developed in (Abramovich et al., 2004) directly. Therefore, beforeperforming the test made the transformation y t − ˆ ρy t − , t = 1 , , . . . , n in the observed data forthe uncorrelated errors. Thus, y t − ˆ ρy t − = f t − ˆ ρf t − + u t ⇐⇒ z t = g t + u t , in which errors u t have approximately distribution N(0 , σ u ). To perform the test replaces the function f by thecurve estimated for 2009. Thus we testedH : z − g ≡ Constant versus H : ( z − g − Constant) ∈ F ( (cid:37) ) , with significance α = 5%. The test result for June and September to the station of Bras´ılia are inTable 4.In all cases, the null hypothesis was rejected. Thus the curves observed in the months of June
10 20900905 time (h) p r e ss u r e p r e ss u r e p r e ss u r e p r e ss u r e p r e ss u r e p r e ss u r e p r e ss u r e p r e ss u r e p r e ss u r e p r e ss u r e p r e ss u r e p r e ss u r e p r e ss u r e p r e ss u r e p r e ss u r e p r e ss u r e p r e ss u r e p r e ss u r e p r e ss u r e p r e ss u r e p r e ss u r e p r e ss u r e Figure 4: Bras´ılia’s estimated and observed air pressure curves for the month of March 2009. t e m pe r a t u r e ( º C ) t e m pe r a t u r e ( º C ) t e m pe r a t u r e ( º C ) t e m pe r a t u r e ( º C ) d1d2d3d4d5d6d7d8d9d10d11d12d13d14d15d16d17d18d19d20d21d22 Figure 5: Bras´ılia’s Daily Estimated temperature curves for March, June, September and Decemberof 2009. d i represents days i . and September 2010 are different curves observed in the same period of 2009. hu m i d i t y ( % ) hu m i d i t y ( % ) hu m i d i t y ( % ) hu m i d i t y ( % ) d1d2d3d4d5d6d7d8d9d10d11d12d13d14d15d16d17d18d19d20d21d22 Figure 6: Bras´ılia’s Daily Estimated humidity curves for March, June, September and December of2009. d i represents days i . a i r p r e ss u r e a i r p r e ss u r e a i r p r e ss u r e a i r p r e ss u r e d1d2d3d4d5d6d7d8d9d10d11d12d13d14d15d16d17d18d19d20d21d22 Figure 7: Bras´ılia’s Daily Estimated atmospheric pressure curves for March, June, September andDecember of 2009. d i represents days i . 10 t e m pe r a t u r e ( º C ) marjunsepdec0 5 10 15 20 25406080100 time (h) hu m i d i t y ( % ) marjunsepdec0 5 10 15 20 25900905910 time (h) a i r p r e ss u r e marjunsepdec Figure 8: Bras´ılia’s Mean Estimated curves for the first 22 days of March, June, September andDecember of 2009. Separate Plots for mean temperature, humidity and pressure curves are presented.Table 4: Test results H : z − g ≡ Constant versus H : ( z − g − Constant) ∈ F ( (cid:37) ) with significance α = 5%. T ( j (6)) + Q ( j (6)) is the value of statistics and (cid:112) v (6) + w (6) z . is the critical valueTemperature Humidity PressureJune T ( j (6)) + Q ( j (6)) 213.43 81,900,00 215.69 (cid:112) v (6) + w (6) z . T ( j (6)) + Q ( j (6)) 466.01 3,396.40 43.94 (cid:112) v (6) + w (6) z . Discussion
The analysis of functional data has become more common in the last decades due to the exponen-tial increase in computing power, which has driven the also increasing large datasets’ acquisitionand the development of appropriate statistical analysis tools. We present a modification of opti-mal wavelet procedures(Abramovich et al., 2004) to deal with dependent errors. The theoreticaladvantages of correctly estimating the error dependence are shown, by simulation and applicationto real data set, to be also quite relevant in practice.
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Ann. Statist. , 24(6):2477–2498. ppendix We show below some figures and tables for the Doppler function simulation results. Results forthe other functions were qualitatively equivalent, and are available as supplementary material.
Figure 9: Box-Plots for ˆ ρ . Doppler function, ρ = 0 .
99, SNR = 1. 1000 replications. 50 initial valuesrandomly chosen from U ( − , n = 512; 1024; 2048; 4096; 8192 are shown from left to right. ’ ρ . Doppler function, ρ = 0 .
99, SNR = 3. 1000 replications. 50 initial valuesrandomly chosen from U ( − , n = 512; 1024; 2048; 4096; 8192 are shown from left to right.Figure 11: Box-Plots for ˆ ρ . Doppler function, ρ = 0 .
99, SNR = 7. 1000 replications. 50 initial valuesrandomly chosen from U ( − , n = 512; 1024; 2048; 4096; 8192 are shown from left to right.14igure 12: Box-Plots for ˆ ρ . Doppler function, ρ = 0 . U ( − , n = 512; 1024; 2048; 4096; 8192 are shown from left to right.Figure 13: Box-Plots for ˆ ρ . Doppler function, ρ = 0 . U ( − , n = 512; 1024; 2048; 4096; 8192 are shown from left to right.15igure 14: Box-Plots for ˆ ρ . Doppler function, ρ = 0 . U ( − , n = 512; 1024; 2048; 4096; 8192 are shown from left to right.Figure 15: Box-Plots for the number of iterations until numerical convergence. Doppler function, ρ = 0 .
99, SNR = 1. 1000 replications. 50 initial values randomly chosen from U ( − , n = 512; 1024; 2048; 4096; 8192 areshown from left to right. 16igure 16: Box-Plots for the number of iterations until numerical convergence. Doppler function, ρ = 0 .
99, SNR = 3. 1000 replications. 50 initial values randomly chosen from U ( − , n = 512; 1024; 2048; 4096; 8192 areshown from left to right. 17igure 17: Box-Plots for for the number of iterations until numerical convergence. Doppler function, ρ = 0 .
99, SNR = 7. 1000 replications. 50 initial values randomly chosen from U ( − , n = 512; 1024; 2048; 4096; 8192 areshown from left to right. 18igure 18: Box-Plots for the number of iterations until numerical convergence. Doppler function, ρ = 0 . U ( − , n = 512; 1024; 2048; 4096; 8192 areshown from left to right. 19igure 19: Box-Plots for the number of iterations until numerical convergence. Doppler function, ρ = 0 . U ( − , n = 512; 1024; 2048; 4096; 8192 areshown from left to right. 20igure 20: Box-Plots for for the number of iterations until numerical convergence. Doppler function, ρ = 0 . U ( − , n = 512; 1024; 2048; 4096; 8192 areshown from left to right.Figure 21: M SE (ˆ ρ ) as a function of sample sizes n = 512; 1024; 2048; 4096; 8192. Doppler function, ρ = 0 .
99. 1000 replications. 50 initial values randomly chosen from U ( − , M SE (ˆ ρ ) as a function of sample sizes n = 512; 1024; 2048; 4096; 8192. Doppler function, ρ = 0 . U ( − , a b l e : I n t e g r a t e dS q u a r e d M e a n E rr o r f o r ˆ f . D o pp l e r f un c t i o n , ρ = . , S N R = , , . r e p li c a t i o n s a nd s a m p l e s i ze s n = ; ; ; ; . i n i t i a l v a l u e s r a nd o m l y c h o s e n f r o m U ( − , ) . W a v e l e t b a s e s :’ db ’, db ’ a nd ’ s y m ’. F un c t i o n a l e s t i - m a t i o n s t e p s a r ec a ll e d ’ L ” ,’ N L ’ o r ’ N B ’i f li n e a r , n o n li n e a r w i t h t e r m b y t e r m t h r e s h o l d i n go r n o n li n e a r w i t hb l o c k t h r e s h o l d i n g i s e m p l o y e d , r e s p ec t i v e l y . I M S E ( x A ) r e p r e s e n tt h e a v e r ag e I M S E o f ˆ f f o r t h e r e p li c a t i o n , i n i t i a l v a l u e s i n t h ec o m b i n a t i o n o f a nS N R = x a nd t h r e s h o l d i n g p r o ce du r e A , w h e r e x = , r nd A = ’ L ’,’ N L ’ o r ’ N B ’. S N R = S N R = S N R = n db I M S E ( L ) . . . . . . . . . . . . . . . I M S E ( L ) . . . . . . . . . . . . . . . I M S E ( L ) . . . . . . . . . . . . . . . I M S E ( N T ) . . . . . . . . . . . . . . . I M S E ( N T ) . . . . . . . . . . . . . . . I M S E ( N T ) . . . . . . . . . . . . . . . I M S E ( N B ) . . . . . . . . . . . . . . . I M S E ( N B ) . . . . . . . . . . . . . . . I M S E ( N B ) . . . . . . . . . . . . . . . db I M S E ( L ) . . . . . . . . . . . . . . . I M S E ( L ) . . . . . . . . . . . . . . . I M S E ( L ) . . . . . . . . . . . . . . . I M S E ( N T ) . . . . . . . . . . . . . . . I M S E ( N T ) . . . . . . . . . . . . . . . I M S E ( N T ) . . . . . . . . . . . . . . . I M S E ( N B ) . . . . . . . . . . . . . . . I M S E ( N B ) . . . . . . . . . . . . . . . I M S E ( N B ) . . . . . . . . . . . . . . . s y m I M S E ( L ) . . . . . . . . . . . . . . . I M S E ( L ) . . . . . . . . . . . . . . . I M S E ( L ) . . . . . . . . . . . . . . . I M S E ( N T ) . . . . . . . . . . . . . . . I M S E ( N T ) . . . . . . . . . . . . . . . I M S E ( N T ) . . . . . . . . . . . . . . . I M S E ( N B ) . . . . . . . . . . . . . . . I M S E ( N B ) . . . . . . . . . . . . . . . I M S E ( N B ) . . . . . . . . . . . . . . . a b l e : I n t e g r a t e dS q u a r e d M e a n E rr o r f o r ˆ f . D o pp l e r f un c t i o n , ρ = . , S N R = , , . r e p li c a t i o n s a nd s a m p l e s i ze s n = ; ; ; ; . i n i t i a l v a l u e s r a nd o m l y c h o s e n f r o m U ( − , ) . W a v e l e t b a s e s :’ db ’, db ’ a nd ’ s y m ’. F un c t i o n a l e s t i m a t i o n s t e p s a r ec a ll e d ’ L ” ,’ N L ’ o r ’ N B ’i f li n e a r , n o n li n e a r w i t h t e r m b y t e r m t h r e s h o l d i n go r n o n li n e a r w i t hb l o c k t h r e s h o l d i n g i s e m p l o y e d , r e s p ec t i v e l y . I M S E ( x A ) r e p r e s e n tt h e a v e r ag e I M S E o f ˆ f f o r t h e r e p li c a t i o n , i n i t i a l v a l u e s i n t h ec o m b i n a t i o n o f a nS N R = x a nd t h r e s h o l d i n g p r o ce du r e A , w h e r e x = , r nd A = ’ L ’,’ N L ’ o r ’ N B ’. S N R = S N R = S N R = n db I M S E ( L ) . . . . . . . . . . . . . . . I M S E ( L ) . . . . . . . . . . . . . . . I M S E ( L ) . . . . . . . . . . . . . . . I M S E ( N T ) . . . . . . . . . . . . . . . I M S E ( N T ) . . . . . . . . . . . . . . . I M S E ( N T ) . . . . . . . . . . . . . . . I M S E ( N B ) . . . . . . . . . . . . . . . I M S E ( N B ) . . . . . . . . . . . . . . . I M S E ( N B ) . . . . . . . . . . . . . . . db I M S E ( L ) . . . . . . . . . . . . . . . I M S E ( L ) . . . . . . . . . . . . . . . I M S E ( L ) . . . . . . . . . . . . . . . I M S E ( N T ) . . . . . . . . . . . . . . . I M S E ( N T ) . . . . . . . . . . . . . . . I M S E ( N T ) . . . . . . . . . . . . . . . I M S E ( N B ) . . . . . . . . . . . . . . . I M S E ( N B ) . . . . . . . . . . . . . . . I M S E ( N B ) . . . . . . . . . . . . . . . s y m I M S E ( L ) . . . . . . . . . . . . . . . I M S E ( L ) . . . . . . . . . . . . . . . I M S E ( L ) . . . . . . . . . . . . . . . I M S E ( N T ) . . . . . . . . . . . . . . . I M S E ( N T ) . . . . . . . . . . . . . . . I M S E ( N T ) . . . . . . . . . . . . . . . I M S E ( N B ) . . . . . . . . . . . . . . . I M S E ( N B ) . . . . . . . . . . . . . . . I M S E ( N B ) . . . . . . . . . . . . . . .713