A Study of the Choice of Model Orders in
Arxstruc-Type Methods for Open-Loop
Time-Delay Estimation in Linear Systems
Svante Bj¨orklund Control & Communication Department of Electrical Engineering
Link¨opings universitet, SE-581 83 Link¨oping, Sweden WWW: http://www.control.isy.liu.se
E-mail: svabj@isy.liu.se 31st August 2003
AUTOMATIC CONTROL
COMMUNICATION SYSTEMS LINKÖPING
Report no.: LiTH-ISY-R-2536
Technical reports from the Control & Communication group in Link¨oping are available at http://www.control.isy.liu.se/publications.
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Automatic Control
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LiTH-ISY-R-1400-3902
http://www.control.isy.liu.seDepartment of Electrical Engineering
2536
A Study of the Choice of Model Orders in Arxstruc-Type Methods for Open-Loop
Time-Delay Estimation in Linear Systems
In this report we study estimation of time-delays in linear dynamical systems with additive
noise. Estimating time-delays is a common engineering problem, e.g. in automatic control,
sys-tem identification and signal processing.
The purpose with this work is to test and evaluate a certain class of methods for time-delay
esti-mation, especially with automatic control applications in mind. The principle of the methods in
the class is to estimate several discrete-time models of a certain model structure with different
explicit time-delays. The estimated time-delay is the time-delay whose model has the lowest
mean square difference between the true and estimated output signal. The methods are evaluated
experimentally with the aid of simulations and plots of RMS error and confidence intervals for
different cases.
The results are: The output error (OE) model structure has the lowest RMS error but is very slow.
Low model orders give the best result. The ARX model structure has a higher RMS error but is
very fast. High model orders give the best result. An ARX model structure with prefiltered input
and output signals was also tested. It has an RMS error that is nearly as good as for the OE
mod-el structure and is fast but not as fast as the unfiltered ARX. The best modmod-el orders are high for
the denominator polynomial and low for the numerator polynomial.
X
2003-08-31
Contents
1 Introduction 1 2 The methods 2 2.1 Arxstruc . . . 2 2.2 Oestruc . . . 2 2.3 Met1struc . . . 2 2.3.1 Principles . . . 3 2.3.2 Implementation . . . 5 3 Simulation setup 7 4 Results 10 4.1 Choice of arxstruc parameters . . . 104.2 Choice of oestruc parameters . . . 11
4.3 Choice of met1struc parameters . . . 11
4.4 Comparison of methods . . . 13
5 Discussion and conclusions 14 5.1 Discussion . . . 14
5.2 Conclusions . . . 14
5.3 Future work . . . 15
References 17 A Analysis by confidence intervals 19 A.1 Choice of arxstruc parameters . . . 19
A.2 Choice of oestruc parameters . . . 19
1
Introduction
The problem we address in this report is estimating time-delays in linear dynamical sys-tems with additive noise. A synonym for time delay is dead-time. Estimating time-delays is a common engineering problem, e.g. in control performance monitoring of industrial processes [Hor00, Swa99], in design and tuning of controllers, in range estimation in radar [KQ92] and in direction estimation by time-delay of arrival in signal intelligence [HR97, FHJ02, Wik02]. Dead-time estimation is also a necessary part in all system iden-tification [Lju99].
The purpose with this work is to test and evaluate a certain class of methods for time-delay estimation, especially with automatic control applications in mind. The principle of the methods in the class is to estimate several discrete-time models of a certain model structure with different explicit integer delays. The estimated delay is the time-delay whose model has the lowest mean square difference between the true and estimated output signal. The methods are tested and evaluated experimentally with the aid of simulations.
In the next chapter, the time-delay estimation methods are briefly described. Then, Chap-ter 3 is about the simulation setup. AfChap-ter that, in ChapChap-ter 4 the analysis of the simulations is conducted. Following, Chapter 5 contains discussion, conclusions and suggestions for further work. Appendix A contains validation of required prerequisites for the analysis.
Algorithm 1 Matlab code for arxstructd using the Matlab System Identification Toolbox. function dtEst = arxstructd(zIn);
zIn = [outSig, inSig]; na = 10;
nb = 5;
nkVec = 1:20;
nkMax = length(nkVec);
nn = [na*ones(nkMax,1), nb*ones(nkMax,1), nkVec’]; V = arxstruc(zIn,zIn,nn);
modelStruc = selstruc(V,0); dtEst = modelStruc(3);
2
The methods
2.1 Arxstruc
In the method arxstruc several ARX models [Lju99]
A(q)y(t) = B(q)u(t− nk) + e(t)
are estimated with PEM (Prediction Error Method) [Lju99] for different time-delays nk.
The delay whose model has the lowest loss function is chosen. The estimation is quick since the ARX model can be written as a linear regression and can be estimated by solving a linear equation system [Lju99]. See Algorithm 1 for Matlab code using the Matlab System Identification Toolbox.
2.2 Oestruc
The principle of the methods oestruc is the same as for arxstruc but OE (output error) models [Lju99]
y(t) = B(q)
F (q)u(t) + e(t)
are estimated instead of ARX models. To estimate OE models is much more computation-ally demanding than ARX models since a multidimensional optimization with a numerical search must be carried out [Lju99]. See Algorithm 2 for Matlab code using the Matlab System Identification Toolbox.
2.3 Met1struc
The method oestruc gives a better result than arxstruc in the simulations presented in this report (see Sections 4.1 and 4.2). On the other hand, arxstruc has a much lower computation time than oestruc. In this section we suggest a new time-delay estimation method with the aim to imitate oestruc but with much lower computational demands.
Algorithm 2 Matlab code for oestructd using the Matlab System Identification Toolbox. function dtEst = oestructd(zIn);
zIn = [outSig, inSig]; nf = 2;
nb = 1;
nkVec = 1:20;
for nnn = 1:length(nkVec),
model = oe(z,[nb nf nkVec(nnn)], ’Covariance’,’None’) lossFunc(nnn) = model.NoiseVariance;
end%&for
[minVal, nnnmin] = min(lossFunc); dtEst = nkVec(nnnmin);
2.3.1 Principles
Assume the true system is given by
y = G0u + H0e ,
where G0 and H0 are rational functions in the delay operator q−1. The noise e is white.
The model structure used to estimate the true system is
y = Gu + He ,
where G and H also are rational functions in the delay operator q−1.
The reason for the difference between OE and ARX model structures can be two-fold:
1. If the model structure G can not exactly describe the true system G0, the estimated
model G will have a bias, even if the number of data N → ∞. The OE and ARX model structures will behave differently :[Lju99, Ex. 8.5, p. 268-269]:
• The ARX model structure will give models with a good fit to the true system at high frequencies.
• The OE model structure will give models with a good fit to the true system at low frequencies.
2. If the model structure G can exactly describe the true system G0, again the OE and
ARX model structures behave differently:
• The ARX model structure will give models G with a bias if the noise model structure 1/A cannot describe the true noise system H0. See Equations 8.63
and 8.69 and page 267 in [Lju99].
• The OE model structure will give estimates without bias. The bias of the OE model structure is thus not dependent on the noise model. See [Lju99, Eq. 8.71, p. 266].
Algorithm 3 Proposed time-delay estimation method.
1. Estimate an ARMAX model A1(q)y(k) = B1(q)u(k) + C1(q)e(k).
2. Prefilter u and y through 1/C1(q) .
3. Arxstruc gives an estimate of nk.
We propose the the time-delay estimation method in Algorithm 3. The motivation for this method is the following. Assume the true system is given by
y = G0u + H0e = B0 F0 u + C0 D0 e ,
where G0 and H0 are rational functions and B0, F0, C0 and D0 are polynomials in the
delay operator q−1. The noise e is white. Estimate a first model
y = B1 F1
u + C1 D1
e (2.1)
where F1, B1, C1 and D1 are polynomials in q−1, i.e. a Box-Jenkins model [Lju99].
The model C1/D1will be an approximation of H0 = C0/D0. Then, filter the output signal
y through D1/C1 giving yF: yF = D1 C1 y = D1 C1 B 0 F0 u + H0e = B0 F0 D1u C1 + H0D1 C1 e .
We notice that uF = uD1/C1 is the input signal u filtered through D1/C1. We rewrite
the equation for yF as
yF = B0 F0 uF + H0 D1 C1 e . (2.2)
Now, if the polynomials F1, B1, C1 and D1 in the model structure 2.1 have high enough
orders so that that they can exactly describe the true system and the input-output data is informative enough, then F1, B1, C1 and D1 will converge to the true values F0, B0, C0
and D0as the number of data N → ∞ [Lju99, p. 273]. When this happens, D1/C1= 1/H0
and Equation 2.2 simplifies to
F0yF = B0uF + e , (2.3)
where e as before is white noise. This true system is obviously an ARX system. This means that if we estimate an ARX model AyF = BuF+ e, there will be no bias due to an
incorrect noise model. If an ARX model has high enough orders, we can get a model for the system G without bias (as in the case with an OE model of high enough orders).
Algorithm 4 met1struc.
1. Estimating a state space model by state space method. 2. Converting to A1(q)y(k) = B1(q)u(k) + C1(q)e(k) .
3. Prefiltering u and y through 1/C1(q) .
4. Arxstruc gives an estimate of nk.
Let us now assume that the true system instead has OE structure: y = G0u + H0e =
B0
F0
u + e ⇒ F0y = B0u + F0e. (2.4)
Since this has ARMAX structure [Lju99] we estimate a first model
A1y = B1u + C1e , (2.5)
where A1, B1 and C1 are polynomials in q−1. A1 and C1 will be approximations of F0. If
we filter the input and output signal through 1/C1 we will get the following true system
F0yF = B0uF + e , (2.6)
which is also of ARX structure and can be approximated by an ARX model of enough model order without bias.
Since the systems simulated in this report (Chapter 3) have OE structure (equation (2.4)), we will filter the input and output signals through 1/C1 where C1 is from equation (2.5).
2.3.2 Implementation
In Algorithm 3 the first step is to estimate an ARMAX model. Unfortunately, the standard way to estimate a model of this model structure also requires a numerical search as with the OE model which we tried to avoid. Another way is to first estimate a state space model and then convert it to an ARMAX model. This conversion will be possible if the order of the state space model and the orders of the polynomials A1, B1 and C1 are high
enough to describe the true system. The state space model can be quickly estimated by a subspace method [Lju99]. See Algorithm 4 for the resulting method, which we call met1struc. Matlab code for met1struc is given in Algorithm 5. The order of the state space model is 10. This will hopefully be enough for most systems. The orders of A1, B1 and
Algorithm 5 Matlab code for met1struc using the Matlab System Identification Toolbox. The function arxstructd is described in Section 2.1.
function dtEst = met1structd(inSig, outSig, Ts) order = 10;
modelSs = n4sid(iddata(outSig, inSig, Ts),order,’cov’,’none’); [A,B,C,D,F] = polydata(idpoly(modelSs));
BFilt = 1; AFilt = C;
uFilt = filter(BFilt,AFilt,inSig); yFilt = filter(BFilt,AFilt,outSig); zIn = [yFilt, uFilt];
na = 10; nb = 1;
nkVec = 1:20;
Time (sec.) Amplitude Impulse response G1 (t130g) 0 14 28 42 56 70 0 0.02 0.04 0.06 0.08 From: U(1) To: Y(1) Time (sec.) Amplitude Impulse response G2 (t130g) 0 14 28 42 56 70 0 0.2 0.4 0.6 0.8 From: U(1) To: Y(1) Time (sec.) Amplitude Impulse response G5 (t130g) 0 14 28 42 56 70 0 0.02 0.04 0.06 0.08 From: U(1) To: Y(1) Time (sec.) Amplitude Impulse response G6 (t130g) 0 14 28 42 56 70 −0.01 0 0.01 0.02 0.03 0.04 0.05 From: U(1) To: Y(1)
Figure 3.1: Impulse response of system G1-G2 and G5-G6. True time-delay after sampling
Td= 10.
3
Simulation setup
The setup for the simulations is the same as in [Bj¨o03a] with the following exceptions: • The number of trials was 2048 for arxstruc, 192 for oestruc and 512 for met1struc.
the reason for the different number of trials is the very different execution times for the methods. The trials were split into four groups and each group was used to compute an estimate of the RMS error (our response variable) of the time-delay estimate. This gave 4 estimates of the RMS error that was used in the calculation of the confidence intervals. See [Bj¨o03a].
• The methods were obviously different. See Chapter 2.
Three environment factors were varied during the simulations: The system, the input signal type and the SNR [Bj¨o03a]. The signal-to-noise ratio (SNR) was either 1 or 100. See [Bj¨o03a] for the definition of the SNR. The impulse responses of the four used systems are depicted in Figure 3.1. Note that for all the systems the time delay will be 10 after the sampling. More information about the systems can be found in [Bj¨o03a].
Figures 3.2-3.4 show the used input signals in the time and frequency domains. Figure 3.2 depicts the signal RBS 10-30% which is a bandpass random signal with frequency contents
0 50 100 150 200 250 300 350 400 450 500 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 samples Time signal RBS10−30% 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 10−2 10−1 100 101 Normalized frequency Power Power spectruml RBS10−30%
Figure 3.2: Time signal (left) and frequency spectrum (right) for a realization of the input signal type RBS 10-30%. 0 50 100 150 200 250 300 350 400 450 500 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 samples Time signal RBS0−100% 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 10−1 100 101 Normalized frequency Power Power spectruml RBS0−100%
Figure 3.3: Time signal (left) and frequency spectrum (right) for a realization of the input signal type RBS 0-100%.
between 10% and 30% of the Nyquist frequency. Figure 3.2 depicts the signal RBS 0-100% which is a wideband random signal with frequency contents between 0% and 100% of the Nyquist frequency. It is thus white noise. Figure 3.4 depicts the signal Steps which is a signal with three steps. It has a frequency contents between 0% and about 5% of the Nyquist frequency. More information about the input signals can be found in [Bj¨o03a]. In addition to the environment factors, three method factors were varied during the sim-ulations: The model orders na (or nf) and nb and using prewhitening (see [Bj¨o03a]) or
0 50 100 150 200 250 300 350 400 450 500 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 samples Time signal steps
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 10−4 10−3 10−2 10−1 100 101 102 Normalized frequency Power
Power spectruml steps
Figure 3.4: Time signal (left) and frequency spectrum (right) for a realization of the input signal type Steps.
nopw*1 nopw*2 nopw*3 nopw*4 nopw*5 pw*1 pw*2 pw*3 pw*4 pw*5 2 4 6 8 10 0 1 2 3 4 2.76 2.75 2.81 2.83 2.78 2.9 3.01 2.85 2.87 3.05 3.15 3.01 2.9 3 3.25 3.12 3.07 Prewhite*nb 3.02 3.21 2.75 3.15 3.19 3.24 2.8 2.7 t161b2:030509 08:50 rms: rms, data(m,m,:,m,m,m,m,m,m,m,m,:,:) 3.25 3.38 2.82 2.75 2.69 . MIN 3.47 . MAX 2.8 2.78 2.75 2.75 2.91 2.77 2.78 2.81 2.9 2.85 2.79 2.85 2.94 2.82 2.87 2.98 2.88 3 nanf 3.06
Figure 4.1: RMS error for arxstruc as a function of the model orders naand nband prewhitening
or not. (t161b2.m)
4
Results
This chapter presents plots of the RMS error of the time-delay estimation on simulated signals. The RMS values have the unit sampling interval.
4.1 Choice of arxstruc parameters
For the method arxstruc we see in Figure 4.1 that there are a lot of values of na, nb and
prewhitening that give approximately the same average RMS error. The lowest RMS error has na = 10, nb = 3 and without prewhitening in this simulation. However, we
prefer na= 10, nb = 5 and without prewhitening because these values gave the best result
in a similar simulation (t102b1.m and t102b2.m). We call this combination arxstruc3. The ANOVA of this simulation required much computing power, both execution time and memory. It is questionable if the ANOVA and confidence intervals are useful because the residuals are very non-Gaussian and the variance is not constant, see Figure A.1-A.2 in Appendix A.
nopw*1 nopw*2 nopw*3 nopw*4 nopw*5 pw*1 pw*2 pw*3 pw*4 pw*5 2 4 6 8 10 0 1 2 3 4 3.89 . MAX 3.78 3.71 3.73 3.6 3.55 3.59 3.53 3.48 3.41 3.76 3.38 3.42 3.33 3.03 3.47 2.6 2.76 3.27 Prewhite*nb 3.25 3.05 3.55 3.19 3.1 2.93 3.49 3.38 t156b2:030508 17:26 rms: rms, data(m,m,:,m,m,m,m,m,m,m,m,:,:) 2.95 3.5 3.37 3.13 3.27 3.21 3.06 2.84 3.37 3.06 2.96 2.75 2.25 3.02 2.88 2.69 2.17 2.66 2.58 2.12 2.29 1.99 nanf 1.79 . MIN
Figure 4.2: RMS error for oestruc as a function of the model orders nf and nband prewhitening
or not. (t156b2.m)
4.2 Choice of oestruc parameters
For the method oestruc it is seen in Figure 4.2 that the best model orders with respect to time-delay estimation are the lowest (nf = 2 and nb = 1) of the tested. If nb > 1 this
would enable more models that give a low optimization criterion value but with different time-delays. It is also seen in the same figure that without prewhitening is the best. The ANOVA of this simulation required much computing power, both execution time (2 h) and memory (590 MB) on a SunBlade 100 computer. If we study Figure A.3 in Appendix A, we will doubt that the ANOVA and the confidence intervals are useful because the residuals are non-Gaussian. On the other hand the result in Figure A.5 is very clear. The confidence interval with the model orders nf and nb and prewhitening (or not) with
the lowest RMS error is clearly separated from the other confidence intervals. This makes the confidence interval analysis robust and the confidence intervals confirm that nf = 2,
nb = 1 and no prewhitening is the best choice. We call this combination oestruc3.
4.3 Choice of met1struc parameters
In Figure 4.3 it is seen that the best model parameters for the method met1struc with respect to time-delay estimation are na = 10, nb = 1 and without prewhitening. We call
nopw*1 nopw*2 nopw*3 nopw*4 nopw*5 pw*1 pw*2 pw*3 pw*4 pw*5 2 4 6 8 10 0 1 2 3 4 3.21 3.22 2.92 3.22 2.97 2.59 3.3 2.95 2.66 2.39 3.66 . MAX 3.06 2.75 2.42 2.31 3.48 2.42 2.55 2.36 2.85 Prewhite*nb 2.51 3.48 3.32 2.61 3.55 3.03 t163b2:030508 13:32 rms: rms, data(m,m,:,m,m,m,m,m,m,m,m,:,:) 3.19 3.51 3.14 2.54 3.05 3.49 3.15 2.66 2.24 3.59 3.21 2.82 2.35 1.96 . MIN 3.44 2.89 2.49 2.09 3.17 2.67 2.22 2.99 2.45 nanf 2.85
Figure 4.3: RMS error for met1struc as a function of the model orders naand nband
It is questionable if the ANOVA and confidence intervals are useful because because the residuals are non-Gaussian and the variance is not constant, see Figure A.6-A.7 in Ap-pendix A.
4.4 Comparison of methods
When we look at Figures 4.1-4.3 and measure the execution time we see that
• oestruc has the lowest RMS error (1.8 sampling intervals) but is very slow (One estimation took 13.0 s on a SunBlade 100 computer).
• met1struc has a RMS error that is nearly as good as for oestruc (met1struc: 2.0 sampling intervals) and is fast (One estimation took 0.741 s ).
• arxstruc has a higher RMS (2.7 sampling intervals) but is very fast (One estimation took 0.143 s).
5
Discussion and conclusions
5.1 Discussion
We find in this report that oestruc is the best method in the tested cases. Also Swanda in [Swa99] consider that oestruc is better than arxstruc. It is not surprising that oestruc is better than arxstruc since the tested systems have OE structure. This also helps met1struc to give good results.
When estimating a discrete-time state space model (zoh sampling) of a system with a long time-delay (longer than the sampling interval) the order of the model will increase with one for each sampling interval of the time-delay [˚AW84, p. 42]. This could indicate that the used order 10 of the state space model in met1struc could be too low for long time-delays. If the continuous-time time-delay is 9, a model order of 10 seems to be on the limit to be too low. Another way for the state space model to handle the time-delay is to to approximate it with non-minimum phase zero(s). In this way a lower model order can be sufficient. This is also what happens in met1struc. However, for longer time-delays than used in this report, it would be advisable to use a higher fixed model order or to chose the model order automatically to give a good model. This can be done by giving ’best’ as the input parameter order to the function n4sid (Algorithm 5) in the Matlab Identification Toolbox.
The advantage of met1struc over oestruc is the higher execution speed. A disadvantage is that it is more complicated. In applications where the time-delay is changing and the noise does not change it should be possible to estimate the noise model once off-line and use it in many subsequent time-delay estimations with a modified met1struc method. It is not necessary to estimate this noise model with a subspace state space method as in Section 2.3 but can by done by a less complicated method.
In Section 4.1 the best choice of model orders for arxstruc was the highest of the tested na and nb. The reason for this is probably that high orders are needed to approximate
the noise system well by 1/A since the true systems are not of ARX structure. An all-pole system 1/A of enough high order should be able to approximate the noise system enough well. Such an approximation is used in [FMS91, p. 655]. See also the discussion in Section 2.3.1 about bias in the model G for different cases of system/model structures and orders.
In Section 4.2 the best choice of model orders for oestruc was the lowest of the tested (na= 2 and nb = 1). These orders are enough to accurately model the true system. The
true systems are either of second or fourth order. This would mean that na= 2 or na= 4
would be appropriate. In average na= 2 is apparently better. If the order nb were higher
than 1 it would introduce an ambiguity for the time-delay. For example, both the true time-delay and one minus the true time-delay (with the the first B-parameter equal to zero) would give good fits to the data. Therefore it is understandable that nb = 1.
The theoretical explanation for the choice of model orders for met1struc in Section 4.3 is not clear.
5.2 Conclusions
• The method oestruc give the best estimates but it is very slow. The lowest model orders (nf = 2 and nb = 1) give the best result since they can describe the
input-output dynamics well.
• The method arxstruc give estimates that is not as good as oestruc in the tested cases (true OE systems) but it is very fast. High model orders (e.g na = 10, nb = 5) give
the best result.
• The method met1struc give nearly as good estimates as oestruc in the tested cases but is much faster. However, it is slower than arxstruc. The best model orders are high na (na= 10) and low nb (nb = 1).
5.3 Future work
Possible future work is:
• Test with other true model structures than output error. • Test with closed-loop.
• Try also filter the input and output signals with D1/C1 to handle arbitrary noise
systems. See Section 2.3.
References
[˚AW84] K. ˚Astr¨om and B. Wittenmark. Computer Controlled Systems. Theory and De-sign. Prentice-Hall, 1984.
[Bj¨o03a] Svante Bj¨orklund. Experimental evaluation of some cross correlation methods for time-delay estimation in linear systems. Technical Report LiTH-ISY-R-2513, De-partment of Electrical Engineering, Link¨oping University, SE-581 83 Link¨oping, Sweden, April 2003.
[Bj¨o03b] Svante Bj¨orklund. Experimental evaluation of some methods using simple process models for estimating continuous time-delays in open-loop. Technical Report LiTH-ISY-R-2526, Department of Electrical Engineering, Link¨oping University, SE-581 83 Link¨oping, Sweden, July 2003.
[Bj¨o03c] Svante Bj¨orklund. Experimental evaluation of some thresholding methods for estimating time-delays in open-loop. Technical Report LiTH-ISY-R-2525, De-partment of Electrical Engineering, Link¨oping University, SE-581 83 Link¨oping, Sweden, July 2003.
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[FMS91] G. Ferretti, C. Maffezzoni, and R. Scattolini. Recursive estimation of time delay in sampled systems. Automatica, 27(4):653–661, 1991.
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[KQ92] Simon Kingsley and Shaun Quegan. Understanding Radar Systems. McGraw-Hill, 1992. ISBN 0-07-707426-2.
[Lju99] Lennart Ljung. System Identification: Theory for the User. Prentice-Hall, Upper Saddle River, N.J. USA, 2nd edition, 1999.
[Mon97] D. C. Montgomery. Design and Analysis of Experiments. Wiley, 1997. ISBN 0-471-15746-5.
[Swa99] Anthony Paul Swanda. PID Controller Performance Assessment Based on Closed-Loop Response Data. Phd thesis, University of California, Santa Bar-bara, California, USA, June 1999.
[Wik02] Maria Wikstr¨om. Utveckling och implementering av ett audiopejlsystem baserat p˚a tidsdifferensm¨atning. Master’s thesis LiTH-ISY-EX-3277-2002, Link¨opings universitet, Link¨oping, Sweden, October 2002. In swedish.
−0.1 0 0.1 0.001 0.0030.01 0.02 0.05 0.10 0.25 0.50 0.75 0.90 0.95 0.98 0.99 0.997 0.999 Data Probability
t161b2: Normal plot of residuals
−0.20 −0.1 0 0.1 0.2 50 100 150 200 250 300 350 t161b2: Histogram of residuals No resids=4800 0 1000 2000 3000 4000 5000 −0.2 −0.1 0 0.1 0.2 t161b2: Residuals vs. time Time Residuals 0 1 2 3 4 −0.2 −0.1 0 0.1
0.2 t161b2: Residuals vs. fitted value
Fitted value
Residuals
Figure A.1: Residual analysis for ANOVA of arxstruc.
A
Analysis by confidence intervals
This appendix contains an attempt to analyze with ANOVA and confidence intervals for pair-wise comparisons. The Analysis was performed in the same way as in [Bj¨o03b, Bj¨o03c]. Since we consider the prerequisites (see [Mon97, Bj¨o03a]) only to be fulfilled for the method oestruc, only for this methods confidence intervals are presented. Common validation graphs [Mon97, Bj¨o03a] are shown for all tested methods.
A.1 Choice of arxstruc parameters
Figure A.1-A.2 shows plots for testing whether the prerequisites for ANOVA and confidence intervals are fulfilled when using arxstruc. The positive transformation xˆ(0.505127) was used. This means ”The lower the better” in a confidence interval plot.
A.2 Choice of oestruc parameters
Figure A.3-A.4 shows plots for testing whether the prerequisites for ANOVA and confidence intervals are fulfilled when using oestruc. Figure A.5 contains confidence intervals for pair-wise comparisons between different model orders and with/without prewhitening. The
1 1.5 2 2.5 3 0 0.005 0.01 0.015 0.02 t161b2: Residuals vs. InType InType Residuals 1 1.2 1.4 1.6 1.8 2 0 0.005 0.01 0.015 0.02 t161b2: Residuals vs. Prewhite Prewhite Residuals 1 1.2 1.4 1.6 1.8 2 0 0.005 0.01 0.015 0.02 0.025 t161b2: Residuals vs. SNR SNR Residuals 1 1.5 2 2.5 3 3.5 4 0 0.005 0.01 0.015 0.02 0.025 0.03 t161b2: Residuals vs. Sys Sys Residuals 1 1.5 2 2.5 3 3.5 4 4.5 5 0 0.005 0.01 0.015 0.02 0.025 t161b2: Residuals vs. nb nb Residuals 1 1.5 2 2.5 3 3.5 4 4.5 5 0 0.005 0.01 0.015 0.02 t161b2: Residuals vs. nanf nanf Residuals
−0.4 −0.2 0 0.2 0.001 0.0030.01 0.02 0.05 0.10 0.25 0.50 0.75 0.90 0.95 0.98 0.99 0.997 0.999 Data Probability
t156b2: Normal plot of residuals
−0.60 −0.4 −0.2 0 0.2 0.4 20 40 60 80 100 120 140 t156b2: Histogram of residuals No resids=4800 0 1000 2000 3000 4000 5000 −0.6 −0.4 −0.2 0 0.2 0.4 t156b2: Residuals vs. time Time Residuals 0 0.5 1 1.5 2 2.5 −0.6 −0.4 −0.2 0 0.2
0.4 t156b2: Residuals vs. fitted value
Fitted value
Residuals
Figure A.3: Residual analysis for ANOVA of oestruc.
positive transformation xˆ(0.4296) was used. This means ”The lower the better” in the confidence interval plot.
A.3 Choice of met1struc parameters
Figure A.6-A.7 shows plots for testing whether the prerequisites for ANOVA and confidence intervals are fulfilled when using met1struc. The positive transformation xˆ(0.4028) was used. This means ”The lower the better” in a confidence interval plot.
1 1.5 2 2.5 3 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 t156b2: Residuals vs. InType InType Residuals 1 1.2 1.4 1.6 1.8 2 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 t156b2: Residuals vs. Prewhite Prewhite Residuals 1 1.2 1.4 1.6 1.8 2 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 t156b2: Residuals vs. SNR SNR Residuals 1 1.5 2 2.5 3 3.5 4 0 0.02 0.04 0.06 0.08 t156b2: Residuals vs. Sys Sys Residuals 1 1.5 2 2.5 3 3.5 4 4.5 5 0 0.02 0.04 0.06 0.08 0.1 t156b2: Residuals vs. nb nb Residuals 1 1.5 2 2.5 3 3.5 4 4.5 5 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 t156b2: Residuals vs. nanf nanf Residuals
0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 50:5*10*pw 49:4*10*pw 48:3*10*pw 47:2*10*pw 46:1*10*pw45:5*8*pw 44:4*8*pw 43:3*8*pw 42:2*8*pw 41:1*8*pw 40:5*6*pw 39:4*6*pw 38:3*6*pw 37:2*6*pw 36:1*6*pw 35:5*4*pw 34:4*4*pw 33:3*4*pw 32:2*4*pw 31:1*4*pw 30:5*2*pw 29:4*2*pw 28:3*2*pw 27:2*2*pw 26:1*2*pw 25:5*10*nopw 24:4*10*nopw 23:3*10*nopw 22:2*10*nopw 21:1*10*nopw20:5*8*nopw 19:4*8*nopw 18:3*8*nopw 17:2*8*nopw 16:1*8*nopw 15:5*6*nopw 14:4*6*nopw 13:3*6*nopw 12:2*6*nopw 11:1*6*nopw 10:5*4*nopw9:4*4*nopw 8:3*4*nopw 7:2*4*nopw 6:1*4*nopw 5:5*2*nopw 4:4*2*nopw 3:3*2*nopw 2:2*2*nopw 1:1*2*nopw t156b2:nb*nanf*Prewhite
49 groups have population marginal means significantly different from Group 1
Figure A.5: oestruc: Confidence intervals (the lines in the circles) for pair-wise comparisons (95% simultaneous confidence level) for different thresholding methods and input signals. Positive transformation: (RMS error)ˆ(0.4296) => ”The lower the better”. (t156b2.m)
−0.4 −0.2 0 0.2 0.001 0.0030.01 0.02 0.05 0.10 0.25 0.50 0.75 0.90 0.95 0.98 0.99 0.997 0.999 Data Probability
t163b2: Normal plot of residuals
−0.60 −0.4 −0.2 0 0.2 0.4 20 40 60 80 100 120 t163b2: Histogram of residuals No resids=4800 0 1000 2000 3000 4000 5000 −0.6 −0.4 −0.2 0 0.2 0.4 t163b2: Residuals vs. time Time Residuals 0 0.5 1 1.5 2 2.5 −0.6 −0.4 −0.2 0 0.2
0.4 t163b2: Residuals vs. fitted value
Fitted value
Residuals
1 1.5 2 2.5 3 0 0.01 0.02 0.03 0.04 t163b2: Residuals vs. InType InType Residuals 1 1.2 1.4 1.6 1.8 2 0 0.01 0.02 0.03 0.04 t163b2: Residuals vs. Prewhite Prewhite Residuals 1 1.2 1.4 1.6 1.8 2 0 0.01 0.02 0.03 0.04 0.05 t163b2: Residuals vs. SNR SNR Residuals 1 1.5 2 2.5 3 3.5 4 0 0.01 0.02 0.03 0.04 0.05 t163b2: Residuals vs. Sys Sys Residuals 1 1.5 2 2.5 3 3.5 4 4.5 5 0 0.01 0.02 0.03 0.04 0.05 0.06 t163b2: Residuals vs. nb nb Residuals 1 1.5 2 2.5 3 3.5 4 4.5 5 0 0.01 0.02 0.03 0.04 0.05 t163b2: Residuals vs. nanf nanf Residuals
