Damien Bouvier
email: damien.bouvier@ircam.fr

Thomas Hélie

David Roze

NONLINEAR HOMOGENEOUS ORDER SEPARATION FOR VOLTERRA SERIES IDENTIFICATION

This article addresses identification of nonlinear systems represented by Volterra series. To improve the robustness of some existing methods, we propose a pre-processing stage that separates nonlinear homogeneous order contributions from which Volterra kernels can be identified independently. The proposed separation method exploits phase relations between test signals rather than amplitude relations that are usually used. This method is compared with standard separation process. Its contribution to identification is illustrated on a simulated loudspeaker with nonlinear suspension.

INTRODUCTION

Volterra series are a representation formalism for dynamical system with memory and weak nonlinearities. In audio, it has been widely used in various applications such as simulation of nonlinear resonators (brass instruments [START_REF] Hélie | Simulation of the weakly nonlinear propagation in a straight pipe: application to a realtime brassy audio effect[END_REF], Moog ladder filter [START_REF] Hélie | Volterra series and state transformation for real-time simulations of audio circuits including saturations: Application to the Moog ladder filter[END_REF], etc), distortion effect [START_REF] Finn | Volterra Series Based Distortion Effect[END_REF][START_REF] Tronchin | The emulation of nonlinear timeinvariant audio systems with memory by means of Volterra series[END_REF], audio transducer analysis [START_REF] Arie | Modeling of the nonlinear response of an electrodynamic loudspeaker by a Volterra series expansion[END_REF]. For any purpose (analysis, simulation or even system control), the use of Volterra series has to begin with the computation of the Volterra kernels, either directly from the system's dynamical equation [START_REF] Hélie | Volterra series and state transformation for real-time simulations of audio circuits including saturations: Application to the Moog ladder filter[END_REF], or by identification from inputs/outputs measurement [START_REF] Goussard | Practical identification of functional expansions of nonlinear systems submitted to non-Gaussian inputs[END_REF].

This paper focuses on the identification problem in blind context, that is, without assuming any particular structures (Hammerstein, Wiener-Hammerstein, etc) [START_REF] Novak | Modeling of nonlinear audio systems using swept-sine signals: Application to audio effects[END_REF][START_REF] Rébillat | Identification of cascade of Hammerstein models for the description of nonlinearities in vibrating devices[END_REF] or parametric model [START_REF] Billings | Nonlinear system identification: NAR-MAX methods in the time, frequency, and spatio-temporal domains[END_REF]. To this end, we propose a pre-processing stage that separates nonlinear homogeneous order contributions from which Volterra kernels can be identified independently. Such an approach is commonly used (see [START_REF] Boyd | Measuring volterra kernels[END_REF][START_REF] Bard | Modeling of Nonlinearities in Electrodynamic Loudspeakers[END_REF][START_REF] Agerkvist | Eliminating transducer distortion in acoustic measurements[END_REF][START_REF] Lambert | Vandermonde Method for Separation of Nonlinear Orders and Measurement of Linear Response[END_REF][14]), but the separation process relies on amplitude discrimination, that rapidly leads to ill-conditioned problems. To improve robustness, a new method is proposed that also exploits phase relations. This pre-processing method is embedded into a kernel identification method proposed in [START_REF] Goussard | Practical identification of functional expansions of nonlinear systems submitted to non-Gaussian inputs[END_REF].

This paper is organized as follows. Section 2 gives an overview of Volterra Series and standard order separation. In Section 3, the proposed separation method is presented, and its advantages and disadvantages are discussed. Finally, in Section 4, simulation of a loudspeaker with nonlinear suspension is used to compare the separation methods and their contribution to identification.

RECALLS ON VOLTERRA SERIES AND ORDER SEPARATION

Overview on Volterra Series

An overview of the Volterra formalism is given here; further and more thorough explanations can be found in [15,16], among the vast literature on Volterra series.

Definition 1 (Volterra series). A nonlinear causal time-invariant system is described by a Volterra series {hn} n≥N * if, for all input signals u such that u ∞ < ρ, the output signal y is given by the following Volterra operator:

y = V [u] = ∞ n=1 yn (1)
where, for continuous-time systems:

yn(t) = R n + hn(τ1, . . . , τn) n i=1 u(t -τi)dτi (2)
and for discrete-time systems:

yn[l] = N n hn[m1, . . . , mn] n i=1 u[l -mi] (3)
and with ρ the convergence radius of the characterising function Φ h (x) = +∞ n=1 hn 1x n . The terms hn are called the Volterra kernels of the system, and terms yn the nonlinear homogeneous order contributions (or more simply the nonlinear orders).

In the following, for sake of notation, continuous-time signals and systems will be used; if not specified otherwise, results are also valid for their discrete-time counterparts.

Remark 1 (Convergence). It has been shown in [17,18] that there exists a large class of well-posed systems for which we know how to compute the convergence radius of the Volterra series. In this work, we will always assume that convergence's conditions are met.

Remark 2 (Order and memory truncation). In numerical implementation for simulation or identification, it will be necessary to truncate both infinite sums (i.e. for nonlinear orders and memory). Thus, in practice, Volterra series can only be used to approximate systems with small nonlinearities (limited to the first few nonlinear orders) and with finite memory [19].

Remark 3 (Non-unicity of kernels). It can easily be seen from ( 2) that kernels are not uniquely defined. To circumvent this problem for identification purpose, uniquely-defined forms can be specified, such as the triangular or symmetric kernels (where the kernel is invariant to any permutation of its arguments).

Remark 4 (Frequency domain kernels). As it is common for linear filters, it is possible to work in the frequency domain by means of a Laplace or Fourier transform.

Volterra operator properties

In the following definition, L p (E, F) denotes the standard Lebesguespace of functions from vector spaces E to F with p-norm.

Definition 2 (Volterra operator of order n). Let hn ∈ L 1 (R n + , R) be a Volterra kernel, for n ∈ N * . Then we introduce the multilinear operator Vn :

L ∞ (R, R) × . . . × L ∞ (R, R) → L ∞ (R, R) such that function Vn[u1, . . . , un] is defined ∀t ∈ R by Vn[u1, . . . , un](t) = R n + hn(τ1, . . . , τn) n i=1 ui(t -τi)dτi (4) If u1 = . . . = un = u, Vn[u, . . . , u] = yn.
Property 1 (Symmetry). Given a symmetric kernel hn, the corresponding Volterra operator Vn is also symmetric, meaning

Vn u π(1) , . . . , u π(n) (t) = Vn[u1, . . . , un](t) (5)
for any permutations π and ∀t ∈ R.

In the following, symmetry of hn and Vn will be supposed. n q λ n-q µ q Vn u1, . . . , u1

n-q u2, . . . , u2

q (t) (6)
This also implies that Vn is a homogeneous operator of degree n, i.e. for any signal u and scalar α, Vn[αu, . . . , αu](t) = α n Vn u, . . . , u (t) (7)

State-of-the-art order separation

Nonlinear homogeneous order separation implies the ability to recover signals yn from the output y of a system described by a Volterra series truncated to order N . From [START_REF] Novak | Modeling of nonlinear audio systems using swept-sine signals: Application to audio effects[END_REF],

V [αu](t) = N n=1 α n yn(t). Consider a collection of input signals u k (t) = α k u(t), with α k ∈ R * , k = 1, . . . , N and note z k (t) = V [u k ](t)
their corresponding output through the system; then, for all time t:

     z1 z2 . . . zN      (t) =       α1 α 2 1 . . . α N 1 α2 α 2 2 . . . α N 2 . . . . . . . . . . . . αN α 2 N . . . α N N       •       y 1 y 2 . . . y N       (t) , α k ∈ R * Z(t) = A • Y (t) (8
) Since A is a Vandermonde matrix, it is invertible if and only if all α k are different; hence it is possible to recover terms yn.

But for real-valued α k , this type of matrix is known for becoming rapidly ill-conditioned when its size grows (meaning small noise in the measured outputs would become a large error in the estimation); so robustness decreases rapidly when the truncation order N increases. In order to circumvent that, it is possible to solve the linear problem Z = AY by using a Newton Recursive or Lagrange Recursive method [20, Algorithm 4.6.1 and 4.6.2].

But in practice, this approach is still very sensitive to the choice of amplitude factors α k . Indeed, for small amplitudes, higher orders will be hidden in measurement noise, while high values of α k will potentially overload the system, or simply lead it out of its Volterra convergence radius.

Despite those disadvantages, this separation method has been used intensively for simplifying identification process [START_REF] Lambert | Vandermonde Method for Separation of Nonlinear Orders and Measurement of Linear Response[END_REF]; it is generally used in frequency domain (the previous equations and remarks remains valid) and jointly with frequency probing methods [START_REF] Boyd | Measuring volterra kernels[END_REF][START_REF] Bard | Modeling of Nonlinearities in Electrodynamic Loudspeakers[END_REF]; recently, a maximum order truncation estimation method has been constructed from it [14]. In the following, this method will be referred to as the Amplitude Separation (AS) method.

PHASE-BASED HOMOGENEOUS SEPARATION METHOD

The starting point of the proposed separation method are the following remarks:

• using the AS method with factor 1 and -1, it is possible to separate odd and even orders by inverting a mixing matrix A with optimum condition number; • multiplying a signal by amplitude factor -1 is equivalent to taking its opposite phase; • thus, for a system truncated to order N = 2, there exists a robust separation method that relies only on phase deconstruction and reconstruction between tests signals. The main idea of this paper is to generalize the use of phase for robust separation method to Volterra systems with truncation N > 2.

Method for complex-valued input

This section proposes a theoretical separation method relying on the use of complex signals u(t) ∈ C as system inputs.

Principle

Using complex signals, factors α k in the AS method are not limited to real scalar. So it would be possible to choose values which only differs by their phase (e.g. are on the unit circle instead of the real axis). Noticing that 1 and -1 are the two square root of unity, a natural extension of the toy-method proposed for order-2 systems would be to take the N -th roots of unity as factors α k . Choosing α k = w k N with wN = e j 2π N , (8) becomes, for all time t:

     z1 z2 . . . zN      (t) =       wN w 2 N . . . w N N w 2 N w 4 N . . . w 2N N . . . . . . . . . . . . w N N w 2N N . . . w N 2 N       •       y 1 y 2 . . . y N       (t) , wN = e j 2π N Z(t) = W N • Y (t) (9
) where W N is the Discrete Fourier Transform (DFT) matrix of order N (after a column and row permutation 1 ). It is important to note that here the DFT does not apply on time but on the homogeneous nonlinear orders.

Since the DFT is invertible, order separation is possible. Furthermore, the DFT matrix is well-conditioned 2 , and the solution 1 It suffices to consider vectors Z(t) = [z N , z 1 , . . . , z N -1 ] T and Y = [y N , y 1 , . . . , y N -1 ] T to recover the usual DFT matrix. 2 Its condition number is even optimum, since it is 1 for any order N . can be computed using the Fast Fourier Transform algorithm 3 . In the following, this method will be referred to as the Phase Separation (PS) method.

Nonlinear order aliasing and rejection factor

Given the N -periodicity of N -th root of unity wn, the ouptut of a Volterra system with no order truncation is:

V [wN u](t) = N n=1 w n N ∞ r=0 yn+rN (t) (10)
By applying the PS method, estimation Y of nonlinear orders 1 to N yields:

Y (t) =      y1 + r=1 y1+rN y2 + r=1 y2+rN . . . yN + r=1 yN+rN      (t) (11)
Equation ( 11) reveals that estimation yn is perturbed by a residual term r=1 yn+rN , which is structured as an aliasing with respect to the nonlinear order: we 4 call this effect the nonlinear order aliasing.

For a band-limited input signal, presence of term y n+k in the estimated signal yn means presence of higher-frequency components than expected; therefore, this artefact can help detect a wrong truncation order N . But moreover, (11) permits to create higherorder rejection by using amplitude as a contrast factor. Taking α k = ρw k N , where ρ is a positive real number less than 1, estimation Y using PS method becomes

Y (t) =      ρ 0 ρ 2 0 . . . ρ N           y1 + r=1 ρ rN y1+rN y2 + r=1 ρ rN y2+rN . . . yN + r=1 ρ rN yN+rN      (t), (12)
creating a ρ N ratio between desired signal yn(t) and the first perturbation yn+N (t). Thus parameters N and ρ enables to reach a required Signal-to-Noise Ratio (SNR).

However, the need of complex input (and output) signals prevents the use of PS method in practice.

Application to real-valued input

Consider the real signal v(t) constructed as follows:

v(t) = w u(t) + w u(t) = 2 Re[w u(t)] ( 13
)
where w is a complex scalar on the unit circle (such that w = w -1 ) and u(t) a complex signal. Therefore, using property (6), and assuming symmetry for the operator Vn (meaning symmetry 3 Even if the gain in time computation is not significant since generally N is not a power of 2 and is not very high. 4 We and the anonymous reviewer that proposed this relevant expression.

for kernel hn), the order n contribution is:

yn(t) = Vn[v, . . . , v](t) = Vn w u + w u, . . . , w u + w u (t) = n q=0 n q
w n-q w q Vn u, . . . , u

(n-q) times , u, . . . , u

q times (t) = n q=0 n q w n-2q Mn,q(t) (14)
with Mn,q(t) = Vn u, . . . , u n-q , u, . . . , u q (t) ∈ C. This term represents the homogeneous contribution of order n for an equivalent multi-input system excited by combinations of u and u.

Remark 5. By symmetry of Vn, there is Mn,q(t) = Mn,n-q(t) and term M n,n/2 (t), for even n, is real. Therefore, from sum over q in (14), realness of yn(t) is recovered.

Remark 6. By their definition, terms Mn,0(t) (respectively Mn,n(t)) are homogeneous contribution of order n of the system excited by the complex signal u(t) (resp. u(t)).

Equation ( 14) shows that, in the output term yn, there is more than one characterising phase factor w. So PS method is not directly exploitable to separate terms yn.

Method for real-valued input

Difficulty analysis on an example: Consider a system truncated to N = 3 with input the real signal described in [START_REF] Lambert | Vandermonde Method for Separation of Nonlinear Orders and Measurement of Linear Response[END_REF]; then, omitting temporal dependency, its nonlinear orders are:

y1 = wM1,0 + w -1 M1,1 y2 = w 2 M2,0 + 2 M2,1 + w -2 M2,2 y3 = w 3 M3,0 + 3 wM3,1 + 3 w -1 M3,2 + w -3 M3,3 (15
) Only 7 different phase terms appears (from w -3 to w 3 ). Consider a collection of

K = 7 real signals v k (t) = w k K u(t) + w k K u(t)
, where wK is the first K-th root of unity, and z k (t) = V [v k ](t) their corresponding output through the system. Then:

         z1 z2 z3 z4 z5 z6 z7          (t) = W K •          M2,1 M1,0 + 3M3,1 M2,0 M3,0 M3,3 M2,2 M1,1 + 3M3,2          (t) , (16)
where W K is the DFT matrix5 of order K. Therefore, by application of the PS algorithm (i.e. inversion of W K ) on this of signals z k , the right-hand side vector in (16) is recovered. Further separation could be made using two different amplitudes to discriminate between M1,0 and M3,1, and thus be able to reconstruct nonlinear orders yn using (14).

Generalization: Using (14), it is possible to generalize phase grouping shown in (15) as follows (see Appendix 7 for detailed computation):

y(t) = -N ≤p≤N p even w p 1<|p|≤n≤N n even n n-p 2 M n, n-p 2 (t) + -N ≤p≤N p odd w p |p|≤n≤N n odd n n-p 2 M n, n-p 2 (t) (17)
So, by applying a PS algorithm of order K = 2N + 1, we can separate the terms

Qp(t) = |p|≤n≤N n≡p mod 2 n n-p 2 M n, n-p 2 (t) (18)
with -N ≤ p ≤ N . Then, application of the AS algorithm on each Qp (with N/2 amplitudes) gives all Mn,q terms; terms yn are reconstructed using ( 14). This concatenation of PS and AS algorithm constitutes the proposed phase-based method, which will be referred to as Phase-Amplitude Separation (PAS) method. As will be pointed out in Section 4.3, it can be more interesting to use directly terms Mn,q for identification, instead of nonlinear orders yn; this alternative process will be referred to as raw-PAS (rPAS) method.

Condition number

In numerical analysis, condition number κ measures the method's sensibility to noise in the measured data; it only depends on the solving method itself (the matrix to invert in linear problems), and not the data it is applied to. In this section, condition number is used to compare AS and PAS robustness 6 .

For AS method, amplitude factors α k are chosen equally-spaced in dB scale, with alternating signs:

α2p = γ p-1 and α2p+1 = -α2p (19)
where γ is a chosen spacing gain.

Because the DFT matrix is optimally conditioned, PAS method overall conditioning only depends on the K applications of AS method. Amplitude factors are also chosen equally-spaced in dB scale; the need to separate terms with same order-parity prevents us from using alternating signs. Only the worst condition number for all K sub-problems is reported.

Figure 1 presents conditioning for AS and PAS methods. For all maximum order truncation and gain spacing, an improvement from amplitude-based to phase-based method is visible. Furthermore, for negative gain spacing, optimum conditioning is divided by a factor 2 between AS and PAS. But, for both methods, the same behaviour when truncation order N increases is remarked (which is unsurprising due to the fact that PAS relies partly on AS).

Those results indicates that PAS robustness to noisy measurements should be better than AS.

APPLICATIONS

In order to test and compare the different separation method, data from simulation of a nonlinear device were used.

Simulation of a loudspeaker with nonlinear suspension

The simulated system was a loudspeaker with nonlinear suspension described by the modified Thiele & Small following equations:

u(t) = Rei(t) + L di(t) dt + Bl d (t) dt (20a) M d 2 (t) dt 2 = Bl i(t) -Rm d (t) dt - 3 n=1 kn n (t) (20b)
where u is the voltage at the loudspeakers terminals, i the current flowing through it and the position of the diaphragm. The term 3 n=1 kn n (t) corresponds to the nonlinear force that the suspension applies on the diaphragm.

Consider the state vector x = i d( )/dt T . Then, using state-space formalism, the system of input u(t) and output i(t) is written:

ẋ = Ax + Bu + K2(x, x) + K3(x, x, x) i = Cx (21)
where ẋ indicates the temporal derivative of x, and:

A =     - Re L 0 - Bl L 0 0 1 Bl M - k1 M - Rm M     , B =    1 L 0 0   , C = 1 0 0
and where K2 et K3 are multilinear functions of x such that: Simulations were made using numerical methods described in Appendix 8, with parameters given in Table 1.

K2(a, b) =    0 0 - k2 M a2b2    , and K3(a, b, c) =    0 0 - k3 M a2b2c2    Re 5, 7 Ω Rm 4, 06 • 10 -1 Nm -1 s L 1, 1 • 10 -1 H k1 912, 3 kg • s -2 Bl 2, 99 NA -1 k2 611.5 kg • m -1 s -2 M 1, 9 • 10 -3 kg k3 8, 0 • 10 7 kg • m -2 s -2

Error separation

Order separation method are compared using the relative error n of each order n for noisy measurements of system outputs, where

n = RMS( yn[k] -yn[k]) RMS(yn[k]) (22)
with yn the true nonlinear homogeneous contribution, yn their estimation, and RMS the standard Root-Mean Square measure.

Simulation were made at a sampling frequency of 20000 Hz, and with a truncation order N = 5. All signals were 1 second long.

Figure 2 shows the total output and first three nonlinear orders of the system. Its linear behaviour is predominant: y1 is one (respectively four) magnitude order above y3 (resp. y2). This large difference of amplitude between signals y (or equivalently y1) and y2 means that the quadratic order could be hidden by noise in relatively high SNR. Figure 3 compares the separation measure n for both AS and PAS methods applied to noisy output data with different SNR. Estimation on clean data output is not shown here; in this case, both methods perform similarly and give the true homogeneous contribution yn (within machine accuracy). The high values of relative errors 2 and 4 is due to the smaller amplitude of signals y2 and y4 in respect to the other orders, which makes their estimation more sensitive to measurement noise.

Furthermore, Figure 3 shows that PAS outperforms AS method in presence of noise. For same order n and SNR, the gain in n is around 6 dB.

Kernel identification using order separation

Identification algorithms

The standard Volterra identification method used in this paper is the Korenberg's algorithm for Least-Squares problem (KLS) as described in [START_REF] Goussard | Practical identification of functional expansions of nonlinear systems submitted to non-Gaussian inputs[END_REF].

It consists of solving the following linear-in-the-parameters problem:

y = Φf (23)
where:

• vector y = [y[0], . . . , y[L -1]]
T is the concatenation of all output samples, with L the signal length (in samples); Furthermore, using rPAS method, ( 23) is separated into the following problems Mn,q = Φn,qfn

• Φ = [φ[0], . . . , φ[L -1]]
for n = 1, . . . , N and q = 1, . . . , n/27 . Identification is then carried on separately for each couple (n, q); kernels coefficients are taken as the mean of the n/2 estimations. This gives the proposed rPAS-KLS identification algorithm.

Identification results

The aforementioned identification methods were tested on the nonlinear loudspeaker, truncated to order N = 3.

Simulation were made at a sampling frequency of 1500 Hz. Gaussian noise signals of amplitude 10 V and 2 second length were used for input test signals. Kernel memory length was supposed to be equal to M = 22 samples (equivalent to 14 ms memory).

First two kernels of the nonlinear loudspeaker are given in Figure 4. Both linear and quadratic kernels shows exponential-like decay; chosen memory length proves to be adequate at this sampling rate.

For all algorithms, relative identification error (computed as ξn = RMS hn -hn /RMS hn ) are given in Figure 5.

When using clean output data, the addition of a prior order separation stage before identification improves greatly overall estimation process: error is reduced of more than 30 dB for the first order and 50 dB for the second. In this case, AS-KLS and PAS-KLS performs similarly; this is due to the similar separation results of AS and PAS methods with clean data.

Likewise, the better separation robustness of PAS over AS (around 6 dB gain) induces PAS-KSL to be more robust to noise than AS-KLS (around 2 dB gain). Furthermore, for second and third-order, rPAS-KLS algorithm offers a little improvement over PAS-KLS. This amelioration is the direct consequence of using terms Mn,q rather than nonlinear orders yn directly; indeed, averaging over all estimated fn,q for identification of kernel hn can be seen as a way to artificially increase the data size used for identification, thus leading to better estimates.

The first two estimated kernels using rPAS-KLS on clean data are shown in Figure 6. We can see that the second-order kernel, which has a relative error ξ2 = -46 dB, is well-identified using this method. Figure 6: First and second-order kernels estimated with rPAS-KLS.

CONCLUSIONS

In this paper, a new method of nonlinear homogeneous order separation for Volterra series was proposed, based on phase dissimilarity. This method has been shown to be better conditioned, and thus more robust to noise, than amplitude-based separation algorithm. Furthermore, two Volterra kernels identification methods were constructed combining the proposed separation process and a Least Squares-based identification algorithm from literature. Those methods have been shown to greatly improves kernel estimation.

Future works will be on the reproduction, with real signals, of the product's complex behaviour that enables usage of the theoretical PS method. An other interest would be to derive another identification method from Korenberg's algorithm that could be used directly on the phase-grouped terms Qp, without needing to use AS method for further separation.

Figure 1 :

 1 Figure 1: Condition number κ = A A -1 (Frobenius norm): evolution w.r.t gain spacing γ and truncation order N (3 to 6, from bottom to top) for AS (solid line, × indicating minima) and PAS (dashed line, • indicating minima) methods.

3 Figure 2 :

 32 Figure 2: Total output and first three nonlinear orders yn of the simulated loudspeaker for a cosinusoidal input of frequency 100 Hz and amplitude 10 V.

Figure 3 :

 3 Figure 3: Relative error n (in dB) of separation w.r.t. nonlinear order n and SNR (80 dB, 60 dB and 40 dB from bottom to top) for AS (solid line with ×) and PAS (dashed line with •) methods. Input signals are linear sweep of amplitude 10 V going from 30 Hz to 200 Hz.

Figure 4 :

 4 Figure 4: First and second-order kernels computed from (30) and (31) in Appendix 8.2.

Figure 5 :

 5 Figure 5: Relative error ξn (in dB) of identification w.r.t. order n for all identification methods using clean and noisy data (SNR = 80 dB)

Table 1 :

 1 Thiele & Small parameters of SICA loudspeaker model Z000900, given by constructor [21] (apart from Bl and the kn, which were measured experimentally).

It is important to notice that the right-hand side vector has hermitian symmetry, due to its Fourier Transform (the left-hand side) being real.

As PAS and rPAS methods share the same steps, their condition number are equal.

We consider only one case for the complex conjugates couples (n, q) and (n, n -q).

Proceedings of the 20 th International Conference on Digital Audio Effects (DAFx-17), Edinburgh, UK, September[START_REF] Arie | Modeling of the nonlinear response of an electrodynamic loudspeaker by a Volterra series expansion[END_REF][START_REF] Goussard | Practical identification of functional expansions of nonlinear systems submitted to non-Gaussian inputs[END_REF][START_REF] Novak | Modeling of nonlinear audio systems using swept-sine signals: Application to audio effects[END_REF][START_REF] Rébillat | Identification of cascade of Hammerstein models for the description of nonlinearities in vibrating devices[END_REF][START_REF] Billings | Nonlinear system identification: NAR-MAX methods in the time, frequency, and spatio-temporal domains[END_REF] 2017

APPENDIX: DETAILED COMPUTATION OF (17)

Consider a system truncated to N = 3 with real input signal described in [START_REF] Lambert | Vandermonde Method for Separation of Nonlinear Orders and Measurement of Linear Response[END_REF]. From (17), we have:

So, using (1), the overall output of the system is:

n q w n-2q Mn,q(t)

by posing p = n -2q. Taking into account the fact that both sum are finite, it is possible to inverse their orders, and thus:

8. APPENDIX: SYSTEM NUMERICAL APPROXIMATION

Numerical simulation

The input signal u(t) is approximated at sampling rate fs with a zero-order holder; then, following the state-space representation in (21), output current i at sample time l is given by:

where xn are the states of nonlinear homogeneous order n; first three orders are given by the following recursive equations:

where Ts = 1/fs is the sampling time and ∆0 = A -1 e ATs -I is a bias term due to sampling.

Discrete-time kernel computation

Discrete-time kernels hn corresponding to the loudspeaker simulation using (28) and (29) are given by: hn[m1, . . . , mn] =

N n=1

Cgn[m1, . . . , mn]

where gn are the input-to-states kernels of order n; first three orders are given by:

with δm,n the Kronecker delta.


Yan Wang
email: yan.wang@etu.u-bordeaux.fr

Grégory Zacharewicz
email: gregory.zacharewicz@u-bordeaux.fr

David Chen
email: david.chen@ims-bordeaux.fr

Mamadou Kaba Traoré

Integrating dependency with DEVS in the process mining

Keywords: DEVS, process mining, dependency, time

Process mining is used as an intelligent technique to improve process model by extracting event log. DEVS is specified as one of the resulting process model for simulation and it is transformed from transition system by using region-based approach in the process mining. However, the discovered DEVS model is pretty lacking and has a negative effect on the overall process flow. This paper reviews current approaches and proposes a time-related method dependency. Using an example of event logs, we calculate the value of the dependency and integrate with DEVS model. We also discuss about the time criteria.

Introduction

Nowadays, the development of the information systems cannot be separated by the operational process. Moreover, more and more events with abundant process information are recorded. Despite the omnipresence of event data, most of the organizations analyze and build models based on expert assumption rather than observed fact. Thanks to process mining [START_REF] Van Der Aalst | Process mining: discovery, conformance and enhancement of business processes[END_REF], it provides the methods to discover monitor and improve actual processes by using event data to extract processrelated information. Process mining combines process model-driven approaches with data mining. It includes process discovery, process conformance and process enhancement.

The Discrete Event System Specification (DEVS) [START_REF] Wainer | Discrete-event modeling and simulation[END_REF] provides a hierarchical and modular formalism to describe the event system. It contains the basic components like explicit time, inputs, outputs, states and functions. DEVS is also specified as the resulting process model in the process discovery [START_REF] Wang | A proposal of using DEVS model for process mining[END_REF]. As DEVS has the advantage of explicit time and separating model simulation, a region-based approach is proposed to convert the transition system directly to DEVS. However, there are still two missing part which need to be enriched for the DEVS model in process mining, DEVS semantic and considered time. The DEVS process model is not strongly controllable to simulate business process, meaning that no controller or methods provide the desired states without choosing some bad states. It presents all the possible business process in the DEVS model without considering how to find the next transition. The objective of this paper is to integrate the dependency relation with DEVS model which has a strong relationship with the time. The new method provides more precise information for transitions.

First we will depict the whole structure of DEVS model in the process mining to provide a general background of this research in section 2. And the previous work region-based approach will also be introduced. Section 3 will give some related researches. Section 4 will explain how to calculate the dpendency in DEVS model and how it works. At last we conclude the paper in section 5. Figure 1 shows the general background of the whole research. We reuse the discovery technique and take the transition system instead of Petri Nets as input. The transition system is discovered by the traces in the event log. In the resource part, we need to provide the DEVS model base, in another words semantic DEVS to support the region-based approach. The process mining tool ProM is used as the development environment for simulation. The representational bias, noise and incompleteness [START_REF] Van Der Aalst | Process mining: discovery, conformance and enhancement of business processes[END_REF] are the main constraints when we try to transform the event log into other formalism with the optimal and desired results. According to the representational bias, DEVS model base provides several solutions. For example, parallel DEVS models with multiple inputs and outputs and time segments can solve the problem of concurrency, a switch network can solve the problem of OR-splits, generator coupled model can solve the problem of loop and duplicate actions. All these models will be designed in the DEVS model base. The noise defines event log contains rare and infrequent behavior not representative for the typical behavior of the process. The incompleteness defines event log contains too few events to be able to discover some of the underlying control-flow structures. Besides, time is another very important constraint which will lead to desired state scheduling. This paper focuses on providing new time-related methods to develop the DEVS models. At last, we wish to implement a structured and meaningful model used for anticipating optimal business process. in the last paragraph. These models will be used as components to support the transformation. The main part of this approach is to implement the transformation between transition system and DEVS model. Table 1 shows the mapping between transition system and DEVS. To implement the transformation, we divide the transition system as several regions. In this case all activities can be classified into entering the region, leaving the region and non-crossing. An activity cannot be entering the region in one part of the transition system and exiting the region in some other part. As DEVS model has the characteristic of separating model simulation, each region can be mapped with one component in the model base, each activity will be recognized as input event or output event. Then all the components will be connected by the internal or external transitions to build a complete DEVS model. The green part of the figure shows the extenstion of ontology based on various perspectives (organizational perspective, time perspective, data perspective, etc.) to be more meaningful and visualized like a map. So customers can simulate business process, identify various components and supervise the operating process. The red part of the figure is the main work of this paper and it will be shown in more detail in section 4.

DEVS in process mining

Related research

In the process mining [START_REF] Van Der Aalst | Process mining: discovery, conformance and enhancement of business processes[END_REF], heuristic mining algorithms try to discovery the dependency graph from the event log. It first offers a list of event log examples under limited time. Then it calculate the frequency between any pair of activities and the value of the dependency relation based on dependency measure. At last it derives the dependency graph using a threshold and the states are connected with the dependency.

In order to mine useful information , many researchers propose new approach and strategy to address the issue based on the dependency graph. Das et al. [START_REF] Das | Attribute Clustering and Dimensionality Reduction Based on In/Out Degree of Attributes in Dependency Graph[END_REF] propose the feature selection approaches to reduce the complexity of the overall process by allowing the data mining system to focus on what is really important. The proposed method encompasses a new strategy in dimensionality reduction by attribute clustering based on the dependency graph of the attributes. The method strongly based on the graph theory and information gain, an established theory of measuring uncertainty and quantified the information contained in the system. Zimmermann et al.. [START_REF] Zimmermann | Predicting Subsystem Failures using Dependency Graph Complexities[END_REF] use the complexity of dependency graphs for predicting failures at statistically significant levels. This helps for resource allocation and decision making.

Some researchers use dependency graph to represent different respectives. Wang et al. [START_REF] Wang | Representing document as dependency graph for document clustering[END_REF] propose a novel document representation model and a new similarity measure for documents which calculates the similarity based on dependency graphs. The similarity measure is applied to the Group average Agglomerative Hierarchical Clustering (GAHC) algorithm and promising results are obtained in the document clustering experiments.

Dependency based DEVS model

The discovered DEVS process model in the process mining is not comprehensive. For example, there are two transitions which start from the same atomic model but point to the different destination. We don't know which transition we can choose so we can't predict the business process based on the current DEVS model. In order to find the desired state and the next anticipating transition, we propose a method based on dependency with the criteria of time. If D i,j is close to 1, then there is a strong positive dependency between i and j. If D i,j is close to -1, then there is a strong negative dependency between i and j. Using dependency, figure 3 provides a more intuitive visualization. The thickness of the arcs corresponds to the frequencies of the corresponding paths. Such visualizations are important to get insight into the desired process flow.

D i,j = δ i,j -δ j,i δ i,j + δ j,i + 1 if i = j ( 1
)
D i,j = δ i,i δ i,i + 1 if i = j (2)
Meanwhile, we can also get the time duration from the event log. We select the earliest event time and the latest event time from the list of event logs as the time duration ta i . We make a criteria of the time that if the state is in the ta i , we will take D i,j into consideration. If not, we will reduce the value of D i,j .

Conclusion

This paper introduces the whole structure of specifying DEVS model as the resulting process model in the process mining. The original DEVS model can be built up based on the region-based approach. Then this paper focuses on the extension of DEVS model on time-related method. In order to predict the desired process flow, it proposes a dependency method with providing a more intuitive visualization based on a study case. Time criteria is also taken into consideration.

Fig. 1 .

 1 Fig. 1. DEVS-based process mining structure

Fig. 2 .

 2 Fig. 2. Main process based on the structure

Figure 2

 2 Figure 2 shows the whole process of Modeling & Simulation by integrating DEVS model with process mining. The blue part of the figure has already done in the last paper. First we need to extract the useful information from the list of event log. The event log is recorded by the XES standard. Using the state abstraction, we can automatically construct a transition system based on the event log. Meanwhile, it is necessary to construct the DEVS model base. It has atomic model and coupled model with special functions which we have talked

Fig. 3 .

 3 Fig. 3. Main process based on the structure

Table 1 .

 1 Mapping between transition system and DEVS

	Transition System	DEVS
	Region	Atomic model
	State	State
	Transition	Internal transition External transition
	Activity	Input event Output event


Yan Wang
email: yan.wang@u-bordeaux.fr

Mamadou Kaba Traoré
email: traore@isima.fr

Grégory Zacharewicz
email: gregory.zacharewicz@u-bordeaux.fr

David Chen
email: david.chen@u-bordeaux.fr

Verification and Validation of D2FD Method

Keywords: system inference, modeling and simulation, D2FD method, verification, validation, ProM, SimStudio

D2FD (Data to Fuzzy-DEVS) method provides a solution for the problem of system inference. This method is well designed and implemented as an available and dedicated plug-in within the process mining framework (ProM). This plug-in is also integrated with the simulation tool SimStudio. However, the last step of the process of inferring models and simulations, which is verification and validation, is missing. This paper proposes a new paradigm of verification and validation in system inference. The case study uses the method of comparing with other models as the main validation technique. Based on the same data source from the Dutch Employee Insurance Agency, it attempts to compare the previous results with other studies.

I. INTRODUCTION (HEADING 1)

System theory [START_REF] Simon | The architecture of complexity, Facets of systems science[END_REF] provides some methods to describe systems. The knowledge of the system can be organized in a 4level hierarchy [START_REF] Klir | Architecture of systems problem solving[END_REF]. It consists the source level, the data level, the generative level and the structure level. Moving between these levels, there are three systems problems: in system analysis, we have an existing system and we are trying to review or create source; in system design, the system doesn't exist yet and we are investigating the alternative structures for a new system; in system inference, the system exists and we are trying to generate the structure. Among them, system inference is a big challenge of system modeling.

D2FD method provides a method to deal with system inference and mine a discrete event simulation model. This method contains three parts. In the first part, it provides a new method to transform the event data to the event logs. The event data is observed from real world and the event logs is recorded based on XES (Extensible Event Stream) standard [START_REF] Van Der Aalst | Process Mining: Discovery, Conformance and Enhancement of Business Processes[END_REF]. In the second part, a method of transforming event logs into transition systems [START_REF] Robert | Formal verification of parallel programs[END_REF] is included. This method is coming from process mining [START_REF] Van Der Aalst | Process Mining: Discovery, Conformance and Enhancement of Business Processes[END_REF]. In the third part, a new method of transforming transition system into Fuzzy-DEVS (Discrete Event System Specification) is proposed. Fuzzy-DEVS model [START_REF] Kwon | Fuzzy-DEVS Formalisme: Concepts, Realization and Application[END_REF] not only provides a general framework to represent dynamic systems, but also represents more information including the modularity, frequency and timing aspects.

The original process of developing models and simulations has three steps. A conceptual model is designed by a subject matter expert (SME) from careful consideration of a problem and its domain. Then, it is realized via a source code simulation through the implementation of interfaces, data structures and algorithms. Finally, the output of the simulation for a set of test cases is validated against historical data or other trusted sources [START_REF] Pace | Ideas about simulation conceptual model development[END_REF]. The process of inferring models and simulations is a little different in the first steps. The first step is that an inference model is mined from the data or the problem in the reality using some mining techniques. However, the last two steps are the same. The inferred model is used for simulation, verification and validation. Model verification is often defined as "ensuring that the computer program of the computerized model and its implementation are correct". Model validation is usually defined to mean "substantiation that a computerized model within its domain of applicability possesses a satisfactory range of accuracy consistent with the intended application of the model" [START_REF] Schlesinger | Terminology for model credibility[END_REF].

Sargent [START_REF] Sargent | Verification and validation of simulation models[END_REF] proposes a paradigm that relates verification and validation to the model development process. This paradigm shows a Real World and a Simulation World. However, this paradigm is based on the process of developing models and simulations. In this paper, we propose a new paradigm of the process of inferring models and simulations. Based on this paradigm, we explain the position of D2FD method and supply this method with one validation method. The validation method is selected from all the validation techniques [START_REF] Sargent | Verification and validation of simulation models[END_REF]. At last, the case study chooses the Dutch Employee Insurance Agency [START_REF]Business Process Intelligence (BPI) Challenge[END_REF] as the data source. We compare the previous results [START_REF] Wang | Use of Fuzzy Clustering for Discrete Event Simulation Model Consruction[END_REF] and the other studies [START_REF] Jalali | Exploring different aspects of users behaviours in the Dutch autonomous administrative authority through process cubes[END_REF] for its validation. This paper is organized as follows. Section 2 introduces some related studies. Section 3 introduces the new paradigm. Section 4 briefly shows the D2FD method and locates it in the paradigm. Section 5 explains the main process of the selected validation method. The case study is shown in Section 6. The final conclusion is in Section 7.

II. RELATED STUDIES

The design of a model that appeared complete and robust can become incoherent, incomplete and potentially invalid during simulation implementation [START_REF] Sargent | Verification and validation of simulation models[END_REF]. Exploring alternative models and alternative modeling questions becomes impossible because SMEs are unable to identify the respective modifications that need to be made to the simulation [START_REF] Law | How to build valid and credible simulation models[END_REF]. It is important to make verification and validation.

A lot of techniques for building valid and credible simulation models are proposed [START_REF] Law | How to build valid and credible simulation models[END_REF][14] [START_REF] Harbola | Infinite Automata And Formal Verification[END_REF]. Gore et al. [START_REF] Gore | The need for usable formal methods in verification and validation[END_REF] provides the approach used in practice with a formal specification languages. They force attention to mathematical details. However, there are no related studies which focus on the process of inferring models and simulations.

III. NEW PARADIGM

The proposed paradigm is based on the paradigm in [START_REF] Sargent | Verification and validation of simulation models[END_REF] and the process of inferring models and simulation. It is shown in Figure 1.There are a Real World and a Simulation World. In the Real World, there some systems or problems. System theories describe the characteristics of the system and its behavior. System data is obtained by conducting experiments on the system. System theories are developed by abstracting what has been observed from the system and by hypothesizing from the system data and results. System theories are validated by performing theory validation. Theory validation involves the comparison of system theories against system data and results over the domain the theory is applicable for to determine if there is agreement. This process requires numerous experiments to be conducted on the real system. We now discuss about the Simulation World which contains four processes. Simulation model is always developed based on objectives. The inference model is the mathematical discipline to represent the system for the objectives of a particular study. The simulation model is the inference model running on the computer system like ProM. The simulation model data and results are the data and results from experiments conducted on the simulation model. The inference model is mined from the data or problem in the Real World. If it is a Fuzzy-DEVS model, the method of inferring is based on D2FD method. Then this model is implemented through a plugin [START_REF] Wang | The subversion server at the Technical University of Eindhoven[END_REF] available in ProM and experimented in SimStudio [START_REF] Traoré | SimStudio: a next generation modeling and simulation framework[END_REF]. Inference model validation is defined as determining that the theories and assumptions underlying the inference model are consistent with those in the system theories. Inference verification is defined as assuring that the software design and the specification for programming and implementing the inference model on the specified computer system is satisfactory. Operational validation is defined as determining that the model's output behavior has sufficient accuracy for the model's intended purpose over the domain of the model's intended applicability.

IV. D2FD METHOD

The D2FD method has three major stages, as presented in Figure 2: (1) from event data to event logs; (2) from event logs to transition system; (3) from transition system to Fuzzy-DEVS model. Here we introduce the general structure of D2FD method.

A. From Event Data to Event Logs

The data is observed from the world in the data level of Figure 1. However, not every data is available for D2FD method. The event data are the starting point and they are selected through several guidelines. The guidelines include the twelve guidelines from Van der Aalst [START_REF] Van Der Aalst | Process Mining: Discovery, Conformance and Enhancement of Business Processes[END_REF] and four more guidelines. When the event data is observed, five steps are proposed to transform event data to event logs as shown in Figure 2. First, we need to set up the goal by the interview of business people. The goal can be the business problem which we are going to solve. Second, System Entity Structure (SES) [START_REF] Zeigler | Modeling & simulation-based data engineering: introducing pragmatics into ontologies for net-centric information exchange[END_REF] defines a ontological framework in the systems theory. We construct SES from event data in order to discover the relationships between the activities and refine the event data. Third, we identify the activities as well as the modularity. The activities can be identified as public activity and private activity. If some activities have a strong relationship with some activities in other documents, we can define their children activities as private activities. Conversely, public activities. Fourth, process instance relates to the object that you will follow throughout the process. As we get the hierarchy of SES structure, we select the interesting level according to the goals. This interesting level is related the one of the attributes and contains several activities. Fifth, the activities of interesting attribute, time, case and instance are transformed into the parameters of the event logs.

B. Form Event Logs to Transition System

The method of transforming from event logs to transition system is coming from Two Phase Approach. State and transition can be mined from event logs. Let S T be the state and T be the transition. In this method, there are different methods in different dimensions to capture states, which lead to different kinds of transition systems. For example, past or future; set abstraction or multi-set abstraction; k-tail method. The constructed transition system is considered as a low-level process model. The method of transforming from transition system to Fuzzy-DEVS model is based on the previous work [START_REF] Wang | A proposal of using DEVS model for process mining[END_REF]. In Fuzzy-DEVS atomic model, an improved regionbased approach is defined to specify state. Let TS = (S T , A, T) be a transition system and R  S T be a subset of states. P a is a period time for each activity a  A. R is a region if for each activity a  A and one of the following conditions holds:

C. From Transition System to Fuzzy-DEVS Model

 All transition (s T 1 , a, s T 2 )  T enter R, i.e. s T 1  R and s T 2  R;  All transition (s T 1 , a, s T 2 )  T exit R, i.e. s T 1  R and s T 2  R;  All transition (s T 1 , a, s T 2 )  T do not cross R, i.e. s T 1 , s T 2  R or, s T 1 , s T 2  R;  For all the transitions a 1  T 1 , a 2  T 2 , …, a n  T n enter R, P a1  P a2  …  P an .
Let pa be the private activity and ua be the public activity. According to Fuzzy-DEVS formalism in chapter 3, the transformation follows the rules:

S R  (1)
Where the state of DEVS atomic model sS.

y x ua   (2)
Where the input value xX and the output value yY.

          I F S S Infinite T S s a t 1 0 0 ~ (3)
Where s 0 is the initial state, T F is the result coming from Adapted Fuzzy Time Controller, S I 1 is the input states of all internal transition. ) , ,

  s s s ua s T T T   (4) ( ) , , ( ~int 2 1 2 1 int
Where s1  R1 and s2  R2,  int is the result coming from dependency method.

) , ( : ~int   y (

1 ext T T ext s x e s s pa s T     (6)                        j i s s F s s F j i s s F s s F s s F s s F s s j i j i i j j i i j j i j i 1 ) ( ) 5) ) , , , , ( ) , , ( ~2 1 2
In Fuzzy-DEVS coupled model, fuzzy cluster is proposed for integration [START_REF] Wang | Use of Fuzzy Clustering for Discrete Event Simulation Model Consruction[END_REF]. Three functions are integrated with membership coefficients. In Equation 8, 9 and 10,  is the membership coefficient calculated based on Dependency Method and e is the elapsed time. While the time elapses, the membership coefficients may change.

V. VALIDATION TECHNIQUES

The operational validation is a big challenge for D2FD method. It is determining whether the simulation model's output behavior has the accuracy required for the model's intended purpose over the domain of the model's intended applicability. According to [START_REF] Sargent | Verification and validation of simulation models[END_REF], a lot of techniques are proposed and collected for the operational validation. In this paper, we choose to use "comparison to other models" as the main technique. This technique focuses on the subject and it is not observable. The simulation model which we construct is compared to other simulation models that have been validated.

VI. CASE STUDY

The case study is from BPI Challenge (2016). It covers the information of the customers and the records of the telephone calls in the category of question and complaints from a Dutch Employee Insurance Agency. The workbook message is contacted by customers through a digital channel (Here the Dutch language is translated into English). For validating the model built by D2FD method, we compare with other simulation models [START_REF] Jalali | Exploring different aspects of users behaviours in the Dutch autonomous administrative authority through process cubes[END_REF] that have been validated. Figure 3 shows the model built from question file by D2FD method. Compared with it, other model which considers the same problem is shown in Figure 4. The process from werkmap to taken is the same but the other processes show different possibilities. The werkmap to taken is not considered as important in Figure 3. Figure 5 shows the model built by D2FD method and Figure 6 shows the validated model. These two models show almost the same process between the channels. However, by using D2FD method, which activity is using channels can be detected. For example, werkmap is using channel 1.

VII. CONCLUSION

Verification and validation are very important for simulation models. The aim is to prove the coherence of the model, to ensure the correct use of the modeling means and account for the description of the requirements that prevailed in the existence of models. Despite the process of developing models and simulations, a new process of inferring models and simulations is proposed. The paradigm provides a different perspective of modeling and simulation. The method of this paradigm is based on D2FD method. It is available, interoperable, feasible and visible. One of the verification and validation techniques is selected to supply D2FD method. The case study illustrates the validation of the real enterprise problem. In the full paper, the case study will be explained more in detail. We anticipate using more verification and validation techniques for the real problems.

Figure 1 :

 1 Figure 1: The structure of the inferring process with verification and validation.

Figure 2 :

 2 Figure 2: The structure of D2FD method.

  Fuzzy-DEVS model is based on the Fuzzy-DEVS formalism which applies fuzzy sets theory into the functions of DEVS formalism. It consists of Fuzzy-DEVS atomic model and Fuzzy-DEVS coupled model. A fuzzy atomic model M ~ is characterized by: M ~ = <X, Y, S, δ ~int , δ ~ext , λ ~, ta ~>  X: the set of input values;  Y: the set of output values;  S: the set of states;  δ ~int : S×S → [0, 1], fuzzy internal transition function;  δ ~ext : Q×X×S×S → [0, 1], fuzzy external transition function, Q = {(s, e) | s  S, 0 ≤ e ≤ ta(s)} where ta(s) is the defuzzified value of ta ~; 1], fuzzy time advance function, A ~ = the set of fuzzy linguistic numbers. A coupled model DN is defined: DN = <X, Y, D, EIC, EOC, IC, SELECT>  X: input event sets;  Y: output events sets;  D: DEVS components set;  EIC  {((N, ip N ), (d, ip d )) | ip N IPorts, dD, ip d IPorts d }.  EOC  {((d, op d ), (N, op N )) | op N OPorts, dD, op d OPorts d }.  IC  {((a, op a ), (b, ip b )) | a, bD, op a D, OPorts a , ip b IPorts b }.  SELECT: 2 D →{}→D, tie-breaking selector.

( 7 )

 7 Where the elapsed time e: 0  e  ta ~,  ext is the result coming from the dependency method. The transition system is first divided into regions then transformed into state. The public activities can be the sets of the input or the output values. If the state is the initial state, the fuzzy time is set as 0. If the state has no internal transition, the fuzzy time is set infinite. Otherwise, the fuzzy time is calculated based on Adapted Fuzzy Time Controller (AFTC). Time duration and remaining time are the inputs of AFTC. They first convert into fuzzy sets by membership function and then defuzzify into crisp value. They activate the final fuzzy time based on the rule base. If the transition T contains the public activity, it is transformed into fuzzy internal transition. If the transition T contains the private activity, it is transformed into fuzzy external transition. Both the fuzzy internal transition and fuzzy external transition have the membership function . It is measured by Dependency Method. The dependency method is based on Equation 7. The fuzzy output function has the same membership function as fuzzy internal transition.

Figure 3 :

 3 Figure 3: Question Model built by D2FD method.

Figure 4 :

 4 Figure 4: Other model compared with Figure 3. Another comparison uses the channel file as experiment.Figure5shows the model built by D2FD method and Figure6shows the validated model. These two models show almost the same process between the channels. However, by using D2FD method, which activity is using channels can be detected. For example, werkmap is using channel 1.

Figure 5 :

 5 Figure 5: Channel Model built by D2FD method.

Figure 6 :

 6 Figure 6: Other model compared with Figure 5.