Root Cause Diagnosis of Plant-Wide Oscillations Based on

Article pubs.acs.org/IECR

Root Cause Diagnosis of Plant-Wide Oscillations Based on Information Transfer in the Frequency Domain Shu Xu,† Michael Baldea,† Thomas F. Edgar,*,† Willy Wojsznis,‡ Terrence Blevins,‡ and Mark Nixon‡ †

McKetta Department of Chemical Engineering, The University of Texas at Austin, Austin, Texas 78712, United States Process Systems and Solutions, Emerson Process Management, Round Rock, Texas 78759, United States

‡

ABSTRACT: Plant-wide oscillations generated in a single unit can negatively affect the overall control performance of the process; thus, it is necessary to detect them and diagnose their root cause. However, the interference of noise and the need for oscillation propagation routes pose more challenges for process engineers. In this paper, the concept spectral transfer entropy is proposed and its connection to the spectral Granger causality is derived. Moreover, an information transfer method incorporating spectral envelope algorithm and spectral transfer entropy is applied to provide new diagnostic guidance, whose feasibility and effectiveness have been demonstrated by both simulated and industrial case studies. Compared with current methods, the new procedure enjoys the following advantages: (a) performing oscillation detection and diagnosis within a targeted frequency range and mitigating the effects of measurement noise outside the bandwidth; (b) provides an nominal causal map reflecting the oscillation propagation pattern. The root cause obtained by the method in the industrial case is further validated by a wavelet power spectrum.

1. INTRODUCTION A modern chemical plant is massively instrumented with IPenabled intelligent sensors, controllers, and actuators collecting online process information. Such enormous amounts of data facilitate the development of data-driven methods such as principal component analysis (PCA)1−3 that provide process engineers with more convenient and intuitive ways to monitor the process than traditional first-principles models. One of the most important issues in process monitoring is to detect and diagnose process disturbances such as plant-wide oscillations.4 Plant-wide oscillations can occur largely due to poorly tuned controllers, actuator nonlinearities, and possible external oscillatory disturbances.5−7 Because the material flow streams between units are highly correlated, oscillations generated at one point will propagate to the unit or even the whole plant through connecting flow streams and may cause poor control performance, inferior quality products, excessive energy consumption, and even plant safety issues.4 A distinguishing feature of oscillations is that, unlike other disturbances, they have similar spectral patterns in affected variables and can be detected using power spectra. A spectral envelope method used by Jiang et al. (2007)8 provides an intuitive and fast way to find the shared oscillation frequency and related contributing variables so that power spectra on individual variables are no longer needed. After detecting the oscillations, the next step is to determine the root causes of such oscillations from contributing variables. Generally, methods for root cause diagnosis of plant-wide oscillations can be categorized into process data-based analysis methods and topology-based methods.5 While the process data-based methods such as the oscillation contribution index (OCI)8 aim to analyze nonlinearity and power spectrum features, the topology-based methods are either based on process flow sheets and qualitative models9,10 or based on a nominal map exhibiting causal relationships. There are two types of methods to construct such © 2016 American Chemical Society

a casual map: time-domain based ones including crosscorrelation analysis,11 nearest neighbors,12 Granger causality,7,13,14 and transfer entropy15,16 and frequency-domain based ones including directed transfer functions,17 partial directed coherence,18 and spectral Granger causality.7,19 Analyzing data within the frequency domain helps the users focus on frequencies containing useful information and provides a shield from unwanted noise. In this paper, we extended the concept of transfer entropy, a key measure of information transfer in the information theory, to the frequency domain, and showed its connection to the spectral Granger causality in section 2. Brief descriptions of the spectral envelope method and related oscillation contribution index (OCI)8 are also included in that section. Moreover, an information transfer method which combines spectral envelope algorithm with spectral transfer entropy is proposed in section 2 and tested in section 3 through simulated and industrial case studies. The diagnosis result is compared with the oscillation contribution index (OCI) method,8 and the effectiveness of the new method is demonstrated: it not only can detect and diagnose the plantwide oscillations at a specific frequency, but also can provide a causal map showing the oscillation propagation routes. To facilitate the visualization of oscillation frequencies, a wavelet power spectrum of the root cause variable is provided.

2. METHODS DESCRIPTION 2.1. Spectral Envelope Method. First proposed by Stoffer et al. (1993),20 the spectral envelope method has been applied to detect common signals in a series of time series,21,22 and Received: Revised: Accepted: Published: 1623

August 20, 2015 January 16, 2016 January 18, 2016 January 19, 2016 DOI: 10.1021/acs.iecr.5b03068 Ind. Eng. Chem. Res. 2016, 55, 1623−1629

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Industrial & Engineering Chemistry Research

variables at frequency ω with the larger contributions at the spectral envelope peak.5 2.3. Spectral Granger Causality. Granger causality,13 originally developed within the context of econometric theory, is broadly applied in determining the ”predictive causality” between two time series in areas such as neurosciences19,23,24 and process modeling.7 A powerful feature of Granger causality is that it can be decomposed by frequency25 so that spectral causality at specific frequencies can be calculated, which is very useful when the signals contain unwanted frequencies. Assume X(t) and Y(t) are two time series describing stationary stochastic processes, a bivariate autoregressive (AR) model can be built as follows:

successfully pinpointed the plant-wide oscillations with common frequencies in industrial applications.8 Assume x(t) = [x1(t), x2(t), ..., xn(t)]T to be a time series on n ℜ , and each variable xi(t) has been autoscaled to have identical power. Define the spectral envelope tobe ⎧ β*Px (ω)β ⎫ ⎬ λ(ω) = sup ⎨ β≠ 0 ⎩ β*Vβ ⎭

(1)

where β is an n-dimensional real or complex vector, * represents the conjugate transpose, Vx and Px (ω) denote the covariance matrix and the power spectral density (PSD) matrix of x respectively, and ω is the normalized frequency satisfying −1/2 ≤ ω ≤ 1/2. λ(ω) in eq 1 denotes the largest portion of power that can be obtained at frequency ω. Since the data have been normalized, λ(ω) is the largest eigenvalue of Px (ω), and β (ω) is the corresponding eigenvector. Assume x (t) has N samples, i.e., t = 0, 1, ..., N − 1. The Fourier frequencies are defined as ωk = k/N, for k = 1, 2, ..., [N/2], where [N/2] is the greatest integer less than or equal to N/2. A simple estimate of Px(ωk) is given by

⎡ X(t )⎤ ⎢ ⎥= ⎢⎣ Y (t ) ⎥⎦

⎡ εx , t ⎤ ⎡ Σxx Σxy ⎤ ⎥ Σ = cov⎢ ⎥ = ⎢ ⎣ εy , t ⎦ ⎢⎣ Σyx Σyy ⎥⎦

hjIN̂ (ωk + j)

∑

(2)

j =−r

⎡ Axx (ω) Axy (ω)⎤⎡ X(ω)⎤ ⎡ E (ω)⎤ x ⎢ ⎥⎢ ⎥ ⎥=⎢ ⎢ A (ω) A (ω)⎥⎢⎣ Y (ω) ⎥⎦ ⎢ Ey(ω)⎥ ⎣ ⎦ yy ⎣ yx ⎦

N−1

1 [ ∑ x(t )exp( −2πitωk)] N t=0

t=0

(3)

Assume that λ1̂ (ω) = λ̂(ω), λ̂2(ω), ..., λ̂n(ω) are the eigenvalues of P̂ x (ω) arranged in decreasing order, and β̂1(ω) = β̂(ω), β̂2(ω), ..., β̂n(ω) are the corresponding eigenvectors. The asymptotic covariance matrix of the sample optimal scaling vector β̂(ω) is given by

⎡ X(ω)⎤ ⎡ Hxx(ω) Hxy(ω)⎤⎡ Ex(ω)⎤ ⎥⎢ ⎥ ⎢ ⎥=⎢ ⎢⎣ Y (ω) ⎥⎦ ⎢⎣ Hyx(ω) Hyy(ω)⎥⎦⎢⎣ Ey(ω)⎥⎦

S(ω) = ⟨X(ω)X *(ω)⟩ = H(ω)ΣH *(ω)

Vβ (̂ ω) = v−2λ1̂ (ω) ∑ λl̂ (ω)[λ1̂ (ω) − λl̂ (ω)] −2βl̂ (ω) (4)

⎡

OCIj(ω) =

hj2)−1/2 .

⎡ X(ω)⎤ ⎡ H̃ xx(ω) H̃ xy(ω)⎤⎡ Ex(ω)⎤ ⎥⎢ ⎥ ⎢ ⎥=⎢ ⎢⎣ Y (ω) ⎥⎦ ⎢⎣ H̃ yx(ω) H̃ yy(ω)⎥⎦⎢⎣ Eỹ (ω)⎥⎦

0⎤

(11)

where Eỹ (ω) = Ey(ω) −

Σxy Σxx

Ex(ω)

(12)

and ⎡ ⎡ H̃ (ω) H̃ (ω)⎤ ⎢ Hxx(ω) + xy ⎢ xx ⎥=⎢ ⎢ H̃ (ω) H̃ (ω)⎥ ⎢ yy ⎣ yx ⎦ ⎢ H (ω) + ⎢⎣ yx

|βj ̂(ω)| 2σ β(̂ ω)

1

premultiplying eq 9 on both sides with ⎢−Σ /Σ 1 ⎥, we ⎣ 12 11 ⎦ obtain

where v = The distribution of 2|β̂j(ω) − 2 βj(ω)| /σj(ω) approximately follows a Chi-squared distribution with 2 degrees of freedom, where σj(ω) is the jth diagonal element of Vβ̂(ω), and β̂j(ω) and βj(ω), j = 1, ..., n are the jth element of the estimated and true optimal scaling vector, respectively. If 2|β̂j(ω)|2/σj(ω) violates the threshold χ22(α), we assume oscillation at frequency ω is contained because the null hypothesis βj(ω) = 0 is rejected with (1 − α) confidence. If βj(ω) ≠ 0, we assume that the corresponding variable j contains such an oscillation frequency. 2.2. Oscillation Contribution Index. The oscillation contribution index of xj(t)8 is defined tobe r (∑ j =−r

(10)

where Σ is the covariance of the full model residuals and X*(ω) is the Hermit transpose of X(ω). For the X1 process,

l=2 l

(9)

Thus, the spectral density matrix S(ω) is derived as

n

β ̂*(ω)

(8)

where the components of the coefficient matrix [Aij(ω)] are Aij(ω) = δij − ∑k p= 1 aij,ke−iωk. Denote the transfer function matrix H(ω) = [Aij(ω)] −1, and eq 8 can be rewritten as follows:

N−1

[ ∑ x(t )exp( −2πitωk)]*

(7)

If we perform Fourier transform on eq 6, we obtain

where hj is symmetric positive weights satisfying hj = h−j ({h0 = 3/9, h±1 = 2/9, h±2 = 1/9 }in this paper), and IN̂ (ωk) =

(6)

where aij,k are the AR coefficients, and the residual covariance matrix as

r

Px̂ (ωk) =

⎡ axx , k axy , k ⎤⎡ X(t − k)⎤ ⎡ εx , t ⎤ ⎥+⎢ ⎥ ⎥⎢ yx , k ayy , k ⎦⎢ ⎣ Y (t − k) ⎥⎦ ⎣ εy , t ⎦ k=1 ⎣ p

∑ ⎢a

(5)

where σβ̂(ω) is the standard deviation of the magnitude of the optimal scalings of all the oscillating variables. Commonly, variables having OCI (ω) > 1 are considered as the root cause

⎤ Hxy(ω) Hxy(ω)⎥ Σxx ⎥ ⎥ Σxy ⎥ Hxx(ω) Hyy(ω)⎥ Σxx ⎦ Σxy

(13) 1624

DOI: 10.1021/acs.iecr.5b03068 Ind. Eng. Chem. Res. 2016, 55, 1623−1629

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Industrial & Engineering Chemistry Research The spectrum of X is represented by an “intrinsic” term and a “causal” term: * (ω) + H̃ xy(ω)Σ̃yyH̃ xy *(ω) Sxx(ω) = H̃ xx(ω)ΣxxHxx

(14)

where Σ̃yy = Σyy − (Σ2xy/Σxx). Therefore, the spectral Granger causality from Y to X at frequency f is ⎛ |S | ⎞ IY → X(ω) = ln⎜ X ⎟ ⎝ |SX | Y | ⎠ ⎞ ⎛ |Sxx(ω)| ⎟ = ln⎜ ̃ (ω)Σ̃ 22H12 ̃ *(ω)| ⎠ ⎝ |Sxx(ω) − H12

(15)

2.4. Spectral Transfer Entropy. The application of information theory has gained an increasing amount of interest recently,26−28 and an typical example is related to the concept of transfer entropy. First proposed by Schreiber (2000), it measures the amount of directed transform of information between two random variables,29 and it has been applied in process industries for diagnosis of root cause of oscillation.5,15,16 In the time domain, the transfer entropy between two variables is defined as TY → X = H(X |X −) − H(X |X − ⊕ Y −)

Figure 1. Combined methodology for oscillation detection and diagnosis in the frequency domain.

method is used for unveiling oscillation propagation routes, and finding the contributing variables (root causes) at the dominant frequency f. The wavelet analysis serves as a confirmation step, like field tests, which is critical in this case, because we have to confirm the cause of oscillation in order to take steps.

(16)

where H(X) is the Shannon entropy of X, H(·|·) is the conditional entropy, ⊕ denotes a joint relation, and X−, Y− stand for the past values of X, Y. Given the past values of X, the transfer entropy can be seen as the reduction of uncertainty in the future values of X by knowing the past values of Y. In the frequency domain, we define the spectral transfer entropy in a similar fashion: TY → X = H(X(ω)) − H(X(ω)|Y (ω))

3. CASE STUDY In this section, the procedure in the frequency domain including the spectral envelope and the spectral transfer entropy calculation will be tested based on a simulated and an industrial data set. 3.1. Simulated Data. Five time series are generated by an autoregressive (AR) model shown in eq 21:18

(17)

since the Fourier transform of a Gaussian is also a Gaussian (for a proof, see Appendix A). For Gaussian variables, we have30 1 1 ln(|Σ(X )|) + n ln(2πe) 2 2 1 1 ⇒ ln(|SX |) + n ln(2πe) 2 2

X1, n = 0.95 2X1, n − 1 − 0.9025X1, n − 2 + w1, n

H (X ) =

1 1 ln(|Σ(X |Y )|) + n ln(2πe) 2 2 1 1 ⇒ ln(|SX | Y |) + n ln(2πe) 2 2

X 2, n = 0.5X1, n − 2 + w2, n X3, n = −0.4X1, n − 3 + w3, n

(18)

X4, n = −0.5X1, n − 2 + 0.25 2X4, n − 1 + 0.25

H (X | Y ) =

2X5, n − 1 + w4, n X5, n = −0.25 2X4, n − 1 + 0.25 2X5, n − 1

(19)

+ w5, n

From eqs 18 and 19, we have TY → X = H(X(ω)) − H(X(ω)|Y (ω)) =

(21)

where w1,n, w2,n, w3,n, w4,n, and w5,n are drawn from Gaussian noise with zero mean and unit variance. As shown in the model, it is clear that X1 is the root cause of the oscillation. The independent power spectrum shown in Figure 2 suggests that the variables share a common oscillation at 0.12 Hz. If we apply the spectral envelope method on the simulated data set, the dominant frequency is clearly revealed, as shown in Figure 3. To find out the root cause of the oscillation, the oscillation contribution index (OCI) was calculated in Table 1. The OCI of X1 has a larger value than others, and it suggests that X1 is likely to be the source. The spectral transfer entropy matrix at f = 0.12 Hz is shown in Figure 4, and the darkness is proportional to the strength of causality. X1 strongly affects X2 and X4, and also exerts moderate causal effects on X3. Moreover, there is a mutual causal relationship, though not very strong, between X4 and X5.

1 ⎛ |SX | ⎞ 1 ln⎜ ⎟ = IY → X(ω) 2 ⎝ |SX | Y | ⎠ 2

(20)

Thus, in the frequency domain, for Gaussian variables, the Granger causality and transfer entropy are linear correlated by a factor of 2. Thus, the spectral entropy can be calculated in the same ways as spectral Granger causality, including AR model identification and Fourier transformation.19 A similar result was obtained in the time domain by Barnett et al. (2009).30 2.5. Methodology for Oscillation Detection and Diagnosis in the Frequency-Domain. A combined methodology for oscillation detection and diagnosis in the frequency domain is shown in Figure 1. The spectral envelope is applied to find the dominant frequency f of oscillations in the data, dampening the effects of noise. The spectral transfer entropy 1625

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Figure 2. Power spectrum of the simulated data.

Figure 4. Spectral transfer entropy of the simulated data.

Figure 5. Process topology of the simulated data based on spectral transfer entropy.

PI, TI, and SI stand for the indicators of flow, level, pressure, temperature, and rotor speed. As shown in Figure 6, there are two decanters, three distillation columns, and numerous recycle streams in the process. A common disturbance with an oscillation period of about 2 h (about 320 samples/cycle) had been identified, namely: valve stiction in the actuator of control loop LC2. The feasibility and effectiveness of the spectral transfer entropy method will be demonstrated. The first 28 h data sampled at every 20s are selected to perform oscillation diagnosis, including 14 controlled process variables (pv’s). The normalized time trends and power spectra of the data are shown in Figures 7 and 8: several variables share a common oscillation at a frequency about 0.003 cycles/sample. Instead of drawing the power spectra for all variables, the spectral envelope method successfully finds a common frequency at 0.0031 cycles/sample (approximately 322 samples/cycle) shown in Figure 9, and related variables with values of 2|β̂j(ω)|2/σj(ω) larger than χ22(0.001) = 13.82 are listed in Table 2. Combining Figures 7 and 8 with Table 2, we can see that the eight variables listed are stationary, and their oscillation contribution indices (OCI) calculated from eq 5 are listed in descending order. The variable LC2.PV has the largest value of OCI, which indicates that it may be the source of oscillation. If one assumes those eight stationary variables are Gaussian, at f = 0.0031 cycles/sample, the resulting spectral transfer entropy matrix and process topology diagram are shown in Figures 10 and 11, respectively:

Figure 3. Spectral envelope of the simulated data.

Table 1. Oscillation Contribution Index variable

OCI

X1 X2 X3 X4 X5

1.6247 0.0408 0.9952 0.1960 1.0071

The process topology diagram was drawn based on the transfer entropy matrix, as shown in Figure 5: In Figure 5, a line with an arrow indicates a unidirectional causality from one variable to the other, a double headed arrow suggests a bidirectional causality relationship. From both figures we can see clearly that variable X1 is the root cause of the oscillation, as validated in the process model shown in eq 21. 3.2. Industrial Data. An industrial data set provided by the Advanced Controls Technology group of Eastman Chemical Company5,8 was used. The process schematic of the plant is shown in Figure 6: In Figure 6, AC, FC, LC, PC, and TC stand for controlled composition, flow, level, pressure, and temperature tags; FI, LI, 1626

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Figure 6. Process schematic. The oscillation variables are marked by circle symbols.

Figure 7. Time trends of process variables(pv’s).

Figure 8. Power spectra of process variables (pv’s).

A causal map representing the interconnected oscillation propagation pathways is shown in Figure 11. We can see that LC1.pv and LC2.pv do not receive any causal effects from other variables. While LC2.pv reaches all other variables directly or indirectly except LC1.pv, LC1.pv only affects FC1.pv. Thus, we may draw a conclusion that LC2.pv is more likely to be the root cause candidate. Figure 11 also shows that oscillations in loop LC2 propagate to loops FC1, TC2, FC8, TC1, and FC5. By combining this information with the process flow sheet shown in Figure 6, one can conclude that oscillations of loop LC2 propagate from the left-hand decanter to columns 1, 2, and 3, which is consistent with material flow pathways in the physical process. Moreover, the causality between LC1.pv and FC1.pv

also matches the cascade control strategy for the liquid level of column 1. The oscillation period changes vs time can be revealed by the wavelet power spectrum of LC2.PV shown in Figure 12. The period of oscillation gradually reaches 320 samples/cycle near the 1000th sample, and remains stable afterward. 3.3. Discussion. From the results of simulated and industrial cases, we can see that the application of the procedure has been demonstrated to be successful. For oscillation detection the spectral envelope method provides a convenient and intuitive way to find the common frequency corresponding to abnormality and related variables. For the 1627

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Figure 11. Process topology of the Eastman Chemical process based on spectral transfer entropy. Figure 9. Spectral envelope of the Eastman Chemical process data.

Table 2. Summary of Test Statistics and Oscillation Contribution Index tag no.

test statistic

OCI

tag no.

test statistic

OCI

LC2.PV TC1.PV FC5.PV PC2.PV

706.1 1067.4 543.2 1370.4

1.62 1.42 1.03 0.14

FC1.PV LC1.PV FC8.PV TC2.PV

941.3 680.9 277.8 261.5

0.36 0.83 0.31 0.64

Figure 12. Wavelet power spectrum of variable LC2.PV.

Gaussian mixture models. Moreover, to enhance the robustness of the method, we can incorporate the minimum covariance determinant (MCD) detector to remove those outliers inconsistent with the majority of the data set which follows a normal distribution.

4. CONCLUSIONS As a common type of process disturbance, plant-wide oscillations propagate between units and negatively affect the process. Thus, it is necessary to detect those oscillations and diagnose the root cause. In this paper, the concept of transfer entropy from the information theory is extended to the frequency domain and its connection between spectral Granger causality is derived. As a result, an information transfer method is successfully applied which combines the spectral envelope with the spectral transfer entropy calculations. In simulated and industrial case studies, such a procedure is compared with the oscillation contribution index (OCI) and enjoys the advantages of extracting the process topology and oscillation propagation pathways at a specific frequency range, which is especially useful when the plant data is contaminated with noise at unwanted frequencies. It is worth pointing out the result of oscillation diagnosis in an industrial case study needs to be confirmed through field tests, and the wavelet power spectrum can be used as a promising tool to analyze the period change over time for root cause candidates.

Figure 10. Spectral transfer entropy of the Eastman Chemical process data.

root cause diagnosis step, both the oscillation contribution index (OCI) and the spectral transfer entropy give satisfactory results. While the OCI method enjoys the advantage of a low computational cost, it may be hard to find a physical explanation for the diagnosis result and the assumption that root cause variables have relatively larger power at the specific frequency is not validated. Although the spectral transfer entropy provides an intuitive process topology to visualize and explain the causal-effect relationship in the frequency domain, it is based on linear AR model identification and the assumption that the variables are Gaussian is often not satisfied. Future research will be focused on spectral transfer entropy calculation in nonlinear cases, one possible solution is incorporating the 1628

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(10) Yang, F.; Shah, S.; Xiao, D. Signed directed graph based modeling and its validation from process knowledge and process data. Int. J. Appl. Math. Comput. Sci. 2012, 22, 41−53. (11) Govindan, R.; Raethjen, J.; Kopper, F.; Claussen, J.; Deuschl, G. Estimation of time delay by coherence analysis. Phys. A 2005, 350, 277−295. (12) Bauer, M.; Cox, J. W.; Caveness, M. H.; Downs, J. J.; Thornhill, N. F. Nearest Neighbors Methods for Root Cause Analysis of Plantwide Disturbances. Ind. Eng. Chem. Res. 2007, 46, 5977−5984. (13) Granger, C. W. J. Investigating Causal Relations by Econometric Models and Cross-spectral Methods. Econometrica 1969, 37, 424−438. (14) Ancona, N.; Marinazzo, D.; Stramaglia, S. Radial basis function approach to nonlinear Granger causality of time series. Phys. Rev. E 2004, 70, 056221. (15) Bauer, M.; Cox, J.; Caveness, M.; Downs, J. J.; Thornhill, N. Finding the Direction of Disturbance Propagation in a Chemical Process Using Transfer Entropy. IEEE Trans. Control Syst. Technol. 2007, 15, 12−21. (16) Duan, P.; Yang, F.; Chen, T.; Shah, S. Direct Causality Detection via the Transfer Entropy Approach. IEEE Trans. Control Syst. Technol. 2013, 21, 2052−2066. (17) Kamiński, M.; Blinowska, K. A new method of the description of the information flow in the brain structures. Biol.Cybern. 1991, 65, 203−210. (18) Baccalá, L. A.; Sameshima, K. Partial directed coherence: a new concept in neural structure determination. Biol.Cybern. 2001, 84, 463− 474. (19) Barnett, L.; Seth, A. K. The MVGC multivariate Granger causality toolbox: A new approach to Granger-causal inference. J. Neurosci. Methods 2014, 223, 50−68. (20) Stoffer, D. S.; Tyler, D. E.; McDougall, A. J. Spectral Analysis for Categorical Time Series: Scaling and the Spectral Envelope. Biometrika 1993, 80, 611−622. (21) Stoffer, D. S. Detecting Common Signals in Multiple Time Series Using The Spectral Envelope. J. Am. Stat. Assoc. 1999, 94, 1341. (22) Stoffer, D. S.; Tyler, D. E.; Wendt, D. A. The Spectral Envelope and Its Applications. Statist. Sci. 2000, 15, 224−253. (23) Seghouane, A.-K.; Amari, S.-i. Identification of directed influence: Granger causality, Kullback-Leibler divergence, and complexity. Neural Comput 2012, 24, 1722−1739. (24) Tana, M. G.; Sclocco, R.; Bianchi, A. M. GMAC: A Matlab toolbox for spectral Granger causality analysis of fMRI data. Comput. Biol. Med. 2012, 42, 943−956. (25) Geweke, J. Measurement of Linear Dependence and Feedback between Multiple Time Series. J. Am. Stat. Assoc. 1982, 77, 304−313. (26) Kullback, S.; Leibler, R. A. On Information and Sufficiency. Ann. Math. Stat. 1951, 22, 79−86. (27) Massey, J. Causality, feedback and directed information. Proc. Int. Symp. Inf. Theory Applic.(ISITA-90). 1990, 303−305. (28) Massey, J.; Massey, P. Conservation of mutual and directed information. ISIT Proceedings. International Symposium on Information Theory, 2005; pp 157−158. (29) Schreiber, T. Measuring Information Transfer. Phys. Rev. Lett. 2000, 85, 461−464. (30) Barnett, L.; Barrett, A. B.; Seth, A. K. Granger Causality and Transfer Entropy Are Equivalent for Gaussian Variables. Phys. Rev. Lett. 2009, 103, 238701.

A. PROOF FOR THE FOURIER TRANSFORM OF GAUSSIAN VARIABLES Assume variable X follows a Gaussian distribution 2 2 1 X ≈ f (x) = σ 2π e−x /2σ , its Fourier transform is given by ⎡ 1 2 2⎤ Fx⎢ e−x /2σ ⎥(ω) = ⎣ σ 2π ⎦ ∞

=

∞

∫−∞ σ 12π e−x /2σ e−iωx dx 2

2

∫−∞ σ 12π e−x /2σ [cos(ωx) − i sin(ωx)] dx =

2

2

∞ 2 2 1 [ e−x /2σ cos(ωx) dx σ 2π ∞ −∞ 2 2 −i e−x /2σ sin(ωx) dx]

∫

∫−∞

(22)

Since the second integrand is odd, integration over a symmetrical range gives zero. Calculating the first integral we have Fx[X ](ω) = 2

1 σ 2π

⎛ ω 2σ 2 ⎞ 1 2 2 2π σ e−σ ω /2 = exp⎜ − ⎟ 2 2 ⎠ ⎝ (23)

Thus, we can see that the Fourier transform of a Gaussian is also Gaussian, and the result can be extended to a multivariate case.

■

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

■ ■

ACKNOWLEDGMENTS The authors gratefully acknowledge financial and technical support from Emerson Process Management. REFERENCES

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DOI: 10.1021/acs.iecr.5b03068 Ind. Eng. Chem. Res. 2016, 55, 1623−1629