A.2 Semi-qualitative, Symbolic Data Analysis
A.2.1 Data Preprocessing
As most of industrial data are multidimensional and have superponed noise elements in the signal, the algorithm can be extended with static and dynamic principal component analysis [115, 165] and a ltering technique to handle these tasks. These preprocessing elements serve as add-ins for a generalized application area of the basic algorithm.
Dynamic Principal Component Analysis
Principal Component Analysis, also called hoteling algorithm, is a widely known and applied method for lowering the dimensionality of a data set from n to q dimensions (n > q) based on its multidimensional structure and nd patterns in data [115]. During an orthogonal linear transformation from n to a lower dimension of q, PCA calculates the eigenvectors and eigenvalues of the n-dimensional preprocessed (zero prospective value) data and selects the largest q eigenvalues, which corresponding eigenvectors create a subspace, on which the original data are projected.
In other words, PCA nds the most signicant directions with the largest variance in the data set. As a formal description, letXbe ann×N dimensional data set, where N means the number of observations. The aim of PCA is to nd orthonormaln×qprojection matrix (P) that fullls the following equation:
P−1 =PT, (A.31)
Y =PTX, (A.32)
whereYis the diagonal covariance matrix of the transformed data. In gen-eral, the principal components of a data set are calculated from the correlation matrix C by the following eigenvalue equation:
Cp=λp, (A.33)
The λ eigenvalues are sorted in decreasing order, the eigenvector corre-sponding to the largest eigenvalue will be the rst principal component and so on. The selected q principal components are collected in P and the projected data are calculated by Eq. A.32. If not all the principal components are se-lected (which is in most cases), the information loss of PCA can be given as two informative variable:
- Projection error, the cumulative sum of the neglected eigenvalues;
εproj = XN
i=q+1
λi, (A.34)
- Recontruction error, the dierence norm between the original and the reconstructed data;
εrecon =kX−PYk, (A.35)
To handle time delays in the input variables, the algorithm has an im-plementation of Dynamic PCA (DPCA), which can extract time-dependent relations in the measurements, because it mimics the concept of an ARMAX (auto-regressive moving average exogenous) time series model by forming the data matrix with the previous observations in each observation vector [165].
When the data are stacked with the current observation vector and the previ-ous d observations, the resulting data matrix is formed as follows:
Xd =
xT(k) xT(k−1) · · · xT(k−d) xT(k−1) xT(k−2) · · · xT(k−1−d)
... ... ... ...
xT(k−K) xT(k−K−1) · · · xT(k−K −d)
(A.36)
where xT(k) = [xk,1, xk,2, . . . , xk,N] the N-dimensional observation in time point k. Obviously, n = K +d equation holds, so d observations are 'lost'
and only K observations can be applied for principal component analysis.
This means that DPCA algorithm has an extra parameterd called time delay parameter, which expresses the time shift between data points: e.g. it shifts the output values by the residence time of a reactor in order to have the corresponding input-output data pairs at the same timepoint. As a result, DPCA can represent a multivariate time-dependent time series as a 1-D signal for further transformations.
Data Filtering
Generally in chemical engineering, process data are generally noisy, some high-frequency disturbances are always superposed to the original signal that need to be neglected for an accurate analysis.
Data ltering is actually a science itself, there are a lot of lter-types and classes in signal processing literature, like analog and digital lters, low-pass, band-pass lters, active and passive lters, etc., but their explanation is not the topic of this work.
Instead of a common moving average lter we applied the one-parametric Gaussian lter, its convolution kernel can be found in Eq. A.37. The σ pa-rameter controls the level of ltering, the larger values result in signals with less high-frequency features.
X(σ, t) = x(t)◦g(σ, t) = Z +∞
−∞
x(u)· { 1 σ√
2π ·exp[−(t−u)2
σ2 ]}du (A.37)
A.2.2 Result table: Comparison to common distance mea-sures
1-NN 1-NN best No
# of train test data Euclidean warping warping Seq.
Name classes set set length distance win. DTW win. DTW Align. σ
Synthetic Control 6 300 300 60 0.12 0.017 0.007 0.6 0
Gun-Point 2 50 150 150 0.087 0.087 0.093 0.0733 2.25
CBF 3 30 900 128 0.148 0.004 0.003 0.1289 4
Face(all) 14 560 1690 131 0.286 0.192 0.192 0.4822 2
OSU Leaf 6 200 242 427 0.483 0.384 0.409 0.5578 0
Swedish Leaf 15 500 625 128 0.213 0.157 0.21 0.505 0
50Words 50 450 455 270 0.369 0.242 0.31 0.6 8
Trace 4 100 100 275 0.24 0.01 0 0.18 0
Two Patterns 4 1000 4000 128 0.09 0.0015 0 0.0765 1
Wafer 2 1000 6174 152 0.005 0.005 0.02 0.00032 0
Face (four) 4 24 88 350 0.216 0.114 0.17 0.4886 0
Lightning-2 2 60 61 637 0.246 0.131 0.131 0.409 2.3
Lightning-7 7 70 73 319 0.425 0.288 0.274 0.6712 0
ECG 2 100 100 96 0.12 0.12 0.23 0.26 0
Adiac 37 390 391 176 0.389 0.391 0.396 0.854 0
Yoga 2 300 3000 426 0.17 0.155 0.164 0.24 0
Fish 7 175 175 463 0.217 0.16 0.167 0.3428 1
Beef 5 30 30 470 0.467 0.467 0.5 0.4667 10
Coee 2 28 28 286 0.25 0.179 0.179 0.107 1
OliveOil 4 30 30 570 0.133 0.167 0.133 0.467 0.5
Table A.2: Comparison of the proposed algorithm with well known methods for classication purposes. Sequence alignment outperforms all measures (bold) and Euclidean distance (italic) for seven data sets.
A.2.3 Introduction to Piecewise Linear Approximation based Dynamic Time Warping
The simplest and most intuitive segmentation is PAA (Piecewise Aggregate Approximation), which was independently proposed for data mining commu-nity by two authors [178, 179]. To reduce the x= [x(1), x(2), . . . , x(N)]data from N to q dimensions, time series are simply divided into q similar sized frames. The vector x, which was generated from the mean values of these frames, represents the segments of the original data. Assuming that q is a factor of N, we get:
x(i) = q N
N qi
X
j=Nq(i−1)+1
x(j) (A.38)
The distance can be chosen freely between two PAA representation. Besides ltering noise, PAA can compensate phase shifts of time axis and dierent sampling rates of time series, but cannot handle "locally elastic" shifts of time axis and it is not tight enough representation for most time series (Fig. A.6).
50 100 150 200
0
Figure A.6: The original signal (upper), PAA (middle) and PLA representation (lower).
To correct these faults, mostly the more sophisticated PLA method is ap-plied. While PAA represents an equi-sized segment with only one value, PLA replaces the original data with not equally sized segments of straight lines, i.e x (the segmented version of x) is a 4-taple:
x(i) = [x(xl)i,x(xr)i,x(yl)i,x(yr)i], (A.39)
where x(xl)i and x(xr)i are the left and right time coordinates of the ith segment. Similarly,x(yl)iandx(yr)iare the values ofx(i)inx(xl)iandx(xr)i. Finding the optimal PLA of time series and a suitable distance for this representation is a dicult task that hardly depends on the application. Its detailed description is not in the scope of this thesis, interested reader is re-ferred to [180].
To align these two sequences rstly we dene a grid D with size N ×M (length of each time series) where each cell represents a distance between the two time series appropriate indices (Fig. A.7). In this step we can choose any application-dependent distance like L1 and L∞ norms, but generally Eu-clidean distance is suggested, because it allows eciently indexing of DTW.
Considering this we have:
D(i, j) =p
(x(i)−y(j))2 (A.40)
Using grid D, we can create arbitrary mappings called warping paths -betweenxandy. The construction of a warping path[wp(1), wp(2), . . . , wp(l)]
has to be restricted with the following constraints:
- Boundary condition: The path has to start inD(1,1)and end inD(N, M). - Monotonicity: The path has to be monotonous, i.e always heading from
D(1,1) to D(N, M). If wp(k) = D(i, j) and wp(k+ 1) = D(i0, j0) then i0−i≥0 and j0−j ≥0.
- Continuity: The path has to be continuous. If wp(k) = D(i, j) and wp(k+ 1) =D(i0, j0) then i0−i≤1and j0−j ≤1.
To nd the optimal warping path (the DTW distance of the two time series), every warping path has an assigned cost which is the sum of values of the aected cells divided by a normalization constant K:
dDT W(x,y) = min
qPk
i=1wp(k) K
(A.41)
The value of K depends on the application, in most cases this is the length of the path, but it can be omitted. More information about the method of dening K and its signicance can be found in [129]. Note that the Euclidean
Figure A.7: The D grid and the optimal warping path of DTW.
distance is a special case of DTW, when we choose the path that is located on the diagonal of grid D and K = 1.
Obviously, the number of paths exponentially grows by the size of time series. Fortunately, the optimal path can be found in O(NM) time with the help of dynamic programming using cumulative distance matrix (Db). Cells of the cumulative matrix contains the sum of appropriate cell value in matrix D and the minimum of the three cell from where the cell can be reached:
D(i, j) =b min
D(ib −1, j) +D(i, j), D(i, jb −1) +D(i, j), D(ib −1, j−1) +D(i, j)
(A.42)
The DTW distance between the two time series can be found in D(n, m)b . Vullings et al. [134] was the rst to use PLA based DTW on ECG data. Keogh and Pazzani [181] gave a generalized distance between PLA segments:
dDT W(x(i), y(j)) =
= ((x(yl)i+x(yr)i)/2−(y(yl)i+y(yr)i)/2)2 (A.43) As we can see this distance arises from PAA. It compares the mean of seg-ments, but uses the original time series, thus it is a tighter PLA representation.
In the following we use this distance in DTW realization.
A.3 Optimal Experiment Design
A.3.1 Evolutionary Strategy
The design variables in ES are represented by n-dimensional vector xj = [xj,1, . . . , xj,n]T ∈Rn, where xj represents the j-th potential solution, i.e. the j-th the member of the population. The mutation operator adds zj,i nor-mal distributed random numbers to the design variables: xj,i = xj,i + zj,i, where zj,i = N(0, σj,i) is a random number with σj,i standard deviation.
To allow a better adaptation to the objective function topology, the design variables are accompanied by these standard deviation variables, which are so-called strategy parameters. Hence the σj strategy variables control the step size of standard deviations in the mutation for j-th individual. So an ES-individual aj = (xj, σj) consists of two components: the design variables xj = [xj,1, . . . , xj,n]T and the strategy variables σj = [σj,1, . . . , σj,n]T. Before the design variables are changed by mutation operator, the standard deviations σj are mutated using a multiplicative normally distributed process:
σj,i(t) =σ(t−1)j,i exp(τ0N(0,1) +τ Ni(0,1)). (A.44) The exp (τ0N(0,1)) is a global factor which allows an overall change of the mutability, and the exp(τ Ni(0,1)) allows individuals to change of their mean step sizes σj,i. So τ0 and τ parameters can be interpreted as global learning rates. Schwefel suggests setting them as [182]:
τ0 = 1
√2n, τ = 1 p2√
n. (A.45)
Discrete recombination of the object variables and intermediate recombination of the strategy parameters were used:
x0j,i =xF,i or xM,i (A.46)
σj,i0 = (σF,i+σM,i)/2, (A.47) where F and M denotes the parents, j is index of the new ospring.
Appendix B
Acronyms and Notations
B.1 Acronyms
Ordered as they appear in text.
Acronyms in Chapter 1
PID controller with Proportional-Intergal-Derivative actions SISO Single-Input-Single-Output system
MIMO Muli-Input-Multi-Output system DCS Distributed Control System
FOPTD First Order plus Time Delay model representation IMC Internal Model Control
LTI Linear Time-Invariant model representation LPV Linear Parameter-Varying model representation MPC Model Predictive Control
APC Advanced Process Control PC Personal Computer OSS Operator Support System OCS Open Control System
SCADA Supervisory Control and Data Acquisition system DW Data Warehouse
KDD Knowledge Discovery in Databases GA Genetic Algorithm
EDA Exploratory Data Analysis PCA Principal Component Analysis PLS Projection to Latent Structure
MDS Multi-dimensional Scaling SOM Self-Organizing Map q-q plot Quantile-Quantile plot QTA Qualitative Trend Analysis DTW Dynamic Time Warping
LCS Longest Common Subsequence technique PLA Piecewise Linear Approximation
SAX Symbolic Aggregate Approximation MEXA Model-based Experiment Analysis OED Optimal Experiment Design
ERP Enterprise Resource Planning system
Acronyms in Chapter 2 OTS Operator Training System R&D Research and Development SP Set Point
PV Process Variable CV Calculated Variable OP Operational set point sSP Simulated Set Point
sPV Simulated Process Variable sCV Simulated Calculated Values sOP Simulated Operational set point PP Polypropylene
MI Melting index
PHD Process History Database RDI Real-Time Data Interface
API Application Programming Interface ODBC Open Database Connectivity PFR Plug Flow Reactor model
CSTR Continuously Stirred Tank Reactor model
RMPCT Robust Multi-Variable Predictive Control Technology HPM High Performance Process Manager
TEAL Tri-Ethyl-Alumina, catalyst component FC Flow Controller
TC Temperature Controller DC Density Controller CWS Cooling Water Supply
CWR Cooling Water Return
F Fluidum
HDPE High-Density Polyethylene RSM Reactor system model FPM First principle model BBM Black-box model
GUI Graphical User Interface
Acronyms in Chapter 3 CS Continuous state of a trend QS Qualitative state of a trend
H-H Homopolymer to homopolymer transition C-C Copolymer to copolymer transition H-C Homopolymer to copolymer transition C-H Copolymer to homopolymer transition
Acronyms in Chapter 4 ES Evolutionary Strategy
SQP Sequential Quadratic Programming NLS Nonlinear Least Squares method
Acronyms in Appendix BMU Best-matching unit
TE Topographic error QE Quantization error
ARMAX Auto-regressive moving average exogenous model NN Nearest neighbor method
PAA Piecewise aggregate approximation
B.2 Notations
Ordered as they appear in text.
Notations in Chapter 2
∆Esystem Energy change of the system
∆U Internal energy change of the system
∆Ekinetic Kinetic energy change of the system
∆Epotential Potential energy change of the system Q Added or removed heat
W Work done on or removed from the system m Weight of reactor and its content
cp Specic heat of reactor and its content
T Temperature
t Time variable
Qf eed sensible heat given up to feed streams Qamb heat lost to environment
Qjw heat removed by jacket water Wpump work from circulation pump Qrxn Polymerization heat
P R Production Rate
∆Hrxn Specic reaction heat
K Proportional parameter of PID TI Integral time parameter of PID U Model input variables
x Model state variables Y Model output variables mP P Polymer inventory RP Pout Polymer output rate
ζ Non-linear mixing coecient M Icurr Instantaneous MI
cmp Specic cost of monomer feed pumps ccp Specic cost of circulating pumps cjw Specic cost of jacket water cC3 Specic cost of propylene cCAT Specic cost of catalyst
cT EAL Specic cost of TEAL component
cDon Specic cost of Donor component cP P Specic cost of polymer product FC3 Monomer owrate
Fjw Jacket water owrate Fmat Flowrate of materials cmat Cost of materials
Notations in Chapter 3 x(t) real-valued function
x0(t) rst derivative of a real-valued function x00(t) second derivative of a real-valued function
∂x(t) sign of rst derivative
∂∂x(t) sign of secind derivative A rst sequence identier B second sequence identier a element of A
b element of B n length of A m length of B
α alignment of sequences w transformation weight
d distance of sequence elements D distance matrix
s similarity score of two sequence elements S score of optimal alignment
σ ltering parameter
X raw industrial data vector Xb reconstructed data vector
Notations in Chapter 4 x state variable
u manipulated input variable y output variable
p model parameters f() non-linear function g() output function
Jmse identication cost function
Q(t) weight matrix of measurement error e output error
b
y model output
F Fisher information matrix Texp time of experiment p0 nominal parameters p∗ true parameters
JD D-optimal cost function JE E-optimal cost function λ eigenvalue of F
S substrate concentration X biomass concentration
V Volume
σ substrate consumption rate µ kinetic rate
CS substrate concentration in inlet feed KS kinetic rate parameter (Monode) KP kinetic rate parameter (Haldane) KI kinetic rate parameter (Haldane) Emse mean square error
µM kinetic rate of Monode kinetics µH kinetic rate of Haldane kinetics
Notations in Appendix C3 Propylene identier C2 Ethylene identier H2 Hydrogen identier SL Slurry identier P P Polymer identier CAT Catalyst identier m Mass
n Mole x Mole ratio t Time (h)
F Flow Rate (kg/h) L Liquid phase identier S Solid phase identier R Reaction rate (kg/h)
ρ density (kg/m3) T Temperature (oC) ARL 1st liq. density coe.
BRL 2nd liq. density coe.
CRL 3rd liq. density coe.
DRL 4th liq. density coe.
ζ0 Product output parameter ERS 1st solid density coe.
F RS 2nd solid density coe.
GRS 3rd solid density coe.
HRS 4th solid density coe.
Vr Reactor volume mr Reactor weight
P R Catalyst productivity (kg/h) AP R 1st productivity coe.
BP R 2nd productivity coe.
RCM Monomer reaction rate coe.
RCE Ethylene reaction rate coe.
RCH Hydrogen reaction rate coe.
cp Specic heat
∆H Reaction heat
αj Heat transfer coe. of jacket Aj Surface of jacket
α Heat transfer coe. of unjacketed surface A Unjacketed reactor surface
X input vector of SOM or PCA mi weight vector of a SOM cell α(t) v learning rate
hi neighborhood kernel around BMU N number of observations
n original dimension of X q projected dimension of X P projection matrix
Y diagonal covariance matrix C correlation matrix
λ eigenvalues of C
εproj projection error εrecon reconstruction error
d number of observation delays x mean value of a segment (PAA)
four-taple of a segment (PLA) x(xl) left hand coordinate of a segment x(xr) right hand coordinate of a segment x(yl) value of x at x(xl)
x(yr) value of x at x(xr) D distance grid matrix wp() warping path vector
Db cumulative distance matrix dDT W distance measure of DTW xj j th member of a population n dimension of xj
zj random number
σj standard deviation of zj τ global learning rate τ0 global learning rate xF elder solution (female) xM elder solution (male)
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