soft sensors

8/10/2019 Soft Sensors

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IntroductionOutlier detection

Process monitoringExamples

Soft Sensors for Monitoring

Antanas Verikas †

†Intelligent Systems Lab, Halmstad University,Box 823, 30118 Halmstad, Sweden

[email protected]

2009 September

Antanas Verikas Soft Sensors for Monitoring





Soft sensor definitionTechniques and tasks

Definitions

Soft sensor is a common name for software where severalmeasurements are processed together. There may be dozens

or even hundreds of measurements. The interaction of thesignals can be used for calculating new quantities that can notbe measured.

Soft sensors or inferential calculators are operators’ virtualeyes. Soft sensors create windows to a process where physical

equivalents are unrealistic or even impossible.

Sensor output can be a control signal, advisory information foroperators, predictions of product quality, information onprocess faults or outliers in data.







Techniques

Neural networks (NN)

Neuro-fuzzy systems

Kernel methods (support vector machines)

Multivariate statistical analysis

Data fusion (Dempster-Shafer theory, for example)Image analysis







Tasks

Outlier detection in process dataProcess and system monitoring

Fault diagnosisDefect detectionTrend detection

Monitoring uncertainty of predictions






Basic informationDistance from the data centerResidual analysis, Influence measuresClassification techniques

Outliers, high leverage pointsThere are two types of outliers: in the X-space, called leveragepoints, and in the y-space. The goal is to eliminate bad leveragepoints and outliers in the y-space. The most difficult problem is to

distinguish between good and bad leverage points.

Figure: Data point 1 is outlier in y -space, data points 2 and 3 areoutliers in x -space. Data point 2 is a good leverage point and data point

3 is a bad leverage point.Antanas Verikas Soft Sensors for Monitoring






Robust statistics

Use robust statistics (scaling, PCA, covariance matrix) to increaseeffectiveness of outlier detection.

A robust estimate of scale can be obtained using the medianabsolute deviation of the median (MADM) defined as

d ≡ med |x − med (x )|0.6745

(1)

The divisor is a correction factor—if the number of samples is largethen d is approximately equal to the actual standard deviation of the true distribution of x .







Robust sample estimates

It is very important to protect sample estimates from outliersinfluence. Below is the list of methods allowing to selectobservations, consistent with the majority.

1 Resampling by half means (RHM)

2 The smallest half volume (SHV)

3 The closest distance to the center (CDC)

4 Minimum volume ellipsoid (MVE)5 Minimum scatter determinant (MSD)

Sample estimates are then computed based on the selected data.These estimates are used for scaling and outlier detection.







Categorization of techniques (1)

Methods based on distance from the data center:

PCA based techniquesTechniques analyzing the projection matrixH = X(XT X)−1XT , where X is an N × K data matrix with K

being the data dimensionality and N the number of observations.

Methods based on the difference between the predictedand actual values of a dependent variable:

ResidualsInfluence measures







Categorization of techniques (2)

Classification techniques. A classifier separates data into

the outlier and normal classes.

Robust estimators are used instead of the common leastsquare estimator.

The least absolute deviation (LAD)

The least median of squares (LMS) are representatives of the group .







PCA (1)The i th transformed variable z i , which is known as PC or score is

z i = xpi (2)

where pi is the i th column of the loading matrix P.Scores are normally distributed as long as the process does notdeviate from the normal operating conditions. Thus, the T 2

statistic can be used to detect deviations from the normality. TheT 2 statistic, based on M components, for a data point i is:

T 2i =M

j =1

z 2ij

s 2 j (3)

where s 2 j

stands for the variance of z j

.







PCA (2)

The variable

T 2i N (N − M )

M (N 2

− 1)

(4)

is F -distributed with M and N − M degrees of freedom. Thethreshold for the T 2 statistic is

T 2α = M (N 2 − 1)

N (N − M )F α(M ,N

− M ) (5)

where F α(M ,N − M ) is the upper 100α% critical point of the Fdistribution with M and N − M degrees of freedom.







PCA (3)

The portion of the input space corresponding to the K − M

smallest eigenvalues can be monitored using the Q statistic (χ2)

Q = eT e (6)

wheree = (I − PPT )x (7)

is the residual vector. The threshold Q α for the Q statistic can becomputed from the eigenvalues and the normal deviatecorresponding to the 1 − α percentile.







H matrixhi = xi (XT X)−1xT i the cut-off value is 2K /N

The cut-off value for the Jackknife residual t ∗i (next slide) is givenby t 0.95(N-K-1)—the 95% quantile of the Student distribution.

0 0.2 0.4 0.6 0.8 1−5

0

5

J a c k k n i f e

r e s i d u a l s

High leverage

Figure: Outliers are denoted by ’o’ and normal data points with ’+’.Lines illustrate the cut-off values.







Residual analysis, Influence measures

Measure based on Formula Cut-off value

Residuals t ∗i = t i

(N −K −1)(N −K −t 2

i )

t (N − K − 1)

Conf. Ellipsoid Volume h∗i = hi + e 2i eT e

2(K −1)N

Influence Function WK i = |t ∗i | hi (1 − hi ) 2 K /N

t i = e i

σ

√ (1−hi )

e i = y i −

y i







Classification techniquesCan be supervised or unsupervised. In the unsupervised case, thedetector is trained by finding a hypersphere of the minimum radiusr ∗ that contains most of the data points from a set X t .

The function f (x) used to categorize x is:

f (x) = Hκ(x, x) − 2N i =1

α∗i κ(xi , x) + T

(8)

where κ(x, x) is a kernel, α∗i

and T are found by maximizing

W (α) =N i =1

αi κ(xi , xi ) −N i =1

N j =1

αi α j κ(xi , x j ) (9)

subject to N

i =1 αi = 1 and 0 ≤ αi ≤ 1/ν N

, i

= 1,...,N

.Antanas Verikas Soft Sensors for Monitoring






Combining different measuresProcess Data

Contradictory Data?

CDC

Data Diagnostic Measures

Yes

Fuzzy Expert Neural Network

Inlier Outlier

AANNRA

x x

y

Figure: (Left): The flowchart of data processing (CDC=closest distanceto centre; RA=regression analysis; AANN=autoassociative neuralnetwork). (Right): The process model network and the AANN.






PCA and PLS based monitoringKernel PCA-based monitoringMonitoring by AANNMonitoring the uncertainty of prediction

PCA based monitoringThe T 2 statistic, based on M components and SPEy are givenby:

T 2M =M

j =1

z 2 j

s 2 j ; SPEy =

K i =M +1

(y i − y i )2 (10)

where s 2 j is the variance of z j .

z j = pT j y is the j th score , where p j is the j th column of the

loading matrix P, and y is the multivariate process output.An upper limit (UL) for the T 2M is:

T 2UL

= M (N − 1)(N + 1)

N (N

− M )

F α(M ,N − M ) (11)







PLS based monitoring

PLS extracts variables that explain the variation in X which ismost predictive of Y.

PLS uses the sample covariance matrix (XT Y)(YT X).The first variable z 1 = wT x maximizes the covariancebetween z 1 and the Y space, w1 is the first eigenvector of (XT Y)(YT X).

w2 is the first eigenvector of XT

2 YYT

X2, whereX2 = X − z1pT 1 is the deflation of X with p1 = Xz1/(zT 1 z1).

Variables z i (scores) can be used for monitoring (via scoreplots for example).







Kernel PCA-based monitoring (1)

The sample covariance matrix in the feature space

ΣΦ = 1N

N i =1

Φ(xi )

Φ(xi )

T

Φ(xi ) = Φ(xi ) − mΦ

mΦ = 1N N

i =1 Φ(xi )


I d i PCA d PLS b d i i






Kernel PCA-based monitoring (2)A Mahalanobis type distance in the feature space can be consideredas a dissimilarity between a point x and the training data

E Φ(x) = Φ(x)T (ΣΦ)−1

Φ(x)

The regularized covariance ΣΦ is obtained by replacing all zero ornear-zero eigenvalues of ΣΦ by a constant λc

ΣΦ = VΛVT + λc (I − VVT )

where Λ is the diagonal matrix of nonzero eigenvaluesλ1 ≤ · · · ≤ λp of ΣΦ and V is the matrix of the correspondingeigenvectors v1, · · · , vp

vk =

N i =1 α

k i

Φ(xi ), 1 ≤ k ≤ p , where αk

i are coefficients found

by solving the eigenvalue problem: λα

= (1/N

)Kα

.Antanas Verikas Soft Sensors for Monitoring

I t d ti PCA d PLS b d it i







The k th projection (score) t k of the centered value Φ(x) isobtained

t k = vk , Φ(x) = N i =1 αk i Φ(xi ), Φ(x)

Using ΣΦ, E Φ(x) can be split into two terms:

E Φ(x) =

p k =1

1λk

vk , Φ(x)2 + 1λc Φ(x)2 −

p k =1

vk , Φ(x)2= tT Λ−1t + (λc )

−1(

K (x, x) − tT t)


Introduction PCA and PLS based monitoring







E Φ(x) represents the estimated statistic for characterizing thelikelihood of x. The first and the second terms of E

Φ(x) represent

the distance in the feature space and the distance from the featurespace, respectively.

Similar to the T 2 and SPE statistics in the linear PCA, thecorresponding statistics T 2

f

= tT Λ−1t and SPEf = K (x, x) −

tT t

can be monitored to detect faults. It is assumed that training dataare multivariate normal in the feature space.








AANN with orthogonal bottleneck layer outputs

Objective function F

F = ηSSE + (1 − η)Φ

Φ =S

i =1

S j =i +1

s ij √ s i s j

s i , s j – variance of u i and u j

s ij – covariance of u i and u j

For the i th data point

SPEi =

R j =1(x ij −

x ij )

2








Confidence limits (1)

Figure: Non-linear scores obtained for the 102 nominal runs (+). Threefaulty runs () were projected on the model (Left : confidence limitsassuming normal distribution, 95 and 99%; Right : confidence limits basedon the kernel density estimate f (x ) = 1

Nh N i =1 K

x −x i h , 95 and 99%).









Figure: Normal trending is observed in both the scores and SPE plots.





Outlier detectionProcess monitoring

Examples

gKernel PCA-based monitoringMonitoring by AANNMonitoring the uncertainty of prediction


Figure: The failure of sensor 11 manifests itself as a different cluster in

the lower left scores plot. The SPE also indicates this event.Antanas Verikas Soft Sensors for Monitoring





Examples

gKernel PCA-based monitoringMonitoring by AANNMonitoring the uncertainty of prediction


Figure: An abnormal trending in the system.






Examples

Kernel PCA-based monitoringMonitoring by AANNMonitoring the uncertainty of prediction

Detecting sensor faults and process upsets (1)At time t when the SPElimit is violated, one can go one step backin each sensor array and calculate the SPE againx1

− = [x 1(t

− 1), x 2(t ),..., x p (t )] SPE1 =

x1

− − x1

−2

x2− = [x 1(t ), x 2(t − 1),..., x p (t )] SPE2 = x2− − x2−2

.....................................

xp − = [x 1(t ), x 2(t ), ..., x p (t − 1)] SPEp = xp − −

xp −2

where x j − is a vector containing measurements for all sensors attime t , but t − 1 sample for sensor j , x j − is the model estimate.

The sensor j = arg mini SPEi is the most probable candidate of being the fault sensor.






Examples


Detecting sensor faults and process upsets (2)Different sensors are usually highly correlated

CC = cov (x i , x j )

σx i σx j (12)

where the vectors xi and x j are taken from a moving windowcontaining the current sample and a number of past samples.

CCs are compared with the ones obtained from the training

data.A failure in one sensor affects readings of only that sensor.

The unavailable sensor values can be reconstructed from theremaining sensors using the initial (calibration) model.


IntroductionO li d i

PCA and PLS based monitoringK l PCA b d i i




Examples


Bayesian approachTwo sources of errors: the noise on the target σ2

t (x), and thevariance of the output due to weights (parameters) uncertaintyσ2w (x). These two terms are independent, thus:

σ2y (x) = σ

2w (x) + σ

2t (x) (13)

σ2t (x) is often assumed to be constant and approximated by

σ2t (x) = 2E D /(N − γ ) (14)

where E D is the training set error and γ is the number of well-determined model parameters.

σ2w (x) = gT (x)H−1g(x) (15)

H is the Hessian matrix and g (x) = ∂ y (x)/∂ w is the gradient of the model output.


IntroductionO tli d t ti

PCA and PLS based monitoringK l PCA b d it i




Examples


NN-based approach

For each x, the local variance is estimated as t (x) − y (x)2 andused as the target for the second network.

σ2(x) is an approximation to the local expected variance.

Figure: The predictive error network.Antanas Verikas Soft Sensors for Monitoring


Chemical industryForest industryTextile industry




Examples

yMachineryElectronics industryPaper-making industryPrinting industry

Predicting online biomass concentration

Recurrent NN to capture dynamic information in the input-outputdata. Inputs: feed rate, liquid volume, and dissolved oxygen.







Examples

MachineryElectronics industryPaper-making industryPrinting industry

Detecting sound and dead knots in wood

A potential defect area is determined by the statistical t -test forequality of average R intensities of the area and a clear wood area.Classification is based on histograms of intensity variance.







Examples


SOM based detection of wood defects (1)

An image is divided into non-overlapping regions.Percentiles of R, G, and B intensity histograms of a region and differences of the percentiles are thefeatures fed into the self-organizing map (SOM).



Chemical industryForest industryTextile industryM hi




Examples



Detection SOM: The dashed lines are manually drawn, pessimistic(upper) and optimistic (lower) approximations of the boundary

between sound-wood and defects.Antanas Verikas Soft Sensors for Monitoring


Chemical industryForest industryTextile industryMachiner






Classification SOM: Defects with similar appearance are close toeach other.



Chemical industryForest industryTextile industryMachinery






b

c

Defects mapped onto 3 SOM nodes: a) dry knots; b) sound knots;

and c) shakes. Antanas Verikas Soft Sensors for Monitoring







MLP-based detection of textile defects

Each pixel is characterized by a feature vector—grey values fromthe neighbourhood. PCs are then calculated and presented to amultilayer perceptron.

(a) (d) (g) (j)

(c) (f) (i) (m)



P i i






Gabor filtering-based detection of textile defects (1)

A 2-D Gabor function is an oriented complex sinusoidalgrating modulated by a 2-D Gaussian. The Gaussian

has an aspect ratio of σx /σy . The sinusoid has aspatial frequency of f o and an orientation θ.

Usually the X -axis of the Gaussian has the sameorientation θ as the sinusoidal wave.

h(x , y ) = 12πσx σy

exp− 12 x 2

σ2x

+ y 2σ2y exp{2π jf o q (x , y )}

q (x , y ) = cos(θ)x + sin(θ)y



P it i







Four scales: p = 4 andorientations: q = 4;

Features—the meanvalue R pq and standarddeviation σpq of thefiltered image energy.

K (x , y ) = [K 12(x , y ) + K 23(x , y ) + K 34(x , y )]/3

K ij (x , y ) = [K i (x , y ) · K j (x , y )]; K p (x , y ) =

4q =1[S pq (x , y )]2



Process monitoring





yElectronics industryPaper-making industryPrinting industry


−

Figure: a) Ideal weaves, b) faulty weaves, c) fused images, d) segmented.



Process monitoring





Electronics industryPaper-making industryPrinting industry

ICA-based detection of textile defects (1)

An image is divided into overlapping windows—x vectors.x = As, where A is the mixing matrix and s is the vector of independent components.

Having A, s is obtained, s = Wx, where W is the pseudo inverseof A obtained from defect-free images. The columns of thede-mixing matrix W are called independent component filters.

Each window is checked for being defective. The defect detectionfor a window i is based on computing the Euclidean distanced i = ||sm − si ||2 and comparing it with a threshold, where sm isthe mean vector obtained from defect-free windows.



Process monitoring






ICA-based detection of textile defects (2)

Figure: The defect detection results for two classes of textures.Antanas Verikas Soft Sensors for Monitoring


Process monitoring

Chemical industryForest industryTextile industryMachineryEl i i d





SOM-based bearing degradation monitoring (1)

Features characterizing the vibration waveform are extracted andused to train the SOM.



Process monitoring

Chemical industryForest industryTextile industryMachineryEl t i i d t






SOM is trained using data from the normal operation. For a givenx the MQE is evaluated as MQE = ||x − mBMU||, where mBMU is theweight vector of the best matching unit of the SOM.



Process monitoring

Chemical industryForest industryTextile industryMachineryElectronics industry



gExamples



SOM is trained using data from all the operation conditions. TheU-matrix (matrix of distances between weights of SOM nodes) isused to visualize the trajectory of the degradation process.



Process monitoring




ExamplesElectronics industryPaper-making industryPrinting industry

Detecting defects on surface of ceramics capacitors (1)

Examples of images of defective surfaces

It is assumed that p quality characteristics are normally distributed.The procedure requires computing the mean of the characteristicsx from a sample of size n.



Process monitoring






An image is divided into blocks and the Hotelling T 2 statistic isused to detect defects in each block.

T 2 = n(x − µ)T Σ−1(x − µ)

Since µ and Σ are unknown, they are estimated from m samplesof size n. Therefore, each block is divided into m sub-blocks. Thestatistic T 2 then is

T 2 = n(x − x)T S−1(x − x)

x and S are estimated from defect free images.



Process monitoringE l






R component of RGB image

DR

AR

Feature extraction of image blocks

by wavelet transform

Approximated

characteristic

Detail

characteristic

Wavelet transformImage

synthesis

Wavelet

characteristics

One multivariate processing unit of size 4 x 4 pixels

Four wavelet processing units of equal size 2 x 2 pixels

: Mean matrix of texture features

: Covariance matrix of texture features

X

S

2 ' 1( ) ( )T n X X S X X −

= – –

0

X : 255

Y : 255

Z : T2

value

T2 distance diagram

Defect locations

(high energy)

Normal location

(low energy)



Process monitoringE l




Examplesy

Paper-making industryPrinting industry








Examplesy


The task

To develop soft sensors for on-line detection of abnormal paperformation variations in machine direction in various frequencyregions.

Forming

section

Press

section

Drying

section

Reel

section

HeadboxCalender

Figure: A schematic drawing of a paper machine.







ExamplesPaper-making industryPrinting industry

Reflection sensor design and arrangement

Figure: Reflection sensor design (left) and arrangement (right).








Processing flowchart

Subdivide into blocks of a given lengthSubdivide into blocks of a given length

On-Line Monitoring




Chemical industryForest industryTextile industryMachineryElectronics industryP ki i d




Multi-resolution time series representation

Compute the Fourier power spectrum of the time series f (x )

P (u ) = F (u )2 = R 2(u ) + I 2(u ) (16)

Divide the frequency axis into several frequency regions R i of

different average frequency:

u ∈ R i if iW ≤ log2(u ) < (i + 1)W (17)

whereW =

{log2[N /(2

√ 2)]

}/

{N r

− 1

} (18)

Characterize the frequency content of the region a by afeature vector xa ∈ Rk .

The partitioning implements the multi-resolution time seriesrepresentation.




Chemical industryForest industryTextile industryMachineryElectronics industryP ki i d t



pPaper-making industryPrinting industry

Feature extraction

E i = 1

N i

u ∈R i

P (u ) (19)

E mi = 1E i

maxu ∈R i

P (u ) (20)

χi = N 2i u ∈R i P i (u )

u ∈R i

P i (u ) − 1

N i 2

(21)

M i = − 1

log N i

u ∈R i

P ni (u )log P ni (u ) (22)




Chemical industryForest industryTextile industryMachineryElectronics industryPaper making industry



pPaper-making industryPrinting industry

Novelty (abnormality) detection

The optimal values of the parameter vector α∗ of the detectorare found by maximizing the following objective function

W (α

) =

N

i =1 αi κ(xa

i ,xa

i ) −

N

i =1

N

j =1 αi α j κ(xa

i ,xa

j ) (23)

subject toN

i =1 αi = 1 and 0 ≤ αi ≤ 1/ν N , i = 1,...,N .

The function f (xa) used to categorize the data point xa is:

f (xa) = Hκ(xa, xa) − 2N i =1

α∗i κ(xai , xa) + T (24)

where α∗i and T are found by maximizing (23) andH[y (xa)] = 1 if y (xa) ≥ 0 and −1 otherwise.




Chemical industry

Forest industryTextile industryMachineryElectronics industryPaper making industry




Data

Data sets A and C have been recorded on production line andcorrespond to 920 and 70 km of newsprint divided intoM = 9200 and 700 blocks, respectively.

The data set A was randomly split into training, validation,and test subsets. Ten different random splits were used.

All data of the set C have been used for testing. In the 700data blocks there were

≈70% and 75% of outliers in the

lowest and highest frequency region, respectively.

The abnormalities screened were located in the region of 0.2 − 0.4 mm and 1.6 − 3.2 m.




Chemical industry

Forest industryTextile industryMachineryElectronics industryPaper-making industry



Paper making industryPrinting industry

Categorization results of the set C data

Table: The expected amount of outliers is approximately 75% and 70%for the highest and the lowest frequency region, respectively.

Set\n 1 2 3 4 5 6 7 8 9 10

Highest frequency region

Test 73.1 74.0 73.4 74.4 73.5 74.7 75.6 75.3 74.9 72.6

Lowest frequency region

Test 66.3 68.9 69.9 68.9 67.3 68.3 69.7 69.2 68.9 67.2




Chemical industry




Paper making industryPrinting industry

Abnormalities detection software

Figure: The detection result in the highest frequency region.




Chemical industry




p g yPrinting industry

Graphical process




Chemical industry




p g yPrinting industry

Printing press

Paper path

Impression cylinder

Blanket cylinder

Plate cylinder

Ink bathDampening




Chemical industry




Printing industry

The problem

Colour deviation and areas to measure the amount of colour.




Chemical industry




Printing industry

Estimating the amount of ink by soft computing

Neural NetNeural Net

RGB

CMYK




Chemical industry

Forest industryTextile industryMachineryElectronics industryPaper-making industryP i i i d



Printing industry

Measuring ink density

400 500 600 7000

0.2

0.4

0.6

0.8

1

Wavelength (nm)

D = − lg

λ

R (λ)S (λ)F (λ)

λ S (λ)F (λ)

(25)




Chemical industry

Forest industryTextile industryMachineryElectronics industryPaper-making industryP i ti g i d t



Printing industry

Measuring ink density by soft computing

400 450 500 550 600 650 700

−0.4

−0.2

0

0.2

0.4

0.6

Wavelength (nm)

A = projections of of spectra on eigenvectors x = RGB tripletB∗ = (K + λIN )

−1A k = κ(xi , x)κ(xi , x j ) = exp{−||xi − x j ||2/σ} a = B∗T kSpectrum is reconstructed using a and eigenvs




Chemical industry

Forest industryTextile industryMachineryElectronics industryPaper-making industryPrinting industry



Printing industry

Spectra from RGB values

— Original; — From RGB; ∆E = 0.7

400 450 500 550 600 650 7000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Wavelength (nm)400 450 500 550 600 650 700

0.1

0.2

0.3

0.4

0.5

0.6

Wavelength (nm)




Chemical industry




Printing industry

Print quality characterization

Print quality attributes

Missing dots (Q 1)

Dot deformation

Dot size deformation Q 2)Dot shape deformation Q 3)

Ink density variation Q 4)

Noise level Q 5)

Quality =

f (Q )g (•) (26)

where g (•) is a measure




Chemical industry




Printing industry

Print quality ranking by manSample A B C D E F G H

Ann 6 7 8 3 4 5 1 2

Anna R 6 7 8 3 4 5 2 1

Sofia 6 8 7 3 4 5 2 1

Peter 6 8 7 3 4 5 2 1

Frank 6 7 8 3 4 5 2 1

Stellan 6 8 7 3 4 5 2 1

Bernd 6 8 7 4 3 5 2 1

Dan 6 8 7 4 3 5 1 2

Mean 6.00 7,63 7,38 3,25 3,75 5 1,75 1,25

Cristofer 6 8 7 2 5 4 1 3

Magnus 6 7 8 3 4 5 2 1

Wolfgang 6 8 7 3 4 5 1 2

Kerstin 6 8 7 2 4 5 1 3

Maria 6 8 7 3 4 5 1 2

Marcus 6 8 7 3 4 5 2 1

Mean 6.00 7.71 7,28 3.0 3.93 4.93 1.57 1.57




Chemical industry




Printing industry

Correlation between quality rankings by man and machine

When ranking prints by the machine, the usefulness of printingdots of different colours was assessed.

Ink\Correlation R p

Magenta 0.9524 0.0003Yellow 0.9524 0.0003

Cyan 0.9762 0.0000Black 0.9762 0.0000




Chemical industry




g y

Assessing banknote printing quality

Line width

Colour

Circle Diameter

Spherical shell clustering

Each cluster resembles a circle. Eachcircle prototype λi consists of: r i and

ci . The distance d ij from an edgepixel x j to a prototype λi is defined:

d ij = (||x j − ci ||2 − r 2i )2 (27)

and used to minimize:

J (λ1,...,λK ) =K i =1

x j ∈λi

d 2ij (28)


soft sensors

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