Statistics for Chemical and Process Engineers. A Modern Approach

453 Pages·2022·12.088 MB·English

by  Yuri A. W. Shardt

Yuri A. W. Shardt
Statistics for
Chemical and
Process Engineers
A Modern Approach
Second Edition
Statistics for Chemical and Process Engineers
Yuri A. W. Shardt
Statistics for Chemical
and Process Engineers
A Modern Approach
Second Edition
YuriA.W.Shardt
DepartmentofAutomationEngineering
TechnicalUniversityofIlmenau
Ilmenau,Germany
ISBN978-3-030-83189-9 ISBN978-3-030-83190-5 (eBook)
https://doi.org/10.1007/978-3-030-83190-5
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Preface
Theneedforthedevelopmentandunderstandingoflarge,complexdatasetsexists
inawiderangeofdifferentfields,includingeconomics,chemistry,chemicalengi-
neering, and control engineering. In all these fields, the common thread is using
thesedatasetsforthedevelopmentofmodelstoforecastorpredictfuturebehaviour.
Furthermore, the availability of fast computers has meant that many of the tech-
niques can now be used and tested even on one’s own computer. Although there
existawealthoftextbooksavailableonstatistics,theyareoftenlackingintwokey
respects: application to the chemical and process industry and their emphasis on
computationally relevant methods. Many textbooks still contain detailed explana-
tionsofhowtomanuallysolveaproblem.Therefore,thegoalofthistextbookisto
provide a thorough mathematical and statistical background to regression analysis
through the use of examples drawn from the chemical and process industries. The
majority of the textbook presents the required information using matrices without
linking to any particular software. In fact, the goal here is to allow the reader to
implement the methods on any appropriate computational device irrespective of
theirspecificavailability.Thus,detailedexamples,thatis,basecases,andsolution
stepsareprovided toeasethistask.Nevertheless, thetextbook contains twochap-
tersdevotedtousingMATLAB®andExcel®,asthesearethemostcommonlyused
toolsbothinacademicsandinindustry.Finally,thetextbookcontainsattheendof
eachchapteraseriesofquestionsdividedintothreeparts:conceptualquestionsto
testthereader’sunderstandingofthematerial;simpleexerciseproblemsthatcanbe
solvedusingpen,paper,andasimple,handheldcalculatortoprovidestraightforward
examplestotestthemechanicsandunderstandingofthematerial;andcomputational
questions that require modern computational software that challenge and advance
thereader’sunderstandingofthematerial.
Thistextbookassumesthatthereaderhascompletedabasicfirst-yearuniversity
course, including univariate calculus and linear algebra. Multivariate calculus, set
theory,andnumericalmethodsareusefulforunderstandingsomeoftheconcepts,but
knowledgeisnotrequired.Basicchemicalengineering,includingmassandenergy
balances,mayberequiredtosolvesomeoftheexamples.
v
vi Preface
The textbook is written so that the chapters flow from the basic to the most
advanced material with minimal assumptions about the background of the reader.
Nevertheless, multiple different course can be organised based on the material
presented here depending on the time and focus of the course. Assuming a single
semestercourseof39h,thefollowingwouldbesomeoptions:
(1) Introductory Course to Statistics and Data Analysis: The foundations of
statisticsandregressionareintroducedandexamined.Themainfocuswould
beonChap.1:IntroductiontoStatisticsandDataVisualisation,Chap.2:Theo-
reticalFoundation forStatisticalAnalysis,andpartsofChap. 3:Regression,
including all of linear regression. This course would prepare the student to
taketheFundamentalsofEngineeringExamintheUnitedStatesofAmerica,
aprerequisiteforbecominganengineerthere.
(2) DeterministicModellingandDesignofExperiments:In-depthanalysisand
interpretation of deterministic models, including design of experiments, is
introduced. The main focus would be on Chap. 3: Regression and Chap. 4:
DesignofExperiments.PartsofChap.2:TheoreticalFoundationforStatistical
Analysismaybeincludedifthereisaneedtorefreshthestudent’sknowledge
ofbackgroundinformation.
(3) StochasticModellingofDynamicProcesses:In-depthanalysisandinterpre-
tation of stochastic models, including both time series and prediction error
methods, is examined. The main focus would be on Chap. 5: Modelling
Stochastic Processes with Time Series Analysis and Chap. 6: Modelling
Dynamic Processes. As necessary, information from Chap. 2: Theoretical
Foundation for Statistical Analysis and Chap. 3: Regression could be used.
The depth in which these concepts would be considered would depend on
the orientation of the course: either a theoretical emphasis can be made, by
focusingonthetheoryandproofs,oranapplicationemphasiscanbemade,by
focusingonthepracticaluseofthedifferentresults.
Asappropriate,materialfromChap.7:UsingMATLAB®forStatisticalAnalysis
and Chap. 8: Using Excel® to do Statistical Analysis could be introduced to show
and explain how the students can implement the proposed methods. It should be
emphasisedthatthismaterialshouldnotoverwhelmthestudentsnorshoulditbecome
themainemphasisandhenceavoidthoughtfulandinsightfulanalysisoftheresulting
data.
The author would liketo thank all those who read and commented on previous
versions of this textbook, especially the members of the process control group at
theUniversityofAlberta,thestudentswhoattendedtheauthor’scourseonprocess
dataanalysisintheSpring/Summer2012semester,themembersoftheInstituteof
ControlEngineeringandComplexSystems(InstitutefürAutomatisierungstechnik
und komplexe Systeme) at the University of Duisburg-Essen, the members of the
DepartmentofAutomationEngineering(FachgebietAutomatisierungstechnik)atthe
TechnicalUniversityofIlmenau(TechnischeUniversita¨tIlmenau),andthestudents
who attended the course “System Identification” at the Technical University of
Ilmenau. The author would specifically wish to thank Profs. Steven X. Ding and
Preface vii
Biao Huang for their support, Oliver Jackson from Springer for his assistance and
support,andtheAlexandervonHumboldtFoundationformonetarysupport.Aswell,
theauthorwouldliketothankYingDeng,MikeEichhorn,BenediktOppeneiger,and
M.P.fortheirhelpinimprovingtheEnglishversion.
Downloadingthedata:Thedatasets,MATLAB®files,andExcel®templatescan
bedownloadedfromhttps://link.springer.com/book/9783030831899.
Ilmenau,Germany YuriA.W.Shardt
Symbols and Abbreviations
Thissectionsummarisesthekeysymbolsandabbreviationsfoundinthebook.Please
notethatdespiteattemptstolimitsymbolstosinglemeaninganduse,itisoccasionally
necessary to have multiple meanings assigned to a given symbol. This cases are
clarifiedbypointingoutwhichmeaningisusedinwhichchapter.Symbolsarealways
writteninitalicsorusingspecialfonts,whileabbreviationsarewritteninuppercase,
notinitalics,andusingnormalLatinletters.Symbolsandabbreviationsareinnormal
alphabeticalorder.
Symbols
(cid:2)·(cid:3) Round-downfunction
A A-polynomial
A Regressionmatrix(Chaps.3,4);statematrix(Chap.5)
B B-polynomial
B Inputmatrix
B(1,q) Bernoullidistribution
B(n,q) Binomialdistribution
C C-polynomial
C Complexnumbers
C Outputmatrix
D D-polynomial;seasonaldifferencingorder(bothChap.5)
d Differencingorder
D Throughputmatrix
D Cook’sDistance
i
E Expectationoperator
e (White,Gaussian)noise
e Disturbancesignal
t
F F-polynomial
f Frequency
ix
x SymbolsandAbbreviations
F Fisherinformationmatrix
F Spaceofallpossibleevents
f(x) Probabilitydensityfunction;probabilitymassfunction
f Marginalprobabilitydensityfunction
X1
f Conditionalprobabilitydensityfunction
Y|x
F Fouriertransform
F(ν1,ν2),Fν1,ν2 F-distribution
f(ω) Powerspectrum;spectraldensity
G Generaltransferfunction
G Actuatormodel
a
G Controllermodel
c
G Disturbancemodel
l
G Processmodel
p
G Sensormodel
s
gθˆ Efficiency
h Impulseresponsecoefficients
H Nullhypothesis
0
H Alternativehypothesis
1
I Identitymatrix
J Jacobianmatrix
J(cid:4) GrandJacobianmatrix
J Kalmansmoothergain
t
k Factor(Chap.4);discretetime(cid:4) N(Chaps.5,6)
l Level
L(θ|x) Likelihoodfunction
(cid:5)(θ|x) Log-likelihoodfunction
m Numberofdatapointsforregression,inatimeseries
m Uncentredmoment
i
m¯ Centredmoment
i
n Numberofsamples,parameters
N Setofnaturalnumbers
N(μ,σ ) Gaussian(normal)distribution
2
n Numberofcentrepointreplicates
C
n Numberofreplicates
R
P Probabilitymeasurefunction;orderoftheseasonalautoregres-
sivepolynomial(Chap.5)
p Orderoftheautoregressivepolynomial(Chap.5)
p(λ) Poissondistribution
P(Y|X) Conditionalprobability
p Leftprobability
l
p Rightprobability
r
q Orderofthemoving-averagepolynomial
Q Orderoftheseasonalmoving-averagepolynomial
r Residual
R Setofrealnumbers
SymbolsandAbbreviations xi
R2 Pearson’scoefficientofregression
r Criticalvalue
critical
r Referencesignal
t
S Sensitivity
s Seasonalorder
S Samplespace
SSE Sumofsquaresduetotheerror
SSR Sumofsquaresduetoregression
SSR Sumofsquaresduetoregressionfortheithparameter
i
t(ν),tν Student’st-distribution
TSS Totalsumofsquares
u Input
u Inputsignal
t
w Weight
X Datapoint;randomvariable
x Regressor;state(Chaps.5,6)
y Observation
yˆ∞ Infinite-horizonpredictor/infinite-step-aheadpredictor
y Outputsignal
t
yˆt+τ|t τ-step-aheadpredictor
z Forwardshiftoperator/z-operator
Z Standardnormaldistribution;Z-score
Z Setofintegers
z−1 Backwardshiftoperator/z−1-operator
α Falsepositiverate;confidencelevel
β Falsenegativerate(Chap.2);parameter(Chaps.3,4)
β Meanresponse
0
γ Autocorrelation;skew(Chap.2)
(cid:12) Autocovariancematrix
γ Experimentalcoefficients(Chap.4)
ij
γ (τ) Cross-covariance
XY
δ Bias
(cid:14) Difference
ε Error
εt+τ|t Predictionerror
θ Regressivecoefficients;timedelay(Chap.6)
κ Basisfunction
μ Mean
ν Degreesoffreedom
ρ Autocovariance
ρ (τ) Partialautocorrelation
X|Z
ρ Correlation
X1,X2
ρ Partialcorrelation
XY|Z
(cid:18) Covariancematrix
σ Standarddeviation