kundu iit kanpur-experimental error.pdf
TRANSCRIPT
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EXPERIM
ENTAL
ERRO
R
Deba
sisKundu
DepartmentofMathematicsa
ndStatistic
s
IndianInstituteo
fTechnologyKanpur
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OUTLINEO
FTHET
ALK
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OUTLINEO
FTHET
ALK
Experimen
t
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OUTLINEO
FTHET
ALK
Experimen
t
Data
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OUTLINEO
FTHET
ALK
Experimen
t
Data
Informatio
n
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OUTLINEO
FTHET
ALK
Experiment
Data
Informatio
n
DifferentTypesofModels
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OUTLINEO
FTHETALK
Experiment
Data
Informatio
n
DifferentTypesofModels
DifferentFormsofErrors
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OUTLINEO
FTHETALK
Experiment
Data
Informatio
n
DifferentTypesofModels
DifferentFormsofErrors
Estimating
Models/M
odelParam
eters
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OUTLINEO
FTHETALK
Experiment
Data
Informatio
n
DifferentTypesofModels
DifferentFormsofErrors
Estimating
Models/M
odelParam
eters
Estimating
Errors
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OUTLINEO
FTHETALK
Experiment
Data
Informatio
n
DifferentTypesofModels
DifferentFormsofErrors
Estimating
Models/M
odelParam
eters
Estimating
Errors
Conclusions
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EXPERIMENT:
.p.
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EXPERIMENT:
Wh
atisanexp
eriment?
.p
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EXPERIMENT:
Wh
atisanexp
eriment?
Exp
erimentisaprocessw
hichprodu
cesoutcom
es/data.
.p
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EXPERIMENT:
We
dividetheexperimentsintotwob
roadclasse
s
.p
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EXPERIMENT:
We
dividetheexperimentsintotwob
roadclasse
s
RandomE
xperiment
.
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EXPERIMENT:
We
dividetheexperimentsintotwob
roadclasse
s
RandomE
xperiment
Non-Rand
omExperiment
.
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EXPERIME
NT:
Wh
atisaRandomExperiment?
.
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EXPERIME
NT:
Wh
atisaRandomExperiment?
Randomexper
imentisan
experimentwhichpro
duces
random/non-p
redictableoutcomes.T
hatmeans
evenif
we
repeattheexperiment
underthesameenvironmental
con
ditions,the
outcomesmaynotbe
thesame,w
hatever
bethereason.
.
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EXPERIME
NT:
Examples:
.
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EXPERIME
NT:
Examples:
Tossacoi
n10times.
.
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EXPERIME
NT:
Examples:
Tossacoi
n10times.
RumfordCoolingexperiment(1798).
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EXPERIME
NT:
Examples:
Tossacoi
n10times.
RumfordCoolingexperiment(1798).
ECGdata
ofaparticularperson.
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EXPERIME
NT:
Examples:
Tossacoin10times.
RumfordCoolingexperiment(1
798).
ECGdata
ofaparticularperson.
Stockpric
eofaparticularbrand
.
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EXPERIME
NT:
Wh
atisaNon
-RandomE
xperiment?
is .p.7/3
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EXPERIME
NT:
Wh
atisaNon
-RandomE
xperiment?
Ifw
eknowtheexactoutc
omeeachtimeanexperiment
per
formed,we
callitano
n-randome
xperiment.
is .p.7/3
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EXPERIME
NT:
Wh
atisaNon
-RandomE
xperiment?
Ifw
eknowtheexactoutc
omeeachtimeanexperiment
per
formed,we
callitano
n-randomexperiment.
Mo
stoftheex
perimentsinnaturearerandom
exp
eriments.
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EXPERIME
NT:
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EXPERIME
NT:
Wearemainlyinterestedinnon-random
experiments.
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EXPERIME
NT:
Wearemainlyinterestedinnon-random
experiments.
Itisassum
edthatthe
outcomesarequantita
tivein
nature.
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EXPERIME
NT:
Wearemainlyintere
stedinnon-random
experiments.
Itisassum
edthatthe
outcomesarequantita
tivein
nature.
Wecallth
eoutcomesasDATA.
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EXPERIME
NT:
Wearemainlyintere
stedinnon-random
experiments.
Itisassum
edthatthe
outcomesarequantita
tivein
nature.
Wecallth
eoutcomesasDATA.
Datacanbeunivariateormultiv
ariate.
n e.p.8/3
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EXPERIME
NT:
Wearemainlyintere
stedinnon-random
experiments.
Itisassum
edthatthe
outcomesarequantitativein
nature.
Wecallth
eoutcomesasDATA.
Datacanbeunivariateormultiv
ariate.
AIM:Tryingtoextractm
aximuminformationfromthe
observedDAT
A
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DA
TA
Exam
ple:Rumfo
rdCooling
Experimen
t
(Amountofheatgenerated
byfriction)
Time
Temperatu
re
Time
Temperatu
re
(min)
(0F)
(min)
(0F)
4
126
24
115
5
125
28
124
7
123
31
113
12
120
34
112
14
119
37.5
111
16
118
41
110
20
116
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DA
TA
Example(ECGData)
200
1000
100
200
300
400
500
600
700
0
100
200
300
400
500
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DA
TA
Example(Tex
tureData)
rm .
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DA
TA:
MostoftheDatacanbeexpressedin
thefollow
ingfo
orm
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DA
TA:
MostoftheD
atacanbeexpressedinthefollow
ingfo
Data=Informatio
n/Signal
+Noise/E
rror
orm
may
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DA
TA:
MostoftheD
atacanbeexpressedinthefollow
ingfo
Data=I
nformatio
n/Signal
+Noise/E
rror
He
re,Informa
tion/Signalisaveryg
enericterm
.Itm
be
completely
determinis
ticorcomp
letelyrandomor
mixed.Ourai
mistoextr
acttheInfo
rmation/Signal
fro
mthe(nois
y)Data.
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INF
ORMAT
ION
Whyisitnece
ssary?
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INF
ORMAT
ION
Whyisitnece
ssary?
Tounderstandthepr
ocessitself.
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INF
ORMAT
ION
Whyisitnecessary?
Tounderstandthepr
ocessitself.
Predictionpurposes.
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INF
ORMAT
ION
Whyisitnecessary?
Tounderstandthepr
ocessitself.
Predictionpurposes.
Storingpurposes.
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INF
ORMAT
ION
Whyisitnecessary?
Tounderstandthepr
ocessitself.
Predictionpurposes.
Storingpurposes.
Designingtheexperiments.
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INF
ORMAT
ION
Whenisitpossible?
much
act
wiseit .p
.14
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INFORMAT
ION
Whenisitpossible?
Intuitivelywe
feelthatif
theInformation/Signalism
morethanthe
noise/erro
r,weshouldbeableto
extra
theInformatio
n/Signalfromthenoisydata.Otherw
willbealmostimpossible.
much
ract
wiseit
rroris .
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INFORMAT
ION
Whenisitpossible?
Intuitivelywe
feelthatif
theInformation/Signalism
morethanthe
noise/erro
r,weshouldbeableto
extr
theInformation/Signalfromthenoisydata.Otherw
willbealmostimpossible.
Th
erefore,iti
sclearthat
understand
ingthenoise/er
veryimportan
tinanyscientificexperiment.
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INFORMAT
ION
Weneedtoqu
antifyInfo
rmation.To
quantify
Informationw
eapproxim
ateitbya
Mathematical
Model,therefore
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INFORMAT
ION
WeneedtoquantifyInfo
rmation.Toquantify
Informationw
eapproxim
ateitbya
Mathematical
Model,therefore I
nforma
tionM
odel
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MO
DELS
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MO
DELS
Data=Model+Error
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MO
DELS
Data=Model+Error
Er
rorhasseveralcomponents
ModelingError
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MO
DELS
Data=Model+Error
Er
rorhasseveralcomponents
ModelingError
ExperimentationError
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MO
DELS
Data=Model+Error
Er
rorhasseveralcomponents
ModelingError
ExperimentationError
DataCollectionError
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MODELS
Data=Model+Error
Er
rorhasseveralcomponents
ModelingError
ExperimentationError
DataCol
lectionError
CalibrationError
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MODELS
DifferentTyp
esofMode
l
MechanisticModel
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MODELS
DifferentTyp
esofMode
l
MechanisticModel
Function
alModel
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MODELS
M
echanisticModel:
ctmech-
plexbut
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MODELS
M
echanisticModel:
A
mechanisticmodelisa
modelbas
edontheexac
an
ismofanexperiment.Itisusuallyquiteco
mp
mostofthetimeitisquiteeffective.
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MODELS
Fu
nctionalM
odel:
need
onal
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MODELS
Fu
nctionalM
odel:
It
ispurelyamathematic
almodel.O
nedoesno
tn
an
yexperime
ntalknowledgetoconstructafunctio
model.
edata
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MODELS
Usingtheme
chanistic/functionalm
odeloften
the
ca
nbewritteninthefollowingform
:
edata
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MODELS
Usingtheme
chanistic/functionalm
odeloften
the
ca
nbewritteninthefollowingform
:
Data=Y
=
f(X)+
Error
edata
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MODELS
Usingtheme
chanistic/f
unctionalm
odeloften
th
ca
nbewritteninthefollowingform
:
Data=Y
=
f(X)+
Error
YandX
areknown
hedata
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MODELS
Usingtheme
chanistic/f
unctionalm
odeloften
th
ca
nbewritteninthefollowingform
:
Data=Y
=
f(X)+
Error
YandX
areknown
Errorisunknown
hedata
estimate .
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MODELS
Usingtheme
chanistic/f
unctionalm
odeloften
th
ca
nbewritteninthefollowingform
:
Data=Y
=
f(X)+
Error
YandX
areknown
Errorisunknown
Aim:From{
X1,
Y1
},..
.,
{X
n,
Yn}
wewanttoe
f().
wing
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EX
AMPLE
S
RumfordCoolingData
MechanisticModel:
BasedonNewtons
lawofcoolingthefoll
ow
modelhasbeenused:
Y(t)=6
0+70et
+(t)
(t)
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EX
AMPLE
S
ECGData
Function
alModel:
Y(t)=
p k=1
{Ak
cos(kt)+Bk
sin(kt)}
+
kn)}+(m,n
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EX
AMPLE
S
TextureData
Function
alModel:
Y
(m,n)=
p
k=1
{Ak
cos(km
+
kn)+B
k
sin(km
+
k
.p.
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MODELS:
ESTIMATION
PossibleApp
roaches:
metric
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MODELS:
ESTIMATION
PossibleApp
roaches:
FromY
andX
estimatef().
(Non-Param
Approac
h)
metric
f()is
erefore,
hasbeen
ref(X,
) .p.24/3
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MODELS:
ESTIMATION
PossibleApp
roaches:
FromY
andX
estimatef().
(Non-Param
Approac
h)
Assume
f(X)=f(X,
),here
theformof
knownb
uttheparameterisu
nknown.The
estimate
assuming
f()isknown.Once
estimatedandsince
f()iskno
wn,therefo
r
becomes
known.(Parametric
Approach)
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ER
RORS
D
ifferentformsofError
s:
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ER
RORS
D
ifferentformsofError
s:
Random
Errors
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ER
RORS
D
ifferentformsofError
s:
Random
Errors
Systema
ticErrors
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ER
RORS
D
ifferentAssumptionsonErrors:
nzeros
t.
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ER
RORS
D
ifferentAssumptionsonErrors:
Errorsarepurelyra
ndom.Theyhavemean
andsamevariances.Theyareindependent
nzeros
t. ces.They
.p
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ER
RORS
D
ifferentAssumptionsonErrors:
Errorsarepurelyra
ndom.Theyhavemean
andsamevariances.Theyareindependent
Theyhavemeanzerosbutunequalvarianc
areindependent.
nzeros
t. ces.They
tare
.p
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ER
RORS
D
ifferentAssumptionsonErrors:
Errorsarepurelyra
ndom.Theyhavemean
andsam
evariances.Theyareindependent
Theyhavemeanzerosbutunequalvarianc
areindependent.
Theyhavemeanzeros,equalv
ariancesbu
correlated.
nzeros
t. ces.They
tare
.p
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ER
RORS
D
ifferentAssumptionsonErrors:
Errorsarepurelyra
ndom.Theyhavemean
andsam
evariances.Theyareindependent
Theyhavemeanzerosbutunequalvarianc
areindependent.
Theyhavemeanzeros,equalv
ariancesbu
correlated.
Spuriouserrors.
nzeros
t. ces.They
tare
stribution.
.p
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ER
RORS
D
ifferentAssumptionsonErrors:
Errorsarepurelyra
ndom.Theyhavemea
andsam
evariances
.Theyareindependen
Theyhavemeanzerosbutunequalvarianc
areindependent.
Theyhavemeanzeros,equalv
ariancesbu
correlated.
Spuriouserrors.
Errorshavesomeparticularprobabilitydi
.p
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ER
RORS
W
hyarethe
assumption
snecessary
?
.p
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ER
RORS
W
hyarethe
assumption
snecessary
?
Choosin
gthecorrectmodel.
meters.
.p
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ER
RORS
W
hyarethe
assumption
snecessary
?
Choosin
gthecorrectmodel.
Effectiveestimationoftheunk
nownpara
meters.
tionsif
.p
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ER
RORS
W
hyarethe
assumption
snecessary
?
Choosin
gthecorrectmodel.
Effectiveestimationoftheunk
nownpara
Effectiveprediction
ofthefutu
reobserva
necessary.
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meters.
tionsif
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ER
RORS
W
hyarethe
assumption
snecessary?
Choosin
gthecorrectmodel.
Effectiveestimationoftheunk
nownpara
Effectiveprediction
ofthefutu
reobserva
necessary.
Constructingerrorbarsoftheestimates.
Derivingproperties
oftheestimates.
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ER
RORS
H
owtoverif
ytheassum
ptions?
fpossible
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ERRORS
H
owtoverif
ytheassum
ptions?
Choose
aparticularf(,
)fromaclasso
models.
fpossible
ror.
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ERRORS
H
owtoverifytheassum
ptions?
Choose
aparticularf(,
)fromaclasso
models.
Makeso
mesimple
assumption
sontheer
fpossible
ror.
.p.28/3
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ERRORS
H
owtoverifytheassum
ptions?
Choose
aparticularf(,
)fromaclasso
models.
Makeso
mesimple
assumption
sontheer
Estimatetheparam
eters.
fpossible
ror.
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ERRORS
H
owtoverifytheassum
ptions?
Choose
aparticularf(,
)fromaclasso
models.
Makesomesimple
assumption
sontheer
Estimatetheparam
eters.
Estimatetheerroralso.
fpossible
ror.
.p.28/3
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ERRORS
H
owtoverifytheassum
ptions?
Choose
aparticularf(,
)fromaclasso
models.
Makesomesimple
assumptionsontheer
Estimatetheparam
eters.
Estimatetheerroralso.
Thenve
rifytheerrorassumptions.
fpossible
ror.
HAPPY
.
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-
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ERRORS
H
owtoverifytheassum
ptions?
Choose
aparticularf(,
)fromaclasso
models.
Makesomesimple
assumptionsontheer
Estimatetheparam
eters.
Estimatetheerroralso.
Thenve
rifytheerrorassumptions.
Iftheer
rorassumptionsaretrue,thenbe
otherwise
ofpossible
eror.
eHAPPY
um
ptionsand
.
p.28/3
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96/116
ERRORS
H
owtoverifytheassum
ptions?
Choose
aparticularf(,
)fromaclasso
models.
Makesomesimple
assumptionsontheer
Estimatetheparam
eters.
Estimatetheerroralso.
Thenve
rifytheerrorassumptions.
Iftheer
rorassumptionsaretrue,thenbe
otherwise
Change
themodel,
changethe
errorassum
continuetheprocess.
ET
ERS
ne
.
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ESTIMATINGMO
DELPA
RAMET
L
eastSquaresEstimato
rs(MostPo
pularOne)
ET
ERS
ne
.
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ESTIMATINGMO
DELPA
RAMET
L
eastSquaresEstimato
rs(MostPo
pularOne)
min
n
i=1
(Yi
f(X
i,
))2
ET
ERS
ne) epe
ndentwith
.
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ESTIMATINGMO
DELPA
RAMET
L
eastSquaresEstimato
rs(MostPo
pularOne)
min
n
i=1
(Yi
f(X
i,
))2
Itworksverywelliftheerrorsareindepe
meanzeroandsam
evariances
.
ET
ERS
ne) epe
ndentwith
en
the
.
p.29/3
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ESTIMATINGMO
DELPA
RAMET
L
eastSquaresEstimato
rs(MostPo
pularOne)
min
n
i=
1
(Yi
f(X
i,
))2
Itworksverywelliftheerrorsareindepe
meanzeroandsam
evariances
.
Iftheerrorassump
tionsareno
tmetthen
estimatescanbequitebad.
ET
ERS
.
p.30/3
-
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ESTIMATINGMO
DELPA
RAMET
G
eneralized
LeastSqua
resEstimators
ET
ERS
.
p.30/3
-
7/23/2019 kundU IIT kanpur-experimental error.pdf
102/116
ESTIMATINGMO
DELPA
RAMET
G
eneralized
LeastSqua
resEstimators
min
n
i=1
(Yi
f(X
i,
))2
Variance(i)
ET
ERS
epe
ndentwith
.
p.30/3
-
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ESTIMAT
INGMO
DELPA
RAMET
G
eneralized
LeastSqua
resEstimators
min
n
i=1
(Yi
f(X
i,
))2
Variance(i)
Itworksverywell
iftheerrorsareindepe
meanzeroanddifferentvarian
ces.
ET
ERS
epe
ndentwith
uares
.
p.30/3
-
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104/116
ESTIMAT
INGMO
DELPA
RAMET
G
eneralized
LeastSqua
resEstimators
min
n
i=1
(Yi
f(X
i,
))2
Variance(i)
Itworksverywell
iftheerrorsareinde
meanzeroanddiff
erentvarian
ces.
Itworksmuchbett
erthanthe
leastsqu
estimators.
-
7/23/2019 kundU IIT kanpur-experimental error.pdf
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MET
ERS
.
p.31/3
-
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ESTIMAT
INGMO
DELPA
RAM
L
eastAbsol
uteDeviations
min
n
i=
1
|Yi
f(Xi,
)|
MET
ERS
urio
usin
.
p.31/3
-
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107/116
ESTIMAT
INGMO
DELPA
RAM
L
eastAbsol
uteDeviations
min
n
i=
1
|Yi
f(Xi,
)|
Itworksverywell
iftheerrorsarespu
nature.
MET
ERS
urio
usin
mparedtothe
.
p.31/3
-
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108/116
ESTIMAT
INGMO
DELPA
RAM
L
eastAbsol
uteDeviations
min
n
i=
1
|Yi
f(Xi,
)|
Itworksverywell
iftheerrorsarespu
nature.
Theestimatesarequiterobustascom
leastsquaresestim
ators.
MET
ERS
urio
usin
mparedtothe
g.
.
p.31/3
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ESTIMAT
INGMO
DELPA
RAM
L
eastAbsol
uteDeviations
min
n
i=1
|Yi
f(Xi,
)|
Itworksverywell
iftheerrorsarespu
nature.
Theestimatesarequiterobustascom
leastsq
uaresestim
ators.
Computationallyitisverydem
anding
.
p.32/3
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CONCLUSIONS
.p.32/3
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C
ONCLUSIONS
A
llexperim
entaldataw
illhaveerrors
edata
.p.32/3
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C
ONCLUSIONS
A
llexperim
entaldataw
illhaveerrors
T
oextractm
aximumin
formationfromthe
hedata
alysisAre
.p.32/3
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C
ONCLUSIONS
A
llexperim
entaldataw
illhaveerrors
T
oextractm
aximumin
formationfromth
ProperErrorAssumptions/ErrorAna
Required
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heda
ta
nalysisAre
Ana
lysisAre
Nee
ded
.p.32/3
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115/116
C
ONCLUSIONS
Allexperim
entaldataw
illhaveerrors
Toextractm
aximumin
formationfromth
Proper
ErrorAssumptions/ErrorAn
Required
Proper
ModelAssumptitons/Model
Required
ProfessionalStatist
iciansHelpis
.p.33/3
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Th
ankYou