kundu iit kanpur-experimental error.pdf

7/23/2019 kundU IIT kanpur-experimental error.pdf

1/116

EXPERIM

ENTAL

ERRO

R

Deba

sisKundu

DepartmentofMathematicsa

ndStatistic

s

IndianInstituteo

fTechnologyKanpur

.p.1/3


2/116

OUTLINEO

FTHET

ALK

.p.2/3


3/116

OUTLINEO

FTHET

ALK

Experimen

t

.p.2/3


4/116

OUTLINEO

FTHET

ALK

Experimen

t

Data

.p.2/3

3


5/116

OUTLINEO

FTHET

ALK

Experimen

t

Data

Informatio

n

.p.2/3

/3


6/116

OUTLINEO

FTHET

ALK

Experiment

Data

Informatio

n

DifferentTypesofModels

.p.2/3

/3


7/116

OUTLINEO

FTHETALK

Experiment

Data

Informatio

n


DifferentFormsofErrors

.p.2/3

2/3


8/116

OUTLINEO

FTHETALK

Experiment

Data

Informatio

n



Estimating

Models/M

odelParam

eters

.p.2

2/3


9/116

OUTLINEO

FTHETALK

Experiment

Data

Informatio

n



Estimating

Models/M

odelParam

eters

Estimating

Errors

.p.2

2/3


10/116

OUTLINEO

FTHETALK

Experiment

Data

Informatio

n



Estimating

Models/M

odelParam

eters

Estimating

Errors

Conclusions

.p.2

.3/3


11/116

EXPERIMENT:

.p.

p.3/3


12/116

EXPERIMENT:

Wh

atisanexp

eriment?

.p

p.3/3


13/116

EXPERIMENT:

Wh

atisanexp

eriment?

Exp

erimentisaprocessw

hichprodu

cesoutcom

es/data.

.p

p.4/3


14/116

EXPERIMENT:

We

dividetheexperimentsintotwob

roadclasse

s

.p

p.4/3


15/116

EXPERIMENT:

We


roadclasse

s

RandomE

xperiment

.

p.4/3


16/116

EXPERIMENT:

We


roadclasse

s

RandomE

xperiment

Non-Rand

omExperiment

.

p.5/3


17/116

EXPERIME

NT:

Wh

atisaRandomExperiment?

.

p.5/3


18/116

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.

.

p.6/3


19/116

EXPERIME

NT:

Examples:

.

.p.6/3


20/116

EXPERIME

NT:

Examples:

Tossacoi

n10times.

.

.p.6/3


21/116

EXPERIME

NT:

Examples:

Tossacoi

n10times.

RumfordCoolingexperiment(1798).

.p.6/3


22/116

EXPERIME

NT:

Examples:

Tossacoi

n10times.

RumfordCoolingexperiment(1798).

ECGdata

ofaparticularperson.

.p.6/3


23/116

EXPERIME

NT:

Examples:

Tossacoin10times.

RumfordCoolingexperiment(1

798).

ECGdata

ofaparticularperson.

Stockpric

eofaparticularbrand

.

.p.7/3


24/116

EXPERIME

NT:

Wh

atisaNon

-RandomE

xperiment?

is .p.7/3


25/116

EXPERIME

NT:

Wh

atisaNon

-RandomE

xperiment?

Ifw

eknowtheexactoutc

omeeachtimeanexperiment

per

formed,we

callitano

n-randome

xperiment.

is .p.7/3


26/116

EXPERIME

NT:

Wh

atisaNon

-RandomE

xperiment?

Ifw

eknowtheexactoutc

omeeachtimeanexperiment

per

formed,we

callitano

n-randomexperiment.

Mo

stoftheex

perimentsinnaturearerandom

exp

eriments.

.p.8/3


27/116

EXPERIME

NT:

.p.8/3


28/116

EXPERIME

NT:

Wearemainlyinterestedinnon-random

experiments.

.p.8/3


29/116

EXPERIME

NT:

Wearemainlyinterestedinnon-random

experiments.

Itisassum

edthatthe

outcomesarequantita

tivein

nature.

.p.8/3


30/116

EXPERIME

NT:

Wearemainlyintere

stedinnon-random

experiments.

Itisassum

edthatthe

outcomesarequantita

tivein

nature.

Wecallth

eoutcomesasDATA.

.p.8/3


31/116

EXPERIME

NT:

Wearemainlyintere

stedinnon-random

experiments.

Itisassum

edthatthe

outcomesarequantita

tivein

nature.

Wecallth

eoutcomesasDATA.

Datacanbeunivariateormultiv

ariate.

n e.p.8/3


32/116

EXPERIME

NT:

Wearemainlyintere

stedinnon-random

experiments.

Itisassum

edthatthe

outcomesarequantitativein

nature.

Wecallth

eoutcomesasDATA.

Datacanbeunivariateormultiv

ariate.

AIM:Tryingtoextractm

aximuminformationfromthe

observedDAT

A

.p.9/

3


33/116

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

.p.10/3


34/116

DA

TA

Example(ECGData)

200

1000

100

200

300

400

500

600

700

0

100

200

300

400

500

.p.11/3


35/116

DA

TA

Example(Tex

tureData)

rm .

p.12/3


36/116

DA

TA:

MostoftheDatacanbeexpressedin

thefollow

ingfo

orm

.p.12/3


37/116

DA

TA:

MostoftheD

atacanbeexpressedinthefollow

ingfo

Data=Informatio

n/Signal

+Noise/E

rror

orm

may

.p.12/3


38/116

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.

.p.13/3


39/116

INF

ORMAT

ION

Whyisitnece

ssary?

.p.13/3


40/116

INF

ORMAT

ION

Whyisitnece

ssary?

Tounderstandthepr

ocessitself.

.p.13

/3


41/116

INF

ORMAT

ION

Whyisitnecessary?

Tounderstandthepr

ocessitself.

Predictionpurposes.

.p.13

/3


42/116

INF

ORMAT

ION

Whyisitnecessary?

Tounderstandthepr

ocessitself.

Predictionpurposes.

Storingpurposes.

.p.13

/3


43/116

INF

ORMAT

ION

Whyisitnecessary?

Tounderstandthepr

ocessitself.

Predictionpurposes.

Storingpurposes.

Designingtheexperiments.

.p.14

/3


44/116

INF

ORMAT

ION

Whenisitpossible?

much

act

wiseit .p

.14

/3


45/116

INFORMAT

ION

Whenisitpossible?

Intuitivelywe

feelthatif

theInformation/Signalism

morethanthe

noise/erro

r,weshouldbeableto

extra

theInformatio

n/Signalfromthenoisydata.Otherw

willbealmostimpossible.

much

ract

wiseit

rroris .

p.14

/3


46/116

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.

.p.15

/3


47/116

INFORMAT

ION

Weneedtoqu

antifyInfo

rmation.To

quantify

Informationw

eapproxim

ateitbya

Mathematical

Model,therefore

.p.15

/3


48/116

INFORMAT

ION

WeneedtoquantifyInfo

rmation.Toquantify

Informationw

eapproxim

ateitbya

Mathematical

Model,therefore I

nforma

tionM

odel

.p.16/3


49/116

MO

DELS

.p.16/3


50/116

MO

DELS

Data=Model+Error


51/116

.p.16/3


52/116

MO

DELS

Data=Model+Error

Er

rorhasseveralcomponents

ModelingError

.p.16/3


53/116

MO

DELS

Data=Model+Error

Er


ModelingError

ExperimentationError

.p.16/3


54/116

MO

DELS

Data=Model+Error

Er


ModelingError


DataCollectionError

.p.16/3


55/116

MODELS

Data=Model+Error

Er


ModelingError


DataCol

lectionError

CalibrationError


56/116

.p.1

7/3


57/116

MODELS

DifferentTyp

esofMode

l

MechanisticModel

.p.1

7/3


58/116

MODELS

DifferentTyp

esofMode

l

MechanisticModel

Function

alModel

.p.1

8/3


59/116

MODELS

M

echanisticModel:

ctmech-

plexbut

.p.1

8/3


60/116

MODELS

M

echanisticModel:

A

mechanisticmodelisa

modelbas

edontheexac

an

ismofanexperiment.Itisusuallyquiteco

mp

mostofthetimeitisquiteeffective.

.p.1

9/3


61/116

MODELS

Fu

nctionalM

odel:

need

onal

.p.1

9/3


62/116

MODELS

Fu

nctionalM

odel:

It

ispurelyamathematic

almodel.O

nedoesno

tn

an

yexperime

ntalknowledgetoconstructafunctio

model.

edata

.p.2

0/3


63/116

MODELS

Usingtheme

chanistic/functionalm

odeloften

the

ca

nbewritteninthefollowingform

:

edata

.p.20/3


64/116

MODELS

Usingtheme

chanistic/functionalm

odeloften

the

ca


:

Data=Y

=

f(X)+

Error

edata

.p.20/3


65/116

MODELS

Usingtheme

chanistic/f

unctionalm

odeloften

th

ca


:

Data=Y

=

f(X)+

Error

YandX

areknown

hedata

.p.20/3


66/116

MODELS

Usingtheme

chanistic/f

unctionalm

odeloften

th

ca


:

Data=Y

=

f(X)+

Error

YandX

areknown

Errorisunknown

hedata

estimate .

p.20/3


67/116

MODELS

Usingtheme

chanistic/f

unctionalm

odeloften

th

ca


:

Data=Y

=

f(X)+

Error

YandX

areknown

Errorisunknown

Aim:From{

X1,

Y1

},..

.,

{X

n,

Yn}

wewanttoe

f().

wing

.p.21/3


68/116

EX

AMPLE

S

RumfordCoolingData

MechanisticModel:

BasedonNewtons

lawofcoolingthefoll

ow

modelhasbeenused:

Y(t)=6

0+70et

+(t)

(t)

.p.22/3


69/116

EX

AMPLE

S

ECGData

Function

alModel:

Y(t)=

p k=1

{Ak

cos(kt)+Bk

sin(kt)}

+

kn)}+(m,n

.p.23/3


70/116

EX

AMPLE

S

TextureData

Function

alModel:

Y

(m,n)=

p

k=1

{Ak

cos(km

+

kn)+B

k

sin(km

+

k

.p.

24/3


71/116

MODELS:

ESTIMATION

PossibleApp

roaches:

metric

.p.24/3


72/116

MODELS:

ESTIMATION

PossibleApp

roaches:

FromY

andX

estimatef().

(Non-Param

Approac

h)

metric

f()is

erefore,

hasbeen

ref(X,

) .p.24/3


73/116

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)

.p.25/3


74/116

ER

RORS

D

ifferentformsofError

s:

.p.25/3


75/116

ER

RORS

D


s:

Random

Errors

.p.25/3


76/116

ER

RORS

D


s:

Random

Errors

Systema

ticErrors

.p.26/3


77/116

ER

RORS

D

ifferentAssumptionsonErrors:

nzeros

t.

.p.26/3


78/116

ER

RORS

D


Errorsarepurelyra

ndom.Theyhavemean

andsamevariances.Theyareindependent

nzeros

t. ces.They

.p

.26/3


79/116

ER

RORS

D


Errorsarepurelyra

ndom.Theyhavemean

andsamevariances.Theyareindependent

Theyhavemeanzerosbutunequalvarianc

areindependent.

nzeros

t. ces.They

tare

.p

.26/3


80/116

ER

RORS

D


Errorsarepurelyra

ndom.Theyhavemean

andsam

evariances.Theyareindependent


areindependent.

Theyhavemeanzeros,equalv

ariancesbu

correlated.

nzeros

t. ces.They

tare

.p

.26/3


81/116

ER

RORS

D


Errorsarepurelyra

ndom.Theyhavemean

andsam

evariances.Theyareindependent


areindependent.


ariancesbu

correlated.

Spuriouserrors.

nzeros

t. ces.They

tare

stribution.

.p

.26/3


82/116

ER

RORS

D


Errorsarepurelyra

ndom.Theyhavemea

andsam

evariances

.Theyareindependen


areindependent.


ariancesbu

correlated.

Spuriouserrors.

Errorshavesomeparticularprobabilitydi

.p

.27/3


83/116

ER

RORS

W

hyarethe

assumption

snecessary

?

.p

.27/3


84/116

ER

RORS

W

hyarethe

assumption

snecessary

?

Choosin

gthecorrectmodel.

meters.

.p

.27/3


85/116

ER

RORS

W

hyarethe

assumption

snecessary

?

Choosin

gthecorrectmodel.

Effectiveestimationoftheunk

nownpara

meters.

tionsif

.p

.27/3


86/116

ER

RORS

W

hyarethe

assumption

snecessary

?

Choosin

gthecorrectmodel.


nownpara

Effectiveprediction

ofthefutu

reobserva

necessary.


87/116

meters.

tionsif

.p.27/3


88/116

ER

RORS

W

hyarethe

assumption

snecessary?

Choosin

gthecorrectmodel.


nownpara

Effectiveprediction

ofthefutu

reobserva

necessary.

Constructingerrorbarsoftheestimates.

Derivingproperties

oftheestimates.

.p.28/3


89/116

ER

RORS

H

owtoverif

ytheassum

ptions?

fpossible

.p.28/3


90/116

ERRORS

H

owtoverif

ytheassum

ptions?

Choose

aparticularf(,

)fromaclasso

models.

fpossible

ror.

.p.28/3


91/116

ERRORS

H

owtoverifytheassum

ptions?

Choose

aparticularf(,

)fromaclasso

models.

Makeso

mesimple

assumption

sontheer

fpossible

ror.

.p.28/3


92/116

ERRORS

H

owtoverifytheassum

ptions?

Choose

aparticularf(,

)fromaclasso

models.

Makeso

mesimple

assumption

sontheer

Estimatetheparam

eters.

fpossible

ror.

.p.28/3


93/116

ERRORS

H

owtoverifytheassum

ptions?

Choose

aparticularf(,

)fromaclasso

models.

Makesomesimple

assumption

sontheer

Estimatetheparam

eters.

Estimatetheerroralso.

fpossible

ror.

.p.28/3


94/116

ERRORS

H

owtoverifytheassum

ptions?

Choose

aparticularf(,

)fromaclasso

models.

Makesomesimple

assumptionsontheer

Estimatetheparam

eters.


Thenve

rifytheerrorassumptions.

fpossible

ror.

HAPPY

.

p.28/3


95/116

ERRORS

H

owtoverifytheassum

ptions?

Choose

aparticularf(,

)fromaclasso

models.

Makesomesimple

assumptionsontheer

Estimatetheparam

eters.


Thenve


Iftheer

rorassumptionsaretrue,thenbe

otherwise

ofpossible

eror.

eHAPPY

um

ptionsand

.

p.28/3


96/116

ERRORS

H

owtoverifytheassum

ptions?

Choose

aparticularf(,

)fromaclasso

models.

Makesomesimple

assumptionsontheer

Estimatetheparam

eters.


Thenve


Iftheer

rorassumptionsaretrue,thenbe

otherwise

Change

themodel,

changethe

errorassum

continuetheprocess.

ET

ERS

ne

.

p.29/3


97/116

ESTIMATINGMO

DELPA

RAMET

L

eastSquaresEstimato

rs(MostPo

pularOne)

ET

ERS

ne

.

p.29/3


98/116

ESTIMATINGMO

DELPA

RAMET

L

eastSquaresEstimato

rs(MostPo

pularOne)

min

n

i=1

(Yi

f(X

i,

))2

ET

ERS

ne) epe

ndentwith

.

p.29/3


99/116

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


100/116

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


101/116

ESTIMATINGMO

DELPA

RAMET

G

eneralized

LeastSqua

resEstimators

ET

ERS

.

p.30/3


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


103/116

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


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.


105/116

MET

ERS

.

p.31/3


106/116

ESTIMAT

INGMO

DELPA

RAM

L

eastAbsol

uteDeviations

min

n

i=

1

|Yi

f(Xi,

)|

MET

ERS

urio

usin

.

p.31/3


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


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


109/116

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


110/116

CONCLUSIONS

.p.32/3


111/116

C

ONCLUSIONS

A

llexperim

entaldataw

illhaveerrors

edata

.p.32/3


112/116

C

ONCLUSIONS

A

llexperim

entaldataw

illhaveerrors

T

oextractm

aximumin

formationfromthe

hedata

alysisAre

.p.32/3


113/116

C

ONCLUSIONS

A

llexperim

entaldataw

illhaveerrors

T

oextractm

aximumin

formationfromth

ProperErrorAssumptions/ErrorAna

Required


114/116

heda

ta

nalysisAre

Ana

lysisAre

Nee

ded

.p.32/3


115/116

C

ONCLUSIONS

Allexperim

entaldataw

illhaveerrors

Toextractm

aximumin

formationfromth

Proper

ErrorAssumptions/ErrorAn

Required

Proper

ModelAssumptitons/Model

Required

ProfessionalStatist

iciansHelpis

.p.33/3


116/116

Th

ankYou

kundu iit kanpur-experimental error.pdf

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