Uji Statistik - [PDF Document] (2024)

Slide 1

Error Analysis - Statistics

Accuracy and Precision Individual Measurement Uncertainty

Distribution of Data Means, Variance and Standard DeviationConfidence Interval

Uncertainty of Quantity calculated from several MeasurementsError Propagation

Least Squares Fitting of Data

Slide 2

Accuracy and Precision

AccuracyCloseness of the data (sample) to the true value.

PrecisionCloseness of the grouping of the data (sample) aroundsome central value.

Slide 3

Accuracy and Precision

Inaccurate & Imprecise Precise but Inaccurate

Rel

ativ

e Fr

eque

ncy

X ValueTrue Value

Rel

ativ

e Fr

eque

ncy

X ValueTrue Value

Slide 4

Accuracy and Precision

Accurate but Imprecise Precise and Accurate

Rel

ativ

e Fr

eque

ncy

X ValueTrue Value

Rel

ativ

e Fr

eque

ncy

X ValueTrue Value

Slide 5

Accuracy and Precision

Q: How do we quantify the concept of accuracy and precision? --How do we characterize the error that occurred in ourmeasurement?

Individual Measurement Statistics

Take N measurements: X1, . . . , XN Calculate mean and standarddeviation:

What to use as the best value and uncertainty so we can say weare Q% confident that the true value lies in the interval xbestx.

Need to know how data is distributed.

N

iiXN

x1

1

N

ixix XN

S1

22 1

Slide 6

Slide 7

Population and Sample

Parent PopulationThe set of all possible measurements.

SampleA subset of the population -measurements actuallymade.

Population

Bag of Marbles

Handful of marbles from the bag

Samples

Slide 8

Histogram (Sample Based)

Histogram A plot of the number of

times a given value occurred.

Relative Frequency A plot of the relative

number of times a given value occurred.

Histogram

5

10

15

20

25

30 35 40 45 50 55 60 65 70 75 80

X Value (Bin)

Num

ber o

f M

easu

rem

ents

Relative Frequency Plot

0.05

0.1

0.15

0.2

0.25

0.3

30 35 40 45 50 55 60 65 70 75 80

X Value (Bin)

Rel

ativ

e Fr

eque

ncy

Slide 9

Probability Distribution Function (P(x))

Probability Distribution Function is the integral of the pdf,i.e.

Q: Plot the probability distribution function vs x.

Q: What is the maximum value of P(x)?

Probability Distribution (Population Based)

Probability Density Function (pdf) (p(x)) Describes theprobability

distribution of all possible measures of x.

Limiting case of the relative frequency.

xX

dxxpxP x Probability Density Function

0.05

0.1

0.15

0.2

0.25

0.3

30 35 40 45 50 55 60 65 70 75 80

x Value (Bin)

Prob

abili

ty p

er u

nit

chan

ge in

x

][ xXPxP Probability that

Slide 10

Ex:

is a probability density function. Find the relationship betweenA and B.

Probability Density Function

The probability that a measurement X takes value between (-) is1.

Every pdf satisfies the above property.

Q: Given a pdf, how would one find the probability that ameasurement is between A and B?

p x dx 1

p xA

xB

12

e

e 2

Hint: - a x dxa

120

Slide 11

Gaussian (Normal) Distribution

where: x = measured valuex = true (mean) valuex = standarddeviationx2 = variance

Q: What are the two parameters that define a Gaussiandistribution?

Common Statistical Distributions

2

2 2 1 e 2

x

x

x

x

p x

Q: How would one calculate the probability of a Gaussiandistribution between x1and x2? ( See Chapter 4, Appendix A )

x Value

p x

Slide 12

Uniform Distribution

where: x = measured valuex1 = lower limitx2 = upper limit

Q: Why do x1 and x2 also define the magnitude of the uniformdistribution PDF?

Common Statistical Distributions

otherwise 0

1 2112

xxxxx

xp

x Value

p x

Slide 13

Common Statistical Distributions

Ex: A voltage measurement has a Gaussian distribution with mean3.4 [V] and a standard deviation of 0.4 [V]. Using Chapter 4,Appendix A, calculate the probability that a measurement isbetween:(a) [2.98, 3.82] [V]

(b) [2.4, 4.02] [V]

Ex: The quantization error of an ADC hasa uniform distributionin the quantization interval Q. What is the probability that theactual input voltage is within Q/8 of the estimated inputvoltage?

Slide 14

Standard Deviation (x and Sx ) Characterize the typicaldeviation of measurements from the mean

and the width of the Gaussian distribution (bell curve). Smallerx , implies better ______________.

Population Based

Sample Based (N samples)

Q: Often we do not know x , how should we calculate Sx ?

Statistical Analysis

x xx p x dx

2

12

N

ixix XN

S1

21

Slide 15

Standard Deviation (x and Sx ) (cont.)

Statistical Analysis

Common Name for"Error" Level

Error Level inTerms of

% That the Deviationfrom the Mean is Smaller

Odds That theDeviation is Greater

Standard Deviation 68.3 about 1 in 3

"Two-Sigma Error" 95 1 in 20

"Three-Sigma Error" 99.7 1 in 370

"Four-Sigma Error" 99.994 1 in 16,000

x x x xZ x Z

Slide 16

Sampled Mean is the best estimate of x .

Sampled Standard Deviation ( Sx ) Use when x is not available.reduce by one degree of freedom.

Q: If the sampled mean is only an estimate of the true mean x ,how do we characterize its error?

Q: If we take another set of samples, will we get a differentsampled mean?Q: If we take many more sample sets, what will be thestatistics of the set of sampled means?

Statistical Analysis

x

dxxpxXEx

N

iiXN

x1

1

Degree of Freedom

Best Estimate

x

N

iix

N

ixix xXN

SXN

S x1

2knownnot When

1

2

11 1

Slide 17

Statistical Analysis

Ex: The inlet pressure of a steam generator was measured 100times during a 12 hour period. The specified inlet pressure is 4.00MPa, with 0.7% allowable fluctuation. The measured data issummarized in the following table:Pressure (P)(MPa) Number ofResults (m)

3.970 13.980 33.990 124.000 254.010 334.020 174.030 64.04024.050 1

(1) Calculate the mean, variance and standard deviation. (2)Given the data, what pressure range will contain 95% of thedata?

Slide 18

Sampled Mean Statistics If N is large, will also have a Gaussiandistribution. (Central Limit Theorem)

Mean of :

is an unbiased estimate.

Standard Deviation of :

is the best estimate of the errorin estimating x .

Q: Since we dont know x , how would we calculate ?

Confidence Interval

x

x xE x x

x

x

xx

N

x

x

x

x

p x( )

p x( )

p x( )

Slide 19

For Large Samples ( N > 60 ), Q% of all the sampled meanswill lie in the interval

Equivalently,

is the Q% Confidence Interval

When x is unknown, Sx will be a reasonable approximation.

Confidence Interval

x

x x xx

N z zQ Q

x

Nx

Nx

xx

x x

z zQ Q

x x

p x

zQ x zQ x

Slide 20

Confidence Interval

Ex: 64 acceleration measurements were taken during anexperiment. The estimated mean and standard deviation of themeasurements were 3.15 m/s2and 0.4 m/s2. (1) Find the 98%confidence interval for the true mean.

(2) How confident are you that the true mean will be in therange from 2.85 to 3.45 m/s2 ?

Slide 21

For Small Samples ( N < 60 ), the Q% Confidence Interval canbe calculated using the Student-T distribution, which is similar tothe normal distribution but depends on N.

with Q% confidence, the true mean x will lie in the followinginterval about any sampled mean:

t,Q is defined in class notes Chapter 4, Appendix B.

Confidence Interval

x S

Nx S

N

N

x

S

xx

Sx x

t t

where

,Q ,Q

Q% confidence interval

1

Slide 22

Confidence Interval

Ex: A simple postal scale is supplied with , 1, 2, and 4 ozbrass weights. For quality check, 14 of the 1 oz weights weremeasured on a precision scale. The results, in oz, are asfollows:

1.08 1.03 0.96 0.95 1.041.01 0.98 0.99 1.05 1.080.97 1.00 0.981.01

Based on this sample and that the parent population of theweight is normally distributed, what is the 95% confidence intervalfor the true weight of the 1 oz brass weights?

Slide 23

Propagation of Error

Q: If you measured the diameter (D) and height (h) of acylindrical container, how would the measurement error affect yourestimation of the volume ( V = D2h/4 )?

Q: What is the uncertainty in calculating the kinetic energy (mv2/ 2 ) given the uncertainties in the measurements of mass (m)and velocity (v)?

How do errors propagate through calculations?

Slide 24

A Simple ExampleSuppose that y is related to two independentquantities X1 and X2 through

To relate the changes in y to the uncertainties in X1 and X2, weneed to find dy = g(dX1, dX2):

The magnitude of dy is the expected change in y due to theuncertainties in x1 and x2:

Propagation of Error

212211 , XXfXCXCy

dy

22212

22

2

11

21 xxyCCx

Xfx

Xfy

Slide 25

General FormulaSuppose that y is related to n independentmeasured variables {X1, X2, , Xn} by a functionalrepresentation:

Given the uncertainties of Xs around some operating points:

The expected value of and its uncertainty y are:

Propagation of Error

nXXXfy ,,, 21

x x x x x xn n1 1 2 2 , , ,

nxxx

nn

n

xXfx

Xfx

Xfy

xxxfy

,,,

22

22

2

11

11

11

,,,

y

Propagation of Error

Proof:Assume that the variability in measurement y is caused byk independent zero-mean error sources: e1, e2, . . . , ek.Then, (y- ytrue)2 = (e1 + e2 + . . . + ek)2

= e12 + e22 + . . . + ek2 + 2e1e2 + 2e1e3 + . . .E[(y - ytrue)2]= E[e12 + e22 + . . . + ek2 + 2e1e2 + 2e1e3 + . . .]

= E[e12 + e22 + . . . + ek2]

y k kE e E e E e 12 2 2 2 12 2 2 2

Slide 26

Slide 27

Example (Standard Deviation of Sampled Mean)Given

Use the general formula for error propagation:

Propagation of Error

NXXXXNx 321

1

N

Xx

Xx

Xx

Xx

xx

xN

xxxx N

22

3

2

2

2

1321

Slide 28

Propagation of Error

Ex: What is the uncertainty in calculating the kinetic energy (mv2/ 2 ) given the uncertainties in the measurements of mass (m)and velocity (v)?

KE KEm

m KEv

v

mv mm

mv vv

mv mm

vv

2 2

22

22

22 2

12

2

12

2

Slide 29

Best Linear FitHow do we characterize BEST?

Fit a linear model (relation)

to N pairs of [xi, yi] measurements.

Given xi, the error between the estimated output and themeasured output yi is:

The BEST fit is the model that minimizes the sum of the___________ of the error

Least Squares Fitting of Data

Input X

Out

put Y best linear

fit yest

measured output yi

y a a xi o i 1

y i

n y yi i i

min minn y yi

i=

N

i ii=

N2

1

2

1

Least Square Error

Slide 30

Let

The two independent variables are?

Q: What are we trying to solve?

Least Squares Fitting of Data

J y y y a a xi ii=

N

i o ii=

N

2

11

2

1

M inim ize Find and such that 1J a a dJo 0

Ja

y a a x

o

i o iiN

2 011

Ja

x y a a xi i o iiN

2 011

Slide 31

Least Squares Fitting of Data

Rewrite the last two equations as two simultaneous equations forao and a1:

ax y x x y

aN x y x y

N x xo

i i i i i

i i i ii i

2

1

2 2

where

a N a x y

a x a x x y

aa

yx y

o i i

o i i i i

o i

i i

1

12

1

Slide 32

Summary: Given N pairs of input/output measurements [xi, yi],the best linear Least Squares model from input xi to output yiis:

where

The process of minimizing squared error can be used for fittingnonlinear models and many engineering applications.

Same result can also be derived from a probability distributionpoint of view (see Course Notes, Ch. 4 - Maximum LikelihoodEstimation ).

Q: Given a theoretical model y = ao + a2 x2 , what are the LeastSquares estimates for ao & a2?

Least Squares Fitting of Data

y a a xi o i 1

a

x y x x y

aN x y x y N x x

oi i i i i

i i i ii i

2

1

2 2

and

Slide 33

Least Squares Fitting of Data

Variance of the fit:

Variance of the measurements in y: y2

Assume measurements in x are precise. Correlationcoefficient:

is a measure of how well the model explains the data.R2 = 1implies that the linear model fits the data perfectly.

RS

n

y

n

y

22

2

2

21 1

,

n N i o iiN y a a x2 1 2 1

21