MEASUREMENT AND INSTRUMENTATION BMCC 4743elektro.umk.ac.id/1qbal/si/lecture3.ppt · PPT file ·...
Transcript of MEASUREMENT AND INSTRUMENTATION BMCC 4743elektro.umk.ac.id/1qbal/si/lecture3.ppt · PPT file ·...
LECTURE 3: ANALYSIS OF EXPERIMENTAL DATA
Mochamad Safarudin
Faculty of Mechanical Engineering, UTeM
2010
MEASUREMENT AND INSTRUMENTATIONBMCC 3743
Introduction Measures of dispersion Parameter estimation Criterion for rejection questionable data
points Correlation of experimental data
2
Needed in all measurements with random inputs, e.g. random broadband sound/noise◦ Tyre/road noise, rain drops, waterfall
Some important terms are:◦ Random variable (continuous or discrete),
histogram, bins, population, sample, distribution function, parameter, event, statistic, probability.
3
Population : the entire collection of objects, measurements, observations and so on whose properties are under consideration
Sample: a representative subset of a population on which an experiment is performed and numerical data are obtained
4
IntroductionMeasures of dispersion Parameter estimation Criterion for rejection questionable data
points Correlation of experimental data
5
Deviation (error) is defined as
Mean deviation is defined as
Population standard deviation is defined as
xxd ii
n
i
i
nd
d1
6
=>Measures of data spreading or variability
N
i
i
Nx
1
2
Sample standard deviation is defined as
◦ is used when data of a sample are used to estimate population std dev.
Variance is defined as
n
i
i
nxxS
1
2
1
sampleaforS
orpopulationthefor
2
2
7
Find the mean, median, standard deviation and variance of this measurement:1089, 1092, 1094, 1095, 1098, 1100, 1104, 1105,
1107, 1108, 1110, 1112, 1115
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Introduction Measures of dispersionParameter estimation Criterion for rejection questionable data
points Correlation of experimental data
10
Generally,Estimation of population mean, is sample mean, .
Estimation of population standard deviation, is sample standard deviation, S.
x
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Confidence interval is the interval between to , where is an
uncertainty. Confidence level is the probability for the
population mean to fall within specified interval:
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xxP
x x
Normally referred in terms of , also called level of significance, where
confidence level If n is sufficiently large (> 30), we can apply
the central limit theorem to find the estimation of the population mean.
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1
1. If original population is normal, then distribution for the sample means’ is normal (Gaussian)
2. If original population is not normal and n is large, then distribution for sample means’ is normal
3. If original population is not normal and n is small, then sample means’ follow a normal distribution only approximately.
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When n is large,
where
Rearranged to get
Or with confidence level
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1/ 2/2/ zn
xzP
nxz
/
nzx 2/ 1
12/2/ nzx
nzxP
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Table z
Confidence Interval
Confidence Level (%)
Level of Significance (%)
3.30 99.9 0.1
3.0 99.7 0.3
2.57 99.0 1.0
2.0 95.4 4.6
1.96 95.0 5.0
1.65 90.0 10.0
1.0 68.3 31.7
Area under 0 to z
When n is small,
where
Rearranged to get
Or with confidence level
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1/ 2/2/ tnS
xtP
nSxt
/
12/2/ nStx
nStxP
nStx 2/ 1
t table
Similarly as before, but now using chi-squared distribution, , (always positive)
where
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2
11 2
2/,2
22
2/1, vvSnP
2
22 1
Sn
Hence, the confidence interval on the population variance is
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2
2/1,
22
22/,
2 11
vv
SnSn
Chi squared table
Introduction Measures of dispersion Parameter estimationCriterion for rejection questionable data points
Correlation of experimental data
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To eliminate data which has low probability of occurrence => use Thompson test.
Example: Data consists of nine values, Dn = 12.02, 12.05, 11.96, 11.99, 12.10, 12.03, 12.00, 11.95 and 12.16.
= 12.03, S = 0.07 So, calculate deviation:
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08.003.1295.11
13.003.1216.12
2
arg1
DD
DD
smallest
estl
D
From Thompson’s table, when n = 9, then
Comparing withwhere then D9 = 12.16 should be discarded.
Recalculate S and to obtain 0.05 and 12.01 respectively.
Hence for n = 8, andso remaining data stay.
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777.1
12.077.107.0 S ,13.01 ,1 S
D
749.1 ,09.0S
Thompson’s ble
Introduction Measures of dispersion Parameter estimation Criterion for rejection questionable data
pointsCorrelation of experimental data
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A) Correlation coefficientB) Least-square linear fitC) Linear regression using data
transformation
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Case I: Strong, linear relationship between x and y
Case II: Weak/no relationship Case III: Pure chance
=> Use correlation coefficient, rxy to determine Case III
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Given as
where +1 means positive slope (perfectly linear
relationship) -1 means negative slope (perfectly linear
relationship) 0 means no linear correlation
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2/1
1 1
22
1
n
i
n
iii
i
n
ii
xy
yyxx
yyxxr
11 xyr
In practice, we use special Table (using critical values of rt) to determine Case III.
If from experimental value of |rxy| is equal or more than rt as given in the Table, then linear relationship exists.
If from experimental value of |rxy| is less than rt as given in the Table, then only pure chance => no linear relationship exists.
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To get best straight line on the plot: Simple approach: ruler & eyes More systematic approach: least squares
◦ Variation in the data is assumed to be normally distributed and due to random causes
◦ To get Y = ax + b, it is assumed that Y values are randomly vary and x values have no error.
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For each value of xi, error for Y values are
Then, the sum of squared errors is
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iii yYe
n
iii
n
iii ybaxyYE
1
2
1
2
Minimising this equation and solving it for a & b, we get
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22
2
22
ii
iiiii
ii
iiii
xxn
yxxyxb
xxn
yxyxna
Substitute a & b values into Y = ax + b, which is then called the least-squares best fit.
To measure how well the best-fit line represents the data, we calculate the standard error of estimate, given by
where Sy,x is the standard deviation of the differences between data points and the best-fit line. Its unit is the same as y.
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2111
2
,
n
yxaybyS i
xy
…Is another good measure to determine how well the best-fit line represents the data, using
For a good fit, must be close to unity.
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2
22 1
yyybax
ri
ii
2r
For some special cases, such as
Applying natural logarithm at both sides, gives
where ln(a) is a constant, so ln(y) is linearly related to x.
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bxaey
abxy lnln
Example Thermocouples are usually approximately linear
devices in a limited range of temperature. A manufacturer of a brand of thermocouple has obtained the following data for a pair of thermocouple wires:T(0C) 20 30 40 50 60 75 100
V(mV) 1.02 1.53 2.05 2.55 3.07 3.56 4.05
Determine the linear correlation between T and V
Solution:Tabulate the data using this table:
2/1
1 1
22
1
n
i
n
iii
i
n
ii
xy
yyxx
yyxxr rxy= 0.980392
No x (0C) y(mV)1 20 1.02 -33.57 1127.04 -1.53 2.33 51.272 30 1.53 -23.57 555.61 -1.02 1.03 23.983 40 2.05 -13.57 184.18 -0.50 0.25 6.754 50 2.55 -3.57 12.76 0.00 0.00 -0.015 60 3.07 6.43 41.33 0.52 0.27 3.366 75 3.56 21.43 459.18 1.01 1.03 21.707 100 4.05 46.43 2155.61 1.50 2.26 69.78
53.572.55
4535.71 7.17 176.82
xy
xx i yy i )( xx i )( yy i 2)( xx i 2yy i
Another example
The following measurements were obtained in the calibration ofa pressure transducer:
Voltage P H2O
0.31 1.960.65 4.200.75 4.900.85 5.480.91 5.911.12 7.301.19 7.731.38 9.001.52 9.90
a. Determine the best fit straight line
b. Find the coefficient ofdetermination for the best fit
xi xi2 yi xiyi yi
2
0.31 0.0961 1.96 0.6076 3.84160.65 0.4225 4.2 2.73 17.640.75 0.5625 4.9 3.675 24.010.85 0.7225 5.48 4.658 30.03040.91 0.8281 5.91 5.3781 34.92811.12 1.2544 7.3 8.176 53.291.19 1.4161 7.73 9.1987 59.75291.38 1.9044 9 12.42 811.52 2.3104 9.9 15.048 98.01
sum () 8.68 9.517 56.38 61.8914 402.503
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2
22
ii
iiiii
ii
iiii
xxn
yxxyxb
xxn
yxyxna
a= 6.560646
b= -0.062934
Y=6.56x-0.06
2
22 1
yyybax
ri
ii
0.999926r2=
xi yi (Yi-yi)2 (yi-y)2
0.31 1.96 0.000118 18.530.65 4.2 0.000002 4.260.75 4.9 0.001802 1.860.85 5.48 0.001130 0.620.91 5.91 0.000008 0.131.12 7.3 0.000225 1.071.19 7.73 0.000203 2.151.38 9 0.000085 7.481.52 9.9 0.000086 13.22
sum () 0.003659 49.31
From the result before we can find coeff of determination r2
by tabulating the following values