4. measurement errors
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14. MEASUREMENT ERRORS
4. MEASUREMENT ERRORS
Practically all measurements of continuums involve errors. Understanding the nature and source of these errors can help in reducing their impact.
In earlier times it was thought that errors in measurement could be eliminated by improvements in technique and equipment, however most scientists now accept this is not the case.
Reference: www.capgo.com
The types of errors include: systematic errors and random errors.
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Systematic error are deterministic; they may be predicted and hence eventually removed from data.
Systematic errors may be traced by a careful examination of the measurement path: from measurement object, via the measurement system to the observer.
Another way to reveal a systematic error is to use the repetition method of measurements.
References: www.capgo.com, [1]
NB: Systematic errors may change with time, so it is important that sufficient reference data be collected to allow the systematic errors to be quantified.
4.1. Systematic errors
4. MEASUREMENT ERRORS. 4.1. Systematic errors
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Example: Measurement of the voltage source value
VS
Temperature sensor
Rs
RinVin
Measurement system
VSVin
VSVin
Rin+ RS
Rin
4. MEASUREMENT ERRORS. 4.1. Systematic errors
4
Random error vary unpredictably for every successive measurement of the same physical quantity, made with the same equipment under the same conditions.
We cannot correct random errors, since we have no insight into their cause and since they result in random (non-predictable) variations in the measurement result.
When dealing with random errors we can only speak of the probability of an error of a given magnitude.
Reference: [1]
4. MEASUREMENT ERRORS. 4.2. Random errors. 4.2.1. Uncertainty and inaccuracy
4.2. Random errors4.2.1. Uncertainty and inaccuracy
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NB: Random errors are described in probabilistic terms, while systematic errors are described in deterministic terms. Unfortunately, this deterministic character makes it more difficult to detect systematic errors.
Reference: [1]
4. MEASUREMENT ERRORS. 4.2. Random errors. 4.2.1. Uncertainty and inaccuracy
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Measurements
t
True value
Example: Random and systematic errors
4. MEASUREMENT ERRORS. 4.2. Random errors. 4.2.1. Uncertainty and inaccuracy
(0.14%)(0.14% )
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Maximum random error
2 Bending point
Amplitude, 0p rms
Inaccuracy
UncertaintySystematic error
f )x(
Measurements
Mean measurement result
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4.2.2. Crest factorOne can define the ‘maximum possible error’ for 100% of the measurements only for systematic errors.
Reference: [1]
For random errors, an maximum random error (error interval) is defined, which is a function of the ‘probability of excess deviations’.
where k is so-called crest* factor )k0(. This inequality accretes that the probability deviations that exceed kis not greater than one over the square of the crest factor.*Crest stands here for ‘peak’.
1P{x xk}
k2
The upper (most pessimistic) limit of the error interval for any shape of the probability density function is given by the inequality of Chebyshev-Bienaymé:
4. MEASUREMENT ERRORS. 4.2. Random errors. 4.2.2. Crest factor
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2
(xx)2f)x(dx+
xk (xx)2f)x(dx
xk
(xx)2f)x(dx xk
xk1
k22
P{x xk} f)x(dx
xk
f)x(dx
xk
Proof:
k22f)x(dx
xk k22f)x(dx
xk
1k22
1k2
k22
k22
4. MEASUREMENT ERRORS. 4.2. Random errors. 4.2.2. Crest factor
(xx)2f)x(dx
xk (xx)2f)x(dx
xk
1k22
x xk)xx(2k22
x xk)xx(2k22
94. MEASUREMENT ERRORS. 4.2. Random errors. 4.2.2. Crest factor
Note that the Chebyshev-Bienaymé inequality can be derived from the Chebyshev inequality
which can be derived from the Markov inequality
2
P{x xa} a2
xP{ x a}a
x
10
Tchebyshev (most pessimistic) limit
any pdf
10 0
10-1
10-2
10-3
10-4
10-5
10-6
0 1 2 3 54
Pro
babi
lity
of e
xces
s de
viat
ions
Crest factor, k
Normal pdf
Illustration: Probability of excess deviations
4. MEASUREMENT ERRORS. 4.2. Random errors. 4.2.2. Crest factor
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4.3. Error sensitivity analysis
The sensitivity of a function to the errors in arguments is called error sensitivity analysis or error propagation analysis.
Reference: [1]
We will discuss this analysis first for systematic errors and then for random errors.
4.3.1. Systematic errors
Let us define the absolute error as the difference between the measured and true values of a physical quantity,
a a a0,
4. MEASUREMENT ERRORS. 4.3. Error propagation
12
Reference: [1]
and the relative error as:
a a a0
a0
If the final result x of a series of measurements is given by:
x = f)a,b,c,…( ,
where a, b, c ,… are independent, individually measured physical quantities, then the absolute error of x is:
x = f)a,b,c,…( f)a0,b0,c0,…(.
aa0
4. MEASUREMENT ERRORS. 4.3. Error propagation. 4.3.2. Systematic errors
13
Reference: [1]
With a Taylor expansion of the first term, this can also be written as:
in which all higher-order terms have been neglected. This is permitted provided that the absolute errors of the arguments are small and the curvature of f)a,b,c,…( at the point )a,b,c,…( is small.
f)a,b,c,…( x = a a
f)a,b,c,…( + b + ,… b
4. MEASUREMENT ERRORS. 4.3. Error propagation. 4.3.2. Systematic errors
(a0,b0,c0…,) (a0,b0,c0…,)
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Reference: [1]
One never knows the actual value of a, b, c,… . Usually the
individual measurements are given as a ± amax, b ± bmax, …
in which amax, bmax are the maximum possible errors.
In this case
f)a,b,c,…(xmax = amaxa
f)a,b,c,…( +bmax + . … b
4. MEASUREMENT ERRORS. 4.3. Error propagation. 4.3.2. Systematic errors
(a0,b0,c0…,) (a0,b0,c0…,)
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Reference: [1]
4. MEASUREMENT ERRORS. 4.3. Error propagation. 4.3.2. Systematic errors
fS x
a , … ,a
Defining the sensitivity factors:
this becomes:
xmax S xaamax S x
b bmax . … +
(a0,b0,c0…,)
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Reference: [1]
This expression can be rewritten to obtain the maximal relative error:
f a amaxxmax = a f0 a +. … +
f b bmax
b f0 bxmax
x0
this becomes:
xmax s xaamax s
xb bmax . … +
f as
xa , … , a f0
fa
Defining the relative sensitivity factors:
4. MEASUREMENT ERRORS. 4.3. Error propagation. 4.3.2. Systematic errors
f/f0
a/a
174. MEASUREMENT ERRORS. 4.3. Error propagation. 4.3.2. Systematic errors
Illustration: The rules that simplify the error sensitivity analysis
2. s xa s
xb s
ba
b = a2 x = ba 6a2a
1. s xa s
xa m
nmn
2ax = a2
a
3. s x1 x2
a s x1
a+ s x2
a
x1 = a2a 2a
x2 = aa
a
a+
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Reference: [1]
4.3.2. Random errors
If the final result x of a series of measurements is given by:
x = f)a,b,c,…( ,
where a, b, c, … are independent, individually measured physical quantities, then the absolute error of x is:
Again, we have neglected the higher order terms of the Taylor expansion.
fa
(a,b,c,…)
fb
(a,b,c,…)
fc
(a,b,c,…)
dx = da + db + dc + ….
4. MEASUREMENT ERRORS. 4.3. Error propagation. 4.3.2. Random errors
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Reference: [1]
Since dx= xx,
fa = )dx(2 = da + db + dc + … .
fb
fc
2
= )da(2 + )db(2 + …+ da db + …fa
2 fb
2 fa
fb
squares cross products )=0(
4. MEASUREMENT ERRORS. 4.3. Error propagation. 4.3.2. Random errors
= )da(2 + )db(2 + … .fa
2 fb
2
(a,b,c,…) (a,b,c,…)
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Reference: [1]
Considering that )da(2 a2 …, the expression for x
2 can be
written as (Gauss’ error propagation rule):
x2 = a
2 + b2 + c
2 + …fa
2 fb
2 fc
2
(a,b,c,…)(a,b,c,…)(a,b,c,…)
x = f)a,b,c,…(
NB: In the above derivation, the shape of the pdf of the individual measurements a, b, c, … does not matter.
4. MEASUREMENT ERRORS. 4.3. Error propagation. 4.3.2. Random errors
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Example A: Let us apply Gauss’ error propagation rule to the case of averaging in which
x = ai :1n
i = 1
n
or for the standard deviation of the end result:
x = a . 1n
Thanks to averaging, the measurement uncertainty decreases with the square root of the number of measurements.
x a .x2 n aa, 1
n21n
4. MEASUREMENT ERRORS. 4.3. Error propagation. 4.3.2. Random errors
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Example B: Let us apply Gauss’ error propagation rule to the case of integration in which
x = ai :i = 1
n
or for the standard deviation of the end result:
Due to integration, the measurement uncertainty increases with the square root of the number of measurements.
x a .x2 na
4. MEASUREMENT ERRORS. 4.3. Error propagation. 4.3.2. Random errors
x = n a .
i =
234. MEASUREMENT ERRORS. 4.3. Error propagation. 4.3.2. Random errors
Illustration: Noise averaging and integration
Gaussian white noise Averaging (10) and integration
Averaging
Integration
x = a1n
x = n a
OutputInput
244. MEASUREMENT ERRORS. 4.3. Error propagation. 4.3.2. Random errors
Illustration: LabView simulation
100
100*x+1
x+1 x*10
Integration
Averaging
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