4. measurement errors

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.

2

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

3

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

5

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]


6

Measurements

t

True value

Example: Random and systematic errors


(0.14%)(0.14% )

6

Maximum random error

2 Bending point

Amplitude, 0p rms

Inaccuracy

UncertaintySystematic error

f )x(

Measurements

Mean measurement result

7

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

8

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


(xx)2f)x(dx

xk (xx)2f)x(dx

xk

1k22

x xk)xx(2k22

x xk)xx(2k22


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


11

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


(a0,b0,c0…,) (a0,b0,c0…,)

14

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


(a0,b0,c0…,) (a0,b0,c0…,)

15

Reference: [1]


fS x

a , … ,a

Defining the sensitivity factors:

this becomes:

xmax S xaamax S x

b bmax . … +

(a0,b0,c0…,)

16

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:


f/f0

a/a


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+

18

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

19

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(


= )da(2 + )db(2 + … .fa

2 fb

2

(a,b,c,…) (a,b,c,…)

20

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.


21

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


22

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


x = n a .

i =


Illustration: Noise averaging and integration

Gaussian white noise Averaging (10) and integration

Averaging

Integration

x = a1n

x = n a

OutputInput


Illustration: LabView simulation

100

100*x+1

x+1 x*10

Integration

Averaging

25Next lecture

Next lecture:

4. measurement errors

Documents