Transcript Chapter 4 - Errors in experimental measurements.

Errors in Experimental
Measurements




Sources of errors
Accuracy, precision, resolution
A mathematical model of errors
Confidence intervals



For means
For proportions
How many measurements are needed for
desired error?
Copyright 2004 David J. Lilja
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Why do we need statistics?
1. Noise, noise, noise, noise, noise!
OK – not really this type of noise
Copyright 2004 David J. Lilja
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Why do we need statistics?
445 446 397 226
388 3445 188 1002
47762 432 54 12
98 345 2245 8839
77492 472 565 999
1 34 882 545 4022
827 572 597 364
2. Aggregate data into
meaningful
information.
x  ...
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What is a statistic?

“A quantity that is computed from a sample
[of data].”
Merriam-Webster
→ A single number used to summarize a larger
collection of values.
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What are statistics?
“A branch of mathematics dealing with the
collection, analysis, interpretation, and
presentation of masses of numerical data.”
Merriam-Webster
→ We are most interested in analysis and
interpretation here.

“Lies, damn lies, and statistics!”

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Goals
Provide intuitive conceptual background for
some standard statistical tools.

•
•
Draw meaningful conclusions in presence of
noisy measurements.
Allow you to correctly and intelligently apply
techniques in new situations.
→ Don’t simply plug and crank from a formula.
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Goals
Present techniques for aggregating large
quantities of data.

•
•
Obtain a big-picture view of your results.
Obtain new insights from complex
measurement and simulation results.
→ E.g. How does a new feature impact the
overall system?
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Sources of Experimental Errors

Accuracy, precision, resolution
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Experimental errors

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Errors → noise in measured values
Systematic errors
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Result of an experimental “mistake”
Typically produce constant or slowly varying bias
Controlled through skill of experimenter
Examples
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Temperature change causes clock drift
Forget to clear cache before timing run
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Experimental errors

Random errors
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Result of
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Unpredictable, non-deterministic
Unbiased → equal probability of increasing or decreasing
measured value
Limitations of measuring tool
Observer reading output of tool
Random processes within system
Typically cannot be controlled

Use statistical tools to characterize and quantify
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Example: Quantization
→ Random error
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Quantization error


Timer resolution
→ quantization error
Repeated measurements
X±Δ
Completely unpredictable
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A Model of Errors
Error
Measured
value
Probability
-E
x–E
½
+E
x+E
½
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A Model of Errors
Error 1
Error 2
Probability
-E
Measured
value
x – 2E
-E
+E
x
¼
+E
-E
x
¼
+E
+E
x + 2E
¼
-E
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¼
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A Model of Errors
Probability
0.6
0.5
0.4
0.3
0.2
0.1
0
x-E
x
x+E
Measured value
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Probability of Obtaining a
Specific Measured Value
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A Model of Errors
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Pr(X=xi) = Pr(measure xi)
= number of paths from real value to xi
Pr(X=xi) ~ binomial distribution
As number of error sources becomes large
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n → ∞,
Binomial → Gaussian (Normal)
Thus, the bell curve
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Frequency of Measuring Specific
Values
Accuracy
Precision
Mean of measured values
Resolution
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True value
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Accuracy, Precision,
Resolution

Systematic errors → accuracy
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Random errors → precision
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How close mean of measured values is to true
value
Repeatability of measurements
Characteristics of tools → resolution

Smallest increment between measured values
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Quantifying Accuracy,
Precision, Resolution

Accuracy
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Hard to determine true accuracy
Relative to a predefined standard
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
Resolution
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E.g. definition of a “second”
Dependent on tools
Precision

Quantify amount of imprecision using statistical
tools
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Confidence Interval for the
Mean
1-α
α/2
c1
c2
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α/2
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Normalize x
xx
z
s/ n
n  number of measurement s
n
x  mean   xi
i 1
2
(
x

x
)
i1 i
n
s  standarddeviation
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n 1
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Confidence Interval for the
Mean

Normalized z follows a Student’s t distribution
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(n-1) degrees of freedom
Area left of c2 = 1 – α/2
Tabulated values for t
1-α
α/2
c1
c2
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α/2
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Confidence Interval for the
Mean

As n → ∞, normalized distribution becomes
Gaussian (normal)
1-α
α/2
c1
c2
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α/2
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Confidence Interval for the
Mean
c1  x  t1 / 2;n 1
c2  x  t1 / 2;n 1
s
n
s
n
T hen,
Pr(c1  x  c2 )  1  
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An Example
Experiment
1
2
3
4
5
6
7
8
Measured value
8.0 s
7.0 s
5.0 s
9.0 s
9.5 s
11.3 s
5.2 s
8.5 s
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An Example (cont.)

x
n
x
i 1 i
 7.94
n
s  samplestandarddeviation 2.14
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An Example (cont.)
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90% CI → 90% chance actual value in interval
90% CI → α = 0.10
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1 - α /2 = 0.95
n = 8 → 7 degrees of freedom
1-α
α/2
c1
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c2
α/2
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90% Confidence Interval
a  1   / 2  1  0.10 / 2  0.95
t a;n 1  t0.95;7  1.895
1.895(2.14)
c1  7.94 
 6.5
8
1.895(2.14)
c2  7.94 
 9.4
8
a
n
0.90
…
5
…
…
…
1.476 2.015 2.571
6
7
1.440 1.943 2.447
1.415 1.895 2.365
…
∞
…
…
…
1.282 1.645 1.960
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0.95
0.975
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95% Confidence Interval
a  1   / 2  1  0.10 / 2  0.975
t a;n 1  t0.975;7  2.365
2.365(2.14)
 6.1
8
2.365(2.14)
c2  7.94 
 9.7
8
c1  7.94 
a
n
0.90
…
5
…
…
…
1.476 2.015 2.571
6
7
1.440 1.943 2.447
1.415 1.895 2.365
…
∞
…
…
…
1.282 1.645 1.960
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0.95
0.975
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What does it mean?
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90% CI = [6.5, 9.4]
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95% CI = [6.1, 9.7]
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90% chance real value is between 6.5, 9.4
95% chance real value is between 6.1, 9.7
Why is interval wider when we are more
confident?
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Higher Confidence → Wider
Interval?
90%
9.4
6.5
95%
6.1
9.7
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Key Assumption
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Measurement errors are
Normally distributed.
Is this true for most
measurements on real
computer systems?
1-α
α/2
c1
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c2
α/2
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Key Assumption
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Saved by the Central Limit Theorem
Sum of a “large number” of values from any
distribution will be Normally (Gaussian)
distributed.
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What is a “large number?”
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Typically assumed to be >≈ 6 or 7.
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How many measurements?
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Width of interval inversely proportional to √n
Want to minimize number of measurements
Find confidence interval for mean, such that:

Pr(actual mean in interval) = (1 – α)
(c1 , c2 )  (1  e) x , (1  e) x 
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How many measurements?
(c1 , c2 )  (1  e) x
s
n
 x  z1 / 2
z1 / 2
s
 xe
n
 z1 / 2 s 
n

 xe 
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How many measurements?



But n depends on knowing mean and
standard deviation!
Estimate s with small number of
measurements
Use this s to find n needed for desired
interval width
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How many measurements?
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Mean = 7.94 s
Standard deviation = 2.14 s
Want 90% confidence mean is within 7% of
actual mean.
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How many measurements?
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Mean = 7.94 s
Standard deviation = 2.14 s
Want 90% confidence mean is within 7% of
actual mean.
α = 0.90
(1-α/2) = 0.95
Error = ± 3.5%
e = 0.035
Copyright 2004 David J. Lilja
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How many measurements?
 z1 / 2 s   1.895(2.14) 
  212.9
n
  
 x e   0.035(7.94) 
2
213 measurements
→ 90% chance true mean is within ± 3.5% interval

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Proportions
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p = Pr(success) in n trials of binomial
experiment
Estimate proportion: p = m/n
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m = number of successes
n = total number of trials
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Proportions
c1  p  z1 / 2
p (1  p )
n
c2  p  z1 / 2
p (1  p )
n
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Proportions
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How much time does processor spend in
OS?
Interrupt every 10 ms
Increment counters
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n = number of interrupts
m = number of interrupts when PC within OS
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Proportions
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How much time does processor spend in
OS?
Interrupt every 10 ms
Increment counters
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n = number of interrupts
m = number of interrupts when PC within OS
Run for 1 minute
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n = 6000
m = 658
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Proportions
(c1 , c2 )  p  z1 / 2
p (1  p )
n
0.1097(1  0.1097)
 0.1097 1.96
 (0.1018,0.1176)
6000
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95% confidence interval for proportion
So 95% certain processor spends 10.2-11.8% of its
time in OS
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Number of measurements for
proportions
(1  e) p  p  z1 / 2
p (1  p )
n
p (1  p )
ep  z1 / 2
n
2
z1 / 2 p (1  p )
n
2
(ep )
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Number of measurements for
proportions



How long to run OS experiment?
Want 95% confidence
± 0.5%
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Number of measurements for
proportions





How long to run OS experiment?
Want 95% confidence
± 0.5%
e = 0.005
p = 0.1097
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Number of measurements for
proportions
z12 / 2 p (1  p )
n
2
(ep )
(1.960) (0.1097)(1  0.1097)

2
0.005(0.1097)
 1,247,102
2
10 ms interrupts
→ 3.46 hours

Copyright 2004 David J. Lilja
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Important Points

Use statistics to



Deal with noisy measurements
Aggregate large amounts of data
Errors in measurements are due to:
Accuracy, precision, resolution of tools
 Other sources of noise
→ Systematic, random errors

Copyright 2004 David J. Lilja
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Important Points: Model errors
with bell curve
Accuracy
Precision
Mean of measured values
Resolution
Copyright 2004 David J. Lilja
True value
51
Important Points


Use confidence intervals to quantify precision
Confidence intervals for



Confidence level


Mean of n samples
Proportions
Pr(actual mean within computed interval)
Compute number of measurements needed
for desired interval width
Copyright 2004 David J. Lilja
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