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Transcript of 0000 - FIT Sample - Module 2 - Statistics - Framatome · excesti quia sanducit qui quis nis pa quam...
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Outline
INTRODUCTION
NORMAL DISTRIBUTION
TOLERANCE INTERVALS
RESPONSE SURFACE MODELS
CE Setpoints – Statistics - p.3
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Learning Objectives
Introduction� Describe the difference between a bias and a random un certainty
Normal Distribution� Explain why “2-sigma” is commonly used to describe 95% coverage of a
normal distribution
Tolerance Intervals� Describe a 95/95 tolerance interval
� Explain the difference between a one-sided tolerance in terval and a two-sided tolerance interval
Response Surface Models� Define a Response Surface Model
CE Setpoints – Statistics - p.4
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INTRODUCTION
Outline
INTRODUCTION
NORMAL DISTRIBUTION
TOLERANCE INTERVALS
RESPONSE SURFACE MODELS
CE Setpoints – Statistics - p.5
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INTRODUCTION
Learning Objectives
Introduction� Describe the difference between a bias and a random u ncertainty
Normal Distribution� Explain why “2-sigma” is commonly used to describe 95% coverage of a
normal distribution
Tolerance Intervals� Describe a 95/95 tolerance interval
� Explain the difference between a one-sided tolerance interval and a two-sided tolerance interval
Response Surface Models� Define a Response Surface Model
CE Setpoints – Statistics - p.6
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INTRODUCTION
Deterministic vs. Statistical
Statistics allows for the treatment of random variability. � Deterministic – Single value used to represent a Param eter.
� Statistical – Distribution of value used to represent a parameter including random variability.
CE Setpoints – Statistics - p.7
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INTRODUCTION
Types of Uncertainty
Bias – Shifts the mean of the distribution to the left or to the right
Random – Impacts the shape of the distribution
CE Setpoints – Statistics - p.8
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INTRODUCTION
Types of Uncertainty
A base assumption throughout the setpoint methodology is that uncertainty parameters may be treated as symmetric and normally distributed.
Example� Tavg uncertainty = +4.0F/-4.8F
Option 1� Apply bounding random uncertainty
Option 2� Apply bias to nominal setting
Pay extra attention in these situations
Not always the case in reality
Bias = -0.4F
Random = +-4.4F
Bias = -0.0F
Random = +-4.8F
Bias = -0.4F
Random = +-4.4F
CE Setpoints – Statistics - p.9
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INTRODUCTION
Visualization
A Histogram is used to visualize how a collection of data points is distributed.Probability Density Functions (PDF) are often used to analytically estimate statistical distributions.
CE Setpoints – Statistics - p.10
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INTRODUCTION
Review Learning Objectives
Introduction� Describe the difference between a bias and a random u ncertainty
Normal Distribution� Explain why “2-sigma” is commonly used to describe 95% coverage of a
normal distribution
Tolerance Intervals� Describe a 95/95 tolerance interval
� Explain the difference between a one-sided tolerance interval and a two-sided tolerance interval
Response Surface Models� Define a Response Surface Model
CE Setpoints – Statistics - p.11
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NORMAL DIST.
Outline
INTRODUCTION
NORMAL DISTRIBUTION
TOLERANCE INTERVALS
RESPONSE SURFACE MODELS
CE Setpoints – Statistics - p.12
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NORMAL DIST.
Learning Objectives
Introduction� Describe the difference between a bias and a random u ncertainty
Normal Distribution� Explain why “2-sigma” is commonly used to describe 95% coverage of a
normal distribution
Tolerance Intervals� Describe a 95/95 tolerance interval
� Explain the difference between a one-sided tolerance interval and a two-sided tolerance interval
Response Surface Models� Define a Response Surface Model
CE Setpoints – Statistics - p.13
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NORMAL DIST.
Normal Distribution
Many of the setpoint calculations use probability distributions to model the real world variability in input parameters.
Simplified models are typically used.� Most common is a Normal distribution.
� � �1
� 2��� �
��
Where,
µ = mean
σ = standard deviation
CE Setpoints – Statistics - p.14
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NORMAL DIST.
Normal Distribution
A normal curve represents probability as the area under the curve.� Typically calculated using
tables because there is not a closed form solution for the integral.
� Student’s T distribution
� Note that the 2-sigma interval covers just over 95% of the data. µ-3σ µ-2σ µ-1σ µ µ+1σ µ+2σ µ+3σ
0.9545
0.9973
0.6827
CE Setpoints – Statistics - p.15
Two-sided Intervals
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NORMAL DIST.
Normal Distribution
A normal curve represents probability as the area under the curve.� Typically calculated using
tables because the is not closed form solution for the integral.
µ-3σ µ-2σ µ-1σ µ µ+1σ µ+2σ µ+3σ
0.9773
0.9987
0.8414
CE Setpoints – Statistics - p.16
One-sided Intervals
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NORMAL DIST.
Review of Learning Objectives
Introduction� Describe the difference between a bias and a random un certainty
Normal Distribution� Explain why “2-sigma” is commonly used to describe 95% coverage of a
normal distribution
Tolerance Intervals� Describe a 95/95 tolerance interval
� Explain the difference between a one-sided tolerance in terval and a two-sided tolerance interval
Response Surface Models� Define a Response Surface Model
CE Setpoints – Statistics - p.17
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TOLERANCE INTV.
Outline
INTRODUCTION
NORMAL DISTRIBUTION
TOLERANCE INTERVALS
RESPONSE SURFACE MODELS
CE Setpoints – Statistics - p.18
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TOLERANCE INTV.
Learning Objectives
Introduction� Describe the difference between a bias and a random u ncertainty
Normal Distribution� Explain why “2-sigma” is commonly used to describe 95% coverage of a
normal distribution
Tolerance Intervals� Describe a 95/95 tolerance interval
� Explain the difference between a one-sided tolerance interval and a two-sided tolerance interval
Response Surface Models� Define a Response Surface Model
CE Setpoints – Statistics - p.19
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TOLERANCE INTV.
Tolerance Interval
A finite data set is inadequate to characterize a parameter’s variability with 100% certainty.
A tolerance interval has two numbers associated with it, namely a confidence level and a coverage level.
The interval is made so that we can have specified confidence that at least the specified portion of the entire population is covered by the interval.
For example, a 99/95 tolerance interval means that there is 99% confidence that 95% of the population will be covered by the given interval.� The order of the terms is not always treated consistently (i.e. 99/95 vs.
95/99).
� Luckily, we usually use 95/95 so it does not matter.
CE Setpoints – Statistics - p.20
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TOLERANCE INTV.
Tolerance Interval
For example, a 99/95 tolerance interval means that there is 99% confidence that 95% of the population will be covered by the given interval.
The 99% refers to the confidence of the interval.
This is typically impacted by the number of samples available. � A large uncertainty factor must be applied if a small number of samples
is available or if a high confidence is desired.
� A small uncertainty factor can be applied is a large number of samples is available or if a low confidence is desired.
� The confidence of a best estimate calculation is 50% .
CE Setpoints – Statistics - p.21
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TOLERANCE INTV.
Tolerance Interval
The 95% is the portion of the population being covered.
Coverage can be one-sided or two-sided.
one-sidedtwo-sided
CE Setpoints – Statistics - p.22
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TOLERANCE INTV.
Tolerance Interval
Tolerance intervals for normal distributions are of the form:
The constants k are typically referred to as “k-factors” and are tabulated.
The K factor for a two-sided 95/95 tolerance interval with infinite samples is 1.96
The K factor for a one-sided 95/95 tolerance interval with infinite samples is 1.645
�� � � ∗ �(two−sided) �� � � ∗ �(lower) �� � � ∗ �(upper)
�� (is the sample population mean) �(is the sample population standard deviation)
CE Setpoints – Statistics - p.23
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TOLERANCE INTV.
Review of Learning Objectives
Introduction� Describe the difference between a bias and a random u ncertainty
Normal Distribution� Explain why “2-sigma” is commonly used to describe 95% coverage of a
normal distribution
Tolerance Intervals� Describe a 95/95 tolerance interval
� Explain the difference between a one-sided tolerance interval and a two-sided tolerance interval
Response Surface Models� Define a Response Surface Model
CE Setpoints – Statistics - p.24
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RSM
Outline
INTRODUCTION
NORMAL DISTRIBUTION
TOLERANCE INTERVALS
RESPONSE SURFACE MODELS
CE Setpoints – Statistics - p.25
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RSM
Learning Objectives
Introduction� Describe the difference between a bias and a random uncertain ty
Normal Distribution� Explain why “2-sigma” is commonly used to describe 95% cover age of a
normal distribution
Tolerance Intervals� Describe a 95/95 tolerance interval
� Explain the difference between a one-sided tolerance interv al and a two-sided tolerance interval
Response Surface Models� Define a Response Surface Model
CE Setpoints – Statistics - p.26
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RSM
Response Surface Model (RSM)
A response surface is a multi-dimensional fit of a particular response to a set of input parameters.
Typically used to estimate complex phenomenon in an efficient way.
CE Setpoints – Statistics - p.27
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RSM
Response Surface Model
To develop an RSM, a set of experimental results characterizing the design space is required. � For DNBR calculation, the experiments are explicit XCO BRA-IIIC runs
� The design space is defined by the min and max of each input parameter
� Typically inputs are varied at integer multiples of t heir standard deviation
One option is to evaluate all possible combinations of each parameter at a given set of levels.� For example, assume each parameter can be at -2σ, -1σ, 0σ, 1σ, or 2σ
� Running all combinations of 10 parameters at 5 level s results in 9,765,625 XCOBRA-IIIC runs
CE Setpoints – Statistics - p.28
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RSM
Response Surface Model
Design of Experiments (DOE)� A design of experiments can be used to limit the num ber of explicit runs
needed to build an RSM while minimizing the loss of information
� Many types of DOE exist• Box-Benhken design• Plackett-Burman• Cubic centered design
CE Setpoints – Statistics - p.29
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RSM
Review of Learning Objectives
Introduction� Describe the difference between a bias and a random uncertainty
Normal Distribution� Explain why “2-sigma” is commonly used to describe 9 5% coverage of a
normal distribution
Tolerance Intervals� Describe a 95/95 tolerance interval
� Explain the difference between a one-sided tolerance interval and a two-sided tolerance interval
Response Surface Models� Define a Response Surface Model
CE Setpoints – Statistics - p.30
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Exercise 2.1Companion Notebook > Statistics Tab > Exercise 2.1� Tolerance Intervals
CE Setpoints – Statistics - p.31
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Review of Learning Objectives
Introduction� Describe the difference between a bias and a random u ncertainty
Normal Distribution� Explain why “2-sigma” is commonly used to describe 95% coverage of a
normal distribution
Tolerance Intervals� Describe a 95/95 tolerance interval
� Explain the difference between a one-sided tolerance interval and a two-sided tolerance interval
Response Surface Models� Define a Response Surface Model
CE Setpoints – Statistics - p.32
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