UQLab User Manual – The Input Module
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UQL AB U SER M ANUAL
T HE I NPUT MODULE
C. Lataniotis, S. Marelli, B. Sudret
C HAIR OF R ISK , S AFETY AND U NCERTAINTY Q UANTIFICATION
S TEFANO -F RANSCINI -P LATZ 5
CH-8093 Z ÜRICH
Risk, Safety &
Uncertainty Quantification
How to cite UQL AB
S. Marelli, and B. Sudret, UQLab: A framework for uncertainty quantification in Matlab, Proc. 2nd Int. Conf. on
Vulnerability, Risk Analysis and Management (ICVRAM2014), Liverpool, United Kingdom, 2014, 2554-2563.
How to cite this manual
C. Lataniotis, S. Marelli and B. Sudret, UQLab user manual – The Input module, Report UQLab-V1.1-102, Chair of
Risk, Safety & Uncertainty Quantification, ETH Zurich, 2018.
BIBTEX entry
@TechReport{UQdoc 11 102,
author = {Lataniotis, C. and Marelli, S. and Sudret, B.},
title = {{UQLab user manual -- I N P U T module}},
institution = {Chair of Risk, Safety & Uncertainty Quantification, ETH Zurich},
year = {2018},
note = {Report # UQLab-V1.1-102},
}
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Document Ref.
Title:
Authors:
Date:
Doc. Version
V1.1
UQL AB-V1.1-102
UQL AB User Manual – The Input module
C. Lataniotis, S. Marelli, B. Sudret
Chair of Risk. Safety and Uncertainty Quantification, ETH Zurich,
Switzerland
01/07/2018
Date
01/07/2018
Comments
UQL AB V1.1 release
• New section in the reference list about uq subsample
V1.0
01/05/2017
UQL AB V1.0 release
• New distributions: triangular, logistic, Laplace
• Updated custom distributions section
• Updated description of several functions
V0.9
01/07/2015
Initial release
Abstract
The UQL AB INPUT module is used to define the probabilistic input model in uncertainty
quantification problems. It offers extensive possibilities to perform operations like drawing
samples of random vectors, or transforming samples of random vectors to samples of different random vectors (isoprobabilistic transforms). The dependence structure between the
components of random vectors is specified with the copula formalism.
This user manual includes a review of the methods that are used to define, draw and transform samples of random vectors. It also contains information about each of the available
probability distributions that can be used in the current version of UQL AB. After introducing
the theoretical aspects, an in-depth example-driven user guide is provided to help new users
to properly set up and use the INPUT module objects. Finally, a comprehensive reference list
of the methods and functions available in the UQL AB INPUT module is given at the end of the
manual.
Keywords: Probabilistic Input Model, Marginals, Copula, Sampling
Contents
1 Theory
1
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
1.2 Representation of common univariate distributions . . . . . . . . . . . . . . .
1
1.2.1 Uniform distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
1.2.2 Gaussian (Normal) . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4
1.2.3 Lognormal distribution . . . . . . . . . . . . . . . . . . . . . . . . . . .
5
1.2.4 Gumbel distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5
1.2.5 Gumbel-min distribution . . . . . . . . . . . . . . . . . . . . . . . . . .
6
1.2.6 Weibull distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8
1.2.7 Gamma distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9
1.2.8 Exponential distribution . . . . . . . . . . . . . . . . . . . . . . . . . .
10
1.2.9 Beta distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11
1.2.10 Triangular distribution . . . . . . . . . . . . . . . . . . . . . . . . . . .
12
1.2.11 Logistic distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13
1.2.12 Laplace distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14
1.3 Truncated distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
15
1.4 Representation of random vectors and joint PDFs . . . . . . . . . . . . . . . .
15
1.4.1 Marginals and copula . . . . . . . . . . . . . . . . . . . . . . . . . . .
15
1.4.2 Copulas currently available in UQL AB . . . . . . . . . . . . . . . . . .
16
1.5 Sampling random vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
18
1.5.1 Isoprobabilistic transform of independent marginals . . . . . . . . . .
18
1.5.2 Generalized Nataf transform . . . . . . . . . . . . . . . . . . . . . . . .
18
1.5.3 Sampling multivariate distributions with the inverse generalized Nataf
transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2 Usage
19
21
2.1 Drawing samples from a distribution . . . . . . . . . . . . . . . . . . . . . . .
21
2.1.1 Introductory example . . . . . . . . . . . . . . . . . . . . . . . . . . .
21
2.1.2 Special cases of distributions . . . . . . . . . . . . . . . . . . . . . . .
22
2.1.3 Using a copula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
23
2.1.4 Selecting an INPUT object and specifying the sampling method . . . . .
24
2.2 Enrichment of an experimental design with new samples . . . . . . . . . . . .
26
2.3 Performing an isoprobabilistic transform . . . . . . . . . . . . . . . . . . . . .
27
2.4 Adding bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
27
2.5 Switching between input objects . . . . . . . . . . . . . . . . . . . . . . . . .
28
2.6 Defining and using custom marginals . . . . . . . . . . . . . . . . . . . . . . .
29
2.6.1 Advanced options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
2.7 Constant variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
3 Reference List
33
3.1 Creating an INPUT object: uq_createInput . . . . . . . . . . . . . . . . . . .
35
3.2 Getting samples from an INPUT object: uq_getSample . . . . . . . . . . . . .
38
3.3 Printing/Visualizing an INPUT object . . . . . . . . . . . . . . . . . . . . . . .
39
3.3.1 Printing information: uq_print . . . . . . . . . . . . . . . . . . . . . .
39
3.3.2 Graphical visualization: uq_display . . . . . . . . . . . . . . . . . . .
39
3.4 Enriching an existing sample set . . . . . . . . . . . . . . . . . . . . . . . . . .
41
3.4.1 Enriching a Latin Hypercube: uq_enrichLHS . . . . . . . . . . . . . .
41
3.4.2 Enriching a Sobol sequence: uq_enrichSobol . . . . . . . . . . . . .
41
3.4.3 Enriching a Halton sequence: uq_enrichHalton . . . . . . . . . . . .
42
3.4.4 Pseudo-LHS enrichment: uq_LHSify . . . . . . . . . . . . . . . . . . .
43
3.5 Sub-sampling an existing sample set: uq_subsample . . . . . . . . . . . . . .
44
3.6 Transforming samples between spaces . . . . . . . . . . . . . . . . . . . . . .
45
3.6.1 uq_GeneralIsopTransform . . . . . . . . . . . . . . . . . . . . . . .
45
3.6.2 uq_IsopTransform . . . . . . . . . . . . . . . . . . . . . . . . . . . .
45
3.6.3 uq_NatafTransform . . . . . . . . . . . . . . . . . . . . . . . . . . . .
46
3.6.4 uq_invNatafTransform . . . . . . . . . . . . . . . . . . . . . . . . .
46
3.7 Additional functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
48
3.7.1 uq_sampleU . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
48
3.7.2 uq_MarginalFields . . . . . . . . . . . . . . . . . . . . . . . . . . . .
49
3.7.3 uq_estimateMoments . . . . . . . . . . . . . . . . . . . . . . . . . . .
49
3.7.4 uq_setDefaultSampling . . . . . . . . . . . . . . . . . . . . . . . . .
50
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
52
Chapter 1
Theory
1.1
Introduction
Identification and modelling of the sources of uncertainty are crucial steps for the solution
of any uncertainty quantification problem. In a probabilistic setting, each uncertain model
parameter can be represented by a random variable and a corresponding probability density
function (PDF) in the form X ∼ fX (x). Several input parameters, including their dependence
structure, can be grouped together in a random vector with joint PDF X ∼ fX (x). The INPUT
module in UQL AB offers a suite of tools to handle the representation, transformation and
sampling of a wide variety of PDFs and joint PDFs.
1.2
Representation of common univariate distributions
A random variable can be represented by its univariate PDF X ∼ fX (x). UQL AB supports a
number of distributions employed in many fields of applied sciences, namely:
• Uniform
• Normal or Gaussian
• Lognormal
• Gumbel (maxima)
• Gumbel (minima)
• Beta
• Gamma
• Exponential
• Weibull
• Triangular
1
UQL AB User Manual
• Logistic
• Laplace
In this section, a brief overview of each distribution and its properties (e.g.PDF, cumulative
distribution function, support, moments,parameters) is given for reference.
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1.2.1
Uniform distribution
Uniform distributions are commonly used to represent variables with unknown moments and
known support. The uniform distribution is the maximum entropy distribution on any closed
support.
Notation
: X ∼ U (a, b)
Parameters : a, b ∈ R ; a < b
Support
PDF
: DX = [a, b]
Moments
1
b−a
if x ∈ [a, b]
0 if x ∈
/ [a, b]
b−a
if x ≤ a
0
x−a
x−a
1 (x) = b−a if x ∈ [a, b]
: FX (x) =
b − a [a,b]
1
if x ≥ b
a+b
: µX =
2
b−a
σX = √
2 3
: fX (x) =
CDF
1[a,b] (x)
=
1
1.2
0.9
0.8
1
0.7
0.8
0.6
0.5
0.6
0.4
0.4
0.3
0.2
0.2
0.1
0
0
-2
-1
0
1
2
-2
-1
0
1
2
Figure 1: PDF and CDF of uniform distributions for various parameter values.
A particularly important uniform distribution in the field of numerical statistics is X = U(0, 1).
In fact, every class of random number generators produces samples from this PDF (or its
multidimensional version), which are then manipulated to produce samples distributed according any other distribution as needed. For details, see Section 1.5.3.
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1.2.2
Gaussian (Normal)
Gaussian distributions are another basic family of PDF that are pervasive throughout any
fields of applied science. They are commonly employed to represent measurement error,
noise terms etc..
Notation
: X ∼ N (µ, σ)
Parameters : µ ∈ R , σ > 0
: DX = R
Support
PDF
CDF
Moments
(x−µ)2
1
: fX (x) = √ e− 2σ2
σ 2π
x−µ
1
√
1 + erf
: FX (x) =
2
σ 2
x−µ
=Φ
σ
: µX = µ
σX = σ
where µ is the mean, σ 2 the variance and erf(·) is the error function, defined by:
Z x
2
2
e−t dt
erf(x) = √
π 0
2
1
1.8
0.9
1.6
0.8
1.4
0.7
1.2
0.6
1
0.5
0.8
0.4
0.6
0.3
0.4
0.2
0.2
0.1
0
-1
0
1
2
3
4
0
-1
0
1
2
3
(1.1)
4
Figure 2: PDF and CDF of Gaussian distributions for various parameter values.
An important distribution belonging to the normal family is the so-called standard normal
distribution N (0, 1), characterized by µ = 0, σ = 1. The notation Φ(x) is used to identify the
standard normal CDF:
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Zx
2
e−t /2
√
dt
2π
Φ(x) =
−∞
(1.2)
All normal distributions can be represented as linear transforms of standard normal distributions as follows:
N (µ, σ) = µ + σN (0, 1)
1.2.3
(1.3)
Lognormal distribution
A lognormal variable is a random variable X ∼ LN (λ, ζ) such that its logarithm is a Gaussian
variable:
X ∼ LN (λ, ζ) ⇔ ln(X) ∼ N (λ, ζ)
Notation
(1.4)
: X ∼ LN (λ, ζ)
Parameters : λ ∈ R , ζ > 0
Support
: DX = (0, +∞)
CDF
1
(ln x − λ)2
: fX (x) = √
exp −
2ζ 2
2πζx
1 1
ln x − λ
√
: FX (x) = + erf
2 2
2ζ
ln x − λ
=Φ
ζ
Moments
: µX = eλ+ζ
2 /2
σX = eλ+ζ
2 /2
PDF
!
p
eζ 2 − 1
where λ and ζ are the mean and standard deviation of the natural logarithm of the variable
and the error function is defined in Eq. (1.1).
Lognormal distributions are commonly used in Engineering to describe parameters which are
positive in nature, such as material or physical properties. An important property of lognormally distributed variables is that their products and ratios are also lognormally distributed.
1.2.4
Gumbel distribution
The Gumbel distribution is also referred to as Extreme Value distribution of type I (EV I). Note
that in the literature the name ’Gumbel distribution’ is used to refer to either the maximum
or the minimum extreme value distribution. In UQL AB “Gumbel” refers to the maximum
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1
4.5
0.9
4
0.8
3.5
0.7
3
0.6
2.5
0.5
2
0.4
1.5
0.3
1
0.2
0.5
0.1
0
0
0
1
2
3
4
0
1
2
3
4
Figure 3: PDF and CDF of lognormal distributions for various parameter values.
Gumbel distribution .
Notation
: X ∼ G (µ, β)
Support
: DX = R
Parameters : µ ∈ R , β > 0
−
1 − x−µ
−e
e β
β
PDF
: fX (x) =
CDF
: FX (x) = e−e
Moments
: µX = µ + βγe
−
x−µ
β
x−µ
β
where γe = 0.577216 . . . is the Euler constant
πβ
σX = √
6
Note that parameter µ coincides with the mode of the distribution.
The Gumbel distribution is used for modelling random variables obtained as the maximum of
identically distributed variables. It is used for instance in hydrology to model flood intensity.
1.2.5
Gumbel-min distribution
The Gumbel-min is also referred to as the Smallest Extreme Value (SEV) distribution or the
Smallest Extreme Value (Type I) distribution.
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1
0.9
2
0.8
0.7
1.5
0.6
0.5
1
0.4
0.3
0.5
0.2
0.1
0
-1
0
1
2
3
4
0
-1
5
0
1
2
3
4
5
Figure 4: PDF and CDF of Gumbel maximum extreme value distributions for various parameter values.
Notation
: X ∼ G (µ, β)
Parameters : µ ∈ R , β > 0
Support
: DX = R
PDF
1 x−µ +e−
: fX (x) = e β
β
CDF
: FX (x) = 1 − e−e
Moments
: µX = µ − βγe
−
x−µ
β
x−µ
β
where γe = 0.577216 . . . is the Euler constant
πβ
σX = √
6
Note that parameter µ coincides with the mode of the distribution. .
The Gumbel-min distribution’s PDF is skewed to the left, unlike the Gumbel-max which is
skewed to the right (see Figures 4 and 5).
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1
0.9
2
0.8
0.7
1.5
0.6
0.5
1
0.4
0.3
0.5
0.2
0.1
0
-1
0
1
2
3
4
5
0
-1
0
1
2
3
4
5
Figure 5: PDF and CDF of Gumbel-min extreme value distributions for various parameter
values.
1.2.6
Weibull distribution
The Weibull distribution is the last type of extreme-value distributions. It is commonly employed to parametrize time-to-failure-type variables.
Notation
: X ∼ W (α, β)
Parameters : α > 0 , β > 0
Support
PDF :
CDF :
Moments
: DX = [0, +∞)
(
β x β−1 −(x/α)β
e
if x ≥ 0
α
α
: fX (x) =
0
,x < 0
(
β
1 − e−(x/α) if x ≥ 0
: FX (x) =
0
if x < 0
: µX = α Γ (1 + 1/β)
q
σX = α Γ (1 + 2/β) − Γ (1 + 1/β)2
Other uses of the Weibull distribution include the parametrization of strength or strengthrelated lifetime parameters, material strength and lifetime parameters for brittle materials
(for which the weakest-link-theory is applicable).
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1
0.9
2
0.8
0.7
1.5
0.6
0.5
0.4
1
0.3
0.2
0.5
0.1
0
0
0
1
2
3
4
0
1
2
3
4
Figure 6: PDF and CDF of Weibull distributions for various parameter values.
1.2.7
Gamma distribution
The Gamma distribution may be used to model variables which are positive in nature, such as
those connected to Poisson processes. Assuming events occur randomly in time in a Poisson
process at a constant rate λ, the time to first occurrence follows an exponential distribution
Γ(λ, 1). The time to k-th occurrence follows a Gamma distribution Γ(λ, k).
Notation
: X ∼ Γ (λ, k)
Parameters : λ > 0 , k > 0
Support
PDF
: DX = [0, +∞)
λk k−1 −λx
x e
Γ(k)
γ (k, λx)
: FX (x) =
Γ(k)
k
: µX =
λ
√
k
σX =
λ
: fX (x) =
CDF
Moments
where Γ(x) is the Gamma function defined by
Z
Γ(x) =
∞
tx−1 e−t dt
(1.5)
0
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and γ(x, y) is the incomplete Gamma function defined by
Z x
γ(k, x) =
tk−1 e−t dt
(1.6)
0
1
1
0.9
0.9
0.8
0.8
0.7
0.7
0.6
0.6
0.5
0.5
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0
0
0
2
4
6
8
0
2
4
6
8
Figure 7: PDF and CDF of Gamma distributions for various parameter values.
A special case of Gamma distribution is X ∼ Γ(λ, 1), which corresponds to the exponential
distribution (see Section 1.2.8).
1.2.8
Exponential distribution
A special case of the Gamma distribution, the exponential distribution is commonly used to
represent the time to first-occurrence of Poissonian-type processes, e.g. radioactive decays.
Notation
: X ∼ E (λ)
Parameters : λ > 0
Support
: DX = [0, +∞)
P DF
: fX (x) = λe−λx
CDF
: FX (x) = 1 − e−λx
Moments
: µX = 1/λ
σX = 1/λ
where λ > 0 is the scale parameter.
The Exponential distribution is also used to model the waiting times in queuing problems
(e.g.for load balancing applications in large computational facilities).
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0.45
1
0.4
0.9
0.35
0.8
0.7
0.3
0.6
0.25
0.5
0.2
0.4
0.15
0.3
0.1
0.2
0.05
0.1
0
0
0
2
4
6
8
0
2
4
6
8
Figure 8: PDF and CDF of exponential distributions for various parameter values.
1.2.9
Beta distribution
The Beta distribution commonly used to model bounded variables.
Notation
: X ∼ B (r, s, a, b)
Parameters : r > 0 , s > 0
Support
PDF :
CDF :
Moments
: DX = [a, b]
(
: fX (x) =
: FX (x) =
(x−a)r−1 (b−x)s−1
(b−a)r+s−1 B (r, s)
if x ∈ [a, b]
0
if x ∈
/ [a, b]
1
(b−a)
r+s−1
B(r, s)
if x ≤ a
Rx 0
r−1
s−1
(b − t)
dt if x ∈ [a, b]
a (t − a)
1
if x ≥ b
: µX = a + (b − a)r/(r + s)
r
b−a
rs
σX =
r+s r+s+1
where B (r, s) is the Beta function:
1
Γ(r)Γ(s)
B(r, s) =
tr−1 (1 − t)s−1 dt =
(1.7)
Γ(r + s)
0
The range of the variable is given by the parameters [a, b]. The shape of the distribution is
Z
related to parameters [r, s] and their ratio:
• When r = s the PDF is symmetrical;
• If r, s > 1 the PDF is unimodal;
• if r, s < 1 the PDF is maximum at the boundaries.
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1
3.5
0.9
0.8
3
0.7
2.5
0.6
2
0.5
0.4
1.5
0.3
1
0.2
0.5
0
0.1
0
0
0.2
0.4
0.6
0.8
1
0
0.2
0.4
0.6
0.8
1
Figure 9: PDF and CDF of Beta distributions for various parameter values (DX = [0, 1]).
1.2.10
Triangular distribution
Notation
: X ∼ Tr(a, b, c)
Parameters : a ∈ R , a < b , a < c < b
Support
PDF
CDF
Moments
: DX = [a, b]
0
if x ≤ a
2(x−a)
(b−a)(c−a) if x ∈ (a, c)
2
if x = c
: fX (x) =
b−a
2(b−x)
if x ∈ (c, b)
(b−a)(c−a)
0
if x ≥ b
0
if x ≤ a
(x−a)2
if x ∈ (a, c]
(b−a)(c−a)
: FX (x) =
2
(b−x)
1 − (b−a)(b−c) if x ∈ (c, b)
1
if x ≥ b
a+b+c
: µX =
√ 3
a2 + b2 + c2 − ab − ac − bc
√
σX =
3 2
The triangular distribution is typically used as a subjective description of a population for
which there is only limited sample data. It is based on knowledge of the minimum and
maximum and an “inspired guess” of the modal value.
UQL AB-V1.1-102
- 12 -
The Input module
1
1
0.9
0.9
0.8
0.8
0.7
0.7
0.6
0.6
0.5
0.5
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0
0
0
2
4
6
0
2
4
6
Figure 10: PDF and CDF of triangular distributions for various parameter values.
1.2.11
Logistic distribution
The logistic distribution resembles the normal distribution in shape but has heavier tails
(higher kurtosis).
Notation
: X ∼ P (µ, s)
Parameters : µ ∈ R , s > 0
UQL AB-V1.1-102
Support
: DX = R
PDF
: fX (x) =
CDF
: FX (x) =
Moments
: µX = µ
sπ
σX = √
3
e−
x−µ
s
s(1 + e−
1
1 + e−
x−µ
s
)2
x−µ
s
- 13 -
UQL AB User Manual
0.3
1
0.9
0.25
0.8
0.7
0.2
0.6
0.15
0.5
0.4
0.1
0.3
0.2
0.05
0.1
0
-10
0
10
20
0
-10
0
10
20
Figure 11: PDF and CDF of logistic distributions for various parameter values.
1.2.12
Laplace distribution
Laplace distribution is also known as double exponential distribution, because it can be
thought of as two exponential distributions (with an additional location parameter) spliced
together back-to-back.
Notation
: X ∼ L(µ, b)
Parameters : µ ∈ R , b > 0
Support
PDF
CDF
Moments
UQL AB-V1.1-102
: DX = R
1
|x − µ|
: fX (x) =
exp −
2b
b
x−µ
1
exp
if x < µ
2
b
: FX (x) =
1 − 1 exp − x−µ
if x ≥ µ
2
b
: µX = µ
√
σX = b 2
- 14 -
The Input module
0.5
1
0.45
0.9
0.4
0.8
0.35
0.7
0.3
0.6
0.25
0.5
0.2
0.4
0.15
0.3
0.1
0.2
0.05
0.1
0
-20
-10
0
10
20
0
-20
-10
0
10
20
Figure 12: PDF and CDF of Laplace distributions for various parameter values.
1.3
Truncated distributions
A truncated distribution results from restricting the domain of some probability distribution.
Assume that a random variable X follows some distribution with CDF FX (X). Of interest in
this section is the derivation of the CDF, FX0 and inverse CDF, FX0−1 of the random variable X
after limiting the support to X ∈ [a, b].
The derivation of these quantities is given below:
FX0 (X)
=
0
FX (X)−FX (a)
FX (b)−FX (a)
,
X≤a
, a
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Author : C. Lataniotis, S. Marelli, B. Sudret
Title : UQLab User Manual – The Input module
Subject : UQLab Input Module Manual
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Keywords : UQLab;Probabilistic, Input, Model;Input;, Uncertainty, Quantification;Marginals;Copula;Isoprobabilistic, Transform;Nataf, Transform;
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