Sampling¶

In UQit, different types of samples can be taken from the parameter space. From one point of view, the parameter samples are divided into training and test. To construct a surrogate or perform a UQ forward problem, we need to take training samples from a mapped or standardized space $\Gamma$ and then map them onto the parameter admissible space $\mathbb{Q}$ . In contrast, the test samples which are, for instance, used to evaluate the constructed surrogates, are taken from $\mathbb{Q}$ and then are mapped onto $\Gamma$ .

Available types of training samples:

GQ: Gauss-Quadrature nodes

can be used with distributions Unif, Norm

GLL: Gauss-Lobatto-Legendre nodes

unifSpaced: Uniformly-spaced samples

unifRand: Uniformly distributed random samples

normRand: Gaussian distributed random samples

Clenshaw: Clenshaw points

Clenshaw-Curtis: Clenshaw-Curtis points

Available types of test samples:

GLL: Gauss-Lobatto-Legendre nodes

unifSpaced: Uniformly-spaced points

unifRand: Uniformly distributed random

normRand: Gaussian distributed random

Note that the argument qInfo appearing in sampling methods:

qInfo=[a,b], if the parameter is Unif over range $[a,b]$ , i.e. $q\sim\mathcal{U}[a,b]$
qInfo=[m,s] contains the mean $m$ and standard-deviation $s$ , if the parameter is Norm, i.e. $q\sim \mathcal{N}(m,s^2)$

Example¶

tr_=trainSample(sampleType='GQ',GQdistType='Unif',qInfo=[2,3],nSamp=10)
tr_=trainSample(sampleType='NormRand',qInfo=[2,3],nSamp=10)
tr_=trainSample(sampleType='GLL',qInfo=[2,3],nSamp=10)

ts_=testSample(sampleType='unifRand',GQdistType='Unif',qBound=[-1,3],nSamp=10)
ts_=testSample(sampleType='unifRand',qBound=[-1,3],nSamp=10)
ts_=testSample(sampleType='normRand',GQdistType='Norm',qBound=[-1,3],qInfo=[0.5,2],nSamp=10)
ts_=testSample(sampleType='unifSpaced',GQdistType='Norm',qBound=[-1,3],qInfo=[0.5,2],nSamp=10)
ts_=testSample(sampleType='unifSpaced',GQdistType='Unif',qBound=[-1,3],nSamp=10)
ts_=testSample(sampleType='GLL',qBound=[-1,3],nSamp=10)

Implementation¶

sampling.LHS_sampling(n, xBound)¶

LHS (Latin Hypercube) sampler from a p-D random variable distributed uniformly. Credits: https://zmurchok.github.io/2019/03/15/Latin-Hypercube-Sampling.html

Args:

n: int: Number of samples to be taken
xBound: list of length p: =[[min(x1),max(x1)],…[min(xp),max(xp)]], where [min(xi),max(xi)] specifies the range of the i-th parameter

Returns:

x: 2D numpy array of size (n,p): Samples taken from the p-D space with ranges xBound

class sampling.testSample(sampleType, qBound, nSamp, GQdistType='Unif', qInfo=[])¶

Generating test samples from a 1D paramter space using different methods. Samples of q in parameter space Q are drawn and then mapped to xi in the mapped space Gamma.

Args:

sampleType: string

Type of sample, chosen from the following list:

‘GLL’: Gauss-Lobatto-Legendre nodes
‘unifSpaced’: Uniformly-spaced
‘unifRand’: Uniformly distributed random
‘normRand’: Gaussian distributed random

GQdistType: string

Type of standard distribution in gPCE; default is ‘Unif’

‘Unif’: Uniform distribution, Gamma=[-1,1]
‘Norm’: Gaussian distribution, Gamma=[-infty,infty]

qInfo: List of length 2 (optional)

qInfo=[mean,sdev] only if GQdistType==’Norm’

qBound: List of length 2

Admissible range of q

nSamp: int

Number of samples to draw

Attributes:

xi: 1D numpy array of size nSamp: Samples on the mapped space Gamma
xiBound: List of length 2: Admissible range of xi
q: 1D numpy array of size nSamp: Samples q from the mapped space Q
qBound: List of length 2: Admissible range of q. It will be the same as the argument qBound if GQdistType==’Unif’

Examples:

ts1=testSample(sampleType=’unifRand’,GQdistType=’Unif’,qBound=[-1,3],nSamp=10) ts2=testSample(sampleType=’unifRand’,qBound=[-1,3],nSamp=10) ts3=testSample(sampleType=’unifSpaced’,GQdistType=’Norm’,qBound=[-1,3],qInfo=[0.5,2],nSamp=10) ts4=testSample(sampleType=’normRand’,GQdistType=’Norm’,qBound=[-1,3],qInfo=[0.5,2],nSamp=10) ts5=testSample(sampleType=’unifSpaced’,GQdistType=’Unif’,qBound=[-1,3],nSamp=10) ts6=testSample(sampleType=’GLL’,qBound=[-1,3],nSamp=10)

q2xi_map()¶: Linearly map q in Q to xi in Gamma

class sampling.trainSample(sampleType='', GQdistType='', qInfo=[], nSamp=0)¶

Generating training samples from a 1D paramter space using different methods. Samples of xi are drawn from the mapped space Gamma and are then mapped to the parameter space Q.

Args:

sampleType: string

Sample type, chosen from the following list:

‘GQ’: Gauss-Quadrature nodes
‘GLL’: Gauss-Lobatto-Legendre nodes
‘unifSpaced’: Uniformly-spaced
‘unifRand’: Uniformly distributed random
‘normRand’: Gaussian distributed random
‘Clenshaw’: Clenshaw points
‘Clenshaw-Curtis’: Clenshaw-Curtis points

GQdistType: string (optional)

Specifies type of gPCE standard distribution if sampleType==’GQ’

‘Unif’: Uniform distribution, Gamma=[-1,1]
‘Norm’: Gaussian distribution, Gamma=[-infty,infty]

qInfo: List of length 2 (optional)

Information about the parameter range or distribution.

If q is Gaussian (‘Norm’ or ‘normRand’) => qInfo=[mean,sdev]
Otherwise, qInfo`=[min(q),max(q)]=admissible range of `q

nSamp: Integer

Number of samples to draw

Attributes:

xi: 1D numpy array of size nSamp: Samples drawn from the mapped space Gamma
xiBound: List of length 2: Admissible range of xi
q: 1D numpy array of size nSamp: Samples over the parameter space Q
qBound: List of length 2: Admissible range of q
w: 1D numpy array of size nSamp: Weights in Gauss-Quadrature rule only if sampleType==’GQ’

Examples:

ts1=trainSample(sampleType=’GQ’,GQdistType=’Unif’,qInfo=[2,3],nSamp=10) ts2=trainSample(sampleType=’NormRand’,qInfo=[2,3],nSamp=10) ts3=trainSample(sampleType=’GLL’,qInfo=[2,3],nSamp=10)

gqPtsWts()¶: Gauss-Quadrature nodes and weights according to the gPCE rule

xi2q_map()¶: Linearly map xi in Gamma to q in Q

Nodes¶

Some of the sampling methods rely on generating nodes from mathematical polynomials, for instance see [Canuto87]. The associated methods are implemented in nodes.py.

nodes.ClenshawCurtis_pts(l)¶

Generates Clenshaw-Curtis nodes at level l over range [0,1]

Args:

l: int: Level

Returns:

x: 1D numpy array of size n: Contains Clenshaw-Curtis points

nodes.Clenshaw_pts(n)¶

Generates Clenshaw points over range [-1,1]

Args:

n: int: Number of nodes

Returns

x_: 1D numpy array of size n: Clenshaw points

nodes.gllPts(n, eps=1e-08, maxIter=1000)¶

Generating Gauss-Lobatto-Legendre (GLL) nodes of order n using the Newton-Raphson iteration.

Args:

n: int: Number of GLL nodes
eps: float (optional): Min error to keep the iteration running
maxIter: float (optional): Max number of iterations

Outputs:

xi: 1D numpy array of size n: GLL nodes
w: 1D numpy array of size n: GLL weights

Reference:

Canuto C., Hussaini M. Y., Quarteroni A., Tang T. A., “Spectral Methods in Fluid Dynamics,” Section 2.3. Springer-Verlag 1987. https://link.springer.com/book/10.1007/978-3-642-84108-8