Other Core Codes¶

Analytical Test Functions¶

Analytical model functions to test and validate implementation of different UQ techniques.

class analyticTestFuncs.fEx1D(q, typ, qInfo=[])¶

Analytical test functions and their exact moments for 1D parameter

Args:

q: List or 1D numpy array of size n

Samples of the parameter

typ: string

Function type, available: ‘type1’, ‘type2’:

‘type1’ for q~Uniform
‘type2’ for q~Normal

qInfo: (optional) List (Required for the moments or if q is Gaussian)

qInfo=[qMin,qMax] if q~U[qMin,qMax]
qInfo=[m,sdev] if q~N(m,sdev^2)

Methods:

eval():: Evaluates f(q) at q
moments(qInfo):: Mean and variance of f(q)

Returns:

val: 1D numpy array of size n,: Values of f(q) at q
mean: float: E[f(q)] for q
var: float: V[f(q)] for q

eval()¶: Value of f(q) at q

moments(qInfo)¶: Analytical Mean and variance of f(q)

class analyticTestFuncs.fEx2D(q1, q2, typ, method)¶

Analytical test functions for 2D parameter

Args:

q1,`q2`: Two lists or 1D numpy arrays of size n1, n2, respectively.

Samples of the parameters q1 and q2

typ: string

Function type, available: ‘type1’, ‘type2’, ‘type3’, ‘Rosenbrock’.

method: string

Method for handling the multi-dimensionality: ‘comp’ or ‘tensorProd’.

If ‘comp’ (component): n1 must be equal to n2 to make pairs of samples: q=(q1,q2).
If tensorProd’ (tensor-product): size of val is n=n1*n2.

Methods:

eval():: Evaluates f(q) at (q1,q2)
sobol(qBound):: Sobol indices, available only for typ`==’type3’. `qBound: List of length 2, qBound=[qBound1,qBound2] where qBound_i is range of qi

Returns:

val: 1D numpy array of size n

Value of f(q) at q=(q1,q2)

If method==’comp’: n=n1=n2
If method==’tensorProd’: n=n1*n2

Si[S1,S2],

where Si is the main Sobol index with respect to the i-th parameter, i=1,2

STi[ST1,ST2],

where Si is the total Sobol index with respect to the i-th parameter, i=1,2

Sij: [S12],

Dual interaction

eval()¶: Evaluates f(q) at given q1, q2

moments(distType, qInfo)¶

Mean and variance of f(q) estimated by the Monte-Carlo approach (These can be used as reference values instead of the analytical values)

Args:

distType: List of length 2

The i-th value (string) specifies the distribution type of the i-th parameter

qInfo: List of length 2

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

Returns:

mean: float: Expected value of f(q) estimated by the Monte-Carlo method
var: float: Variance of f(q) estimated by the Monte-Carlo method

sobol(qBound)¶

Sobol sensitivity indices of f(q) with respect to q1 and q2

Args:

qBound: =[qBound1,qBound2] admissible range of q1, q2

Returns:

Si: =[S1,S2],: where Si is the main Sobol index wrt the i-th parameter, i=1,2
STi[ST1,ST2],: where Si is the total Sobol index wrt the i-th parameter, i=1,2
Sij: =[S12],: dual interaction

class analyticTestFuncs.fEx3D(q1, q2, q3, typ, method, opts)¶

Analytical test functions for 3D parameter

Args:

q1, q2, q3: Three lists or 1D numpy arrays of size n1, n2, n3, respectively

Samples of the parameters q1, q2 and q3

typ: string

Function type, available: ‘Ishigami’

method: string

Method for handling the multi-dimensionality: ‘comp’ or ‘tensorProd’

If ‘comp’ (component): n1 must be equal to n2. Pair of samples: q=(q1,q2,q3)
If ‘tensorProd’ (tensor-product): size of val is n=n1*n2*n3.

opts: function fixed parameters

If ‘Ishigami’: opts=[‘a’:a_val,’b’:b_val]

Methods:

eval():: Evaluates f(q) at (q1,q2,q3)
moments(qInfo):: Analytical values of mean and variance of f(q)
sobol(qBound):: Analytical Sobol indices

Returns:

val: 1D numpy array of size n

Values of f(q) at q

If ‘comp’: n=n1=n2=n3
If ‘tensorProd’: n=n1*n2*n3

moments(qInfo)¶

Analytical mean and variance of f(q)

Args:

qInfo: List of length 3: qInfo=[qBound_1,qBoun_2,qBound_3] where qBound_i: admissible range of the i-th parameter

Returns:

mean:: Expected value of f(q)
var:: Variance of f(q)

sobol(qBound)¶

Sobol sensitivity indices of f(q) with respect to q1, q2, and q3

Args:

qBound: List of length 3: =[qBound1,qBound2,qBound3] admissible range of q1, q2, q3

Returns:

Si: =[S1,S2,S3]: where Si is Sobol indices with respect to qi
Sij: =[S12,S13,S23]: dual interactions

Surrogate to Surrogate¶

Interpolate values from one surrogate to another surrogate.

surr2surr.lagIntAtGQs(fValM1, qM1, spaceM1, nM2, spaceM2, distType)¶

Given response values fValM1 at nM1 arbitrary samples over the p-D spaceM1, the values at nM2 Gauss quadrature points over spaceM2 are computed using Lagrange interpolation.

Both spaceM1 and spaceM2 have the same dimension p.

At each of the p dimensions, ||`spaceM2`||<||`spaceM1`|| at each dimension.

The Gauss quadrature nodes over ‘spaceM2’ can be the abscissa of different types of polynomials based on the distType and the gPCE rule.

A tensor-product grid from the GQ nodes on spaceM2 is created

args:

qM1: List of length p: List of samples on spaceM1; qM1=[qM1_1,qM1_2,…,qM1_p] where qM1_i is a 1D numpy array of size nM1_i, for i=1,2,…,p.
fValM1: numpy p-D array of shape (nM1_1,nM1_2,…,nM1_p).: Response values at qM1
spaceM1: List of length p.: =[spaceM1_1,spaceM1_2,…,spaceM1_p] where spaceM1_i is a list of two elements, specifying the admissible range of qM1_i, for i=1,2,…,p.
nM2 List of length p,: Containing the number of Gauss quadrature samples qM2 in each parameter dimension, nM2=[nM2_1,nM2_2,…,nM2_p]
spaceM2: List of length p.: =[spaceM2_1,spaceM2_2,…,spaceM2_p] where spaceM2_i is a list of two elements, specifying the admissible range of qM2_i, for i=1,2,…,p.
distType: List of length p with string members: The i-th element specifies the distribution type of the i-th parameter according to the gPCE rule.

Returns:

qM2: List of length p: List of samples on spaceM2; qM2=[qM2_1,qM2_2,…,qM2_p] where qM2_i is a 1D numpy array of size nM2_i, for i=1,2,…,p.
xiM2: numpy array of shape (nM2_1*nM2_2*…*nM2_p,p): Tensor-product grid of Gauss-quadrature nodes on the mapped space of spaceM2
fValM2: 1D numpy array of size (nM1_1*nM1_2*…*nM1_p).: Interpolated response values at xiM2

Statistical Tools¶

stats.pdfFit_uniVar(f, doPlot, pwOpts)¶

Fits a PDF to samples f and plots both histogram and the fitted PDF. As an option, the plots and data can be saved on disk.

Args:

f: 1D numpy array of size n

Samples

doPlot: bool

Whether or not plotting the PDF

pwOpts: dict (optional)

Options for plotting and dumping the data with the following keys:

‘figDir’: string, Directory to save the figure and dump the data
‘figName’: string, Name of the figure
‘header’: string, the header of the dumped file
‘iLoc’: int, After converting to string will be added to the figName

stats.pdfPredict_uniVar(f, fTest, doPlot)¶

Evaluates the continuous PDF fitted to f at fTest.

Args:

f: 1D numpy array

fTest: List of length m

Returns:

pdfPred: 1D numpy array of size m

Linear Algebra¶

Tools for linear algebra.

To solve a linear system which is under-determined, the compressed sensing method is used. The required optimization is handled by cxvpy [Diamond16]. Different solvers can be used for this purpose, a list of which can be obtained by cvxpy.installed_solvers(). The required options for each solver can be found in this cvxpy page. Note that the default solver is directly specified in linAlg.myLinearRegress().

linAlg.myLinearRegress(A, R, L_=1, solver_='OSQP', max_iter_=100000)¶

Solves the linear system of equations Af=R in its normalized form A’Af=A’R

This solver works for uniquely-, over-, and under-determined linear systems. If system is under-determined, the compressed sensing method with L1 or L2 regularization is used. For this purpose, the library cvxpy is used (https://www.cvxpy.org).

Args:

A: numpy array of shape (n,K)

R: numpy array of size n

L_: int (optional): Specifies the regularization order, L_=1 or 2
solver_: string (optional): A cvxpy solver for the optimization problem. To see the available list of solvers print cvxpy.installed_solvers(). Also see https://www.cvxpy.org/tutorial/advanced/index.html?highlight=installed_solvers
max_iter_: int (optional): Maximum number of iterations to find the optimal solution when doing compressed sensing

Returns:

f: 1D numpy array of size K: The solution of the linear system A.f=R

Reshaping Tools¶

Tools for converting and reshaping arrays and lists.

reshaper.lengthVector(x)¶: Returns the length of vector x which can be a list or a numpy array

reshaper.vecs2grid(x)¶

Makes a p-D (p>1) tensor-product grid from a list of length p containg 1D numpy arrays of points in each dimension.

Args:

x: A list of length p>1: x=[x1,x2,…,xp] where xi is a 1D numpy array of size ni

Returns:

‘z’: A numpy array of shape (n1*n2*…*np,p)

reshaper.vecsGlue(*x)¶

Makes a set by gluing p>1 1D numpy arrays x0,x1,…,xp of the same size (=n)

Args:

x: 1D numpy arrays each having the size of n

Return:

z: numpy array of shape (n,p): z[:,i]=xi where i=1,2,…,p

Tools for Printing and Writing¶

Tools for printing or writing data in file

write.printRepeated(string_to_expand, length)¶: Repeats the string string_to_extend, length times