14.9. Checking algorithm “SamplingTest”¶

Description¶

This algorithm allows to calculate the values, linked to a $\mathbf{x}$ state, of a general error function $J$ of type $L^1$ , $L^2$ or $L^{\infty}$ , with or without weights, and of the observation operator, for an priori given states sample. The default error function is the augmented weighted least squares function, classically used in data assimilation.

It is useful to test the sensitivity, of the error function $J$ , in particular, to the state $\mathbf{x}$ variations. When a state is not observable, a “NaN” value is returned.

The sampling of the states $\mathbf{x}$ can be given explicitly or under the form of hyper-cubes, explicit or sampled using classic distributions. Be careful to the size of the hyper-cube (and then to the number of calculations) that can be reached, it can be big very quickly.

To be visible by the user, the results of sampling has to be explicitly asked for. One use for that, on the desired variable, the final saving through “UserPostAnalysis” or the treatment during the calculation by “observer”.

To perform distributed or more complex sampling, see OPENTURNS module available in SALOME.

Optional and required commands¶

The general required commands, available in the editing user graphical or textual interface, are the following:

CheckingPoint: Vector. The variable indicates the vector used as the state around which to perform the required check, noted $\mathbf{x}$ and similar to the background $\mathbf{x}^b$ . It is defined as a “Vector” or “VectorSerie” type object. Its availability in output is conditioned by the boolean “Stored” associated with input.

BackgroundError: Matrix. This indicates the background error covariance matrix, previously noted as $\mathbf{B}$ . Its value is defined as a “Matrix” type object, a “ScalarSparseMatrix” type object, or a “DiagonalSparseMatrix” type object, as described in detail in the section Requirements to describe covariance matrices. Its availability in output is conditioned by the boolean “Stored” associated with input.

Observation: List of vectors. The variable indicates the observation vector used for data assimilation or optimization, and usually noted $\mathbf{y}^o$ . Its value is defined as an object of type “Vector” if it is a single observation (temporal or not) or “VectorSeries” if it is a succession of observations. Its availability in output is conditioned by the boolean “Stored” associated in input.

ObservationError: Matrix. The variable indicates the observation error covariance matrix, usually noted as $\mathbf{R}$ . It is defined as a “Matrix” type object, a “ScalarSparseMatrix” type object, or a “DiagonalSparseMatrix” type object, as described in detail in the section Requirements to describe covariance matrices. Its availability in output is conditioned by the boolean “Stored” associated with input.

ObservationOperator: Operator. The variable indicates the observation operator, usually noted as $H$ , which transforms the input parameters $\mathbf{x}$ to results $\mathbf{y}$ to be compared to observations $\mathbf{y}^o$ . Its value is defined as a “Function” type object or a “Matrix” type one. In the case of “Function” type, different functional forms can be used, as described in the section Requirements for functions describing an operator. If there is some control $U$ included in the observation, the operator has to be applied to a pair $(X,U)$ .

The general optional commands, available in the editing user graphical or textual interface, are indicated in List of commands and keywords for an ADAO checking case. Moreover, the parameters of the command “AlgorithmParameters” allow to choose the specific options, described hereafter, of the algorithm. See Description of options of an algorithm by “AlgorithmParameters” for the good use of this command.

The options are the following:

QualityCriterion

Predefined name. This key indicates the quality criterion, minimized to find the optimal state estimate. The default is the usual data assimilation criterion named “DA”, the augmented weighted least squares. The possible criteria has to be in the following list, where the equivalent names are indicated by the sign “<=>”: [“AugmentedWeightedLeastSquares” <=> “AWLS” <=> “DA”, “WeightedLeastSquares” <=> “WLS”, “LeastSquares” <=> “LS” <=> “L2”, “AbsoluteValue” <=> “L1”, “MaximumError” <=> “ME” <=> “Linf”].

Example: {"QualityCriterion":"DA"}

SampleAsExplicitHyperCube

List of list of real values. This key describes the calculations points as an hyper-cube, from a given list of explicit sampling of each variable as a list. That is then a list of lists, each of them being potentially of different size.

Example : {"SampleAsExplicitHyperCube":[[0.,0.25,0.5,0.75,1.], [-2,2,1]]} for a state space of dimension 2

SampleAsIndependantRandomVariables

List of triplets [Name, Parameters, Number]. This key describes the calculations points as an hyper-cube, for which the points on each axis come from a independent random sampling of the axis variable, under the specification of the distribution, its parameters and the number of points in the sample, as a list ['distribution', [parameters], number] for each axis. The possible distributions are ‘normal’ of parameters (mean,std), ‘lognormal’ of parameters (mean,sigma), ‘uniform’ of parameters (low,high), or ‘weibull’ of parameter (shape). That is then a list of the same size than the one of the state.

Example : {"SampleAsIndependantRandomVariables":[ ['normal',[0.,1.],3], ['uniform',[-2,2],4]] for a state space of dimension 2

SampleAsMinMaxStepHyperCube

List of triplets of real values. This key describes the calculations points as an hyper-cube, from a given list of implicit sampling of each variable by a triplet [min,max,step]. That is then a list of the same size than the one of the state. The bounds are included.

Example : {"SampleAsMinMaxStepHyperCube":[[0.,1.,0.25],[-1,3,1]]} for a state space of dimension 2

SampleAsnUplet

List of states. This key describes the calculations points as a list of n-uplets, each n-uplet being a state.

Example : {"SampleAsnUplet":[[0,1,2,3],[4,3,2,1],[-2,3,-4,5]]} for 3 points in a state space of dimension 4

SetDebug

Boolean value. This variable leads to the activation, or not, of the debug mode during the function or operator evaluation. The default is “False”, the choices are “True” or “False”.

Example: {"SetDebug":False}

SetSeed

Integer value. This key allow to give an integer in order to fix the seed of the random generator used in the algorithm. By default, the seed is left uninitialized, and so use the default initialization from the computer, which then change at each study. To ensure the reproducibility of results involving random samples, it is strongly advised to initialize the seed. A simple convenient value is for example 123456789. It is recommended to put an integer with more than 6 or 7 digits to properly initialize the random generator.

Example: {"SetSeed":123456789}

StoreSupplementaryCalculations

List of names. This list indicates the names of the supplementary variables, that can be available during or at the end of the algorithm, if they are initially required by the user. Their avalability involves, potentially, costly calculations or memory consumptions. The default is then a void list, none of these variables being calculated and stored by default (excepted the unconditionnal variables). The possible names are in the following list (the detailed description of each named variable is given in the following part of this specific algorithmic documentation, in the sub-section “Information and variables available at the end of the algorithm”): [ “CostFunctionJ”, “CostFunctionJb”, “CostFunctionJo”, “CurrentState”, “InnovationAtCurrentState”, “SimulatedObservationAtCurrentState”, ].

Example : {"StoreSupplementaryCalculations":["BMA", "CurrentState"]}

Information and variables available at the end of the algorithm¶

At the output, after executing the algorithm, there are information and variables originating from the calculation. The description of Variables and informations available at the output show the way to obtain them by the method named get, of the variable “ADD” of the post-processing in graphical interface, or of the case in textual interface. The input variables, available to the user at the output in order to facilitate the writing of post-processing procedures, are described in the Inventory of potentially available information at the output.

Permanent outputs (non conditional)

The unconditional outputs of the algorithm are the following:

CostFunctionJ

List of values. Each element is a value of the chosen error function $J$ .

Example: J = ADD.get("CostFunctionJ")[:]

CostFunctionJb

List of values. Each element is a value of the error function $J^b$ , that is of the background difference part. If this part does not exist in the error function, its value is zero.

Example: Jb = ADD.get("CostFunctionJb")[:]

CostFunctionJo

List of values. Each element is a value of the error function $J^o$ , that is of the observation difference part.

Example: Jo = ADD.get("CostFunctionJo")[:]

Set of on-demand outputs (conditional or not)

The whole set of algorithm outputs (conditional or not), sorted by alphabetical order, is the following:

CostFunctionJ

List of values. Each element is a value of the chosen error function $J$ .

Example: J = ADD.get("CostFunctionJ")[:]

CostFunctionJb

List of values. Each element is a value of the error function $J^b$ , that is of the background difference part. If this part does not exist in the error function, its value is zero.

Example: Jb = ADD.get("CostFunctionJb")[:]

CostFunctionJo

List of values. Each element is a value of the error function $J^o$ , that is of the observation difference part.

Example: Jo = ADD.get("CostFunctionJo")[:]

CurrentState

List of vectors. Each element is a usual state vector used during the iterative algorithm procedure.

Example: Xs = ADD.get("CurrentState")[:]

InnovationAtCurrentState

List of vectors. Each element is an innovation vector at current state before analysis.

Example: ds = ADD.get("InnovationAtCurrentState")[-1]

SimulatedObservationAtCurrentState

List of vectors. Each element is an observed vector simulated by the observation operator from the current state, that is, in the observation space.

Example: hxs = ADD.get("SimulatedObservationAtCurrentState")[-1]

14.9. Checking algorithm “SamplingTest”¶

Description¶

Optional and required commands¶

Information and variables available at the end of the algorithm¶

See also¶