13.12. Calculation algorithm “NonLinearLeastSquares”¶

13.12.1. Description¶

This algorithm realizes a state estimation by variational minimization of the classical $J$ function of weighted “Least Squares”:

$J(\mathbf{x})=(\mathbf{y}^o-\mathbf{H}.\mathbf{x})^T.\mathbf{R}^{-1}.(\mathbf{y}^o-\mathbf{H}.\mathbf{x})$

It is similar to a Calculation algorithm “3DVAR”, without its background part. The background, required in the interface, is only used as an initial point for the variational minimization.

This mono-objective optimization algorithm is naturally written for a single estimate, without any dynamic or iterative notion (there is no need in this case for an incremental evolution operator, nor for an evolution error covariance). In ADAO, it can also be used on a succession of observations, placing the estimate in a recursive framework partly similar to a Kalman Filter. A standard estimate is made at each observation step on the state predicted by the incremental evolution model.

In all cases, it is recommended to prefer a Calculation algorithm “3DVAR” for its stability as for its behavior during optimization.

13.12.2. Optional and required commands¶

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

Background: Vector. The variable indicates the background or initial vector used, previously noted as $\mathbf{x}^b$ . Its value is defined as a “Vector” or “VectorSerie” type object. 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 data assimilation or optimization case. Moreover, the parameters of the command “AlgorithmParameters” allows 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:

Bounds

List of pairs of real values. This key allows to define pairs of upper and lower bounds for every state variable being optimized. Bounds have to be given by a list of list of pairs of lower/upper bounds for each variable, with a value of None each time there is no bound. The bounds can always be specified, but they are taken into account only by the constrained optimizers.

Example: {"Bounds":[[2.,5.],[1.e-2,10.],[-30.,None],[None,None]]}

CostDecrementTolerance

Real value. This key indicates a limit value, leading to stop successfully the iterative optimization process when the cost function decreases less than this tolerance at the last step. The default is 1.e-7, and it is recommended to adapt it to the needs on real problems. One can refer to the section describing ways for Convergence control for calculation cases and iterative algorithms for more detailed recommendations.

Example: {"CostDecrementTolerance":1.e-7}

EstimationOf

Predefined name. This key allows to choose the type of estimation to be performed. It can be either state-estimation, with a value of “State”, or parameter-estimation, with a value of “Parameters”. The default choice is “Parameters”.

Example: {"EstimationOf":"Parameters"}

GradientNormTolerance

Real value. This key indicates a limit value, leading to stop successfully the iterative optimization process when the norm of the gradient is under this limit. It is only used for non-constrained optimizers. The default is 1.e-5 and it is not recommended to change it.

Example: {"GradientNormTolerance":1.e-5}

InitializationPoint

Vector. The variable specifies one vector to be used as the initial state around which an iterative algorithm starts. By default, this initial state is not required and is equal to the background $\mathbf{x}^b$ . Its value must allow to build a vector of the same size as the background. If provided, it replaces the background only for initialization.

Example : {"InitializationPoint":[1, 2, 3, 4, 5]}

MaximumNumberOfIterations

Integer value. This key indicates the maximum number of internal iterations allowed for iterative optimization. The default is 15000, which is very similar to no limit on iterations. It is then recommended to adapt this parameter to the needs on real problems. For some optimizers, the effective stopping step can be slightly different of the limit due to algorithm internal control requirements. One can refer to the section describing ways for Convergence control for calculation cases and iterative algorithms for more detailed recommendations.

Example: {"MaximumNumberOfIterations":100}

Minimizer: Predefined name. This key allows to choose the optimization minimizer. The default choice is “LBFGSB”, and the possible ones are “LBFGSB” (nonlinear constrained minimizer, see [Byrd95], [Morales11], [Zhu97]), “TNC” (nonlinear constrained minimizer), “CG” (nonlinear unconstrained minimizer), “BFGS” (nonlinear unconstrained minimizer), It is strongly recommended to stay with the default.

ProjectedGradientTolerance

Real value. This key indicates a limit value, leading to stop successfully the iterative optimization process when all the components of the projected gradient are under this limit. It is only used for constrained optimizers. The default is -1, that is the internal default of each minimizer (generally 1.e-5), and it is not recommended to change it.

Example: {"ProjectedGradientTolerance":-1}

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 availability 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 unconditional 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”): [ “Analysis”, “BMA”, “CostFunctionJ”, “CostFunctionJAtCurrentOptimum”, “CostFunctionJb”, “CostFunctionJbAtCurrentOptimum”, “CostFunctionJo”, “CostFunctionJoAtCurrentOptimum”, “CurrentIterationNumber”, “CurrentOptimum”, “CurrentState”, “CurrentStepNumber”, “ForecastState”, “IndexOfOptimum”, “Innovation”, “InnovationAtCurrentAnalysis”, “InnovationAtCurrentState”, “OMA”, “OMB”, “SimulatedObservationAtBackground”, “SimulatedObservationAtCurrentOptimum”, “SimulatedObservationAtCurrentState”, “SimulatedObservationAtOptimum”, ].

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

Tips for this algorithm:

As the “BackgroundError” command is required for ALL the calculation algorithms in the graphical interface ADAO EFICAS, you have to provide a value, even if this command is not required for this algorithm, and will not be used. The simplest way is to give “1” as a STRING.

13.12.3. 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 information 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:

Analysis

List of vectors. Each element of this variable is an optimal state $\mathbf{x}^*$ in optimization, an interpolate or an analysis $\mathbf{x}^a$ in data assimilation.

Example: xa = ADD.get("Analysis")[-1]

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:

Analysis

List of vectors. Each element of this variable is an optimal state $\mathbf{x}^*$ in optimization, an interpolate or an analysis $\mathbf{x}^a$ in data assimilation.

Example: xa = ADD.get("Analysis")[-1]

BMA

List of vectors. Each element is a vector of difference between the background and the optimal state.

Example: bma = ADD.get("BMA")[-1]

CostFunctionJ

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

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

CostFunctionJAtCurrentOptimum

List of values. Each element is a value of the error function $J$ . At each step, the value corresponds to the optimal state found from the beginning.

Example: JACO = ADD.get("CostFunctionJAtCurrentOptimum")[:]

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")[:]

CostFunctionJbAtCurrentOptimum

List of values. Each element is a value of the error function $J^b$ . At each step, the value corresponds to the optimal state found from the beginning. If this part does not exist in the error function, its value is zero.

Example: JbACO = ADD.get("CostFunctionJbAtCurrentOptimum")[:]

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")[:]

CostFunctionJoAtCurrentOptimum

List of values. Each element is a value of the error function $J^o$ , that is of the observation difference part. At each step, the value corresponds to the optimal state found from the beginning.

Example: JoACO = ADD.get("CostFunctionJoAtCurrentOptimum")[:]

CurrentIterationNumber

List of integers. Each element is the iteration index at the current step during the iterative algorithm procedure. There is one iteration index value per assimilation step corresponding to an observed state.

Example: cin = ADD.get("CurrentIterationNumber")[-1]

CurrentOptimum

List of vectors. Each element is the optimal state obtained at the usual step of the iterative algorithm procedure of the optimization algorithm. It is not necessarily the last state.

Example: xo = ADD.get("CurrentOptimum")[:]

CurrentState

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

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

CurrentStepNumber

List of integers. Each element is the index of the current step in the iterative process, driven by the series of observations, of the algorithm used. This corresponds to the observation step used. Note: it is not the index of the current iteration of the algorithm even if it coincides for non-iterative algorithms.

Example: csn = ADD.get("CurrentStepNumber")[-1]

ForecastState

List of vectors. Each element is a state vector forecasted by the model during the iterative algorithm procedure.

Example: xf = ADD.get("ForecastState")[:]

IndexOfOptimum

List of integers. Each element is the iteration index of the optimum obtained at the current step of the iterative algorithm procedure of the optimization algorithm. It is not necessarily the number of the last iteration.

Example: ioo = ADD.get("IndexOfOptimum")[-1]

Innovation

List of vectors. Each element is an innovation vector, which is in static the difference between the optimal and the background, and in dynamic the evolution increment.

Example: d = ADD.get("Innovation")[-1]

InnovationAtCurrentAnalysis

List of vectors. Each element is an innovation vector at current analysis. This quantity is identical to the innovation vector at analysed state in the case of a single-state assimilation.

Example: da = ADD.get("InnovationAtCurrentAnalysis")[-1]

InnovationAtCurrentState

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

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

OMA

List of vectors. Each element is a vector of difference between the observation and the optimal state in the observation space.

Example: oma = ADD.get("OMA")[-1]

OMB

List of vectors. Each element is a vector of difference between the observation and the background state in the observation space.

Example: omb = ADD.get("OMB")[-1]

SimulatedObservationAtBackground

List of vectors. Each element is a vector of observation simulated by the observation operator from the background $\mathbf{x}^b$ . It is the forecast from the background, and it is sometimes called “Dry”.

Example: hxb = ADD.get("SimulatedObservationAtBackground")[-1]

SimulatedObservationAtCurrentOptimum

List of vectors. Each element is a vector of observation simulated from the optimal state obtained at the current step the optimization algorithm, that is, in the observation space.

Example: hxo = ADD.get("SimulatedObservationAtCurrentOptimum")[-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]

SimulatedObservationAtOptimum

List of vectors. Each element is a vector of observation obtained by the observation operator from simulation on the analysis or optimal state $\mathbf{x}^a$ . It is the observed forecast from the analysis or the optimal state, and it is sometimes called “Forecast”.

Example: hxa = ADD.get("SimulatedObservationAtOptimum")[-1]

13.12.4. Python (TUI) use examples¶

Here is one or more very simple examples of the proposed algorithm and its parameters, written in [DocR] Textual User Interface for ADAO (TUI/API). Moreover, when it is possible, the information given as input also allows to define an equivalent case in [DocR] Graphical User Interface for ADAO (GUI/EFICAS).

This example describes the calibration of parameters $\mathbf{x}$ of a quadratic observation model $H$ . This model is here represented as a function named QuadFunction. This function get as input the coefficients vector $\mathbf{x}$ , and return as output the evaluation vector $\mathbf{y}$ of the quadratic model at the predefined internal control points. The calibration is done using an initial coefficient set (background state specified by Xb in the code), and with the information $\mathbf{y}^o$ (specified by Yobs in the code) of 5 measures obtained in these same internal control points. We set twin experiments (see To test a data assimilation chain: the twin experiments) and the measurements are supposed to be perfect.

The adjustment is carried out by displaying intermediate results during iterative optimization.

# -*- coding: utf-8 -*-
#
from numpy import array, ravel
def QuadFunction( coefficients ):
    """
    Quadratic simulation in x: y = a x^2 + b x + c
    """
    a, b, c = list(ravel(coefficients))
    x_points = (-5, 0, 1, 3, 10)
    y_points = []
    for x in x_points:
        y_points.append( a*x*x + b*x + c )
    return array(y_points)
#
Xb   = array([1., 1., 1.])
Yobs = array([57, 2, 3, 17, 192])
#
print("Resolution of the calibration problem")
print("-------------------------------------")
print("")
from adao import adaoBuilder
case = adaoBuilder.New()
case.setBackground( Vector = Xb, Stored=True )
case.setObservation( Vector = Yobs, Stored=True )
case.setObservationError( ScalarSparseMatrix = 1. )
case.setObservationOperator( OneFunction = QuadFunction )
case.setAlgorithmParameters(
    Algorithm='NonLinearLeastSquares',
    Parameters={
        'MaximumNumberOfIterations': 100,
        'StoreSupplementaryCalculations': [
            'CurrentState',
            ],
        },
    )
case.setObserver(
    Info="  Intermediate state at the current iteration:",
    Template='ValuePrinter',
    Variable='CurrentState',
    )
case.execute()
print("")
#
#-------------------------------------------------------------------------------
#
print("Calibration of %i coefficients in a 1D quadratic function on %i measures"%(
    len(case.get('Background')),
    len(case.get('Observation')),
    ))
print("----------------------------------------------------------------------")
print("")
print("Observation vector.................:", ravel(case.get('Observation')))
print("A priori background state..........:", ravel(case.get('Background')))
print("")
print("Expected theoretical coefficients..:", ravel((2,-1,2)))
print("")
print("Number of iterations...............:", len(case.get('CurrentState')))
print("Number of simulations..............:", len(case.get('CurrentState'))*4)
print("Calibration resulting coefficients.:", ravel(case.get('Analysis')[-1]))
#
Xa = case.get('Analysis')[-1]
import matplotlib.pyplot as plt
plt.rcParams['figure.figsize'] = (10, 4)
#
plt.figure()
plt.plot((-5,0,1,3,10),QuadFunction(Xb),'b-',label="Simulation at background")
plt.plot((-5,0,1,3,10),Yobs,            'kX',label='Observation',markersize=10)
plt.plot((-5,0,1,3,10),QuadFunction(Xa),'r-',label="Simulation at optimum")
plt.legend()
plt.title('Coefficients calibration', fontweight='bold')
plt.xlabel('Arbitrary coordinate')
plt.ylabel('Observations')
plt.savefig("simple_NonLinearLeastSquares.png")

The execution result is the following:

Resolution of the calibration problem
-------------------------------------

  Intermediate state at the current iteration: [1. 1. 1.]
  Intermediate state at the current iteration: [1.99739508 1.07086406 1.01346638]
  Intermediate state at the current iteration: [1.83891966 1.04815981 1.01208385]
  Intermediate state at the current iteration: [1.8390702  1.03667176 1.01284797]
  Intermediate state at the current iteration: [1.83967236 0.99071957 1.01590445]
  Intermediate state at the current iteration: [1.84208099 0.8069108  1.02813037]
  Intermediate state at the current iteration: [ 1.93711599 -0.56383147  1.12097995]
  Intermediate state at the current iteration: [ 1.99838848 -1.00480576  1.1563713 ]
  Intermediate state at the current iteration: [ 2.0135905  -1.04815936  1.16155285]
  Intermediate state at the current iteration: [ 2.01385679 -1.03874812  1.16129658]
  Intermediate state at the current iteration: [ 2.01377856 -1.03700048  1.16157611]
  Intermediate state at the current iteration: [ 2.01338903 -1.02943739  1.16528951]
  Intermediate state at the current iteration: [ 2.01265633 -1.01708474  1.17793974]
  Intermediate state at the current iteration: [ 2.01124871 -0.99745512  1.21485092]
  Intermediate state at the current iteration: [ 2.00863696 -0.96943287  1.30917045]
  Intermediate state at the current iteration: [ 2.00453385 -0.94011718  1.51021885]
  Intermediate state at the current iteration: [ 2.00013539 -0.93313894  1.80539445]
  Intermediate state at the current iteration: [ 1.95437219 -0.76890307  2.04566901]
  Intermediate state at the current iteration: [ 1.99797363 -0.92538077  1.81674454]
  Intermediate state at the current iteration: [ 1.99760514 -0.9592967   2.01402117]
  Intermediate state at the current iteration: [ 1.99917565 -0.99152673  2.03171823]
  Intermediate state at the current iteration: [ 1.99990376 -0.99963125  2.00671607]
  Intermediate state at the current iteration: [ 1.99999841 -1.00005288  2.00039727]
  Intermediate state at the current iteration: [ 2.00000014 -1.00000309  2.00000249]
  Intermediate state at the current iteration: [ 2.         -0.99999995  2.00000015]

Calibration of 3 coefficients in a 1D quadratic function on 5 measures
----------------------------------------------------------------------

Observation vector.................: [ 57.   2.   3.  17. 192.]
A priori background state..........: [1. 1. 1.]

Expected theoretical coefficients..: [ 2 -1  2]

Number of iterations...............: 25
Number of simulations..............: 100
Calibration resulting coefficients.: [ 2.         -0.99999995  2.00000015]

The figures illustrating the result of its execution are as follows:

_images/simple_NonLinearLeastSquares.png

13.12.5. See also¶

References to other sections:

Bibliographical references: