selregparam

Optimal selection of a regularization parameter value according to different model selection criteria.

Syntax

alpha = selregparam(V,K,r,'type','method')
alpha = selregparam(V,K,r,'type','all')
alpha = selregparam(V,K,r,'type',{'method1','method2',___})
alpha = selregparam({V1,V2,___},{K1,K2,___},r,'type','method')
alpha = selregparam(___,'Property',Value)
[alpha,selfcn,alphas,res,pen] = selregparam(___)
Parameters
  • V - Input signal (N-element array)
  • K - Dipolar kernel (NxM-element array)
  • r - Distance axis (M-element array)
  • type - Regularization type (string)
  • method - Model selection type (string)
Returns
  • alpha - Optimal regularization parameter (scalar)
  • selfcn - Model selection functional for given alphas (scalar)
  • alphas - Evaluated candidate regularization parameters (array)
  • res - Evaluated regularization residual term (array)
  • pen - Evaluated regularization penalty term (array)

Description

[alpha] = selregparam(V,K,r,'type','method')

Returns the optimal regularization parameter alpha from a range of regularization parameter candidates alphas. The parameter for the regularization type given by 'type' is computed based on the input signal V and the dipolar kernel K. The method employed for the selection of the regularization parameter can be specified as the 'method' input argument. The available regularization models specified by 'type' are

  • 'tikhonov' - Tikhonov regularization
  • 'tv' - Total variation regularization
  • 'huber' - Pseudo-Huber regularization
[alpha,selfcn,alphas] = selregparam(V,K,r,'type',{'method1','method2',___})

If multiple selection methods are passed as a cell array of strings, the function returns alpha as an array of optimal regularization parameters corresponding to the input methods. The selection models functionals selfcn are also returned as a cell array of arrays containing the evaluated functionals of the requested models. The order of the output parameters corresponds to the order of the model strings in the input.

[alpha,selfcn,alphas,res,pen] = selregparam(V,K,r,'type',{'method1','method2',___})

The vector of evaluated residual res and penalty pen terms can be requested as additional outputs. They can be used, e.g. to build the L-curve.

[alpha,selfcn,alphas] = selregparam(V,K,r,'type','all')

Alternatively, the argument 'all' can be passed, which will compute the optimal regularization parameter based on all the selection methods implemented in the function.

alpha = selregparam({V1,V2,___},{K1,K2,___},r,'type','method')

Passing multiple signals/kernels enables selection of the regularization parameter for global fitting of the regularization model to a single distribution. The global fit weights are automatically computed according to their contribution to ill-posedness. The multiple signals are passed as a cell array of arrays of sizes N1, N2,… and a cell array of Kernel matrices with sizes N1xM, N2xM,… must be passed as well.

Available Model Selection Criteria
String Acronym Model Selection Method
'aic' AIC Akaike information criterion
'aicc' AICc Corrected Akaike information criterion
'bic' BIC Bayesian information criterion
'cv' CV Cross-validation
'gcv' GCV Generalized cross-validation
'rgcv' rGCV Robust generalized cross-validation
'srgcv' srGCV Strong-robust generalized cross-validation
'dp' DP Discrepancy principle
'ee' EE Extrapolated error
'gml' GML Generalized maximum-likelihood
'lc' Lc L-curve (curvature-based)
'lr' Lr L-curve (radius-based)
'mcl' MCL Mallows’ C_L
'ncp' NCP Normalized cumulative periodogram
'rm' RM Residual method

Additional Settings

Additional settings can be specified via name-value pairs. All property names are case insensitive and the property-value pairs can be passed in any order after the required input arguments have been passed.

alpha = selregparam(___,'Property1',Value1,'Property2',Value2,___)
  • 'Range' - Regularization parameter search range

    Array of regularization parameter candidates to evaluate.

    Default: [empty] - Computes an optimal range automatically with regparamrange

    Example:

    alpha = selregparam(___,'Range',logspace(-3,4,100))
  • 'Search' - Regularization parameter search algorithm

    Specifies the type of algorithm used for searching the optimal regularization parameter. The possible settings are:

    • 'fminbnd' - MATLAB’s built-in function minimizer, based on golden section search with parabolic interpolation, over the interval specified in 'Range'.
    • 'grid' - Systematic search over a grid of regularization parameter values, using the grid specified in 'Range'.
    • 'golden' - Manually implemented golden section search algorithm over the interval specified in 'Range' (not compatible with the lc or lr selection methods), more primitive than 'fminbnd'.

    Default: fminbnd

    Example:

    alpha = selregparam(___,'Search','grid')
  • 'NonNegConstrained' - Non-negativity constraint

    Specifies whether the distance distribution P is to be computed under the non-negativity constraint. If the constraint is lifted, the distance distribution is computed according to the analytical solution of the inverse problem.

    Default: true

    Example:

    alpha = selregparam(___,'NonNegConstrained',false)
  • 'HuberParam' - Huber parameter value

    Value of the super-parameter used in pseudo-Huber regularization.

    Default: 1.35

    Example:

    alpha = selregparam(___,'HuberParam',2.5)
  • 'GlobalWeights' - Weights for global analysis

    Array of weighting coefficients for the individual signals in global fitting regularization. If not specified, the global fit weights are automatically computed according to their contribution to ill-posedness. Weight values do not need to be normalized. The same number of weights as number of input signals is required.

    Default: [empty]

    Example:

    alpha = selregparam(alphas,{V1,V2,V3},{K1,K2,K3},r,L,'tikhonov','aic','GlobalWeights',[0.1 0.6 0.3]])
  • 'TolFun' - Optimizer tolerance value

    Optimizer function tolerance. The solver stops once the regularization functional evaluation reaches a value lower than this tolerance. Lower values increase the precision of the result, albeit at the cost of longer computation times.

    Default: 1e-9

    Example:

    alpha = selregparam(___,'TolFun','1e-20')
  • 'RegOrder' - Regularization matrix order

    Order of the regularization operator (0,1, 2 or 3).

    Default: 2

    Example:

    alpha = selregparam(___,'RegOrder',3)
  • 'NoiseLevel' - Estimation of the noise level

    Level (standard deviation) of the noise in the input signal(s). If not specified, it is automatically computed via noiselevel. If multiple signals are passed (global fitting), the same number of noise levels must be specified. Required only for the 'dp' and 'mcl' selection methods.

    Default: [empty]

    Example:

    alpha = selregparam(___,'NoiseLevel',0.05)