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stk_param_estim


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 STK_PARAM_ESTIM estimates the parameters of a covariance function

 CALL: PARAM = stk_param_estim (MODEL, XI, YI, PARAM0)
 CALL: [PARAM, LNV] = stk_param_estim (MODEL, XI, YI, PARAM0)

   estimates the parameters PARAM of the covariance function in MODEL
   from the data (XI, YI) using the restricted maximum likelihood (ReML)
   method. A starting point PARAM0 must be provided.

   The observations are assumed to be noisy if MODEL.lognoisevariance is
   not -inf. In this case, the variance of the noise is estimated if
   MODEL.lognoisevariance is nan, and assumed known otherwise. The
   estimated log-variance is returned as the second output argument LNV
   (equal to MODEL.lognoisevariance when it is assumed to be known).

 CALL: [PARAM, LNV] = stk_param_estim (MODEL, XI, YI, PARAM0, LNV0)

   additionally provides an initial guess LNV0 for the logarithm of the
   noise variance. In this case the observations are automatically assumed
   to be noisy, and the value of MODEL.lognoisevariance is ignored.

 CALL: PARAM = stk_param_estim (MODEL, XI, YI, PARAM0, [], CRIT)
 CALL: [PARAM, LNV] = stk_param_estim (MODEL, XI, YI, PARAM0, LNV0, CRIT)

   uses the estimation criterion CRIT instead of the default ReML criterion.

 EXAMPLES: see, e.g., stk_example_kb02, stk_example_kb03, stk_example_kb04,
           stk_example_kb06, stk_example_misc02



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 STK_PARAM_ESTIM estimates the parameters of a covariance function



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stk_param_getdefaultbounds


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 STK_PARAM_GETDEFAULTBOUNDS [STK internal]



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 STK_PARAM_GETDEFAULTBOUNDS [STK internal]




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stk_param_gls


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 STK_PARAM_GLS computes a generalised least squares estimate

 CALL: BETA = stk_param_gls (MODEL, XI, ZI)

   computes the generalised least squares estimate BETA of the vector of
   coefficients for the linear part of MODEL, where XI and ZI stand for
   the evaluation points and observed responses, respectively.

 CALL: [BETA, SIGMA2] = stk_param_gls (MODEL, XI, ZI)

   also returns the associated unbiased estimate SIGMA2 of sigma^2, assu-
   ming that the actual covariance matrix of the Gaussian process part of
   the model is sigma^2 K, with K the covariance matrix built from MODEL.

   SIGMA2 is actually the "best" unbiased estimate of sigma^2 :

                 1
      SIGMA2 = ----- * || ZI - P BETA ||^2_{K^{-1}}
               n - r

   where n is the number of observations, r the length of BETA, P the
   design matrix for the linear part of the model, and || . ||_{K^{-1}}
   the norm associated to the positive definite matrix K^{-1}. It is the
   best estimate with respect to the quadratic risk, among all unbiased
   estimates which are quadratic in the residuals.



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 STK_PARAM_GLS computes a generalised least squares estimate



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stk_param_init


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 STK_PARAM_INIT provides a starting point for stk_param_estim

 CALL: PARAM = stk_param_init (MODEL, XI, YI)

   provides a quick and dirty estimate of the parameters of MODEL based on the
   data (XI, YI), that can be used as a starting point for stk_param_estim. It
   selects the maximizer of the ReML criterion out of a list of possible values
   given data (XI, YI). This syntax is appropriate for noiseless observations
   and for noisy observations with known noise variance (i.e., when the
   'lognoisevariance' field in MODEL is either -Inf or has a finite value).

 CALL: [PARAM, LNV] = stk_param_init (MODEL, XI, YI)

   also returns a value for the 'lognoisevariance' field. In the case of
   noiseless observations or noisy observations with known noise variance, this
   is simply the value that was provided by the user in MODEL.lognoisevariance.
   In the case where MODEL.lognoisevariance is NaN (noisy observation with
   unknown noise variance), LNV is estimated by stk_param_init together with
   PARAM.

 CALL: [PARAM, LNV] = stk_param_init (MODEL, XI, YI, BOX)

   takes into account the (hyper-rectangular) domain on which the model is
   going to be used. It is used in the heuristics that determines the list of
   parameter values mentioned above. BOX should be a 2 x DIM matrix with BOX(1,
   j) and BOX(2, j) being the lower- and upper-bound of the interval on the
   j^th coordinate, with DIM being the dimension of XI, DIM = size(XI,2). If
   provided,  If missing or empty, the BOX argument defaults to [min(XI);
   max(XI)].

 CALL: [PARAM, LNV] = stk_param_init (MODEL, XI, YI, BOX, DO_ESTIM_LNV)

   with DO_ESTIM_LNV = TRUE forces the estimation of the variance of the noise,
   regardless of the value of MODEL.lognoisevariance. If FALSE, it prevents
   estimation of the variance of the noise, which is only possible if the
   'lognoisevariance' field in MODEL is either -Inf or has a finite value.

 See also stk_example_kb02, stk_example_kb03, stk_example_misc03



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 STK_PARAM_INIT provides a starting point for stk_param_estim



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stk_param_init_lnv


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 STK_PARAM_INIT_LNV provides a rough estimate of the variance of the noise

 CALL: LNV = stk_param_init_lnv (MODEL, XI, YI)

   returns a rough estimate of the log of the noise variance computed using
   the given MODEL and data (XI, YI), using the restricted maximum likelihood
   (ReML) method. It selects the maximizer of the ReML criterion out of a
   list of possible values.

 NOTE: assumption on the model

   The model is assumed to be a stationary Gaussian process, with
   model.param(1) corresponding to the log of the Gaussian process variance.
   This assumption is currently fulfilled by all the models shipped with STK.

 See also: stk_param_estim, stk_param_init



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 STK_PARAM_INIT_LNV provides a rough estimate of the variance of the noise



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stk_param_relik


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 STK_PARAM_RELIK computes the restricted likelihood of a model given data

 CALL: [ARL, dARL_dtheta, dARL_dLNV] = stk_param_relik (MODEL, XI, YI)

   computes the opposite of the restricted likelihood (denoted by ARL for
   Anti-Restricted Likelihood) of MODEL given the data (XI, YI). The function
   also returns the gradient dARL_dtheta of ARL with respect to the parameters
   of the covariance function and the derivative dARL_dLNV of ARL with respect
   to the logarithm of the noise variance.

 EXAMPLE: see paramestim/stk_param_estim.m



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 STK_PARAM_RELIK computes the restricted likelihood of a model given data





