quad_spline_kn {npbr}

AIC and BIC criteria for choosing the optimal number of inter-knot segments in quadratic spline fits

Description

Computes the optimal number k_n of inter-knot segments in the quadratic spline fits proposed by Daouia, Noh and Park (2016).

Usage

quad_spline_kn(xtab, ytab, method, krange = 1:20, type = "AIC", 
 control = list("tm_limit" = 700))

Arguments

xtab

a numeric vector containing the observed inputs x_1,\ldots,x_n.

ytab

a numeric vector of the same length as xtab containing the observed outputs y_1,\ldots,y_n.

method

a character equal to "u" (unconstrained estimator), "m" (under the monotonicity constraint) or "mc" (under simultaneous monotonicity and concavity constraints).

krange

a vector of integers specifying the range in which the optimal number of inter-knot segments is to be selected.

type

a character equal to "AIC" or "BIC".

control

a list of parameters to the GLPK solver. See *Details* of help(Rglpk_solve_LP).

Details

For the implementation of the unconstrained quadratic spline smoother \tilde\varphi_n (see quad_spline_est), based on the knot mesh \{t_j = x_{[j n/k_n]}: j=1,\ldots,k_n-1\}, the user has to employ the option method="u". Since the number k_n determines the complexity of the spline approximation, its choice may be viewed as model selection via the minimization of the following Akaike (option type="AIC") or Bayesian (option type="BIC") information criteria:

A\tilde{I}C(k) = \log \left( \sum_{i=1}^{n} (\tilde \varphi_n(x_i)- y_i) \right) + (k+2)/n,

B\tilde{I}C(k) = \log \left( \sum_{i=1}^{n} (\tilde \varphi_n(x_i) - y_i) \right) + \log n \cdot (k+2)/2n.

For the implementation of the monotone (option method="m") quadratic spline smoother \hat\varphi_n (see quad_spline_est), the authors first suggest using the set of knots \{ t_j = {\mathcal{X}_{[j \mathcal{N}/k_n]}},~j=1,\ldots,k_n-1 \} among the FDH points (\mathcal{X}_{\ell},\mathcal{Y}_{\ell}), \ell=1,\ldots,\mathcal{N} (function quad_spline_est). Then, they propose to choose k_n by minimizing the following AIC (option type="AIC") or BIC (option type="BIC") information criteria:

A\hat{I}C(k) = \log \left( \sum_{i=1}^{n} (\hat \varphi_n(x_i)- y_i) \right) + (k+2)/n,

B\hat{I}C(k) = \log \left( \sum_{i=1}^{n} (\hat \varphi_n(x_i) - y_i) \right) + \log n \cdot (k+2)/2n.

A small number of knots is typically needed as elucidated by the asymptotic theory.

For the implementation of the monotone and concave (option method="mc") spline estimator \hat\varphi^{\star}_n, just apply the same scheme as above by replacing the FDH points (\mathcal{X}_{\ell},\mathcal{Y}_{\ell}) with the DEA points (\mathcal{X}^*_{\ell},\mathcal{Y}^*_{\ell}) (see dea_est).

Value

Returns an integer.

Author(s)

Hohsuk Noh.

References

Akaike, H. (1973). Information theory and an extension of the maximum likelihood principle, in Second International Symposium of Information Theory, eds. B. N. Petrov and F. Csaki, Budapest: Akademia Kiado, 267–281.

Daouia, A., Noh, H. and Park, B.U. (2016). Data Envelope fitting with constrained polynomial splines. Journal of the Royal Statistical Society: Series B, 78(1), 3-30. doi:10.1111/rssb.12098.

Schwartz, G. (1978). Estimating the dimension of a model, Annals of Statistics, 6, 461–464.

See Also

quad_spline_est

Examples

data("green")
## Not run: 
# BIC criteria for choosing the optimal number of 
# inter-knot segments in:   
# a. Unconstrained quadratic spline fits
(kn.bic.green.u <- quad_spline_kn(log(green$COST), 
 log(green$OUTPUT), method = "u", type = "BIC"))
# b. Monotone quadratic spline smoother
(kn.bic.green.m <- quad_spline_kn(log(green$COST), 
 log(green$OUTPUT), method = "m", type = "BIC"))  
# c. Monotone and concave quadratic spline smoother
(kn.bic.green.mc<-quad_spline_kn(log(green$COST), 
 log(green$OUTPUT), method = "mc", type = "BIC"))

## End(Not run)