shape.constrained.smooth.terms: Shape preserving smooth terms in SCAM

Description

As in mgcv(gam), shape preserving smooth terms are specified in a scam formula using s terms. All the shape constrained smooth terms are constructed using the B-splines basis proposed by Eilers and Marx (1996) with a discrete penalty on the basis coefficients. The univariate single penalty built in shape constrained smooth classes are summarized as follows

Monotone increasing P-splines

{ bs="mpi". To achieve monotone increasing smooths these reparametrize the coefficients so that they form an increasing sequence. For details see smooth.construct.mpi.smooth.spec.} Monotone decreasing P-splines{ bs="mpd". To achieve monotone decreasing smooths these reparametrize the coefficients so that they form a decreasing sequence. A first order difference penalty applied to the basis coefficients starting with the second is used for the monotone increasing and decreasing cases.} Convex P-splines{ bs="cx". These reparametrize the coefficients so that the second order differences of the basis coefficients are greater than zero. For details see smooth.construct.cx.smooth.spec.} Concave P-splines{ bs="cv". These reparametrize the coefficients so that the second order differences of the basis coefficients are less than zero. For details see smooth.construct.cv.smooth.spec.} Monotone increasing and convex P-splines{ bs="micx". These reparametrize the coefficients so that the first and the second order differences of the basis coefficients are greater than zero. For details see smooth.construct.micx.smooth.spec.} Monotone increasing and concave P-splines{ bs="micv". These reparametrize the coefficients so that the first order differences of the basis coefficients are greater than zero while the second order difference are less than zero.} Monotone decreasing and convex P-splines{ bs="mdcx". These reparametrize the coefficients so that the first order differences of the basis coefficients are less than zero while the second order difference are greater. For details see smooth.construct.mdcx.smooth.spec.} Monotone decreasing and concave P-splines{ bs="mdcv". These reparametrize the coefficients so that the first and the second order differences of the basis coefficients are less than zero.} For all four types of the mixed constrained smoothing a first order difference penalty applied to the basis coefficients starting with the third one is used.

Arguments

itemize

Double monotone increasing P-splines

code

bs="tesmd2"

item

Double monotone decreasing P-splines
Single monotone increasing P-splines along the first covariate direction
Single monotone increasing P-splines along the second covariate direction
Single monotone decreasing P-splines along the first covariate direction
Single monotone decreasing P-splines along the second covariate direction

References

Pya, N. and Wood, S.N. (2015) Shape constrained additive models. Statistics and Computing, 25(3), 543-559 Pya, N. (2010) Additive models with shape constraints. PhD thesis. University of Bath. Department of Mathematical Sciences Eilers, P.H.C. and B.D. Marx (1996) Flexible Smoothing with B-splines and Penalties. Statistical Science, 11(2):89-121 Wood S.N. (2006a) Generalized Additive Models: An Introduction with R. Chapman and Hall/CRC Press. Wood, S.N. (2006b) Low rank scale invariant tensor product smooths for generalized additive mixed models. Biometrics 62(4):1025-1036

Examples

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