Package 'aspline'

Fit B-splines with automatic knot selection.

Description

Fit B-splines with automatic knot selection.

Usage

aspline(
  x,
  y,
  knots = seq(min(x), max(x), length = 42)[-c(1, 42)],
  pen = 10^seq(-3, 3, length = 100),
  degree = 3L,
  family = c("gaussian", "binomial", "poisson"),
  maxiter = 1000,
  epsilon = 1e-05,
  verbose = FALSE,
  tol = 1e-06
)

aridge_solver(
  x,
  y,
  knots = seq(min(x), max(x), length = 42)[-c(1, 42)],
  pen = 10^seq(-3, 3, length = 100),
  degree = 3L,
  family = c("gaussian", "binomial", "poisson"),
  maxiter = 1000,
  epsilon = 1e-05,
  verbose = FALSE,
  tol = 1e-06
)

Arguments

x, y	Input data, numeric vectors of same length
knots	Knots
pen	A vector of positive penalty values. The adaptive spline regression is performed for every value of pen
degree	The degree of the splines. Recommended value is 3, which corresponds to natural splines.
family	A description of the error distribution and link function to be used in the model. The "gaussian", "binomial", and "poisson" families are currently implemented, corresponding to the linear regression, logistic regression, and Poisson regression, respectively.
maxiter	Maximum number of iterations in the main loop.
epsilon	Value of the constant in the adaptive ridge procedure (see Frommlet, F., Nuel, G. (2016) An Adaptive Ridge Procedure for L0 Regularization.)
verbose	Whether to print details at each step of the iterative procedure.
tol	The tolerance chosen to diagnostic convergence of the adaptive ridge procedure.

Value

A list with the following elements:

• sel: list giving for each value of lambda the vector of the knot selection weights (a knot is selected if its weight is equal to 1.)

• knots_sel: list giving for each value of lambda the vector of selected knots.

• model: list giving for each value of lambda the fitted regression model.

• par: parameters of the models for each value of lambda.

• sel_mat: matrix of booleans whose columns indicate whether each knot is selected.

• aic, bic, and ebic: Akaike Information Criterion (AIC), Bayesian Information Criterion (BIC), and Extended BIC (EBIC) scores, for each value of lambda.

• dim: number of selected knots for each value of lambda.

• loglik: log-likelihood of the selected model, for each value of lambda.

Functions

• aridge_solver: Alias for aspline, for backwards compatibility.

---

Create the penalty matrix

Description

Create the penalty matrix

Usage

band_weight(w, diff)

Arguments

w	Vector of weights
diff	Order of the differences to be applied to the parameters. Must be a strictly positive integer

Value

Weighted penalty matrix $D^T diag(w) D$ where D <- diff(diag(length(w) + diff), differences = diff). Only the non-null superdiagonals of the weight matrix are returned, each column corresponding to a diagonal.

---

bandsolve

Description

Main function to solve efficiently and quickly a symmetric bandlinear system. Theses systems are solved much faster than standards system, dropping from complexity O(n³) to O(0.5*nk²), where k is the number of sub diagonal.

Usage

bandsolve(A, b = NULL, inplace = FALSE)

Arguments

A	Band square matrix in rotated form. The rotated form can be obtained with the function as.rotated: it's the visual rotation by 90 degrees of the matrix, where subdiagonal are discarded.
b	right hand side of the equation. Can be either a vector or a matrix. If not supplied, the function return the inverse of A.
inplace	Should results overwrite pre-existing data? Default set to false.

Value

Solution of the linear problem.

Examples

A = diag(4)
A[2,3] = 2
A[3,2] = 2
R = mat2rot(A)
solve(A)
bandsolve(R)

set.seed(100)

n = 1000
D0 = rep(1.25, n)
D1 = rep(-0.5, n-1)
b = rnorm(n)

---

Bladder Cancer aCGH profile data

Description

A dataset of 500 observations corresponding to 500 probes of the aCGH profile of a bladder cancer patient. The original data are provided by Stransky et al. (2006). This dataset consists of probes 1 through 500 of individual 1.

Usage

bladder

Format

A data frame with 500 observations and 2 variables:

x

probe number

y

aCGH profile value

Source

Stransky, N., Vallot, C., Reyal, F., Bernard-Pierrot, I., de Medina, S. G. D., Segraves, R., de Rycke, Y., Elvin, P., Cassidy, A., Spraggon, C., Graham, A., Southgate, J., Asselain, B., Allory, Y., Abbou, C. C., Albertson, D. G., Thiery, J. P., Chopin, D. K., Pinkel, D. and Radvanyi, F. (2006). Regional Copy Number Independent Deregulation of Transcription in Cancer', Nature Genetics 38(12), 1386-1396.

---

Transform a Spline Design Matrix in block compressed form

Description

Transform a Spline Design Matrix in block compressed form

Usage

block_design(X, degree)

Arguments

X	The design matrix, as given by splines2::bSpline.
degree	Degree of the spline regression, as used in function splines2::bSpline.

Value

A matrix B with all non-zero entries of X and a vector of indices alpha representing the positions of the non-zero blocks of X.

---

Yearly number of coal mine disasters in Britain

Description

A data of 112 observations registering the yearly number of coal mine disasters in Britain from 1851 to 1962. The data comes from Diggle et al. (1988) and has been used for spline regression by Eilers et al. (1996).

Usage

coal

Format

A data frame with 112 observations and 2 variables:

year

year

n

number of coal mine disasters

Source

Diggle, P. and Marron, J. S. (1988). 'Equivalence of Smoothing Parameter Selectors in Density and Intensity Estimation', Journal of the American Statistical Association 83(403), 793-800.

References

Eilers, P. H. C. and Marx, B. D. (1996). ‘Flexible Smoothing with B-splines and Penalties’, Statistical Science 11(2), 89-102.

---

Fossil data

Description

A dataset with 106 observations on fossil shells from the SemiPar package (https://CRAN.R-project.org/package=SemiPar).

Usage

fossil

Format

A data frame with 106 observations and 2 variables:

age

The age of fossils, in millions of years

strontium.ratio

Ratio of strontium isotopes

...

Source

Bralower, T.T, Fullagar, P.D., Paull, C.K, Dwyer, G.S. and Leckie, R.M. (1997). Mid-cretaceous strontium-isotope stratigraphy of deep-sea sections. Geological Society of America Bulletin, 109, 1421-1442.

References

Ruppert, D., Wand, M.P. and Carroll, R.J. (2003). Semiparametric Regression, Cambridge University Press.

---

Testing Crash Helmets

Description

A dataset containing the acceleration and time after impact of helmets from a simulated motorcycle accident.

Usage

helmet

Format

A data frame with 132 rows and 2 variables:

x

Time after impact, in milliseconds

y

Head acceleration, in units of $g$

...

Source

Dataset number $338$ of Hand, D. et al. (1993) A Handbook of Small Datasets.

---

Inverse the hessian and multiply it by the score

Description

Inverse the hessian and multiply it by the score

Usage

hessian_solver(par, XX_band, Xy, pen, w, diff)

Arguments

par	The parameter vector
XX_band	The matrix $X^T X$ where X is the design matrix. This argument is given in the form of a band matrix, i.e., successive columns represent superdiagonals.
Xy	The vector of currently estimated points $X^T y$, where $y$ is the y-coordinate of the data.
pen	Positive penalty constant.
w	Vector of weights. Has to be of length
diff	The order of the differences of the parameter. Equals degree + 1 in adaptive spline regression.

Value

The solution of the linear system:

$(X^T X + pen D^T diag(w) D) ^ {-1} X^T y - par$

---

LDL

Description

Fast inplace LDL decomposition of symmetric band matrix of length k.

Arguments

D	Rotated row-wised matrix of dimensions n*k, with first column corresponding to the diagonal, the second to the first super-diagonal and so on.

Value

List with D as solution of our LDL decomposition.

Examples

n=10;
D0=1:10;
D1=exp(-c(1:9));
D=cbind(D0,c(D1,0))
sol=LDL(D)

---

Lidar data

Description

Data from a light detection and ranging (LIDAR) experiment

Usage

lidar

Format

range

distance travelled before the light is reflected back to its source

logratio

logarithm of the ratio of received light from two laser sources

Source

• Sigrist, M. (Ed.) (1994). Air Monitoring by Spectroscopic Techniques (Chemical Analysis Series,vol. 197). New York: Wiley

• The R package https://CRAN.R-project.org/package=SemiPar

References

Ruppert, D., Wand, M.P. and Carroll, R.J. (2003). Semiparametric Regression, Cambridge University Press.

---

Rotate a band matrix to get the rotated row-wised matrix associated.

Description

Rotate a symmetric band matrix to get the rotated matrix associated. Each column of the rotated matrix correspond to a diagonal. The first column is the main diagonal, the second one is the upper-diagonal and so on. Artificial 0 are placed at the end of each column if necessary.

Usage

mat2rot(M)

Arguments

M	Band square matrix or a list of diagonal.

Value

Rotated matrix.

Examples

A = diag(4)
A[2,3] = 2
A[3,2] = 2

## Original Matrix
A
## Rotated version
R = mat2rot(A)
R

rot2mat(mat2rot(A))

---

Montreal Temperature Data

Description

A dataset containing the tempature in Montreal for two years

Usage

montreal

Format

A data frame with 730 rows and 2 variables:

day

The day of the year from January 1, 1961, to December 31, 1962

temp

Temperature in Celsius

...

Source

fda::"MontrealTemp"

---

Nuclear Magnetic Resonance data

Description

A signal of nuclear magnetic resonance.

Usage

nmr

Format

Data farme of 1024 rows and two columns: the index x and the signal y.

Source

• Data from https://web.stanford.edu/~hastie/ElemStatLearn/datasets/nmr1.csv.

• See also The Elements of Statisical Learning (2001, 2nd Ed.), Hastie, T., Friedman, J., and Tibshirani, R.J, p. 176

.

---

Get back a symmetric square matrix based on his rotated row-wised version.

Description

Get back a symmetric square matrix based on his rotated row-wised version. The rotated form of the input is such each column correspond to a diagonal, where the first column is the main diagonal and next ones are the upper/lower-diagonal. To match dimension, last element of these columns are discarded.

Usage

rot2mat(R)

Arguments

R	Rotated matrix.

Value

Band square matrix.

Examples

D0 = 1:5;
D1 = c(0,1,0,0);
D2 = rep(2,3);

A = rot2mat(cbind(D0,c(D1,0),c(D2,0,0)))
A
mat2rot(rot2mat(cbind(D0,c(D1,0),c(D2,0,0))))

---

Titanium heat data

Description

A data set of 49 samples expressing the thermal property of titanium

Usage

titanium

Format

49 observations and two variables:

x

temperature

y

physical property

Source

• de Boor, C., and Rice, J. R. (1986), Least-squares cubic spline approximation. II: variable knots. Report CSD TR 21, Purdue U., Lafayette, IN.

• Dierckx, P. (1993), Curve and Surface Fitting with Splines, Springer.

• Jupp, D. L. B. (1975), Approximation to data by splines with free knots, SIAM Journal on Numerical Analysis, 15: 328-343.

---

Fast computation of weighted design matrix for generalized linear model

Description

Fast computation of weighted design matrix for generalized linear model

Usage

weight_design_band(w, alpha, B)

Arguments

w	Vector of weights.
alpha	Vector of indexes representing the start of blocks of the design matrix, as given by block_design.
B	Design matrix in compressed block format, as given by block_design.

Value

Weighted design matrix $X^T diag(w) X$ where X is the design matrix and W = diag(w) is a diagonal matrix of weights.

---

Fit B-Splines with weighted penalization over differences of parameters

Description

Fit B-Splines with weighted penalization over differences of parameters

Usage

wridge_solver(
  XX_band,
  Xy,
  degree,
  pen,
  w = rep(1, nrow(XX_band) - degree - 1),
  old_par = rep(1, nrow(XX_band)),
  maxiter = 1000,
  tol = 1e-08
)

Arguments

XX_band	The matrix $X^T X$ where X is the design matrix. This argument is given in the form of a band matrix, i.e., successive columns represent superdiagonals.
Xy	The vector of currently estimated points $X^T y$, where y is the y-coordinate of the data.
degree	The degree of the B-splines.
pen	Positive penalty constant.
w	Vector of weights. The case $\mathbf w = \mathbf 1$ corresponds to fitting P-splines with difference #' order degree + 1 (see Eilers, P., Marx, B. (1996) Flexible smoothing with B-splines and penalties.)
old_par	Initial parameter to serve as starting point of the iterating process.
maxiter	Maximum number of Newton-Raphson iterations to be computed.
tol	The tolerance chosen to diagnostic convergence of the adaptive ridge procedure.

Value

The estimated parameter of the spline regression.

---