Nonparametric Regression and Bandwidth Selection for Spatial Models

Description

Nonparametric smoothing techniques for data on a lattice and functional time series. Smoothing is done via kernel regression or local polynomial regression, a bandwidth selection procedure based on an iterative plug-in algorithm is implemented. This package allows for modeling a dependency structure of the error terms of the nonparametric regression model. Methods used in this paper are described in Feng/Schaefer (2021) <https://ideas.repec.org/p/pdn/ciepap/144.html>, Schaefer/Feng (2021) <https://ideas.repec.org/p/pdn/ciepap/143.html>.

Package Content

Index of help topics:

DCSmooth-package        Nonparametric Regression and Bandwidth
                        Selection for Spatial Models
dcs                     Nonparametric Double Conditional Smoothing for
                        2D Surfaces
kernel.assign           Assign a Kernel Function
kernel.list             Print a list of available kernels in the
                        DCSmooth package
plot.dcs                Contour Plot for the Double Conditional
                        Smoothing
print.dcs               Summarize Results from Double Conditional
                        Smoothing
print.dcs_options       Print and Summarize Options for Double
                        Conditional Smoothing
print.summary_dcs       Print the Summary of a DCS estimation
print.summary_sarma     Print the Summary of a "sarma"/"sfarima" object
residuals.dcs           Residuals of "dcs"-object
returns.alv             Returns of Allianz SE
sarma.est               Estimation of an SARMA-process
sarma.sim               Simulation of a SARMA(p, q)-process
set.options             Set Options for the DCS procedure
sfarima.est             Estimation of a SFARIMA-process
sfarima.sim             Simulation of a SFARIMA(p, q, d)-process
summary.dcs             Summarizing Results from Double Conditional
                        Smoothing
summary.dcs_options     Print and Summarize Options for Double
                        Conditional Smoothing
summary.sarma           Summarizing SARMA/SFARIMA Estimation or
                        Simulation
surface.dcs             3D Surface Plot of "dcs"-object or numeric
                        matrix
temp.nunn               Temperatures from Nunn, CO
temp.yuma               Temperatures from Yuma, AZ
volumes.alv             Volumes of Allianz SE
wind.nunn               Wind Speed from Nunn, CO
wind.yuma               Wind Speed from Yuma, AZ
y.norm1                 Single Gaussian Peak
y.norm2                 Double Gaussian Peak
y.norm3                 Double Gaussian Ridges

Maintainer

Bastian Schaefer <bastian.schaefer@uni-paderborn.de>

Author(s)

Bastian Schaefer [aut, cre], Sebastian Letmathe [ctb], Yuanhua Feng [ths]

Nonparametric Double Conditional Smoothing for 2D Surfaces

Description

dcs provides a double conditional nonparametric smoothing of the expectation surface of a functional time series or a random field on a lattice. Bandwidth selection is done via an iterative plug-in method.

Usage

dcs(Y, dcs_options = set.options(), h = "auto", parallel = FALSE, ...)

Arguments

Value

dcs returns an object of class "dcs", including

Details

See the vignette for a more detailed description of the function.

References

Schäfer, B. and Feng, Y. (2021). Fast Computation and Bandwidth Selection Algorithms for Smoothing Functional Time Series. Working Papers CIE 143, Paderborn University.

See Also

set.options

Examples

# See vignette("DCSmooth") for examples and explanation

y <- y.norm1 + matrix(rnorm(101^2), nrow = 101, ncol = 101)
dcs(y)

Assign a Kernel Function

Description

Assign a Kernel Function

Usage

kernel.assign(kernel_id)

Arguments

Value

kernel.assign returns an object of class "function". This function takes two arguments, a numeric vector in the first argument and a single number in the second. The function itself will return a matrix with one column and the same number of rows as the input vector.

Details

kernel.assign sets a pointer to a specified kernel function available in the DCSmooth package. The kernels are boundary kernels of the form K(u,q), where u \in [-1, q] and q \in [0, 1] q = [0, 1]. Kernels are of the Müller-Wang type ("MW"), Müller type ("M") or truncated kernels ("TR").

References

Müller, H.-G. and Wang, J.-L. (1994). Hazard rate estimation under random censoring with varying kernels and bandwidths. Biometrics, 50:61-76.

Müller, H.-G. (1991). Smooth optimum kernel estimators near endpoints. Biometrika, 78:521-530.

Feng, Y. and Schäfer B. (2021). Boundary Modification in Local Regression. Working Papers CIE 144, Paderborn University.

See Also

kernel.list

Examples

 # See vignette("DCSmooth") for further examples and explanation

u <- seq(from = -1, to = 0.5, length.out = 151)
kern_MW220 <- kernel.assign("MW_220")
k <- kern_MW220(u, 0.5)
plot(u, k, type = "l") 

Print a list of available kernels in the DCSmooth package

Description

Print a list of available kernels in the DCSmooth package

Usage

kernel.list(print = TRUE)

Arguments

Value

If print = FALSE, a list is returned containing the kernel identifiers

Details

kernel.list is used to get a list of available kernels in the DCSmooth package.

kernel.list prints a list of identifiers kernel_id of available kernels in the DCSmooth package. The available kernel types are "T": truncated, "MW": Müller-Wang boundary correction, "M": Müller boundary correction.

References

Müller, H.-G. and Wang, J.-L. (1994). Hazard rate estimation under random censoring with varying kernels and bandwidths. Biometrics, 50:61-76.

Müller, H.-G. (1991). Smooth optimum kernel estimators near endpoints. Biometrika, 78:521-530.

Feng, Y. and Schäfer B. (2021). Boundary Modification in Local Regression. Working Papers CIE 144, Paderborn University.

See Also

kernel.assign

Examples

 # See vignette("DCSmooth") for further examples and explanation
 
 kernel.list()

Contour Plot for the Double Conditional Smoothing

Description

plot method for class "dcs"

Usage

## S3 method for class 'dcs'
plot(x, ...)

Arguments

Value

No return value.

Details

plot.dcs provides a contour plot of either the original data (1), smoothed surface (2) or residuals (3).

See Also

surface.dcs to plot the surface.

Examples

## Contour plot of smoothed surface
y <- y.norm1 + matrix(rnorm(101^2), nrow = 101, ncol = 101)
dcs_object <- dcs(y)
plot(dcs_object, plot_choice = 2)

Summarize Results from Double Conditional Smoothing

Description

print method for class "dcs"

Usage

## S3 method for class 'dcs'
print(x, ...)

Arguments

Value

No return value.

Details

print.dcs prints a short summary of an object of class dcs, only including bandwidths and the estimated variance coefficient (only if automatic bandwidth selection is used).

See Also

plot.dcs, print.dcs_options

Examples

y <- y.norm1 + matrix(rnorm(101^2), nrow = 101, ncol = 101)
dcs_object <- dcs(y)
print(dcs_object)
dcs_object

Print and Summarize Options for Double Conditional Smoothing

Description

print method for class "dcs_options"

Usage

## S3 method for class 'dcs_options'
print(x, ...)

Arguments

Value

No return value.

Details

print.dcs_options prints the main options and summary.dcs_options prints main and advanced (IPI) options used for the dcs function. Arguments should be an object of class "dcs_options".

See Also

print.dcs, summary.dcs_options

Examples

## Default options
myOpt <- set.options()
print(myOpt)
summary(myOpt)

## Use Kernel regression
myOpt <- set.options(type = "KR")
print(myOpt)
summary(myOpt)

Print the Summary of a DCS estimation

Description

print method for class "summary_dcs"

Usage

## S3 method for class 'summary_dcs'
print(x, ...)

Arguments

Value

No return value.

See Also

summary.dcs

Print the Summary of a "sarma"/"sfarima" object

Description

print methods for class "summary_sarma"/ "summary_sfarima"

Usage

## S3 method for class 'summary_sarma'
print(x, ...)

## S3 method for class 'summary_sfarima'
print(x, ...)

Arguments

Value

No return value.

See Also

summary.sarma summary.sfarima

Residuals of "dcs"-object

Description

Returns the residuals of an object of class "dcs".

Usage

## S3 method for class 'dcs'
residuals(x, ...)

Arguments

Value

Returns the n_x \times n_t-matrix of residuals.

See Also

dcs

Examples

y = y.norm1 + matrix(rnorm(101^2), nrow = 101, ncol = 101)
dcs_object = dcs(y)
residuals(dcs_object)

Returns of Allianz SE

Description

The (log-) returns of the shares of the German insurance company Allianz SE from 2007-01-02 to 2010-12-30 aggregated to 5-minute observations. The data is adjusted to matrix form for direct use with the DCSmooth-functions.

Usage

returns.alv

Format

A numeric matrix with 1016 rows representing the days and 101 columns representing the intraday time points.

Estimation of an SARMA-process

Description

Parametric Estimation of an SARMA(p, q)-process on a lattice.

Usage

sarma.est(Y, method = "HR", model_order = list(ar = c(1, 1), ma = c(1, 1)))

qarma.est(Y, model_order = list(ar = c(1, 1), ma = c(1, 1)))

Arguments

Value

The function returns an object of class "sarma" including

Details

The MA- and AR-parameters of a top-left quadrant ARMA process are estimated by the specified method. The lag-orders of the SARMA(p, q) are given by p = (p_1, p_2), q = (q_1, q_2), where p_1, q_1 are the lags over the rows and p_2, q_2 are the lags over the columns. The estimation process is based on the model

\phi(B_{1}B_{2})X_{i,j} = \theta(B_{1}B_{2})u_{i,j}

.

See Also

sarma.sim, sfarima.est

Examples

# See vignette("DCSmooth") for examples and explanation

## simulation of SARMA process
ma <- matrix(c(1, 0.2, 0.4, 0.1), nrow = 2, ncol = 2)
ar <- matrix(c(1, 0.5, -0.1, 0.1), nrow = 2, ncol = 2)
sigma <- 0.5
sarma_model <- list(ar = ar, ma = ma, sigma = sigma)
sarma_simulated <- sarma.sim(100, 100, model = sarma_model)
sarma_simulated$model

## estimation of SARMA process
sarma.est(sarma_simulated$Y)$model
sarma.est(sarma_simulated$Y, 
           model_order = list(ar = c(1, 1), ma = c(1, 1)))$model

Simulation of a SARMA(p, q)-process

Description

sarma.sim simulates a specified SARMA-model on a lattice with normally distributed innovations.

Usage

sarma.sim(n_x, n_t, model)

qarma.sim(n_x, n_t, model)

Arguments

Value

The function returns an object of class "sarma", consisting of

Details

Simulation of a top-left dependent spatial ARMA process (SARMA). This function returns an object of class "sarma". The simulated innovations are created from a normal distribution with specified variance \sigma^2.

see the vignette for further details.

See Also

sarma.est, sfarima.est

Examples

# See vignette("DCSmooth") for examples and explanation
 
ma <- matrix(c(1, 0.2, 0.4, 0.1), nrow = 2, ncol = 2)
ar <- matrix(c(1, 0.5, -0.1, 0.1), nrow = 2, ncol = 2)
sigma <- 0.5
sarma_model <- list(ar = ar, ma = ma, sigma = sigma)

sarma_sim <- sarma.sim(100, 100, model = sarma_model)
summary(sarma_sim)

Set Options for the DCS procedure

Description

Set Options for the DCS procedure

Usage

set.options(
  type = "LP",
  kerns = c("MW_220", "MW_220"),
  drv = c(0, 0),
  var_model = "iid",
  ...
)

Arguments

Value

An object of class "dcs_options".

Details

This function is used to set the options for bandwidth selection in the dcs function. Detailed information can be found in the vignette.

See Also

dcs

Examples

# See vignette("DCSmooth") for examples and explanation

set.options()

myOpt <- set.options(type = "KR", var_model = "iid")
y <- y.norm1 + matrix(rnorm(101^2), nrow = 101, ncol = 101)
dcs(y, dcs_options = myOpt)

Estimation of a SFARIMA-process

Description

Parametric Estimation of a SFARIMA(p, q, d)-process on a lattice.

Usage

sfarima.est(Y, model_order = list(ar = c(1, 1), ma = c(1, 1)))

Arguments

Value

The function returns an object of class "sfarima" including

Details

The MA- and AR-parameters as well as the long-memory parameters

d

of a SFARIMA process are estimated by minimization of the residual sum of squares RSS. Lag-orders of SFARIMA(p, q, d) are given by p = (p_1, p_2), q = (q_1, q_2), where p_1, q_1 are the lags over the rows and p_2, q_2 are the lags over the columns. The estimated process is based on the (separable) model

\varepsilon_{ij} = \Psi_1(B) \Psi_2(B) \eta_{ij}

, where

\Psi_i = (1 - B_i)^{-d_i}\phi^{-1}_i(B_i)\psi_i(B_i), i = 1,2

.

See Also

sarma.est, sfarima.sim

Examples

# See vignette("DCSmooth") for examples and explanation

## simulation of SFARIMA process
ma <- matrix(c(1, 0.2, 0.4, 0.1), nrow = 2, ncol = 2)
ar <- matrix(c(1, 0.5, -0.1, 0.1), nrow = 2, ncol = 2)
d <- c(0.1, 0.1)
sigma <- 0.5
sfarima_model <- list(ar = ar, ma = ma, d = d, sigma = sigma)
sfarima_sim <- sfarima.sim(50, 50, model = sfarima_model)

## estimation of SFARIMA process
sfarima.est(sfarima_sim$Y)$model
sfarima.est(sfarima_sim$Y, 
           model_order = list(ar = c(1, 1), ma = c(0, 0)))$model

Simulation of a SFARIMA(p, q, d)-process

Description

sfarima.sim simulates a specified SFARIMA-model on a lattice with normally distributed innovations.

Usage

sfarima.sim(n_x, n_t, model)

Arguments

Value

The function returns an object of class "sfarima", consisting of

Details

Simulation of a separable spatial fractionally ARIMA process (SFARIMA). This function returns an object of class "sfarima". The simulated innovations are created from a normal distribution with specified variance \sigma^2.

see the vignette for further details.

See Also

qarma.est

Examples

# See vignette("DCSmooth") for examples and explanation
 
ma <- matrix(c(1, 0.2, 0.4, 0.1), nrow = 2, ncol = 2)
ar <- matrix(c(1, 0.5, -0.1, 0.1), nrow = 2, ncol = 2)
d <- c(0.1, 0.1)
sigma <- 0.5
sfarima_model <- list(ar = ar, ma = ma, d = d, sigma = sigma)

sfarima_sim <- sfarima.sim(100, 100, model = sfarima_model)
surface.dcs(sfarima_sim$Y)

Summarizing Results from Double Conditional Smoothing

Description

summary method for class "dcs"

Usage

## S3 method for class 'dcs'
summary(object, ...)

Arguments

Value

The function summary.dcs returns an object of class summary_dcs including

Details

summary.dcs strips an object of class "dcs" from all large matrices (Y, X, T, M, R), allowing for easier handling of meta-statistics of the bandwidth selection procedure.

print.summary_dcs returns a list of summary statistics from the dcs procedure. The output depends on the use of the dcs- function. If automatic bandwidth selection is chosen, summary.dcs prints detailed statistics of the type of regression, the estimated bandwidths h_x, h_t, the variance coefficient c_f and performance statistics such as the number of iterations of the IPI-algorithm and the time used for bandwidth selection.

The method used for estimation of the variance coefficient is printed and the results of an SARMA/SFARIMA-estimation, if available.

If bandwidths are supplied to dcs, summary.dcs only prints the given bandwidths.

Examples

y <- y.norm1 + matrix(rnorm(101^2), nrow = 101, ncol = 101)
dcs_object <- dcs(y)
summary(dcs_object)

Print and Summarize Options for Double Conditional Smoothing

Description

summary method for class "dcs_options"

Usage

## S3 method for class 'dcs_options'
summary(object, ...)

Arguments

Value

No return value.

Details

print.dcs_options prints the main options and summary.dcs_options prints main and advanced (IPI) options used for the dcs function. Arguments should be an object of class "dcs_options".

See Also

print.dcs, print.dcs_options

Examples

## Default options
myOpt <- set.options()
print(myOpt)
summary(myOpt)

## Use Kernel regression
myOpt <- set.options(type = "KR")
print(myOpt)
summary(myOpt)

Summarizing SARMA/SFARIMA Estimation or Simulation

Description

summary method for class "sarma" or "sfarima"

Usage

## S3 method for class 'sarma'
summary(object, ...)

## S3 method for class 'sfarima'
summary(object, ...)

Arguments

Value

The function summary.sarma/summary.sfarima returns an object of class summary_sarma including

Details

summary.sarma/summary.sfarima strips an object of class "sarma"/"sfarima" from all large matrices (Y, innov), allowing for easier handling of meta-statistics of the bandwidth selection procedure.

print.summary_sarma/print.summary_sarma returns a list of summary statistics from the estimation or simulation procedure.

See Also

sarma.est, sfarima.est, sarma.sim, sfarima.sim

Examples

# SARMA Simulation and Estimation
ma = matrix(c(1, 0.2, 0.4, 0.1), nrow = 2, ncol = 2)
ar = matrix(c(1, 0.5, -0.1, 0.1), nrow = 2, ncol = 2)
sigma = 0.5
sarma_model = list(ar = ar, ma = ma, sigma = sigma)
sarma_sim = sarma.sim(100, 100, model = sarma_model)
summary(sarma_sim)
sarma_est = sarma.est(sarma_sim$Y)
summary(sarma_est)

# SFARIMA Simulation and Estimation
ma = matrix(c(1, 0.2, 0.4, 0.1), nrow = 2, ncol = 2)
ar = matrix(c(1, 0.5, -0.1, 0.1), nrow = 2, ncol = 2)
d = c(0.1, 0.1)
sigma = 0.5
sfarima_model = list(ar = ar, ma = ma, d = d, sigma = sigma)
sfarima_sim = sfarima.sim(100, 100, model = sfarima_model)
summary(sfarima_sim)
sfarima_est = sfarima.est(sfarima_sim$Y)
summary(sfarima_est)

3D Surface Plot of "dcs"-object or numeric matrix

Description

3D Surface Plot of "dcs"-object or numeric matrix

Usage

surface.dcs(Y, trim = c(0, 0), plot_choice = "choice", ...)

Arguments

Value

dcs.3d returns an object of class "plotly" and "htmlwidget".

Details

surface.dcs uses the plotly device to plot the 3D surface of the given "dcs"-object or matrix. If a "dcs"-object is passed to the function, it can be chosen between plots of the original data (1), smoothed surface (2) and residuals (3).

See Also

plot.dcs

Examples

# See vignette("DCSmooth") for examples and explanation

smth <-  dcs(y.norm1 + rnorm(101^2))
surface.dcs(smth, trim = c(0.05, 0.05), plot_choice = 2)

Temperatures from Nunn, CO

Description

This dataset contains the 5-minute observations of the 2020 temperature in Nunn, CO. The data is from the U.S. Climate Reference Network database at www.ncdc.noaa.gov. (see Diamond et al., 2013). The observations were adjusted matrix form for direct use with the DCSmooth-functions.

Usage

temp.nunn

Format

A numeric matrix with 366 rows and 288 columns containing the temperatures in Celsius.

Temperatures from Yuma, AZ

Description

This dataset contains the 5-minute observations of the 2020 temperature in Yuma, AZ. The data is from the U.S. Climate Reference Network database at www.ncdc.noaa.gov. (see Diamond et al., 2013). The observations were adjusted matrix form for direct use with the DCSmooth-functions.

Usage

temp.yuma

Format

A numeric matrix with 366 rows and 288 columns containing the temperatures in Celsius.

Volumes of Allianz SE

Description

The trading volumes of the shares of the German insurance company Allianz SE from 2007-01-02 to 2010-09-30 aggregated to 5-minute observations. The data is adjusted to matrix form for direct use with the DCSmooth-functions.

Usage

volumes.alv

Format

A numeric matrix with 1016 rows representing the days and 102 columns representing the intraday time points.

Wind Speed from Nunn, CO

Description

This dataset contains the 5-minute observations of the 2020 wind speed in Nunn, CO. The data is from the U.S. Climate Reference Network database at www.ncdc.noaa.gov. (see Diamond et al., 2013). The observations were adjusted matrix form for direct use with the DCSmooth-functions.

Usage

wind.nunn

Format

A numeric matrix with 366 rows and 288 columns containing the wind speed in m/s.

Wind Speed from Yuma, AZ

Description

This dataset contains the 5-minute observations of the 2020 wind speed in Yuma, AZ. The data is from the U.S. Climate Reference Network database at www.ncdc.noaa.gov. (see Diamond et al., 2013). The observations were adjusted matrix form for direct use with the DCSmooth-functions.

Usage

wind.yuma

Format

A numeric matrix with 366 rows and 288 columns containing the wind speeds in m/s.

Single Gaussian Peak

Description

Example data for using the DCSmooth functions. Data resembles a single gaussian peak on the interval [0,1] \times [0,1] with maximum at (0.5, 0.5) and variance matrix 0.1 \cdot {\bf I}, where {\bf I} represents the 2 \times 2 identity matrix.

Usage

y.norm1

Format

A numeric matrix with 101 rows and 101 columns.

Double Gaussian Peak

Description

Example data for using the DCSmooth functions. Data resembles two gaussian peaks on the interval [0,1] \times [0,1] with maxima at (0.5, 0.3) with variance matrix 0.1 \cdot {\bf I} and at (0.2, 0.8) with variance matrix 0.05 \cdot {\bf I}, where {\bf I} represents the 2 \times 2 identity matrix.

Usage

y.norm2

Format

A numeric matrix with 101 rows and 101 columns.

Double Gaussian Ridges

Description

Example data for using the DCSmooth functions. Data resembles two gaussian ridges on the interval [0,1] \times [0,1] with maxima at (0.25, 0.75) with variance matrix (0.01, -0.1) \cdot {\bf I} and at (0.75, 0.5) with variance matrix (0.01, -0.1) \cdot {\bf I}, where {\bf I} represents the 2 \times 2 identity matrix.

Usage

y.norm3

Format

A numeric matrix with 101 rows and 101 columns.