| Title: | Dimension Reduction Techniques for Conditional Quantiles |
|---|---|
| Description: | An implementation of dimension reduction techniques for conditional quantiles. Nonparametric estimation of conditional quantiles is also available. |
| Authors: | Eliana Christou [aut, cre] |
| Maintainer: | Eliana Christou <[email protected]> |
| License: | GPL-2 | GPL-3 |
| Version: | 1.2.2 |
| Built: | 2025-03-10 05:35:54 UTC |
| Source: | https://github.com/elianachristou/quantdr |
cqs estimates the directions of the central quantile subspace.
cqs(x, y, tau = 0.5, dtau = NULL)cqs(x, y, tau = 0.5, dtau = NULL)
x |
A design matrix (n x p). The rows represent observations and the columns represent predictor variables. |
y |
A vector of the response variable. |
tau |
A quantile level, a number strictly between 0 and 1. |
dtau |
An optional dimension of the central quantile subspace. If
specified, it should be an integer between 1 and p, the number of columns
of the design matrix |
The function computes the directions that span the th central
quantile subspace, i.e., the directions that define linear combinations of
the predictor x that contain all the information available on the
conditional quantile function.
The function starts by estimating the initial vector, which is defined as the
least-squares estimator from regressing the conditional quantile on the
predictor variable x. Then, if the dimension of the central quantile
subspace is one, the algorithm stops and reports that vector as the basis of
the central quantile subspace. Otherwise, the algorithm continues by creating
more vectors and applying an eigenvalue decomposition to extract linearly
independent vectors.
cqs computes the directions of the central quantile subspace
and returns:
qvectors: The estimated directions of the
th central quantile subspace.
qvalues: The eigenvalues resulting from the eigenvalue decomposition
of the matrix with column vectors that span the central quantile subspace.
If dtau is one, the qvalues output is not produced.
dtau: Suggested dimension of the central quantile subspace. If
dtau is specified by the user then the algorithm outputs the
user-defined value. If dtau is not specified by the user then the
algorithm outputs a suggested dimension using the modified-BIC type
criterion of Zhu et al. (2010). Note that this is one suggested method to
estimate the structural dimension and is not necessarily a perfect one. The
user has the option to use the eigenvalues qvalues on other
criteria, like cross-validation, and determine the estimated dimension of
the subspace.
Zhu, L.-P., Zhu, L.-X., Feng, Z.-H. (2010) Dimension reduction in regression through cumulative slicing estimation. Journal of the American Statistical Association, 105, 1455-1466.
# estimate the directions of a single-index model set.seed(1234) n <- 100 p <- 10 x <- matrix(rnorm(n * p), n, p) error <- rnorm(n) y <- 3 * x[, 1] + x[, 2] + error tau <- 0.5 out <- cqs(x, y, tau, dtau = 1) out # without specifying dtau out <- cqs(x, y, tau) out out$qvectors[, 1:out$dtau]# estimate the directions of a single-index model set.seed(1234) n <- 100 p <- 10 x <- matrix(rnorm(n * p), n, p) error <- rnorm(n) y <- 3 * x[, 1] + x[, 2] + error tau <- 0.5 out <- cqs(x, y, tau, dtau = 1) out # without specifying dtau out <- cqs(x, y, tau) out out$qvectors[, 1:out$dtau]
llqr estimates the th conditional quantile of y given
x based on a local linear fit. The estimation is performed at each of
the design points or, if specified, at a single observation point x0.
llqr(x, y, tau = 0.5, h = NULL, method = "rule", x0 = NULL)llqr(x, y, tau = 0.5, h = NULL, method = "rule", x0 = NULL)
x |
A design matrix (n x p). The rows represent observations and the columns represent predictor variables. |
y |
A vector of the response variable. |
tau |
A quantile level, a number strictly between 0 and 1. |
h |
A univariate bandwidth. If not specified, the bandwidth is estimated
using either " |
method |
A character string specifying the method to select the
bandwidth, if it is missing. Use " |
x0 |
A single observation for which to perform the estimation. It needs
to be a scalar (for a univariate predictor) or a vector (for a
multivariate predictor). If |
The function computes the local linear quantile regression fit for a specified
quantile level at the design points of the matrix x or at a
pre-specified point x0. The estimation is based on a standard normal
kernel and a univariate bandwidth. The bandwidth, if not specified by the
user, is defined using either the rule-of-thumb given by Yu and Jones (1994)
or the cross-validation criterion.
The estimation applies to univariate and multivariate predictor variables. For
the latter, the local linear fit uses the multivariate standard normal kernel.
Note that if the estimation is performed at a pre-specified point x0,
then x0 should be a scalar (for univariate predictor) or a vector (for
multivariate predictor).
llqr computes the local linear th conditional
quantile function of y given x and returns:
ll_est: The estimated function value at the design points x
or, if specified, at the point x0.
h: The bandwidth for the local linear quantile regression fit. If
not specified by the user, h is estimated using either the
rule-of-thumb given by Yu and Jones (1994) or the cross-validation
criterion.
The user needs to be careful about the bandwidth selection. When the dimension of the predictor variable is large compared to the sample size, local linear fitting meets the 'curse of dimensionality' problem. In situations like that, the bandwidth selected by the rule-of-thumb or the cross- validation criterion might be small and lead to a sparse neighborhood. This will cause the function to fail. For these cases, we advice the user to specify a bandwidth in the function. See the last example below.
Yu, K., and Jones, M.C. (1998), Local linear quantile regression. Journal of the American Statistical Association, 93, 228-237.
# Example 1 # estimate the function at a specific quantile level for simulated data set.seed(1234) n <- 100 x <- rnorm(n) error <- rnorm(n) y <- (x + 1)^3 + 0.1 * (x - 2)^3 + error tau <- 0.5 plot(x, y, main = tau) points(x, llqr(x, y, tau = tau)$ll_est, col = 'red', pch = 16) # Example 2 # estimate the function at a point x0 set.seed(1234) n <- 100 x <- rnorm(n) error <- rnorm(n) y <- (x + 1)^3 + 0.1 * (x - 2)^3 + error tau <- 0.5 x0 <- 1 llqr(x, y, tau = tau, x0 = x0) # Example 3 # estimate the function for different quantile levels data(mcycle, package = "MASS") attach(mcycle) plot(times, accel, xlab = "milliseconds", ylab = "acceleration") taus <- c(0.1, 0.25, 0.5, 0.75, 0.9) for(i in 1:length(taus)) { fit <- llqr(times, accel, tau = taus[i])$ll_est lines(times, fit, lty = i) } legend(45, -50, c("tau=0.1","tau=0.25","tau=0.5","tau=0.75", "tau=0.9"), lty=1:length(taus)) # Example 4 # demonstrate a situation where the dimension of the predictor is large and # the local linear fitting meets the 'curse of dimensionality' problem set.seed(1234) n <- 100 p <- 10 x <- matrix(rnorm(n * p), n, p) error <- rnorm(n) y <- 3 * x[, 1] + x[, 2] + error tau <- 0.5 # use the following instead of llqr(x, y, tau = tau) fit.alt <- llqr(x, y, tau = tau, h=1) fit.alt# Example 1 # estimate the function at a specific quantile level for simulated data set.seed(1234) n <- 100 x <- rnorm(n) error <- rnorm(n) y <- (x + 1)^3 + 0.1 * (x - 2)^3 + error tau <- 0.5 plot(x, y, main = tau) points(x, llqr(x, y, tau = tau)$ll_est, col = 'red', pch = 16) # Example 2 # estimate the function at a point x0 set.seed(1234) n <- 100 x <- rnorm(n) error <- rnorm(n) y <- (x + 1)^3 + 0.1 * (x - 2)^3 + error tau <- 0.5 x0 <- 1 llqr(x, y, tau = tau, x0 = x0) # Example 3 # estimate the function for different quantile levels data(mcycle, package = "MASS") attach(mcycle) plot(times, accel, xlab = "milliseconds", ylab = "acceleration") taus <- c(0.1, 0.25, 0.5, 0.75, 0.9) for(i in 1:length(taus)) { fit <- llqr(times, accel, tau = taus[i])$ll_est lines(times, fit, lty = i) } legend(45, -50, c("tau=0.1","tau=0.25","tau=0.5","tau=0.75", "tau=0.9"), lty=1:length(taus)) # Example 4 # demonstrate a situation where the dimension of the predictor is large and # the local linear fitting meets the 'curse of dimensionality' problem set.seed(1234) n <- 100 p <- 10 x <- matrix(rnorm(n * p), n, p) error <- rnorm(n) y <- 3 * x[, 1] + x[, 2] + error tau <- 0.5 # use the following instead of llqr(x, y, tau = tau) fit.alt <- llqr(x, y, tau = tau, h=1) fit.alt
llqrcv estimates the bandwidth necessary for the local linear fit of
the th conditional quantile of y given x. The
estimation is performed using the Cross-Validation criterion.
llqrcv(x, y, tau = 0.5)llqrcv(x, y, tau = 0.5)
x |
A design matrix (n x p). The rows represent observations and the columns represent predictor variables. |
y |
A vector of the response variable. |
tau |
A quantile level, a number strictly between 0 and 1. |
A grid of bandwidth values is created and the local linear fit is estimated
using all the data points except for one point, which is used to make the
prediction. This procedure is repeated n times, where n is the
number of observations. Then, the bandwidth is selected as the one with the
smallest average error.
When the dimension of the predictor variable is large compared with the sample
size, local linear fitting meets the 'curse of dimensionality' problem. In
situations like that, the grid bandwidth values might be too small and cause
the function to fail. For these cases, we advice the user to directly use the
llqr function of the package and specify a bandwidth in the function.
llqrcv returns the optimal bandwidth selected using
Cross-Validation criterion for the local linear fit of the th
conditional quantile of y given x.
set.seed(1234) n <- 100 x <- rnorm(n) error <- rnorm(n) y <- x^2 + error tau <- 0.5 llqrcv(x, y, tau = tau)set.seed(1234) n <- 100 x <- rnorm(n) error <- rnorm(n) y <- x^2 + error tau <- 0.5 llqrcv(x, y, tau = tau)
ValAR estimates the one-step ahead th Value-at-Risk for a
vector of returns.
ValAR(y, p, tau, movwind = NULL, chronological = TRUE)ValAR(y, p, tau, movwind = NULL, chronological = TRUE)
y |
A vector of returns (1 x n). |
p |
An integer for the number of past observations to be used as predictor variables. This will form the n x p design matrix. |
tau |
A quantile level, a number strictly between 0 and 1. Commonly used choices are 0.01, 0.025, and 0.05. |
movwind |
An optional integer number for the moving window. If not specified, a default value of min(250, n - p) will be used. If specified, the moving window should be an integer between p and n - p. Typical values for moving windows correspond to one or two years of return values. If the user wants to use all observations to fit the model, then the moving window should be equal to n - p. Note that, the number n - p comes from the fact that the full data set starts from the (p + 1)th observation since we use the last p observations for prediction. |
chronological |
A logical operator to indicate whether the returns are in standard chronological order (from oldest to newest). The default value is TRUE. If the returns are in reverse chronological order, the function will rearrange them. |
The function calculates the th Value-at-Risk of the next time occurrence,
i.e., that number such that the probability that the returns fall below its
negative value is . The parameter is typically chosen to be a
small number such as 0.01, 0.025, or 0.05. By definition, the negative value of the
th Value-at-Risk is the th conditional quantile. Therefore,
the estimation is performed using a local linear conditional quantile estimation.
However, prior to this nonparametric estimation, a dimension reduction technique
is performed to select linear combinations of the predictor variables.
Specifically, the user provides a vector of returns y (usually log-returns)
and an integer p for the number of past observations to be used as the
predictor variables. The function then forms the m x p design matrix x, where m is
the number of used observations. For example, m can be n - p if the user wants to
use all observations, or m can be equal to the moving window (default value is
min(250, n - p)). Value-at-Risk is then defined as the negative value of the
th conditional quantile of y given x. However, to aid the nonparametric
estimation of the th conditional quantile, the cqs function is
applied to estimate the fewest linear combinations of the predictor x that
contain all the information available on the conditional quantile function. Finally,
the llqr function is applied to estimate the local linear conditional quantile
of y using the extracted directions as the predictor variables.
For more details on the method and for an application to the Bitcoin data, see Christou (2020). Also, see Christou and Grabchak (2019) for a thorough comparison between the proposed methodology and commonly used methods.
ValAR returns the one-step ahead th Value-at-Risk.
Christou, E. (2020) Central quantile subspace. Statistics and Computing, 30, 677–695.
Christou, E., Grabchak, M. (2019) Estimation of value-at-risk using single index quantile regression. Journal of Applied Statistics, 46(13), 2418–2433.
# estimate the one-step ahead Value-at-Risk with default moving window data(edhec, package = "PerformanceAnalytics") y <- as.vector(edhec[, 1]) # Convertible Arbitrage p <- 5 # use the 5 most recent observations as predictor variables tau <- 0.05 ValAR(y, p, tau) # the data is already in standard chronological order # compare it with the historical Value-at-Risk calculation PerformanceAnalytics::VaR(y, 0.95, method = 'historical')# estimate the one-step ahead Value-at-Risk with default moving window data(edhec, package = "PerformanceAnalytics") y <- as.vector(edhec[, 1]) # Convertible Arbitrage p <- 5 # use the 5 most recent observations as predictor variables tau <- 0.05 ValAR(y, p, tau) # the data is already in standard chronological order # compare it with the historical Value-at-Risk calculation PerformanceAnalytics::VaR(y, 0.95, method = 'historical')