This function calculates maximum-similarity cross-validated bandwidths for the
distance using kernel summation similarity. This implementation uses the method
described in Ghashti and Thompson (2023) for mixed-type data that includes any
of numeric (continuous), factor (nominal), and ordered factor (ordinal)
variables. mscv.dkps calculates the bandwidths associated with each
kernel function for variable types and returns a numeric vector of bandwidths
that can be used with the dkps pairwise distance calculation.
mscv.dkps(df, nstart = NULL, ckernel = "c_gaussian", ukernel = "u_aitken",
okernel = "o_wangvanryzin", verbose = FALSE)mscv.dkps returns a list object, with the
following components:
a \(p\)-variate vector of bandwidth values, intended to be used for
the dkps pairwise distance calculation
a numeric value of the MSCV objective function, obtained using
the optim function for constrained multivariate optimization
a \(p\)-variate data frame. The data types may be continuous (numeric),
nominal (factor), ordinal (ordered), or any
combination thereof. Columns of df should be set to the appropriate
data type class.
integer number of restarts for the process of finding extrema of the mscv
function from random initial bandwidth parameters (starting points). If the
default of NULL is used, then the number of restarts will be
\(min(3,\text{ncol(df)})\).
character string specifying the continuous kernel function. Options include
c_gaussian, c_epanechnikov, c_uniform, c_triangle,
c_biweight, c_triweight, c_tricube, c_cosine,
c_logistic, c_sigmoid, and c_silverman. Note that if
using np for bw selection above, continuous kernel types are
restricted to either c_gaussian, c_epanechnikov, or
c_uniform. Defaults to c_gaussian. See details.
character string specifying the nominal kernel function for unordered factors.
Options include u_aitken and u_aitchisonaitken. Defaults to
u_aitken. See details.
character string specifying the ordinal kernel function for ordered factors.
Options include o_aitken, o_aitchisonaitken, o_habbema,
o_wangvanryzin, and o_liracine. Note that if using np
for bw selection above, ordinal kernel types are restricted to either
o_wangvanryzin or o_liracine. Defaults to o_wangvanryzin.
See details.
a logical value which specifies whether to output the \(i\)-th iteration of
the total number of nstarts, and output if the optimization procedure
converges. Defaults to TRUE.
John R. J. Thompson john.thompson@ubc.ca, Jesse S. Ghashti jesse.ghashti@ubc.ca
mscv.dkps implements the maximum-similarity cross-validation (MSCV)
technique for bandwidth selection pertaining to the dkps function,
as described by Ghashti and Thompson (2023). This approach uses product kernels
for continuous variables, and summation kernels for nominal and ordinal data,
which are then summed over all variable types to return the pairwise distance
between mixed-type data.
The maximization procedure for bandwidth selection is based on the objective \(\text{arg}\max_{\boldsymbol{\lambda}}\left\{\frac{1}{n}\sum_{i=1}^n\log\left(\frac{1}{(n-1)}\sum_{\substack{j=1 \\ j \ne i}}^n\psi_{\boldsymbol{\lambda}}(\textbf{x}_i,\textbf{x}_j)\right)\right\},\) where
\(\psi(\textbf{x}_i, \textbf{x}_j \ | \boldsymbol{\lambda}) = \prod_{k=1}^{p_c}\frac{1}{\lambda_k^c}K(x_{i,k}^c, x_{j,k}^c, \lambda_k^c) + \sum_{k=1}^{p_u}L(x_{i,k}^u,x_{j,k}^u,\lambda_k^u) + \sum_{k=1}^{p_o}\ell(x_{i,k}^o,x_{j,k}^o,\lambda_k^o).\)
\(K(\cdot)\), \(L(\cdot)\), and \(\ell(\cdot)\) are the continuous,
nominal, and ordinal kernel functions, repectively, with \(\lambda_k\)'s
representing kernel specifical bandwiths for the \(k\)-th variable, and
\(p_c\), \(p_u\), \(p_o\) the number of continuous, nominal, and ordinal
variables in the data frame df. The resulting bw vector returned
is the bandwidths that yield the highest objective function value.
Data contained in the data frame df may constitute any combinations of
continuous, nominal, or ordinal data, which is to be specified in the data
frame df using numeric for continuous data, factor
for nominal data, and ordered for ordinal data. Data can be
entered in an arbitrary order and data types will be detected automatically.
User-inputted vectors of bandwidths bw must be defined in the same
order as the variables in the data frame df, as to ensure they sorted
accordingly by the routine.
The are many kernels which can be specified by the user. Continuous kernel functions may be found in Cameron and Trivedi (2005), Härdle et al. (2004) or Silverman (1986). Nominal kernels use a variation on Aitchison and Aitken's (1976) kernel. Ordinal kernels use a variation of the Wang and van Ryzin (1981) kernel. All nominal and ordinal kernel functions can be found in Li and Racine (2007), Li and Racine (2003), Ouyan et al. (2006), and Titterington and Bowman (1985).
Aitchison, J. and C.G.G. Aitken (1976), “Multivariate binary discrimination by the kernel method”, Biometrika, 63, 413-420.
Cameron, A. and P. Trivedi (2005), “Microeconometrics: Methods and Applications”, Cambridge University Press.
Ghashti, J.S. and J.R.J Thompson (2023), “Mixed-type Distance Shrinkage and Selection for Clustering via Kernel Metric Learning. Journal of Classification, Accepted.”
Härdle, W., and M. Müller and S. Sperlich and A. Werwatz (2004), “Nonparametric and Semiparametric Models”, (Vol. 1). Berlin: Springer.
Li, Q. and J.S. Racine (2007), “Nonparametric Econometrics: Theory and Practice”, Princeton University Press.
Li, Q. and J.S. Racine (2003), “Nonparametric estimation of distributions with categorical and continuous data”, Journal of Multivariate Analysis, 86, 266-292.
Ouyang, D. and Q. Li and J.S. Racine (2006), “Cross-validation and the estimation of probability distributions with categorical data”, Journal of Nonparametric Statistics, 18, 69-100.
Silverman, B.W. (1986), “Density Estimation”, London: Chapman and Hall.
Titterington, D.M. and A.W. Bowman (1985), “A comparative study of smoothing procedures for ordered categorical data”, Journal of Statistical Computation and Simulation, 21(3-4), 291-312.
Wang, M.C. and J. van Ryzin (1981), “A class of smooth estimators for discrete distributions”, Biometrika, 68, 301-309.
mscv.dkss, dkss, dkps
# \donttest{
# example data frame with mixed numeric, nominal, and ordinal data.
levels = c("Low", "Medium", "High")
df <- data.frame(
x1 = runif(100, 0, 100),
x2 = factor(sample(c("A", "B", "C"), 100, TRUE)),
x3 = factor(sample(c("A", "B", "C"), 100, TRUE)),
x4 = rnorm(100, 10, 3),
x5 = ordered(sample(c("Low", "Medium", "High"), 100, TRUE), levels = levels),
x6 = ordered(sample(c("Low", "Medium", "High"), 100, TRUE), levels = levels))
# minimal implementation requires just the data frame, with defaults
bw <- mscv.dkps(df = df)
# specify number of starts and kernel functions
bw2 <- mscv.dkps(df = df, nstart = 5, ckernel = "c_triangle",
ukernel = "u_aitken", okernel = "o_liracine")
# }
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