kdml (version 1.1.1)

kss: Kernel Summation Similarity Function (KSS) for Mixed-type Data

Description

This function calculates the pairwise similarities between mixed-type observations consisting of continuous (numeric), nominal (factor), and ordinal (ordered) variables using the method described in Ghashti (2024). This kernel similarity learning methodology calculates a kernel sum similarity function, with a variety of options for kernel functions associated with each variable type and returns a distance matrix that can be used in any distance-based algorithm.

Usage

kss(df, bw = "np", npmethod = NULL, cFUN = "c_gaussian", 
    uFUN = "u_aitken", oFUN = "o_wangvanryzin", nstart = NULL, 
    stan = TRUE, verbose = FALSE)

Value

kss returns a list object, with the following components:

similarities

an \(n \times n\) numeric matrix containing pairwise similarities between observations

bandwidths

a \(p\)-variate vector of bandwidth values returned based on the bw bandwidth specification method, sorted by variable type

Arguments

df

a \(p\)-variate data frame for which the pairwise similarities between observations will be calculated. The data types may be continuous (numeric), nominal (factor), and ordinal (ordered), or any combination thereof. Columns of df should be of appropriate variable type prior to running the function.

bw

numeric bandwidth vector of length \(p\), with each element \(i\) corresponding to the bandwidth for column \(i\) in df. Alternatively, a character strings may be inputted for bandwidth selection methods. np specifies this techniques which calculate bandwidths using npudensbw in package np. Defaults to np with npmethod set to cv.ml for maximum likelihood cross-validation. See details.

npmethod

character value specifying the np bandwidth selection to be used for calculating bandwidths. Options include cv.ml for maximum likelihood cross-validation, cv.ls for least squares cross-validation, and normal-reference for normal reference. If left as NULL, defaults to cv.ml.

cFUN

character value 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.

uFUN

character value specifying the nominal kernel function for unordered factors. Options include u_aitken and u_aitchisonaitken. Defaults to u_aitken. See details.

oFUN

character value 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.

nstart

integer value specifying the number of random starts for the kmeans algorithm. Defaults to 10.

stan

a logical value which specifies whether to scale the resulting distance matrix between 0 and 1 using min-max normalization. If set to FALSE, distances are unscaled. Defaults to TRUE.

verbose

a logical value which specifies whether to print procedural steps to the console. If set to FALSE, no output is printed to the console. Defaults to FALSE.

Author

John R. J. Thompson , Jesse S. Ghashti

Details

kss implements the kernel summation similarity function (KSS) as described by Ghashti (2024). This approach uses summation kernels for continuous, nominal and ordinal data, which are then summed over all variable types to return the pairwise similarities between mixed-type data.

There are several kernels to select from. The 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, while ordinal kernels use a variation of the Wang and van Ryzin (1981) kernel. Both 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).

Each kernel requires a bandwidth specification, which can either be a user defined numeric vector of length \(p\) from alternative methodologies for bandwidth selection, or through one bandwidth selection method can be specified. The np bandwidth selection methods follow three techniques (cv.ml, cv.ls and normal-reference) described by Li and Racine (2007) and Li and Racine (2003) for kernel density estimation of mixed-type data.

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 factor for nominal data, and ordered for ordinal data. Data types can be in any order and will be detected automatically. User-inputted vectors of bandwidths bw must be specified in the same order as the variables in the data frame df, as to ensure they sorted accordingly by the routine.

References

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. (2024), “Similarity Maximization and Shrinkage Approach in Kernel Metric Learning for Clustering Mixed-type Data (T)”, University of British Columbia.

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.

See Also

mscv.dkps, dkps, mscv.dkss, dkss, link{spectral.clust}

Examples

# \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, and will automatically be
# defaulted to the mscv bandwidth specification technique and default kernel 
# function
s1 <- kss(df = df)
# s$bandwidths to see the mscv obtained bandwidths
# s$similarities to see the similarity matrix


# try using the np package, which has few continuous and ordinal kernels 
# to choose from. Recommended using default kernel functions
s2 <- kss(df = df, bw = "np") #defaults to npmethod "cv.ml"


# precomputed bandwidth example
# note that continuous variables requires bandwidths > 0
# ordinal variables requires bandwidths in [0,1]
# for nominal variables, u_aitken requires bandwidths in [0,1] 
# and u_aitchisonaitken in [0,(c-1)/c]
# where c is the number of unique values in the i-th column of df.
# any bandwidths outside this range will result in a warning message
bw_vec <- c(1.0, 0.5, 0.5, 5.0, 0.3, 0.3) 
s3 <- kss(df = df, bw = bw_vec)


# user-specific kernel functions example with "cv.ls" from np.
s4 <- kss(df = df, bw = "np", npmethod = "cv.ls", cFUN = "c_epanechnikov", 
    uFUN = "u_aitken", oFUN = "o_wangvanryzin")
# }