npcdistbw {np} | R Documentation

## Kernel Conditional Distribution Bandwidth Selection with Mixed Data Types

### Description

`npcdistbw` computes a `condbandwidth` object for estimating a p+q-variate kernel conditional cumulative distribution estimator defined over mixed continuous and discrete (unordered `xdat`, ordered `xdat` and `ydat`) data using either the normal-reference rule-of-thumb or least-squares cross validation method of Li and Racine (2008) and Li, Lin and Racine (2013).

### Usage

    npcdistbw(...)

    ## S3 method for class 'formula'
    npcdistbw(formula, data, subset, na.action, call, gdata = NULL,...)

    ## S3 method for class 'NULL'
    npcdistbw(xdat = stop("data 'xdat' missing"),
              ydat = stop("data 'ydat' missing"),
              bws, ...)

    ## S3 method for class 'condbandwidth'
    npcdistbw(xdat = stop("data 'xdat' missing"),
              ydat = stop("data 'ydat' missing"),
              gydat = NULL,
              bws,
              bandwidth.compute = TRUE,
              nmulti,
              remin = TRUE,
              itmax = 10000,
              do.full.integral = FALSE,
              ngrid = 100,
              ftol = 1.490116e-07,
              tol = 1.490116e-04,
              small = 1.490116e-05,
              memfac = 500.0,
              lbc.dir = 0.5,
              dfc.dir = 3,
              cfac.dir = 2.5*(3.0-sqrt(5)),
              initc.dir = 1.0,
              lbd.dir = 0.1,
              hbd.dir = 1,
              dfac.dir = 0.25*(3.0-sqrt(5)),
              initd.dir = 1.0,
              lbc.init = 0.1,
              hbc.init = 2.0,
              cfac.init = 0.5,
              lbd.init = 0.1,
              hbd.init = 0.9,
              dfac.init = 0.375,
              scale.init.categorical.sample = FALSE,
              ...)

    ## Default S3 method:
    npcdistbw(xdat = stop("data 'xdat' missing"),
              ydat = stop("data 'ydat' missing"),
              gydat,
              bws,
              bandwidth.compute = TRUE,
              nmulti,
              remin,
              itmax,
              do.full.integral,
              ngrid,
              ftol,
              tol,
              small,
              memfac,
              lbc.dir,
              dfc.dir,
              cfac.dir,
              initc.dir,
              lbd.dir,
              hbd.dir,
              dfac.dir,
              initd.dir,
              lbc.init,
              hbc.init,
              cfac.init,
              lbd.init,
              hbd.init,
              dfac.init,
              scale.init.categorical.sample,
              bwmethod,
              bwscaling,
              bwtype,
              cxkertype,
              cxkerorder,
              cykertype,
              cykerorder,
              uxkertype,
              oxkertype,
              oykertype,
              ...)

### Arguments

`formula` | a symbolic description of variables on which bandwidth selection is to be performed. The details of constructing a formula are described below.
`data` | an optional data frame, list or environment (or object coercible to a data frame by `as.data.frame`) containing the variables in the model. If not found in data, the variables are taken from `environment(formula)`, typically the environment from which the function is called.
`subset` | an optional vector specifying a subset of observations to be used in the fitting process.
`na.action` | a function which indicates what should happen when the data contain `NA`s. The default is set by the `na.action` setting of options, and is `na.fail` if that is unset. The (recommended) default is `na.omit`.
`call` | the original function call. This is passed internally by `np` when a bandwidth search has been implied by a call to another function. It is not recommended that the user set this.
`gdata` | a grid of data on which the indicator function for least-squares cross-validation is to be computed (can be the sample or a grid of quantiles).
`xdat` | a p-variate data frame of explanatory data on which bandwidth selection will be performed. The data types may be continuous, discrete (unordered and ordered factors), or some combination thereof.
`ydat` | a q-variate data frame of dependent data on which bandwidth selection will be performed. The data types may be continuous, discrete (ordered factors), or some combination thereof.
`gydat` | a grid of data on which the indicator function for least-squares cross-validation is to be computed (can be the sample or a grid of quantiles for `ydat`).

`bws`

a bandwidth specification. This can be set as a `condbandwidth` object returned from a previous invocation, or as a p+q-vector of bandwidths, with each element i up to i=q corresponding to the bandwidth for column i in `ydat`, and each element i from i=q+1 to i=p+q corresponding to the bandwidth for column i-q in `xdat`. In either case, the bandwidth supplied will serve as a starting point in the numerical search for optimal bandwidths. If specified as a vector, then additional arguments will need to be supplied as necessary to specify the bandwidth type, kernel types, selection methods, and so on. This can be left unset.

`...` | additional arguments supplied to specify the bandwidth type, kernel types, selection methods, and so on, detailed below.

`bwmethod`

which method to use to select bandwidths. `cv.ls` specifies least-squares cross-validation (Li, Lin and Racine (2013), and `normal-reference` just computes the ‘rule-of-thumb’ bandwidth h[j] using the standard formula h[j] = 1.06*sigma[j]*n^(-1.0/(2.0*P+l)), where sigma[j] is an adaptive measure of spread of the jth continuous variable defined as min(standard deviation, mean absolute deviation/1.4826, interquartile range/1.349), n the number of observations, P the order of the kernel, and l the number of continuous variables. Note that when there exist factors and the normal-reference rule is used, there is zero smoothing of the factors. Defaults to `cv.ls`.

`bwscaling`

a logical value that when set to `TRUE` the supplied bandwidths are interpreted as ‘scale factors’ (c[j]), otherwise when the value is `FALSE` they are interpreted as ‘raw bandwidths’ (h[j] for continuous data types, lambda[j] for discrete data types). For continuous data types, c[j] and h[j] are related by the formula h[j] = c[j]*sigma[j]*n^(-1/(2*P+l)), where sigma[j] is an adaptive measure of spread of continuous variable j defined as min(standard deviation, mean absolute deviation/1.4826, interquartile range/1.349), n the number of observations, P the order of the kernel, and l the number of continuous variables. For discrete data types, c[j] and h[j] are related by the formula h[j] = c[j]*n^(-2/(2*P+l)), where here [j] denotes discrete variable j. Defaults to `FALSE`.

`bwtype` | character string used for the continuous variable bandwidth type, specifying the type of bandwidth to compute and return in the `condbandwidth` object. Defaults to `fixed`. Option summary: `fixed`: compute fixed bandwidths `generalized_nn`: compute generalized nearest neighbors `adaptive_nn`: compute adaptive nearest neighbors
`bandwidth.compute` | a logical value which specifies whether to do a numerical search for bandwidths or not. If set to `FALSE`, a `condbandwidth` object will be returned with bandwidths set to those specified in `bws`. Defaults to `TRUE`.
`cxkertype` | character string used to specify the continuous kernel type for `xdat`. Can be set as `gaussian`, `epanechnikov`, or `uniform`. Defaults to `gaussian`.
`cxkerorder` | numeric value specifying kernel order for `xdat` (one of `(2,4,6,8)`). Kernel order specified along with a `uniform` continuous kernel type will be ignored. Defaults to `2`.
`cykertype` | character string used to specify the continuous kernel type for `ydat`. Can be set as `gaussian`, `epanechnikov`, or `uniform`. Defaults to `gaussian`.
`cykerorder` | numeric value specifying kernel order for `ydat` (one of `(2,4,6,8)`). Kernel order specified along with a `uniform` continuous kernel type will be ignored. Defaults to `2`.
`uxkertype` | character string used to specify the unordered categorical kernel type. Can be set as `aitchisonaitken` or `liracine`. Defaults to `aitchisonaitken`.
`oxkertype` | character string used to specify the ordered categorical kernel type. Can be set as `wangvanryzin` or `liracine`. Defaults to `liracine`.
`oykertype` | character string used to specify the ordered categorical kernel type. Can be set as `wangvanryzin` or `liracine`.
`nmulti` | integer number of times to restart the process of finding extrema of the cross-validation function from different (random) initial points
`remin` | a logical value which when set as `TRUE` the search routine restarts from located minima for a minor gain in accuracy. Defaults to `TRUE`.
`itmax` | integer number of iterations before failure in the numerical optimization routine. Defaults to `10000`.
`do.full.integral` | a logical value which when set as `TRUE` evaluates the moment-based integral on the entire sample.
`ngrid` | integer number of grid points to use when computing the moment-based integral. Defaults to `100`.
`ftol` | fractional tolerance on the value of the cross-validation function evaluated at located minima (of order the machine precision or perhaps slightly larger so as not to be diddled by roundoff). Defaults to `1.490116e-07` (1.0e+01*sqrt(.Machine$double.eps)).
`tol` | tolerance on the position of located minima of the cross-validation function (tol should generally be no smaller than the square root of your machine's floating point precision). Defaults to `1.490116e-04 (1.0e+04*sqrt(.Machine$double.eps))`.
`small` | a small number used to bracket a minimum (it is hopeless to ask for a bracketing interval of width less than sqrt(epsilon) times its central value, a fractional width of only about 10-04 (single precision) or 3x10-8 (double precision)). Defaults to `small = 1.490116e-05 (1.0e+03*sqrt(.Machine$double.eps))`.
`lbc.dir,dfc.dir,cfac.dir,initc.dir` | lower bound, chi-square degrees of freedom, stretch factor, and initial non-random values for direction set search for Powell's algorithm for `numeric` variables. See Details
`lbd.dir,hbd.dir,dfac.dir,initd.dir` | lower bound, upper bound, stretch factor, and initial non-random values for direction set search for Powell's algorithm for categorical variables. See Details
`lbc.init, hbc.init, cfac.init` | lower bound, upper bound, and non-random initial values for scale factors for `numeric` variables for Powell's algorithm. See Details
`lbd.init, hbd.init, dfac.init` | lower bound, upper bound, and non-random initial values for scale factors for categorical variables for Powell's algorithm. See Details
`scale.init.categorical.sample` | a logical value that when set to `TRUE` scales `lbd.dir`, `hbd.dir`, `dfac.dir`, and `initd.dir` by n^{-2/(2P+l)}, n the number of observations, P the order of the kernel, and l the number of `numeric` variables. See Details

`memfac`

The algorithm to compute the least-squares objective function uses a block-based algorithm to eliminate or minimize redundant kernel evaluations. Due to memory, hardware and software constraints, a maximum block size must be imposed by the algorithm. This block size is roughly equal to memfac*10^5 elements. Empirical tests on modern hardware find that a memfac of around 500 performs well. If you experience out of memory errors, or strange behaviour for large data sets (>100k elements) setting memfac to a lower value may fix the problem.

### Details

`npcdistbw` implements a variety of methods for choosing bandwidths for multivariate distributions (p+q-variate) defined over a set of possibly continuous and/or discrete (unordered `xdat`, ordered `xdat` and `ydat`) data. The approach is based on Li and Racine (2004) who employ ‘generalized product kernels’ that admit a mix of continuous and discrete data types.

The cross-validation methods employ multivariate numerical search algorithms (direction set (Powell's) methods in multidimensions).

Bandwidths can (and will) differ for each variable which is, of course, desirable.

Three classes of kernel estimators for the continuous data types are available: fixed, adaptive nearest-neighbor, and generalized nearest-neighbor. Adaptive nearest-neighbor bandwidths change with each sample realization in the set, x[i], when estimating the cumulative distribution at the point x. Generalized nearest-neighbor bandwidths change with the point at which the cumulative distribution is estimated, x. Fixed bandwidths are constant over the support of x.

`npcdistbw` may be invoked either with a formula-like symbolic description of variables on which bandwidth selection is to be performed or through a simpler interface whereby data is passed directly to the function via the `xdat` and `ydat` parameters. Use of these two interfaces is mutually exclusive.

Data contained in the data frame `xdat` may be a mix of continuous (default), unordered discrete (to be specified in the data frames using `factor`), and ordered discrete (to be specified in the data frames using `ordered`). Data contained in the data frame `ydat` may be a mix of continuous (default) and ordered discrete (to be specified in the data frames using `ordered`). Data can be entered in an arbitrary order and data types will be detected automatically by the routine (see `np` for details).

Data for which bandwidths are to be estimated may be specified symbolically. A typical description has the form `dependent data ~ explanatory data`, where `dependent data` and `explanatory data` are both series of variables specified by name, separated by the separation character '+'. For example, `y1 + y2 ~ x1 + x2` specifies that the bandwidths for the joint distribution of variables `y1` and `y2` conditioned on `x1` and `x2` are to be estimated. See below for further examples.

A variety of kernels may be specified by the user. Kernels implemented for continuous data types include the second, fourth, sixth, and eighth order Gaussian and Epanechnikov kernels, and the uniform kernel. Unordered discrete data types use a variation on Aitchison and Aitken's (1976) kernel, while ordered data types use a variation of the Wang and van Ryzin (1981) kernel.

The optimizer invoked for search is Powell's conjugate direction method which requires the setting of (non-random) initial values and search directions for bandwidths, and, when restarting, random values for successive invocations. Bandwidths for `numeric` variables are scaled by robust measures of spread, the sample size, and the number of `numeric` variables where appropriate. Two sets of parameters for bandwidths for `numeric` can be modified, those for initial values for the parameters themselves, and those for the directions taken (Powell's algorithm does not involve explicit computation of the function's gradient). The default values are set by considering search performance for a variety of difficult test cases and simulated cases. We highly recommend restarting search a large number of times to avoid the presence of local minima (achieved by modifying `nmulti`). Further refinement for difficult cases can be achieved by modifying these sets of parameters. However, these parameters are intended more for the authors of the package to enable ‘tuning’ for various methods rather than for the user themselves.

### Value

`npcdistbw` returns a `condbandwidth` object, with the following components:

`xbw` | bandwidth(s), scale factor(s) or nearest neighbours for the explanatory data, `xdat`
`ybw` | bandwidth(s), scale factor(s) or nearest neighbours for the dependent data, `ydat`
`fval` | objective function value at minimum

if `bwtype` is set to `fixed`, an object containing bandwidths (or scale factors if `bwscaling = TRUE`) is returned. If it is set to `generalized_nn` or `adaptive_nn`, then instead the kth nearest neighbors are returned for the continuous variables while the discrete kernel bandwidths are returned for the discrete variables.

The functions `predict`, `summary` and `plot` support objects of type `condbandwidth`.

### Usage Issues

If you are using data of mixed types, then it is advisable to use the `data.frame` function to construct your input data and not `cbind`, since `cbind` will typically not work as intended on mixed data types and will coerce the data to the same type.

Caution: multivariate data-driven bandwidth selection methods are, by their nature, computationally intensive. Virtually all methods require dropping the ith observation from the data set, computing an object, repeating this for all observations in the sample, then averaging each of these leave-one-out estimates for a given value of the bandwidth vector, and only then repeating this a large number of times in order to conduct multivariate numerical minimization/maximization. Furthermore, due to the potential for local minima/maxima, restarting this procedure a large number of times may often be necessary. This can be frustrating for users possessing large datasets. For exploratory purposes, you may wish to override the default search tolerances, say, setting ftol=.01 and tol=.01 and conduct multistarting (the default is to restart min(5, ncol(xdat,ydat)) times) as is done for a number of examples. Once the procedure terminates, you can restart search with default tolerances using those bandwidths obtained from the less rigorous search (i.e., set `bws=bw` on subsequent calls to this routine where `bw` is the initial bandwidth object). A version of this package using the `Rmpi` wrapper is under development that allows one to deploy this software in a clustered computing environment to facilitate computation involving large datasets.

### Author(s)

Tristen Hayfield tristen.hayfield@gmail.com, Jeffrey S. Racine racinej@mcmaster.ca

### References

Aitchison, J. and C.G.G. Aitken (1976), “Multivariate binary discrimination by the kernel method,” Biometrika, 63, 413-420.

Hall, P. and J.S. Racine and Q. Li (2004), “Cross-validation and the estimation of conditional probability densities,” Journal of the American Statistical Association, 99, 1015-1026.

Li, Q. and J.S. Racine (2007), Nonparametric Econometrics: Theory and Practice, Princeton University Press.

Li, Q. and J.S. Racine (2008), “Nonparametric estimation of conditional CDF and quantile functions with mixed categorical and continuous data,” Journal of Business and Economic Statistics, 26, 423-434.

Li, Q. and J. Lin and J.S. Racine (2013), “Optimal bandwidth selection for nonparametric conditional distribution and quantile functions”, Journal of Business and Economic Statistics, 31, 57-65.

Pagan, A. and A. Ullah (1999), Nonparametric Econometrics, Cambridge University Press.

Scott, D.W. (1992), Multivariate Density Estimation. Theory, Practice and Visualization, New York: Wiley.

Silverman, B.W. (1986), Density Estimation, London: Chapman and Hall.

Wang, M.C. and J. van Ryzin (1981), “A class of smooth estimators for discrete distributions,” Biometrika, 68, 301-309.

### See Also

`bw.nrd`, `bw.SJ`, `hist`, `npudens`, `npudist`

### Examples

    ## Not run:
    # EXAMPLE 1 (INTERFACE=FORMULA): For this example, we compute the
    # cross-validated bandwidths (default) using a second-order Gaussian
    # kernel (default). Note - this may take a minute or two depending on
    # the speed of your computer.

    data("Italy")
    attach(Italy)

    bw <- npcdistbw(formula=gdp~ordered(year))

    # The object bw can be used for further estimation using
    # npcdist(), plotting using plot() etc. Entering the name of
    # the object provides useful summary information, and names() will also
    # provide useful information.

    summary(bw)

    # Note - see the example for npudensbw() for multiple illustrations
    # of how to change the kernel function, kernel order, and so forth.

    detach(Italy)

    # EXAMPLE 1 (INTERFACE=DATA FRAME): For this example, we compute the
    # cross-validated bandwidths (default) using a second-order Gaussian
    # kernel (default). Note - this may take a minute or two depending on
    # the speed of your computer.

    data("Italy")
    attach(Italy)

    bw <- npcdistbw(xdat=ordered(year), ydat=gdp)

    # The object bw can be used for further estimation using npcdist(),
    # plotting using plot() etc. Entering the name of the object provides
    # useful summary information, and names() will also provide useful
    # information.

    summary(bw)

    # Note - see the example for npudensbw() for multiple illustrations
    # of how to change the kernel function, kernel order, and so forth.

    detach(Italy)

    ## End(Not run)

[Package np version 0.60-9 Index]