npudens {np} | R Documentation

## Kernel Density Estimation with Mixed Data Types

### Description

`npudens` computes kernel unconditional density estimates on evaluation data, given a set of training data and a bandwidth specification (a `bandwidth` object or a bandwidth vector, bandwidth type, and kernel type) using the method of Li and Racine (2003).

### Usage

    npudens(bws, ...)

    ## S3 method for class 'formula'
    npudens(bws, data = NULL, newdata = NULL, ...)

    ## S3 method for class 'bandwidth'
    npudens(bws,
            tdat = stop("invoked without training data 'tdat'"),
            edat,
            ...)

    ## S3 method for class 'call'
    npudens(bws, ...)

    ## Default S3 method:
    npudens(bws, tdat, ...)

### Arguments

`bws` | a bandwidth specification. This can be set as a `bandwidth` object returned from an invocation of `npudensbw`, or as a p-vector of bandwidths, with an element for each variable in the training data. If specified as a vector, then additional arguments will need to be supplied as necessary to change them from the defaults to specify the bandwidth type, kernel types, training data, and so on.
`...` | additional arguments supplied to specify, the training data, the bandwidth type, kernel types, and so on. This is necessary if you specify bws as a p-vector and not a `bandwidth` object, and you do not desire the default behaviours. To do this, you may specify any of `bwscaling`, `bwtype`, `ckertype`, `ckerorder`, `ukertype`, `okertype`, as described in `npudensbw`.
`tdat` | a p-variate data frame of sample realizations (training data) used to estimate the density. Defaults to the training data used to compute the bandwidth object.
`edat` | a p-variate data frame of density evaluation points. By default, evaluation takes place on the data provided by `tdat`.
`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(bws)`, typically the environment from which `npudensbw` was called.
`newdata` | An optional data frame in which to look for evaluation data. If omitted, the training data are used.

### Details

Typical usages are (see below for a complete list of options and also the examples at the end of this help file)

        Usage 1: first compute the bandwidth object via npudensbw and then
        compute the density:

        bw <- npudensbw(~y)
        fhat <- npudens(bw)

        Usage 2: alternatively, compute the bandwidth object indirectly:

        fhat <- npudens(~y)

        Usage 3: modify the default kernel and order:

        fhat <- npudens(~y, ckertype="epanechnikov", ckerorder=4)

        Usage 4: use the data frame interface rather than the formula
        interface:

        fhat <- npudens(tdat = y, ckertype="epanechnikov", ckerorder=4)

`npudens` implements a variety of methods for estimating multivariate density functions (p-variate) defined over a set of possibly continuous and/or discrete (unordered, ordered) data. The approach is based on Li and Racine (2003) who employ ‘generalized product kernels’ that admit a mix of continuous and discrete data types.

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 density at the point x. Generalized nearest-neighbor bandwidths change with the point at which the density is estimated, x. Fixed bandwidths are constant over the support of x.

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

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.

### Value

`npudens` returns a `npdensity` object. The generic accessor functions `fitted`, and `se`, extract estimated values and asymptotic standard errors on estimates, respectively, from the returned object. Furthermore, the functions `predict`, `summary` and `plot` support objects of both classes. The returned objects have the following components:

`eval` | the evaluation points.
`dens` | estimation of the density at the evaluation points
`derr` | standard errors of the density estimates
`log_likelihood` | log likelihood of the density estimates

### 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.

### 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.

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.

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

`npudensbw` , `density`

### Examples

    ## Not run:
    # EXAMPLE 1 (INTERFACE=FORMULA): For this example, we load Giovanni
    # Baiocchi's Italian GDP panel (see Italy for details), then create a
    # data frame in which year is an ordered factor, GDP is continuous,
    # compute bandwidths using likelihood cross-validation, then create a
    # grid of data on which the density will be evaluated for plotting
    # purposes.

    data("Italy")
    attach(Italy)

    # Compute bandwidths using likelihood cross-validation (default).

    bw <- npudensbw(formula=~ordered(year)+gdp)

    # At this stage you could use npudens() to do a variety of
    # things. Here we compute the npudens() object and place it in fhat.

    fhat <- npudens(bws=bw)

    # Note that simply typing the name of the object returns some useful
    # information. For more info, one can call summary:

    summary(fhat)

    # Next, we illustrate how to create a grid of `evaluation data' and feed
    # it to the perspective plotting routines in R, among others.

    # Create an evaluation data matrix

    year.seq <- sort(unique(year))
    gdp.seq <- seq(1,36,length=50)
    data.eval <- expand.grid(year=year.seq,gdp=gdp.seq)

    # Generate the estimated density computed for the evaluation data

    fhat <- fitted(npudens(bws=bw, newdata=data.eval))

    # Coerce the data into a matrix for plotting with persp()

    f <- matrix(fhat, length(unique(year)), 50)

    # Next, create a 3D perspective plot of the PDF f, and a 2D
    # contour plot.

    persp(as.integer(levels(year.seq)), gdp.seq, f, col="lightblue",
          ticktype="detailed", ylab="GDP", xlab="Year", zlab="Density",
          theta=300, phi=50)

    # Sleep for 5 seconds so that we can examine the output...

    Sys.sleep(5)

    contour(as.integer(levels(year.seq)),
            gdp.seq,
            f,
            xlab="Year",
            ylab="GDP",
            main = "Density Contour Plot",
            col=topo.colors(100))

    # Sleep for 5 seconds so that we can examine the output...

    Sys.sleep(5)

    # Alternatively, you could use the plot() command (<ctrl>-C will
    # interrupt on *NIX systems, <esc> will interrupt on MS Windows
    # systems).

    plot(bw)

    detach(Italy)

    # EXAMPLE 1 (INTERFACE=DATA FRAME): For this example, we load Giovanni
    # Baiocchi's Italian GDP panel (see Italy for details), then create a
    # data frame in which year is an ordered factor, GDP is continuous,
    # compute bandwidths using likelihood cross-validation, then create a
    # grid of data on which the density will be evaluated for plotting
    # purposes.

    data("Italy")
    attach(Italy)

    data <- data.frame(year=ordered(year), gdp)

    # Compute bandwidths using likelihood cross-validation (default).

    bw <- npudensbw(dat=data)

    # At this stage you could use npudens() to do a variety of
    # things. Here we compute the npudens() object and place it in fhat.

    fhat <- npudens(bws=bw)

    # Note that simply typing the name of the object returns some useful
    # information. For more info, one can call summary:

    summary(fhat)

    # Next, we illustrate how to create a grid of `evaluation data' and feed
    # it to the perspective plotting routines in R, among others.

    # Create an evaluation data matrix

    year.seq <- sort(unique(year))
    gdp.seq <- seq(1,36,length=50)
    data.eval <- expand.grid(year=year.seq,gdp=gdp.seq)

    # Generate the estimated density computed for the evaluation data

    fhat <- fitted(npudens(edat = data.eval, bws=bw))

    # Coerce the data into a matrix for plotting with persp()

    f <- matrix(fhat, length(unique(year)), 50)

    # Next, create a 3D perspective plot of the PDF f, and a 2D
    # contour plot.

    persp(as.integer(levels(year.seq)), gdp.seq, f, col="lightblue",
          ticktype="detailed", ylab="GDP", xlab="Year", zlab="Density",
          theta=300, phi=50)

    # Sleep for 5 seconds so that we can examine the output...

    Sys.sleep(5)

    contour(as.integer(levels(year.seq)),
            gdp.seq,
            f,
            xlab="Year",
            ylab="GDP",
            main = "Density Contour Plot",
            col=topo.colors(100))

    # Sleep for 5 seconds so that we can examine the output...

    Sys.sleep(5)

    # Alternatively, you could use the plot() command (<ctrl>-C will
    # interrupt on *NIX systems, <esc> will interrupt on MS Windows
    # systems).

    plot(bw)

    detach(Italy)

    # EXAMPLE 2 (INTERFACE=FORMULA): For this example, we load the old
    # faithful geyser data and compute the density and distribution
    # functions.

    library("datasets")
    data("faithful")
    attach(faithful)

    # Note - this may take a few minutes depending on the speed of your
    # computer...

    bw <- npudensbw(formula=~eruptions+waiting)

    summary(bw)

    # Plot the density function (<ctrl>-C will interrupt on *NIX systems,
    # <esc> will interrupt on MS Windows systems). Note that we use xtrim =
    # -0.2 to extend the plot outside the support of the data (i.e., extend
    # the tails of the estimate to meet the horizontal axis).

    plot(bw, xtrim=-0.2)

    detach(faithful)

    # EXAMPLE 2 (INTERFACE=DATA FRAME): For this example, we load the old
    # faithful geyser data and compute the density and distribution
    # functions.

    library("datasets")
    data("faithful")
    attach(faithful)

    # Note - this may take a few minutes depending on the speed of your
    # computer...

    bw <- npudensbw(dat=faithful)

    summary(bw)

    # Plot the density function (<ctrl>-C will interrupt on *NIX systems,
    # <esc> will interrupt on MS Windows systems). Note that we use xtrim =
    # -0.2 to extend the plot outside the support of the data (i.e., extend
    # the tails of the estimate to meet the horizontal axis).

    plot(bw, xtrim=-0.2)

    detach(faithful)

    ## End(Not run)

[Package np version 0.60-9 Index]