Kernel Conditional Density Estimation with Mixed Data Types

npcdens computes kernel conditional density estimates on $p+q$-variate evaluation data, given a set of training data (both explanatory and dependent) and a bandwidth specification (a conbandwidth object or a bandwidth vector, bandwidth type, and kernel type) using the method of Hall, Racine, and Li (2004). The data may be continuous, discrete (unordered and ordered factors), or some combination thereof.

Usage

npcdens(bws, ...)

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


# S3 method for class 'conbandwidth'
npcdens(bws,
        txdat = stop("invoked without training data 'txdat'"),
        tydat = stop("invoked without training data 'tydat'"),
        exdat,
        eydat,
        gradients = FALSE,
        gradient.order = 1L,
        proper = FALSE,
        proper.method = c("project"),
        proper.control = list(),
        ...,
        se = FALSE)

# Default S3 method
npcdens(bws, txdat, tydat, nomad = FALSE, ...)

Arguments

Data, Bandwidth Inputs And Formula Interface

These arguments identify the bandwidth specification, formula/data interface, and training data.

bws

a bandwidth specification. This can be set as a conbandwidth object returned from a previous invocation of npcdensbw, or as a $p+q$-vector of bandwidths, with each element $i$ up to $i=q$ corresponding to the bandwidth for column $i$ in tydat, and each element $i$ from $i=q+1$ to $i=p+q$ corresponding to the bandwidth for column $i-q$ in txdat. If specified as a vector, then additional arguments will need to be supplied as necessary to specify the bandwidth type, kernel types, training data, and so on.

data

This argument belongs to the formula interface (including supported formula-bearing bandwidth-object refits). Native calls must supply their vectors, matrices or data frames directly, or use with(data, ...). Supplying data, including explicit data = NULL, to a native route that does not consume it is an error; see np-data-interfaces.

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 npcdensbw was called.

txdat

a $p$-variate data frame of sample realizations of explanatory data (training data). Defaults to the corresponding training data recoverable from the bandwidth object in the current session.

tydat

a $q$-variate data frame of sample realizations of dependent data (training data). Defaults to the corresponding training data recoverable from the bandwidth object in the current session.

Local-Polynomial Degree And Bandwidth Search

This argument controls the recommended automatic local-polynomial NOMAD route, which jointly selects continuous polynomial degree and bandwidths when these are computed inside npcdens.

nomad: logical shortcut passed through to npcdensbw when bandwidths are computed inside npcdens. When TRUE, the bandwidth route fills any missing values among regtype, search.engine, degree.select, bernstein.basis, degree.min, degree.max, degree.verify, and bwtype with the recommended automatic local-polynomial degree-and-bandwidth NOMAD preset documented in npcdensbw. Additional NOMAD tuning arguments such as nomad.nmulti may also be supplied through ...; nmulti remains the outer restart count while nomad.nmulti controls inner native crs NOMAD API multistarts within each outer restart. After fitting, inspect fit$bws$nomad.shortcut on the returned object fit to see the normalized shortcut metadata.

Evaluation Data And Returned Quantities

These arguments control where the fitted conditional density is evaluated and which estimates are returned.

exdat: a $p$-variate data frame of explanatory data on which conditional densities will be evaluated. By default, evaluation takes place on the data provided by txdat.
eydat: a $q$-variate data frame of dependent data on which conditional densities will be evaluated. By default, evaluation takes place on the data provided by tydat.
se: logical scalar, default FALSE. If TRUE, compute and retain available asymptotic standard errors of fitted values and, when gradients = TRUE, of the requested gradients. If FALSE, skip uncertainty-only computation; point estimates and gradients are unchanged. Unavailable derivative errors remain NA when requested. To obtain errors without repeating bandwidth selection, refit using npcdens(bws = fit$bws, se = TRUE); also supply gradients = TRUE for gradient errors. Explicit training data may be supplied if they are no longer recoverable from the bandwidth object. The argument must be named.
gradients: a logical value specifying whether to return estimates of the gradients at the evaluation points. Defaults to FALSE.
gradient.order: derivative order for continuous explanatory-variable gradients when gradients = TRUE and regtype = "lp". A scalar is recycled across continuous explanatory variables, or one value may be supplied per continuous explanatory variable. The default 1L returns first derivatives. Higher orders are available for continuous explanatory variables only and must not exceed the corresponding local-polynomial degree, apart from the established all-degree-zero first-derivative compatibility route. An unavailable continuous component is returned as NA; plotting retains it as an empty panel when another continuous derivative or categorical effect is available. A request for which no component is available is rejected. Unordered and ordered predictors retain their usual first-order discrete effects.
newdata: An optional data frame in which to look for evaluation data. If omitted, the training data are used.

Fit Properization Controls

These arguments control optional post-estimation properization of the fitted conditional density.

proper: a logical value specifying whether to apply post-estimation properization to the conditional density estimate. Defaults to FALSE. Local-constant or degree-zero fits bypass correction only when their kernels and bandwidth geometry guarantee nonnegative, unit-mass response components and nonnegative predictor weights. Signed kernels, generalized-nearest-neighbour response bandwidths and target-centred bounded or beta response densities do not have this guarantee. Requested correction then uses the existing supported grid or slice procedure, with its additional evaluation cost and support requirements.
proper.control: a list of controls for properization. Supported entries are tol, grid.check, store.raw, and fail.on.unsupported. Resolved controls are retained in fitted objects and inherited by predict() and plot() unless explicitly replaced. An explicit control list replaces the retained list; unspecified entries then use their defaults. Prediction still errors on unsupported correction unless fail.on.unsupported=FALSE is explicitly supplied.
proper.method: a character string specifying the properization method. Currently "project" is supported.

Additional Arguments

Further arguments are passed to npcdensbw when bandwidths are computed internally, or used to interpret a numeric bws vector.

...

additional arguments supplied to npcdensbw when npcdens computes bandwidths internally, or arguments needed to interpret a numeric bws vector. This is where bandwidth selection controls such as bwmethod, bwtype, kernel/support controls such as cxkertype, cykertype, cxkerbound, and cykerbound, search controls such as nmulti, scale.factor.search.lower, and nomad.nmulti, and local-polynomial controls such as regtype, degree, basis, and bernstein.basis are supplied. See npcdensbw for the complete bandwidth-selection argument surface.

Details

Conditional fitting and replay use the literal categorical kernels for both explanatory and response variables. Near-upper categorical bandwidths are not replaced by approximate constant factors. The same target is used for levels, gradients, standard errors, conditional hats, and reconstructed plot uncertainty. This conditional-family policy does not change direct regression or unconditional kernel-sum evaluation and does not modify the caller's np.largelambda option.

With generalized nearest-neighbor bandwidths, query identity matters when selecting a radius: a training observation excludes its own occurrence, whereas explicit evaluation data remain external queries even if their values equal training observations. Consequently, predict(fit, newdata = training.data) can differ from fitted(fit). This does not omit the training contribution from the fitted estimator. See np.kernels for the radius convention.

With a formula-derived bandwidth object, supplying only tydat replaces the evaluated response in retained-training-sample order: one row per observation retained after the original subset and missing-value selection. Original excluded rows are not restored; row names or equal lengths do not imply original-data order. The effective na.action is then applied jointly to the retained predictors and replacement response; any new omission indices refer to this retained sample. The selected bandwidths remain fixed, and the returned object retains the replacement sample for subsequent refits. Use data for full formula-data replacement, not together with a partial response replacement. Explicit native training arguments retain their separate positional contract.

For native (non-formula) objects, named newdata columns are matched to the retained variable names and reordered as needed. Extra columns are ignored; missing, duplicate or ambiguous names are errors. An unnamed matrix or vector remains positional for a single evaluation role. Explicit native evaluation arguments retain their positional contract and take precedence over newdata.

For a native bandwidth object, newdata is split into exdat and eydat using the retained predictor and response column names. Names must identify both roles unambiguously; explicitly supplied native evaluation arguments take precedence. This evaluates at the requested rows without repeating bandwidth selection.

For explicit external evaluation rows, a completed explanatory-kernel row whose weights are all zero has no conditional estimate. Such rows return NA for the fitted value and requested uncertainty or derivatives, with one notice. A missing categorical counterfactual affects only its contrast; it does not invalidate an otherwise supported base estimate. A zero response-kernel contribution is not a missing explanatory row. Supported rows retain their usual computation. Required internal evaluations remain strict. With proper = TRUE, complete unsupported fixed-X slices retain NA; projection still applies to supported slices.

Documentation guide: see npcdensbw for bandwidth selection and search controls, np.kernels for kernels, np.options for global options, and plot, plot.np for plotting options.

When bws is omitted, the formula and default methods call npcdensbw first and pass bandwidth-selection arguments from ... to that call. When bws is already a conbandwidth object, npcdens estimates with the stored bandwidth metadata in that object.

Beta associated kernels may be selected independently for explanatory X coordinates (cxkertype="beta") and response Y coordinates (cykertype="beta"). Orders 2, 4, 6, and 8, fixed/generalized-NN/ adaptive-NN bandwidths, explicit finite bounds, sample-range bounds, and automatic cv.ml/cv.ls bandwidth selection are supported. Beta kernels act on continuous coordinates and can coexist with supported categorical X or Y coordinates using their respective categorical kernels. Local-constant, local-linear and local-polynomial fits are supported. For local-constant fits, analytic first derivatives with respect to the continuous X coordinates are available with gradients=TRUE, including beta-X/legacy-Y, legacy-X/beta-Y, fixed, generalized-NN, and adaptive-NN combinations. NN bandwidths are held fixed while differentiating. Exact beta-X endpoint jumps are cancelled in the normalized ratio before a gradient is classified; genuine jumps are signed infinities, while indeterminate cancellation is NA, both with an explanatory warning. Returned infinities retain that extended-real meaning. Plot methods convert them to missing geometry only in a temporary drawing copy, leaving fitted and plot-data objects unchanged. For ordinary conditional-density level plots, bootstrap intervals support different continuous X- and Y-side kernel families or orders, including beta on either side. Beta-containing bootstrap levels use the same signed log-domain conditional ratio as the fitted estimator. With non-fixed bandwidths, np_boot_control(nonfixed="exact") reconstructs each duplicate-row resample and recomputes its nearest-neighbour bandwidths, while nonfixed="frozen" keeps the original fitted bandwidths. In both modes, resolved beta "range" support bounds are replayed from the bandwidth object rather than recomputed from bootstrap counts. This mixed-family level-bootstrap support does not alter the separate restrictions on gradient, quantile, or bias-corrected centers. Higher-order fits are signed and need not be proper densities; no positivity repair is applied. Bounded beta-response cv.ls integrates the actual conditional estimator over the Y support and does not replace that integral by the public observation-coordinate beta overlap.

Argument groups for bandwidth selection are documented on npcdensbw. The most common workflow is to choose data and bandwidth inputs first, then bandwidth criterion and representation, then kernel/support controls, numerical search controls, bounded cv.ls quadrature controls if relevant, and finally local-polynomial/NOMAD controls for polynomial-adaptive fits.

For S3 plotting help, see plot.np. You can list available plot methods with methods("plot").

npcdens implements a variety of methods for estimating multivariate conditional distributions ($p+q$-variate) defined over a set of possibly continuous and/or discrete (unordered, ordered) 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.

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 $x$.

Training and evaluation input data 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 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.

For practitioners who want the recommended automatic local-polynomial degree-and-bandwidth NOMAD route without spelling out all LP tuning arguments, npcdens(..., nomad=TRUE) and npcdensbw(..., nomad=TRUE) expand missing settings to the same documented preset. Explicit incompatible settings fail fast rather than being silently rewritten.

With a raw polynomial basis, high powers of poorly scaled continuous predictors (such as uncentred calendar years) can produce severely ill-conditioned systems. Conditional local-polynomial fits do not apply the design-conditioning rejection used by npreg; they may return inaccurate finite values without a warning, with or without a ridge adjustment. Centre or scale such predictors before fitting and apply the same transformation to all evaluation or prediction data. The explicit bernstein.basis = TRUE representation is another option for improving conditioning. Check numerical stability for the chosen data, degree and bandwidths. proper = TRUE enforces probability-shape constraints; it does not repair an ill-conditioned polynomial solve.

With regtype = "lp", gradients = TRUE and gradient.order greater than one expose higher-order derivative estimates with respect to continuous explanatory variables. These derivatives use the same local-polynomial basis, degree, Bernstein option, bandwidths, and kernels as the fitted conditional density. See npreg for the complete heterogeneous GLP multi-index and Bernstein-representation definitions shared by these routes. With se = TRUE, available higher-order derivative columns for fixed and adaptive nearest-neighbour bandwidths also have standard errors. These combine the existing local pseudo-response variance with the squared norm of the actual derivative weighting row, including its accepted ridge adjustment. They condition on the selected bandwidths, neighbour radii and local-polynomial map; they do not account for their selection uncertainty. Generalized nearest-neighbour higher-order point derivatives remain subject to the existing hat-operator restriction. Bootstrap inference remains recommended.

With positive-degree local-polynomial fits (including local-linear fits), numerical ridging follows the policies described in npreg (Seifert and Gasser, 2000). The conditional pseudo-response uncertainty formulas here are distinct from ordinary regression's residual normalization.

With positive-degree local-polynomial fits (including local-linear fits), categorical first-difference standard errors use paired endpoint influence calculations. The covariance between the two category endpoints is retained, not replaced by the sum of their marginal variances. These calculations condition on the selected basis, smoothing parameters and realized neighbour radii for all three bandwidth types. They run only when both gradients = TRUE and se = TRUE request them. A nonsmooth regularization boundary or nonfinite influence gives NA uncertainty with a notice, without changing the point contrast.

For ordinary scalar fits, leading interior standard errors use the squared X-kernel moment; conditional density errors additionally use the independent Y-kernel moment. Analytic first-derivative errors for fixed and generalized nearest-neighbour bandwidths with Gaussian or Epanechnikov kernels use the squared derivative-kernel moment and the X bandwidth. These formulas do not establish uniform-kernel derivative inference or full adaptive-NN inference with estimated random radii. Positive-degree local-polynomial fits use their separate covariance calculation.

For fixed and generalized nearest-neighbour scalar fits with categorical variables, these leading moments retain squared categorical kernel factors and the joint training-category frequencies. Conditional distributions, and conditional densities with no continuous response, retain the numerator–denominator covariance of the ratio estimator. In particular, a constant response-kernel contribution has zero conditional variance. Purely categorical fits use this ratio covariance without a continuous bandwidth factor, including nearest-neighbour-labelled bandwidth objects. Beta-kernel and positive-degree local-polynomial covariance calculations retain their separate targets.

For scalar adaptive nearest-neighbour fits with any continuous variable, requested level and available continuous first-derivative standard errors use the complete empirical influence of the conditional ratio. They retain the actual normalized X weights and continuous/categorical Y contributions, including numerator and denominator derivatives. This working covariance holds the selected model, observed X geometry and realized response nearest-neighbour radii fixed; it does not account for their sampling or selection uncertainty. Local-constant-equivalent degree-zero fits share this calculation, including continuous responses with categorical-only X. Positive-degree local-polynomial, beta-kernel and paired categorical- contrast errors retain their separate covariance calculations. With se = FALSE, the additional uncertainty pass is not performed.

Value

The se component records whether uncertainty was requested. conderr and congerr are NULL when se = FALSE. se() and gradients(..., se = TRUE) only extract stored results and give a no-search refit message when the errors were omitted. predict(..., se.fit = TRUE) explicitly requests the needed errors.

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

xbw: bandwidth(s), scale factor(s) or nearest neighbours for the explanatory data, txdat
ybw: bandwidth(s), scale factor(s) or nearest neighbours for the dependent data, tydat
xeval: the evaluation points of the explanatory data
yeval: the evaluation points of the dependent data
condens: estimates of the conditional density at the evaluation points. An evaluation row whose required explanatory normalization is exactly zero or non-finite is returned as NA; other rows remain available.
conderr: when se = TRUE, standard errors of the conditional density estimates. Undefined normalization rows are returned as NA.
congrad: if invoked with gradients = TRUE, estimates of the gradients at the evaluation points. For local-polynomial fits, continuous-coordinate derivative orders are controlled by gradient.order. Undefined normalization rows are returned as NA.
congerr: if invoked with gradients = TRUE and se = TRUE, standard errors of the gradients at the evaluation points, including available higher-order continuous derivatives under the working-covariance assumptions described above. Undefined normalization or uncertainty rows are returned as NA.
log_likelihood: aggregate fitted conditional-density log likelihood. Every represented strictly positive estimate, including a positive subnormal value, contributes its literal logarithm. An exact zero or negative estimate has no mathematical logarithm and is represented in this aggregate by the established computational placeholder log(DBL_MIN). Consequently this component is not a literal log likelihood if any fitted density is nonpositive.

Book And Method Pointers

The conditional density target is $f(y\mid x)$. In the product-kernel formulation this can be viewed as estimating the joint mixed-data density of $(Y,X)$ and normalizing by the marginal density of $X$, with local-polynomial routes using the selected continuous-coordinate polynomial degree in the conditional fit. The requested standard errors are asymptotic standard errors for the fitted conditional density values, and gradient standard errors are returned when gradients are requested and defined.

Scalar conditional fits involving beta kernels use the empirical observation-influence covariance. For normalized explanatory weights $a_i$, response-kernel contributions $z_i$, and fitted value $m$, the level variance is $\{n/(n-1)\}\sum_i [a_i(z_i-m)]^2$, using the full training count $n$. Supported continuous-derivative errors use the corresponding differentiated influence. Categorical contrasts use the difference of the two endpoints' observation influences inside the same squared norm, thereby retaining their covariance; self-contrasts have zero variance. These calculations hold the selected bandwidth state fixed; they do not add bandwidth-selection or random-NN-radius uncertainty, and do not replace positive-degree LP covariance.

For book-length derivations, see Li and Racine (2007), Chapter 5 Conditional Density Estimation, especially Sections 5.1-5.4, and Chapter 4 Kernel Estimation with Mixed Data. The later workflow treatment is Racine (2019), Chapter 4 Conditional Probability Density and Cumulative Distribution Functions.

References

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

Chen, S.X. (1999), “Beta kernel estimators for density functions,” Computational Statistics & Data Analysis, 31(2), 131–145. doi:10.1016/S0167-9473(99)00010-9.

Igarashi, G. and Kakizawa, Y. (2020), “Higher-order bias corrections for asymmetric kernel density estimators,” Journal of Nonparametric Statistics, 32(3), 716–752. doi:10.1080/10485252.2020.1770754.

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.

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.

Author

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

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.

For formula fits, character-valued missing-data policies are resolved through the stats model-frame owner; pass a caller-local policy as a function object rather than its name. With replacement training data, an explicit na.action, including NULL, takes precedence over the saved policy. Native fitting still requires complete rows, so a permissive policy does not guarantee the same retained sample as stats::model.frame.

When reusing a bandwidth object without explicit training or evaluation data, newly constructed bandwidth objects use retained training values. A fit with explicit replacement training data retains that sample in its returned bandwidth object; evaluation grids do not replace training data. Older objects without retained samples use their stored call environment when available and may require explicit training inputs after saveRDS() and readRDS(). Custom formula transformations used on new evaluation data must still be available.

Examples

if (FALSE) { # \dontrun{
# EXAMPLE 1 (INTERFACE=FORMULA): For this example, we load Giovanni
# Baiocchi's Italian GDP panel (see Italy for details), and compute the
# likelihood 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")
Italy <- Italy[seq_len(min(300, nrow(Italy))), ]
with(Italy, {

# First, compute the bandwidths... note that this may take a minute or
# two depending on the speed of your computer.

bw <- npcdensbw(formula=gdp~ordered(year), nmulti=1)

# Next, compute the condensity object...

fhat <- npcdens(bws=bw)

# The object fhat now contains results such as the estimated conditional
# density function (fhat$condens) and so on...

summary(fhat)

# Call the plot() function to visualize the results (<ctrl>-C will
# interrupt on *NIX systems, <esc> will interrupt on MS Windows
# systems).

if (interactive()) plot(bw)

})

# EXAMPLE 1 (INTERFACE=DATA FRAME): For this example, we load Giovanni
# Baiocchi's Italian GDP panel (see Italy for details), and compute the
# likelihood 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")
Italy <- Italy[seq_len(min(300, nrow(Italy))), ]
with(Italy, {

# First, compute the bandwidths... note that this may take a minute or
# two depending on the speed of your computer.

# Note - we cast `X' and `y' as data frames so that plot() can
# automatically grab names (this looks like overkill, but in
# multivariate settings you would do this anyway, so may as well get in
# the habit).

X <- data.frame(year=ordered(year))
y <- data.frame(gdp)

bw <- npcdensbw(xdat=X, ydat=y, nmulti=1)

# Next, compute the condensity object...

fhat <- npcdens(bws=bw)

# The object fhat now contains results such as the estimated conditional
# density function (fhat$condens) and so on...

summary(fhat)

# Call the plot() function to visualize the results (<ctrl>-C will
# interrupt on *NIX systems, <esc> will interrupt on MS Windows systems).

if (interactive()) plot(bw)

})

# EXAMPLE 2 (INTERFACE=FORMULA): For this example, we load the old
# faithful geyser data from the R `datasets' library and compute the
# conditional density function.

library("datasets")
data("faithful")
with(faithful, {

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

bw <- npcdensbw(formula=eruptions~waiting, nmulti=1)

summary(bw)

# Plot the density function (<ctrl>-C will interrupt on *NIX systems,
# <esc> will interrupt on MS Windows systems).

if (interactive()) plot(bw)

})

# EXAMPLE 2 (INTERFACE=DATA FRAME): For this example, we load the old
# faithful geyser data from the R `datasets' library and compute the
# conditional density function.

library("datasets")
data("faithful")
with(faithful, {

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

# Note - we cast `X' and `y' as data frames so that plot() can
# automatically grab names (this looks like overkill, but in
# multivariate settings you would do this anyway, so may as well get in
# the habit).

X <- data.frame(waiting)
y <- data.frame(eruptions)

bw <- npcdensbw(xdat=X, ydat=y, nmulti=1)

summary(bw)

# Plot the density function (<ctrl>-C will interrupt on *NIX systems,
# <esc> will interrupt on MS Windows systems)

if (interactive()) plot(bw)

})

# EXAMPLE 3 (INTERFACE=FORMULA): Replicate the DGP of Klein & Spady
# (1993) (see their description on page 405, pay careful attention to
# footnote 6 on page 405).

set.seed(123)

n <- 300

# x1 is chi-squared having 3 df truncated at 6 standardized by
# subtracting 2.348 and dividing by 1.511

x <- rchisq(n, df=3)
x1 <- (ifelse(x < 6, x, 6) - 2.348)/1.511

# x2 is normal (0, 1) truncated at +- 2 divided by 0.8796

x <- rnorm(n)
x2 <- ifelse(abs(x) < 2 , x, 2) / 0.8796

# y is 1 if y* > 0, 0 otherwise.

y <- ifelse(x1 + x2 + rnorm(n) > 0, 1, 0)

data.train <- data.frame(y = factor(y), x1 = x1, x2 = x2)

# Generate data-driven bandwidths (likelihood cross-validation). Note -
# this may take a few minutes depending on the speed of your computer...

bw <- npcdensbw(formula=y~x1+x2, data=data.train, nmulti=1)

# Next, create the evaluation data in order to generate a perspective
# plot

x1.seq <- seq(min(x1), max(x1), length=30)
x2.seq <- seq(min(x2), max(x2), length=30)
X.eval <- expand.grid(x1=x1.seq,x2=x2.seq)

data.eval <- data.frame(
  y=factor(rep(1, nrow(X.eval)), levels=levels(data.train$y)),
  x1=X.eval[,1],
  x2=X.eval[,2]
)

# Now evaluate the conditional probability for y=1 and for the
# evaluation Xs

fit <- fitted(npcdens(bws=bw,newdata=data.eval))

# Finally, coerce the data into a matrix for plotting with persp()

fit.mat <- matrix(fit, 30, 30)

# Generate a perspective plot similar to Figure 2 b of Klein and Spady
# (1993)

persp(x1.seq,
      x2.seq,
      fit.mat,
      col="white",
      ticktype="detailed",
      expand=0.5,
      axes=FALSE,
      box=FALSE,
      main="Estimated Nonparametric Probability Perspective",
      theta=310,
      phi=25)

# EXAMPLE 3 (INTERFACE=DATA FRAME): Replicate the DGP of Klein & Spady
# (1993) (see their description on page 405, pay careful attention to
# footnote 6 on page 405).

set.seed(123)

n <- 300

# x1 is chi-squared having 3 df truncated at 6 standardized by
# subtracting 2.348 and dividing by 1.511

x <- rchisq(n, df=3)
x1 <- (ifelse(x < 6, x, 6) - 2.348)/1.511

# x2 is normal (0, 1) truncated at +- 2 divided by 0.8796

x <- rnorm(n)
x2 <- ifelse(abs(x) < 2 , x, 2) / 0.8796

# y is 1 if y* > 0, 0 otherwise.

y <- ifelse(x1 + x2 + rnorm(n) > 0, 1, 0)

# Create the X matrix

X <- cbind(x1, x2)
y.factor <- factor(y)

# Generate data-driven bandwidths (likelihood cross-validation). Note -
# this may take a few minutes depending on the speed of your computer...

bw <- npcdensbw(xdat=X, ydat=y.factor, nmulti=1)

# Next, create the evaluation data in order to generate a perspective
# plot

x1.seq <- seq(min(x1), max(x1), length=30)
x2.seq <- seq(min(x2), max(x2), length=30)
X.eval <- expand.grid(x1=x1.seq,x2=x2.seq)

# Now evaluate the conditional probability for y=1 and for the
# evaluation Xs

fit <- fitted(npcdens(exdat=X.eval,
               eydat=factor(rep(1, nrow(X.eval)), levels=levels(y.factor)),
               bws=bw))

# Finally, coerce the data into a matrix for plotting with persp()

fit.mat <- matrix(fit, 30, 30)

# Generate a perspective plot similar to Figure 2 b of Klein and Spady
# (1993)

persp(x1.seq,
      x2.seq,
      fit.mat,
      col="white",
      ticktype="detailed",
      expand=0.5,
      axes=FALSE,
      box=FALSE,
      main="Estimated Nonparametric Probability Perspective",
      theta=310,
      phi=25)

# EXAMPLE 4: Variations on local polynomial conditional density
# estimation with proper = TRUE.

data("Italy")

# Work in years relative to 1975 to condition the raw polynomial basis.
# Calendar year = model year + year.offset, also for prediction data.
year.offset <- 1975
Italy2 <- within(Italy, {
  year <- as.numeric(as.character(year)) - year.offset
})

# Plot only: make the plotted surface proper on the plot evaluation grid.

fhat <- npcdens(gdp ~ year, data = Italy2,
                regtype = "lp", degree = 3, nmulti = 1)

plot(fhat, proper = TRUE)

# Fit an object whose fitted values are themselves proper.

ctrl_fit <- list(
  mode = "slice",
  apply = "fitted",
  slice.grid.size = 101L,
  slice.extend.factor = 0.1
)

fhat_fit <- npcdens(
  gdp ~ year,
  data = Italy2,
  regtype = "lp",
  degree = 3,
  nmulti = 1,
  proper = TRUE,
  proper.control = ctrl_fit
)

fit_proper <- fitted(fhat_fit)
fit_raw <- fhat_fit$condens.raw

# Display the repaired and raw fitted values for cases where the raw
# fitted density is negative.

head(cbind(fit_proper, fit_raw)[which(fit_raw < 0), ])

# Predict on a common explicit y-grid for several years, and render
# those predictions proper.

g.grid <- seq(min(Italy2$gdp), max(Italy2$gdp), length.out = 200)

nd_grid <- expand.grid(
  gdp = g.grid,
  year = c(1955, 1975, 1995) - year.offset
)

pred_grid <- predict(fhat, newdata = nd_grid, proper = TRUE)

# Predict on paired rows with different gdp grids by year, and still
# make the predictions proper via slice mode.

g1 <- seq(quantile(Italy2$gdp, 0.10),
          quantile(Italy2$gdp, 0.60), length.out = 60)
g2 <- seq(quantile(Italy2$gdp, 0.30),
          quantile(Italy2$gdp, 0.90), length.out = 35)

nd_slice <- rbind(
  data.frame(gdp = g1, year = rep(1960 - year.offset, length(g1))),
  data.frame(gdp = g2, year = rep(1985 - year.offset, length(g2)))
)

pred_slice <- predict(
  fhat,
  newdata = nd_slice,
  proper = TRUE,
  proper.control = list(mode = "slice")
)

# One object that carries properization for fitted values and for later
# predict() calls.

ctrl_both <- list(
  mode = "slice",
  apply = "both",
  slice.grid.size = 101L,
  slice.extend.factor = 0.1
)

fhat_both <- npcdens(
  gdp ~ year,
  data = Italy2,
  regtype = "lp",
  degree = 3,
  nmulti = 1,
  proper = TRUE,
  proper.control = ctrl_both
)

fit_both <- fitted(fhat_both)
pred_both <- predict(
  fhat_both,
  newdata = nd_slice,
  proper.control = ctrl_both
)
} # }