rhohat: Nonparametric Estimate of Intensity as Function of a Covariate

Description

Computes a nonparametric estimate of the intensity of a point process, as a function of a (continuous) spatial covariate.

Usage

rhohat(object, covariate, ...)
# S3 method for ppp
rhohat(object, covariate, ...,
       baseline=NULL, weights=NULL,
       method=c("ratio", "reweight", "transform"),
       horvitz=FALSE,
       smoother=c("kernel", "local", "decreasing", "increasing",
       "mountain", "valley", "piecewise"),
       subset=NULL,
       do.CI=TRUE, 
       jitter=TRUE, jitterfactor=1, interpolate=TRUE,
       dimyx=NULL, eps=NULL,
       n = 512, bw = "nrd0", adjust=1, from = NULL, to = NULL,
       bwref=bw,
       covname, confidence=0.95, positiveCI, breaks=NULL)
# S3 method for quad
rhohat(object, covariate, ...,
       baseline=NULL, weights=NULL,
       method=c("ratio", "reweight", "transform"),
       horvitz=FALSE,
       smoother=c("kernel", "local", "decreasing", "increasing",
                  "mountain", "valley", "piecewise"),
       subset=NULL, 
       do.CI=TRUE, 
       jitter=TRUE, jitterfactor=1, interpolate=TRUE,
       dimyx=NULL, eps=NULL,
       n = 512, bw = "nrd0", adjust=1, from = NULL, to = NULL,
       bwref=bw,
       covname, confidence=0.95, positiveCI, breaks=NULL)

Value

A function value table (object of class "fv") containing the estimated values of $\rho$

(and confidence limits) for a sequence of values of $Z$. Also belongs to the class "rhohat"

which has special methods for print, plot

and predict.

Arguments

object: A point pattern (object of class "ppp" or "lpp"), a quadrature scheme (object of class "quad") or a fitted point process model (object of class "ppm", "slrm" or "lppm").
covariate: Either a function(x,y) or a pixel image (object of class "im") providing the values of the covariate at any location. Alternatively one of the strings "x" or "y" signifying the Cartesian coordinates.
weights: Optional weights attached to the data points. Either a numeric vector of weights for each data point, or a pixel image (object of class "im") or a function(x,y) providing the weights.
baseline: Optional baseline for intensity function. A function(x,y) or a pixel image (object of class "im") providing the values of the baseline at any location.
method: Character string determining the estimation method. See Details.
horvitz: Logical value indicating whether to use Horvitz-Thompson weights. See Details.
smoother: Character string determining the smoothing algorithm and the type of curve that will be estimated. See Details.
subset: Optional. A spatial window (object of class "owin") specifying a subset of the data, from which the estimate should be calculated.
do.CI: Logical value specifying whether to calculate standard errors and confidence bands.
jitter: Logical value. If jitter=TRUE (the default), the values of the covariate at the data points will be jittered (randomly perturbed by adding a small amount of noise) using the function jitter. If jitter=FALSE, the covariate values at the data points will not be altered. See the section on Randomisation and discretisation.
jitterfactor: Numeric value controlling the scale of noise added to the covariate values at the data points when jitter=TRUE. Passed to the function jitter as the argument factor.
interpolate: Logical value specifying whether to use spatial interpolation to obtain the values of the covariate at the data points, when the covariate is a pixel image (object of class "im"). If interpolate=FALSE, the covariate value for each data point is simply the value of the covariate image at the pixel centre that is nearest to the data point. If interpolate=TRUE, the covariate value for each data point is obtained by interpolating the nearest pixel values using interp.im.
dimyx,eps: Arguments controlling the pixel resolution at which the covariate will be evaluated. See Details.
bw: Smoothing bandwidth or bandwidth rule (passed to density.default).
adjust: Smoothing bandwidth adjustment factor (passed to density.default).
n, from, to: Arguments passed to density.default to control the number and range of values at which the function will be estimated.
bwref: Optional. An alternative value of bw to use when smoothing the reference density (the density of the covariate values observed at all locations in the window).
...: Additional arguments passed to density.default or locfit.
covname: Optional. Character string to use as the name of the covariate.
confidence: Confidence level for confidence intervals. A number between 0 and 1.
positiveCI: Logical value. If TRUE, confidence limits are always positive numbers; if FALSE, the lower limit of the confidence interval may sometimes be negative. Default is FALSE if smoother="kernel" and TRUE if smoother="local". See Details.
breaks: Breakpoints for the piecewise-constant function computed when smoother='piecewise'. Either a vector of numeric values specifying the breakpoints, or a single integer specifying the number of equally-spaced breakpoints. There is a sensible default.

Smooth estimates

Smooth estimators of $\rho(z)$ were proposed by Baddeley and Turner (2005) and Baddeley et al (2012). Similar estimators were proposed by Guan (2008) and in the literature on relative distributions (Handcock and Morris, 1999).

The estimated function $\rho(z)$ will be a smooth function of $z$.

The smooth estimation procedure involves computing several density estimates and combining them. The algorithm used to compute density estimates is determined by smoother:

If smoother="kernel", the smoothing procedure is based on fixed-bandwidth kernel density estimation, performed by density.default.
If smoother="local", the smoothing procedure is based on local likelihood density estimation, performed by locfit.

The argument method determines how the density estimates will be combined to obtain an estimate of $\rho(z)$:

If method="ratio", then $\rho(z)$ is estimated by the ratio of two density estimates, The numerator is a (rescaled) density estimate obtained by smoothing the values $Z(y_i)$ of the covariate $Z$ observed at the data points $y_i$. The denominator is a density estimate of the reference distribution of $Z$. See Baddeley et al (2012), equation (8). This is similar but not identical to an estimator proposed by Guan (2008).
If method="reweight", then $\rho(z)$ is estimated by applying density estimation to the values $Z(y_i)$ of the covariate $Z$ observed at the data points $y_i$, with weights inversely proportional to the reference density of $Z$. See Baddeley et al (2012), equation (9).
If method="transform", the smoothing method is variable-bandwidth kernel smoothing, implemented by applying the Probability Integral Transform to the covariate values, yielding values in the range 0 to 1, then applying edge-corrected density estimation on the interval $[0,1]$, and back-transforming. See Baddeley et al (2012), equation (10).

If horvitz=TRUE, then the calculations described above are modified by using Horvitz-Thompson weighting. The contribution to the numerator from each data point is weighted by the reciprocal of the baseline value or fitted intensity value at that data point; and a corresponding adjustment is made to the denominator.

Pointwise confidence intervals for the true value of $\rho(z)$ are also calculated for each $z$, and will be plotted as grey shading. The confidence intervals are derived using the central limit theorem, based on variance calculations which assume a Poisson point process. If positiveCI=FALSE, the lower limit of the confidence interval may sometimes be negative, because the confidence intervals are based on a normal approximation to the estimate of $\rho(z)$. If positiveCI=TRUE, the confidence limits are always positive, because the confidence interval is based on a normal approximation to the estimate of $\log(\rho(z))$. For consistency with earlier versions, the default is positiveCI=FALSE for smoother="kernel" and positiveCI=TRUE for smoother="local".

Monotone estimates

The nonparametric maximum likelihood estimator of a monotone function $\rho(z)$ was described by Sager (1982). This method assumes that $\rho(z)$ is either an increasing function of $z$, or a decreasing function of $z$. The estimated function will be a step function, increasing or decreasing as a function of $z$.

This estimator is chosen by specifying smoother="increasing" or smoother="decreasing". The argument method is ignored this case.

To compute the estimate of $\rho(z)$, the algorithm first computes several primitive step-function estimates, and then takes the maximum of these primitive functions.

If smoother="decreasing", each primitive step function takes the form $\rho(z) = \lambda$ when $z \le t$, and $\rho(z) = 0$ when $z > t$, where and $\lambda$ is a primitive estimate of intensity based on the data for $Z \le t$. The jump location $t$ will be the value of the covariate $Z$ at one of the data points. The primitive estimate $\lambda$ is the average intensity (number of points divided by area) for the region of space where the covariate value is less than or equal to $t$.

Confidence intervals are not available for the monotone estimators.

Unimodal estimators

If smoother="valley" then we estimate a U-shaped function. A function $\rho(z)$ is U-shaped if it is decreasing when $z < z_0$ and increasing when $z > z_0$, where $z_0$ is called the critical value. The nonparametric maximum likelihood estimate of such a function can be computed by profiling over $z_0$. The algorithm considers all possible candidate values of the critical value $z_0$, and estimates the function $\rho(z)$ separately on the left and right of $z_0$ using the monotone estimators described above. These function estimates are combined into a single function, and the Poisson point process likelihood is computed. The optimal value of $z_0$ is the one which maximises the Poisson point process likelihood.

If smoother="mountain" then we estimate a function which has an inverted U shape. A function $\rho(z)$ is inverted-U-shaped if it is increasing when $z < z_0$ and decreasing when $z > z_0$. The nonparametric maximum likelihood estimate of such a function can be computed by profiling over $z_0$ using the same technique mutatis mutandis.

Confidence intervals are not available for the unimodal estimators.

Randomisation

By default, rhohat adds a small amount of random noise to the data. This is designed to suppress the effects of discretisation in pixel images.

This strategy means that rhohat does not produce exactly the same result when the computation is repeated. If you need the results to be exactly reproducible, set jitter=FALSE.

By default, the values of the covariate at the data points will be randomly perturbed by adding a small amount of noise using the function jitter. To reduce this effect, set jitterfactor to a number smaller than 1. To suppress this effect entirely, set jitter=FALSE.

Author

Smoothing algorithm by Adrian Baddeley Adrian.Baddeley@curtin.edu.au, Ya-Mei Chang, Yong Song, and Rolf Turner r.turner@auckland.ac.nz.

Nonparametric maximum likelihood algorithm by Adrian Baddeley Adrian.Baddeley@curtin.edu.au.

Details

This command estimates the relationship between point process intensity and a given spatial covariate. Such a relationship is sometimes called a resource selection function (if the points are organisms and the covariate is a descriptor of habitat) or a prospectivity index (if the points are mineral deposits and the covariate is a geological variable). This command uses nonparametric methods which do not assume a particular form for the relationship.

If object is a point pattern, and baseline is missing or null, this command assumes that object is a realisation of a point process with intensity function $\lambda(u)$ of the form $$\lambda(u) = \rho(Z(u))$$ where $Z$ is the spatial covariate function given by covariate, and $\rho(z)$ is the resource selection function or prospectivity index. A nonparametric estimator of the function $\rho(z)$ is computed.

If object is a point pattern, and baseline is given, then the intensity function is assumed to be $$\lambda(u) = \rho(Z(u)) B(u)$$ where $B(u)$ is the baseline intensity at location $u$. A nonparametric estimator of the relative intensity $\rho(z)$ is computed.

If object is a fitted point process model, suppose X is the original data point pattern to which the model was fitted. Then this command assumes X is a realisation of a Poisson point process with intensity function of the form $$ \lambda(u) = \rho(Z(u)) \kappa(u) $$ where $\kappa(u)$ is the intensity of the fitted model object. A nonparametric estimator of the relative intensity $\rho(z)$ is computed.

The nonparametric estimation procedure is controlled by the arguments smoother, method and horvitz.

The argument smoother selects the type of estimation technique.

If smoother="kernel" (the default), the nonparametric estimator is a kernel smoothing estimator of $\rho(z)$ (Guan, 2008; Baddeley et al, 2012). The estimated function $\rho(z)$ will be a smooth function of $z$ which takes nonnegative values. If do.CI=TRUE (the default), confidence bands are also computed, assuming a Poisson point process. See the section on Smooth estimates.
If smoother="local", the nonparametric estimator is a local regression estimator of $\rho(z)$ (Baddeley et al, 2012) obtained using local likelihood. The estimated function $\rho(z)$ will be a smooth function of $z$. If do.CI=TRUE (the default), confidence bands are also computed, assuming a Poisson point process. See the section on Smooth estimates.
If smoother="increasing", we assume that $\rho(z)$ is an increasing function of $z$, and use the nonparametric maximum likelihood estimator of $\rho(z)$ described by Sager (1982). The estimated function will be a step function, that is increasing as a function of $z$. Confidence bands are not computed. See the section on Monotone estimates.
If smoother="decreasing", we assume that $\rho(z)$ is a decreasing function of $z$, and use the nonparametric maximum likelihood estimator of $\rho(z)$ described by Sager (1982). The estimated function will be a step function, that is decreasing as a function of $z$. Confidence bands are not computed. See the section on Monotone estimates.
If smoother="mountain", we assume that $\rho(z)$ is a function with an inverted U shape, with a single peak at a value $z_0$, so that $\rho(z)$ is an increasing function of $z$ for $z < z_0$ and a decreasing function of $z$ for $z > z_0$. We compute the nonparametric maximum likelihood estimator. The estimated function will be a step function, which is increasing and then decreasing as a function of $z$. Confidence bands are not computed. See the section on Unimodal estimates.
If smoother="valley", we assume that $\rho(z)$ is a function with a U shape, with a single minimum at a value $z_0$, so that $\rho(z)$ is a decreasing function of $z$ for $z < z_0$ and an increasing function of $z$ for $z > z_0$. We compute the nonparametric maximum likelihood estimator. The estimated function will be a step function, which is decreasing and then increasing as a function of $z$. Confidence bands are not computed. See the section on Unimodal estimates.
If smoother="piecewise", the estimate of $\rho(z)$ is piecewise constant. The range of covariate values is divided into several intervals (ranges or bands). The endpoints of these intervals are the breakpoints, which may be specified by the argument breaks; there is a sensible default. The estimate of $\rho(z)$ takes a constant value on each interval. The estimate of $\rho(z)$ in each interval of covariate values is simply the average intensity (number of points per unit area) in the relevant sub-region. If do.CI=TRUE (the default), confidence bands are computed assuming a Poisson process.

See Baddeley (2018) for a comparison of these estimation techniques (except for "mountain" and "valley").

If the argument weights is present, then the contribution from each data point X[i] to the estimate of $\rho$ is multiplied by weights[i].

If the argument subset is present, then the calculations are performed using only the data inside this spatial region.

This technique assumes that covariate has continuous values. It is not applicable to covariates with categorical (factor) values or discrete values such as small integers. For a categorical covariate, use intensity.quadratcount applied to the result of quadratcount(X, tess=covariate).

The argument covariate should be a pixel image, or a function, or one of the strings "x" or "y" signifying the cartesian coordinates. It will be evaluated on a fine grid of locations, with spatial resolution controlled by the arguments dimyx,eps which are passed to as.mask.

References

Baddeley, A., Chang, Y.-M., Song, Y. and Turner, R. (2012) Nonparametric estimation of the dependence of a point process on spatial covariates. Statistics and Its Interface 5 (2), 221--236.

Baddeley, A. and Turner, R. (2005) Modelling spatial point patterns in R. In: A. Baddeley, P. Gregori, J. Mateu, R. Stoica, and D. Stoyan, editors, Case Studies in Spatial Point Pattern Modelling, Lecture Notes in Statistics number 185. Pages 23--74. Springer-Verlag, New York, 2006. ISBN: 0-387-28311-0.

Baddeley, A. (2018) A statistical commentary on mineral prospectivity analysis. Chapter 2, pages 25--65 in Handbook of Mathematical Geosciences: Fifty Years of IAMG, edited by B.S. Daya Sagar, Q. Cheng and F.P. Agterberg. Springer, Berlin.

Guan, Y. (2008) On consistent nonparametric intensity estimation for inhomogeneous spatial point processes. Journal of the American Statistical Association 103, 1238--1247.

Handcock, M.S. and Morris, M. (1999) Relative Distribution Methods in the Social Sciences. Springer, New York.

Sager, T.W. (1982) Nonparametric maximum likelihood estimation of spatial patterns. Annals of Statistics 10, 1125--1136.

Examples

Run this code

  X <-  rpoispp(function(x,y){exp(3+3*x)})
  rho <- rhohat(X, "x")
  rho <- rhohat(X, function(x,y){x})
  plot(rho)
  curve(exp(3+3*x), lty=3, col=4, lwd=2, add=TRUE)

  rhoB <- rhohat(X, "x", method="reweight")
  rhoC <- rhohat(X, "x", method="transform")

  rhoI <- rhohat(X, "x", smoother="increasing")
  rhoM <- rhohat(X, "x", smoother="mountain")

  plot(rhoI, add=TRUE, .y ~ .x, col=6)
  legend("top", lty=c(3, 1), col=c(4, 6), lwd=c(2, 1),
         legend=c("true", "increasing"))

  rh <- rhohat(X, "x", dimyx=32)

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