# rhohat.lpp: Nonparametric Estimate of Intensity as Function of a... In spatstat.linnet: Linear Networks Functionality of the 'spatstat' Family

## Description

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

## Usage

 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23``` ```## S3 method for class 'lpp' rhohat(object, covariate, ..., weights=NULL, method=c("ratio", "reweight", "transform"), horvitz=FALSE, smoother=c("kernel", "local", "decreasing", "increasing", "piecewise"), subset=NULL, nd=1000, eps=NULL, random=TRUE, n = 512, bw = "nrd0", adjust=1, from = NULL, to = NULL, bwref=bw, covname, confidence=0.95, positiveCI, breaks=NULL) ## S3 method for class 'lppm' rhohat(object, covariate, ..., weights=NULL, method=c("ratio", "reweight", "transform"), horvitz=FALSE, smoother=c("kernel", "local", "decreasing", "increasing", "piecewise"), subset=NULL, nd=1000, eps=NULL, random=TRUE, n = 512, bw = "nrd0", adjust=1, from = NULL, to = NULL, bwref=bw, covname, confidence=0.95, positiveCI, breaks=NULL) ```

## Arguments

 `object` A point pattern on a linear network (object of class `"lpp"`), or a fitted point process model on a linear network (object of class `"lppm"`). `covariate` Either a `function(x,y)` or a pixel image (object of class `"im"` or `"linim"`) 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. `method` Character string determining the smoothing method. See Details. `horvitz` Logical value indicating whether to use Horvitz-Thompson weights. See Details. `smoother` Character string determining the smoothing algorithm. See Details. `subset` Optional. A spatial window (object of class `"owin"`) specifying a subset of the data, from which the estimate should be calculated. `eps,nd,random` 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::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.

## 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) or `smoother="local"`, the nonparametric estimator is a smoothing estimator of rho(z), effectively a kind of density estimator (Baddeley et al, 2012). The estimated function rho(z) will be a smooth function of z. Confidence bands are also computed, assuming a Poisson point process. See the section on Smooth estimates.

• If `smoother="increasing"` or `smoother="decreasing"`, we use the nonparametric maximum likelihood estimator of rho(z) described by Sager (1982). This 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. See the section on Monotone 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 length) in the relevant sub-region of the network.

See Baddeley (2018) for a comparison of these estimation techniques for two-dimensional point patterns.

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 `eps,nd,random`. The argument `nd` specifies the total number of test locations on the linear network, `eps` specifies the linear separation between test locations, and `random` specifies whether the test locations have a randomised starting position.

## 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`.

## 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::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 ρ(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 ρ(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(ρ(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 ≤ t, and rho(z) = 0 when z > t, where and lambda is a primitive estimate of intensity based on the data for Z <= 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.

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.

Confidence intervals are not available for the monotone estimators.

## Author(s)

Smoothing algorithm by \adrian, Ya-Mei Chang, Yong Song, and \rolf.

Nonparametric maximum likelihood algorithm by \adrian.

## 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.

`rho2hat`, `methods.rhohat`, `parres`.
See `lppm` for a parametric method for the same problem.
 ```1 2 3 4 5 6``` ``` Y <- runiflpp(30, simplenet) rhoY <- rhohat(Y, "y") ## do spiders prefer to be in the middle of a segment? teepee <- linfun(function(x,y,seg,tp){ tp }, domain(spiders)) rhotee <- rhohat(spiders, teepee) ```