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spatstat (version 1.17-6)

Spatial Point Pattern analysis, model-fitting, simulation, tests

Description

A package for analysing spatial data, mainly Spatial Point Patterns, including multitype/marked points and spatial covariates, in any two-dimensional spatial region. Also supports three-dimensional point patterns. Contains functions for plotting spatial data, exploratory data analysis, model-fitting, simulation, spatial sampling, model diagnostics, and formal inference. Data types include point patterns, line segment patterns, spatial windows, pixel images and tessellations. Point process models can be fitted to point pattern data. Cluster type models are fitted by the method of minimum contrast. Very general Gibbs point process models can be fitted to point pattern data using a function ppm similar to lm or glm. Models may include dependence on covariates, interpoint interaction and dependence on marks. Fitted models can be simulated automatically. Also provides facilities for formal inference (such as chi-squared tests) and model diagnostics (including simulation envelopes, residuals, residual plots and Q-Q plots).

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Version

Install

install.packages('spatstat')

Monthly Downloads

16,015

Version

1.17-6

License

GPL (>= 2)

Maintainer

Adrian Baddeley

Last Published

February 9th, 2010

Functions in spatstat (1.17-6)

Gdot

Multitype Nearest Neighbour Distance Function (i-to-any)
G3est

Nearest Neighbour Distance Distribution Function of a Three-Dimensional Point Pattern
Kest

K-function
Kmulti

Marked K-Function
OrdThresh

Ord's Interaction model
Saturated

Saturated Pairwise Interaction model
area.owin

Area of a Window
bronzefilter

Bronze gradient filter data
Extract.splitppp

Extract or Replace Sub-Patterns
cells

Biological Cells Point Pattern
concatxy

Concatenate x,y Coordinate Vectors
Iest

Estimate the I-function
anova.ppm

ANOVA for Fitted Point Process Models
im.object

Class of Images
K3est

K-function of a Three-Dimensional Point Pattern
Pairwise

Generic Pairwise Interaction model
Jest

Estimate the J-function
as.polygonal

Convert a Window to a Polygonal Window
DiggleGratton

Diggle-Gratton model
Lest

L-function
bei

Tropical rain forest trees
SatPiece

Piecewise Constant Saturated Pairwise Interaction Point Process Model
blur

Apply Gaussian Blur to a Pixel Image
as.hyperframe

Convert Data to Hyperframe
F3est

Empty Space Function of a Three-Dimensional Point Pattern
clarkevans

Clark and Evans Aggregation Index
Extract.fv

Extract Subset of Function Values
progressreport

Print Progress Reports
alltypes

Calculate Summary Statistic for All Types in a Multitype Point Pattern
Kest.fft

K-function using FFT
qqplot.ppm

Q-Q Plot of Residuals from Fitted Point Process Model
quadratresample

Resample a Point Pattern by Resampling Quadrats
chorley

Chorley-Ribble Cancer Data
as.hyperframe.ppx

Extract coordinates and marks of multidimensional point pattern
affine.owin

Apply Affine Transformation To Window
by.ppp

Apply a Function to a Point Pattern Broken Down by Factor
as.matrix.im

Convert Pixel Image to Matrix
rNeymanScott

Simulate Neyman-Scott Process
as.data.frame.ppp

Coerce Point Pattern to a Data Frame
rstrat

Simulate Stratified Random Point Pattern
crossdist.ppp

Pairwise distances between two different point patterns
bermantest

Berman's Tests for Point Process Model
Lcross

Multitype L-function (cross-type)
coef.ppm

Coefficients of Fitted Point Process Model
eroded.areas

Areas of Morphological Erosions
append.psp

Combine Two Line Segment Patterns
closing

Morphological Closing
bdist.pixels

Distance to Boundary of Window
Extract.fasp

Extract Subset of Function Array
Gest

Nearest Neighbour Distance Function G
Extract.psp

Extract Subset of Line Segment Pattern
bdist.tiles

Distance to Boundary of Window
bind.fv

Combine Function Value Tables
crossdist.psp

Pairwise distances between two different line segment patterns
diameter

Diameter of a Window
diagnose.ppm

Diagnostic Plots for Fitted Point Process Model
Kinhom

Inhomogeneous K-function
complement.owin

Take Complement of a Window
heather

Diggle's Heather Data
fv

Create a Function Value Table
Softcore

The Soft Core Point Process Model
ants

Harkness-Isham ants' nests data
is.convex

Test Whether a Window is Convex
Kcross.inhom

Inhomogeneous Cross K Function
lut

Lookup Tables
betacells

Beta Ganglion Cells in Cat Retina
colourmap

Colour Lookup Tables
dummy.ppm

Extract Dummy Points Used to Fit a Point Process Model
PairPiece

The Piecewise Constant Pairwise Interaction Point Process Model
affine.ppp

Apply Affine Transformation To Point Pattern
affine

Apply Affine Transformation
inforder.family

Infinite Order Interaction Family
diameter.box3

Geometrical Calculations for Three-Dimensional Box
bounding.box

Bounding Box of a Window or Point Pattern
connected

Connected components of an image or window
data.ppm

Extract Original Data from a Fitted Point Process Model
demopat

Artificial Data Point Pattern
levelset

Level Set of a Pixel Image
clickppp

Interactively Add Points
shift.im

Apply Vector Translation To Pixel Image
duplicated.ppp

Determine Duplicated Points in a Spatial Point Pattern
bounding.box.xy

Convex Hull of Points
setcov

Set Covariance of a Window
fitted.ppm

Fitted Conditional Intensity for Point Process Model
disc

Circular Window
nbfires

Point Patterns of New Brunswick Forest Fires
Extract.ppp

Extract or Replace Subset of Point Pattern
shift.psp

Apply Vector Translation To Line Segment Pattern
Ldot

Multitype L-function (i-to-any)
MultiStraussHard

The Multitype/Hard Core Strauss Point Process Model
nndist.pp3

Nearest neighbour distances in three dimensions
Hest

Spherical Contact Distribution Function
longleaf

Longleaf Pines Point Pattern
density.splitppp

Kernel Smoothed Intensity of Split Point Pattern
Fest

Estimate the empty space function F
spokes

Spokes pattern of dummy points
ewcdf

Weighted Empirical Cumulative Distribution Function
fitin.ppm

Extract the Interaction from a Fitted Point Process Model
finpines

Pine saplings in Finland.
allstats

Calculate four standard summary functions of a point pattern.
Ldot.inhom

Inhomogeneous Multitype L Dot Function
owin

Create a Window
ord.family

Ord Interaction Process Family
eval.fasp

Evaluate Expression Involving Function Arrays
fryplot

Fry Plot of Point Pattern
im

Create a Pixel Image Object
dilated.areas

Areas of Morphological Dilations
chop.tess

Subdivide a Window or Tessellation using a Set of Lines
nnwhich.pp3

Nearest neighbours in three dimensions
collapse.fv

Collapse Several Function Tables into One
clickpoly

Interactively Define a Polygon
pixellate

Convert Spatial Object to Pixel Image
eval.im

Evaluate Expression Involving Pixel Images
Jmulti

Marked J Function
StraussHard

The Strauss / Hard Core Point Process Model
distmap.owin

Distance Map of Window
Replace.im

Reset Values in Subset of Image
lansing

Lansing Woods Point Pattern
gpc2owin

Convert Polygonal Region into Different Format
amacrine

Hughes' Amacrine Cell Data
letterR

Window in Shape of Letter R
predict.kppm

Prediction from a Fitted Cluster Point Process Model
ppp.object

Class of Point Patterns
rlabel

Random Re-Labelling of Point Pattern
distmap.ppp

Distance Map of Point Pattern
rknn

Theoretical Distribution of Nearest Neighbour Distance
rcell

Simulate Baddeley-Silverman Cell Process
as.ppp

Convert Data To Class ppp
interp.im

Interpolate a Pixel Image
clip.infline

Intersect Infinite Straight Lines with a Window
harmonic

Basis for Harmonic Functions
lgcp.estK

Fit a Log-Gaussian Cox Point Process by Minimum Contrast
logLik.ppm

Log Likelihood for Poisson Point Process Model
areaGain

Difference of Disc Areas
pcfdot

Multitype pair correlation function (i-to-any)
as.owin

Convert Data To Class owin
nnwhich

Nearest neighbour
crossdist

Pairwise distances
pixelquad

Quadrature Scheme Based on Pixel Grid
lurking

Lurking variable plot
selfcrossing.psp

Crossing Points in a Line Segment Pattern
dilation

Morphological Dilation
is.owin

Test Whether An Object Is A Window
default.dummy

Generate a Default Pattern of Dummy Points
pcfcross

Multitype pair correlation function (cross-type)
nnclean

Nearest Neighbour Clutter Removal
delaunay

Delaunay Triangulation of Point Pattern
Lcross.inhom

Inhomogeneous Cross Type L Function
box3

Three-Dimensional Box
rMaternI

Simulate Matern Model I
plot.im

Plot a Pixel Image
as.mask

Pixel Image Approximation of a Window
scanpp

Read Point Pattern From Data File
matclust.estK

Fit the Matern Cluster Point Process by Minimum Contrast
pcfinhom

Inhomogeneous Pair Correlation Function
multiplicity.ppp

Count Multiplicity of Duplicate Points
plot.plotppm

Plot a plotppm Object Created by plot.ppm
methods.ppx

Methods for Multidimensional Space-Time Point Patterns
pcf.fasp

Pair Correlation Function obtained from array of K functions
plot.psp

plot a Spatial Line Segment Pattern
spruces

Spruces Point Pattern
convexhull.xy

Convex Hull of Points
spatstat-internal

Internal spatstat functions
eem

Exponential Energy Marks
Kmeasure

Reduced Second Moment Measure
model.images

Compute Images of Constructed Covariates
murchison

Murchison gold deposits
plot.fasp

Plot a Function Array
pp3

Three Dimensional Point Pattern
rotate

Rotate
crossdist.default

Pairwise distances between two different sets of points
crossing.psp

Crossing Points of Two Line Segment Patterns
with.fv

Evaluate an Expression in a Function Table
pairsat.family

Saturated Pairwise Interaction Point Process Family
pairdist.default

Pairwise distances
Linhom

L-function
print.quad

Print a Quadrature Scheme
summary.splitppp

Summary of a Split Point Pattern
plot.splitppp

Plot a List of Point Patterns
plot.colourmap

Plot a Colour Map
nncorr

Nearest-Neighbour Correlation Indices of Marked Point Pattern
ppm.object

Class of Fitted Point Process Models
plot.tess

Plot a tessellation
AreaInter

The Area Interaction Point Process Model
as.data.frame.psp

Coerce Line Segment Pattern to a Data Frame
rmpoispp

Generate Multitype Poisson Point Pattern
pppdist

Distance Between Two Point Patterns
iplot

Point and Click Interface for Displaying a Point Pattern
pcfdot.inhom

Inhomogeneous Multitype Pair Correlation Function (Type-i-To-Any-Type)
rshift.ppp

Randomly Shift a Point Pattern
pcf.fv

Pair Correlation Function obtained from K Function
cut.ppp

Classify Points in a Point Pattern
expand.owin

Expand Window By Factor
applynbd

Apply Function to Every Neighbourhood in a Point Pattern
matchingdist

Distance for a Point Pattern Matching
summary.ppm

Summarizing a Fitted Point Process Model
predict.ppm

Prediction from a Fitted Point Process Model
Geyer

Geyer's Saturation Point Process Model
pointsOnLines

Place Points Evenly Along Specified Lines
is.multitype

Test whether Object is Multitype
summary.psp

Summary of a Line Segment Pattern Dataset
rMatClust

Simulate Matern Cluster Process
rStrauss

Perfect Simulation of the Strauss Process
pppmatching.object

Class of Point Matchings
is.multitype.ppm

Test Whether A Point Process Model is Multitype
rescale.ppp

Convert Point Pattern to Another Unit of Length
midpoints.psp

Midpoints of Line Segment Pattern
psp.object

Class of Line Segment Patterns
Kdot

Multitype K Function (i-to-any)
Poisson

Poisson Point Process Model
BadGey

Hybrid Geyer Point Process Model
nncross

Nearest Neighbours Between Two Patterns
is.subset.owin

Determine Whether One Window is Contained In Another
rmhmodel.list

Define Point Process Model for Metropolis-Hastings Simulation.
as.rectangle

Window Frame
ponderosa

Ponderosa Pine Tree Point Pattern
Gcross

Multitype Nearest Neighbour Distance Function (i-to-j)
quadrat.test

Chi-Squared Dispersion Test for Spatial Point Pattern Based on Quadrat Counts
rescale.psp

Convert Line Segment Pattern to Another Unit of Length
reduced.sample

Reduced Sample Estimator using Histogram Data
pcf.ppp

Pair Correlation Function of Point Pattern
shapley

Galaxies in the Shapley Supercluster
rpoislinetess

Poisson Line Tessellation
Strauss

The Strauss Point Process Model
localK

Neighbourhood density function
contour.im

Contour plot of pixel image
methods.box3

Methods for Three-Dimensional Box
residuals.ppm

Residuals for Fitted Point Process Model
convexhull

Convex Hull
compatible.fasp

Test Whether Two Function Arrays Are Compatible
plot.ppm

plot a Fitted Point Process Model
pixellate.ppp

Convert Point Pattern to Pixel Image
rmh.ppm

Simulate from a Fitted Point Process Model
pixellate.owin

Convert Window to Pixel Image
perimeter

Perimeter Length of Window
quadscheme

Generate a Quadrature Scheme from a Point Pattern
rmh.default

Simulate Point Process Models using the Metropolis-Hastings Algorithm.
Gmulti

Marked Nearest Neighbour Distance Function
compatible.im

Test Whether Two Pixel Images Are Compatible
adaptive.density

Intensity Estimate of Point Pattern Using Tessellation
as.tess

Convert Data To Tessellation
quadrats

Divide Region into Quadrats
rmhmodel.ppm

Interpret Fitted Model for Metropolis-Hastings Simulation.
rotate.ppp

Rotate a Point Pattern
rpoispp3

Generate Poisson Point Pattern in Three Dimensions
coords

Extract Coordinates of a Spatial or Spatiotemporal Point Pattern
print.ppm

Print a Fitted Point Process Model
hyperframe

Hyper Data Frame
rmhcontrol

Set Control Parameters for Metropolis-Hastings Algorithm.
distmap.psp

Distance Map of Line Segment Pattern
envelope

Simulation envelopes of summary function
nndist

Nearest neighbour distances
rSSI

Simulate Simple Sequential Inhibition
nndist.psp

Nearest neighbour distances between line segments
rshift.splitppp

Randomly Shift a List of Point Patterns
model.matrix.ppm

Extract Design Matrix from Point Process Model
simplify.owin

Approximate a Polygon by a Simpler Polygon
suffstat

Sufficient Statistic of Point Process Model
rThomas

Simulate Thomas Process
discpartarea

Area of Part of Disc
rescue.rectangle

Convert Window Back To Rectangle
cut.im

Convert Pixel Image from Numeric to Factor
Jcross

Multitype J Function (i-to-j)
rthin

Random Thinning
shift.owin

Apply Vector Translation To Window
rmhmodel.default

Build Point Process Model for Metropolis-Hastings Simulation.
Jdot

Multitype J Function (i-to-any)
plot.listof

Plot a List of Things
rescale

Convert dataset to another unit of length
rmhmodel

Define Point Process Model for Metropolis-Hastings Simulation.
rmhstart

Determine Initial State for Metropolis-Hastings Simulation.
effectfun

Compute Fitted Effect of a Spatial Covariate in a Point Process Model
as.box3

Convert Data to Three-Dimensional Box
gridweights

Compute Quadrature Weights Based on Grid Counts
spatstat-package

The Spatstat Package
anemones

Beadlet Anemones Data
owin.object

Class owin
tiles

Extract List of Tiles in a Tessellation
nearestsegment

Find Line Segment Nearest to Each Point
copper

Berman-Huntington points and lines data
rpoispp

Generate Poisson Point Pattern
affine.psp

Apply Affine Transformation To Line Segment Pattern
is.marked

Test Whether Marks Are Present
deltametric

Delta Metric
union.quad

Union of Data and Dummy Points
summary.quad

Summarizing a Quadrature Scheme
setmarks

Set or Reset the Marks in a Point Pattern
rmpoint

Generate N Random Multitype Points
pairdist.pp3

Pairwise distances in Three Dimensions
inside.owin

Test Whether Points Are Inside A Window
gridcentres

Rectangular grid of points
crossdist.pp3

Pairwise distances between two different three-dimensional point patterns
infline

Infinite Straight Lines
unique.ppp

Extract Unique Points from a Spatial Point Pattern
discretise

Safely Convert Point Pattern Window to Binary Mask
Extract.tess

Extract or Replace Subset of Tessellation
shift

Apply Vector Translation
plot.kstest

Plot a Spatial Kolmogorov-Smirnov Test
solutionset

Evaluate Logical Expression Involving Pixel Images and Return Region Where Expression is True
intersect.owin

Intersection, Union or Set Subtraction of Two Windows
pcfcross.inhom

Inhomogeneous Multitype Pair Correlation Function (Cross-Type)
square

Square Window
humberside

Humberside Data on Childhood Leukaemia and Lymphoma
rshift

Random Shift
split.ppp

Divide Point Pattern into Sub-patterns
Ord

Generic Ord Interaction model
is.ppp

Test Whether An Object Is A Point Pattern
vcov.ppm

Variance-Covariance Matrix for a Fitted Point Process Model
whist

Weighted Histogram
print.im

Print Brief Details of an Image
superimposePSP

Superimpose Several Line Segment Patterns
smooth.ppp

Spatial smoothing of observations at irregular points
runifpointOnLines

Generate N Uniform Random Points On Line Segments
tess

Create a Tessellation
incircle

Find Largest Circle Inside Window
angles.psp

Orientation Angles of Line Segments
is.im

Test Whether An Object Is A Pixel Image
vertices

Vertices of a Window
dirichlet.weights

Compute Quadrature Weights Based on Dirichlet Tessellation
Extract.listof

Extract or Replace Subset of a List of Things
is.marked.ppp

Test Whether A Point Pattern is Marked
istat

Point and Click Interface for Exploratory Analysis of Point Pattern
japanesepines

Japanese Pines Point Pattern
project2segment

Move Point To Nearest Line
areaLoss

Difference of Disc Areas
swedishpines

Swedish Pines Point Pattern
plot.ppp

plot a Spatial Point Pattern
eval.fv

Evaluate Expression Involving Functions
endpoints.psp

Endpoints of Line Segment Pattern
nearest.raster.point

Find Pixel Nearest to a Given Point
summary.ppp

Summary of a Point Pattern Dataset
hist.im

Histogram of Pixel Values in an Image
by.im

Apply Function to Image Broken Down by Factor
centroid.owin

Centroid of a window
integral.im

Integral of a Pixel Image
unmark

Remove Marks
kppm

Fit cluster point process model
compatible.fv

Test Whether Two Function Objects Are Compatible
Kdot.inhom

Inhomogeneous Multitype K Dot Function
intersect.tess

Intersection of Two Tessellations
rotate.owin

Rotate a Window
bramblecanes

Hutchings' Bramble Canes data
latest.news

Print News About Latest Version of Package
rlinegrid

Generate grid of parallel lines with random displacement
fasp.object

Function Arrays for Spatial Patterns
plot.hyperframe

Plot Entries in a Hyperframe
miplot

Morishita Index Plot
kstest.ppm

Kolmogorov-Smirnov Test for Point Process Model
periodify

Make Periodic Copies of a Spatial Pattern
markcorrint

Mark Correlation Integral
ppx

Multidimensional Space-Time Point Pattern
corners

Corners of a rectangle
is.marked.ppm

Test Whether A Point Process Model is Marked
hamster

Aherne's hamster tumour data
Extract.quad

Subset of Quadrature Scheme
plot.fv

Plot Function Values
pppmatching

Create a Point Matching
distfun

Distance Map as a Function
quad.object

Class of Quadrature Schemes
print.owin

Print Brief Details of a Spatial Window
rmh

Simulate point patterns using the Metropolis-Hastings algorithm.
lengths.psp

Lengths of Line Segments
is.empty

Test Whether An Object Is Empty
kaplan.meier

Kaplan-Meier Estimator using Histogram Data
LennardJones

The Lennard-Jones Potential
trim.rectangle

Cut margins from rectangle
distmap

Distance Map
markconnect

Mark Connection Function
quadrat.test.splitppp

Chi-Squared Test of CSR for Split Point Pattern
Kcross

Multitype K Function (Cross-type)
persp.im

Perspective Plot of Pixel Image
erosion

Morphological Erosion
rMosaicSet

Mosaic Random Set
print.psp

Print Brief Details of a Line Segment Pattern Dataset
ganglia

Beta Ganglion Cells in Cat Retina, Old Version
rMaternII

Simulate Matern Model II
reach

Interaction Distance of a Point Process
quad.ppm

Extract Quadrature Scheme Used to Fit a Point Process Model
bdist.points

Distance to Boundary of Window
residualspaper

Data and Code From JRSS Discussion Paper on Residuals
markcorr

Mark Correlation Function
summary.owin

Summary of a Spatial Window
redwood

California Redwoods Point Pattern (Ripley's Subset)
rpoint

Generate N Random Points
rpoisppOnLines

Generate Poisson Point Pattern on Line Segments
Extract.im

Extract Subset of Image
density.ppp

Kernel Smoothed Intensity of Point Pattern
fv.object

Data Frames of Function Values
rjitter

Random Perturbation of a Point Pattern
spatstat-deprecated

Deprecated spatstat functions
marktable

Tabulate Marks in Neighbourhood of Every Point in a Point Pattern
plot.owin

Plot a Spatial Window
plot.quad

plot a Spatial Quadrature Scheme
simdat

Simulated Point Pattern
ppp

Create a Point Pattern
rescale.im

Convert Pixel Image to Another Unit of Length
dirichlet

Dirichlet Tessellation of Point Pattern
is.multitype.ppp

Test Whether A Point Pattern is Multitype
rescale.owin

Convert Window to Another Unit of Length
MultiStrauss

The Multitype Strauss Point Process Model
simulate.ppm

Simulate a Fitted Gibbs Point Process Model
as.im

Convert to Pixel Image
runifpoint

Generate N Uniform Random Points
as.psp

Convert Data To Class psp
is.ppm

Test Whether An Object Is A Fitted Point Process Model
rshift.psp

Randomly Shift a Line Segment Pattern
update.ppm

Update a Fitted Point Process Model
update.kppm

Update a Fitted Cluster Point Process Model
print.ppp

Print Brief Details of a Point Pattern Dataset
psp

Create a Line Segment Pattern
mincontrast

Method of Minimum Contrast
markvario

Mark Variogram
simulate.kppm

Simulate a Fitted Cluster Point Process Model
summary.listof

Summary of a List of Things
with.hyperframe

Evaluate an Expression in Each Row of a Hyperframe
thomas.estK

Fit the Thomas Point Process by Minimum Contrast
border

Border Region of a Window
quadratcount

Quadrat counting for a point pattern
runifdisc

Generate N Uniform Random Points in a Disc
pairdist

Pairwise distances
summary.im

Summarizing a Pixel Image
density.psp

Kernel Smoothing of Line Segment Pattern
unitname

Name for Unit of Length
quantile.im

Sample Quantiles of Pixel Image
identify.ppp

Identify Points in a Point Pattern
redwoodfull

California Redwoods Point Pattern (Entire Dataset)
ripras

Estimate window from points alone
ppm

Fit Point Process Model to Data
Emark

Diagnostics for random marking
plot.bermantest

Plot Result of Berman Test
profilepl

Profile Maximum Pseudolikelihood
runifpoint3

Generate N Uniform Random Points in Three Dimensions
raster.x

Cartesian Coordinates for a Pixel Raster
km.rs

Kaplan-Meier and Reduced Sample Estimator using Histograms
markstat

Summarise Marks in Every Neighbourhood in a Point Pattern
spatstat.options

Internal Options in Spatstat Package
mean.im

Mean, Median and Range of Pixel Values in an Image
pairdist.ppp

Pairwise distances
rotate.psp

Rotate a Line Segment Pattern
rGaussPoisson

Simulate Gauss-Poisson Process
methods.pp3

Methods for three-dimensional point patterns
opening

Morphological Opening
rpoisline

Generate Poisson Random Line Process
pairdist.psp

Pairwise distances between line segments
superimpose

Superimpose Several Point Patterns
pairwise.family

Pairwise Interaction Process Family
split.im

Divide Image Into Sub-images
urkiola

Urkiola Woods Point Pattern
pcf

Pair Correlation Function
nztrees

New Zealand Trees Point Pattern
plot.kppm

Plot a fitted cluster point process
plot.pp3

Plot a three-dimensional point pattern
rMosaicField

Mosaic Random Field
rsyst

Simulate systematic random point pattern
shift.ppp

Apply Vector Translation To Point Pattern
stieltjes

Compute Integral of Function Against Cumulative Distribution