A function to simulate a continuous time Markov chain.

Description

This function simulates a continuous time finite state space Markov chain with known rate matrix Q, state space 0,1,..,n and initial state i for the time period (0,T). If plotflag is TRUE it also produces a plot.

Usage

CMCSimulation(Q,i,Tend,plotflag = FALSE)

Arguments

Details

We assume that Q is well defined rate matrix.

Value

Returns the matrix (statehist,timehist) containing the realisation of the chain for the specified period. The function also produces a plot of the realisation. \

References

Jones, O.D., R. Maillardet, and A.P. Robinson. 2009. An Introduction to Scientific Programming and Simulation, Using R. Chapman And Hall/CRC.

See Also

MCSimulation

Examples

Q <- matrix(c(-24,24,0, 2,-14,12, 0,4,-4), 
            nrow = 3, ncol = 3, byrow = TRUE)
CMCSimulation(Q,0,1)

A function to estimate the transition matrix for a discrete time Markov chain.

Description

This function estimates the transition matrix for a discrete time Markov chain with state space 0,1,..,n given a realisation. The chain has n+1 states.

Usage

MCEstimation(statehist,n)

Arguments

Details

We assume that the state space is 0,1,2...,n. n is assumed known as it cannot be reliably infered from the realisation.

Value

Returns the empirical transition matrix obtained by calculating the observed frequencies of actual transitions in the realisation.\

References

Jones, O.D., R. Maillardet, and A.P. Robinson. 2009. An Introduction to Scientific Programming and Simulation, Using R. Chapman And Hall/CRC.

See Also

MCSimulation

Examples

P <- matrix(c(0.5,0.5,0,0,0.7,0.1,0.2,0,0,0.1,0.1,0.8,0,0,0.7,0.3),
            nrow = 4, ncol = 4, byrow = TRUE)
statehist<-MCSimulation(P, 0, 3000)
MCEstimation(statehist, 3)

A function to simulate a discrete time Markov chain.

Description

This function simulates a discrete time Markov chain with transition matrix P, state space 0,1,..,n and and initial state i for nsteps transitions.

Usage

MCSimulation(P,i,nsteps)

Arguments

Details

We assume that P is well defined transition matrix with rows summing to 1.

Value

Returns the vector statehist containing the realisation of the chain for nsteps transitions.\

References

Jones, O.D., R. Maillardet, and A.P. Robinson. 2009. An Introduction to Scientific Programming and Simulation, Using R. Chapman And Hall/CRC.

See Also

MCEstimation, CMCSimulation

Examples

P <- matrix(c(0.5,0.5,0,0,0.7,0.1,0.2,0,0,0.1,0.1,0.8,0,0,0.7,0.3),
            nrow = 4, ncol = 4, byrow = TRUE)
MCSimulation(P, 0, 250)

A function which uses the Fourth order Runge-Kutta method with adaptive step size to solve a system of ODE's.

Description

This function simulates a discrete time Markov chain with transition matrix P, state space 0,1,..,n and and initial state i for nsteps transitions.

Usage

RK4adapt(dydt, t0, y0, t1, h0 = 1, tol = 1e-10, ...)

Arguments

Details

We assume that P is well defined transition matrix with rows summing to 1.

Value

Returns a list with elements t, a vector giving times, and y, a matrix whose rows give the solution at successive times.

References

Jones, O.D., R. Maillardet, and A.P. Robinson. 2009. An Introduction to Scientific Programming and Simulation, Using R. Chapman And Hall/CRC.

Examples

LV <- function(t=NULL, y, a, b, g, e, K=Inf)
  c(a*y[1]*(1 - y[1]/K) - b*y[1]*y[2], g*b*y[1]*y[2] - e*y[2])

xy <- RK4adapt(LV, 0, c(100, 50), 200, 1, tol=1e-3, 
               a=0.05, K=Inf, b=0.0002, g=0.8, e=0.03)

par(mfrow = c(2,1))
plot(xy$y[,1], xy$y[,2], type='p', 
     xlab='prey', ylab='pred', main='RK4, adaptive h')
plot(xy$t, xy$y[,1], type='p', xlab='time', 
     ylab='prey circles pred triangles', main='RK4, adaptive h')
points(xy$t, xy$y[,2], pch=2)
par(mfrow=c(1,1))

A function of the bisection algorithm.

Description

Applies the bisection algorithm to find x such that ftn(x) == x.

Usage

bisection(ftn, x.l, x.r, tol = 1e-09)

Arguments

Details

We assume that ftn is a function of a single variable.

Value

Returns the value of x at which ftn(x) == x. If the function fails to converge within max.iter iterations, returns NULL.

References

Jones, O.D., R. Maillardet, and A.P. Robinson. 2009. An Introduction to Scientific Programming and Simulation, Using R. Chapman And Hall/CRC.

See Also

newtonraphson, fixedpoint

Examples

ftn5 <- function(x) return(log(x)-exp(-x))
bisection(ftn5, 1, 2, tol = 1e-6)

A function to simulate the harassed booking clerk Markov chain.

Description

Simulates the harassed booking clerk Markov chain with given arrival and service rates up to t.end. The state space is (C(t),X(t),Y(t)), where C(t) represents the status of the clerk, X(t) the number of people waiting, and Y(t) the number of calls waiting. C(t) is 0 if clerk is idle, 1 if clerk is serving a person and 2 if clerk is serving a call.

Usage

booking_clerkMC(personArrRate, 
                callArrRate,
                personServRate,
                callServRate,
                t.end)

Arguments

Details

We assume that all given rates are finite and positive.

Value

Returns the matrix (t.hist, state.hist) containing the realisation of the chain.

References

Jones, O.D., R. Maillardet, and A.P. Robinson. 2009. An Introduction to Scientific Programming and Simulation, Using R. Chapman And Hall/CRC.

Examples

booking_clerkMC(3,6,5,8,1)

Function to fit a model to seed transect distance/count data.

Description

This function uses maximum likelihood to fit a nominated probability density function to the data of a seedtrap transect holder.

Usage

fitDistances(x, family)

Arguments

Value

The function returns the parameter estimates for the nominated family.

References

Jones, O.D., R. Maillardet, and A.P. Robinson. 2009. An Introduction to Scientific Programming and Simulation, Using R. Chapman And Hall/CRC.

See Also

fitdistr, trapTransect

Examples

library(MASS)
s1 <- trapTransect(distances = 1:4, seed.counts = c(4, 3, 2, 0))
allTraps <- transectHolder(s1, family="Weibull")
fitDistances(allTraps, "exponential")

A function of the fixed point algorithm.

Description

Applies the fixed point algorithm to find x such that ftn(x) == x.

Usage

fixedpoint(ftn, x0, tol = 1e-09, max.iter = 100)

Arguments

Details

We assume that ftn is a function of a single variable.

Value

Returns the value of x at which ftn(x) == x. If the function fails to converge within max.iter iterations, returns NULL.

References

Jones, O.D., R. Maillardet, and A.P. Robinson. 2009. An Introduction to Scientific Programming and Simulation, Using R. Chapman And Hall/CRC.

See Also

newtonraphson, bisection

Examples

ftn1 <- function(x) return(exp(exp(-x)))
fixedpoint(ftn1, 2, tol = 1e-6)

A function of the fixed point algorithm.

Description

Applies the fixed point algorithm to find x such that ftn(x) == x, and plots the process.

Usage

fixedpoint_show(ftn, x0, xmin = x0 - 1, xmax = x0 + 1)

Arguments

Details

We assume that ftn is a function of a single variable.

Value

Returns the value of x at which ftn(x) == x. If the function fails to converge within max.iter iterations, returns NULL.

References

Jones, O.D., R. Maillardet, and A.P. Robinson. 2009. An Introduction to Scientific Programming and Simulation, Using R. Chapman And Hall/CRC.

See Also

fixedpoint

303 years of monthly rainfall data from Kew Gardens, London, U.K.

Description

The monthly rainfall at Kew Gardens, London, U.K., from 1697 to 1999, in mm.

Usage

data(kew)

Format

A wide-format data frame with 303 observations. Each month has its own column.

Source

Data obtained from the U.S. National Climatic Data Centre, Global Historical Climatology Network data base (GHCN-Monthly Version 2, NB: not Version 3) https://www.ncei.noaa.gov/products/land-based-station/global-historical-climatology-network-monthly.

References

Jones, O.D., R. Maillardet, and A.P. Robinson. 2014. An Introduction to Scientific Programming and Simulation, Using R. 2nd Ed. Chapman And Hall/CRC.

Examples

data(kew)

Function to compute the mean dispersal distance along a transect of seed traps.

Description

This function computes the mean dispersal distance along a transect of seed traps.

Usage

## S3 method for class transectHolder 
## S3 method for class 'transectHolder'
mean(x, ...)

Arguments

Value

The mean seed dispersal distance is returned.

References

Jones, O.D., R. Maillardet, and A.P. Robinson. 2009. An Introduction to Scientific Programming and Simulation, Using R. Chapman And Hall/CRC.

See Also

transectHolder

Examples

transect.1 <- trapTransect(distances = 1:4,
                        seed.counts = c(4, 3, 2, 0))
transect.2 <- trapTransect(distances = 1:3,
                        seed.counts = c(3, 2, 1))
transect.3 <- trapTransect(distances=(1:5)/2,
                        seed.counts = c(3, 4, 2, 3, 1))
allTraps <- transectHolder(transect.1, transect.2, transect.3,
                           family="Weibull")

mean(allTraps)

Function to compute the mean dispersal distance along a transect of seed traps.

Description

This function computes the mean dispersal distance along a transect of seed traps.

Usage

## S3 method for class trapTransect 
## S3 method for class 'trapTransect'
mean(x, ...)

Arguments

Value

The mean seed dispersal distance is returned.

References

Jones, O.D., R. Maillardet, and A.P. Robinson. 2009. An Introduction to Scientific Programming and Simulation, Using R. Chapman And Hall/CRC.

See Also

trapTransect

Examples

s1 <- trapTransect(distances = 1:4, seed.count = c(4, 3, 2, 0))
mean(s1)

A function of the Newton-Raphson algorithm.

Description

Applies the Newton-Raphson algorithm to find x such that ftn(x)[1] == 0.

Usage

newtonraphson(ftn, x0, tol = 1e-09, max.iter = 100)

Arguments

Value

Returns the value of x at which ftn(x)[1] == 0. If the function fails to converge within max.iter iterations, returns NULL.

References

Jones, O.D., R. Maillardet, and A.P. Robinson. 2009. An Introduction to Scientific Programming and Simulation, Using R. Chapman And Hall/CRC.

See Also

fixedpoint, bisection

Examples

ftn4 <- function(x) {
  # returns function value and its derivative at x
  fx <- log(x) - exp(-x)
  dfx <- 1/x + exp(-x)
  return(c(fx, dfx))
}
newtonraphson(ftn4, 2, 1e-6)

A function of the Newton-Raphson algorithm, plotting the path.

Description

Applies the Newton-Raphson algorithm to find x such that ftn(x)[1] == 0, and plots the trace of the estimate.

Usage

newtonraphson_show(ftn, x0, xmin = x0 - 1, xmax = x0 + 1)

Arguments

Value

Returns the value of x at which ftn(x)[1] == 0. If the function fails to converge within max.iter iterations, returns NULL.

References

Jones, O.D., R. Maillardet, and A.P. Robinson. 2009. An Introduction to Scientific Programming and Simulation, Using R. Chapman And Hall/CRC.

See Also

newtonraphson

Function to assess whether or not an integer is prime.

Description

An inefficient, brute-force algorithm to assess whether or not an integer is prime.

Usage

prime(n)

Arguments

Details

The function assumes that n is a positive integer.

Value

The function returns a logical object that is TRUE if the integer is prime.

References

Jones, O.D., R. Maillardet, and A.P. Robinson. 2009. An Introduction to Scientific Programming and Simulation, Using R. Chapman And Hall/CRC.

See Also

primesieve

Examples

prime(10)
prime(7)

Function to identify all the primes in a vector of positive integers.

Description

This function uses the Sieve of Eratosthenes to find all the primes less than or equal to a given integer.

Usage

primesieve(sieved, unsieved)

Arguments

Details

The function assumes that unsieved is a vector of positive integers.

Value

Returns a vector of primes sieved (selected) from the input vector.

References

Jones, O.D., R. Maillardet, and A.P. Robinson. 2009. An Introduction to Scientific Programming and Simulation, Using R. Chapman And Hall/CRC.

See Also

prime

Examples

primesieve(c(), 2:200)

Function to print a transectHolder object usefullly.

Description

This function prints the details of a transectHolder object.

Usage

## S3 method for class transectHolder 
## S3 method for class 'transectHolder'
print(x, ...)

Arguments

Details

The print function simply uses str on the transectHolder object.

Value

This function is called for its side-effect, which is to print the object informatively.

References

Jones, O.D., R. Maillardet, and A.P. Robinson. 2009. An Introduction to Scientific Programming and Simulation, Using R. Chapman And Hall/CRC.

See Also

transectHolder

Examples

transect.1 <- trapTransect(distances = 1:4,
                        seed.counts = c(4, 3, 2, 0))
transect.2 <- trapTransect(distances = 1:3,
                        seed.counts = c(3, 2, 1))
transect.3 <- trapTransect(distances=(1:5)/2,
                        seed.counts = c(3, 4, 2, 3, 1))
allTraps <- transectHolder(transect.1, transect.2, transect.3,
                           family="Weibull")
allTraps

Function to print a trapTransect object usefullly.

Description

This function prints the details of a trapTransect object.

Usage

## S3 method for class trapTransect 
## S3 method for class 'trapTransect'
print(x, ...)

Arguments

Details

The print function simply uses str on the trapTransect object.

Value

This function is called for its side-effect, which is to print the object informatively.

References

Jones, O.D., R. Maillardet, and A.P. Robinson. 2009. An Introduction to Scientific Programming and Simulation, Using R. Chapman And Hall/CRC.

See Also

trapTransect

Examples

s1 <- trapTransect(distances = 1:4, seed.count = c(4, 3, 2, 0))
s1

Function to compute the sd dispersal distance along a transect of seed traps.

Description

This function computes the standard deviation of the dispersal distances along a transect of seed traps.

Usage

sd.transectHolder(transectHolder)

Arguments

Value

The standard deviation of the seed dispersal distances is returned.

References

Jones, O.D., R. Maillardet, and A.P. Robinson. 2009. An Introduction to Scientific Programming and Simulation, Using R. Chapman And Hall/CRC.

See Also

transectHolder

Examples

transect.1 <- trapTransect(distances = 1:4,
                        seed.counts = c(4, 3, 2, 0))
transect.2 <- trapTransect(distances = 1:3,
                        seed.counts = c(3, 2, 1))
transect.3 <- trapTransect(distances=(1:5)/2,
                        seed.counts = c(3, 4, 2, 3, 1))
allTraps <- transectHolder(transect.1, transect.2, transect.3,
                           family="Weibull")

mean(allTraps)

sd.transectHolder(allTraps)

Function to simulate a modelled seed rain from a transectHolder

Description

This function simulates a two-dimensional seed rain according to the model stored in a transectHolder object. The angle of the seed location from the parent plant is uniformly distributed on [0, 2 pi).

Usage

## S3 method for class transectHolder 
## S3 method for class 'transectHolder'
simulate(object, nsim=1, seed=NULL, ...)

Arguments

Value

A dataframe with n rows with the following components:

References

Jones, O.D., R. Maillardet, and A.P. Robinson. 2009. An Introduction to Scientific Programming and Simulation, Using R. Chapman And Hall/CRC.

See Also

transectHolder

Examples

transect.1 <- trapTransect(distances = 1:4,
                        seed.counts = c(4, 3, 2, 0))
transect.2 <- trapTransect(distances = 1:3,
                        seed.counts = c(3, 2, 1))
transect.3 <- trapTransect(distances=(1:5)/2,
                        seed.counts = c(3, 4, 2, 3, 1))
allTraps <- transectHolder(transect.1, transect.2, transect.3,
                           family="Weibull")
allTraps
simulate(allTraps, nsim=5, seed=123)

Function to construct an object representing a collection of trapTransect objects.

Description

This function constructs a transectHolder object given a collection of trapTransect objects and a nominated probability density function to fit to the seed count profile.

Usage

transectHolder(..., family = "exponential")

Arguments

Details

This function is a constructor.

The nominated distribution, which must be one of those distributions that can be fit by fitdistr of the MASS package.

Value

A transectHolder object, which is a list comprising

References

Jones, O.D., R. Maillardet, and A.P. Robinson. 2009. An Introduction to Scientific Programming and Simulation, Using R. Chapman And Hall/CRC.

See Also

trapTransect

Examples

transect.1 <- trapTransect(distances = 1:4,
                        seed.counts = c(4, 3, 2, 0))
transect.2 <- trapTransect(distances = 1:3,
                        seed.counts = c(3, 2, 1))
transect.3 <- trapTransect(distances=(1:5)/2,
                        seed.counts = c(3, 4, 2, 3, 1))
allTraps <- transectHolder(transect.1, transect.2, transect.3,
                           family="Weibull")
allTraps

Function to construct an object representing a transect of seedtraps.

Description

This function constructs a trapTransect object given a vector of trap distances from the parent plant, a vector of trap seed counts corresponding to the trap distances, and a single trap area.

Usage

 trapTransect(distances, seed.counts, trap.area = 0.0001) 

Arguments

Details

This function is a constructor.

Value

A trapTransect object, which is a list comprising three objects:

References

Jones, O.D., R. Maillardet, and A.P. Robinson. 2009. An Introduction to Scientific Programming and Simulation, Using R. Chapman And Hall/CRC.

See Also

mean.trapTransect, print.trapTransect

Examples

s1 <- trapTransect(distances = 1:4, seed.counts = c(4, 3, 2, 0))
s1
mean(s1)

Grand fir tree growth data from northern and central Idaho, USA.

Description

A sample of 66 grand fir (Abies grandis) trees was selected from national forests around northern and central Idaho. The trees were selected to be dominant in their environment, with no visible evidence of crown damage, forks, broken tops, etc. For each tree the habitat type and the national forest from which it came were recorded. We have data from nine national forests and six different habitat types.

Usage

data(treeg)

Format

A data frame with 542 observations on the following 6 variables.

tree.ID

Tree number.

forest

National forest number.

habitat

Habitat code (see Details).

dbh.in

Bole diameter at 1.37 m, in inches

height.ft

Tree height, in feet.

age

Age at which measurement was taken.

Details

For each tree the height, diameter and age were measured (age is measured using tree rings), then the tree was split lengthways, which allows you to determine the height and diameter of the tree at any age. In this instance height and diameter were recorded for the age the tree was felled and then at ten year periods going back in time. The diameter of the tree was measured at a height of 1.37 m (4'6”), which is called breast height in forestry. The height refers to the height of the main trunk only.

The habitats corresponding to codes 1 through 5 are: Ts/Pach; Ts/Op; Th/Pach; AG/Pach and PA/Pach. These codes refer to the climax tree species, which is the most shade-tolerant species that can grow on the site, and the dominant understorey plant, respectively. Ts refers to Thuja plicata and Tsuga heterophylla, Th refers to just Thuja plicata, AG is Abies grandis, PA is Picea engelmanii and Abies lasiocarpa, Pach is Pachistima myrsinites, and Op is the nasty Oplopanaz horridurn. Grand fir is considered a major climax species for AG/Pach, a major seral species for Th/Pach and PA/Pach, and a minor seral species for Ts/Pach and Ts/Op. Loosely speaking, a community is seral if there is evidence that at least some of the species are temporary, and climax if the community is self-regenerating (Daubenmire, 1952).

Source

These data were kindly supplied by Dr Al Stage, Principal Mensurationist (retired), USDA Forest Service Foresct Sciences Laboratory, Moscow, ID, USA.

References

R. Daubenmire, 1952. Forest Vegetation of Northern Idaho and Adjacent Washington, and Its Bearing on Concepts of Vegetation Classification, Ecological Monographs 22, 301–330.

A. R. Stage, 1963. A mathematical approach to polymorphic site index curves for grand fir. Forest Science 9, 167–180.

Examples

data(treeg)

von Guttenberg Norway spruce tree measurement data

Description

These are a subset of the von Guttenberg data, a set of measurements on Norway spruce (Picea abies [L.] Karst) in several different locations and site categories.

Usage

data(trees)

Format

A data frame with 1200 observations on the following 3 variables.

ID

A factor identifying the tree by location, site, and tree number.

Age

The age at which the tree was measured.

Vol

The bole volume of the tree, in cubic dm.

Source

These data were kindly provided by Professor Boris Zeide, University of Arkanasa, Monticello, AK, USA, and are further documented in Zeide (1993).

References

A.R. von Guttenberg. 1915. Growth and yield of spruce in Hochgebirge. Franz Deuticke, Wien. (In German)

B. Zeide, 1993. Analysis of growth equations. Forest Science 39 594–616.

Examples

data(trees)

Upper Flat Creek forest cruise tree data

Description

These are a subset of the tree measurement data from the Upper Flat Creek unit of the University of Idaho Experimental Forest, which was measured in 1991.

Usage

data(ufc)

Format

A data frame with 336 observations on the following 5 variables.

plot

plot label

tree

tree label

species

species kbd with levels DF, GF, WC, WL

dbh.cm

tree diameter at 1.37 m. from the ground, measured in centimetres.

height.m

tree height measured in metres

Details

The inventory was based on variable radius plots with 6.43 sq. m. per ha. BAF (Basal Area Factor). The forest stand was 121.5 ha. This version of the data omits errors, trees with missing heights, and uncommon species. The four species are Douglas-fir, grand fir, western red cedar, and western larch.

Source

The data are provided courtesy of Harold Osborne and Ross Appelgren of the University of Idaho Experimental Forest.

References

Jones, O.D., R. Maillardet, and A.P. Robinson. 2009. An Introduction to Scientific Programming and Simulation, Using R. Chapman And Hall/CRC.

See Also

ufc.plots

Examples

data(ufc)

Upper Flat Creek forest cruise plot data

Description

These are a subset of the plot measurement data from the Upper Flat Creek unit of the University of Idaho Experimental Forest, which was measured in 1991.

Usage

data(ufc.plots)

Format

A data frame with 144 observations on the following 6 variables.

plot

plot label

north.n

northerly plot count

east.n

easterly plot count

north

northerly coordinate

east

easterly coordinate

vol.m3.ha

total above-ground merchantable volume, in cubic metres per hectare.

Source

The data are provided courtesy of Harold Osborne and Ross Appelgren of the University of Idaho Experimental Forest.

References

Jones, O.D., R. Maillardet, and A.P. Robinson. 2009. An Introduction to Scientific Programming and Simulation, Using R. Chapman And Hall/CRC.

See Also

ufc

Examples

data(ufc.plots)

Function to compute the volume of a tree bole assuming a particular shape.

Description

This function computes the volume of a tree bole given its basal diameter and length, assuming that the bole is a frustum of a geometric solid.

Usage

vol.m3(dbh.cm, height.m, multiplier = 0.5)

Arguments

Details

Commonly-used shapes are:

1/3

conoid

1/2

second-degree parabaloid

1

cylinder

Value

The volume is returned, in units of cubic metres.

Examples

vol.m3(30, 30)
vol.m3(30, 30, 1)