simple_gif {PtProcess}

Non-Homogeneous Poisson Processes

Description

The functions listed here are intensity functions that are not conditional on the history of the process. Each has exactly the same “Usage” and calling format (see section “Value”) as the function simple_gif. They are: expfourier_gif, exppoly_gif, fourier_gif, poly_gif, and simple_gif.

Usage

simple_gif(data, evalpts, params, TT=NA, tplus=FALSE)

Arguments

data

NULL or a data frame. The contents of this object are not used by these functions, though they retain this argument for consistency with other gif functions.

evalpts

a vector, matrix or data.frame. If a vector, the elements will be assumed to represent the required evaluation times. Other objects must include a column named "time" that can be referred to as evalpts[,"time"], at which the intensity function will be evaluated.

params

vector of parameter values as required by the particular intensity function, see Details below.

TT

vector of length 2, being the time interval over which the integral of the intensity function is to be evaluated.

tplus

logical, \lambda_g(t|{\cal H}_t) is evaluated as \lambda_g(t^+|{\cal H}_t) if TRUE, else \lambda_g(t^-|{\cal H}_t). Included for compatibility with others conditional intensity functions.

Details

The models are parameterised as follows.

expfourier_gif

The vector of parameters is

(p, a_0, a_1, a_2, \cdots, a_n, b_1, b_2, \cdots, b_n)

and the intensity function is

\lambda_g(t) = \exp\left\{a_0 + \sum_{j=1}^n a_j\cos\left(\frac{2j\pi t}{p}\right) + \sum_{j=1}^n b_j\sin\left(\frac{2j\pi t}{p}\right)\right\}.

The length of params is 2n + 2, and determines the order of the fitted Fourier series. The numbers of specified sine and cosine coefficients must be the same. The integral is evaluated using numerical integration, using the R function integrate.

exppoly_gif

The vector of parameters is (b_0, b_1, b_2, \cdots, b_n) and the intensity function is

\lambda_g(t) = \exp\left\{b_0 + \sum_{j=1}^n b_j t^j \right\}.

The length of params determines the order of the fitted polynomial. The integral is evaluated using numerical integration, using the R function integrate.

fourier_gif

The Fourier intensity function is the same as expfourier_gif, except the intensity function omits the exponential, and the integration is performed explicitly.

poly_gif

The polynomial intensity function is the same as exppoly_gif, except the intensity function omits the exponential, and the integration is performed explicitly.

simple_gif

The intensity function is \lambda_g(t) = a + b t^g and the vector of parameters is (a, b, g).

Value

Two usages are as follows.

simple_gif(data, evalpts, params, tplus=FALSE)
simple_gif(data, evalpts=NULL, params, TT=NA)

The first usage returns a vector containing the values of \lambda_g(t) evaluated at the specified points. In the second usage, it returns the value of the integral.

Function Attributes

rate

is "bounded".

See Also

General details about the structure of conditional intensity functions are given in the topic gif.

Examples

expfourier_gif(NULL, c(1.1,1.2,1.3), c(2,3,1,2,3,4), TT=NA)
#  Evaluates:  lambda_g(t) = exp(3 + 1*cos(2*pi*t/2) + 2*cos(4*pi*t/2) +
#                                3*sin(2*pi*t/2) + 4*sin(4*pi*t/2))
#  lambda_g(1.1) = 162.56331
#  lambda_g(1.2) = 127.72599
#  lambda_g(1.3) =  23.83979

expfourier_gif(NULL, NULL, c(2,3,1,2,3,4), TT=c(3,4))
#  Let:  lambda_g(t) = exp(3 + 1*cos(2*pi*t/2) + 2*cos(4*pi*t/2) +
#                              3*sin(2*pi*t/2) + 4*sin(4*pi*t/2))
#  Evaluates: integral_3^4 lambda_g(t) dt = 46.21920


#--------------------------------------------------------
#   Plot intensity function: lambda(t) = 3 + 3*sin(t)
#   on interval (0, 6*pi), no marks

params <- c(2*pi, 3, 0, 3)
TT <- c(0, 6*pi)
x <- seq(TT[1], TT[2], length.out=500)

plot(x, fourier_gif(NULL, x, params, TT=NA),
     ylim=c(0, 6), type="l", axes=FALSE,
     xlab="t",
     ylab=expression(lambda(t) == 3 + 3*phantom(.)*plain(sin)*phantom(.)*t),
     main="Sinusoidal Intensity Function", font.main=1)
abline(h=params[2], lty=2, col="red")
box()
axis(2)
axis(1, at=0, labels=0)
axis(1, at=2*pi, labels=expression(2*pi))
axis(1, at=4*pi, labels=expression(4*pi))
axis(1, at=6*pi, labels=expression(6*pi))

#   Now define a model object
#   note NULL "marks" argument, see manual page for "mpp"
z <- mpp(data=NULL,
         gif=fourier_gif,
         marks=list(NULL, NULL),
         params=params,
         gmap=expression(params),
         mmap=NULL,
         TT=TT)

#   Simulate event times
z <- simulate(z, seed=3, max.rate=6)

#   Plot simulated times on sine curve
x <- z$data$time
points(x, fourier_gif(NULL, x, params, TT=NA), col="blue", lwd=5)

#   Number of simulated events
print(nrow(z$data))

#   Estimate parameters based on simulated data
parmap <- function(y, p){
    #    fix parameters 1 and 3
    y$params <- c(2*pi, p[1], 0, p[2])
    return(y)
}

initial <- c(3, 3)
y <- nlm(neglogLik, initial, object=z, pmap=parmap,
         print.level=2, iterlim=20, stepmax=0.1)
print(y$estimate)