IC {yuima}

Information criteria for the stochastic differential equation

Description

Information criteria BIC, Quasi-BIC (QBIC) and CIC for the stochastic differential equation.

Usage

IC(drif = NULL, diff = NULL, jump.coeff = NULL, data = NULL, Terminal = 1, 
   add.settings = list(), start, lower, upper, ergodic = TRUE,
   stepwise = FALSE, weight = FALSE, rcpp = FALSE, ...)

Arguments

drif

a character vector that each element presents the candidate drift coefficient.

diff

a character vector that each element presents the candidate diffusion coefficient.

jump.coeff

a character vector that each element presents the candidate scale coefficient.

data

the data to be used.

Terminal

terminal time of the grid.

add.settings

details of model settings(see setModel).

start

a named list of the initial values of the parameters for optimization.

lower

a named list for specifying lower bounds of the parameters.

upper

a named list for specifying upper bounds of the parameters.

ergodic

whether the candidate models are ergodic SDEs or not(default ergodic=TRUE).

stepwise

specifies joint procedure or stepwise procedure(default stepwise=FALSE).

weight

calculate model weight? (default weight=FALSE)

rcpp

use C++ code? (default rcpp=FALSE)

...

passed to qmle

Details

Calculate the information criteria BIC, QBIC, and CIC for stochastic processes. The calculation and model selection are performed by joint procedure or stepwise procedure.

Value

BIC

values of BIC for all candidates.

QBIC

values of QBIC for all candidates.

AIC

values of AIC-type information criterion for all candidates.

model

information of all candidate models.

par

quasi-maximum likelihood estimator for each candidate.

weight

model weights for all candidates.

selected

selected model number and selected drift and diffusion coefficients

Note

The function IC uses the function qmle with method="L-BFGS-B" internally.

Author(s)

The YUIMA Project Team

Contacts: Shoichi Eguchi [e-mail]

References

## AIC, BIC

Akaike, H. (1973). Information theory and an extension of the maximum likelihood principle. In Second International Symposium on Information Theory (Tsahkadsor, 1971), 267-281. doi:10.1007/978-1-4612-1694-0_15

Schwarz, G. (1978). Estimating the dimension of a model. The Annals of Statistics, 6(2), 461-464. doi:10.1214/aos/1176344136

## BIC, Quasi-BIC

Eguchi, S. and Masuda, H. (2018). Schwarz type model comparison for LAQ models. Bernoulli, 24(3), 2278-2327. doi:10.3150/17-BEJ928.

## CIC

Uchida, M. (2010). Contrast-based information criterion for ergodic diffusion processes from discrete observations. Annals of the Institute of Statistical Mathematics, 62(1), 161-187. doi:10.1007/s10463-009-0245-1

## Model weight

Burnham, K. P. and Anderson, D. R. (2002). Model Selection and Multimodel Inference. Springer-Verlag, New York.

Examples

## Not run: 
### Ex.1 
set.seed(123)

N <- 1000   # number of data
h <- N^(-2/3)  # sampling stepsize
Ter <- N*h  # terminal sampling time

## Data generate (dXt = -Xt*dt + exp((-2*cos(Xt) + 1)/2)*dWt)
mod <- setModel(drift="theta21*x", diffusion="exp((theta11*cos(x)+theta12)/2)")
samp <- setSampling(Terminal=Ter, n = N)
yuima <- setYuima(model=mod, sampling=setSampling(Terminal=Ter, n=50*N))
simu.yuima <- simulate(yuima, xinit=1, true.parameter=list(theta11=-2, theta12=1, 
                       theta21=-1), subsampling=samp)
Xt <- NULL
for(i in 1:(N+1)){
  Xt <- c(Xt, simu.yuima@[e-mail][50*(i-1)+1])
}

## Candidate coefficients
diffusion <- c("exp((theta11*cos(x)+theta12*sin(x)+theta13)/2)", 
               "exp((theta11*cos(x)+theta12*sin(x))/2)", 
               "exp((theta11*cos(x)+theta13)/2)", "exp((theta12*sin(x)+theta13)/2)")
drift <- c("theta21*x + theta22", "theta21*x")

## Parameter settings
para.init <- list(theta11=runif(1,max=5,min=-5), theta12=runif(1,max=5,min=-5), 
                  theta13=runif(1,max=5,min=-5), theta21=runif(1,max=-0.5,min=-1.5),
                  theta22=runif(1,max=-0.5,min=-1.5))
para.low <- list(theta11=-10, theta12=-10, theta13=-10, theta21=-5, theta22=-5)
para.upp <- list(theta11=10, theta12=10, theta13=10, theta21=-0.001, theta22=-0.001)

## Ex.1.1 Joint
ic1 <- IC(drif=drift, diff=diffusion, data=Xt, Terminal=Ter, start=para.init, lower=para.low, 
          upper=para.upp, stepwise = FALSE, weight = FALSE, rcpp = TRUE)
ic1

## Ex.1.2 Stepwise
ic2 <- IC(drif=drift, diff=diffusion, data=Xt, Terminal=Ter, 
          start=para.init, lower=para.low, upper=para.upp,
          stepwise = TRUE, weight = FALSE, rcpp = TRUE)
ic2


### Ex.2 (multidimansion case) 
set.seed(123)

N <- 3000   # number of data
h <- N^(-2/3)  # sampling stepsize
Ter <- N*h  # terminal sampling time

## Data generate
diff.coef.matrix <- matrix(c("beta1*x1+beta3", "1", "-1", "beta1*x1+beta3"), 2, 2)
drif.coef.vec <- c("alpha1*x1", "alpha2*x2")
mod <- setModel(drift = drif.coef.vec, diffusion = diff.coef.matrix, 
                state.variable = c("x1", "x2"), solve.variable = c("x1", "x2"))
samp <- setSampling(Terminal = Ter, n = N)
yuima <- setYuima(model = mod, sampling = setSampling(Terminal = N^(1/3), n = 50*N))
simu.yuima <- simulate(yuima, xinit = c(1,1), true.parameter = list(alpha1=-2, alpha2=-1, 
                       beta1=-1, beta3=2), subsampling = samp)
Xt <- matrix(0,(N+1),2)
for(i in 1:(N+1)){
  Xt[i,] <- simu.yuima@[e-mail][50*(i-1)+1,]
}

## Candidate coefficients
diffusion <- list(matrix(c("beta1*x1+beta2*x2+beta3", "1", "-1", "beta1*x1+beta2*x2+beta3"), 2, 2),
                  matrix(c("beta1*x1+beta2*x2", "1", "-1", "beta1*x1+beta2*x2"), 2, 2),
                  matrix(c("beta1*x1+beta3", "1", "-1", "beta1*x1+beta3"), 2, 2),
                  matrix(c("beta2*x2+beta3", "1", "-1", "beta2*x2+beta3"), 2, 2),
                  matrix(c("beta1*x1", "1", "-1", "beta1*x1"), 2, 2),
                  matrix(c("beta2*x2", "1", "-1", "beta2*x2"), 2, 2),
                  matrix(c("beta3", "1", "-1", "beta3"), 2, 2))
drift <- list(c("alpha1*x1", "alpha2*x2"), c("alpha1*x2", "alpha2*x1"))
modsettings <- list(state.variable = c("x1", "x2"), solve.variable = c("x1", "x2"))

## Parameter settings
para.init <- list(alpha1 = runif(1,min=-3,max=-1), alpha2 = runif(1,min=-2,max=0),
                  beta1 = runif(1,min=-2,max=0), beta2 = runif(1,min=0,max=2), 
                  beta3 = runif(1,min=1,max=3))
para.low <- list(alpha1 = -5, alpha2 = -5, beta1 = -5, beta2 = -5, beta3 = 1)
para.upp <- list(alpha1 = 0.01, alpha2 = -0.01, beta1 = 5, beta2 = 5, beta3 = 10)

## Ex.2.1 Joint
ic3 <- IC(drif=drift, diff=diffusion, data=Xt, Terminal=Ter, add.settings=modsettings, 
          start=para.init, lower=para.low, upper=para.upp, 
          weight=FALSE, rcpp=FALSE)
ic3

## Ex.2.2 Stepwise
ic4 <- IC(drif=drift, diff=diffusion, data=Xt, Terminal=Ter, add.settings=modsettings, 
             start=para.init, lower=para.low, upper=para.upp,
             stepwise = TRUE, weight=FALSE, rcpp=FALSE)
ic4


## End(Not run)