Information criteria for the stochastic differential equation
Information criteria BIC, Quasi-BIC (QBIC) and CIC for the stochastic differential equation.
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, ...)
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 qmleCalculate the information criteria BIC, QBIC, and CIC for stochastic processes. The calculation and model selection are performed by joint procedure or stepwise procedure.
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
Akaike, H. (1973). Information theory and an extension of the maximum likelihood principle. In Second International Symposium on Information Theory (Tsahkadsor, 1971), 267-281. tools:::Rd_expr_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. tools:::Rd_expr_doi("10.1214/aos/1176344136")
Eguchi, S. and Masuda, H. (2018). Schwarz type model comparison for LAQ models. Bernoulli, 24(3), 2278-2327. tools:::Rd_expr_doi("10.3150/17-BEJ928") .
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. tools:::Rd_expr_doi("10.1007/s10463-009-0245-1")
Burnham, K. P. and Anderson, D. R. (2002). Model Selection and Multimodel Inference. Springer-Verlag, New York.
The YUIMA Project Team
Contacts: Shoichi Eguchi shoichi.eguchi@oit.ac.jp
The function IC uses the function qmle with method="L-BFGS-B" internally.
## 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@data@original.data[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@data@original.data[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)