IC | R Documentation |
Information criteria BIC, Quasi-BIC (QBIC) and CIC for the stochastic differential equation.
IC(drif = NULL, diff = 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. |
data |
the data to be used. |
Terminal |
terminal time of the grid. |
add.settings |
details of model settings(see |
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 |
stepwise |
specifies joint procedure or stepwise procedure(default |
weight |
calculate model weight? (default |
rcpp |
use C++ code? (default |
... |
Calculate 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. |
CIC |
values of CIC 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 |
The function IC
uses the function qmle
with method="L-BFGS-B"
internally.
The YUIMA Project Team
Contacts: Shoichi Eguchi eguchi@sigmath.es.osaka-u.ac.jp
## 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.
## 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)
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