Stem.Model {Stem} | R Documentation |
The function Stem.Model
is used to create an object of class “Stem.Model”.
Stem.Model(...) is.Stem.Model(x)
... |
List with named elements: phi , K , z , coordinates , covariates and, optionally, p (default equal to 1). See the model details and notation
below. |
x |
an object of class Stem.Model |
The hierarchical spatio-temporal model is given by
z_t = X_t * β + K * y_t + e_t , e_t ~ N(0, Σ_e )
y_t = G * y_{t-1} + eta_t , eta_t ~ N(0,Σ_{eta})
for t=1,...,n.
The initialization is given by y_0 ~ N(m0,C0).
Note that z_t has dimension d by 1, where d is the number of spatial locations and y_t has dimension p by 1, where p
is the dimension of the latent process. The matrix X_t is the known covariate matrix and has dimension d by r, where r is the number of covariates.
Moreover, the d-dimensional square matrix Σ_e is given by σ^2_ε+σ^2_omega in the diagonal (for spatial distance h equal to 0), while the off-diagonal entries are given by
σ^2_omega * C(h,theta), where C(h,theta) is the spatial covariance function. Using the default
exponential spatial covariance function, it is C(h,theta)=exp(-theta* h).
So the parameter vector phi is composed by β, σ^2_ε, σ^2_omega, theta, G,
Σ_eta and m0 (C0 is supposed fixed).
The elements required by the function must have the following characteristics:
beta
(matrix r * 1), sigma2eps
(scalar), sigma2omega
(scalar),
theta
(scalar), G
(matrix p * p), Sigmaeta
(matrix p * p), m0
(matrix p * 1), C0
(matrix p*p). Note that these values will be used as the true parameter values in the Stem.Simulation
function and as initial values for the EM algorithm in the Stem.Estimation
functionThe function returns a list which is given by:
skeleton |
a list with components phi , p , K as given in the input. |
data |
a list with components z , coordinates , covariates , as given in the input, r , n and d . |
No missing values are admitted in the observation matrix z
, in the covariates matrix covariates
and in the coordinates
matrix.
Michela Cameletti michela.cameletti@unibg.it
Amisigo, B.A., Van De Giesen, N.C. (2005) Using a spatio-temporal dynamic state-space model with the EM algorithm to patch gaps in daily riverflow series. Hydrology and Earth System Sciences 9, 209–224.
Fasso', A., Cameletti, M., Nicolis, O. (2007) Air quality monitoring using heterogeneous networks. Environmetrics 18, 245–264.
Fasso', A., Cameletti, M. (2007) A general spatio-temporal model for environmental data. Tech.rep. n.27 Graspa - The Italian Group of Environmental Statistics - http://www.graspa.org .
Fasso', A., Cameletti, M. (2009) A unified statistical approach for simulation, modelling, analysis and mapping of environmental data. Accepted for publication by Simulation: transaction of the Society for Modeling and Simulation International.
Mc Lachlan, G.J., Krishnan, T. (1997) The EM Algorithm and Extensions. Wiley, New York.
Shumway, R.H., Stoffer, D.S. (2006) Time Series Analysis and Its Applications: with R Examples. Springer, New York.
Xu, K., Wikle, C.K. (2007) Estimation of parameterized spatio-temporal dynamic models. Journal of Statistical Inference and Planning 137, 567–588.
#load the data data(pm10) names(pm10) #extract the data coordinates <- pm10$coords covariates <- pm10$covariates z <- pm10$z #build the parameter list phi <- list(beta=matrix(c(3.65,0.046,-0.904),3,1), sigma2eps=0.1, sigma2omega=0.2, theta=0.01, G=matrix(0.77,1,1), Sigmaeta=matrix(0.3,1,1), m0=as.matrix(0), C0=as.matrix(1)) K <-matrix(1,ncol(z),1) mod1 <- Stem.Model(z=z,covariates=covariates, coordinates=coordinates,phi=phi,K=K) class(mod1) is.Stem.Model(mod1)