TSSS (version 1.3.4-5)

tsmooth: Prediction and Interpolation of Time Series

Description

Predict and interpolate time series based on state space model by Kalman filter.

Usage

tsmooth(y, f, g, h, q, r, x0 = NULL, v0 = NULL, filter.end = NULL,
        predict.end = NULL, minmax = c(-1.0e+30, 1.0e+30), missed = NULL,
        np = NULL, plot = TRUE, ...)

Value

An object of class "smooth", which is a list with the following components:

mean.smooth

mean vectors of the smoother.

cov.smooth

variance of the smoother.

esterr

estimation error.

llkhood

log-likelihood.

aic

AIC.

Arguments

y

a univariate time series \(y_n\).

f

state transition matrix \(F_n\).

g

matrix \(G_n\).

h

matrix \(H_n\).

q

system noise variance \(Q_n\).

r

observational noise variance \(R\).

x0

initial state vector \(X(0\mid0)\).

v0

initial state covariance matrix \(V(0\mid0)\).

filter.end

end point of filtering.

predict.end

end point of prediction.

minmax

lower and upper limits of observations.

missed

start position of missed intervals.

np

number of missed observations.

plot

logical. If TRUE (default), mean vectors of the smoother and estimation error are plotted.

...

graphical arguments passed to plot.smooth.

Details

The linear Gaussian state space model is

$$x_n = F_n x_{n-1} + G_n v_n,$$ $$y_n = H_n x_n + w_n,$$

where \(y_n\) is a univariate time series, \(x_n\) is an \(m\)-dimensional state vector.

\(F_n\), \(G_n\) and \(H_n\) are \(m \times m\), \(m \times k\) matrices and a vector of length \(m\) , respectively. \(Q_n\) is \(k \times k\) matrix and \(R_n\) is a scalar. \(v_n\) is system noise and \(w_n\) is observation noise, where we assume that \(E(v_n, w_n) = 0\), \(v_n \sim N(0, Q_n)\) and \(w_n \sim N(0, R_n)\). User should give all the matrices of a state space model and its parameters. In current version, \(F_n\), \(G_n\), \(H_n\), \(Q_n\), \(R_n\) should be time invariant.

References

Kitagawa, G. (2020) Introduction to Time Series Modeling with Applications in R. Chapman & Hall/CRC.

Kitagawa, G. and Gersch, W. (1996) Smoothness Priors Analysis of Time Series. Lecture Notes in Statistics, No.116, Springer-Verlag.

Examples

## Example of prediction (AR model)
data(BLSALLFOOD)
BLS120 <- BLSALLFOOD[1:120]
z1 <- arfit(BLS120, plot = FALSE)
tau2 <- z1$sigma2

# m = maice.order, k=1
m1 <- z1$maice.order
arcoef <- z1$arcoef[[m1]]
f <- matrix(0.0e0, m1, m1)
f[1, ] <- arcoef
if (m1 != 1)
  for (i in 2:m1) f[i, i-1] <- 1
g <- c(1, rep(0.0e0, m1-1))
h <- c(1, rep(0.0e0, m1-1))
q <- tau2[m1+1]
r <- 0.0e0
x0 <- rep(0.0e0, m1)
v0 <- NULL

s1 <- tsmooth(BLS120, f, g, h, q, r, x0, v0, filter.end = 120, predict.end = 156)
s1

plot(s1, BLSALLFOOD)

## Example of interpolation of missing values (AR model)
z2 <- arfit(BLSALLFOOD, plot = FALSE)
tau2 <- z2$sigma2

# m = maice.order, k=1
m2 <- z2$maice.order
arcoef <- z2$arcoef[[m2]]
f <- matrix(0.0e0, m2, m2)
f[1, ] <- arcoef
if (m2 != 1)
  for (i in 2:m2) f[i, i-1] <- 1
g <- c(1, rep(0.0e0, m2-1))
h <- c(1, rep(0.0e0, m2-1))
q <- tau2[m2+1]
r <- 0.0e0
x0 <- rep(0.0e0, m2)
v0 <- NULL

tsmooth(BLSALLFOOD, f, g, h, q, r, x0, v0, missed = c(41, 101), np = c(30, 20))