car: Fit Continuous Time AR Models to Irregularly Sampled Time...

In cts: Continuous Time Autoregressive Models

Description

Fit a continuous AR model to an irregularly sampled univariate time series with the Kalman filter

Usage

car(x, y=NULL, scale = 1.5, order = 3, ctrl=car_control())
## S3 method for class 'car'
print(x, digits = 3, ...)
## S3 method for class 'car'
summary(object, ...)
## S3 method for class 'car'
plot(x, type=c("spec", "pred", "diag"),...) 
## S3 method for class 'car'
predict(object, se.fit = TRUE, digits = 3, plot.it=TRUE,...)
## S3 method for class 'car'
spectrum(object, frmult=1, n.freq, plot.it = TRUE, na.action = na.fail, ...)
## S3 method for class 'car'
AIC(object, ..., k=NULL)
## S3 method for class 'car'
tsdiag(object, gof.lag = 10, ...)
## S3 method for class 'car'
kalsmo(object)

Arguments

x	two column data frame or matrix with the first column being the sampled time and the second column being the observations at the first column; otherwise x is a numeric vector of sampled time. It can be a car object for S3 methods
y	not used if x has two columns; otherwise y is a numeric vector of observations at sampled time x.
scale	The kappa value referred to in the paper by Belcher et a. (1994). We now recommend selection of kappa along with the model order by using AIC. Also, it is suggested to choose kappa close to 2pi times 1/mean.delta (reciprocal of the mean time between observations), though it is a good idea to explore somewhat lower and higher values to see whether the spectrum estimates were sensitive to this choice. Choosing kappa lower increases the risk of trying to estimate the spectrum beyond the effective Nyquist frequency of the data - though this does depend on the distribution of intersample times.
order	order of autoregression.
ctrl	control parameters used in predict and numerical optimization.
object	object of class car
type	in plot a character indicating the type of plot. type="spec", call spec; type="pred", call predict; type="diag", call diag
se.fit	Logical: should standard errors of prediction be returned?
digits	return value digits
plot.it	Logical: plot the forecast values?
gof.lag	the maximum number of lags for a Portmanteau goodness-of-fit test
frmult	numerical value, can be used to multiply the frequency range
n.freq	number of frequency
k	penalty, not used
na.action	NA action function.
...	further arguments to be passed to particular methods

Details

spectrum returns (and by default plots) the spectral density of the fitted model.

tsdiag is a generic diagnostic function for continuous AR model. It will generally plot the residuals, often standadized, the autocorrelation function of the residuals, and the p-values of a Portmanteau test for all lags up to gof.lag. The method for car object plots residuals scaled by the estimate of their (individual) variance, and use the Ljung–Box version of the portmanteau test.

AIC For continuous CAR model selection, t-statistic and AIC are calculated based on reparameterized coefficients phi and covariance matrix ecov. From the t-statistic, the final model is chosen such that if the true model order is less than the large value used for model estimation then for i > order the deviations of the estimated parameters phi from their true value of 0 will be small. From the AIC, the final model is chosen based on the smallest AIC value. A table with t-statistic and AIC for the corresponding model order.

factab calculate characteristic roots and system frequency from the estimated reparameterized coefficients of CAR fits.

smooth computes components corresponding to the diagonal transition matrix with the Kalman smoother. This may not be stable for some data due to numerical inversion of matrix.

Value

A list of class "car" with the following elements:

n.used	The number of observations used in fitting
order	The order of the fitted model. This is chosen by the user.
np	The number of parameters estimated. This may include the mean and the observation noise ratio.
scale	The kappa value referred to in the paper of Belcher et al.
vri	If vri=1, estimate the observation noise ratio.
vr	The estimated observation noise ratio.
sigma2	The estimated innovation variance.
phi	The estimated reparameterized autoregressive parameters.
x.mean	The estimated mean of the series used in fitting and for use in prediction.
b	All estimated parameters, which include phi, and possibly x.mean and vr.
delb	The estimated standard error of b
essp	The estimated correlation matrix of b
ecov	The estimated covariance matrix of phi. See also AIC
rootr	The real part of roots of phi. See also AIC
rooti	The imaginary part of roots of phi. See also AIC
tim	The numeric vector of sampled time.
ser	The numeric vector of observations at sampled time tim.
filser	The filtered time series with the Kalman filter.
filvar	The estimated variance of Kalman filtered time series filser
sser	The smoothed time series with the Kalman smoother.
svar	The estimated variance of smoothed time series sser
stdred	The standardized residuals from the fitted model.
pretime	Time of predictions.
pred	Predictions for the pretime.
prv	Prediction variance of pred
pre2	Fitted values including pred for all the time series. See also fty.
prv2	Variance of fitted values including prv for all the time series. See also fty.
fty	Forecast type
tnit	Numeric vector: iteration
ss	Numeric vector: sum of squares for each tnit
bit	Matrix with rows for tnit and columns for parameter estimates
aic	AIC value for the fitted model
bic	BIC value for the fitted model

Author(s)

G. Tunnicliffe Wilson and Zhu Wang

References

Belcher, J. and Hampton, J. S. and Tunnicliffe Wilson, G. (1994). Parameterization of continuous time autoregressive models for irregularly sampled time series data. Journal of the Royal Statistical Society, Series B, Methodological,56,141–155

Jones, Richard H. (1981). Fitting a continuous time autoregression to discrete data. Applied Time Series Analysis II, 651–682

Wang, Zhu (2004). The Application of the Kalman Filter to Nonstationary Time Series through Time Deformation. PhD thesis, Southern Methodist University

Wang, Zhu and Woodward, W. A. and Gray, H. L. (2009). The Application of the Kalman Filter to Nonstationary Time Series through Time Deformation. Journal of Time Series Analysis, 30(5), 559-574.

Wang, Zhu (2013). cts: An R Package for Continuous Time Autoregressive Models via Kalman Filter. Journal of Statistical Software, Vol. 53(5), 1–19. http://www.jstatsoft.org/v53/i05

See Also

car_control for predict and numerical optimization parameters, and AIC for model selection

Examples

data(V22174)
fit <- car(V22174,scale=0.2,order=7, ctrl=car_control(trace=TRUE))
summary(fit)
spectrum(fit)
tsdiag(fit)
AIC(fit)
factab(fit)
###fitted values vs observed values
ntim <- dim(V22174)[1]
plot(V22174[,1], V22174[,2], type="l")
points(V22174[,1], fit$pre2[1:ntim], col="red")
### alternatively
fit2 <- car(V22174,scale=0.2,order=7, ctrl=car_control(fty=3))
plot(V22174[,1], V22174[,2], type="l")
points(V22174[,1], fit2$pre2, col="red")

data(asth)
fit <- car(asth,scale=0.25,order=4, ctrl=car_control(n.ahead=10))
kalsmo(fit)

Example output

Attaching package: 'cts'

The following objects are masked from 'package:stats':

    spectrum, tsdiag


READING OF MODEL PARAMETER PARAMETER SUCCESSFUL
READING OF CONTROL PARAMETER SUCCESSFUL 
 
ROOT EQUALITY SWITCH: 
[1] 1
MAIN LOOP IN RESG1 BEGINS
ITERATION 0:
LAMBDA = 
[1] 0.01
   INITIAL SUM OF SQUARES = 
[1] 12.92737
   INITIAL PARAMETER VALUES
[1] 0 0 0 0 0 0 0
   INITIAL VALUE OF CONSTANT TERM = 
[1] 0.1053049
MAIN LOOP IN RESG1 BEGINS
MAIN LOOP IN RESG1 BEGINS
MAIN LOOP IN RESG1 BEGINS
MAIN LOOP IN RESG1 BEGINS
MAIN LOOP IN RESG1 BEGINS
MAIN LOOP IN RESG1 BEGINS
ROOT EQUALITY SWITCH: 
[1] 0
 
ITERATION :
[1] 1
   LAMBDA: 
[1] 0.001
   SUM OF SQUARES = 
[1] 9.107596
   PARAMETER VALUES
[1] -0.30940142  0.12339492  0.08810410  0.04348192  0.41661288 -0.10084202
[7]  0.20167791  0.17401878
ROOT EQUALITY SWITCH: 
[1] 0
ROOT EQUALITY SWITCH: 
[1] 0
ROOT EQUALITY SWITCH: 
[1] 0
 
ITERATION :
[1] 2
   LAMBDA: 
[1] 0.01
   SUM OF SQUARES = 
[1] 8.853833
   PARAMETER VALUES
[1] -0.394234150  0.214279964  0.099421439  0.007241027  0.531324736
[6] -0.242421399  0.334272521  0.172746273
ROOT EQUALITY SWITCH: 
[1] 0
 
ITERATION :
[1] 3
   LAMBDA: 
[1] 0.001
   SUM OF SQUARES = 
[1] 8.754946
   PARAMETER VALUES
[1] -0.49817173  0.33311175  0.09249703 -0.02483595  0.60470553 -0.37474650
[7]  0.46502435  0.17300766
ROOT EQUALITY SWITCH: 
[1] 0
 
ITERATION :
[1] 4
   LAMBDA: 
[1] 1e-04
   SUM OF SQUARES = 
[1] 8.745203
   PARAMETER VALUES
[1] -0.49787052  0.35157496  0.08335691 -0.01990383  0.60243629 -0.36674995
[7]  0.47582980  0.17321285
ROOT EQUALITY SWITCH: 
[1] 0
 
ITERATION :
[1] 5
   LAMBDA: 
[1] 1e-05
   SUM OF SQUARES = 
[1] 8.744615
   PARAMETER VALUES
[1] -0.50211925  0.35494164  0.08588158 -0.02213610  0.60544366 -0.37311695
[7]  0.48349319  0.17331639
ROOT EQUALITY SWITCH: 
[1] 0
 
PROCESSING COMPLETED
ITERATIONS COMPLETED:
[1] 6
CONVERGENCE ACHIEVED
SEARCH PROGRESSED
 
FINAL SUM OF SQUARES: 
[1] 8.74455
MEAN SUM OF SQUARES : 
[1] 0.05605481
INNOVATION PROCESS VARIANCE ESTIMATE: 
[1] 1.369916e-09
GEOMETRIC MEAN VARIANCE MULTIPLIER:   
[1] 40918427
FINAL PARAMETER VALUES:
[1] -0.50100015  0.35535886  0.08542050 -0.02177307  0.60469577 -0.37050651
[7]  0.48268799  0.17330672
 
MAIN LOOP IN RESG1 BEGINS

Call:
car(x = V22174, scale = 0.2, order = 7, ctrl = car_control(trace = TRUE))

Order of model = 7, sigma^2 = 1.37e-09 

Estimated coefficients (standard errors):
      phi_1 phi_2 phi_3  phi_4 phi_5  phi_6 phi_7
coef -0.501 0.355 0.085 -0.022 0.605 -0.371 0.483
S.E.  0.108 0.111 0.060  0.071 0.084  0.124 0.112

Estimated mean (standard error):
[1] 0.173
[1] 0.022

Call:
car(x = V22174, scale = 0.2, order = 7, ctrl = car_control(trace = TRUE))

Model selection statistics 

 order t.statistic     AIC
     1       -4.77  -20.78
     2       -4.45  -38.57
     3        3.25  -47.15
     4        2.37  -50.76
     5        6.11  -86.05
     6       -0.76  -84.63
     7        4.32 -101.27

Call:
factab(object = fit)

Characteristic root of original parameterization in alpha 

             1               2               3               4               5  
 -0.006+0.058i   -0.006-0.058i   -0.029+0.300i   -0.029-0.300i   -0.030+0.135i
             6               7  
 -0.030-0.135i  -11.921+0.000i

Frequency 

    1      2      3      4      5      6      7  
0.009  0.009  0.048  0.048  0.022  0.022  0.000  

cts documentation built on May 2, 2019, 2:46 a.m.