remotePARTS (version 1.0.4)

fitAR: AR regressions by REML

Description

fitAR is used to fit AR(1) time series regression analysis using restricted maximum likelihood

Usage

fitAR(formula, data = NULL)
AR_fun(par, y, X, logLik.only = TRUE)

Value

fitAR returns a list object of class "remoteTS", which contains the following elements.

call

the function call

coefficients

a named vector of coefficients

SE

the standard errors of parameter estimates

tstat

the t-statistics for coefficients

pval

the p-values corresponding to t-tests of coefficients

MSE

the model mean squared error

logLik

the log-likelihood of the model fit

residuals

the residuals: response minus fitted values

fitted.values

the fitted mean values

rho

The AR parameter, determined via REML

rank

the numeric rank of the fitted model

df.residual

the residual degrees of freedom

terms

the stats::terms object used

Output is structured similarly to an "lm" object.

When logLik.only == F, AR_fun returns the output described in ?fitAR. When logLik.only == T, it returns a quantity that is linearly and negatively related to the restricted log likelihood (i.e., partial log-likelihood).

Arguments

formula

a model formula, as used by stats::lm()

data

optional data environment to search for variables in formula. As used by lm()

par

AR parameter value

y

vector of time series (response)

X

model matrix (predictors)

logLik.only

logical: should only the partial log-likelihood be computed

Details

This function finds the restricted maximum likelihood (REML) to estimate parameters for the regression model with AR(1) random error terms

$$y(t) = X(t) \beta + \varepsilon(t)$$ $$\varepsilon(t) = \rho \varepsilon(t-1) + \delta(t)$$

where \(y(t)\) is the response at time \(t\);

\(X(t)\) is a model matrix containing covariates;

\(\beta\) is a vector of effects of \(X(t)\); \(\varepsilon(t)\) is the autocorrelated random error;

\(\delta \sim N(0, \sigma)\) is a temporally independent Gaussian random variable with mean zero and standard deviation \(\sigma\);

and \(\rho\) is the AR(1) autoregression parameter

fitAR estimates the parameter via mathematical optimization of the restricted log-likelihood function.

AR_fun is the work horse behind fitAR that is called by optim to estimate the autoregression parameter \(\rho\).

References

Ives, A. R., K. C. Abbott, and N. L. Ziebarth. 2010. Analysis of ecological

time series with ARMA(p,q) models. Ecology 91:858-871.

See Also

fitAR_map to easily apply fit_AR to many pixels; fitCLS and fitCLS_map for conditional least squares time series analyses.

Other remoteTS: fitAR_map(), fitCLS_map(), fitCLS()

Other remoteTS: fitAR_map(), fitCLS_map(), fitCLS()

Examples

# simulate dummy data
t = 1:30 # times series
Z = rnorm(30) # random independent variable
x = .2*Z + (.05*t) # generate dependent effects
x[2:30] = x[2:30] + .2*x[1:29] # add autocorrelation

# fit the AR model, using Z as a covariate
(AR = fitAR(x ~ Z))

# get specific components
AR$residuals
AR$coefficients
AR$pval

# now using time as a covariate
(AR.time <- fitAR(x ~ t))

# source variable from a dataframe
df = data.frame(y = x, t.scaled = t/30, Z = Z)
fitAR(y ~ t.scaled + Z, data = df)

## Methods
summary(AR)
residuals(AR)
coefficients(AR)