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<table width="100%" summary="page for corAR1 {nlme}"><tr><td>corAR1 {nlme}</td><td style="text-align: right;">R Documentation</td></tr></table>

<h2>AR(1) Correlation Structure</h2>

<h3>Description</h3>

<p>This function is a constructor for the <code>corAR1</code> class,
representing an autocorrelation structure of order 1. Objects
created using this constructor must later be initialized using the
appropriate <code>Initialize</code> method. 
</p>

<h3>Usage</h3>

<pre>
corAR1(value, form, fixed)
</pre>

<h3>Arguments</h3>

<table summary="R argblock">
<tr valign="top"><td><code>value</code></td>
<td>
<p>the value of the lag 1 autocorrelation, which must be
between -1 and 1. Defaults to 0 (no autocorrelation).</p>
</td></tr>
<tr valign="top"><td><code>form</code></td>
<td>
<p>a one sided formula of the form <code>~ t</code>, or <code>~ t |
     g</code>, specifying a time covariate <code>t</code> and,  optionally, a
grouping factor <code>g</code>. A covariate for this correlation structure
must be integer valued. When a grouping factor is present in
<code>form</code>, the correlation structure is assumed to apply only
to observations within the same grouping level; observations with
different grouping levels are assumed to be uncorrelated. Defaults to
<code>~ 1</code>, which corresponds to using the order of the observations
in the data as a covariate, and no groups.</p>
</td></tr>
<tr valign="top"><td><code>fixed</code></td>
<td>
<p>an optional logical value indicating whether the
coefficients should be allowed to vary in the optimization, or kept
fixed at their initial value. Defaults to <code>FALSE</code>, in which case
the coefficients are allowed to vary.</p>
</td></tr>
</table>

<h3>Value</h3>

<p>an object of class <code>corAR1</code>, representing an autocorrelation
structure of order 1. 
</p>

<h3>Author(s)</h3>

<p>José Pinheiro and Douglas Bates <a href="mailto:bates@stat.wisc.edu">bates@stat.wisc.edu</a></p>

<h3>References</h3>

<p>Box, G.E.P., Jenkins, G.M., and Reinsel G.C. (1994) &quot;Time Series
Analysis: Forecasting and Control&quot;, 3rd Edition, Holden-Day.
</p>
<p>Pinheiro, J.C., and Bates, D.M. (2000) &quot;Mixed-Effects Models
in S and S-PLUS&quot;, Springer, esp. pp. 235, 397.  
</p>

<p><code><a href="ACF.lme.html">ACF.lme</a></code>,
<code><a href="corARMA.html">corARMA</a></code>, 
<code><a href="corClasses.html">corClasses</a></code>,
<code><a href="Dim.corSpatial.html">Dim.corSpatial</a></code>, 
<code><a href="Initialize.corStruct.html">Initialize.corStruct</a></code>, 
<code><a href="summary.corStruct.html">summary.corStruct</a></code>
</p>

<h3>Examples</h3>

<pre>
## covariate is observation order and grouping factor is Mare
cs1 &lt;- corAR1(0.2, form = ~ 1 | Mare)

# Pinheiro and Bates, p. 236
cs1AR1 &lt;- corAR1(0.8, form = ~ 1 | Subject)
cs1AR1. &lt;- Initialize(cs1AR1, data = Orthodont)
corMatrix(cs1AR1.)

# Pinheiro and Bates, p. 240
fm1Ovar.lme &lt;- lme(follicles ~ sin(2*pi*Time) + cos(2*pi*Time),
                   data = Ovary, random = pdDiag(~sin(2*pi*Time)))
fm2Ovar.lme &lt;- update(fm1Ovar.lme, correlation = corAR1())

# Pinheiro and Bates, pp. 255-258:  use in gls
fm1Dial.gls &lt;-
  gls(rate ~(pressure + I(pressure^2) + I(pressure^3) + I(pressure^4))*QB,
      Dialyzer)
fm2Dial.gls &lt;- update(fm1Dial.gls,
                 weights = varPower(form = ~ pressure))
fm3Dial.gls &lt;- update(fm2Dial.gls,
                    corr = corAR1(0.771, form = ~ 1 | Subject))

# Pinheiro and Bates use in nlme:  
# from p. 240 needed on p. 396
fm1Ovar.lme &lt;- lme(follicles ~ sin(2*pi*Time) + cos(2*pi*Time),
                   data = Ovary, random = pdDiag(~sin(2*pi*Time)))
fm5Ovar.lme &lt;- update(fm1Ovar.lme,
                corr = corARMA(p = 1, q = 1))
# p. 396
fm1Ovar.nlme &lt;- nlme(follicles~
     A+B*sin(2*pi*w*Time)+C*cos(2*pi*w*Time),
   data=Ovary, fixed=A+B+C+w~1,
   random=pdDiag(A+B+w~1),
   start=c(fixef(fm5Ovar.lme), 1) )
# p. 397
fm2Ovar.nlme &lt;- update(fm1Ovar.nlme,
         corr=corAR1(0.311) )
</pre>

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