zy.G - 5 months ago 21

R Question

I'm doing a non-linear regression in R and want to add one moving-average term to my model to eliminate the autocorrelations in residuals.

Basically, here is the model:

`y[n] = a + log((x1[n])^g + (x2[n])^g) + c*e[n-1] + e[n]`

where

`[e]`

I plan to use

`ARIMA(0, 0, 1)`

More information: I know how to use

`nls`

`a`

`g`

`e[n]`

I know that

`xreg`

`arima`

Thank you for the help in advance!

Answer

`nlme`

has such capability, as it is fitting non-linear mixed models. You can think of it an extension to `nls`

(a fixed-effect only non-linear regression), by allowing random effect and correlated errors.

`nlme`

can handle ARMA correlation, by something like `correlation = corARMA(0.2, ~ 1, p = 0, q = 1, fixed = TRUE)`

. This means, that residuals are MA(1) process, with initial guess of coefficient 0.2, but to be updated during model fitting. The `~ 1`

suggests that `MA(1)`

is on intercept and there is no further grouping structure.

I am not an expert in `nlme`

, but I know `nlme`

is what you need. I produce the following example, but since I am not an expert, I can't get `nlme`

work at the moment. I post it here to give a start / flavour.

```
set.seed(0)
x1 <- runif(100)
x2 <- runif(100)
## MA(1) correlated error, with innovation standard deviation 0.1
e <- arima.sim(model = list(ma = 0.5), n = 100, sd = 0.1)
## a true model, with `a = 0.2, g = 0.5`
y0 <- 0.2 + log(x1 ^ 0.5 + x2 ^ 0.5)
## observations
y <- y0 + e
## no need to install; it comes with R; just `library()` it
library(nlme)
fit <- nlme(y ~ a + log(x1 ^ g + x2 ^ g), fixed = a + g ~ 1,
start = list(a = 0.5, g = 1),
correlation = corARMA(0.2, form = ~ 1, p = 0, q = 1, fixed = FALSE))
```

Similar to `nls`

, we have an overall model formula `y ~ a + log(x1 ^ g + x2 ^ g)`

, and starting values are required for iteration process. I have chosen `start = list(a = 0.5, g = 1)`

. The `correlation`

bit has been explained in the beginning.

`fixed`

and `random`

arguments in `nlme`

specify what should be seen as fixed effects and random effects in the overall formula. Since we have no random effect, we leave it unspecified. We want `a`

and `g`

as fixed effect, so I tried something like `fixed = a + g ~ 1`

. Unfortunately it does not quite work, for some reason I don't know. I read the `?nlme`

, and thought this formula means that we want a common `a`

and `g`

for all observations, but later `nlme`

reports an error saying this is not a valid group formula.

I am also investing at this; as I said, the above gives us a start. We are already fairly close to the final answer.

Thanks to user20650 for point out my awkward error. I should use `gnls`

function rather than `nlme`

. By design nature of `nlme`

package, functions `lme`

and `nlme`

have to take a `random`

argument to work. Luckily, there are several other routines in `nlme`

package for extending linear models and non-linear models.

`gls`

and`gnls`

extend`lm`

and`nls`

by allowing non-diagonal variance functions.

So, I should really use `gnls`

instead:

```
## no `fixed` argument as `gnls` is a fixed-effect only
fit <- gnls(y ~ a + log(x1 ^ g + x2 ^ g), start = list(a = 0.5, g = 1),
correlation = corARMA(0.2, form = ~ 1, p = 0, q = 1, fixed = FALSE))
#Generalized nonlinear least squares fit
# Model: y ~ a + log(x1^g + x2^g)
# Data: NULL
# Log-likelihood: 92.44078
#
#Coefficients:
# a g
#0.1915396 0.5007640
#
#Correlation Structure: ARMA(0,1)
# Formula: ~1
# Parameter estimate(s):
# Theta1
#0.4184961
#Degrees of freedom: 100 total; 98 residual
#Residual standard error: 0.1050295
```