Repeated Observations Linear Mixed Model with Outside-Group Spatial Correlation

Hi,

I'm new to the listserv.

A shortened version of my dataset is below. I am developing a model to
forecast monthly ridership at Bikeshare stations. I want to predict 'Cnts'
as a function of 'Population' - 'Temperature'. The dataset is unbalanced
(unequal number of observations for each station) and most of covariates do
not vary over time, but a few do.

I have successfully used lmer() and lme() in R to capture the dependency
between the error terms for repeated observations from a given station
('id').

model_spatial = lme(log(counts) ~ log(Population)
                 +Drive +Med_Income + Buff2 +Rain + Temperature
                 , data = Data, random = ~1|id, method = "ML" )

model_All = lmer(log(counts) ~ log(Population)
                 +Drive +Med_Income + Buff2 +Rain + Temperature
                 + (1|id)
                  , data = Data )

However, a Moran's I test of the residuals suggests that the residuals are
spatially correlated.
station.dists <- as.matrix(dist(cbind(Data$longitude, Data$latitude)))
station.dists.inv <- 1/station.dists
station.dists.inv[is.infinite(station.dists.inv)] <- 0   #Distance value is
inf for repeated observations from the same station
Data$resid_all = resid(model_spatial)
Moran.I(Data$resid_all, station.dists.inv)


Hence, I need to develop a model in R that accounts for spatial correlation
across stations, while simultaneously capturing correlations between
observations from a single station.  I've tried the following updates to
the lme() model, but have been unsuccessful.
model_spatial.gau <- update(model_spatial, correlation = corGaus(form = ~
latitude +longitude ), method = "ML")
model_spatial.gau <- update(model_spatial, correlation = corGaus(form = ~
latitude +longitude| id ), method = "ML")

Is there a way to formulate the correlation matrix in lme() or lmer() such
that the correlation between repeated obvservations of a given station *and*
the spatial autocorrelation between stations is accounted for?


year month id Cnts latitude longitude Population Drive Med_Income Buff2 Rain
Temperature
2015 8 2 2597 41.87264 -87.62398 4256 0.3418054 76857 127 0.07 71.8
2015 9 2 2772 41.87264 -87.62398 4256 0.3418054 76857 128 0.15 69
2015 10 2 684 41.87264 -87.62398 4256 0.3418054 76857 128 0.07 54.7
2015 11 2 394 41.87264 -87.62398 4256 0.3418054 76857 128 0.15 44.6
2015 12 2 148 41.87264 -87.62398 4256 0.3418054 76857 129 0.16 39
2016 1 2 44 41.87264 -87.62398 4256 0.3418054 76857 129 0.03 24.7
2015 5 3 2303 41.86723 -87.61536 16735 0.4312349 75227 90 0.15 60.4
2015 6 3 3323 41.86723 -87.61536 16735 0.4312349 75227 98 0.24 67.4
2015 7 3 5920 41.86723 -87.61536 16735 0.4312349 75227 98 0.09 72.3
2015 8 3 4405 41.86723 -87.61536 16735 0.4312349 75227 98 0.07 71.8
2015 9 3 3638 41.86723 -87.61536 16735 0.4312349 75227 99 0.15 69
2015 10 3 2061 41.86723 -87.61536 16735 0.4312349 75227 99 0.07 54.7
2015 11 3 1074 41.86723 -87.61536 16735 0.4312349 75227 99 0.15 44.6
2015 12 3 374 41.86723 -87.61536 16735 0.4312349 75227 100 0.16 39
2016 1 3 188 41.86723 -87.61536 16735 0.4312349 75227 100 0.03 24.7
2016 2 3 474 41.86723 -87.61536 16735 0.4312349 75227 100 0.04 30.4
2015 6 4 2968 41.85627 -87.61335 16735 0.4312349 75227 68 0.24 67.4
2015 7 4 4266 41.85627 -87.61335 16735 0.4312349 75227 68 0.09 72.3
2015 8 4 3442 41.85627 -87.61335 16735 0.4312349 75227 68 0.07 71.8
2015 9 4 2552 41.85627 -87.61335 16735 0.4312349 75227 69 0.15 69
2015 10 4 1301 41.85627 -87.61335 16735 0.4312349 75227 69 0.07 54.7


Thanks,
Mike Hyland

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