# examples of parametric modeling using R

#  I extracted july values and year info from 
#   co2, supplied with the R system

# fit a linear model
co2.trend <- lm(co2.july~co2.year)

# or a quadratic
co2.year2 <- co2.year^2
co2.trend <- lm(co2.july~co2.year + co2.year2)

summary(co2.trend)

# plot the autocorrelation function
acf(resid(co2.trend))
# or partial autocorrelation function
acf(resid(co2.trend),type='partial')

#lme function with correlation=corAR1() is supposed to fit
#  a regression with ar(1) errors, but lme() and I don't
#   get along well.  
#  I think the problem lme() expects subgroups of correl. obs
#   here there is only one group.
# co2.ar1 <- lme(co2.july ~ co2.year + co2.year2, correlation=corAR1() )

# fit loess regression
co2.loess <- loess(co2.july ~ co2.year)
#  span = specifies span (f in my notation), default 0.75
#  degree = specifies degree, default 2

summary(co2.loess)

plot(co2.year,co2.july)
lines(co2.year,predict(co2.loess))

# use loess to test lack of fit of a linear model
# anova.loess can only compare multiple loess fits
#  here's how to approximate a quadratic model by a loess fit
co2.loess2 <- loess(co2.july ~ co2.year, span = 100)

anova(co2.loess,co2.loess2)

# a linear model would be obtained by adding degree=1