#MULTIPLE LINEAR REGRESSION

#DATA INPUT -  "volume.txt" - v-volume of tree in m^3, 
# d - BHD in cm, h - height in m, lc - length of a tree crown in m

volume=read.table("http://user.mendelu.cz/drapela/Forest_Biometry/Data/volume.txt",sep="\t",header=T)
head(volume)

attach(volume)

#exploratory data analysis and plots
library(car)
scatterplotMatrix(~v+d+h+lc,diagonal="boxplot",lwd=2)
cor(cbind(v,d,h,lc))
vif(lm(v~d+h+lc))
avPlots(lm(v~d+h+lc))


# full linear model
lm_vol1=lm(v~d+h+lc)
summary(lm_vol1)
confint(lm_vol1)

#SIMPLIFIED MODEL
lm_vol2=lm(v~d)
summary(lm_vol2)
scatterplot(d,v, id.n=5, lwd =2,lty.smooth = 5, lty.spread =3,lwd.smooth=1,lwd.spread=1)


#polynomial regression

lm_vol3= lm(v~d+I(d^2))
summary(lm_vol3)

lm_vol4= lm(v~-1+d+I(d^2))
summary(lm_vol4)

AIC(lm_vol2)
AIC(lm_vol3)
AIC(lm_vol4)

anova(lm_vol3,lm_vol2)
anova(lm_vol4,lm_vol3)

plot(d,v, xlab="diameter (cm)", ylab="volume(m^3)")
xx=(seq(min(d),max(d),1))
lines(xx,coef(lm_vol4)[1]*xx+coef(lm_vol4)[2]*I(xx^2),col="red")
lines(xx,coef(lm_vol2)[1]+coef(lm_vol2)[2]*xx,col="blue")