options(repr.plot.width=10, repr.plot.height=6.5)   # this command just formats the size of the figures. Adapt to view them nicely
                                                    # in your browser.

# Linear regression

# Cell culture growth at low temperatues

# You found a new cell that you think grows well around freezing (T=0°C)


Temperature = seq(-4,4,by=0.02)    # Set up some temperature points

# Simulate data with error
GrowthRate = 0.2 +0.6*Temperature^2+  rnorm(length(Temperature),mean = 0,sd=1)        # Outcome of your measured data. Ideal slope + random error

plot(Temperature,GrowthRate)

# hypothesis: Growth rate depends on Temperature^2
#
# model: Y = alpha + beta X^2
# Transform to linear by variable transformation f(X)=Z

T2 = Temperature^2

plot(T2,GrowthRate)

# Linear regression computed in class:

n=length(T2)

beta = (n*mean(GrowthRate)*mean(T2)- sum(GrowthRate*T2)) / (n*mean(T2)^2-sum(T2^2))
alpha= mean(GrowthRate)-beta*mean(T2)

plot(T2,GrowthRate)
lines(T2,alpha+beta*T2,col="red",lwd=4)

paste('alpha: ', alpha)
paste('beta: ', beta)

# Now know regression coefficients, replot original variable
plot(Temperature,GrowthRate)
lines(Temperature,alpha+beta*Temperature^2,col="red",lwd=4)

# How good is the fit: Compare the sum of squares
SSTotal= sum(  (GrowthRate-mean(GrowthRate))^2 )
SSE = sum( (GrowthRate -alpha- beta*Temperature^2 )^2   )
SST = SSTotal-SSE

#Goodness of fit parameter R^2
RSQ=1-SSE/SSTotal

paste('SSTotal: ',SSTotal)
paste('SSE: ',SSE)
paste('SST: ',SST)
paste('R^2: ',RSQ)

# Let us do this in R directly
# First generate a dataframe from your columns, if you dont already have one.

mydata <- data.frame(Temperature,GrowthRate)

quadraticmodel <- lm(GrowthRate ~ I(Temperature^2), data=mydata)
predicted <- predict(quadraticmodel,Temperature=mydata$Temperature)

plot(Temperature,GrowthRate)
lines(Temperature,predicted)

predicted