View source for Least squares modelling (2013)

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| date = 08 February 2013 to 07 March 2013
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<span style="color:#900000">{{Huge|This page is out of date.}}</span> {{Huge|Please see the [[Least_squares_modelling |latest version of these notes]].}}


== Course notes and slides ==

* [http://learnche.mcmaster.ca/pid/?source=Least-squares Course textbook] (print chapter 4)
* [[Image:Nuvola_mimetypes_pdf.png|20px|link=Media:2013-4C3-Overheads-Least-squares-modelling.pdf]] [[Media:2013-4C3-Overheads-Least-squares-modelling.pdf|Slides for class]] 


== Software source code ==

Take a look at [[Software_tutorial|the software tutorial]].


== Code used in class ==

Least squares demo
<syntaxhighlight lang="sas">
x <- c(10, 8, 13, 9, 11, 14, 6, 4, 12, 7, 5)
y <- c(8.04, 6.95, 7.58, 8.81, 8.33, 9.96, 7.24, 4.26, 10.84, 4.82, 5.68)
plot(x,y)
model.ls <- lm(y ~ x)
summary(model.ls)

coef(model.ls)
confint(model.ls)

names(model.ls)
model.ls$resisduals
resid(model.ls)

plot(x, y)
abline(model.ls)
</syntaxhighlight>


Thermocouple data
<syntaxhighlight lang="sas">
V <- c(0.01, 0.12, 0.24, 0.38, 0.51, 0.67, 0.84, 1.01, 1.15, 1.31)
T <- c(273, 293, 313, 333, 353, 373, 393, 413, 433, 453)
plot(V, T)
model <- lm(T ~ V)
summary(model)
coef(model)
confint(model)   # get the coefficient confidence intervals
resid(model)     # model residuals
library(car)
qqPlot(resid(model)) # q-q plot of the residuals to check normality

plot(V, T)
v.new <- seq(0, 1.5, 0.1)
t.pred <- coef(model)[1] + coef(model)[2] * v.new
lines(v.new, t.pred, type="l", col="blue")

# Plot x against the residuals to check for non-linearity
plot(V, resid(model))
abline(h=0)

# Plot the raw data and the regression line in red
plot(V, T)
abline(model, col="red")
</syntaxhighlight>

Multiple linear regression (manually and automatically with R)
<syntaxhighlight lang="sas">
# Calculate the model manually
x1.raw <- c(1,3,4,7,9,9)
x2.raw <- c(9,9,6,3,1,2)
y.raw  <- c(3,5,6,8,7,10)
n <- length(x1.raw)

x1 <- x1.raw - mean(x1.raw)
x2 <- x2.raw - mean(x2.raw)
y <- y.raw - mean(y.raw)

X <- cbind(x1, x2)

# Calculate:  b = inv(X'X) X'y
XTX <- t(X) %*% X    # compare this to cov(X)*(n-1)
XTY <- t(X) %*% y
XTX.inv <- solve(XTX)
b <- XTX.inv %*% XTY
b

# Let R calculate the model:
model <- lm(y.raw ~ x1.raw + x2.raw)
summary(model)
</syntaxhighlight>