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# Least squares modelling (2012)

Class date(s): 09 February, 01 March 2012
Video material (part 1)

Video material(part 2)

Video material (part 3)

Video material (part 4)

Video material (part 5)

## Materials

• Audio: 09 February and 01 March
• Video: from 09 February: class 6A (partial), class 6B not recorded (battery ran out; full audio is available though), 6C
• Video: from 01 March: class 7A, 7B, 7C
• Course notes (print chapter 4)
• Slides:
 Class date: 09 February 2012 (only slides 1 to 62)01 March 2012 (slides 63 to the end) I want my notes with: 1x1 (landscape) 2x1 (portrait) 3x1 (portrait) 3x1 (but with space for notes) 2x2 (landscape) 3x2 (portrait) pages per physical page Use page frames?

##  Code used in class

Thermocouple data (09 February)

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)

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")

Thermocouple data (01 March)

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)
confint(model)   # get the coefficient confidence intervals
resid(model)     # model residuals
qqPlot(resid(model)) # q-q plot of the residuals to check normality

# 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")

Example from the notes on multiple linear regression (MLR) (01 March)

# 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)