Difference between revisions of "Least squares modelling (2013)"
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__NOTOC__{{ClassSidebar | __NOTOC__{{ClassSidebar | ||
| date = 08 February 2013 | | date = 08 February 2013 to 07 March 2013 | ||
| dates_alt_text = | | dates_alt_text = | ||
| vimeoID1 = 59609654 | | vimeoID1 = 59609654 | ||
| vimeoID2 = 59763486 | | vimeoID2 = 59763486 | ||
| vimeoID3 = | | vimeoID3 = 60592349 | ||
| vimeoID4 = | | vimeoID4 = 60688909 | ||
| vimeoID5 = | | vimeoID5 = 60862565 | ||
| vimeoID6 = | | vimeoID6 = 61138805 | ||
| vimeoID7 = | | vimeoID7 = 61211690 | ||
| vimeoID8 = | | vimeoID8 = | ||
| course_notes_PDF = 2013-4C3-Overheads-Least-squares-modelling.pdf | | course_notes_PDF = 2013-4C3-Overheads-Least-squares-modelling.pdf | ||
| Line 22: | Line 22: | ||
| video_notes2 = | | video_notes2 = | ||
| video_download_link3_MP4 = http://learnche.mcmaster.ca/media/4C3-2013-Class-07A.mp4 | | video_download_link3_MP4 = http://learnche.mcmaster.ca/media/4C3-2013-Class-07A.mp4 | ||
| video_download_link3_MP4_size = M | | video_download_link3_MP4_size = 352 M | ||
| video_notes3 = | | video_notes3 = | ||
| video_download_link4_MP4 = http://learnche.mcmaster.ca/media/4C3-2013-Class-07B.mp4 | | video_download_link4_MP4 = http://learnche.mcmaster.ca/media/4C3-2013-Class-07B.mp4 | ||
| video_download_link4_MP4_size = M | | video_download_link4_MP4_size = 290 M | ||
| video_notes4 = | | video_notes4 = | ||
| video_download_link5_MP4 = | | video_download_link5_MP4 =http://learnche.mcmaster.ca/media/4C3-2013-Class-07C.mp4 | ||
| video_download_link5_MP4_size = | | video_download_link5_MP4_size = 375 M | ||
| video_notes5 = | | video_notes5 = | ||
| video_download_link6_MP4 = | | video_download_link6_MP4 = http://learnche.mcmaster.ca/media/4C3-2013-Class-08A.mp4 | ||
| video_download_link6_MP4_size = M | | video_download_link6_MP4_size = 372 M | ||
| video_notes6 = | | video_notes6 = | ||
| video_download_link7_MP4 = | | video_download_link7_MP4 = http://learnche.mcmaster.ca/media/4C3-2013-Class-08B.mp4 | ||
| video_download_link7_MP4_size = M | | video_download_link7_MP4_size = 380 M | ||
| video_notes7 = | | video_notes7 = | ||
| video_download_link8_MP4 = | | video_download_link8_MP4 = | ||
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| video_notes8 = | | video_notes8 = | ||
}} | }} | ||
<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{{Huge|This page is out of date.}}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 == | == Course notes and slides == | ||
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* [http://learnche.mcmaster.ca/pid/?source=Least-squares Course textbook] (print chapter 4) | * [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]] | * [[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 == | == 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> | |||
Latest revision as of 07:18, 4 January 2017
| Class date(s): | 08 February 2013 to 07 March 2013 | ||||
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Course slides | ||||
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This page is out of date. Please see the latest version of these notes.
Course notes and slides
- Course textbook (print chapter 4)
- Slides for class
Software source code
Take a look at the software tutorial.
Code used in class
Least squares demo
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)
Thermocouple data
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")
Multiple linear regression (manually and automatically with R)
# 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)