04/19/23 330 Lecture 2 1

STATS 330: Lecture 2

04/19/23 330 Lecture 2 2

Housekeeping matters

• STATS 762– An extra test (details to be provided). An extra

assignment• Class rep – ????

• Office hours– Alan: 10:30 – 12:00 Tuesday and Thursday in Rm

265, Building 303S– Tutors: TBA

• Assignment 1: Due 2 August

04/19/23 330 Lecture 2 3

Today’s lecture: Exploratory graphics

Aim of the lecture:

To give you a quick overview of the kinds

of graphs that can be helpful in exploring data. Some of this material has been covered in 201/8. (We will discuss the R code used to make the plots in Lecture 4)

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Exploratory Graphics: topics

• Exploratory Graphics for a single variable: aim is to show distribution of values– Histograms– Kernel density estimators– Qq plots

• For 2 variables: aim is to show relationships– Both continuous: scatter plots– One of each: side by side boxplots– Both categorical: mosaic plots – see Ch 5

• 3 variables– Pairs plot– Rotating plot– coplots– 3D plots, contour plots

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Single variable:Exchange rate data

• The data of interest: daily changes in log(exchange rate) for the US$/Kiwi.– Monthly date from June 1986 to may 2012– Source: Reserve Bank

• Questions:– What is the distribution of the daily changes

in the logged exchange rate?– Is it normal? If not, how is it different?

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yt = exchange rate at time t

Difference in logs is log(yt) – log(yt-1) = log(yt/yt-1)

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Data AnalysisSuppose we have the data (3374 differences in

the logs), in an R vector, Diff.in.Logs

hist(Diff.in.Logs,nclass=100,freq=FALSE)# add density estimatelines(density(Diff.in.Logs),col="blue", lwd=2)# add fitted normal densityxvec = seq(-0.1, 0.1, length=100)lines(xvec,dnorm(xvec,mean=mean(Diff.in.Logs), sd=sd(Diff.in.Logs)),col="red", lwd=2)

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Normal plot> qqnorm(Diff.in.Logs)

Normal data?

No –

QQ plot indicates that the differences have longer tails than normal, since the plotted points are lower than the line for small values and higher for big ones

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Two Variables: Rats!Of interest: growth rates of 16 rats

i.e. relationship between weight and time

• Want to explore the relationship graphically.

• Each rat was measured (roughly) every week for 11 weeks

• For weeks 1-6, all rats were on a fixed diet. Diet was changed after week 6.

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Two Variables: Rats!Data set rats.df has variables

– rat (1-16)– growth (weight in grams)– day (day since start of study, 11 values, at

approximately weekly intervals– group (litter, one of 3)– change (has values 1 or 2 - diet was changed

after 6 weeks, diet 1 for weeks 1-6, diet 2 for weeks 7-11

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Rats: the data> rats.df growth group rat change day1 240 1 1 1 12 250 1 1 1 83 255 1 1 1 154 260 1 1 1 225 262 1 1 1 296 258 1 1 1 367 266 1 1 2 438 266 1 1 2 449 265 1 1 2 5010 272 1 1 2 5711 278 1 1 2 6412 225 1 2 1 112 230 1 2 1 8

... More data

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Rats (cont)

• Could plot weight (i.e. the variable growth) versus the variable day:

plot(day,growth)

BUT….

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0 10 20 30 40 50 60

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wth

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Criticisms• Can’t tell which points belong to which rat

• Seem to be 2 groups of points

• In actual fact, the rats came from 3 different litters, is this relevant?

• Could do better

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More rats: improvements

• Join points representing the same rat with a line

• Use different colours (or different line types e.g. dashed or dotted) for the different litters

• Use a legend

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Growth rate of rats

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Litter 1 Litter 2 Litter 3

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More improvements• Plot is too cluttered

• Could plot each rat on a different graph – important to use same scales (axes) for each graph

• This leads to the idea of “Trellis graphics”

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day

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day

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04/19/23 330 Lecture 2 22

Two variables: one continuous, one

categorical• Insurance data: data on 14,000 insurance

claims. Want to explore relationship between the amount of the claim (a continuous variable) and the type of car

(a categorical variable).

• Use side-by side boxplots.

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11 22 33 44 55 66 77 88 99 1111 1313 1515 1717

46

8

CARGROUP

Log(

AD

INC

UR

)Car Group

Loesssmooth

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More than 2 variables:

• If all variables are continuous, we can explore the relationships between them using a pairs plot

• If we have 3 variables, a rotating plot is a very useful tool

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Example: Cherry trees

> cherry.df diameter height volume1 8.3 70 10.32 8.6 65 10.33 8.8 63 10.24 10.5 72 16.45 10.7 81 18.86 10.8 83 19.77 11.0 66 15.68 11.0 75 18.29 11.1 80 22.610 11.2 75 19.9

... more data – 31 trees in all

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Cherry trees: pairs plots

> pairs(cherry.df)

diameter

65 70 75 80 85

810

1214

1618

20

6570

7580

85

height

8 10 12 14 16 18 20 10 20 30 40 50 60 70

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volume

3-d Rotating plots• The challenge: to represent a 3-

dimensional object on a 2-dimensional surface (a TV screen, computer screen etc)

• Traditional method uses projection, perspective

• A powerful idea is to use motion, looking at the 3-d scene from different angles

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Perspective

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Diameter -volume view

Diameter -height view

Volume -height view

Arbitrary view

Projection

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Cherry trees: rotating plot

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Dynamic motion• By dynamically changing the angle of

view, we get a better impression of the 3-dimensional structure of the data

• “Dynamic graphics” is a very powerful tool

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A powerful idea: Coplots

• Coplot shows relationship between x and y for selected values of z (usually a narrow range of z’s)

• By showing separate plots for different z ranges, we can see how the relationship between x and y changes as z changes

• Coplot: conditioning plot, shows relationship between x and y conditional on z (ie for fixed z)

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Cherry trees: coplots

• To show the relationship between height and volume for different values of diameter:

• Divide the range of diameter (8.3 to 20.6) up into 6 subranges 8-11, 10.5 -11.5 etc

• Draw 6 plots, the first using all data whose diameter is between 8 and 11, the second using all data whose diameter is between 10.5 and 11.5, and so on

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65 70 75 80 85 65 70 75 80 85

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me

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Given : diameter

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Interpretation• Note that the lines are not of the same slope• This implies that the point configuration is not

“planar”

rowcolumn

z

-60

-40

-20

0

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Other 3-d graphs

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plot of surface3-d scatter plot

Both can be rotated