univariate eda (exploratory data analysis). eda john tukey (1970s) data –two components: smooth +...
TRANSCRIPT
![Page 1: Univariate EDA (Exploratory Data Analysis). EDA John Tukey (1970s) data –two components: smooth + rough patterned behaviour + random variation resistant](https://reader035.vdocuments.us/reader035/viewer/2022062321/56649e215503460f94b0df92/html5/thumbnails/1.jpg)
Univariate EDA
(Exploratory Data Analysis)
![Page 2: Univariate EDA (Exploratory Data Analysis). EDA John Tukey (1970s) data –two components: smooth + rough patterned behaviour + random variation resistant](https://reader035.vdocuments.us/reader035/viewer/2022062321/56649e215503460f94b0df92/html5/thumbnails/2.jpg)
![Page 3: Univariate EDA (Exploratory Data Analysis). EDA John Tukey (1970s) data –two components: smooth + rough patterned behaviour + random variation resistant](https://reader035.vdocuments.us/reader035/viewer/2022062321/56649e215503460f94b0df92/html5/thumbnails/3.jpg)
![Page 4: Univariate EDA (Exploratory Data Analysis). EDA John Tukey (1970s) data –two components: smooth + rough patterned behaviour + random variation resistant](https://reader035.vdocuments.us/reader035/viewer/2022062321/56649e215503460f94b0df92/html5/thumbnails/4.jpg)
EDA• John Tukey (1970s)
• data– two components:
• smooth + rough
• patterned behaviour + random variation
• resistant measures/displays– little influenced by changes in a small proportion of the total
number of cases
– resistant to the effects of outliers
– emphasizes smooth over rough components
• concepts apply to statistics and to graphical methods
![Page 5: Univariate EDA (Exploratory Data Analysis). EDA John Tukey (1970s) data –two components: smooth + rough patterned behaviour + random variation resistant](https://reader035.vdocuments.us/reader035/viewer/2022062321/56649e215503460f94b0df92/html5/thumbnails/5.jpg)
Tree Ring dates (AD)
1255 1239 1162 1239 1240 1243 1241 1241 1271
• 9 dendrochronology dates
• what do they mean????
• usually helps to sort the data…
![Page 6: Univariate EDA (Exploratory Data Analysis). EDA John Tukey (1970s) data –two components: smooth + rough patterned behaviour + random variation resistant](https://reader035.vdocuments.us/reader035/viewer/2022062321/56649e215503460f94b0df92/html5/thumbnails/6.jpg)
Stem-and-Leaf Diagram
1162 1239 1239 1240 1241 1241 1243 1255 1271
11|62
12|39,39,40,41,41,43,55,71
• original values preserved
• no rounding, no loss of information…
![Page 7: Univariate EDA (Exploratory Data Analysis). EDA John Tukey (1970s) data –two components: smooth + rough patterned behaviour + random variation resistant](https://reader035.vdocuments.us/reader035/viewer/2022062321/56649e215503460f94b0df92/html5/thumbnails/7.jpg)
can simplify in various ways…
11|6
12|44444467
– ‘leaves’ rounded to nearest decade
– ‘stem’ based on centuries
![Page 8: Univariate EDA (Exploratory Data Analysis). EDA John Tukey (1970s) data –two components: smooth + rough patterned behaviour + random variation resistant](https://reader035.vdocuments.us/reader035/viewer/2022062321/56649e215503460f94b0df92/html5/thumbnails/8.jpg)
1162 1239 1239 1240 1241 1241 1243 1255 1271
116|2117|118|119|120|121|122|123|99124|0113125|5126|127|1
‘stem’ based on decades…
![Page 9: Univariate EDA (Exploratory Data Analysis). EDA John Tukey (1970s) data –two components: smooth + rough patterned behaviour + random variation resistant](https://reader035.vdocuments.us/reader035/viewer/2022062321/56649e215503460f94b0df92/html5/thumbnails/9.jpg)
1162 1239 1239 1240 1241 1241 1243 1255 1271
116|2117|118|119|120|121|122|123|99124|0113125|5126|127|1
highlights existence of gaps in the distribution of dates, groups of dates…
![Page 10: Univariate EDA (Exploratory Data Analysis). EDA John Tukey (1970s) data –two components: smooth + rough patterned behaviour + random variation resistant](https://reader035.vdocuments.us/reader035/viewer/2022062321/56649e215503460f94b0df92/html5/thumbnails/10.jpg)
R• stem()
• vuround(runif(25, 0, 50),0); stem(vu)
• vnround(rnorm(25, 25, 10),0); stem(vn)
• stem(vn, scale=2)
![Page 11: Univariate EDA (Exploratory Data Analysis). EDA John Tukey (1970s) data –two components: smooth + rough patterned behaviour + random variation resistant](https://reader035.vdocuments.us/reader035/viewer/2022062321/56649e215503460f94b0df92/html5/thumbnails/11.jpg)
unit 1 unit 2
12.6 16.2
11.6 16.4
16.3 13.8
13.1 13.2
12.1 11.3
26.9 14
9.7 9
11.5 12.5
14.8 15.6
13.5 11.2
12.4 12.2
13.6 15.5
11.7
9 26
25
24
23
22
21
20
19
18
17
3 16 24
15 56
8 14 0
651 13 28
641 12 25
65 11 237
10
7 9 0
unit 1 unit 2
Back-to-back stem-and-leaf plot
rimdiameterdata (cm)
![Page 12: Univariate EDA (Exploratory Data Analysis). EDA John Tukey (1970s) data –two components: smooth + rough patterned behaviour + random variation resistant](https://reader035.vdocuments.us/reader035/viewer/2022062321/56649e215503460f94b0df92/html5/thumbnails/12.jpg)
percentiles
• useful for constructing various kinds of EDA graphics
• don’t confuse percentile with percent or proportion
Note:• frequency = count• relative frequency = percent or proportion
![Page 13: Univariate EDA (Exploratory Data Analysis). EDA John Tukey (1970s) data –two components: smooth + rough patterned behaviour + random variation resistant](https://reader035.vdocuments.us/reader035/viewer/2022062321/56649e215503460f94b0df92/html5/thumbnails/13.jpg)
percentiles
“the pth percentile of a distribution: number such that approximately p percent of the
values in the distribution are equal or less than that number…”
• can be calculated for numbers that actually exist in the distribution, and interpolated for numbers than don’t…
![Page 14: Univariate EDA (Exploratory Data Analysis). EDA John Tukey (1970s) data –two components: smooth + rough patterned behaviour + random variation resistant](https://reader035.vdocuments.us/reader035/viewer/2022062321/56649e215503460f94b0df92/html5/thumbnails/14.jpg)
percentiles
• sort the data so that x1 is the smallest value, and xn is the largest (where n=total number of cases)
• xi is the pith percentile of a dataset of n members where:
n
ipi
5.0100
![Page 15: Univariate EDA (Exploratory Data Analysis). EDA John Tukey (1970s) data –two components: smooth + rough patterned behaviour + random variation resistant](https://reader035.vdocuments.us/reader035/viewer/2022062321/56649e215503460f94b0df92/html5/thumbnails/15.jpg)
original data:
5 1 9 3 14 9 7
sorted data:
x i 1 3 5 7 9 9 14
i 1 2 3 4 5 6 7p i (calculate, using equation [1], as shown below…)
p1 = 100(1 - 0.5) / 7 = 7.1p2 = 100(2 - 0.5) / 7 = 21.4p3 = 100(3 - 0.5) / 7 = 35.7p4 = 100(4 - 0.5) / 7 = 50etc…
x i 1 3 5 7 9 9 14
i 1 2 3 4 5 6 7p i 7.1 21.4 35.7 50 64.3 78.6 92.9
n
ipi
5.0100
[1]
![Page 16: Univariate EDA (Exploratory Data Analysis). EDA John Tukey (1970s) data –two components: smooth + rough patterned behaviour + random variation resistant](https://reader035.vdocuments.us/reader035/viewer/2022062321/56649e215503460f94b0df92/html5/thumbnails/16.jpg)
n
ipi
5.0100
5.0
100 inp
i
x i 1 3 5 7 9 9 14
i 1 2 3 4 5 6 7p i 7.1 21.4 35.7 50 64.3 78.6 92.9
25
?
85
?
50
50th percentile:i=(7*50)/100 + .5i=4, xi=7
25th percentile:i=(7*25)/100 + .5i=2.25, 3<xi<5
![Page 17: Univariate EDA (Exploratory Data Analysis). EDA John Tukey (1970s) data –two components: smooth + rough patterned behaviour + random variation resistant](https://reader035.vdocuments.us/reader035/viewer/2022062321/56649e215503460f94b0df92/html5/thumbnails/17.jpg)
x i 1 3 5 7 9 9 14
i 1 2 3 4 5 6 7p i 7.1 21.4 35.7 50 64.3 78.6 92.9
?
if i < > integer, then…k = integer part of i; f = fractional part of ixint = interpolated value of xxint = (1-f)xk + fxk+1
xint= (1-.25)*3+.25*5xint= 3.5
25th percentile:i=(7*25)/100 + .5i=2.25, 3<xi<5
25
![Page 18: Univariate EDA (Exploratory Data Analysis). EDA John Tukey (1970s) data –two components: smooth + rough patterned behaviour + random variation resistant](https://reader035.vdocuments.us/reader035/viewer/2022062321/56649e215503460f94b0df92/html5/thumbnails/18.jpg)
use R!!
• test<-c(1,3,5,7,9,9,14)
• quantile(test, .25, type=5)
![Page 19: Univariate EDA (Exploratory Data Analysis). EDA John Tukey (1970s) data –two components: smooth + rough patterned behaviour + random variation resistant](https://reader035.vdocuments.us/reader035/viewer/2022062321/56649e215503460f94b0df92/html5/thumbnails/19.jpg)
75th25th 50thpercentiles:
interquartilerange
(midspread)
upper hingelower hinge inner fenceinner fence
“boxplot”63 5885 4795 3344 393 117 11
80 526 1962 320 4286 3752 9055 8664 283 27
65 6046 4129 5596 8982 9066 6399 8326 3295 7276 9746 6765 8184 75
(1.5 x midspread)
![Page 20: Univariate EDA (Exploratory Data Analysis). EDA John Tukey (1970s) data –two components: smooth + rough patterned behaviour + random variation resistant](https://reader035.vdocuments.us/reader035/viewer/2022062321/56649e215503460f94b0df92/html5/thumbnails/20.jpg)
Figure 6.25: Internal diversity of neighbourhoods used to define N-clusters, measured by the 'evenness' statistic H/Hmax on the basis of counts of various A-clusters, and broken down by N-cluster and phase. [Boxes encompass the midspread; lines inside boxes indicate the median, while whiskers show the range of cases that fall within 1.5-times the midspread, above or below the limits of the box.]
![Page 21: Univariate EDA (Exploratory Data Analysis). EDA John Tukey (1970s) data –two components: smooth + rough patterned behaviour + random variation resistant](https://reader035.vdocuments.us/reader035/viewer/2022062321/56649e215503460f94b0df92/html5/thumbnails/21.jpg)
Cleveland, W. S. (1985) The Elements of Graphing Data.
![Page 22: Univariate EDA (Exploratory Data Analysis). EDA John Tukey (1970s) data –two components: smooth + rough patterned behaviour + random variation resistant](https://reader035.vdocuments.us/reader035/viewer/2022062321/56649e215503460f94b0df92/html5/thumbnails/22.jpg)
Histograms
• divide a continuous variable into intervals called ‘bins’
• count the number of cases within each bin
• use bars to reflect counts
• intervals on the horizontal axis
• counts on the vertical axis
![Page 23: Univariate EDA (Exploratory Data Analysis). EDA John Tukey (1970s) data –two components: smooth + rough patterned behaviour + random variation resistant](https://reader035.vdocuments.us/reader035/viewer/2022062321/56649e215503460f94b0df92/html5/thumbnails/23.jpg)
“bins”
Histogram
coun
ts percent63 5885 4795 3344 393 117 11
80 526 1962 320 4286 3752 9055 8664 283 27
65 6046 4129 5596 8982 9066 6399 8326 3295 7276 9746 6765 8184 75
![Page 24: Univariate EDA (Exploratory Data Analysis). EDA John Tukey (1970s) data –two components: smooth + rough patterned behaviour + random variation resistant](https://reader035.vdocuments.us/reader035/viewer/2022062321/56649e215503460f94b0df92/html5/thumbnails/24.jpg)
• useful for illustrating the shape of the distribution of a batch of numbers
• may be helpful for identifying modes and modal behaviour
Histograms
![Page 25: Univariate EDA (Exploratory Data Analysis). EDA John Tukey (1970s) data –two components: smooth + rough patterned behaviour + random variation resistant](https://reader035.vdocuments.us/reader035/viewer/2022062321/56649e215503460f94b0df92/html5/thumbnails/25.jpg)
mode
mode?
mode!
• the distribution is clearly bimodal
• may be multimodal…
![Page 26: Univariate EDA (Exploratory Data Analysis). EDA John Tukey (1970s) data –two components: smooth + rough patterned behaviour + random variation resistant](https://reader035.vdocuments.us/reader035/viewer/2022062321/56649e215503460f94b0df92/html5/thumbnails/26.jpg)
![Page 27: Univariate EDA (Exploratory Data Analysis). EDA John Tukey (1970s) data –two components: smooth + rough patterned behaviour + random variation resistant](https://reader035.vdocuments.us/reader035/viewer/2022062321/56649e215503460f94b0df92/html5/thumbnails/27.jpg)
![Page 28: Univariate EDA (Exploratory Data Analysis). EDA John Tukey (1970s) data –two components: smooth + rough patterned behaviour + random variation resistant](https://reader035.vdocuments.us/reader035/viewer/2022062321/56649e215503460f94b0df92/html5/thumbnails/28.jpg)
![Page 29: Univariate EDA (Exploratory Data Analysis). EDA John Tukey (1970s) data –two components: smooth + rough patterned behaviour + random variation resistant](https://reader035.vdocuments.us/reader035/viewer/2022062321/56649e215503460f94b0df92/html5/thumbnails/29.jpg)
important variables in histogram constuction:
• bin width• bin starting point
![Page 30: Univariate EDA (Exploratory Data Analysis). EDA John Tukey (1970s) data –two components: smooth + rough patterned behaviour + random variation resistant](https://reader035.vdocuments.us/reader035/viewer/2022062321/56649e215503460f94b0df92/html5/thumbnails/30.jpg)
smoothing histograms
• may want to accentuate the ‘smooth’ in a data distribution…
• calculate “running averages” on bin counts• level of smoothing is arbitrary…
1 3 5 2 4 2 0 1
2 3 3.3 3.6 2.6 2 1 0.5
![Page 31: Univariate EDA (Exploratory Data Analysis). EDA John Tukey (1970s) data –two components: smooth + rough patterned behaviour + random variation resistant](https://reader035.vdocuments.us/reader035/viewer/2022062321/56649e215503460f94b0df92/html5/thumbnails/31.jpg)
0
1
2
3
4
5
6
0
0.5
1
1.5
2
2.5
3
3.5
4
![Page 32: Univariate EDA (Exploratory Data Analysis). EDA John Tukey (1970s) data –two components: smooth + rough patterned behaviour + random variation resistant](https://reader035.vdocuments.us/reader035/viewer/2022062321/56649e215503460f94b0df92/html5/thumbnails/32.jpg)
histogram / barchart variations
• 3d
• stacked
• dual
• frequency polygon
• kernel density methods
![Page 33: Univariate EDA (Exploratory Data Analysis). EDA John Tukey (1970s) data –two components: smooth + rough patterned behaviour + random variation resistant](https://reader035.vdocuments.us/reader035/viewer/2022062321/56649e215503460f94b0df92/html5/thumbnails/33.jpg)
bear
carib
ou
muskox
seal
walrus
FAUNA
0
10
20
30
40
50
60
70
80
Co
un
t
21
SITE
dual barchart
![Page 34: Univariate EDA (Exploratory Data Analysis). EDA John Tukey (1970s) data –two components: smooth + rough patterned behaviour + random variation resistant](https://reader035.vdocuments.us/reader035/viewer/2022062321/56649e215503460f94b0df92/html5/thumbnails/34.jpg)
1
bear
caribou
muskoxseal
walrus
FAUNA
0
10
20
30
40
50
60
70
Cou
nt
2
bear
caribou
muskoxseal
walrus
FAUNA
0
10
20
30
40
50
60
70
Cou
nt
Site 1 Site 2
![Page 35: Univariate EDA (Exploratory Data Analysis). EDA John Tukey (1970s) data –two components: smooth + rough patterned behaviour + random variation resistant](https://reader035.vdocuments.us/reader035/viewer/2022062321/56649e215503460f94b0df92/html5/thumbnails/35.jpg)
01020304050607080
Co
un
t
bear
carib
ou
muskox
seal
walrus
FAUNA
01020304050607080
Co
un
t
21
SITE
‘mirror’ barchart
![Page 36: Univariate EDA (Exploratory Data Analysis). EDA John Tukey (1970s) data –two components: smooth + rough patterned behaviour + random variation resistant](https://reader035.vdocuments.us/reader035/viewer/2022062321/56649e215503460f94b0df92/html5/thumbnails/36.jpg)
0
10
20
30
40
50
60
70
80
bear caribou muskox seal walrus
Site 2
Site 1
stacked barchart
![Page 37: Univariate EDA (Exploratory Data Analysis). EDA John Tukey (1970s) data –two components: smooth + rough patterned behaviour + random variation resistant](https://reader035.vdocuments.us/reader035/viewer/2022062321/56649e215503460f94b0df92/html5/thumbnails/37.jpg)
bearcaribou
muskoxseal
walrus
Site 1
Site 20
10
20
30
40
50
60
70
Site 1
Site 2
3d barchart
![Page 38: Univariate EDA (Exploratory Data Analysis). EDA John Tukey (1970s) data –two components: smooth + rough patterned behaviour + random variation resistant](https://reader035.vdocuments.us/reader035/viewer/2022062321/56649e215503460f94b0df92/html5/thumbnails/38.jpg)
![Page 39: Univariate EDA (Exploratory Data Analysis). EDA John Tukey (1970s) data –two components: smooth + rough patterned behaviour + random variation resistant](https://reader035.vdocuments.us/reader035/viewer/2022062321/56649e215503460f94b0df92/html5/thumbnails/39.jpg)
frequency polygon
![Page 40: Univariate EDA (Exploratory Data Analysis). EDA John Tukey (1970s) data –two components: smooth + rough patterned behaviour + random variation resistant](https://reader035.vdocuments.us/reader035/viewer/2022062321/56649e215503460f94b0df92/html5/thumbnails/40.jpg)
Histogram of vol
vol
De
nsi
ty
100 200 300 400
0.0
00
0.0
02
0.0
04
0.0
06
0.0
08
kernel density modelHistogram of vol
vol
De
nsi
ty
100 200 300 400
0.0
00
0.0
02
0.0
04
0.0
06
0.0
08
![Page 41: Univariate EDA (Exploratory Data Analysis). EDA John Tukey (1970s) data –two components: smooth + rough patterned behaviour + random variation resistant](https://reader035.vdocuments.us/reader035/viewer/2022062321/56649e215503460f94b0df92/html5/thumbnails/41.jpg)
![Page 42: Univariate EDA (Exploratory Data Analysis). EDA John Tukey (1970s) data –two components: smooth + rough patterned behaviour + random variation resistant](https://reader035.vdocuments.us/reader035/viewer/2022062321/56649e215503460f94b0df92/html5/thumbnails/42.jpg)
controlling kernel density plots…
• hd <- density(XX)• hh <- hist(XX, plot=F)
• maxD <- max(hd$y)• maxH <- max(hh$density)• Y <- c(0, max(c(maxD, maxH)))
• hist(XX, freq=F, ylim=Y)• lines(density(XX))
![Page 43: Univariate EDA (Exploratory Data Analysis). EDA John Tukey (1970s) data –two components: smooth + rough patterned behaviour + random variation resistant](https://reader035.vdocuments.us/reader035/viewer/2022062321/56649e215503460f94b0df92/html5/thumbnails/43.jpg)
1 2 3 4 5 6 7 8 9 10VAR00003
0
1
2
3
4
5
6
7
8
Cou
nt
Dot Plot [R: dotchart()]
![Page 44: Univariate EDA (Exploratory Data Analysis). EDA John Tukey (1970s) data –two components: smooth + rough patterned behaviour + random variation resistant](https://reader035.vdocuments.us/reader035/viewer/2022062321/56649e215503460f94b0df92/html5/thumbnails/44.jpg)
bear
carib
ou
muskox
seal
walrus
FAUNA
0
10
20
30
40
50
60
70
80
Co
un
t
21
SITE
![Page 45: Univariate EDA (Exploratory Data Analysis). EDA John Tukey (1970s) data –two components: smooth + rough patterned behaviour + random variation resistant](https://reader035.vdocuments.us/reader035/viewer/2022062321/56649e215503460f94b0df92/html5/thumbnails/45.jpg)
Dot Histogram [R: stripchart()]
1 2 3 4 5 6 7 8 9 10VAR00003
1 2 3 4 5 6 7 8 9 10VAR00003
1 2 3 4 5 6 7 8 9 10VAR00003
method = “stack”
![Page 46: Univariate EDA (Exploratory Data Analysis). EDA John Tukey (1970s) data –two components: smooth + rough patterned behaviour + random variation resistant](https://reader035.vdocuments.us/reader035/viewer/2022062321/56649e215503460f94b0df92/html5/thumbnails/46.jpg)
cooking/service service ritual
line plot
![Page 47: Univariate EDA (Exploratory Data Analysis). EDA John Tukey (1970s) data –two components: smooth + rough patterned behaviour + random variation resistant](https://reader035.vdocuments.us/reader035/viewer/2022062321/56649e215503460f94b0df92/html5/thumbnails/47.jpg)
cooking/service service ritual
![Page 48: Univariate EDA (Exploratory Data Analysis). EDA John Tukey (1970s) data –two components: smooth + rough patterned behaviour + random variation resistant](https://reader035.vdocuments.us/reader035/viewer/2022062321/56649e215503460f94b0df92/html5/thumbnails/48.jpg)
bear
carib
ou
muskox
seal
walrus
FAUNA
0
10
20
30
40
50
60
70
80
Cou
nt
21
SITE
![Page 49: Univariate EDA (Exploratory Data Analysis). EDA John Tukey (1970s) data –two components: smooth + rough patterned behaviour + random variation resistant](https://reader035.vdocuments.us/reader035/viewer/2022062321/56649e215503460f94b0df92/html5/thumbnails/49.jpg)
bear
caribou
cat
elk
moose
20%
19%
18%
21%
22%
bear
caribou
cat
elk
moose
pie chart
![Page 50: Univariate EDA (Exploratory Data Analysis). EDA John Tukey (1970s) data –two components: smooth + rough patterned behaviour + random variation resistant](https://reader035.vdocuments.us/reader035/viewer/2022062321/56649e215503460f94b0df92/html5/thumbnails/50.jpg)
1
bear
caribou
catelk
moose
2
bear
caribou
cat
elk
moose
![Page 51: Univariate EDA (Exploratory Data Analysis). EDA John Tukey (1970s) data –two components: smooth + rough patterned behaviour + random variation resistant](https://reader035.vdocuments.us/reader035/viewer/2022062321/56649e215503460f94b0df92/html5/thumbnails/51.jpg)
10
20
30
40
50
60
70
80
90
100
perc
ent
10
20
30
40
50
60
70
80
90
100
cum
ulat
ive
perc
ent
Cumulative Percent Graph
![Page 52: Univariate EDA (Exploratory Data Analysis). EDA John Tukey (1970s) data –two components: smooth + rough patterned behaviour + random variation resistant](https://reader035.vdocuments.us/reader035/viewer/2022062321/56649e215503460f94b0df92/html5/thumbnails/52.jpg)
10
20
30
40
50
60
70
80
90
100
cum
ulat
ive
perc
ent
• some useful statistical measures
(ordinal or ratio scale)
• can be misleading when used with nominal data
• good for comparing data sets
Cumulative Percent Graph
![Page 53: Univariate EDA (Exploratory Data Analysis). EDA John Tukey (1970s) data –two components: smooth + rough patterned behaviour + random variation resistant](https://reader035.vdocuments.us/reader035/viewer/2022062321/56649e215503460f94b0df92/html5/thumbnails/53.jpg)
PercentagesSitesA B C
Types 1 5 5 52 45 0 303 5 48 54 5 5 55 5 5 56 5 5 57 20 5 358 5 22 59 5 5 5
100 100 100
Cumulative PercentsSitesA B C
Types 1 5 5 52 50 5 353 55 53 404 60 58 455 65 63 506 70 68 557 90 73 908 95 95 959 100 100 100
0
20
40
60
80
100
120
1 2 3 4 5 6 7 8 9
A
B
C
![Page 54: Univariate EDA (Exploratory Data Analysis). EDA John Tukey (1970s) data –two components: smooth + rough patterned behaviour + random variation resistant](https://reader035.vdocuments.us/reader035/viewer/2022062321/56649e215503460f94b0df92/html5/thumbnails/54.jpg)
0
20
40
60
80
100
120
1 2 3 4 5 6 7 8 9
A
B
C
0
20
40
60
80
100
120
1 5 3 4 2 6 7 8 9
A
B
C