7 bivariate eda

7/28/2019 7 Bivariate Eda

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bivariate EDA and regression

analysis


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length

width


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distance from quarry

weight

of core


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-4 -3 -2 -1 0 1 2 3 4 5

AG_C1_1

-5

-4

-3

-2

-1

0

1

2

3

AG_

C1

_2


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-4 -3 -2 -1 0 1 2 3 4 5

AG_C1_1

-5

-4

-3

-2

-1

0

1

2

3

AG

_C

1_2


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-4 -3 -2 -1 0 1 2 3 4 5

AG_C1_1

-5

-4

-3

-2

-1

0

1

2

3

AG_

C1

_2


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AG_C1_2

AG

_C1

_1

AG_C2_2 AG_C3_2 AG_C4_2

AG_C1_1

AG

_C2

_1

AG_C

2_1

AG

_C3

_1

AG_C3_1

AG_C1_2

AG

_C4

_1

AG_C2_2 AG_C3_2 AG_C4_2

A

G_C4_1

scatterplot matrix


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AG_C1_1

AG

_C1

_1

AG_C2_1 AG_C3_1 AG_C4_1 AG_C1_2 AG_C2_2 AG_C3_2 AG_C4_2

AG_C1_1

AG

_C2

_1

AG_C2_1

AG

_C3

_1

AG_C3_1

A

G_

C4

_1

AG_C4_1

AG

_C1

_2

AG_C1_2

AG

_C2

_2

AG_C2_2

AG

_C3

_2

AG_C3_2

AG_C1_1

AG

_C4

_2

AG_C2_1 AG_C3_1 AG_C4_1 AG_C1_2 AG_C2_2 AG_C3_2 AG_C4_2

AG_C4_2


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-4 -3 -2 -1 0 1 2 3 4 5

AG_C1_1

-10

-5

0

5

10

AG

_

C2

_1


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scatterplots

scatterplots provide the most detailed summary of abivariate relationship, but they are not concise, andthere are limits to what else you can do with them

simpler kinds of summaries may be useful

more compact; often capture less detail

may support more extended mathematical analyses

may reveal fundamental relationships

-4 -3 -2 -1 0 1 2 3 4 5

AG_C1_1

-5

-4

-3

-2

-1

0

1

2

3

AG

_C

1_

2


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y = a + bx


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y = a + bx

1 2 3 4 5 6

1

2

3

4

5

6

a = y intercept

y

x

(x2,y2)

(x1,y1)

b = slope

b = y/x

b = (y2-y1)/(x2-x1)


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y = a + bx

we can predict values ofy from values ofx

predicted values ofyare called y-hat

the predicted values (y) are often regardedas dependent on the (independent) x

values try to assign independent values to x-axis,

dependent values to the y-axis

bxay


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y = a + bx

becomes a concise summary of a point

distribution, and a model of a relationship

may have important explanatory andpredictive value


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how do we come up with these lines?

various options:

by eye

calculating a Tukey Line (resistant to

outliers)

locally weighted regression LOWESSleast squares regression


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linear regression

linear regression and correlation analysis

are generally concerned with fitting lines to

real data

least squares regression is one of the main

tools

attempts to minimize deviation of observed

points from the regression line

maximizes its potential for prediction


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standard approach minimizes the squared

variation in y

Note:

these are the vertical deviations this is a sum-squared-error approach

n

i

ii yy1

2)(


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regressing x on y would involve defining

the line

by minimizing

ii dycx

2

ii xx


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calculating a line that minimizes this value

is called regressingy onx

appropriate when we are trying to predictyfromx

this is also called Model I Regression


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start by calculating the slope (b):

n

i

i

n

i

ii

xx

yyxx

b

1

2

1

)(

))(( covariance


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once you have the slope, you can calculate

the y-intercept (a):

n

xbyxbya

ii


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regression pathologies

things to avoid in regression analysis


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Tukey Line

resistant to outliers

divide cases into thirds, based onx-axis

identify the medianx andy values in upper

and lower thirds

slope (b)= (My3-My1)/(Mx3-Mx1)

intercept (a) = median of all values yi-b*xi


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Correlation

regression concerns fitting a linear model to

observed data

correlation concerns the degree of fitbetween observed data and the model...

if most points lie near the line:

the fit of the model is good

the two variables are strongly correlated

values of y can be well predicted from x


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Pearsonsr

this is assessed using the product-moment

correlation coefficient:

= covariance (the numerator), standardizedby a measure of variation in both x and y

22 )()(

))((yyxx

yyxxr

ii

ii


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y

x

22 )()(

))((

yyxx

yyxxr

ii

ii

+

+

-

-

(xi,yi)


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unlike the covariance, r is unit-less

ranges between1 and 1 0 = no correlation

-1 and 1 = perfect negative and positive

correlation (respectively) r is symmetrical

correlation betweenx andy is the same as

betweeny andx no question of independence or dependence

recall, this symmetry is not true of regression


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regression/correlation

one can assess the strength of a relationship by

seeing how knowledge of one variable

improves the ability to predict the other


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if you ignorex, the best predictor ofy will

be the mean of ally values (y-bar)

if they measurements are widely scattered,

prediction errors will be greater than if they

are close together

we can assess the dispersion ofy values

around their mean by:

2)( yyi

y


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y

iy

2

)( yyi

2)( ii yy


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2)( ii yy

2)( yyir2=

coefficient of determination (r2)

describes the proportion of variation that is

explained or accounted for by the regression line

r2=.5

half of the variation is explained by the regression

half of the variation iny is explained by variation inx


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y

iy


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correlation and percentages

much of what we want to learn aboutassociation between variables can belearned from counts

ex: are high counts of bone needles associatedwith high counts of end scrapers?

sometimes, similar questions are posed ofpercent-standardized data

ex: are highproportions of decorated potteryassociated with highproportions of copper

bells?


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caution

these are different questions and have

different implications for formal regression

percents will show at least some level ofcorrelation even if the underlying counts do

not

spurious correlation (negative) closed-sum effect


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case C_v1 C_v2 C_v3 C_v4 C_v5 C_v6 C_v7 C_v8 C_v9 C_v10

1 15 14 94 59 76 13 8 97 10 952 35 1 89 95 23 77 14 9 27 43

3 20 96 73 31 90 65 74 60 85 27

4 23 59 7 52 33 83 71 35 57 90

5 36 90 86 15 97 54 52 41 34 3

6 79 2 26 5 11 68 74 44 13 87

7 40 99 28 66 77 23 69 22 63 36

8 95 36 22 75 21 48 95 58 74 68

9 27 0 58 99 32 30 5 5 100 75

10 67 93 98 61 62 94 3 16 43 48

10 vars.

5 vars.

3 vars.

2 vars.


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-1.0 -0.5 0.0 0.5 1.0

r

original counts

-1.0 -0.5 0.0 0.5 1.0

r

percents (10 vars.)

-1.0 -0.5 0.0 0.5 1.0

r

percents (5 vars.)

-1.0 -0.5 0.0 0.5 1.0r

percents (3 vars.)

-1.0 -0.5 0.0 0.5 1.0

r

percents (2 vars.)


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0 20 40 60 80 100

C_V1

0

20

40

60

80

100

C_ V

2

0 5 10 15 20

P10_V1

0

5

10

15

20

P 1 0_ V 2

0 10 20 30 40 50 60 70

T5_V1

0

10

20

30

40

T 5

_ V 2

10 20 30 40 50 60 70 80

T3_V1

0

10

20

30

40

50

60

70

T 3

_ V 2

10 20 30 40 50 60 70 80 90 100

T2_V1

0

10

20

30

40

50

60

70

80

90

T2

_V2


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regression assumptions

both variables are measured at the interval

scale or above

variation is the same at all points along theregression line (variation is homoscedastic)


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residuals

vertical deviations of points around the regression

for case i, residual = yi-y-hati [yi-(a+bxi)]

residuals iny should not show patterned variationeither withx ory-hat

normally distributed around the regression line

residual error should not beautocorrelated

(errors/residuals in y are independent)


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standard error of the regression

recall: standard error of an estimate (SEE) is like

a standard deviation

can calculate an SEE for residuals associated witha regression formula

n

yyS

ii

iyyi

2


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to the degree that the regression assumptions

hold, there is a 68% probability that true

values of y lie within 1 SEE of y-hat 95% within 2 SEE

can plot lines showing the SEE

y-hat = a+bx +/- SEE


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data transformations and

regression

read Shennan, Chapter 9 (esp. pp. 151-173)


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0 50 100 150 200

VAR1

0

50

100

150

200

V A R 2

0 50 100 150 200

VAR1

0

50

100

150

200

V A R 2


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40 80 120 160

VAR1

0

50

100

150

200

V A R 2


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0 5 10 15

VAR1T

0

50

100

150

200

VAR2

let VAR1T = sqr(VAR1)


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distribution and fall-off models

ex: density of obsidian vs. distance from thequarry:

0 10 20 30 40 50 60 70 80

DIST

0

1

2

3

4

5

6

D

E N S I T Y


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0 10 20 30 40 50 60 70 80DIST

0

1

2

3

4

5

6

DENSITY

Plot of Residuals against Predicted Values

-1 0 1 2 3 4ESTIMATE

-1

0

1

2

RESIDUAL


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0 10 20 30 40 50 60 70 80

DIST

1

2

3456

DENSITY

0 10 20 30 40 50 60 70 80

DIST

-3

-2

-1

0

1

2

LG

_DENS

LG_DENS log(DENSITY)


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0 10 20 30 40 50 60 70 80

DIST

-3

-2

-1

0

1

2

L G

_ D E N

S

y = 1.70-.05x

[remembery

is logged

density]


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0 10 20 30 40 50 60 70 80

DISTANCE

0

1

2

3

4

5

6

D E N S I T Y

0 800

6

0 10 20 30 40 50 60 70 80

DISTANCE

0

1

2

3

4

5

6

DENSITY

logy = 1.70-.05x

fploty = exp(1.70-.05*x)


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begin

PLOT DENSITY*DISTANCE / FILL=1,0,0

fplot y = exp(1.70-.05*x) ; XLABEL='' YLABEL=''

XTICK=0 XPIP=0 YTICK=0 YPIP=0 XMIN=0

XMAX=80 YMIN=0 YMAX=6

end

transformation summary


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transformation summary

correcting left skew:x4 stronger

x3 strong

x2 mild

correcting right skew:

x weak

log(x) mild

-1/x strong

-1/x2 stronger

7 bivariate eda

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