Presentation of statistical concepts with dynamic ... - Andrej Blejec

Presentation of statistical concepts with dynamic graphics and simulations in R Andrej Blejec [email protected]

National Institute of Biology and

University of Ljubljana, Slovenia IASE2017, July , 2017

A. Blejec: Animations in R ...

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Some examples


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Empirical distribution function and quartiles 1 100

80

60

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40

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Projection to the first principal component ●

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Move points to median then expand to mid-quintiles

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80

A tribute to Hans Rosling (1948-2017)

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60

70

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50

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1952 ●

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30

GDP per capita

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0.2

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5.0

10.0

20.0

50.0

Life expectancy


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Matrices


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Matrix multiplication: points move

[1,] [2,]

[,1] [,2] 1 2 2 1

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Matrix multiplication: change of direction and magnitude

[1,] [2,]


[,1] [,2] 1 2 2 1

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Matrix multiplication: find the eigenvectors

[1,] [2,]


[,1] [,2] 1 2 2 1

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How far are we from the truth?


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P(a < X ≤ b) =?

a


b

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Parameter estimation

Ways of estimation Point estimation Least squares estimation (LSE) Maximum likelihood estimation (MLE)

Interval estimaton


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Method of least squares

./clp/capture-LSE1.jpg


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Method of least squares

./clp/capture-LSE2.jpg


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Minimal sum of squared deviations

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0

5

10

15

20

Least squares estimation: mean

5

10

15

600 400 200 0

Sum of squares

x

0

1

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6

x


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0.08 0.04 0.00

Density

0.12

Maximum likelihood estimation: mean

5

10

15

−60 −100 −140

log Likelihood

x

0

1

2

3

4

5

6

x


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0.08 0.04 0.00

Density

0.12

Maximum likelihood estimation: sd

5

10

15

−60 −100 −140

log Likelihood

x

0

1

2

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5

6

x


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Inference errors


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Hypothesis testing | H0 = µ0 Larger sample → smaller SE → smaller β specificity

false positive

H0 : µ = µ0

α µ0

H1 : µ > µ0

β

false negative A. Blejec: Animations in R ...

X

sensitivity 20 / 57

What will you do on Sunday?


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What will you do on Sunday? Depends ...


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What is the plan?

Hmmm ... hiking cleaning the flat reading running staying in bed biking


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What is the plan? Hmmm ... hiking cleaning the flat reading running staying in bed biking

Sunny

Rainy

hiking

cleaning the flat

running

reading

biking

staying in bed


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Things are related ... Different conditions yield different distributions (expectations)


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Height and weight


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Correlation


quantitative ∼ quantitative

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Analysis of variance


quantitative ∼ qualitative

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Contingency


descriptive ∼ descriptive

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2

4

6

R 2 goodness of fit Changing intercept

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0 A. Blejec: Animations in R ...

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model1

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2

3

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R 2 goodness of fit Changing slope

R2model2 = 0.22 4

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0 A. Blejec: Animations in R ...

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model1

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Joint and marginal distribution


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From bivariate distribution to marginal distributions


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From marginal distributions to bivariate ?


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IASE 2017


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Animation in R


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Animation solutions for R :

the main idea is: plot frame by frame package animation provides an organized environment for subsequent plotting, see AniWiki animator takes care of smooth transitions


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Drawbacks

plotting everything can be slow for complex figures smooth movements require intermediate plots, thus lot of frames to plot


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Moving a point, line traces the movement. t = 0.91

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Moving a point ...

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Linear homotopy

Start coordinate x0 End coordinate x1 For parameter t ∈ [0, 1] xt = h(x0 , x1 , t) = (1 − t) · x0 + t · x1


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Linear homotopy when: start, end, enter, exit ... 0,1

x0

0.3 , 0.7

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x1

0.2

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h

0.0

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t

t

0.3 , 0.7 , 0.1 , 0.9

0.4 , 0.6 , 0.1 , 0.9

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0.6 t


x0

1.0

x1

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x0

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0.8

x0 1.0

1.0

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h

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x1

life

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t

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Timed functions

tpoints tlines tsegments tarrows tpolygon

(x0, y0, x1 = x0, y1 = y0, t, when, trace = FALSE (x0, y0, x1, y1, t, when, ...) (x0, y0, x1, y1, t, when, fixed = 1, ...)

(x0, y0, x1, y1, t, when, fixed = 1, length = 0.1 (x0, y0, x1, y1, t, when, ...)

trect

(xleft0, ybottom0, xright0, ytop0, xleft1,

ttext

(x0, y0, x1 = x0, y1 = y0, t, when, text =


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Timed parameters ...

tcex

(cex0 = 1, cex1 = 1, t, when, ...)

trgb

(x0, x1 = x0, t, when = c(0, 1), alpha0 = 1,

tmatrix


(X1, X0 = diag(nrow(X1)), t, when, ...)

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Function animator()

animator

(block, life = 1, fps = 25, pause = 0.5, verbose

Accepts argument block as a character string animator(' newplot() tpoints(2, 2, 9, 5, trace = TRUE, pch = 16, cex = 2) ')


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Function animator()

Argument block as an expression > block animator(block)


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Class animator A block of commands (expression or character version) can be changed to a class animator > x x expression({ newplot() tpoints(2, 2, 9, 5, trace = TRUE, pch = 16, cex = 2) }) attr(,"class") [1] "animator" attr(,"life") [1] 2 > is.animator(x) [1] TRUE > plot(x) A. Blejec: Animations in R ...

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Animated graphics in a PDF file use package animate in LaTeX > plot(as.animator(block)) > includeLatex(vspace = "-2cm")

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Moving points

tpoints()

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Lines

tlines() ●

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Growing segments


tsegments()

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Growing arrows


tarrows()

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Polygons


tpolygon()

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Rectangles


trect()

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Final remarks

Graphically supported simulations can - to some extent - replace the proofs, usually not understandable to non-mathematics majors.

Maybe they are the answer to some questions:

If an audience is not convinced by the proof, why do proof? (Moore, 1996) Do [students] need to know the theory or do they need to understand the concepts? (J Brown and I David, ICOTS8, 2010)


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Thank you !


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Presentation of statistical concepts with dynamic ... - Andrej Blejec

Presentation of statistical concepts with dynamic ... - Andrej Blejec

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