Computational statistics, lecture3

Resampling and the bootstrap

Generating random processes

The bootstrap

Some examples of bootstrap techniques

Computational statistics, lecture3

Process-based model of the flow of nitrogenfrom land to sea

CoastalmodelAnthropo-

genicinputs

Primary outputs(nutrient concentrations,chlorophyll, oxygen, etc.)

Open-seaboundaryconditions

Watershedmodel

Physio-graphicinputs

Meteoro-logical

forcings

Atmos-pheric inputs

Physio-graphicinputs

Waterborne inputs

Derived outputs

Meteoro-logical

forcings

Computational statistics, lecture3

Decomposing outputs of process-based modelsdriven by meteorological inputs

Observed forcing Weather-dependent model output

Synthetic forcing Synthetic model output

Weather-normalisedmean output

Weather-specific(random)

component of the model output

0

2

4

6

8

10

12

14

16

1 9 17 25 33 41 49 57 65 73 81 89 97 105 113

0

2

4

6

8

10

12

14

16

1 9 17 25 33 41 49 57 65 73 81 89 97 105 1130

2

4

6

8

10

12

14

16

1 9 17 25 33 41 49 57 65 73 81 89 97 105 113

-8

-6

-4

-2

0

2

4

6

8

1 10 19 28 37 46 55 64 73 82 91 100 109 118

0

2

4

6

8

10

12

14

16

1 9 17 25 33 41 49 57 65 73 81 89 97 105 113

0

2

4

6

8

10

12

14

16

1 9 17 25 33 41 49 57 65 73 81 89 97 105 113

0

2

4

6

8

10

12

14

16

1 9 17 25 33 41 49 57 65 73 81 89 97 105 113

0

2

4

6

8

10

12

14

16

1 9 17 25 33 41 49 57 65 73 81 89 97 105 113

How can we use resampling to better understand model outputs?

Computational statistics, lecture3

Resampling daily temperatures

Split observed data into periods of duration one month

Generate new temperature series by resampling 1-month pieces and combining them so that the seasonal pattern is preserved-15

-10

-5

0

5

10

15

20

25

01/01/1994

01/01/1995

01/01/1996

31/12/1996

31/12/1997

31/12/1998

31/12/1999

30/12/2000

Air

te

mp

era

ture

(oC

)

Computational statistics, lecture3

Observed and resampled daily temperatures

-15

-10

-5

0

5

10

15

20

25

01/01/2000

31/12/2000

Air

te

mp

era

ture

(oC

)

-15

-10

-5

0

5

10

15

20

25

01/01/2000

31/12/2000

Air

te

mp

era

ture

(oC

)

Observed data Resampled data

Computational statistics, lecture 3

Data-driven inference- inference based on resampling observed data

**2

*1 ,...,, Nxxx

3467

798839

41

8570

62

90 58 4460

73

22

587988

41

88

8570

90

22 34 4460

41

60Sampling with replacement

Resampled dataObserved data

x **2

*1 ...,,, Nxxx

Computational statistics, lecture3

Nonparametric bootstrap - empirical cdf

0.00

0.20

0.40

0.60

0.80

1.00

1.20

0.00 1.00 2.00 3.00 4.00 5.00

Em

pir

ica

l cd

f (F

*)

n

ii xXI

nxF

1

)(1

)(*

Obs. nr Observed value1 0.572 0.083 0.204 0.355 0.726 0.107 0.648 4.269 3.07

10 1.2611 0.4912 0.1213 0.4514 2.9315 2.6016 0.1217 1.0118 2.9719 0.6120 1.64

Computational statistics, lecture3

The bootstrap

Let (X1, …, Xn) be a sample and a parameter of the underlying distribution

Suppose is estimated by

The underlying idea of the bootstrap is to first use the sample to estimate the unknown distribution F of the data.

Then this estimated distribution F* is used in place of the unknown true distribution in calculating the distribution of

)...,,(ˆˆ1 nXX

Computational statistics, lecture3

Nonparametric bootstrap

- histogram of sample means of bootstrap samples

Obs. nr Observed value1 0.572 0.083 0.204 0.355 0.726 0.107 0.648 4.269 3.07

10 1.2611 0.4912 0.1213 0.4514 2.9315 2.6016 0.1217 1.0118 2.9719 0.6120 1.64

1.61.41.21.00.8

25

20

15

10

5

0

Sample mean

Fre

quency

Mean 1.237StDev 0.1937N 100

Histogram of Sample meanNormal

Computational statistics, lecture3

Nonparametric bootstrap

- histogram of sample means of bootstrap samples

Obs. nr Observed value1 0.572 0.083 0.204 0.355 0.726 0.107 0.648 4.269 3.07

10 1.2611 0.4912 0.1213 0.4514 2.9315 2.6016 0.1217 1.0118 2.9719 0.6120 1.64

2.241.961.681.401.120.840.56

600

500

400

300

200

100

0

Sample mean

Fre

quency

Histogram of bootstrap sample mean (B=10000)

Computational statistics, lecture3

Nonparametric bootstrap

- histogram of standard deviations of bootstrap samples

Obs. nr Observed value1 0.572 0.083 0.204 0.355 0.726 0.107 0.648 4.269 3.07

10 1.2611 0.4912 0.1213 0.4514 2.9315 2.6016 0.1217 1.0118 2.9719 0.6120 1.64

1.761.541.321.100.880.660.440.22

500

400

300

200

100

0

Sample st.dev.

Fre

quency

Histogram of bootstrap standard deviations

Computational statistics, lecture3

Nonparametric bootstrap

- confidence intervals by computing percentiles

Obs. nr Observed value1 0.572 0.083 0.204 0.355 0.726 0.107 0.648 4.269 3.07

10 1.2611 0.4912 0.1213 0.4514 2.9315 2.6016 0.1217 1.0118 2.9719 0.6120 1.64

1.761.541.321.100.880.660.440.22

500

400

300

200

100

0

Sample st.dev.

Fre

quency

Histogram of bootstrap standard deviations

Sample st.dev. Rank

0.78 250

1.56 9751

Computational statistics, lecture3

Parametric bootstrap - empirical cdf

Obs. nr Observed value1 0.572 0.083 0.204 0.355 0.726 0.107 0.648 4.269 3.07

10 1.2611 0.4912 0.1213 0.4514 2.9315 2.6016 0.1217 1.0118 2.9719 0.6120 1.64

Assume that a sample is drawn from an exponential distribution with cdf F(, x) = 1 – exp(- x)

Use the estimator

Determine the distribution of using the estimated distribution

X

1ˆ

),ˆ()(ˆ xFxF

Computational statistics, lecture3

Residual resampling

Consider the linear regression model

Estimate the beta coefficients and determine the residuals

Generate new bootstrap samples

Make inference about the model parameters by fitting linear regression models to bootstrap samples

nixYe iii ...,,1,ˆˆ10

BbniexY bii

bi ...,,1,...,,1,ˆˆ )(

10)(

nixY iii ...,,1,10