cherry bud workshop, keio university, 25-28 march 2008 - 1 · cherry bud workshop, keio university,...

Cherry Bud Workshop, Keio University, 25-28 March 2008 - 1 -


Outline


E

Eruptions of Old Faithful geyser

Photo by Ansel Adams

0 50 100 150 200 250 300


B

Markov Chain


B

Two-state Bernoulli-Hidden Markov Model


B

Two-state Bernoulli HMM


B

Two-state Bernoulli HMM

hidden


C

Likelihood of an homogeneous HMM


L

Likelihood of a two-state Bernoulli-HMM


L

The likelihood of a three-state Poisson-HMM


E

Models for the Old Faithful series

0 1 2 3 4 5 6 7 8 9

-1.0

-0.5

0.0

0.5

1.0

lag

acf

sample acf1st order MC2nd order MC2-state HMM


E

Additional issues and model selection


E

Interval-censored durations


E

Marginal distribution of duration

m = 2

0 1 2 3 4 5 6

0.0

0.5

1.0

1.5

2.0 m = 3

0 1 2 3 4 5 6

0.0

0.5

1.0

1.5

2.0

m = 4

0 1 2 3 4 5 6

0.0

0.5

1.0

1.5

2.0 m = 2, 3, 4

0 1 2 3 4 5 6

0.0

0.5

1.0

1.5

2.0


E

Bivariate model for durations and intervals between eruptions


P

Wind direction at Koeberg


M

Wind direction at Koeberg


2

Two-state multinomial-HMM


2

Two-state multinomial-HMM – Estimates of state-dependent distributions

0.0

0.1

0.2

0.3

0.4

0.5

lag

acf

N NE E SE S SW W NW N

State 1

0.0

0.1

0.2

0.3

0.4

0.5

acf

N NE E SE S SW W NW N

State 2


3

Three-state multinomial-HMM


2

Three-state multinomial-HMM – Estimates of state-dependent distributions


R

Wind direction at Koeberg by time of day


R

Wind direction at Koeberg (23:00 – 24:00)


2

Two-state seasonal multinomial-HMM


E

Two-state seasonal multinomial-HMM - estimates


2

Model 1 for change in direction: von Mises-HMM


Model 2 for change in direction: von Mises-HMM – wind speed affects G


Model 3 for change in direction: von Mises-HMM – wind speed affects k


Model 3(b) – marginal distributions for increasing numbers of states

-180 -140 -100 -60 -20 20 60 100 140 180

0

1

2

3

4

5

6

7

Change of direction

m = 1

1500 cm/s

m = 1

1000 cm/s

m = 1

500 cm/s

m = 1

250 cm/s

m = 1

50 cm/s

-180 -140 -100 -60 -20 20 60 100 140 180

0

1

2

3

4

5

6

7

Change of direction

density

1500 cm/sm = 2

1000 cm/s

m = 2

500 cm/s

m = 2

250 cm/s

m = 2

50 cm/s

m = 2

-180 -140 -100 -60 -20 20 60 100 140 180

0

1

2

3

4

5

6

7

1500 cm/sm = 3

1000 cm/s

m = 3

500 cm/s

m = 3

250 cm/s

m = 3

50 cm/s

m = 3

-180 -140 -100 -60 -20 20 60 100 140 180

0

1

2

3

4

5

6

7

density

1500 cm/sm = 4

1000 cm/s

m = 4

500 cm/s

m = 4

250 cm/s

m = 4

50 cm/s

m = 4


Model 3(b) estimates – four-state Von-Mises-HMM


Model 3(b) – state-dependent distributions for four-state von-Mises-HMM

-180 -120 -80 -40 0 40 80 120 160

0

2

4

6

8

10

12

Change of direction

density

1500 cm/s

1000 cm/s

500 cm/s

250 cm/s

50 cm/s

state = 1 delta[ 1 ] = 0.397

-180 -120 -80 -40 0 40 80 120 160

0

2

4

6

8

10

12

Change of direction

density

state = 2 delta[ 2 ] = 0.321

-180 -120 -80 -40 0 40 80 120 160

0

2

4

6

8

10

12

density

state = 3 delta[ 3 ] = 0.195

-180 -120 -80 -40 0 40 80 120 160

0

2

4

6

8

10

12

density

state = 4 delta[ 4 ] = 0.087


Financial Series


Volatility Clustering

Periods of high volatility


S

Stochastic volatility models


S

Stochastic volatility without leverage


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

S

Stochastic volatility without leverage


S

Stochastic volatility with leverage


A

Stochastic volatility with leverage


1

Homicides and suicides in Cape Town


2

Time series examined


3

Weekly counts of firearm homicides

0

5

10

15

count

1986 1987 1988 1989 1990 1991

De Klerk speech taxi wars

287


4

Two-state Poisson-HMMs


5

Two-state Poisson-HMM with quadratic trends

0

5

10

15

co

un

t

1986 1987 1988 1989 1990 1991


6

Firearm homicides as a proportion of all homicides and suicides

0.0

0.1

0.2

0.3

0.4

0.5

pro

po

rtio

n

1986 1987 1988 1989 1990 1991


7

Two-state binomial-HMMs


8

Two-state binomial-HMMs with change-point

0.0

0.1

0.2

0.3

0.4

0.5

pro

po

rtio

n

1986 1987 1988 1989 1990 1991

taxi wars

287


1

Multinomial-HMM for all five categories of homicide


1

Parameter estimates for a multinomial-HMM with change point on day 287


0.0

0.2

0.4

0.6

0.8

1.0

pro

b

State 1: Multinomial probabilities weeks 1-287 (left), 288-313 (right)

FH NFH FS NFS LIH

P(State 1) = 0.174

0.0

0.2

0.4

0.6

0.8

1.0

pro

b

State 2: Multinomial probabilities weeks 1-287 (left), 288-313 (right)

FH NFH FS NFS LIH

P(State 2) = 0.826

0.0

0.2

0.4

0.6

0.8

1.0

pro

b

Unconditional probabilities weeks 1-287 (left), 288-313 (right)

FH NFH FS NFS LIH

1

Homicides and suicides in Cape Town


Summary


References

cherry bud workshop, keio university, 25-28 march 2008 - 1 · cherry bud workshop, keio university,...

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