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Some Examples of Stochastic Differential Equations in Forestry Research by Haiganoush K Preisler

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Some Examples of Stochastic Differential Equations in Forestry Research by Haiganoush K Preisler Mathematical Statistician Pacific Southwest Research Station USDA Forest Service. 11 October 2005. Black-Scholes formula. 1997 Nobel Prize in Economic Sciences. - PowerPoint PPT Presentation
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Some Examples of Stochastic Differential Equations in Forestry Research by Haiganoush K Preisler Mathematical Statistician Pacific Southwest Research Station USDA Forest Service 11 October 2005
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Page 1: Some Examples of Stochastic Differential Equations in Forestry Research by Haiganoush K Preisler

Some Examples of Stochastic DifferentialEquations in Forestry Research

byHaiganoush K Preisler

Mathematical StatisticianPacific Southwest Research Station

USDA Forest Service

11 October 2005

Page 2: Some Examples of Stochastic Differential Equations in Forestry Research by Haiganoush K Preisler

Black-Scholes formula

)()()()( tdBtSdttStdS

S(t) is value of a stock at time tB(t) is a Brownian process (random noise)

Stochastic differential equation

1997 Nobel Prize in Economic Sciences

Page 3: Some Examples of Stochastic Differential Equations in Forestry Research by Haiganoush K Preisler

Heat Equation

H x t

t

H x t

x

( , ) ( , )

2

2

H(x,t)= temperature at depth x and time t.

Joseph Fourier (1768-1830)

French Mathematician

Page 4: Some Examples of Stochastic Differential Equations in Forestry Research by Haiganoush K Preisler

Fourier used heat equation to predict temperature of the ground at depth x due to the sun’s heating.

Concluded • At certain depths temperatures are out of step with surface temperature (phase lag).

• Six month lag at 2-3 m depth.(i.e., hotter in winter, cooler in summer).Good depth for cellars.

• Permafros

Page 5: Some Examples of Stochastic Differential Equations in Forestry Research by Haiganoush K Preisler

Risks

• Increase in mortality rates observed in soil organisms at 34 - 40o C.

• Tissue cannot survive temperatures greater than 45 - 65o C.

Page 6: Some Examples of Stochastic Differential Equations in Forestry Research by Haiganoush K Preisler

0 20 40 60 80 100 120Time (hrs)

020

4060

8010

012

014

0T

empe

ratu

re (

Cel

sius

)

Temperature Profiles at 10cm Depth Around Sugar Pine Trees

0 20 40 60 80 100 120Time (hrs)

020

4060

8010

012

014

0T

empe

ratu

re (

Cel

sius

)

Temperature Profiles at 10 cm Depth Around Sequoia Trees

Different curves due to different soil conditions, some of which (e.g. soil moisture) are measured.

Page 7: Some Examples of Stochastic Differential Equations in Forestry Research by Haiganoush K Preisler
Page 8: Some Examples of Stochastic Differential Equations in Forestry Research by Haiganoush K Preisler

Rocky Mountain elk(Cervus elaphus)

Page 9: Some Examples of Stochastic Differential Equations in Forestry Research by Haiganoush K Preisler

Km

Km

0 2 4 6 8 10

0

2

4

6

8

10

12

14 a

Km

Km

0 2 4 6 8 10

0

2

4

6

8

10

12

14 b

Fenced area of Starkey Experimental Station in OregonTracks of four elk

Page 10: Some Examples of Stochastic Differential Equations in Forestry Research by Haiganoush K Preisler

Differential equations

r(t) = { x(t), y(t) } = location of elk at time t.

)( ; )( tvdt

dytv

dt

dxyx

Stochastic differential equations

)( ; )( noisetvdt

dynoisetv

dt

dxyx

Page 11: Some Examples of Stochastic Differential Equations in Forestry Research by Haiganoush K Preisler
Page 12: Some Examples of Stochastic Differential Equations in Forestry Research by Haiganoush K Preisler
Page 13: Some Examples of Stochastic Differential Equations in Forestry Research by Haiganoush K Preisler

Km

Km

4 5 6 7

1.5

2.0

2.5

3.0

3.5

4.0

4.5

Page 14: Some Examples of Stochastic Differential Equations in Forestry Research by Haiganoush K Preisler

References

1. Preisler, H.K., Haase, A.M. and Sackett, S.S. (2000). Modeling and risk assessment for soil temperatures beneath prescribed forest fires. Environmental and Ecological Statistics 7, 239-254.

2. Preisler, H.K., Ager, A.A., Johnson, B.K., and Kie, J.G. (2004) Modeling animal movements using stochastic differential equations. Environmetrics 15: 643-657.


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