model weight output(s) - oregon state...

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Modeling for Non-Modelers


A Gentle Introduction to

Mathematical Models


What is a Model?


Input(s) ? Output(s)

Model

A model, at it’s most basic, takes one or more

inputs, and processes those inputs into one

or more outputs


10

15

20

25

30

35

40

1500 2000 2500 3000 3500 4000 4500 5000

MP

G

Vehicle Weight

MPG as it relates to Vehicle Weight

Individual Observation

A (really simple) Model

MPG = 0.0084*Weight + 48.8

r2 = 0.82


Another (simple) model

What is the

impact of

elevated CO2

levels on air

temperature?

Individual

Observation

The model


Also a model

2

Modeling for Non-Modelers Modeling for Non-Modelers

Causal Loop Diagram of a Model Examining the Growth or Decline of a

Life Insurance Company

Source: Robert A. Taylor, U.S. Department of EnergyC = negative (counteracting) feedback loop, || = delays


Things to do with models

• Generate new, testable hypotheses

about a system.

• Explore emergent properties of a

system.

• Understand our ignorance.

• Predict the future.

• Perform virtual experiments.

• Optimize decisionmaking about

the system.


And some downsides…• They can be difficult to understand. This complexity can pose significant

barriers to understanding by end-users.

• Models contain many "hidden" assumptions. Know what these

assumptions are before using the model!!!

.

• The results of model runs

may be believed to easily.

Be a skeptic!

• Models, particularly complex

models, can be expensive

and time consuming to

develop, implement and

validate.


A Hierarchy of Approaches

Math

em

atical M

odels

Empirical (Statistical – Data-

Driven)

Regression

Linear

Logistic (classification)

NonlinearMachine Learning

Process(Theory-driven)

Analytical Models(closed-form

solution)

Simulation Models (computer-based)

Point (1-D)

Systems Dynamics

Semi-distributed(2/3-D)

Fully Distributed (2/3-D)

Inc

rea

sin

g c

om

ple

xity

/ha

ss

le/p

ow

er


The Process of Modeling

Real World Mathematical World

Real-world problemRepresent

SystemState assumptions

Identify system,

mathematical

representation,

parameters,

structure (e.g.

equations)

Interpret

solutions/

results with data

Real-world

solutions

http://upload.wikimedia.org/wikipedia/commons/5/52/Causal_Loop_Diagram_of_a_Model.gif

3


Is your model any good?

• You need data – as much as possible

• Partition your data into two

independent sets – one for

calibration, one for

validation

Verify ValidateCalibrate


Tools• Excel – great for simpler models, regression – a bunch of

built-in tools for simple data analysis, statistical modeling

• R, Python – simple, interpreted

open source (free) programming

languages, rich modeling support

• Visual Tools – Stella, Simile,

PowerSim, GoldSim…

• MatLab – OSU site license,

simple programming, rich math

modeling support

• Traditional Programming Languages – C++, FORTRAN

– powerful, computationally efficient, but more hassle to

work with – often used for large, complex models.


A Tale of Four Models

Next, we will (briefly) explore four models and

modeling approaches:

1. A simple regression model (in Excel, Python)

2. A classification model (Logistic Regression) in

Python

3. A population model

in Python

4. A systems dynamics

model using Python


To access the Jupyter Notebooks

• https://bee-jupyter.bee.oregonstate.edu

• User: jupyter2

• Password: robot


Model 1 – Empirical Model

Goal: Identify relationship between two

variables based on data

This is an example of an ‘empirical’ (or

“statistical”) model:

1) data driven (not physically-based)

2) assume a model form (e.g. Y=aX+b)

3) use regression to find best fit parameters


Linear Regression - Excel

Dataset available at: http://explorer.bee.oregonstate.edu/Topic/Wodel/Data/StreamTemp.x

lsx

Approach:

1. Load Data into Excel

2. Plot the data of

interest

3. From the plot, add

regression lines

4. Generate summary

statistics

https://bee-jupyter.bee.oregonstate.edu/

http://explorer.bee.oregonstate.edu/Topic/Wodel/Data/StreamTemp.xlsx

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Model 2 - Classification

• Classification models categorize sets of

inputs in two or more discrete bins.

A common example:

predicting species

presence/absence based

on environmental factors

We want to understand

the probability that a given

combination of factors will

result in a “positive” result


Logistic Regression - Python

• Jupyter Notebook available at: https://bee-

jupyter.bee.oregonstate.edu/user/jupyter2/notebooks/Classificatio

n%20and%20Logistic%20Regression.ipynb

• Approach:

1. Load Data into

Python

2. Take advantage of

pre-existing package

(sklearn) for doing all

the work


Model 3 – Population DynamicsProblem: Predator-Prey Interactions - The Lotka-

Volterra Equations

The Lotka-Volterra equations were developed in the early part of the 20th century as a simple description of the dynamics of two populations, one predating on the other.

An example might be arctic foxes and arctic hares.


The MathProblem: Predator-Prey Interactions - The Lotka-Volterra

Equations

This is arguably the simplest possible model of predation. Translating to a

mathematical description results in the following:

• Rate of Change of Prey H = [ growth rate of Prey ] - [ loss due to predation

• Rate of Change of Predator P = - [losses due to mortality ] + [ growth due to

predation ]

bHPPrdt

dP

aHPHrdt

dH

P

H

Questions:

1. Under what circumstances might this model be

reasonable?

2. How do we solve the resulting system of

equations?

3. Under what conditions (if any) are the populations

in equilibrium?

4. Do the equilibrium point imply stability?


Model 4 – Zombie Attacks!

Our final model looks at three

interacting populations –

Zombies, Humans, and the

Dead.

Assumptions (pretty simple!):

• Humans kill Zombies

• Zombies resurrect from the

dead and infect Humans


Zombie Math

𝑑𝐻

𝑑𝑡= (kBirth*H) – (kTrans*H*Z) – (kDeath*H)

𝑑𝑍

𝑑𝑡= (kTrans*H*Z) + (kRes*D) – (kRem*H*Z)

𝑑𝐷

𝑑𝑡= (kDeath*H) + (kRem*H*Z) –-(kRes*D)

Human Birth Infection Natural Mortality

Natural Mortality

Infection

Resurrection

Death by Human

Death by Human

Resurrection

https://bee-jupyter.bee.oregonstate.edu/user/jupyter2/notebooks/Classification and Logistic Regression.ipynb

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Things to think about

• All models are wrong,

some models are useful.

Be sure yours is in the

latter group.

• Keep it as simple as possible,

but not too simple…

• Model iteratively – start

simple, add complexity as needed

• Modeling is a great way to generate insights,

explore and understand the world…


Extras


Uses of Models

Three Primary Uses of Models:

• Analysis

• Prediction

• Control

Additional Secondary Uses

include:

• Conceptual framework for organizing or coordinating empirical research

• Mechanism to summarize or synthesize large quantities of data

• Mechanism to identify areas of ignorance

• Gaming - providing insight to managers by performing "what-if" simulations


A few more benefits...

• Results in a description that is precise and unambiguous – good as a

means of communication.

• Models enable predictions

to be made in such a way

that these predictions can be

checked against reality by

experiment.

• Capture knowledge about a

system in a durable,

distributable form.

• Models can be Fun!


Additional considerations…

• Realism: the degree to which model structure mimics the real world - how completely does the model capture reality

• Precision: the accuracy of the model predictions - how well does the model generate results?

• Generality: the ability of the model to adequately represent diverse systems - is the model broadly useful?

• Often, these properties trade off against each other.


An ExampleObjective

• Construct a description of the dynamics of the worlds population such that the time when the population size will double can be computed:

Assumptions (Hypotheses)

• We will model the population as whole, not individuals, ages, sexes,etc.

• The population grows at a constant per capita rate – not influenced by internal or external factors

rteNNrNdt

dN0

where r is the intrinsic per capita growth rate

To calibrate the model, we solve for r

Resulting Model

time N k= 0.1

0 0.01 N0 0.01

1 0.011052

2 0.012214

3 0.013499

4 0.014918

5 0.016487

6 0.018221

7 0.020138

8 0.022255

9 0.024596

10 0.027183

11 0.030042

12 0.033201

13 0.036693

14 0.040552

15 0.044817

16 0.04953

17 0.054739

18 0.060496

20 0.073891

22 0.09025

24 0.110232

26 0.134637

28 0.164446

30 0.2008550

0.05

0.1

0.15

0.2

0.25

0 10 20 30 40

Time

Exponential Growth

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A slight tweak

Mathematical (Re) Formulation

Assume that the per-capita growth rate is limited by a carry capacity

Which model is better?

rteN

K

KNSo

K

NrN

dt

dN

11

,

)1(

0

where r is the intrinsic per capita growth rateK is the carrying capacity

To calibrate the model, we solve for r and K


Do you need to capture variability?

Deterministic Models

• known inputs and unique outputs.

• no components of the model are random

• multiple model runs with the same input will produce the same

output.

Stochastic Models

• either inputs or parameters in the model are uncertain or variable,

and the statistical distribution of these variables is defined.

• multiple model runs result in a distribution of outputs.

• Examples: Precipitation; flow in the Willamette River.

Modeling for Non-Modelers Modeling for Non-Modelers

Modeling Process Steps

Identify

Rationale/

Key Question(s)

to be

Addressed

System

ConceptualizationAbstract Critical System

Features Relevant to

Question(s):

Diagrammatic ModelIdentify1.System Boundary

2.System Components (States)

3.Linkages (flows/rates)

4.Exogenous drivers

Mathematical ModelIdentify:

1. System Boundary

2. System Components (states)

3. Linkages (flows/rates)4. Exogenous drivers

Calibration

Validation

Testing

Application


Systems Basics

Stock

Inflow Outflow

Systems consist of:

• Stocks

• Flows

• Influences

• System Boundary

External Source External Sink

A Simple Example

𝑑𝑦

𝑑𝑡= 𝐼𝑛𝑓𝑙𝑜𝑤 − 𝑂𝑢𝑡𝑓𝑙𝑜𝑤


Systems Basics (continued)

Mineral

DepositMining

Example: Consumption

𝑑𝑦

𝑑𝑡= −𝑂𝑢𝑡𝑓𝑙𝑜𝑤

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Systems Basics (continued)

Stock of water in a reservoir

𝑑𝑊

𝑑𝑡= 𝑃𝑟𝑒𝑐𝑖𝑝 + 𝐼𝑛𝑓𝑙𝑜𝑤 − 𝐸𝑣𝑎𝑝𝑜𝑟𝑎𝑡𝑖𝑜𝑛 − 𝐷𝑖𝑠𝑐ℎ𝑎𝑟𝑔𝑒

Precipitation EvaporationWater in

Reservoir

(W)

River Inflow Discharge

Note: A Stock can be

increased by decreasing it’s

outflows, as well as

increasing it’s inflow


Feedbacks

Systems of information-feedback control are

fundamental to all life and human endeavor, from

the slow pace of biological evolution to the

launching of the latest space satellite…Everything

we do as individuals, as an industry, or as a

society is done in the context of an information-

feedback system.» Jay Forrester

Feedback loops form when changes in a stock

affect the flows into or out of that same stock


Balancing (Stabilizing) Feedbacks

Balancing (Stabilizing) Loops

Mortality

Population

(P)

𝑑𝑃

𝑑𝑡= −𝑘𝑃

k

Balancing feedback loops

are equilibrating or goal-

seeking structures in

systems and are source of

both stability and

resistance to change

Influences

(Information flow)

k= -0.2Time P

0 100

1 80

2 64

3 51.2

4 40.96

5 32.768

6 26.2144

7 20.97152

8 16.77722

9 13.42177

10 10.73742

11 8.589935

12 6.871948

13 5.497558

14 4.398047

15 3.518437

16 2.81475

17 2.2518

18 1.80144

19 1.441152

20 1.152922

21 0.922337

22 0.73787

23 0.590296

24 0.472237

25 0.377789

0

20

40

60

80

100

120

0 10 20 30

P

Time


Reinforcing (Runaway) Feedbacks

Reinforcing (Runaway)

Loops

Growth

New Population Population

(P)

𝑑𝑃

𝑑𝑡= +𝑘𝑃

k

Self-Reinforcing feedback

loops are self-enhancing,

leading to exponential

growth or to runaway

collapse over time – they

tend to destabilize systems

k= 0.2Time P

0 100

1 120

2 144

3 172.8

4 207.36

5 248.832

6 298.5984

7 358.3181

8 429.9817

9 515.978

10 619.1736

11 743.0084

12 891.61

13 1069.932

14 1283.918

15 1540.702

16 1848.843

17 2218.611

18 2662.333

19 3194.8

20 3833.76

21 4600.512

22 5520.614

23 6624.737

24 7949.685

25 9539.622

0

2000

4000

6000

8000

10000

0 10 20 30

P

Time


Example – Algal Dynamics- Where are the feedbacks?

Growth Mortality

• What do the dynamics look

like?

• What are the key interactions?

• What’s controlling the system?

Amax

Sh

Amax = Max Algal population

Sh = “Effective” shade

k = per-capita growth rate

kd = per-capita mortality

kd

k

𝑑𝐴

𝑑𝑡= 𝐺𝑟𝑜𝑤𝑡ℎ − 𝑀𝑜𝑟𝑡𝑎𝑙𝑖𝑡𝑦 overall dynamics

𝑆ℎ = (1 −𝐴

𝐴𝑚𝑎𝑥) shading factor

𝐺𝑟𝑜𝑤𝑡ℎ = 𝑘 𝐴 𝑆ℎ = 𝑘𝐴(1 −𝐴

𝐴𝑚𝑎𝑥) ;

Exponential

Growth

Density-

Dependence

Algal

Biomass

(A)


Mathematics of an

Unsustainable SystemAssumptions:

• There are two pools: Population and Resource

• Resources are consumed at a constant per-capita rate, and are non-renewable

• The Population is influenced by birth and death: birth is a constant per-capita rate; the size of the Resource base influences the death rate of the population according to: Death Rate = 1-( R(t)/R0 )

So:

cPdt

dR

PR

Rb

dt

dP

0

1Where P = Population

R = Resource

b = per capita birth rate

c = per capita consumption rate

R0 = initial resource base

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cPdt

dR

PR

Rb

dt

dP

0

1

ResourcePopulation

deaths

births

ConsumptionReinforcing

(Positive)

Feedback

Loop

Counteractive

(Negative)

Feedback

LoopCounteractive

(Negative)

Feedback

Loop

Population Overshoot and Collapse


Overshoot and Collapse

What are the possible equilibrium (steady-state) states?

• We find these by setting the derivative(s) equal to zero.

0

010

cPdt

dR

PR

Rb

dt

dP

Where P = Population

R = Resource

b = per capita birth rate

c = per capita consumption rate

R0 = initial resource base

0

,0010

Por

candR

Rb

model weight output(s) - oregon state...

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