Philosophy of Science
Pythagorists-world rational, math is perfect
Epistemic Mediation
Planck, Heisenberg,Schrödinger (and the cat) and the uncertainty principle .
Gödel 1931
Starting with the cracks in the wall of the platonic heaven
Incommensurability: The diagonal of a square can't be measured by the same units used to measure the sides.
Infinitesimals: If you go half way every time you will never get there.
The set of all sets contains more subsets than it contains sets. This is impossible.
"The Gödel Debacle"
Gödel's Theorem
All consistent axiomatic formulations of number theory include undecidable propositions
Desireable Features of Science
Internal Consistency
Beauty
Simplicity
Parsimony
Power
What distinguishes science from religion?????
It must be.. !!!
What is Ecology? "the body of knowledge concerning the economy of nature-the investigation of the total relationships of the animal both to its inorganic and its organic environment; including, above all, its friendly and inimical relations with those animals and plants with which it comes directly or indirectly into contact-in a word ecology is the study of all those complex interrelations referred to by Darwin as the conditions for the struggle for existence."
(Ernst Haeckel 1866)
Greek Oikos = household
"the study of the structure and function of nature"
(Odum 1971:3)
"the scientific study of the relationship between organisms and their environments."
(Mc Naughton and Wolfe 1979)
"the study of the relationship between organisms and the totality of the physical and biological factors affecting them or influenced by them" (Pianka 1988:4)
"the study of the adaptation of organisms to their environment"
(Emlen 1973:1).
"the study of the relationship between organisms and their physical and biological environments"
(Ehrlich and Roughgarden 1987:3)
"the science of the universe"
(Hutchinson)
"the study of how the world works"
(Colinvaux)
What is the environment?
Dictionary Definition of Environment
(Merriam-Webster 1974)
1. The circumstances, objects or conditions by which one is surrounded;
2. The complex of climatic, edaphic and biotic factors that act upon an organism or an ecological community and ultimately determine its form and survival;
3. The aggregate of social and cultural conditions that influence the life of an individual or community;
4. An artistic or theatrical work that involves or encompases the spectator.
This is a diagram of environment. What is wrong?
Environment as circular causal nexus
(von Uexkull 1926)
The entire function circle formed from inner world and surrounding world constitutes a whole.
Continuity of the complete whole must never be lost sight of.
Causation in Environments
Everything is lawfully produced ...and lawfully
produces something else.
"Every object (H) defines two environments: an input environment and its associated causal nexus (H'), and an output environment and its associated causal nexus (H''). The prerogative of environment definition is that of the object" (Patten 1988).
What is a system? (White, Mottershead and Harrison1992)
a set of elements with a set of properties:
1. All systems have structure or organization
2. All function in some way
3. There are functional and structural relationships between parts
4. Function implies that flows and transfers occurr
5. Function requires a driving force or energy source
6. All systems have some degree of integration
Systems possess boundaries, surroundings, elements, states
The state of the system is defined when each of its properties has a definite value.
Systems can be isolated,closed or open.
In any closed system, the final state is determined by the initial condition.
In open systems the final state may be acheived with different initial conditions and in different ways.
Because systems contain elements systems are decomposable.
Everything is a Vector (Whitehead)
Horn 1981
A STATE VECTOR
|
1= Gray Birch |
|
5 |
|
2= Blackgum |
|
36 |
|
3= Red Maple |
|
50 |
|
4= American Beech |
|
9 |
TRANSITION MATRIX (50 year tree-by-tree transition matrix).
|
|
|
1 |
|
2 |
|
3 |
|
4 |
|
1 |
|
.05 |
|
.36 |
|
.50 |
|
.09 |
|
2 |
|
.01 |
|
.57 |
|
.25 |
|
.17 |
|
3 |
|
0 |
|
.14 |
|
.55 |
|
.31 |
|
4 |
|
0 |
|
.01 |
|
.03 |
|
.96 |
Transition Matrix * State Vector= New State Vector
NEW STATE VECTOR (Year 50)
|
1= Gray Birch |
|
1 |
|
2= Blackgum |
|
29 |
|
3=Red Maple |
|
39 |
|
4= American Beech |
|
31 |
NEW STATE VECTOR (Year 100)
|
1= Gray Birch |
|
1 |
|
2= Blackgum |
|
29 |
|
3= Red Maple |
|
39 |
|
4= American Beech |
|
31 |
Example of matrix addition:
|
A= |
1 3 |
|
4 1 |
|
1+4 |
3+1 |
5 4 |
|
2 5 |
+ |
3 7 |
= |
2+3 |
5+7 = |
5 12 |
Examples of Matrix and vector Multiplication
Rows of first * collumns of 2nd
|
A= |
3 1 |
1 |
3*1 + 1*2 |
5 |
||
|
|
2 4 * |
2 |
= |
1*2 + 4*2 |
= |
10 |
So you can multiply a 2*2 * 2*1
BUT YOU CAN'T MULTIPLY 2*1 * 2*2
Result- dimensions of result=dimemsions of the 2nd matrix
Why bother? It can be used to map flows, map changes, calculate utilities, and evaluate direct and indirect effects.
Patten 1985
Adjacency Matrix (A ij) Matrix denotes numbers of paths from column compartments j to row compartments I.
|
|
|
0 |
0 |
0 |
0 |
0 |
0 |
|
|
|
1 |
0 |
0 |
1 |
1 |
1 |
|
Ao= A1 |
= |
0 |
1 |
0 |
0 |
0 |
0 |
|
|
|
0 |
1 |
1 |
0 |
0 |
0 |
|
|
|
0 |
1 |
1 |
1 |
0 |
0 |
|
1 |
0 |
0 |
0 |
1 |
0 |
The product matrix Al gives the number
of paths of length l.
The Matrix A2 gives the number of paths of length 2.
|
|
0 |
0 |
0 |
0 |
0 |
0 |
|
|
1 |
2 |
2 |
1 |
1 |
0 |
|
A2= |
1 |
0 |
0 |
1 |
1 |
1 |
|
|
1 |
1 |
0 |
1 |
1 |
1 |
|
|
1 |
2 |
1 |
1 |
1 |
1 |
|
|
0 |
1 |
1 |
1 |
0 |
0 |
More Examples of Matrix Multiplication
What is an ecosystem?= An open system (exchanges energy and matter with its environment.
1st law of thermodynamics-
energy can neither be created nor destroyed.
2nd law-
Spontaneous transformation of energy is not 100% efficient.
Hierarchies of scale and control.
Within hierarchically structured environments, the behavior of one level is strongly influenced by the behavior of the two adjacent levels (O'Neill).
analysis and synthesis
Models, cognition, and epistemology,
Model=a nonunique, homomorphic mapping or representation
ho*mo*mor*phism (noun)=imperfect representations of reality
[International Scientific Vocabulary]
First appeared 1935
: a mapping of a mathematical set (as a group, ring, or vector space) into or onto another set or itself in such a way that the result obtained by applying the operations to elements of the first set is mapped onto the result obtained by applying the corresponding operations to their respective images in the second set
isomorphic (equal)
Conceptual Models
Statistical models y=a+bx
Analytical Models eg. dN/dt=rN(K-N)/K :the equation can be solved!
Simulation Models lack simple mathematical solutions
deterministic, probabilistic, stochastic
The method used to understand a system is a function of the relative levels of understanding and data available (Holling 1978, Starfield and Bleloch 1986, Grant et al. 1997).
|
System Description |
Effective Approach |
|
Systems with few, highly connected components. |
Analytical Equations Eg. Physics and Mechanics |
|
Systems with many, loosely connected components. |
Statistics |
|
Systems with many, tightly connected components. "organized complexity" |
Simulation and Systems Analysis |
Validation: Does new data give results similar to that acheived by the data initially used to frame the model. This helps us determine the confidence we have in the model.
A tradeoff exists between the ability to predict, the precision, and the generality of a model.
Hypotheses and Ecology
Inductive Method-specific observation to general conclusion
Deductive Method-general understanding yields specific prediction
Data Collection:direct observation or natural experiments, experimental approach(dependent and independent variables
Replicates=systems receiving treatments, replicates should be independent
Hurlbert(1984) warns against pseudo-replication
Testing the Hypothesis
null hypothesis and alternative hypothesis
Beauty, Simplicity, Parsimony, Variance accounted for.
Distributions
Normal Distribution
Mean
Mean=Xi/n
Variance=s2 =S(xi-x)2/n-1
Standard Deviation= sqrt( s2 )
Types of Error
Type 1 error Alpha=probabiliy of rejecting a correct null hypothesis
Type 2 error Beta= probabilty of failing to reject a correct null hypothesis
Choosing a Statistical Test , from Chapter 37 of Intuitive Biotatistics by Harvey Motulsky.
Interactive Statistical Analysis.
Examples of Statistical Analysis Output
The TTEST Procedure Statistics Lower CL Upper CL Lower CL Upper CL Variable N Mean Mean Mean Std Dev Std Dev Std Dev Std Err write 200 51.453 52.775 54.097 8.6318 9.4786 10.511 0.6702 T-Tests Variable DF t Value Pr > |t| write 199 4.14 <.0001
The ANOVA Procedure Class Level Information Class Levels Values Group 3 1 2 3 Number of observations 12 Simple one-way anova 2 The ANOVA Procedure Dependent Variable: Response Sum of Source DF Squares Mean Square F Value Pr > F Model 2 127.1666667 63.5833333 7.76 0.0110 Error 9 73.7500000 8.1944444 Corrected Total 11 200.9166667
The REG Procedure
Model: MODEL1
Dependent Variable: weight
Analysis of Variance
Sum of Mean
Source DF Squares Square F Value Pr > F
Model 1 526.39286 526.39286 14.55 0.0034
Error 10 361.85714 36.18571
Corrected Total 11 888.25000
Root MSE 6.01546 R-Square 0.5926
Dependent Mean 62.75000 Adj R-Sq 0.5519
Coeff Var 9.58638
Parameter Estimates
Parameter Standard
Variable DF Estimate Error t Value Pr > |t|
Intercept 1 30.57143 8.61371 3.55 0.0053
age 1 3.64286 0.95512 3.81 0.0034