CS 365 Introduction to Scientific Modeling Fall Semester, 2011 Review

Transcription

1 CS 365 Introduction to Scientific Modeling Fall Semester, 2011 Review

2 Topics" What is a model?" Styles of modeling" How do we evaluate models?" Aggregate models vs. individual models." Cellular automata" Power laws and data analysis:" Statistical distributions" Testing for power laws and other distributions" Maximum likelihood estimates" Power laws in nature" Predator Prey Models:" Brief introduction to dynamical systems and chaos" Lotka-Volterra equations (2 species only) "

3 Modeling" How do we use models?" How do we evaluate models?" Different approaches to modeling" Examples of different kinds of models and what they are used for. " Pros and cons of different modeling methods" Limitations of modeling?"

4 Examples" Blueprint of a bridge" 2-dimensional projection of a 3-dimensional image" Crash dummies (model humans)" Lotka-Volterra equations" Forest fire simulation"

5 Cellular Automata" 1-D and 2-D" Space-time plots" Neighborhood, Update rules" Wolframʼs classification and dynamical regimes" Forest fire model and the game of Life" Bocavirus model"

6 Power Laws and Scaling" What is a power law?" Why is it important?" How do power laws arise?" How are power laws related to scaling?" How do I know if my data shows a power law?" Fitting curves to data and testing for significance."

7 Power Law Distribution" Polynomial:" Scale invariant:" p(x) = ax b p(cx) = a(cx) b = c b p(x) p(x) Distribution can range over many orders of magnitude" Ratio of largest to smallest sample " Plotted on log-log axes" Slope of line gives scaling exponent" Y-intercept gives the constant" log(p(x)) = log(ax b ) = blog x + loga Heavy tailed (right skewed)" Universality"

8 Exponential Distributions:" aka single-scale" Have form P(x) = e -ax " Use Gaussian to approximate exponential because differentiable at 0." Plot on log-linear scale to see straight line." Power-law Distributions:" aka scale-free or polynomial" Have form P(x) = x -a " Fat tail is associated with power law because it decays more slowly." Plot on log-log scale to see straight line." Scaling Relations"

9 Measuring Power Laws" Plot histogram of samples on log-log axes (a):" Test for linear form of data on plot" Measure slope of best-fit line to determine scaling exponent" Maximum Likelihood Estimate" Problem: Noise in right-hand side of distribution (b)" Each bin on the right-hand side of plot has few samples" Correct with logarithmic binning (c)" Divide #samples in each bin by width of bin (count per unit interval of x)" Cumulative distribution function (d)" P(x) = x p(y)dy Probability P(x) that x has a value greater than y (1 - CDF)" Also follows power law but with the exponent b-1" No need to use logarithmic binning" Sometimes called rank/frequency plots" For power laws" P(x) = p(y)dy = a y b dy = a b 1 x (b 1) = 0 x( b +1) (b 1) = x( b +1) (b 1) x x

10 M. Newman Power laws, Pareto distributions and Zipfʼs Law (2006) 1 million random numbers, with b=2.5"

11 Linear Empirical Models" An empirical model is a function that captures the trend of observed data:" It predicts but does not explain the system that produced the data." A common technique is to fit a line through the data:" y = mx + b Assume Gaussian distributed errors." Note: For logged data, we assume that the errors are log -normally distributed.! Image downloaded from Wikipedia Sept. 11, 2007"

12 Dynamical Systems" State spaces" Trajectories and time series" Attractors" Dynamical regimes" Stability analysis" Examples"

13 Lotka-Volterra Model Rabbit and Lynx Population" The change in the rabbit population is equal to how many rabbits are born minus the number eaten by lynxes:" dx dt = Ax By The change in lynx population is equal to how fast they reproduce (depends on how many rabbits are available to eat) minus their death rate:" dy dt = Cy + Dxy

14 Paramecium Data"

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16 Agent-based (Individual) Version (3 species)" Plants can:" Spread into contiguous empty space." Be eaten by herbivores." Herbivores can:" Die (by starving to death or being eaten by carnivores). " Move into contiguous locations." Eat plants." Have babies (if they have stored enough energy)." Carnivores can:" Die (by starving to death)." Move to a contiguous location." Eat herbivores." Have babies (if they have stored enough energy)."

17 Individual-based Models"

18 Differences Between Continuous and Discrete Lotka-Volterra Models" Continuous Version: Doesnʼt consider competition among prey or predators (e.g., carrying capacity):" Prey population may grow infinitely without any resource limits (the rabbits never run out of food)." Predators have no saturation: Their consumption rate is unlimited (the lynxes never get full)." Only considers two interacting species." Nanofoxes?" Agent-based Version?" Potential for non-uniform mixing (because space is represented explicitly)" Non-deterministic movement into adjacent spaces." Discrete time." Discrete population values." Discrete threshhold for reproduction." How are they the same?"