8: Chapter Data-Driven Models
8.2 Function Tutorial
Background: Data-Driven Models
So far we've been given a model (dP/dt = kP) and used it to generate data. Research often requires us to build a model based on (limited) data. To do this, it helps to be familiar with the standard mathematical functions used for building models.
Linear Functions
Straight line is the simplest model Human beings are biased toward viewing patterns as straight lines with positive slope (Busemeyer et al. 1997)
Linear Functions
y = mx + b
y b x x y
y -y y 2 1 m= = x -x x 2 1
m: slope b: intercept
Quadratic Functions
f x= x+x a () a a + 2 1 0
2
y
x
a special case of ...
Polynomial Functions
f x a + x ++x a () n a =x . a+ . 1 0 .
n n - 1 n - 1
f(x ax )= i
f(x)
i=0
n
i
x
N: degree
Square Root Function
Useful when n - 1 f x a +-quantities a+ () n ax ++x =x a n . are 0 . 1 . 1 already squared: e.g. distance between two points (x1,y1 ) and (x2,y2 ) =
n
f(x)
x
(1 x 2+ 1 y 2 x 2 ( -2 -) y )
Exponential Function
f x a + x ++x a () n a =x . a+ . 1 0 .
n n - 1 n - 1
P) Pa ( =0 t k
r t
P(t)
t
Logarithmic Function
Useful when n - 1 f x a +-dealing. a+ () n ax ++x a =x n .with . 1 0 1 inherently exponential measures, e.g. Richter scale for earthquakes. x
n
y
Log/Log Plot for Power Laws
f x a + x ++x a () n a =x . a+ . 1 0 .
n n - 1 n - 1
Logistic Function
f ...