Chapter 8: Data-Driven Models 8.2 Function Tutorial.
-
Upload
polly-waters -
Category
Documents
-
view
218 -
download
0
Transcript of Chapter 8: Data-Driven Models 8.2 Function Tutorial.
![Page 1: Chapter 8: Data-Driven Models 8.2 Function Tutorial.](https://reader035.fdocuments.net/reader035/viewer/2022081809/5697bfac1a28abf838c9b59e/html5/thumbnails/1.jpg)
Chapter 8: Data-Driven Models
8.2 Function Tutorial
![Page 2: Chapter 8: Data-Driven Models 8.2 Function Tutorial.](https://reader035.fdocuments.net/reader035/viewer/2022081809/5697bfac1a28abf838c9b59e/html5/thumbnails/2.jpg)
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.
![Page 3: Chapter 8: Data-Driven Models 8.2 Function Tutorial.](https://reader035.fdocuments.net/reader035/viewer/2022081809/5697bfac1a28abf838c9b59e/html5/thumbnails/3.jpg)
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)
![Page 4: Chapter 8: Data-Driven Models 8.2 Function Tutorial.](https://reader035.fdocuments.net/reader035/viewer/2022081809/5697bfac1a28abf838c9b59e/html5/thumbnails/4.jpg)
Linear Functions
x
y
x
yb
m: slopeb: intercept
![Page 5: Chapter 8: Data-Driven Models 8.2 Function Tutorial.](https://reader035.fdocuments.net/reader035/viewer/2022081809/5697bfac1a28abf838c9b59e/html5/thumbnails/5.jpg)
Quadratic Functions
x
y
a special case of …
![Page 6: Chapter 8: Data-Driven Models 8.2 Function Tutorial.](https://reader035.fdocuments.net/reader035/viewer/2022081809/5697bfac1a28abf838c9b59e/html5/thumbnails/6.jpg)
Polynomial Functions
x
f(x)
n: degree
![Page 7: Chapter 8: Data-Driven Models 8.2 Function Tutorial.](https://reader035.fdocuments.net/reader035/viewer/2022081809/5697bfac1a28abf838c9b59e/html5/thumbnails/7.jpg)
Square Root Function
f (x) an x n an 1xn 1 ... a1x a0
x
f(x)
Useful when quantities are already squared: e.g. distance between two points (x1,y1 ) and (x2,y2 ) =
![Page 8: Chapter 8: Data-Driven Models 8.2 Function Tutorial.](https://reader035.fdocuments.net/reader035/viewer/2022081809/5697bfac1a28abf838c9b59e/html5/thumbnails/8.jpg)
Exponential Function
f (x) an x n an 1xn 1 ... a1x a0
t
P(t)
![Page 9: Chapter 8: Data-Driven Models 8.2 Function Tutorial.](https://reader035.fdocuments.net/reader035/viewer/2022081809/5697bfac1a28abf838c9b59e/html5/thumbnails/9.jpg)
Logarithmic Function
f (x) an x n an 1xn 1 ... a1x a0
x
y
Useful when dealing with inherently exponential measures, e.g. Richter scale for earthquakes.
![Page 10: Chapter 8: Data-Driven Models 8.2 Function Tutorial.](https://reader035.fdocuments.net/reader035/viewer/2022081809/5697bfac1a28abf838c9b59e/html5/thumbnails/10.jpg)
Log/Log Plot for Power Laws
f (x) an x n an 1xn 1 ... a1x a0
![Page 11: Chapter 8: Data-Driven Models 8.2 Function Tutorial.](https://reader035.fdocuments.net/reader035/viewer/2022081809/5697bfac1a28abf838c9b59e/html5/thumbnails/11.jpg)
Logistic Function
f (x) an x n an 1xn 1 ... a1x a0
t
P(t)
![Page 12: Chapter 8: Data-Driven Models 8.2 Function Tutorial.](https://reader035.fdocuments.net/reader035/viewer/2022081809/5697bfac1a28abf838c9b59e/html5/thumbnails/12.jpg)
Trigonometric Functions
f (x) an x n an 1xn 1 ... a1x a0
x
cos(x)
• Useful for for modeling cyclic processes (ocean temperature, blood sugar level)
• Because sin(θ) = cos(θ-/2), we generally only need cos.