Concept of Mathematical Models

1
Concept of Mathematical Models

• describe regularities
• It is experience, rather than logic, that has
  identified such regularities.
• e.g. the sun will rise in the morning and set in
  the evening
• a good model captures enough important features
  of the object that it
• represents to be useful for a specific purpose.
• types of mathematical models
• deterministic e.g. mechanistic models,
  biological models.
• stochastic (random) e.g. regression models,
  stochastic differential equations.

2
Example of a Deterministic Model

• Starting from rest above the earth's surface, a
  rock falls for a period of time. The formula for
  the distance traveled is dgt2/2
• this is a mechanistic model with theoretical
  basis in physics
• it is deterministic since there is no allowance
  for statistical errors.
• neglected aspects
• effect of air resistance
• measurement inaccuracy, etc.
• an inaccurate empirical model

3
A Stochastic Model(Regression Model)

• A statistical model can be used to model these
  inaccuracies based on data.
• It is believed that writing score is positively
  associated with the reading scores.
• WabRnoise.

4
Overview of Regression Analysis

• A regression model is a statistical model that
  investigates the relationship between variables
  of interest.
• Application of Regression Models
• Data description Describe relationship between
  variables
• Estimation Estimate the parameters
• Prediction and forecasting
• Control
• Regression analysis can be applied to data from
• Insurance (Actuarial Sciences), Biology,
  Economics (Econometrics), Engineering, Finance,
  Health Sciences, Psychology, Sociology

5
Overview of Regression Analysis

• A general regression model is as follows
• Y g(X1, ,Xp ß) e
• Y is called the response variable.
• X1, , Xp are the explanatory variables,
  predictors, regressors.
• Functional form of the g(., .) are known.
• ß is a vector of unknown.
• e is the error term.

6
Types of Regression Models

• linear regression models
• nonlinear regression models
• generalized linear models
• nonparametric regression models
• time series models

7
Linear Regression Models

• g() is a linear function of the parameters
• simple linear regression models
• Yß0 xß e
• multiple linear regression models
• Y ß0 x1 ß1, ,xp ßp e
• polynomial regression models
• Y ß0x ß1x2 ß2, ,xq ßq e

8
Nonlinear Regression Models

• g() is a nonlinear function of the parameters
• examples
• Yr ß0 x1 ß1x2 ß2 e
• Y (ß0 x1 ß1x2 ß2)r e
• Y exp(ß0 x1 ß1x2 ß2) e

9
Nonparametric Regression Models

• no parameters involved
• Y g(x1, ,xp)e
• g() is an unknown function
• data driven techniques