Statistical Modelling Relationships Distributions

Modelling Process IDENTIFICATION     ESTIMATION    ITERATION   ESTIMATION    ITERATION VALIDATION   APPLICATION  

Relationships Simple Regression Models Multiple Regression Models Logistic Regression Models Other functional Models Lagged Models

Simple Regression Assumes one variable (x) relates to another (y) Assumes errors cancel out Assumes errors have constant variance Assumes errors are independent of each other Assumes errors are normally distributed (for testing theories)

Multiple Regression Assumes several variables (xi) relates to another (y) Assumes errors cancel out Assumes errors have constant variance Assumes errors are independent of each other Assumes xi are independent of one another Assumes errors are normally distributed (for testing theories)

Logistic Regression Like multiple regression but variable to be predicted (y) is binary. Estimates odds and log odds rather than direct effects.

Other Models Could be almost anything, common ones are: Log of (some) variables Polynomials Trignometric Power functions

Lagged Models Usually associated with time series data Assume carry-over effects Carry-over of variable Carry-over of error Tend to use simple forms

Distribution Models Discrete Continuous

Discrete Distributions UNIFORM Equal chance of each and every outcome Often a starting hypothesis

Discrete Distributions BINOMIAL n trials Equal chance of success in each trial (p) Gives probability of r successes in n trials

Discrete Distributions POISSON Random events Fixed average (mean) rate Gives probability that r events will occur in a fixed time, distance, space etc

Discrete Distributions GEOMETRIC Constant probability of success (p) Gives probability of r trials before first success

Continuous Distributions UNIFORM Constant density of probability for all measurement values Limited range of possible values

Continuous Distributions NORMAL Commonest distribution assumption Intuitive Characterised by two parameters, mean and standard deviation Arises from a number of theoretical perspectives

Continuous Distributions EXPONENTIAL Complementary to Poisson Assumes events occur randomly, at fixed mean rate Gives probability density for time, distance, space etc until event occurs

Continuous Distributions EXTREME VALUE DISTRIBUTIONS Weibull Double exponential Gumbel (or Extreme Value)