Some Examples

Example: daily auto accidents in Saskatchewan to 1984 to 1992 Data collected: 1.Date 2.Number of Accidents Factors we want to consider: 1.Trend 2.Yearly Cyclical Effect 3.Day of the week effect 4.Holiday effects

Trend This will be modeled by a Linear function : Y =  0 +  1 X (more generally a polynomial) Y =  0 +  1 X +  2 X 2 +  3 X 3 + …. Yearly Cyclical Trend This will be modeled by a Trig Polynomial – Sin and Cos functions with differing frequencies(periods) : Y =  1 sin(2  f 1 X) +  1 cos(2  f 2 X)  1 sin(2  f 2 X) +  2 cos(2  f 2 X) + …

Day of the week effect: This will be modeled using “dummy”variables :  1 D 1 +  2 D 2 +  3 D 3 +  4 D 4 +  5 D 5 +  6 D 6 D i = (1 if day of week = i, 0 otherwise) Holiday Effects Also will be modeled using “dummy”variables :

Independent variables X = day,D1,D2,D3,D4,D5,D6,S1,S2,S3,S4,S5, S6,C1,C2,C3,C4,C5,C6,NYE,HW,V1,V2,cd,T1, T2. Si=sin( *i*day). Ci=cos( *i*day). Dependent variable Y = daily accident frequency

Independent variables ANALYSIS OF VARIANCE SUM OF SQUARES DF MEAN SQUARE F RATIO REGRESSION RESIDUAL VARIABLES IN EQUATION FOR PACC. VARIABLES NOT IN EQUATION STD. ERROR STD REG F. PARTIAL F VARIABLE COEFFICIENT OF COEFF COEFF TOLERANCE TO REMOVE LEVEL. VARIABLE CORR. TOLERANCE TO ENTER LEVEL (Y-INTERCEPT ). day E E IACC D Dths D S D S D S D C D V S V S cd S T C C C C C NYE HW T ***** F LEVELS( 4.000, 3.900) OR TOLERANCE INSUFFICIENT FOR FURTHER STEPPING

D D D D D D Day of the week effects

NYE HW T Holiday Effects

S S S C C C C C Cyclical Effects

Example 2 Data on the reaction rate of the catalytic isomerization of л-pentane to isopentane versus the partial pressures of hydrogen, л- pentane, and isopentane are reproduced in the Table on the following slide

Isomerization is a chemical process in which a complex chemical is converted into more simple units, called isomers: catalytic isomerization employs catalysts to speed the reaction. The reaction rate depends on various factors, such as partial pressures of the products and the concentration of the catalyst.

The differential reaction rate was expressed as grams of isopentane produced per gram of catalyst per hour (hr -1 ), and the instantaneous partial pressure of a component was calculated as the mole fraction of the component times the total pressure, in pounds per square inch absolute (psia).

A common form of model for the reaction rate is the Hougen-Watson model: Fit this model