Mathematical Models

I. Deterministic Models

1) Statistical Models

What methods do we use to estimate the parameters of the underlying relationship?

## Notations

1)   Dependent variable (y)

2)   Independent variable (x)

Equation of a straight Line =

y  =   bo  +  b1 x

bo = y-intercept

b1 = slope

(no error in reading y – for a given value of x we can predict y exactly using the above equation)

Simple to use – unrealistic

## What to do

1)   Plot a scatter diagram

A model that allows for the possibility that the observations do not all lie on a straight line is

y  =  bo  +  b1 x  + e

where e  is a random error

the average value of e for a given value of x  =  0

Thus, the expected value of (y) (average value) for a fixed value of x is the line

E(y)  =  bo  +  b1 x

The observed values of y deviate above or below the line by a random error e

To estimate the values of the constants bo and b1

## The Method of Least Squares

Let y` denote the predicted value of y for a given value of x

Then the error of prediction (residual) is y-y` (the difference between the actual value of y and what we predict it to be)

The least squares method chooses the prediction line

y`  = b`o + b`1 x        that minimizes the sum of the squared errors of prediction

å (y-y`)2 for all sample points

Thus, the method of least squares consists of finding those estimates b`o and b`1 that minimize å (y-y`)2

The estimates:

b`1  =  Sxy /  Sxx  and b`o = Y - b`1 X

Where,

Sxx  = å (x – X )2  = å x2 – (åx)2/n

Sxy  = å (x – X ) ( y – Y ) =  å xy – ( ( åx ) ( å y ) ) /n

Example

 x y 4.2 2.8 3.8 2.5 4.8 3.1 3.4 2.1 4.5 2.9 4.6 2.8 4.3 2.6 3.7 2.4 3.9 2.5

Calculate

Sxx  and Sxy

 X x2 y y2 xy 4.2 17.64 2.8 7.84 11.76 3.8 14.44 2.5 6.25 9.50 4.8 23.04 3.1 9.61 14.88 3.4 11.56 2.1 4.41 7.14 4.5 20.25 2.9 8.41 13.05 4.6 21.16 2.8 7.84 12.88 4.3 18.49 2.6 6.76 11.18 3.7 13.69 2.4 5.76 8.88 3.9 15.21 2.5 6.25 9.75 37.2 155.48 23.7 63.13 99.02

## Correlation coefficient

A measure of the strength of the relationship between two variables x and y

r  =  Sxy / sqr ( SxxSyy)

where

Syy =  å y2  - (åy)2 / n

Or

r =  b`1 sqr ( Sxx / Syy )

Properties of r

1)   r lies between –1 and +1.  r > 0 indicates a positive linear relationship and r < 0 a negative linear relationship between x and y. r = 0 indicates no linear relationship between x and y

2)   r2 gives the proportion of the total variability of the y values that can be accounted for by the independent variable x

r2 = å( y` - Y )2 / Syy  =  S2xy / SxxSyy

1       -   r2  =  å ( y  -  y` )2 / Syy

Example

 Y X 1.4 1 2.3 2 3.1 3 4.2 4 5.1 5 5.8 6 6.8 7 7.6 8 8.7 9 9.5 10

Calculate Syy ,  Sxx , and Sxy

Calculate r

## Lack of fit in linear regression

1)   scatter plot

2)   plot of the residuals ( y – y`) versus y` may give an indication of the need for higher order terms in the model

Linear = no apparent pattern

Higher order model = apparent pattern

# Inferences about bo and b1

The expected value of bo = mbo and the expected value of

b1 = m b1

The standard error of bo = s bo  = s e sqr ( å x2/nSxx )

The standard error of b1  =   s b1  =  s e / sqr ( Sxx )

Estimates of  s b1 and  s bo

S2e = å(y- Y)2 / n-2

Se  =  (S2e )

a) 100( 1- a )% confidence intervals

bo ± ta/2 Se sqr ( å x2 / nSxx )

b1 ± ta/2  Se / sqr ( Sxx )

Note :  d.f. = n –2

b) Statistical tests

Ho: bo = 0

Ha : bo > 0

b0 < 0

bo ¹ 0

T.S.:    t  =  bo` /  Se sqr ( å x2 / nSxx )

R.R.:  for a given value of a and d.f. = n –2

1.       Reject   Ho    if     t > ta

2.       Reject   Ho    if     t < - ta

3.      Reject    Ho    if    | t | > ta/2

Ho: b1 = 0

Ha: b1 > 0

b1 < 0

b1 ¹ 0

T.S. :  t  =  b1` /  ( Se / sqr( Sxx)  )

R.R.:  for a given value of a and d.f. = n – 2

Same as for bo