Maths stats246
Introduction
Models
playa vitally important roleinthesolutions to problems inpractice
In particular linear models are often used in decision processes
Linearstatisticmodels have a stochasticcomponent and are
typically multidimensional
These models are described
using vectors and matrices
classification of models s Physicalmodels
Abstract models
Quantitativeand qualitative mathematicalModels
Thestatisticalmodel
Thestatisticalmodel will be of theform
Y ft E
Where Y is the dependent variable f is a function of one or more quantifiablevariables
and E is the error in the model
Yis the variable of primaryinterest and can be seento be explained bythe
Variables in f so that Ely f
However it is generally the case that for numerical values of Y and variables in f
that fi the difference
can be attributed to measurement error
yi
specification
ie fi te
yi
,Exampleof LinearstatisticalModel s
A researcherexamines the impact of the number of hoursworked intelligence of
student abilityof lecturer and standard of test on how well a student performs
Test performance and hours worked can be measured but the other variables
cannot
The researcherdecides to use a statisticalmodeland include the qualitative
variables as partof the specification error inthe stochasticemerterm
X hoursstudied
Model I Yx f x Ex errorterm
testmark
Yxand Ex are stochastic variables
Nowtheresearcher mustdecide ontheform of f and wether X will be a stochastic
variable or not
Researcherassumes
feat at Bx and Ex Normal o 02
Researchertakestosamples
, cannow predict mark
NotationandBasicDefinitions
Asinglecolumn matrixwith n news is called a vector with n components
Let ai an beelementsfrom a field t then an n component vector
d
realentries
a
9s which is indicated also
by a nx
under
naigffr
an line column vector nrows x icolumn
The transpose of a nxt is a new vector consisting of n components and
m
transpose
If ei n Xi is a vector withthe ith componentequal to one and all other
components o thenthe identity matrix is
g g g
Nxnidentitymatrix
, An n component vector with all zeros is called the zerovector and canbe
written as o
Considertwo n component vectors X and Y The scalar
transposemustalwaysbeonleftside
X'y Y x É XiYi is called theinner product scalarproduct of
twovectors
Thetwo vectors X and Y are orthogonal if X'y Y x 0 The vector x
is normal if and only if
xx It Xi
Thelengthof x is defined as 1 11 X X É Xi
makelength 1
It followsthat the vector Y can be normalised
by dividing Y
by its
length
y
1141 thishaslengthofone unit
yy
Schwarzinequality
the absolute valueof the innerproduct of x and y
x y 11 11 my is smaller or equal to the product ofthe lengths of
X andY
Introduction
Models
playa vitally important roleinthesolutions to problems inpractice
In particular linear models are often used in decision processes
Linearstatisticmodels have a stochasticcomponent and are
typically multidimensional
These models are described
using vectors and matrices
classification of models s Physicalmodels
Abstract models
Quantitativeand qualitative mathematicalModels
Thestatisticalmodel
Thestatisticalmodel will be of theform
Y ft E
Where Y is the dependent variable f is a function of one or more quantifiablevariables
and E is the error in the model
Yis the variable of primaryinterest and can be seento be explained bythe
Variables in f so that Ely f
However it is generally the case that for numerical values of Y and variables in f
that fi the difference
can be attributed to measurement error
yi
specification
ie fi te
yi
,Exampleof LinearstatisticalModel s
A researcherexamines the impact of the number of hoursworked intelligence of
student abilityof lecturer and standard of test on how well a student performs
Test performance and hours worked can be measured but the other variables
cannot
The researcherdecides to use a statisticalmodeland include the qualitative
variables as partof the specification error inthe stochasticemerterm
X hoursstudied
Model I Yx f x Ex errorterm
testmark
Yxand Ex are stochastic variables
Nowtheresearcher mustdecide ontheform of f and wether X will be a stochastic
variable or not
Researcherassumes
feat at Bx and Ex Normal o 02
Researchertakestosamples
, cannow predict mark
NotationandBasicDefinitions
Asinglecolumn matrixwith n news is called a vector with n components
Let ai an beelementsfrom a field t then an n component vector
d
realentries
a
9s which is indicated also
by a nx
under
naigffr
an line column vector nrows x icolumn
The transpose of a nxt is a new vector consisting of n components and
m
transpose
If ei n Xi is a vector withthe ith componentequal to one and all other
components o thenthe identity matrix is
g g g
Nxnidentitymatrix
, An n component vector with all zeros is called the zerovector and canbe
written as o
Considertwo n component vectors X and Y The scalar
transposemustalwaysbeonleftside
X'y Y x É XiYi is called theinner product scalarproduct of
twovectors
Thetwo vectors X and Y are orthogonal if X'y Y x 0 The vector x
is normal if and only if
xx It Xi
Thelengthof x is defined as 1 11 X X É Xi
makelength 1
It followsthat the vector Y can be normalised
by dividing Y
by its
length
y
1141 thishaslengthofone unit
yy
Schwarzinequality
the absolute valueof the innerproduct of x and y
x y 11 11 my is smaller or equal to the product ofthe lengths of
X andY
