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Shalabh and S. S. Dhar
2.6 Prediction of Values of Study Variable
Any model is prepared with an objective to use further in other applications. One
important application is the prediction. The meaning of prediction in the contest
of regression modelling is to know the value of study variable for given values of
explanatory variables. The predictions are obtained by first fitting a model based on
a given set of data on study and explanatory variables and then finding the value(s)
of study variable for given values of explanatory variables.
We aim to predict the unknown value y 0 of y at a given value of explanatory
variables x 0 = (x 01 , x 02 , . . . , x 0k )
T . The predictions can be made at a point as well
as in an interval. The predictor as a point predictor is given by
p f = x
T
0
ˆ
β = ˆ
β 0 + x 01 ˆ
β 1 + x 02 ˆ
β 2 + · · · + x 0k ˆ
β k ,
(21)
and its variance is estimated by
V ar
( p f ) = ˆ
σ
2
1 + x
T
0 (X
T X )
−1 x 0
.
(22)
The prediction in an interval is obtained by finding the prediction interval. The
100(1 − α)% prediction interval of y 0 at the point x 0 is given by
p f − t α
2 ,n−k−1
ˆ
σ 2 [1 + x T
0 (X T X ) −1 x 0 ], p f + t α
2 ,n−k−1
ˆ
σ 2 [1 + x T
0 (X T X ) −1 x 0 ]
,
(23)
which means that the predicted value will lie in the interval (23) with 100(1 − α)%
chances in the sense that
P
p f − t α
2 ,n−k−1
ˆ
σ 2 [1 + x T
0 (X T X ) −1 x 0 ] ≤ y 0 ≤ p f + t α
2 ,n−k−1
ˆ
σ 2 [1 + x T
0 (X T X ) −1 x 0 ]
= 1 − α.
3 Data-Based Example
Consider a data-based example to understand the steps, computation, and interpretation of different quantities involved in deriving a multiple regression model. Another
objective is to explain how to read the outcome of software. The format of outcome
in different software may vary but the reported quantities are more or less the same.
The rainfall during the monsoon season in a region depends upon several variables
and we consider three such variables for illustration viz., wind speed, precipitation
and relative humidity. The hypothetical data on monthly rainfall (in cms.), wind
speed (in km per hour), precipitation (in %) and relative humidity (in %) are given
in Table 1.
Shalabh and S. S. Dhar
2.6 Prediction of Values of Study Variable
Any model is prepared with an objective to use further in other applications. One
important application is the prediction. The meaning of prediction in the contest
of regression modelling is to know the value of study variable for given values of
explanatory variables. The predictions are obtained by first fitting a model based on
a given set of data on study and explanatory variables and then finding the value(s)
of study variable for given values of explanatory variables.
We aim to predict the unknown value y 0 of y at a given value of explanatory
variables x 0 = (x 01 , x 02 , . . . , x 0k )
T . The predictions can be made at a point as well
as in an interval. The predictor as a point predictor is given by
p f = x
T
0
ˆ
β = ˆ
β 0 + x 01 ˆ
β 1 + x 02 ˆ
β 2 + · · · + x 0k ˆ
β k ,
(21)
and its variance is estimated by
V ar
( p f ) = ˆ
σ
2
1 + x
T
0 (X
T X )
−1 x 0
.
(22)
The prediction in an interval is obtained by finding the prediction interval. The
100(1 − α)% prediction interval of y 0 at the point x 0 is given by
p f − t α
2 ,n−k−1
ˆ
σ 2 [1 + x T
0 (X T X ) −1 x 0 ], p f + t α
2 ,n−k−1
ˆ
σ 2 [1 + x T
0 (X T X ) −1 x 0 ]
,
(23)
which means that the predicted value will lie in the interval (23) with 100(1 − α)%
chances in the sense that
P
p f − t α
2 ,n−k−1
ˆ
σ 2 [1 + x T
0 (X T X ) −1 x 0 ] ≤ y 0 ≤ p f + t α
2 ,n−k−1
ˆ
σ 2 [1 + x T
0 (X T X ) −1 x 0 ]
= 1 − α.
3 Data-Based Example
Consider a data-based example to understand the steps, computation, and interpretation of different quantities involved in deriving a multiple regression model. Another
objective is to explain how to read the outcome of software. The format of outcome
in different software may vary but the reported quantities are more or less the same.
The rainfall during the monsoon season in a region depends upon several variables
and we consider three such variables for illustration viz., wind speed, precipitation
and relative humidity. The hypothetical data on monthly rainfall (in cms.), wind
speed (in km per hour), precipitation (in %) and relative humidity (in %) are given
in Table 1.