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Load forecasting for practical power systems by…
III.
ESTIMATION OF PARAMETERS IN LINEAR REGRESSION MODELS SIMPLE
REGRESSION:
Where X is independent variable and Y is dependent variable. Where “a” indicates intercept
a y bx
.Where “b” indicates
xy nxy x y
b wherex andy =
2 2
x nx
n n
slope……………..(4)
Multiple Linear Regressions
The method of least squares is typically used to estimate the regression coefficients in a multiple linear
regression model. Suppose that n>k observations on the response variable are available say, y 1 , y 2 … y n . Along
with each observed response y i , we will have an observation on each regressed variable and let x ij denote the ith
level of variable x j . We assume that the error term € in the model has E(€) =0 and V(€) = σ 2 and that the {€ i}
are uncorrelated random variables. We may write the model equation
y i = β 0 + β 1 x i1 + β 2 x i2 + …+ β k x ik + € I ……………..(5)
y
i
o
k
j1
x
j
ij
i
…………………………..(6)
i = 1, 2… n
The method of least squares chooses the β‟s in equation (6) so that the sum of the squares of errors, € I , is
minimized. The least squares function is
L
n
i1
2
i
n
k
( yi
o
j
xij
i1
j1
www.ijceronline.com ||May ||2013|| Page 58
)
2
……………..(7)
The function L is to be minimized with respect to β 0 , β 1 … β k . The least square estimators, say β 0 , β 1 … β k , must
satisfy
n
k
L
2
( yi ˆ0 ˆ
jxij)
0 ……………..(8)
i1 j1
n
k
L
2(
yi ˆ0 ˆ
jxij)
xij 0 ……………..(9)
i1 j1
j = 1, 2, ------, k.
Simplifying equation (9), we obtain
n ˆ ˆ
ˆ
ˆ
0
n
0
i1
n
0
i1
i1
n
x
ˆ
1 i1
2
i1 i1
x ˆ
x
ik
ˆ
n
x
n
n
2
x ˆ
1 i1
2
xi
1
i1 i1
n
x x ˆ
1 ik i1
2
i1 i1
n
i2
x
x
i2
..... ˆ
.....
ˆ
x
ik i2
n
x
n
k ik
i1 i1
n
.....
ˆ
x
x
k i1
ik
i1 i1
n
ik
n
n
2
k
xik
xik
i1 i1
y
i
i
……………..(10)
x y ……………..(11)
y
i
……………..(12)
These equations are called the least squares normal equations. Note that there are p = k + 1 normal equations,
one for each of the unknown regression
Defining Problem:
The problem of load forecasting is approached by making the forecasts for one whole day at a time.
The approach is static in the sense that the forecast is not updated during the day. The forecasting of the load on
the daily basis with neural network techniques has been reported in many variations. Single feed forward
artificial neural network architecture is used in forecasting the daily peak, valley, and average loads.
These equations are called the least squares normal equations. Note that there are p = k + 1 normal
equations, one for each of the unknown regression coefficients. The solution to the normal equations will be the
least squares estimators of the regression coefficients ˆ , ˆ
0
1,.....,ˆ
k.
It is simpler to solve the normal equations if they are expressed in matrix notation. We now give a
matrix development of the normal equations that parallels the development of equation (10, 11, 12). The model
in terms of the observations, of the above equation may be written in matrix notation as Y = X β + €
Where,