the least square number is called______
Answers
The minimum of the sum of squares is found by setting the gradient to zero. Since the model contains m parameters, there are m gradient equations:
{\displaystyle {\frac {\partial S}{\partial \beta _{j}}}=2\sum _{i}r_{i}{\frac {\partial r_{i}}{\partial \beta _{j}}}=0,\ j=1,\ldots ,m,}
∂β
j
∂S
=2
i
∑
r
i
∂β
j
∂r
i
=0, j=1,…,m,
and since {\displaystyle r_{i}=y_{i}-f(x_{i},{\boldsymbol {\beta }})}r
i
=y
i
−f(x
i
,β), the gradient equations become
{\displaystyle -2\sum _{i}r_{i}{\frac {\partial f(x_{i},{\boldsymbol {\beta }})}{\partial \beta _{j}}}=0,\ j=1,\ldots ,m.}−2
i
∑
r
i
∂β
j
∂f(x
i
,β)
=0, j=1,…,m.
The gradient equations apply to all least squares problems. Each particular problem requires particular expressions for the model and its partial derivatives.
A regression model is a linear one when the model comprises a linear combination of the parameters, i.e.,
{\displaystyle f(x,\beta )=\sum _{j=1}^{m}\beta _{j}\phi _{j}(x),}f(x,β)=
j=1
∑
m
β
j
ϕ
j
(x),
where the function {\displaystyle \phi _{j}}ϕ
j
is a function of {\displaystyle x}x.
Letting {\displaystyle X_{ij}=\phi _{j}(x_{i})}X
ij
=ϕ
j
(x
i
) and putting the independent and dependent variables in matrices {\displaystyle X}X and {\displaystyle Y}Y we can compute the least squares in the following way, note that {\displaystyle D}D is the set of all data.
{\displaystyle L(D,{\vec {\beta }})=||X{\vec {\beta }}-Y||^{2}=(X{\vec {\beta }}-Y)^{T}(X{\vec {\beta }}-Y)=Y^{T}Y-Y^{T}X{\vec {\beta }}-{\vec {\beta }}^{T}X^{T}Y+{\vec {\beta }}^{T}X^{T}X{\vec {\beta }}}L(D,
β
)=∣∣X
β
−Y∣∣
2
=(X
β
−Y)
T
(X
β
−Y)=Y
T
Y−Y
T
X
β
−
β
T
X
T
Y+
β
T
X
T
X
β
Finding the minimum can be achieved through setting the gradient of the loss to zero and solving for {\displaystyle {\vec {\beta }}}
β
Answer:
We know that the smallest 4 digit number is 1000. But it is not a perfect square number. So, the smallest four digit number which is a perfect square must be near to 1000. Hence, The smallest four digit number which is a perfect square is 1024