suppose that we have two input vector and the variables in each vectors are independent and uncorrelated from each other,just only there is relationship between two vector,but not itself in each vector's variable,let say
x=rand(1,10);
>> x
x =
Columns 1 through 9
0.8147 0.9058 0.1270 0.9134 0.6324 0.0975 0.2785 0.5469 0.9575
Column 10
0.9649
y=10.*x+15
y =
Columns 1 through 9
23.1472 24.0579 16.2699 24.1338 21.3236 15.9754 17.7850 20.4688 24.5751
Column 10
24.6489
that means that variables inside $x$ are uncorrelated,but $y$ and $x$ are correlated,my question is which regression analysis is related to this problem?
plot(x,y)
does it means that linear regression is best?
The regression procedure aims to find the most suitable parameters for the following problem: $$ (\alpha_1,\dots,\alpha_n)\ \mbox{s.t.}\ F((x;\alpha_1,\dots,\alpha_n))\approx f(x), $$
where $F(\cdot)$ is the regression function and $f(x)$ is the function that you want to approximate.
Therefore, since you shown a "perfect" linear relation between $x$ and $y$ you may want to use $$ F(x;\alpha,\beta)=\alpha x+\beta. $$
The goodness of fit will be measured by $\rho^2$, coefficient of determination.
Moreover, be aware that even if you'd had a parabolic relation, i.e. $y=ax^2$, you could have used: $$ x^*=x^2, $$ and your relation would have been $$ y=ax^*, $$ so that you could have used a linear regression function again (be aware that error should be coped with differently).