I have four sets of linear regression:
$y_1(x) = m_1x + c_1$
$y_2(x) = m_2x + c_2$
$y_3(x) = m_3x + c_3$
$y_4(x) = m_4x + c_4$
Additionally, the four sets of equations are related as:
$y_1(x) + y_2(x) + y_3(x) + y_4(x) = 1$
I have 70 data points set for each $y_1(x)$, $y_2(x)$, $y_3(x)$, $y_4(x)$ and $x$. I want to solve the system of equations to determine $m_1$, $m_2$, $m_3$, $m_4$, $c_1$, $c_2$, $c_3$ and $c_4$. One way I can approach this problem is to solve the four equations individually and then adjust $y_1(x)$, $y_2(x)$, $y_3(x)$ and $y_4(x)$ proportionally to add to 1. However, I would like to solve the system of equations combined. Any suggestion pointing in that direction would be helpful. Thanks.