Basics of Paleomagnetism and Rock Magnetism

Line fitting of y = a + bx: Errors in both y and x

An observation and a point of minimum distance to the line. Quantity to be minimized is, \[ t = \sum_{i=1}^n \{(x_i - p_i)^2 + (y_i - q_i)^2\}. \] It is easily shown that, \[ t = \frac{1}{b^2 + 1} \sum_{i=1}^n (y_i - a - bx_i)^2. \] From \(\partial t/\partial a = 0\), we obtain \(a = \bar y - b\bar x\), where \(\bar x = \sum_{i=1}^n x_i/n\) and \(\bar y = \sum_{i=1}^n y_i/n\). Hence, \(t\) is given by, \begin{eqnarray*} t & = & \frac{1}{b^2+1}\sum_{i=1}^n \{(y_i - \bar y) - b(x_i - \bar x)\}^2, \\ & = & \frac{1}{b^2+1}\sum_{i=1}^n (v_i - bu_i)^2, \end{eqnarray*} where \(u_i = x_i - \bar x\), \(v_i = y_i - \bar y\). Setting \(\partial t/\partial b = 0\) and denoting \(S_{uu} = \sum_{i=1}^n u_i^2, S_{vv} = \sum_{i=1}^n v_i^2, S_{uv} = \sum_{i=1}^n u_i v_i\), we obtain, \[ S_{uv}b^2 - (S_{vv} - S_{uu})b - S_{uv} = 0. \] Solving this equation, \(b\) is given by, \begin{equation} b = \frac{(S_{vv} - S_{uu}) \pm \sqrt{(S_{vv} - S_{uu})^2 + 4S_{uv}^2}}{2S_{uv}}. \label{eq01} \end{equation}

However, to analyze an Arai plot in paleointensity studies, we usually do not use this method or the one presented in the previous page. Instead, a special solution of York (1966) is used as described in the next page.