Non-linear least-square systems isochrons

A particular experiment provides observations of n independent variables Xj (j = 1. n) and of a single dependent scalar variable Y which we suspect to be related through a linear relationship such as 

In order to take advantage of a matrix formulation, we define the vector X of the n variables X and the vector a of the n unknowns ctj and write 

We now proceed to m observations. The ith observation provides the estimates xi of the independent variables Xj and the estimate y, of the dependent variable Y. The n estimates xtj of the variables Xj provided by this ith observation are lumped together into the vector xt. We assume that the set of the (n+1) data (i/,y,) associated with the ith observation represent unbiased estimates of the mean ( yf) of a random (n + 1)-vector distributed as a multivariate normal distribution. The unbiased character of the estimates is equivalent to 

Si is the (n + l)x(n + l) covariance matrix of the ith measurement (i y,). The jth diagonal term is the variance of xip while the (n + l)th diagonal term is the variance of yt. The off-diagonal terms are the corresponding covariance terms. 

In order to illustrate how the maximum-likelihood expression can be built, let us consider the case n = 1 (only one. Y), which is the case of a straight line relating X and Y. The expression of c2 is given by 

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