Least-square method
Let us have the set of values yi, every one of which corresponds to any moment of time ti (i = 1, 2,…, N). We need to find dependency y = f(t) if the sum of squared divergences of points of curve from the appropriate point yi is the minimal in the class of approximating functions. That is
(1) Σ(f(ti) - yi)2 à min
We will find the solution in the class of the linear functions (straight lines): y = at + b. Then the term (1) we can write as the following
Σ(a*ti +b - yi)2 à min
The necessary condition of the existence of minimum is the equality to zero of two partial derivatives on a and b accordingly:
Σ((a*ti +b - yi)*ti) =0
Σ(a*ti +b - yi) = 0
In designations
Σti2 = stt
Σ(yi*ti) = syt
Σ(ti) = st
Σ(yi) = sy
Σb = bN
the system can be rewritten as the following
astt + bst - syt = 0
ast +bN - sy = 0
Solving this system of two linear equations for a and b we will get
a = (sytN - syst)/(Nstt - stst)
b = (sttsy - sytst)/(Nstt - stst)