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Least-square method |
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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 a*stt + b*st - syt = 0 a*st +b*N - sy = 0 Solving this system of two linear equations for a and b we will get a = (syt*N - sy*st)/(N*stt - st*st) b = (stt*sy - syt*st)/(N*stt - st*st) |
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