Least-square method

Let us have the set of values yiy_i, every one of which corresponds to any moment of time ti(i=1,2,,N)t_i (i = 1, 2,…, N). We need to find dependency y=f(t)y = f(t), if the sum of squared divergences of points of curve from the appropriate point yiy_i is the minimal in the class of approximating functions. That is

(1)i=1n(f(ti)yi)2min (1) \sum_{i=1}^n (f(t_i) - y_i)^2 \rarr \min

We will find the solution in the class of the linear functions (straight lines): y=at+by = at + b. Then the term (1) we can write as the following

i=1n(ati+byi)2min \sum_{i=1}^n(a*t_i + b - y_i)^2 \rarr \min

The necessary condition of the existence of minimum is the equality to zero of two partial derivatives on a and b accordingly:

i=1n((ati+byi)ti)=0 \sum_{i=1}^n((a*t_i + b - y_i)*t_i) = 0

i=1n(ati+byi)=0 \sum_{i=1}^n(a*t_i + b - y_i) = 0

In designations

i=1nti2=stt \sum_{i=1}^nt_i^2 = stt

i=1n(yiti)=syt \sum_{i=1}^n(y_i*t_i) = syt

i=1n(ti)=st \sum_{i=1}^n(t_i) = st

i=1n(yi)=sy \sum_{i=1}^n(y_i) = sy

Noting that

i=1nb=bN, \sum_{i=1}^nb = b*N,

the system can be rewritten as the following

astt+bstsyt=0 a*stt + b*st - syt = 0

ast+bNsy=0 a*st +b*N - sy = 0

Solving this system of two linear equations for aa and bb we will get

a=(sytNsyst)/(Nsttstst) a = (syt*N - sy*st)/(N*stt - st*st)

b=(sttsysytst)/(Nsttstst) b = (stt*sy - syt*st)/(N*stt - st*st)