least squares function
i varies from 1 to N.
L2 = ∑[ yi - f(yi, pi)]2 (1)
∂[L2]/∂pi = 0 (2)
For a linear function f(yi, pi) = a xi + b
Thus:
L2 = ∑[ yi - (a xi + b) ]2 =
∑ [ yi2 - 2 yi(a xi + b) + (a xi + b) 2]=
∑ [yi2 - 2 a yi xi - 2b yi + a2 xi2 + b2 + 2ab xi]
The condition (2) gives:
∂[L2]/∂a = 0
∂[L2]/∂b = 0
The first leads to:
∑ [a xi2 + b xi - yixi] = 0
The second to:
∑ [a xi - yi + b] = 0
Then:
a ∑xi2 + b ∑xi - ∑yixi = 0
a ∑xi - ∑yi + N b = 0
The latter gives:
b = (1/N) [∑yi - a ∑xi]
This value, substituted to the former leads to:
a ∑xi2 + [(1/N) [∑yi - a ∑xi]] ∑xi = ∑yixi.
Thus:
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a = [N ∑xiyi - ∑xi ∑yi]/[N ∑xi2 - (∑xi)2]
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and
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b = [∑yi ∑xi2 - ∑xi ∑xiyi]/[N ∑xi2 - (∑xi)2]
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