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: ----------------------------------------- a = [N ∑xiyi - ∑xi ∑yi]/[N ∑xi2 - (∑xi)2] ----------------------------------------- and ----------------------------------------- b = [∑yi ∑xi2 - ∑xi ∑xiyi]/[N ∑xi2 - (∑xi)2] -----------------------------------------