Statistics for Environmental Engineers

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The simplest problem is considered here. If we assumed that all n variables have been measured and all constraint equations are linear, the problem is to:


minimize S = Xwt(x, — mt)2    i = 1,2,…,N


subject to linear constraints of the form:


X a^x, = bj    j = 1,2,., M


For this case of linear constraint equations, one method of solution is Lagrange multipliers, which are used to create a new objective function:


N    M


minimize G = Xwt(x, — mi)2+    aijxi — bj)


i=1    j=1


There are now N + M variables, the original N unknowns (the x;’s) plus the M Lagrange multipliers (A/s). The minimum of this function is located at the values where the partial derivatives with respect to each unknown variable vanish simultaneously. The derivative equations have two forms:


dG


dx,


dG


dAj


M


2w,(x, —


— mi) + XAjaij = 0


j=1


i = 1,2,.


,N


N


X aijxi


i=1


bj = 0


j = 1,2,.


,M

The quantities m, w, and aji are known and the resulting set of linear equations are solved to compute x’s and the Я/s:


M


2wixi + X а/Я/ = 2miwi    i = 1,2,…, N


j=1


M


X aijXi = bj    j = 1,2,., M


i= 1


If the constraint equations are nonlinear, such as XaijXn = bj or Xai;— exp (-xi) = bj, the problem can be solved using methods in Madron (1992) and Schrage (1999).

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