# 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 i = 1,2,. ,N j=1 N X aijxi — bj = 0 j = 1,2,. ,M i=1

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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