Statistics for Environmental Engineers

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Y = в0 + в1 (x1 x) + в2 (x2 — x)


This expansion can be used to linearize any continuous function у = /(x1,x2,…,xn). The approximation is centered at (x1, x2) and у = /(x1, x2) = в0. The linear coefficients are the derivatives of у with respect to x1 and x2, evaluated at x1 and x2:


в1



dp    and в2 = (dx.


dx 1) x 1    Vdx2



2



Y — в0 + в1 (x1 -x1) + в2 (x2 -X2)

FIGURE 49.1 A nonlinear function у = /(в, x) can be approximated by a linear function centered at the expected values xx and x2.


Following the example of the linear function presented earlier, the variance of у is:


Cov( x1 x2)


Var( Y) = (4-^1 Var(x1) + (41 1 Var(x2) + 2(-^ У


OX1Jx1    Vdx2 J%2    oX1OX2


This can be generalized to n measured variables:


Var(Y) = X(d») Var(xl)-



n



n



■2Ц


1=1 j=i-1



_i2y_


dxldx



Cov( xlxi)


There will be n(n — 1)/2 covariance terms. If xb x2,…,xn are independent of each other, all covariance terms will be zero. It is tempting to assume independence automatically, but making this simplification needs some justification. Because Cov(xl, x) can be positive or negative, correlated variables might increase or decrease the variance of Y.


If taking derivatives analytically is too unwieldy, the linear approximation can be derived from numerical estimates of the derivatives at x1, x2, etc. This approach may be desirable for even fairly simple functions. The region over which a linearization is desired is specified by x1 + A xj and x2 + Ax2. The function values y0 = f(x1, x2), y1 = f(x1 + Ax1,x2), y2 = f(x1,x2 + Ax2), etc. are used to compute the coefficients of the approximating linear model:

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