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Joint confidence region defined

The joint confidence region defines the region of joint parameter uncertainty, and it is obtained when all the parameters vary simultaneously. All parameter sets p that satisfy the relation... [Pg.116]

The joint confidence region is the region of joint parameter uncertainty accounting for variation of all the parameters. It is defined as... [Pg.548]

The boundary of the joint confidence region is defined by all combinations g that satisfy... [Pg.548]

Determination of confidence limits for non-linear models is much more complex. Linearization of non-linear models by Taylor expansion and application of linear theory to the truncated series is usually utilized. The approximate measure of uncertainty in parameter estimates are the confidence limits as defined above for linear models. They are not rigorously valid but they provide some idea about reliability of estimates. The joint confidence region for non-linear models is exactly given by Eqn. (B-34). Contrary to ellipsoidal contours for linear models it is generally banana-shaped. [Pg.548]

Approximate inference regions for nonlinear models are defined by analogy to the linear models. In particular, the (I-a)I00% joint confidence region for the parameter vector k is described by the ellipsoid,... [Pg.178]

Since the parameters are in fact correlated, a joint confidence region of the estimates b can be defined, which accounts for the simultaneous variation of all the parameters ... [Pg.315]

Here F(p, n — p, 1 — a) is the a percentage point of the Fischer s F distribution at p and ( - p) degrees of freedom. The boundary of the joint confidence region is defined by all values of P which satisfy the hyperellipsoid in the / -dimensional parameter space around b,... [Pg.315]

Once a model is selected it is often important to improve the precision of the estimated parameters. The cornerstone of the theory is the convariance matrix of the parameter estimates. The convariance matrix defines a hyperellipsoid around the optimal parameter combination the joint confidence region (eq 46) can be written as... [Pg.321]

Let be an estimate of the experimental error variance with r degrees of freedom let B be the vector of estimated model parameters let p be the number of estimated model parameters and let be the critical F value with (p,r) degrees of freedom and the significance level a. The boundaries of the joint confidence region is defined by the equation... [Pg.117]

Joint confidence regions With two model parameters the confidence limits are defined by elliptic contours. With three parameters these limits are defined by ellipsoidic shells. With many parameters, these limits are defined by hypereUipsoids. [Pg.118]

Cij 0 are likely to fall anywhere in the rectangle defined by Xi 26X,. Xj 2SXj, whereas the true values of two parameters with Ctj 1 are likely to fall along the 8Xi SXj diagonal. Thus the joint confidence region for two strongly correlated constants is much smaller than the naive estimate from the uncorrelated standard errors. [Pg.251]

The proofs of Eqs. (9.59) and (9.60) explicitly rely on the linear character of the model. The above relations are thus only correct under the same conditions. Therefore, one speaks of the joint confidence region under linear assumptions. If the model is not linear in its parameters, the surface in the parameter space defined by Eq. (9.63) no longer is a contour of constant residual sums of squares. Although the probability level is correct, the contour itself has only been approximated. This property provides a qualitative measure of the degree of nonlinearity of the model it is rather simple to determine the coordinates of... [Pg.299]

The confidence intervals defined for a single random variable become confidence regions for jointly distributed random variables. In the case of a multivariate normal distribution, the equation of the surface limiting the confidence region of the mean vector will now be shown to be an n-dimensional ellipsoid. Let us assume that X is a vector of n normally distributed variables with mean n-column vector p and covariance matrix Ex. A sample of m observations has a mean vector x and an n x n covariance matrix S. [Pg.212]

As has been described, the correlation of the parameter estimates is also of great interest. This correlation is measured by the off-diagonal terms of p(b), completely ignored by the above equations. If an analysis of variance is conducted, one finds that a joint (1 — a) 100% confidence region is defined by... [Pg.126]


See other pages where Joint confidence region defined is mentioned: [Pg.316]    [Pg.309]    [Pg.484]   
See also in sourсe #XX -- [ Pg.178 ]




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