Input estimation

De Nicolao, G., Sparacino, G., and Cobelli, C., Nonparametric input estimation in physiological systems, problems, methods, and case studies, Automatica, 33, 851-870, 1997. 

Note that the proposed check must be perfomed after having obtained the estimation of u. In contrast, in the Kalman filter technique (jt), the corresponding values of e and Eg may be recursively calculated along with the input estimate. 

The input It may be wrong. Since the input estimation is based on PCB mixtures and typical relative congener compositions, an error of 30% is not unlikely and could thus explain the discrepancy found for the trichlorobiphenyl. For the heavier congener, the discrepancy seems to be too large to be solely explained by an input error. 

Fig. 8. The relationship between annual total system net production during the day and consumption during the dark in the MERL mesocosms during two years in which they were receiving different levels of inorganic nutrient (N, P, Si) input. Estimates are based on dawn-dusk changes in dissolved oxygen and have been corrected for air-sea gas exchange. The solid line represents a P/R ratio of 1 with no net organic accumulation in, or export from, the tanks. The broken line represents a net accumulation and/or export of 100 g C m-2 y-1 assuming a PQ of 1. Data from C. Oviatt. A description of the mesocosms and the enrichment experiment is given in Nixon et al. (198 ).

In Table 6.2 the amount of Ch removed from seawater by evaporation has been set to balance the input estimate. This is acceptable because there are no other major Cl sinks, after allowing for sea-to-air fluxes and burial of pore water (Section 6.4.8). The Cl removal term dictates that the same amount of Na+ is also removed to match the equal ratios of these ions in NaCl. The figure for S()4 removal by evaporation (Table 6.2) is plausible, albeit poorly constrained. Again, the S()4 estimate dictates an equal removal of Ca2+ ions to form CaS04.2H20. 

When you are performing an analysis in rtilch uncertain information from different sources must be combined (e.g. when the mean of the output estimate cannot be obtained simply by operating on the means of all the input estimates). 

A particular observed value of an input quantity Xi is called an input estimate. If we denote the input estimates by lowercase variables Xi, X2,. .., x r, then the output estimate, y, is calculated as follows ... 

Each input estimate X has an uncertainty, m(x, ). In principle the standard uncertainty of an input estimate can be evaluated in many ways. For example, if the value of the input quantity can be measured repeatedly, then a series of observations of... 

Another way to perform a Type B evaluation is to estimate the maximum possible absolute error b in an input estimate jc, and to assume that the distribution of the estimator is either rectangular or triangular, with half-width equal to To evaluate the standard deviation of the estimator, if the distribution is rectangular with halfwidth b, the standard deviation (i.e., the standard uncertainty of x, ) equals A/3. If the distribution is triangular, the standard deviation equals b/. ... 

Sometimes a pair of input estimates x,- and Xj may be correlated, because they are not determined independently of each other or because there is some effect in the measurement process that influences the observed value of each. The estimated covariance of x, and Xj is denoted by u(Xi,Xj). A Type A evaluation of covariance may be performed in some cases by making a series of paired observations of the two input quantities, (Xij, X ,i), (x ,2, , (Xi, , Xy ), and performing the... 

All the uncertainties u (x ) and covariances u (x , Xj) of the input estimates combine to produce the total uncertainty of the output estimate, y. The mathematical operation of combining the standard uncertainties and covariances of the input estimates x to obtain the standard uncertainty of the output estimate y is called propagation of uncertainty. The standard uncertainty of y obtained by uncertainty propagation is called the combined standard uncertainty of y and is denoted by Uc(y). The following general equation, which the GUM calls the law of... 

In this equation, dfjdxi, called a sensitivity coefficient, denotes the partial derivative of the function f X, X2,..., X ) with respect to Xi, evaluated at Xi = xi, X2 = X2,. .Xn = x . A sensitivity coefficient df/dxi represents the ratio of the change in the output estimate y to a small change in one input estimate... 

If the input estimates are uncorrelated, then all the covariance terms, which are those terms that involve m(x , xj), are zero, and the uncertainty equation reduces to the simpler form shown in Eq. (10.10) below ... 

The following applications show how Eq. (10.10) can be applied to some simple models when the input estimates are uncorrelated. In these equations, the variable Xi denotes input estimates and the variable y denotes the calculated output estimate a and b are constants. 

The preceding uncertainty equations presume that all pairs of input estimates are uncorrelated, which may or may not be true. One of the most common examples of correlated input estimates in radioanalytical chemistry occurs when the chemical yield Y is calculated from an equation involving the counting efficiency e. This happens in measurements by alpha-particle spectrometry with an isotopic tracer. In this case, the uncertainty equation can be simplified by treating the product e x T as a single variable. What happens in effect is that the efficiency cancels out of the activity equation and for this reason its uncertainty can be considered to be zero ... 

Another example of correlated input estimates occurs when both the counting efficiency e and the yield Y depend on the mass of a precipitate or residue on the prepared sample source. In this case, dealing with the correlation is less simple. It may be necessary to replace the variables e and Y in the activity equation by the expressions used to calculate them, or to include the covariance term for e and Y in the uncertainty equation. 

Step 2 Estimate Average Portfolio Loss. Using the rating of each asset and the tenor of the transaction as inputs, estimate the expected average loss over the period of the deal by reviewing the Moody s table of Idealized Cumulative Loss Rates. 

Figure 16.32 Longitudinal profile analyser - APL. (From Imine et al., road profile input estimation in vehicle dynamics simulation. Vehicle System Dynamics, Vol. 44, No. 4, Taylor Francis, 2006.)...

Disturbing torques d and ds are considered as known inputs, estimated by (3.35), and represented on the bond graph model of Fig. 3.21 by two modulated inputs, with multiphcative uncertainties 8d = 8k-8j + 5a + 70 8d, =... 

The system under consideradmi is described by Eqs. 3 and 4. The noise processes W[jt] and V[jt] are assumed to be zero mean and white, with known covariance matrices Q, R, and S, defined by Eq. 8. Joint input-state estimatimi consists of estimating the forces pj ] and states X[jt] from a set of response measurements d[ t]. A state estimate X, t / is defined as an estimate of X[ tj, given the output sequence d[ ], with n = 0,1,...,/. The corresponding error covariance matrix, denoted as P[ /], is defined in Eq. 10. An input estimate... 

X[o -i and its error covariance matrix P[o -i], both assumed known. Hereafter, it propagates by computing the force and state estimates recursively in three steps, i.e., the input estimation step, the measurement update, and the time update ... 

The gain matrices Mpt] and Lptj are determined such that both the input estimates p i and the state estimates X i i ij are minimum variance and unbiased (i.e M i = argmuiM tr Ppjj jj, ... 

INPUT Estimate spatial components in the current frame Sort the components according to increasing values of FD. Let us denote the sorted components by s t),... 

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