4.1.3. Applications in UR

The case of linear unfolding is represented by several equations of the type

\(x_{i} = \sum_{k = 1}^{n}{A_{k,i}y_{k}}\) ,

or, in matrix notation (A contains the \(X_{i}\left( t_{k} \right)\) terms):

\(\mathbf{x} = \mathbf{A\ y}\)

Now, the desired output quantities Y are found in the right side of this equation, the input quantities on the left side. As it is well known, the linear least squares-method yields the solution to this problem:

\[\mathbf{y =}\mathbf{U}_{\mathbf{y}}\mathbf{A}^{\mathbf{T}}\mathbf{U}_{\mathbf{x}}^{- 1}\left( \mathbf{x} \right)\mathbf{\ x}\]

\[\mathbf{U}_{\mathbf{y}}\mathbf{=}\left( \mathbf{A}^{\mathbf{T}}\mathbf{U}_{\mathbf{x}}^{\mathbf{- 1}}\left( \mathbf{x} \right)\mathbf{\ A} \right)^{- 1}\]

In a next step it is assumed that the elements of the design matrix A, which e.g. consist of decay corrections when analyzing a decay curve, may contain further parameters with associated uncertainties (given as a vector p with covariance matrix \(\mathbf{U}_{\mathbf{p}}\)). Using a transformation matrix Q containing numerically calculated partial derivatives \(Q_{i,k} = \partial Y_{i}/\partial p_{k}\), an extended version of the covariance matrix \(\mathbf{U}_{\mathbf{y}}\) can be calculated (this does not change the value of y):

\[\mathbf{U}_{\mathbf{y}}\mathbf{=}\left( \mathbf{A}^{\mathbf{T}}\mathbf{U}_{\mathbf{x}}^{\mathbf{- 1}}\left( \mathbf{x} \right)\mathbf{\ A} \right)^{- 1}\mathbf{+}\mathbf{Q\ }\mathbf{U}_{\mathbf{p}}\mathbf{\ }\mathbf{Q}^{\mathbf{T}}\]

For every combination \(y_{i}\left( p_{k} \pm \mathrm{\Delta}p_{k} \right)\) (with \(\mathrm{\Delta}p_{k} = 1 \bullet \ 10^{- 6}p_{k}\)) of numerical approximations of the partial derivatives

\[Q_{ik} = \frac{\partial y_{i}}{\partial p_{k}} \approx \frac{y_{i}\left( p_{k} + \mathrm{\Delta}p_{k} \right) - y_{i}\left( p_{k} \right)}{\mathrm{\Delta}p_{k}}\]

the matrix A, the output vector y (its elements being considered as functions) and the covariance matrix \(\mathbf{U}_{\mathbf{p}}\) must be recalculated.

Alternatively, by using the chain rule for derivatives the second term in Eq. (11) can be expressed as follows:

\[\mathbf{C}_{\mathbf{P}}\mathbf{\ }\mathbf{U}_{\mathbf{p}}\mathbf{\ }\mathbf{C}_{\mathbf{p}}^{\mathbf{T}}\mathbf{=}\left( \mathbf{C}_{\mathbf{A}}\mathbf{D}_{\mathbf{P}} \right)\mathbf{U}_{\mathbf{p}}\left( \mathbf{C}_{\mathbf{A}}\mathbf{D}_{\mathbf{P}} \right)^{\mathbf{T}}\mathbf{\ =}\mathbf{C}_{\mathbf{A}}\left( \mathbf{D}_{\mathbf{P}}\mathbf{\ }\mathbf{U}_{\mathbf{p}}\mathbf{\ }\mathbf{D}_{\mathbf{p}}^{\mathbf{T}} \right)\mathbf{\ }\mathbf{C}_{\mathbf{A}}^{\mathbf{T}}\mathbf{=}\mathbf{C}_{\mathbf{A}}\mathbf{\ }\mathbf{U}_{\mathbf{A}}\mathbf{\ }\mathbf{C}_{\mathbf{A}}^{\mathbf{T}}\]

Herein are: \(\mathbf{C}_{\mathbf{A}}\) : matrix of partial derivatives of y_i with respect to the elements of the matrix A, Dp : matrix of partial derivatives of the elements of A with respect to parameters p; UA : covariance matrix of the elements of A, which are functions of p. The chain rule for partial derivatives takes the form Cp= CA·Dp, with the property

\[\mathbf{C}_{\mathbf{p}}^{\mathbf{T}}\mathbf{=}{\mathbf{D}_{\mathbf{p}}^{\mathbf{T}}\mathbf{C}}_{\mathbf{A}}^{\mathbf{T}}\]

Especially the expression .. math:: mathbf{C}_{mathbf{A}}mathbf{}mathbf{U}_{mathbf{A}}mathbf{}mathbf{C}_{mathbf{A}}^{mathbf{T}} is required within the WTLS fitting procedure; it is derived in a subroutine which is used by both fitting methods, WLS and WTLS, respectively. The matrix UA is calculated as indicated in the preceding sub-chapter.

Note on the covariance matrix UA:

For linear unfolding with the WTLS procedure a test had been implemented with the previous version, which by using the Cholesky decomposition tests whether the input covariance matrix is positive definite. This has been improved by testing in advance of the Cholesky decomposition the Cauchy-Schwarz inequality:

\(cov(i,k)^{2} \leq var(i)var(k)\) .

For pairs (i, k), for which equality is found, the associated cov(i, k) is multiplied by the factor (1-δ) with δ=1·10^-09. Should the covariance matrix after these tests again be not positive definite (it cannot be inverted) all non-diagonal elements are multiplied with this factor. Should this not help, the program evaluation is interrupted with giving a hint on this problem.

This primary result (i.e., the values of the fit parameters; they often represent activity values referred to the time of measurement) of linear may need further treatment, if the fit parameters values obtained must be inserted into to further equations. This is the case, if the activity value at measurement (fit parameter) needs to be divided by mass/volume and requires and a further decay correction and these new parameters (parameter vector q) associated with these corrections have uncertainties (embedding linear unfolding). This often may not require matrix algebra, if the corrections of the fit parameters \(y_{i}\) consist in simply multiplying them with factors \(\varphi_{i}\ \)which results in the “final” output quantities \(y_{i}^{'}\):

\[y_{i}^{'} = y_{i}\ \ \varphi_{i}\ \left( q,y_{k} \right)\]

Nevertheless, these functions may be arranged into a vector

\[\mathbf{y}^{\mathbf{'}} = \left( y_{1}\left( \mathbf{q},\mathbf{y} \right),\ {\ y}_{2}\left( \mathbf{q},\mathbf{y} \right),\ \ldots,\ \ y_{n}\left( \mathbf{q},\mathbf{y} \right), \right)^{T}\]

Calculating the covariance matrix of \(\mathbf{y}^{\mathbf{'}}\) now requires, by analogy to Eq. (11), an (n x n)-matrix \(\mathbf{J}\) of partial derivatives \(J_{i,k} = \partial y_{i}^{'}/\partial y_{k}\) of the functions \(y_{i}^{'}\) with respect to the fitted values \(y_{i}\). For taking into account the uncertainties of the n_q values q requires to calculate the correspondent (n x n_q)-matrix \(\mathbf{Q'}\) of partial derivatives \({Q'}_{i,k} = \partial y_{i}^{'}/\partial q_{k}\).

If there are no covariances between y and q, the extended version of Eq. (11) yields the covariance associated with the finally desired output quantities \(\mathbf{y}^{\mathbf{'}}\):

\[\mathbf{U}_{\mathbf{y'}}\mathbf{=}\mathbf{J\ }\mathbf{U}_{\mathbf{y}}\mathbf{\ }\mathbf{J}^{\mathbf{T}}\mathbf{+}\mathbf{Q'\ }\mathbf{U}_{\mathbf{q}}\mathbf{\ }\mathbf{Q'}^{\mathbf{T}}\]

where the covariance matrix Uy is the one from (11), which then results in:

\[\mathbf{U}_{\mathbf{y'}}\mathbf{=}\mathbf{J\ }\left\{ \left( \mathbf{A}^{\mathbf{T}}\mathbf{U}_{\mathbf{x}}^{\mathbf{- 1}}\left( \mathbf{x} \right)\mathbf{\ A} \right)^{- 1}\mathbf{+}\mathbf{Q\ }\mathbf{U}_{\mathbf{p}}\mathbf{\ }\mathbf{Q}^{\mathbf{T}} \right\}\mathbf{\ }\mathbf{J}^{\mathbf{T}}\mathbf{+}\mathbf{Q'\ }\mathbf{U}_{\mathbf{q}}\mathbf{\ }\mathbf{Q'}^{\mathbf{T}}\]

Note: If the partial derivatives \(\partial y_{i}/\partial p_{k}\) being the elements of the matrix Q (Eq. 11) are referred to the vector \(\mathbf{y}^{\mathbf{'}}\), i.e. replaced by \(\partial{y'}_{i}/\partial p_{k}\), the term \(\mathbf{Q\ }\mathbf{U}_{\mathbf{p}}\mathbf{\ }\mathbf{Q}^{\mathbf{T}}\) in Eq. (16) associated with them, hast to be removed from Eq. (16) and moved to Eq. (15) as an additional term.

For further reading, see: Cox et al., 2004.