Patrick Lang

Selectivity and Related Measures for nth-Order Data.

Analytical figures of merit are often used as criteria to decide whether or not a given instrumental method is suitable for attacking an analytical problem. To date, figures of merit primarily exist for analytical instruments producing data indexed by one variable, i.e., first-order instruments and first-order data. Almost none exist for instruments that generate data indexed by two variables, i.e., second-order instruments and data, and none exist for instruments supplying data indexed by three or more variables, i.e., nth-order instruments and data. This paper develops practical mathematical tools that can be used to create several figures of merit for nth-order instrumentation, namely, selectivity, net analyte signal, and sensitivity. In particular, the paper fully develops a local selectivity measure for second-order instrumentation and tests its performance using simulated second-order data and real second-order data obtained by gas chromatography with Fourier transform infrared detection and liquid chromatography with photodiode array detection. Also included in the paper is a brief discussion on practical uses of nth-order figures of merit.

Spectral theory of two-point differential operators determined by −D2. II. Analysis of cases

Abstract In Part I the necessary tools are developed for the complete resolution of the spectral theory for all linear two-point differential operators L in L 2 [0, 1] determined by τ = − D 2 and by boundary values B 1 , B 2 . The development emphasizes six numerical parameters defined in terms of the coefficients of B 1 , B 2 . In this paper we state and prove all results pertaining to the spectrum σ ( L ) of L , to the algebraic multiplicities of the eigenvalues λ ϵ σ ( L ) and the ascents of the operators λI − L , to the boundedness of the family of all finite sums of the projections associated with L , to the denseness of the generalized eigenfunctions, and to the existence of bases consisting of generalized eigenfunctions; we utilize the tools developed in Part I to establish these results. In particular, this paper provides either the explicit or the asymptotic form for the eigenvalues of L , provides explicit numbers for the algebraic multiplicities and ascents, provides the explicit or the asymptotic form for the projections, and provides explicit bounds on the family of all finite sums of these projections or shows that the family is unbounded.

Interrelationships between sensitivity and selectivity measures for spectroscopic analysis

Abstract Quantitative analysis based on spectroscopic data often uses the matrix equation R = CK + E, which is the K-matrix form of the Beer-Lambert law. This equations utility depends on the character of Ks numbers. The quality of these numbers is often discussed by referring to how strongly components respond to measured wavelengths and the extent one component responds toward measured wavelengths compared to other components, i.e. by referring to the sensitivity and selectivity of the used wavelengths. Over the years, various measures for quantifying sensitivity and selectivity at the local (analyte specific) and global (all analytes simultaneously) levels have been put forth with varied degrees of success. This tutorial introduces and discusses many of the commonly used local and global measures and establishes interrelationships between them. It is hoped that this presentation will give the user a better understanding of their utility and drawbacks, as well as provide a background for future work in the assessment of spectroscopic information.

Spectral theory of two-point differential operators determined by −D2. I. Spectral properties

Abstract The necessary tools for the complete resolution of the spectral theory for all linear two-point differential operators L in L2[0, 1] determined by τ = −D2 and by boundary values B1, B2 are developed. All work is based on six numerical parameters defined in terms of the coefficients of B1, B2. These quantities are utilized in the development of the characteristic determinant of L and the Greens function of λI − L, and in the development of the projections of L2[0, 1] onto the generalized eigenspaces of L; they are used in the study of the decay rates of the resolvent operator R λ (L) as ¦λ¦ → ∞ ; and they are used to identify the 13 cases treated in developing the spectral theory for τ = −D2. All results pertaining to the spectrum σ(L) of L, to the algebraic multiplicities of the eigenvalues λ ϵ σ(L), to the ascents of the operators λI − L, to the boundedness of the family of all finite sums of the projections associated with L, to the denseness of the generalized eigenfunctions, and to bases consisting of generalized eigenfunctions are summarized in a table. The proofs of these results are the subject of the sequel (Part II).

Spectral decomposition of a Hilbert space by a Fredholm operator

Abstract Let T be a Fredholm operator in a Hilbert space H with spectrum σ ( T ) = { λ i } i = 1 ∞ . Let T ( λ i ) = T λ i m i , where m i is the ascent of the operator T λ i = λ i I − T , and let P i denote the projection of H onto the generalized eigenspace N ( T (γ i )) along R ( T (γ i )). In this paper it is shown that S ∞ = S ∞ and H = S ∞ ⊕ M ∞ (topological direct sum) iff there exists M > 0 such that ∥∑ i − 1 N P i ∥ ⩽ M , N = 1, 2, …, where S ∞ = {xϵH| = ∑ ∞ i=1 x i , x i ϵ N (T(γ i ))} and M ∞ is the zero or infinite-dimensional subspace ⋃ ∞ i=1 R (T(γ i )). This result is then applied to a differential operator L in L 2 [0, 1], showing that S ∞ ≠ S ∞ = L 2 [0, 1] for this L . Also, an example with M ∞ ≠ {0} is presented.

Denseness of the generalized eigenvectors of an H-S discrete operator

Let T be a closed densely defined linear operator in a Hilbert space H, and assume there exists ξ0 ϵ p(T) such that Rξ0(T) is a Hilbert-Schmidt operator. The operator T is a special type of discrete operator, a so-called H-S discrete operator, which is shown to be a Fredholm operator in H with Fredholm set equal to the whole complex plane. Let σ(T) = {λi}i = 1∞where the operator Tλi has ascent mi, let Pi, i = 1, 2,…, be the projection of H onto the generalized eigenspace N[[Tλi]mi] along R [[Tλi]mi], and let S∞ and M∞ be the subspaces of H consisting of all x ϵ H such that x = ∑i = 1∞ Pix and such that Pix = 0 for i = 1, 2,…, respectively. Sufficient conditions are introduced which guarantee that S∝ = H and M∞ = {0}. These conditions require that ∥Rλ(T)∥ be bounded on certain rays in the complex plane and ∥Rλ(T)∥ → 0 as λ → ∞ on at least one of the rays, but specific decay rates for ∥Rλ(T)∥ are not necessary.

Stabilization of cyclic subspace regression

Abstract Developments that produced a stable numerical algorithm for cyclic subspace regression (CSR) are described. This simple algorithm produces solutions for principal component regression, partial least squares, least squares, and other related intermediate regression methodologies by exactly the same procedure. The development begins with a theoretical CSR algorithm that should produce accurate results. However, when used in numerical form, it does not produce accurate results because numbers are generated which are too small for most computational tools to accurately represent. Several strategies to deal with this numerical instability are described. Results obtained using each approach are reported as applied to two data sets. The development ends with presentation of the stable algorithm as well as MATLAB code for the algorithm.

Response to “comments on interrelationships between sensitivity and selectivity measures for spectroscopic analysis” by K. Faber et al.

Abstract A recent Tutorial [J.H. Kalivas, P.M. Lang, Chemom. Intell. Lab. Syst. 32 (1996) 135–149] discussed interrelationships between several sensitivity and selectivity measures set out in the literature with respect to the Beer-Lambert law. Recommendations by Kalivas and Lang were given for what they believed best captured local, i.e., analyte specific, sensitivity and selectivity information. In particular, the multivariate sensitivity measure proposed by Bergmann et al. [G. Bergmann, B. von Oepen, P. Zinn, Anal. Chem. 59 (1987) 2522–2526] was discerned as most suitable. Faber et al. [K. Faber, A. Lorber, B.R. Kowalski, Chemom. Intell. Lab. Syst., preceding article] give a different opinion on what they believe to be a better suited definition of sensitivity, namely, Lorbers definition [A. Lorber, A. Harel, Z. Goldbart, I.B. Brenner, Anal. Chem. 59 (1987) 1260–1266]. Unfortunately, Faber et al. do not accurately express our line of reasoning in support of the sensitivity definition proposed by Bergmann et al. Additionally, Faber et al. discuss specific calibration requirements for computation of Lorbers sensitivity measure and state that conditions presented in the article of Kalivas and Lang are incorrect. This paper considers ideologies for sensitivity measures for the Beer-Lambert law, addresses comments by Faber et al. regarding our reasoning for adoption of sensitivity as proposed by Bergmann et al., and further discusses computational conditions for Lorbers sensitivity. Some brief comments are given regarding N th-order sensitivity.

Spectral representation of the resolvent of a discrete operator

Let T be a discrete linear operator in a Hilbert space H with spectrum σ(T) = λii = 1∞, let Rλ(T) denote the resolvent of T, and let Pi denote the projection of H onto the generalized eigenspace N((γi,I − T)mj) along R((γi, I − T)mj), where mi is the ascent of the operator λiI − T. In this paper it is shown that Rγ = ∑∞i=1∑mjj=1(−Ni)j−1Pi(γ − γ)j+∑∞j=0(γo − γ)jRγ0(T∞)j+1(I− P∞) in B(H) for all λ ϵ ϱ(T), where Ni is the restriction of λiI − T to is the restriction of T to N((γi,I − T)mj), T∞ D(T)⋃⋃∞i=1R((γiI−T)mj), P∞=∑∞i=1Pi (strong convergence), and λ0 is a fixed but arbitrary point in C. This spectral representation is valid provided there exists M > 0 such that ∥∑i = 1N Pi ∥ ⩽ M, N = 1, 2, …, and generalizes results that apply to self-adjoint, normal, and spectral operators. The results of this paper are applied to represent the resolvent of a differential operator L in L2[0, 1] having infinitely many eigenvalues with ascent mi = 2 and are also applied to represent the resolvent of an operator T with P∞≠I.

Spectral theory for a differential operator: Characteristic determinant and Green's function

Abstract For a two-point differential operator L in L2[a, b], it is shown that the Greens function has the representation G(t, s; λ) = H(t, s; λ) D(λ) for λ belonging to the resolvent set θ(L), where D(λ) is the characteristic determinant and H(t, s; λ) is an entire function in the λ variable admitting a power series expansion about any point λ0∈ C . This representation is given several applications: first, to calculate the coefficient operators in the Laurent series for the resolvent Rλ(L) about each point λ0 in the spectrum σ(L), and second, to relate the algebraic multiplicity v(λ0) of an eigenvalue λ0 to the ascent m0 of the operator λ0I − L.