A note on D-optimal chemical balance weighing designs with autocorrelated observations

In this paper, D-optimal chemical balance weighing designs with three objects are considered. The error terms are assumed to form a first-order autoregressive process, which implies that the covariance matrix of the vector of errors depends on the known parameter $$\rho $$ρ. It is shown that the designs constructed by Katulska and Smaga (Metrika 76:393–407, 2013) are still D-optimal weighing designs with three objects under a wider interval of possible values for parameter $$\rho $$ρ than that considered in that paper. Those designs are also proved to be highly D-efficient designs, when D-optimal design is not known.

The matrix X S −1 X is the information matrix of the design X. In this paper, we only consider nonsingular designs, i.e. the designs with nonsingular information matrix. An example of experiment (chemical process) described by the aforementioned model can be found in Jenkins and Chanmugam (1962). There are also papers in the literature relating to designs with other forms of Cov(e) (see, for example, Ceranka et al. 2006;Graczyk 2009;Masaro and Wong 2008) and spring balance weighing designs with (0, 1)-design matrices (see, for instance, Graczyk 2012; Lopez and Neubauer 2010). The applications of weighing designs (in chemistry, medicine, economics, operations research etc.) are given in the following papers: Angelis et al. (2001); Banerjee (1975);Beckman (1973); Cheng (2014) and Graczyk (2013).
Our purpose is to find the best design with respect to D-optimality criterion. The designX is D-optimal in the class of designs with three objects M n×3 (±1) if det(X S −1X ) = max det(X S −1 X) : X ∈ M n×3 (±1) , i.e. it maximizes the determinant of the information matrix among all designs with three objects. The inverse of S is equal to 1/(1 − ρ 2 )A, where ρ ∈ (−1, 1) and A = I n + ρ 2 K 1 − ρK 2 , I n is the identity matrix of size n, K 1 = diag(0, 1 n−2 , 0) and K 2 is a tridiagonal n × n matrix with 0 on the diagonal, 1 on the first superdiagonal and on the first subdiagonal. Since When ρ = 0, a lot of work has been done on developing optimal weighing design theory (see, for instance, Banerjee 1975;Cheng 2014;Jacroux et al. 1983;Neubauer 1997). In the case ρ = 0, it is harder to deal with optimality problems. Some results concerning optimal designs in this case are given for example in Angelis et al. (2001); Smaga (2012, 2013); Li and Yang (2005); Smaga (2014) and Yeh and Lo Huang (2005), but many problems are still unsolved. In this paper, we extend the results of Katulska and Smaga (2013), where D-optimality of certain designs was proven under ρ ∈ [0, 1/(n − 2)].
The remainder of this paper is organized as follows. In Sect. 2, we show that the D-optimal designs for ρ ∈ [0, 1/(n − 2)] constructed by Katulska and Smaga (2013) are still D-optimal under ρ belonging to wider interval than [0, 1/(n − 2)]. In Sect. 3, we give the lower bound for D-efficiency of weighing designs and we use it to show that the mentioned designs are highly D-efficient in cases, which are not included in Sect. 2. Some preliminary technical results are outlined in the Appendix.

D-optimal designs
In this section, we extend the following theorem.
Unfortunately, it can be easily observed that 1/(n − 2) → 0 as n → ∞. However, Theorem 1 in Katulska and Smaga (2013) implies the design X given by (2) cons(x) = 0, cons(y) = 1, cons(z) = 1 . Furthermore, simulation studies suggest the design X is D-better than any design in C for all ρ ∈ [0, 1). But, it is not easy to show this. We solve partially this problem in the following theorem.
Theorem 1 in Katulska and Smaga (2013) and Theorem 2 imply the following important corollary.
Corollary 1 Under the assumptions of Theorem 2, the design X given by (2) is Doptimal in the class M n×3 (±1).
For n = 4, 8, . . . , 28, we significantly extend Theorem 1 for all ρ ∈ [0, 1) that is important, because we often do not need a large number of observations for three objects. When n = 32, 36, . . ., Lemma 3 in the Appendix shows that the root η of the polynomial p given by (3) belongs to (1/(n − 2), 1), and hence Theorem 1 is also improved for such n. The root η can be derived numerically. It is also worth mentioning that slight changes in the proofs of the necessary and sufficient conditions given in Katulska and Smaga (2013) under which the design with three objects is D-optimal actually show that those conditions still hold under the assumptions of Theorem 2. Unfortunately, it failed to prove D-optimality of the design X given by (2) for all ρ ∈ [0, 1). However, in Sect. 3 we show that the design X is highly D-efficient, when the assumptions of Theorem 2 do not hold.
In the following proof of Theorem 2, we use Budan-Fourier Theorem and an inequality given in Li (2013) which we recall in the Appendix.
We consider f as a quadratic function of the variable a. The discriminant of f is equal Using Budan-Fourier Theorem (see Theorem 4 in the Appendix), we can show that all coefficients of the polynomial w(ρ), considered as polynomials of variable n, do not have any roots in the interval (7, ∞) (for more detailed argumentation of this kind we refer to the proof of Lemma 3 in the Appendix). Therefore all these coefficients are positive for n ∈ (7, ∞). Hence the discriminant of f is negative. So, f (a) > 0 for all a, since the coefficient of a 2 in f is positive. This finishes the proof.

D-efficiency of design X
In this section, we show that the design X of the form (2) is highly D-efficient, when D-optimal designs in M n×3 (±1) are not known.
Following the definition of Bulutoglu and Ryan (2009), the D-efficiency of a design X ∈ M n× p (±1) is defined by the formula under Cov(e) = 1/(1 − ρ 2 )S, where S is given by (1). Under the assumptions of Theorem 2, max Y∈M n×3 (±1) det(Y AY) = det( X A X) by Corollary 1, and hence D-eff( X) = 1. However, when the assumptions of Theorem 2 do not hold, value of the denominator of D-eff(X) is unknown and we can not compute the D-efficiency of X ∈ M n×3 (±1). But, in the following lemma, we give the lower bound for it. Next we use this lower bound for the D-efficiency of weighing design to show that the design X of the form (2) has high D-efficiency when n 32 and ρ η, where η is defined in Theorem 2.
By Lemma 4 in Katulska and Smaga (2013), we have y i Ay i = −4ξρ for i = 1, 2, 3, where ξ = cons(y i ). For a ∈ M n×1 (±1), it is easy to see that a Aa = if and only if a is the vector of ones or the vector of minus ones. Since Y is nonsingular, there exists at most one i such that y i Ay i = . Therefore det(Y AY) ( − 4ρ) 2 , and hence So, by (4), We now show that the equality in (5) never holds. Suppose that the equality in (5) holds. Then det(Y AY) = ( − 4ρ) 2 for some nonsingular Y = [y 1 , y 2 , y 3 ] in the class M n×3 (±1). By Hadamard's inequality, the last equality holds if and only if Y AY is diagonal matrix and its diagonal elements are equal to , − 4ρ, − 4ρ, in any order (see Theorem 7.8.1 in Horn and Johnson 2013, p.505). Without loss of generality, we can assume that y 1 Ay 1 = and y 2 Ay 2 = y 3 Ay 3 = −4ρ. By Lemma 4 in Katulska and Smaga (2013), cons(y 1 ) = 0 and cons(y i ) = 1 for i = 2, 3. Hence the equalities y 1 Ay 2 = 0 and y 1 Ay 3 = 0 imply that fcons(y i ) = n/2 for i = 2, 3 (see Lemma 5 (c) in Katulska and Smaga 2013). But, then the design Y is singular. Contradiction.
Theorem 3 implies the lower bound for the D-efficiency of X given by (6) becomes smaller with an increase in ρ. However, the decrease of it is at most 1.2 %, because D-eff( X) > 0.988. So, the design X is highly D-efficient design in the class M n×3 (±1), when n 32 and ρ η, where η is defined in Theorem 2. This result is the next reason, which together with results of Sect. 2, confirms the conjecture that the design X is D-optimal in M n×3 (±1), when n ≡ 0 (mod 4), Cov(e) = 1/(1 − ρ 2 )S, where S is given by (1) and ρ ∈ [0, 1). We now consider the behavior of the root η of the polynomial p given by (3).