New variance expressions for systematic sampling: the filtering approach

Authors

  • XIMO GUAL-ARNAU,

    1. Department of Mathematics, Campus Riu Sec, University Jaume I, E-12071 Castellon, Spain
      *Department of Mathematics, Statistics and Computation, Faculty of Sciences, University of Cantabria, Avda. Los Castros s/n, E-39005 Santander, Spain
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  • and * LUIS M. CRUZ-ORIVE

    1. Department of Mathematics, Campus Riu Sec, University Jaume I, E-12071 Castellon, Spain
      *Department of Mathematics, Statistics and Computation, Faculty of Sciences, University of Cantabria, Avda. Los Castros s/n, E-39005 Santander, Spain
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Luis M. Cruz–Orive. Tel: +34 942 20 14 24; fax: +34 942 20 14 02; e-mail: Lcruz@matesco.unican.es

Summary

We present a collection of variance models for estimators obtained by geometric systematic sampling with test points, quadrats, and n-boxes in general, on a bounded domain in n-dimensional Euclidean space ℝn, n = 1, 2, ... , and for systematic rays and sectors on the circle. The approach adopted − termed the filtering approach − is new and different from the current transitive approach. This report is only preliminary, however, because it includes only variance models in terms of the covariogram of the measurement function. The estimation step is in preparation.

Introduction

In design-based stereology, systematic sampling plays a prominent role. It is therefore important to develop error variance predictors for the different kinds of systematic probes involved in a stereological design. Such predictors are indispensable to plan a stereological design in an objective manner; see, for instance, Cruz-Orive et al. (2004).

The Cavalieri sections design is a popular systematic design, and it is equivalent to sampling at systematic points on the real axis ℝ; for a recent review, see Cruz-Orive & García-Fiñana (2005). On ℝ2, the grid of test points to estimate planar areas is also classic; for general references, see Weibel (1979), Baddeley & Jensen (2004) and Howard & Reed (2005). The conceptual extension to ℝn is straightforward; see, for instance, Cruz-Orive (1989). Furthermore, systematic sampling at point locations on the circle �� (Gual-Arnau & Cruz-Orive, 2000; Cruz-Orive & Gual-Arnau, 2002; Hobolth & Jensen, 2002) and on the sphere (Gual-Arnau & Cruz-Orive, 2000, 2002) have also been studied.

Besides Cavalieri planes, the Cavalieri slabs design is also popular because it is a prerequisite for estimating particle number (West et al., 1991, 1996; Cruz-Orive & Geiser, 2004). Cavalieri slabs are actually equivalent to systematic segments on ℝ (Gual-Arnau & Cruz-Orive, 1998). The natural extension is systematic sampling by rectangular quadrats on ℝ2 (e.g. Baddeley & Jensen, 2005, and references therein), or by rectangular ‘n-boxes’ on ℝn.

Although the properties of Cavalieri sections and slabs have been studied in considerable depth, no satisfactory variance predictors have been hitherto available for systematic quadrats, or for n-boxes in general. Related results can be found, for instance, in Matheron (1962, 1965, 1971), or in Journel & Huijbregts (1978), but it is not obvious how to exploit these results for design stereology. This is regrettable, because in most stereological designs the intermediate steps usually involve systematic quadrats, or systematic ‘boxes’ such as optical disectors within slices; see, for instance, West et al. (1991, 1996).

The approach adopted in the present paper creates a new scenario that seems to yield variance approximations for systematic boxes in a natural and relatively simple manner (sections 4 and 5). It also yields analogous results for systematic sectors and points on the circle (section 6). We term it the ‘filtering approach’ because it involves the measurement function regularized or ‘filtered’ by the indicator function of the basic probe − this idea is presented in section 2.2. The choice of the term ‘filtering’ is explained further at the end of section 2.

For Cavalieri sections and slices (section 3), the new variance approximations do not quite coincide with the known ones; in fact, the target is different from the extension term of the variance; for a brief description of this concept, see, for instance, Cruz-Orive & García-Fiñana (2005). Furthermore, our models do not require isotropy of the covariogram for n ≥ 2, as was usually the case in the previous theory (e.g. Cruz-Orive, 1989; Eq. 8.2), or, equivalently, an isotropic random design (Baddeley & Jensen, 2005; p. 310). This paper, however, may be regarded as a preliminary report, because it only includes variance representations and models involving the covariogram of the measurement function. The further step will be to choose suitable covariogram models that, inserted into the formulae given here, will yield the desired variance predictors from sampled data. This final step is in preparation.

2. Systematic sampling on ℝn: preliminaries

2.1. Target quantity, sampling and estimation

The quantity of interest can be expressed as the integral of a function f : ℝn → ℝ+, called the measurements function, which is square integrable in a bounded domain D ⊂ ℝn, and is equal to zero outside D. Thus,

image(1)

Examples:

  • (a) Consider a compact subset Y ⊂ ℝ3 of volume Q and fix a sampling axis. Then D ⊂ ℝ1 is the orthogonal lineal projection of Y on this axis, whereas f(x) is the area of the intersection between Y and a plane normal to the sampling axis at a point of abscissa x.
  • (b) In the preceding example, fix a sampling plane. Then D ⊂ ℝ2 is the orthogonal projection (i.e. the orthogonal ‘shadow’) of Y on the sampling plane, whereas f(x) is the length of the intersection between Y and a straight line normal to the sampling plane at a point x of rectangular coordinates (x1x2).

Our purpose is to estimate Q by systematic sampling on the domain D, and then to predict the variance of the corresponding unbiased estimator from the observations of a single systematic sample. The basic probe will be either a point, or a rectangular n-box, namely a segment of length t1 if D ⊂ ℝ1 (which corresponds to a slab of thickness t1, normal to the sampling axis), or a rectangular quadrat if D ⊂ ℝ2, etc.

In general, we consider a partition of ℝn into a denumerable set of translates of a fundamental tile

image(2)

namely a rectangular prism of finite volume V(J0) = T1T2  ... Tn. Thus, any tile Jk, k ∈ Ζ can be made to coincide with J0 by a translation –τk which takes the whole partition onto itself (Santaló, 1976; Gual-Arnau & Cruz-Orive, 1996, 1998).

Sampling with test points.  The test system is a uniform random grid of test points, namely

image(3)

By z ~ UR( J0), we mean that the point z is uniformly random in the fundamental tile J0, i.e.

z = (U1T1U2T2, ... , UnTn),(4)

where (U1U2, ... , Un) denote n independent uniform random numbers in the interval [0, 1). The corresponding unbiased estimator of Q reads

image(5)

Examples:

Sampling with test n-boxes.  The sampling tool is a uniform random test system of n-boxes, namely

image(6)

where T0 ⊂ J0 is an n-box representing the basic probe. The function

image(7)

may be regarded as the regularization of the measurement function f by the indicator function of the basic n-box T0, and we have

image(8)

(Gual-Arnau & Cruz-Orive, 1998). Now we may sample inline image by the uniform grid of test points (Eq. 3), whereby

image(9)

is also an unbiased estimator of Q corresponding to systematic sampling with n-boxes. This idea has already been presented in Matheron (1965), p. 70.

Examples:

2.2. Basic variance identity of the filtering approach

Consider the estimator in Eq. (5), obtained with the test system of points in Eq. (3). The classic representation of its variance reads

image(10)

where

image(11)

is the transitive covariogram of f (Matheron, 1965, 1971; see also Cruz-Orive, 1989). On the other hand, the estimator in Eq. (9) is obtained by sampling the regularized version inline image of f, see Eq. (7), with the same system (Eq. 3) of test points. Therefore, Eq. (10) holds also with g replaced with inline image. Side-by-side subtraction of both variance representations yields

image(12)

where

image(13)

is the covariogram of inline image. In the preceding expression, ‘*’ denotes ‘convolution’, and K is the geometric covariogram of T0, namely

image(14)

(Gual-Arnau & Cruz-Orive, 1998). Equation (12) constitutes the basis of the new approach. The classic approach consists of applying the Euler–MacLaurin summation formula to Eq. (10); see García-Fiñana & Cruz-Orive (2004) for details. In Eq. (12), however, the integral in the right-hand side of Eq. (10) has disappeared, and therefore it is no longer necessary to resort to the Euler–MacLaurin summation formula. On the contrary, it suffices to apply Taylor expansions. In the next section we start with the case n = 1.

Remark on the term ‘filtering approach’.  The regularized measurement function inline image (see Eq. 7) will usually be smoother than f, see, for instance, McNulty et al. (2000), Fig. 2. This effect is similar to that of applying the classic ‘moving average’ operation to a series of data. In the image analysis context, e.g. Glasbey & Horgan (1995), Chapter 3, inline image would be the result of applying a ‘filter’T0 to the measurement function f. For Cavalieri slabs of thickness t, for instance, T0 is a straight-line segment of length t. The key identity in Eq. (12) involves inline image, and for this reason our approach is termed the ‘filtering approach’.

3. Systematic sampling with points and segments on ℝ

3.1. Variance representation using the filtering approach

Purpose and prerequisites.  Our purpose is to expand the summation terms in the right-hand side of Eq. (12) when n = 1 into a generalized Taylor series given in Kiêu (1997) for piecewise smooth functions. To do this, we need the following assumptions.

(a1) The covariogram g of the measurement function f, and its successive derivatives, satisfy

image(15)

where g(0)(x) := g(x), g(k)(x) := dkg(x)/dxk, k = 1, 2, ... , and CK denotes the class of Lebesgue measurable functions with compact support.

(a2) For any function g : ℝ → ℝ+, define

image(16)

namely the jump function of g(k) at the abscissa x. Further, let

image(17)

represent the set of abscissas where g(k) exhibits a jump. We assume that all jumps of the covariogram g of the measurement function f are bounded whenever they exist, namely |Sg(k)(x)| < ∞, x ∈ℝ, k = 0, 1, ... . Moreover, we assume that the set Dg(k) is finite for all k = 0, 1, ... .

The filtering expansion on ℝ.  Let inline image(z) represent the Cavalieri slices estimator of Q with systematic slabs of thickness t and period T (namely the distance between slab midplanes), so that 0 ≤ t ≤ T < ∞. Further, let (z) represent the corresponding estimator with t = 0, namely the Cavalieri sections estimator of Q. In either case, unbiasedness is guaranteed by taking z = UT, where U is a uniform random number in the interval [0, 1). The particular version for ℝ of the basic identity in Eq. (12) reads

image(18)

As shown in Appendix A, term-by-term Taylor expansion of the right-hand side of the preceding expression leads to

image(19)

3.2. Premises for variance modelling

The next step is to construct variance models for the variances of inline image(z) and (z) from the expansion in Eq. (19). To do this, we resort to the following considerations and further assumptions.

  • (p1) The measurement function f is said to be (mp)-piecewise smooth if its support is bounded and if
    •  (i) Df(k) = ∅, (k = 0, 1, … , m − 1), that is, all derivatives of f of order less than m are continuous (i.e. m is the order of the first non-continuous derivative of f).
    • (ii) For k = mm + 1, … , m + p, f(k) may have only a finite number of jumps, and they have to be finite.

If f is (mp)-piecewise smooth, then its covariogram g is (2m+ 1, p)-piecewise smooth (Kiêu, 1997). As a consequence, Sg(k)(x) = 0, x ∈ ℝ, k = 0, 1, ... , 2m. Furthermore, since g(x) = g(–x), it follows that Sg(2j−1)(0) = 2g(2j−1)(0+), j = 1, 2, ... , whereby

g(2j−1)(0) = 0, j = 1, 2, ... , m.(20)
  • (p2) We assume that the grid points {iT, i ∈ ℤ} do not contain singularities. Thus, for ij = 1, 2, … ,
    image(21)
  • (p3) The set Dg(k)k = 1, 2, ... , will generally be unknown a priori, and it cannot be identified from the data. Hence, the summations involving Dg(k) will not be used in the model.

3.3. Basic model identity for systematic sampling on  ℝ

The incorporation of the preceding premises into the right-hand side of Eq. (19) yields the model relationship

image(22)

where the subscripts ‘M’ in the left-hand side stand for ‘model version’. The upper limits of the summations are set so that the orders of the derivatives of g never exceed p.

3.4. Variance model for systematic points (Cavalieri planes) and segments (Cavalieri slabs) on ℝ

If the slab thickness reaches the sampling period, namely if t = T, then clearly inline image(z) = Q for any z, and therefore Var{inline image(z)} = 0. Thus, setting t = T in Eq. (22), we get the variance model for Cavalieri section planes of period T when the measurement function f is (mp)-piecewise smooth, namely

image(23)

On the other hand, replacing VarM{(z)} with its expression (23) in Eq. (22), we readily obtain the variance model for Cavalieri slabs of thickness t and period T, 0 < t ≤ T < ∞, when the measurement function f is (mp)-piecewise smooth, namely

image(24)

4. Systematic sampling with points or with rectangular quadrats on the plane ℝ2

4.1. Preliminaries

The basic setup is a particular case of that in section 2.1 for n = 2. Thus the measurement function f : ℝ2 → ℝ+ is square integrable in a bounded domain D ⊂ ℝ2 and vanishes outside D. The target parameter is

image(25)

To estimate Q, the domain D is probed by a test system with a rectangular fundamental tile

image(26)

and a rectangular basic probe (i.e. the basic quadrat) that can be contained in J0, namely,

image(27)

The test system should be uniform random, but its orientation may be arbitrary because the domain D is of full dimension. For an illustration, see, for instance, Cruz-Orive & Geiser (2004), Fig. 3.

The corresponding point and quadrat estimators of Q are the straightforward particularizations for n = 2 of Eqs (5) and (9), respectively. The estimators depend on a uniform random point z = (z1z2) in J0. Thus, zi = UiTi, i = 1, 2, where U1U2 are independent uniform random numbers in the interval [0, 1).

4.2. Variance approximations: two basic cases

For n = 2, the basic variance identity in Eq. (12) reads

image(28)

To proceed with the Taylor expansions prior to modelling, we consider two cases. We retain only expansion terms containing partial derivatives of the covariogram up to order 2. The resulting variance expressions, derived in Appendix B, are regarded as variance models and thereby subscripted M, rather than ‘approximations’, because we omit the remainder terms (see the general case for ℝn in section 5). Throughout, the following notation is used for the partial derivatives:

image(29)

Further, we say that g ∈C if g(a,b)(h1h2) exists and it is continuous for all ab = 0, 1, … , in a specified bounded domain of ℝ2.

Case 1: g ∈ C for h1 ∈ ℝ, h2 ∈  ℝ. 

image(30)

Case 2: g ∈ C, either for h1 ∈ ℝ, h2 ≥  0, or for h1 ≥ 0, h2 ∈ ℝ. 

We have worked out each of the two cases separately, and then we have obtained the following model identity by taking the average of the corresponding two expressions.

image(31)

Case 2 (continued): Sampling with a rectangular grid of test points:

image(32)

If the basic quadrat T0 is increased to coincide with the fundamental tile J0, namely if we set t1 = T1, t2 = T2, then clearly Var{inline image(z1, z2)}= 0 and the preceding model identity yields the following point counting variance model:

image(33)

Case 2 (continued): Sampling with systematic rectangular quadrats:

image(34)

From Eqs (31) and (33), we readily obtain

image(35)

5. Extension to systematic points and rectangular boxes in ℝn

We briefly revisit the general case described in section 2.1, in which g : ℝn → ℝ+. Suppose that g ∈C  ∞ whenever hj ∈ℝ, j ∈{1, 2, ... , n}∖{i} and hi ≥ 0, i ∈{1, 2, ... , n}. Then, as in Case 2 above, we may use Taylor expansions term by term in the right-hand side of the variance identity in Eq. (12) for each i, and then we may average the corresponding n expressions. By retaining up to the second-order partial derivatives of g, we obtain the generalized version of Eq. (31), namely

image(36)

where kT := (k1T1k2T2, ... , knTn), and 0i represents the vector kT with the component kiTi replaced with zero. Further, k := {k1, k2, ... , kn} and inline image.

The expression of VarM{(z)} is readily obtained by setting ti = Ti, i = 1, 2, ... , n in the right-hand side of Eq. (36), because in this case VarM{inline image(z)} = 0. Once VarM{(z)} is known, the expression of VarM{inline image(z)} is obtained directly from Eq. (36).

6. Systematic sampling with rays and with sectors on the circle ��

6.1. Target quantity, sampling and estimation

Systematic sampling on the circle ��, or the semicircle ��+, is inherent in many stereological designs; for various examples, see Cruz-Orive & Gual-Arnau (2002). Systematic sampling on �� is in fact a special case of systematic sampling on ℝ with sampling period

image(37)

Thus, the sample size n is constant, and this enables us to use periodic extensions that do not hold in the general Cavalieri case.

The measurement function is therefore regarded as a periodic function of period 2π, namely

image(38)

In addition, f is assumed to be (mp)-piecewise smooth; see section 3.2. The target parameter is the integral

image(39)

Sampling with n test rays.  This is equivalent to sampling on �� at n systematic points:

image(40)

The corresponding estimator

image(41)

is unbiased for Q (Gual-Arnau & Cruz-Orive, 2000).

Sampling with n test sectors.  Consider a partition of �� into a fixed number n of sector ‘tiles’

image(42)

where J0 = [0, T) represents a fundamental tile of angle T; see Eq. (37). The basic probe is a sector T0 = [0, t), 0 < t ≤ T, i.e. T0 ⊂ J0, and the test system is a set of n systematic sector probes, namely

image(43)

The regularized version of f by the indicator function of the basic sector probe T0 is

image(44)

whereby the target parameter can be represented as

image(45)

and its unbiased estimator is

image(46)

6.2. Basic variance identity of the filtering approach on ��

In the present case, the basic filtering identity in Eq. (12) takes the form

image(47)

where g and inline image are the (circular) covariograms of f and of inline image, respectively. The expression of the former is

image(48)

(Gual-Arnau & Cruz-Orive, 2000). On the other hand, in Appendix C we show that

image(49)

where

image(50)

is the geometric covariogram of the sector probe T0.

6.3. Basic filtering expansion and variance models for systematic rays and sectors on ��

The basic expansion.  Term-by-term Taylor expansion of the right-hand side of Eq. (47) leads to the following expression:

image(51)

The main difference between the preceding expansion and that for the ℝ case (see Eq. 19) is the factor ‘2’ multiplying g(k)(0+) but not the last two terms led by the factor T/t2. The reason for this is that the second term in the right-hand side of Eq. (18) contains a factor ‘2’, whereas that of Eq. (47) does not.

Variance models for systematic rays and sectors on ��.  To construct a model identity from the expansion in Eq. (51), we invoke the same premises (p1)–(p3) listed in section 3.2. Thus, for an (mp)-piecewise smooth measurement function f : �� → ℝ+ with circular covariogram g, sampled by systematic ray and sector probes on ��, we obtain the model identity

image(52)

where the upper limits of the summations must be set so that no derivative orders of g exceed p.

The usual technique of setting t = T, etc. leads to the following explicit variance models for systematic sampling with rays, and with sector probes of angle t, respectively:

image(53)
image(54)

Case of infinitely differentiable measurement functions on ��.  The present approach enables us to consider the case of a measurement function f ∈C, i.e. an infinitely differentiable periodic function of period 2π; see Gual-Arnau & Cruz-Orive (2000), Proposition 2.2. In this case, g(2j+1)(0+) = 0, j = 0, 1, ... , and consequently Eq. (53) yields

image(55)

for systematic rays of period T = 2π/n.

7. Discussion

The present article constitutes a preliminary report of an entirely new approach for handling the problem of predicting the precision of systematic sampling on a fixed and bounded domain of ℝn, n = 1, 2, … , and on the circle ��. Some of the special features of our approach are as follows:

  • (i) The variance representations obtained here for Cavalieri sampling on ℝ do not resemble the traditional ones obtained via the transitive approach (Matheron, 1965, 1971). The latter are led by a term, called the extension term, whose graph (as a function of the mean number of sections) shows no oscillations, followed by a term called the Zitterbewegung, whose graph exhibits oscillations, and higher-order terms; for a brief review, see Cruz-Orive & García-Fiñana (2005). On the contrary, under the present approach we obtain expressions that do not seem to show any obvious analogies with their transitive counterparts. In particular, the graph of our variance models may show oscillations.
  •  (ii) Under the transitive theory, the target variance term is the extension term. For an (mp)-piecewise smooth measurement function f : ℝ → ℝ+, the extension term is proportional to Sg(2m+1)(0) (Souchet, 1995; Kiêu, 1997; Kiêu et al., 1999). Thus, under the transitive approach, only the behaviour of g at the origin contributes to the variance model. On the contrary, under the filtering approach, the even-order derivatives of g at all points of the grid − which did not contribute to the variance under the transitive approach − will now contribute; see, for instance, Eq. (23).
  • (iii) As a direct consequence of the preceding point, we may consider the following two alternative strategies to model the covariogram: to choose a global model for g on its entire range, in which case the model should be a positive definite function; and to choose proper models piecewise near each point of the grid, which enable us to estimate the relevant derivatives of g at each of these points.
  • (iv) The variance prediction problem in ℝn, n ≥ 2, which under the transitive approach presented considerable difficulties − especially for systematic quadrats or boxes − seems to yield relatively easily with the filtering approach, not only for systematic test points but also for systematic rectangular quadrats or n-boxes in general. Moreover, the isotropy assumption (either of the covariogram, or of the sampling design), standard under the transitive approach, is not necessary. We are now seeking proper covariogram models that, inserted for instance into the right-hand side of Eqs (33) and (35), will yield variance predictors ready to apply to a systematic sample of point- or quadrat-based observations on the plane, respectively.
  •  (v) The two cases considered in section 4.2 to handle systematic sampling on ℝ2 are by no means unique. One can impose different boundary conditions on the covariogram g, such as g ∈C for h1 ≥ 0, h2 ≥ 0, and thereby obtain different variance models. The preceding condition, by the way, requires the additional, generally unwarranted symmetry assumption g(h1h2) = g(–h1h2) = g(h1, –h2). Similar remarks apply for ℝn, n > 2.
  • (vi) For the circle case, the filtering approach yields as a plus the model in Eq. (55) for infinitely differentiable periodic functions. Up to now, for this case, the only available representation has been Eq. (2.9) from Gual-Arnau & Cruz-Orive (2000).

Acknowledgements

This research was supported by the Spanish Ministry of Education and Science I+D Projects no. MTM2005-08689-C02-01 and 02.

Appendix A

The filtering expansion on ℝ

Our purpose is to prove the expansion in Eq. (19). We do this from Eq. (18) by means of the following basic result.

Taylor expansion for piecewise smooth functions

Consider a function g : ℝ → ℝ satisfying the condition in Eq. (15). Then,

image(56)

where

image(57)

for some ɛ ∈[0, 1] and some ξ ∈[h, h +y]. When g is (p + 1) times continuously differentiable in [hh + y], then Eq. (56) becomes the standard Taylor expansion.

A proof of Eq. (56) is given in Kiêu (1997), Appendix A.

Proof of Eq. (19)

The idea is to expand the right-hand side of Eq. (18) term by term using Eq. (56). First, we need the expression of inline image, which is given by Eq. (13) for n = 1, with K given by Eq. (14) for n = 1 (i.e. Eq. 50). Thus

image(58)

(Gual-Arnau & Cruz-Orive, 1998). For the first term in the right-hand side of Eq. (18), we can now write

image(59)

because g(–y) = g(y) and also inline image. Now, setting h = 0 in Eq. (56), we obtain

image(60)

whereby Eq. (59) becomes

image(61)

Next we deal with the second term in the right-hand side of Eq. (18). Direct application of Eqs (56) and (58) yields

image(62)

Substitution of the results (61) and (62) into the right-hand side of Eq. (18) yields the required result (Eq. 19).

Appendix B

The filtering expansions on ℝ2

To prove the results in section 4.2, we start from Eq. (28). First, we apply Eqs (13) and (14) with n = 2 to express inline image in terms of g, as we have done for n = 1, and then we apply standard Taylor expansions term by term, bearing the boundary conditions of each of Cases 1 and 2 in mind. In particular, we will use

image(63)

Case 1: g ∈C for h1 ∈ℝ, h2 ∈ℝ.  The general term in the right-hand side of Eq. (28) can be expressed as

image(64)

because inline image. Now, applying Eq. (56) with h1 = k1T1h2 = k2T2k1k2 ∈ℤ, we obtain

image(65)

where

image(66)

In particular, we have inline image , whereby

image(67)

Substituting these results into the right-hand side of Eq. (65), and then into Eq. (28), we obtain Eq. (30).

Case 2: g ∈C for h1 ∈ℝ, h2 ≥ 0.  Recalling the symmetry property g(h1h2) = g(–h1, –h2), the basic identity in Eq. (28) can be written

image(68)

Next, we evaluate each term in the right-hand side of the preceding identity by means of Eq. (64), and then we use term-by-term Taylor expansions via Eq. (63).

Making use of Eq. (64) and recalling the symmetry properties g(h1h2) = g(–h1, –h2) and K(y1y2) = K(–y1, –y2), we obtain the following result, valid at the points (k1T1, 0), k1 ∈ℤ:

image(69)

Making the change of variables y1 = −x1, y2 = −x2 in the preceding double integral, we obtain

image(70)

Now we are ready to apply the Taylor expansion in Eq. (63) for h1 = k1T1, h2 = 0+, k1 ∈ Ζ, whereby

image(71)

Finally, at the points (k1T1k2T2), k1 ∈ℤ, k2 ∈ℤ, always bearing the aforementioned symmetry conditions in mind, we have no special restrictions, and consequently we obtain Eq. (67), as in Case 1. Bringing the preceding results together into Eq. (68), we obtain the desired model for the case g ∈C, h1 ∈ℝ, h2 ≥ 0. Then, averaging this with the analogous result for h1 ≥ 0, h2 ∈ℝ, we obtain Eq. (31).

Appendix C

The filtering expansion on ��

As pointed out in section 6.3, the differences in detail between the ℝ and the �� cases are minimal, and therefore no further derivations are necessary to justify Eq. (51). We just give a derivation of Eq. (49). First note that inline image (see Eq. 44) may be expressed alternatively in terms of the indicator function of T0 := [0, t) as follows:

image(72)

Now,

image(73)

as claimed.

Ancillary