A prediction framework

by consideration of the linear maps [Y x_j], j ≤q, one after the other. Alternatively, a minor extension of the arguments in this section leads to the appropriate generalization.

4.2.3. (Sub-)Optimality analysis on superspaces This section reconsiders the case of an oblique projectorP_{V /V}^⊥

∗ as defined in section4.2.1.

More specifically,V and U symbolize subspaces of a Euclidean space W. The former is spanned by a nontrivial sequence y₁, . . . , y_k with Y = [y₁ · · · y_k]. The composition ˆY and ˜Y of Y with the orthogonal projectors P_U and P_U^⊥, respectively, allow the con-struction of Y∗ = ˆY + ˜Y∗ with ˜Y∗ = ˜Y P_{ker ˆY}. Proposition 4.3 ensures that the oblique projectorP_{V /V∗}^⊥ ontoV and along the orthogonal complementV_∗^⊥ of V∗ = imgY∗ is well defined and quantifies its k^•k-performance relative to the orthogonal projector P_V.

Section4.2.2identifies the restriction ofP_{V /V∗}^⊥ toW⁰ = [Y x],x∈W, with the orthog-onal projector ontoV defined onW⁰ with respect to an alternative inner producth^•, ^•i_∗. If only projectionsP_{V /V}^⊥

∗ z,z ∈W⁰, are of concern, then this identification facilitates the computations. Moreover, the inequalities <4.4> apply with the principal h^•, ^•i-angles between the complementsV^⊥andV^⊥∗inW⁰. The bounds resulting from<4.4>together with the principalh^•, ^•i-angles between the complementsV_∗^⊥andV^⊥inW remain valid, but may offer only a rough characterization of the k^•k-performance of P_{V /V∗}^⊥ on W⁰.

The following example justifies this claim. It considers a Euclidean space W spanned byy⁰₁, . . . , y⁰_k, x⁰, y⁰⁰₁, . . . , y_k⁰⁰, and x⁰⁰, wherein imgY⁰, Y⁰ = [y₁⁰ · · · y⁰_k], and imgY⁰⁰, Y⁰⁰ = [y₁⁰⁰ · · · y⁰⁰_k], are nontrivial. The geometry of W is such that

Y⁰ x⁰

and

Y⁰⁰ x⁰⁰ share their Gramian G and U = img

Y⁰ x⁰

= (img

Y⁰⁰ x⁰⁰

)^⊥. Herein, the goal is to projectx=x⁰+x⁰⁰onto the imageV ofY =Y⁰+Y⁰⁰. These elements satisfy ˆx_U = ˆx=x⁰, Yˆ_U = ˆY = Y⁰, and the Gramian of [Y x] equals 2G. Hence, the equality ker ˆY = kerY = ker ˜Y holds and implies img ˜Y∗ = {0} as well as img ˆY× = {0}. In addition, V_∗ = imgY_∗ = imgY⁰ 6=V, and the key ratio in proposition4.3 equalskY⁰⁰ck/kY⁰ck= 1 for all c∈(kerY)^⊥, that is, θmin,6=0(V_∗^⊥, V^⊥) = π/4 = θ_max(V_∗^⊥, V^⊥).

Moreover, if z ∈ img [Y x] ∩U^⊥, then z = [Y x]c for some c ∈ R^k+1 and 0 = hh[Y⁰ x⁰], zii = Gc, which implies z = 0. Thus, the present setting amounts to an instance of the special case img [Y x]∩U^⊥ = {0} considered in section 4.2.2. As a consequence, the equality P_{V /V∗}^⊥z = P_{V /V}⊥∗z holds for all z ∈ W⁰, wherein h^•, ^•i_∗ denotes the inner product induced by G. The two inner products h^•, ^•i and h^•, ^•i_∗ onW⁰ satisfy h^•,^•i= 2h^•, ^•i_∗; in particular, the equality P_{V /V}^⊥z =P_{V /V}⊥∗z holds for allz ∈W⁰. In summary, the space img [Y x] containing all elements of interest amounts to a subspace of ker(P_{V /V}^⊥

∗ −P_V). However, the elements ofV^⊥∩(V^⊥∩V_∗^⊥)^⊥ = img(Y⁰− Y⁰⁰) are needed to apply proposition 4.3, which requires U to be a subspace ofW.

the index (t, j) ranges over I_x = {1, . . . , n} × {1, . . . , m}, m, n ∈ N. Herein,

predic-tion refers to the pointwise (with respect to ω) evaluation of random variables of the prediction

typer_t,j+P_{V /V∗}^⊥(x_t,j−r_t,j), whereinr_t,j, (t, j)∈I_x, represent an additional characteris-ticr of themlocations, and V, V_∗^⊥ denote complementary subspaces. In this section, all coordinates are considered known; their estimation—as in section3.5—is not addressed.

These predictions inherit two properties from the underlying projector P_{V /V∗}^⊥. Firstly, linearity guarantees that a prediction of P

(t,j)∈I_xc_t,jx_t,j, C ∈ R^n×m, in form of this linear combination of the predictions of x_t,j exhibits the same structure as its ingredi-ents but with r_t,j replaced by P

(t,j)∈Ixc_t,jr_t,j. Secondly, idempotence ensures that such predictions of linear combinations equal their known value if observed without error.

The overall setting amounts to the span W of P-square integrable random variables z_i,t,j , (i, t, j)∈I_z ={1, . . . , s} ×I_x , v_t,j , (t, j)∈I_x, v¯_t,i , (t, i)∈I_obs , and the constant function ω 7→ 1 defined on a probability space (Ω,F,P). Therein, the index set I_obs ⊂ N× N is finite and nonempty, and s ∈ N. The sequences v_t,j, (t, j) ∈ I_x, and ¯v_t,i, (t, i) ∈ I_obs, form bases of their spans U_v = span{v_t,j|(t, j) ∈ I_x} and Uv¯ = span{¯vt,i|(t, i) ∈Iobs}, respectively. The intersection of the span Uz of zi,t,j, (i, t, j)∈I_z, with span{1} equals{0}; the symbol U_1,z = span{1}+U_z denotes the joint span of these variables. The random variablesz_i,t,j, (i, t, j)∈I_z, are such that the kernel ker [1Zt,j], wherein Zt,j = [z1,t,j · · · zs,t,j], amounts to {0} for all (t, j) ∈Ix. The same applies to the intersections of pairs of subspacesU⁰ 6=U⁰⁰ with U⁰, U⁰⁰ ∈ {U_1,z, U_v, U_v¯}.

The definition hx, yi = R

x(ω)y(ω)P(dω) = Exy for every pair x, y ∈ W endows this linear space with a Euclidean space structure. Herein, the inner product is such that the sequence ¯v_t,i, (t, i)∈I_obs, provides an orthonormal basis of its span U_¯v. In addition, the three subspacesU_1,z, U_v, and U_v¯ satisfy U_1,z^⊥ =U_v+U_¯v, U_v^⊥ =U_1,z+U_¯v, and U_v¯^⊥ = U1,z +Uv. In particular, this specification implies the equalities h1, vt,ji = Evt,j = 0 as well asEv¯_t,i = 0 andEv¯²_t,i = 1 for all (t, j)∈I_xand (t, i)∈I_obs, respectively. Finally, the formal model is chosen relative to h^•, ^•i as explained towards the end of appendix 2.a.

The random variables zi,t,j represent s additional numerical characteristic of the m locations at n points in time. A given 1≤s⁰ ≤s splits these variables into two disjoint subsets z_i,t,j, i < s⁰, and z_i,t,j, i ≥ s⁰, wherein (t, j) ranges over I_x. The case s⁰ = 1 is possible. Then, the first set is empty and all summands consisting only of elements of that set vanish. These two groups play two different roles in the following development.

The random variables x_t,j representing the numerical characteristic xare given by xt,j =αt,j+ [Za,t,j Zb,t,j]

βa

β_b

+vt,j =αt,j +Zt,jβ+vt,j , (t, j)∈Ix , <4.10>

Z_a,t,j = [z_1,t,j · · · z_s⁰−1,t,j] , Z_b,t,j = [z_s⁰_,t,j · · · z_s,t,j] ,

for some β ∈ R^s and αt,j ∈ R. The equalities ker [1 Zt,j] = {0}, (t, j) ∈ Ix, imply that the coordinatesα_t,j, βare uniquely characterized by<4.10>. The subsequent discussion mostly focuses on the modification ¯x_t,j =x_t,j −r_t,j, (t, j)∈I_x, with r_t,j =Z_a,t,jβ_a.

Linear independence of vt,j, (t, j) ∈ Ix, and U1,z ∩Uv = {0} imply ker 1 ¯X

= {0},

wherein ¯X = X¯_n . . . X¯₁

with ¯X_t = [¯x_t,1 · · · x¯_t,m]. The following argument requires that this kernel equality continues to hold if the columns of ¯X are replaced by their orthogonal projections P_Ux¯_t,j = ˆx¯_t,j, (t, j) ∈ I_x, onto U = U_1,z +U_¯v = U_v^⊥. These projections equal ˆx¯_t,j =α_t,j+Z_b,t,jβ_b. Thus, the equality ker

1 ˆX¯

={0}, wherein ˆX¯ = Xˆ¯_n . . . Xˆ¯₁

with ˆX¯_t= ˆ¯

x_t,1 · · · xˆ¯_t,m

, requires some additional restrictions. Lemma4.4 contains an appropriate condition. A proof starts on page 123 in appendix4.a.

Lemma 4.4. The equality ker 1 ˆX¯

={0} holds if and only if β_b 6∈B_b =∪_C∈R^n×m

kCk=1

ker

X

(t,j)∈Ix

c_t,jZ_b,t,j

⊂R^s−s

0+1 .

The kernels corresponding to indexes C =B_t,j equal kerZ_b,t,j ={0}for all (t, j)∈I_x, whereinB_t,j represents the t, j-th standard basis element of R^n×m as in example (b) in section2.1.1. However, the analogous equality in case of a general unit length matrixC∈ R^n×m necessitates some additional linear independence conditions. For example, if the sequence z_i,t,j, s⁰ ≤ i, (t, j) ∈ I_x exhibits linear independence, then B_b = {0} and the requirement reduces to βb 6= 0. The latter scenario places a strong requirement on the random variables z_i,t,j, s⁰ ≤ i, (t, j) ∈ I_x, and therefore requires only the minimal condition β_b 6= 0 onβ_b. The remainder of this section assumes that β_b 6∈B_b holds.

The givens of the initially mentioned prediction task include the imageszi,t,j(ω) under the random variablesz_i,t,j,i < s⁰, (t, j)∈I_x, as well as the imagesy_t,i(ω) ofω ∈Ω under y_t,i =Xa_t,i+ρ_t,iv¯_t,i , (t, i)∈I_obs . <4.11>

The adjustment of the formal representation to h^•, ^•i as explained in appendix 2.a ensures that the just mentioned images reflect the geometry of the space (W,h^•, ^•i).

Subsequently, these images and the corresponding random variables are referred to as observations (or data) and observables, respectively. The random variables in <4.11>

amount to linear combinations of x_t,j, (t, j) ∈ I_x, and ¯v_t,i, (t, i) ∈ I_obs, with coordi-nates 06=a_t,i ∈R^nmandρ_t,i ≥0. The index setI_obs has the form∪t≤n {t}×{1, . . . , k_t} withk_t∈N∪ {0}being the number of observations att. Ifk_t= 0, then{t} × {1, . . . , k_t} equals the empty set; however,I_obs 6=∅is assumed below. The final summand in<4.11>

embodies an observation error with varianceρ²_t,i. The individual observation errorsρ_t,i¯v_t,i are pairwise orthogonal. Consequently, superfluous observables of the type <4.11> in form of linear combinations of other observables of the same type are possible only if all corresponding error variances ρ²_t,i equal zero. The case a_t,i = a_t,j for some i < j ≤ k_t together with max{ρ_t,i, ρ_t,j} > 0 represents the availability of different observations of the same element Xa_t,i. Subsequently, the focus is on the modified observables ¯y_t,i = Xa¯ _t,i+ρ_t,iv¯_t,i, (t, i)∈I_obs, which amount to linear combinations of the observables y_t,i, (t, i)∈I_obs, and z_i,t,j, i < s⁰,(t, j)∈I_x; thus, their images of ω are available, too.

The modified observables ¯y_t,i, (t, i)∈I_obs, together with the constant function 1 span the image V of the projector underlying the predictions. Initially, this projector is only defined on the superspace W⁰ = img

1 ¯Y X¯

of V, wherein ¯Y =Y¯_n . . . Y¯₁ with

Y¯_t = [¯y_t,1 · · · y¯_t,kt], 1 ≤ t ≤ n. If k_t = 0 for some t ≤ n, then the respective ¯Y_t is not defined and missing from ¯Y. The inequality I_obs 6= ∅ ensures that at least one ¯Y_t is present. The linear maps ˆY¯ and ¯V are defined in analogy with respect to ˆy¯_t,i =P_Uy_t,i =

ˆ¯

Xa_t,i+ρ_t,iv¯_t,iwithU =U_1,z+U_¯vand ¯v_t,i, (t, i)∈I_obs. The equalities ker

1 ¯X V¯

={0}= ker

1 ˆX¯ V¯

, which are due to the assumption of the conditionβ_b 6∈B_b in lemma4.4 and the equality U1,z ∩Uv¯ ={0}, imply that the kernels of the two linear maps

1 ¯Y X¯

=

1 ¯X V¯



 1

A^T I S^T



 and

1 ˆY¯ Xˆ¯

=

1 ˆX¯ V¯



 1

A^T I S^T



 <4.12>

coincide with the kernel of their identical second factor. Therein,I symbolizes thenm× nm identity matrix, whose columns amount to the standard basis e1, . . . , enm of R^nm. The number of columnsk=P

t≤nk_t>0 ofY equals the overall number of observations of the typey_t,i. Moreover, the rows ofA∈R^k×mnand the diagonal entries of the diagonal matrix S ∈R^k×k amount toa_t,i and ρ_t,i, respectively, arranged in appropriate order.

Lemma2.3 ensures that the Gramian ˆGof

1 ˆY¯ Xˆ¯

induces an additional inner prod-uct h^•,^•i_∗ on W⁰ by identification of ˆG with the h^•, ^•i_∗-Gramian of

1 ¯Y X¯ . The second equality in<4.12>shows that knowledge of the aggregation matrix A, the obser-vation error matrix S, and the Gramian of

1 ˆX¯

suffices to construct the Gramian ˆG and therebyh^•, ^•i_∗. The latter leads to an additional orthogonal complement V^⊥∗ ofV inW⁰, which in turn defines the projector P_{V /V}⊥∗ (on W⁰) underlying the predictions.

These predictions are of the form rt,j(ω) + P_{V /V}⊥∗x¯t,j

(ω), (t, j)∈Ix, wherein ¯xt,j = x_t,j−r_t,j andr_t,j =Z_a,t,jβ_a. This construct enjoys the initially mentioned properties. In fact, a prediction of a linear combination P

(t,j)∈Ixc_t,jx_t,j—obtained as the same linear combination of the individual predictions—amounts to the image ofω under

X

t,jct,jZa,t,j

βa+P_{V /V}⊥∗

X

t,jct,jxt,j−

X

t,jct,jZa,t,j

βa

and thus equalsy_t,iifc_t,j coincide with the entries of a rowa_t,iofAin<4.12>andρ_t,i = 0.

Calculation of the predictions requires evaluation (at ω) of the h^•, ^•i_∗-orthogonal projection P_{V /V}⊥∗x¯_t,j of ¯x_t,j onto V = img

1 ¯Y

for all (t, j) ∈ I_x. The techniques of section2.2 and2.4 are crucial to this endeavor. More specifically, lemma 2.2 guarantees the existence of a h^•, ^•i_∗-representation of the columns of

1 ¯Y X¯

in form of the columns of a coordinate matrix R as in figure 4.6. Therein, the presence of 1 in the unitary map Q—in form of the function ω 7→ 1—and R—as a number, respectively, follows from the equality k1k∗ = 1, wherein k^•k∗ denotes the norm induced by h^•, ^•i_∗ on W⁰. As a consequence, the vectors r_y,1 ∈ R^k and r_x,1 ∈ R^nm in the first row of R consists of entriesh1,y¯_t,ii_∗ =h1,yˆ¯_t,ii=Eyˆ¯_t,i andh1,x¯_t,ji_∗ =h1,xˆ¯_t,ji=Exˆ¯_t,j, respectively.

The latter equals h1,xˆ¯_t,ji = h1, α_t,j +Z_b,t,jβ_bi = α_t,j +Ps

i=s⁰β_iEz_i,t,j, wherein β_b = (β_s⁰, . . . , β_s). In particular, the entries ofr_x,1 usually differ fromα_t,j as defined in<4.10>. Furthermore, the columns q₁^∗, . . . , q_k^∗0 of Q_y form an h^•, ^•i_∗-orthonormal basis of the

1 R . . . R R R . . . R R . . . R R R . . . R . .. ... ... . .. ...

R R R . . . R R . . . R . .. ...

R



































































 r_y,1^T r_x,1^T

Ry Rx,y

R_x

coords. of

¯

yt,i, (t, i)∈Iobs, with resp. to 1

coords. of

¯

xt,j, (t, j)∈Ix, with resp. to 1

coords. of

¯

yt,i, (t, i)∈Iobs, with resp.

toq^∗₁, . . . ,q^∗_k0

coords. of

¯

x_t,j, (t, j)∈I_x, with resp. to q₁^∗, . . . , q_k^∗0

coords. of

¯

xt,j, (t, j)∈Ix, with resp. to q_k^∗0+1, . . . , q_k^∗0+h

1 Q_y Q_x

=[q∗ 1···q∗ k0] = q∗ k0+1···q∗ k0+h

h^•,^•i_∗-unitary map QfromR^1+k

0+h

toW⁰

1 ¯Y X¯

= ,

coordinate matrixR

Figure 4.6

The figure visualizes the structure of a representation of

1 Y¯ X¯

as considered in lemma 2.2 and with respect to h^•,^•i_∗. Herein, the rank of P_(span{1})⊥∗/span{1}Y¯ = ¯Y −r^T_y,1 and P_V⊥∗/VX¯ = ¯X−r_x,1^T −QyRx,y withV = img

1 ¯Y

is denoted by k⁰ and h, respectively.

image of P_(span{1})⊥∗/span{1}Y¯ = ¯Y −r^T_y,1, and Q_yR_x,y contains the h^•, ^•i_∗-orthogonal projections of the columns ofP_(span{1})⊥∗/span{1}X¯ = ¯X−r^T_x,1 onto that space. The basis elementsq₁^∗, . . . , q_k^∗0 can be evaluated (at ω) based on ¯Y −r_y,1^T =QyRy, which implies—

due to the choice of the formal model—the pointwise (with respect toω) equality

R^T_y





 q₁^∗(ω)

... q_k^∗0(ω)





=







¯ y_n,1(ω)

...

¯ y1,k1(ω)





−r_y,1 . <4.13>

Herein, the number of observationsk₁andk_nof the first andn-th time point are assumed to be nonzero to simplify the presentation. The row echelon matrix R_y ∈ R^k

0×k, k⁰ = rk( ¯Y−r^T_y,1)≤k, provides a Cholesky factor of theh^•, ^•i_∗-Gramian of ¯Y−r^T_y,1; its rows are linearly independent—as elements of R^k. Consequently, the equality <4.13> uniquely determines the row q_y = Q_y(ω) = q₁^∗(ω), . . . , q_k^∗0(ω)

of Q_y. Finally, the columns of P_{V /V}⊥∗X¯ =r_x,1^T +Q_yR_x,y equal theh^•, ^•i_∗-orthogonal projectionsP_{V /V}⊥∗x¯_t,j, (t, j)∈I_x. Thus, the entries ofr_x,1+R^T_x,yq_y supply the required images P_{V /V}⊥∗x¯_t,j

(ω).

By virtue of section 2.4.2, these computations require only the data and the h^•, ^•i_∗ -Gramian of

1 ¯Y X¯

, which may be recovered via <4.12> from A, S, the h^•, ^• i-Gramian of P_(span{1})^⊥X, andˆ¯ r_x,1 whose entries equal h1,xˆ¯_t,ji = Exˆ¯_t,j, (t, j) ∈ I_x. Section4.4.3 shows how to exploit a particular structure of A and the h^•, ^•i_∗-Gramian of P_(span{1})⊥∗/span{1}x¯_t,j, (t, j) ∈ I_x when evaluating the h^•, ^•i_∗-orthogonal projections of P_(span{1})⊥∗/span{1}x¯_t,j, (t, j)∈I_x, onto span{P_(span{1})⊥∗/span{1}y¯_t,i|(t, i)∈I_obs}.

4.3.2. (Sub-)optimality of predictions

An alternative characterization ofP_{V /V}⊥∗ facilitates its comparison in terms of k^•kwith theh^•, ^•i-orthogonal projectorP_{V /V}^⊥. More specifically, the current framework fits the scenario considered in section 4.2.1. In particular, the composition of Y⁰ =

1 ¯Y with the orthogonal projector P_U onto the subspace U =U_1,z +U_¯v of W equals ˆY⁰ =

1 ˆY¯ . As observed below <4.12>, the linear map ˆY⁰ shares its kernel with Y⁰. Consequently, Y_∗⁰ = ˆY⁰ +P_U^⊥Y⁰P_{ker ˆY}0 = ˆY⁰, and proposition 4.3 ensures that there exists an oblique projectorP_{V /V∗}^⊥ (onW) ontoV = imgY⁰ and along the orthogonal complement ofV∗ = imgY_∗⁰ = img ˆY⁰. The restriction of the latter to W⁰ coincides with P_{V /V}⊥∗ as defined in section4.3.1. In fact, the linear mapYˆ⁰ Xˆ¯

amounts to the compositionP_U Y⁰ X¯

and shares its kernel with

Y⁰ X¯

. Consequently, ifz⁰ ∈img Y⁰ X¯

∩U^⊥, thenz⁰ = Y⁰ X¯

c for some c ∈ R^1+k+nm, and its projection onto U equals 0 = P_Uz⁰ = Yˆ⁰ Xˆ¯

c. Thus, c∈kerYˆ⁰ Xˆ¯

= ker Y⁰ X¯

, and therefore z⁰ = 0. The resulting equality img Y⁰ X¯

∩ U^⊥ = {0} identifies the present setting as an instance of the corresponding special case considered in section 4.2.2. Hence, the restriction of the oblique projector P_{V /V∗}^⊥ toW⁰ = img

1 ¯Y X¯

equals the h^•, ^•i_∗-orthogonal projector P_{V /V}⊥∗.

In particular, proposition 4.3 becomes applicable to P_{V /V}⊥∗ and is relevant as xt,j −

rt,j +P_{V /V}⊥∗(xt,j −rt,j)

2 =k¯xt,j −P_{V /V}⊥∗x¯t,jk²

=kP_V^⊥x¯_t,jk² +k(P_V −P_{V /V∗}^⊥)¯x_t,jk² ≤

1 + tan²θ_max(V^⊥, V^⊥∗)

kP_V^⊥x¯_t,jk², <4.14>

wherein the final inequality is due to <4.4>. Corollary 4.5 allows the quantification of the multiplier 1+tan²θ_max(V^⊥, V_∗^⊥) in<4.14>. A proof of this results starts on page124 in appendix4.a. Its statement uses the notion of a variance matrix var(z) of a random vectorz = (z₁, . . . , z_j), j ∈N, which is defined in example(e) in section 2.4.1.

Corollary 4.5. In the above setting, in particular, with I_obs 6= ∅ and β_b 6∈ B_b, one has imgA^T 6= {0}. Moreover, the kernel of the variance matrix var(ˆx)¯ of the random vectorx, which contains the columns of the linear mapˆ¯ Xˆ¯ (in the same order), equals{0}.

Finally, the two subspaces V and V_∗ differ and the inequalities sup

c∈imgA,c6=0

hA^Tc,var(˜x)A¯ ^Tci hA^Tc, var(ˆx) +¯ S²

A^Tci ≤tan²θ_max(V^⊥, V_∗^⊥)≤ sup

a∈imgA^T, a6=0

ha,var(˜x)ai¯ ha,var(ˆx)ai¯ ,

c∈imginfA,c6=0

hA^Tc,var(˜x)A¯ ^Tci hA^Tc, var(ˆx) +¯ S²

A^Tci ≤tan²θ_min,6=0(V^⊥, V_∗^⊥)≤ inf

a∈imgA^T, a6=0

ha,var(˜x)ai¯ ha,var(ˆx)ai¯ hold, whereinx˜¯ is defined in analogy with xˆ¯ but with respect to X˜¯ =P_U^⊥X.¯

If kerA^T ⊂kerS, in particular, ifkerA^T={0}, then the lower bounds for the squared tangents tan²θmax(V^⊥, V_∗^⊥) and tan²θmin,6=0(V^⊥, V_∗^⊥) hold with equality.

A few comments on corollary 4.5 are in order. Firstly, the interest lies in the predic-tions, that is, the values of the translated—by r_t,j—projections of ¯x_t,j, (t, j) ∈ I_x, at a

single ω ∈ Ω. However, the bounds resulting from corollary 4.5 refer to the norm k^•k, which merely provides an average (acrossω) distance measure. Secondly, the findings of section 4.2.3 apply as solely projections in W⁰ are of concern, that is, the bounds may be rather “conservative”. Finally, if kerA^T = {0}, then an increase of the error vari-ancesρ²_t,i, (t, i)∈I_obs, that is, the diagonal elements ofS, decreases the first factor in the final term in <4.14>, but also increases the k^•k-length of the residuals resulting from orthogonal projection ontoV, that is, the second factor of the final term in <4.14>.

Im Dokument A framework for spatiotemporal prediction with small and heterogeneous data - and an application to consumer price indexes - (Seite 105-111)