I. INTRODUCTION
Up to this day, the theory of the other than ARMA-modelled stationary time series has been enriched with some decent attempts: the direct aim of Lomnicki and Zaremba (1955) or Keenan (1982), was on the strictly stationary processes by reducing to categorical 0 – 1 dependencies; the Discrete ARMA models of Jacobs and Lewis (1978) have placed the focus on general mixtures; the bilinear model with deterministic coefficients offers a net solution, by writing the variable as a sum of products of its lagged and error lagged values raised to natural powers, but suffers from a lack of interpretation and demands some immediate action regarding measurability; both the INARMA model (for instance, find a primary form in McKenzie (1985) and then Du and Li (1991)) and the suggestion of Cui and Lund (2009) work for count data, with the first one crossing the variables of interest with errors, and the second one only conveniently combining information from discrete time renewal processes.
Evidently, blending the different equations in order to escape the linear series modelling, has been a common mentality: for example, one could catch up on switching between different ARMA equations in Francq and Zakoïan (2001). The threshold models, an excellent account of which resides in Tong (1990), may be considered as a world popular illustration of non-linearity that caused a lot of this. Finally, the latest suggestion of a TARMA model by Dimitriou-Fakalou (2019) is to fulfil the promise of encompassing the strictly stationary dependence, via a 'non-parametric' assignment of the variables' values into a fixed number of ranges: under causality, Markov chain conditional probability approaches become its special case. Since this paper will be translating the TARMA merits for the spatial series in the line transect, a reminder of existing transitions from time to space must follow.
For the linear series, two main streams developed in the past: the simultaneous bilateral auto-regressive models of Whittle (1954) and the conditional auto-normal processes of Besag (1974). The first exhibited what happens when the standard ARMA model uses its variables from both past and future: the high risks for the inference were exposed and the causal solution has been treasured as the right way to go, subject to the unilateral (more than one) index ordering of Whittle (1954). Besag (1974), on the other hand, worked with conditional arguments and proposed a Gaussian model that equally treats both ends of the axis; that second approach would often relate to a parameterization with intrinsic restrictions. Remaining in the 'best linear predictor' series department, those authentic definitions triggered Dimitriou-Fakalou (2010) to introduce the term 'auto-linear', combining the simultaneous (rather than conditional) element from Whittle (1954) together with the spatial (rather than unilateral) element from Besag (1974).
Trying to record some attempts for the spatial modelling outside the Gaussian distribution, Besag (1974) also introduced the auto-logistic schemes (for a fixated number of sites in space rather than a process in the transect) applied on Bernoulli variables that do, in general, exhibit a higher than second-order dependence: nevertheless, those definitions were accompanied by the assumption that the quantities relating more than two variables were set as null (leaving the analysis only to cliques consisting of single sites and pairs of sites) and shattered any hope of going further than the relevant pairwise elements. The reader is encouraged to look at Cressie (1993) for a beautiful description of the different perspectives in spatial statistics, which is rich in diverse material: nevertheless when working in space, the dilemma of either sticking to linearity for the infinite lattice processes, or resorting to conditional probability arguments and under the knowledge of a basic distribution for the finite lattice variables, remains.
Hence a discretized strictly stationary series indexed regularly in the straight line is to be modelled: a two-sided, random coefficient equation is the proposed solution satisfying the spatial needs. Due to the severe difficulties that arise when scrutinizing the spatial dependencies, the assumption of causality that frees the 'future' errors from any dependence with 'present' and 'past' variables will still be wrapped inside the new definitions. Section 3 introduces a special case of the model, which results in the spatial pth order conditional probabilities, and Section 3.1 explains the thinking that supports the new models: the arguments relate to writing the latent equalizer predictor, as this has been explained already in Section 2. Section 3.2 is looking for ways to get to the restrictions that tie Bernoulli variables in space, and Section 3.3 examines whether time reversal can be achieved when one goes higher than the Gaussianity of random variables and there is no moment generating function to ease the derivations; this is versus the spectral density of an ARMA series that reveals how different ARMA equations share the same auto-covariance function (Brockwell and Davis (1991)). Finally, Definition 1 in Section 4 introduces the Table Auto-Linear Moving-Average model and perfects the previous suggestions, by approximating the infinite spatial Markovian dependence.
II. REMINDERS: THE 'LATENT EQUALIZER PREDICTOR' APPROACH
Suppose that is a strictly stationary time series which, either naturally or by discretization, takes the values 0, with probability one. For a fixed positive integer p, this could be modelled as
\[X _ {t} := I _ {t} ^ {(X _ {t - 1}, \dots , X _ {t - p})},\tag{1}\]
where , are sequences of independent in time and identically distributed random vectors with marginal probability masses
\[\pi_ {w} ^ {(i _ {1}, \dots , i _ {p})} := \mathbb {P} (I _ {t} ^ {(i _ {1}, \dots , i _ {p})} = w), w = 0, v _ {1}, \dots , v _ {k}.\]
If the index functions
\[f _ {x} (y) = \frac {\prod_ {x ^ {*} = 0 , v _ {1} , \ldots , v _ {k} , x ^ {*} \neq x} (y - x ^ {*})}{\prod_ {x ^ {*} = 0 , v _ {1} , \ldots , v _ {k} , x ^ {*} \neq x} (x - x ^ {*})} = \left\{ \begin{array}{l l} 1, & \text {if y = x} \\ 0, & \text {if y = 0, v_{1}, \ldots, v_{k}, y \neq x} \end{array} \right.\]
are considered, then (1) is equivalent to
\[f _ {x} (X _ {t}) = \sum_ {i _ {1}, \ldots , i _ {p} = 0, v _ {1}, \ldots , v _ {k}} f _ {x} (I _ {t} ^ {(i _ {1}, \ldots , i _ {p})}) \cdot f _ {i _ {1}} (X _ {t - 1}) \dots f _ {i _ {p}} (X _ {t - p}), x = 0, v _ {1}, \ldots , v _ {k}.\tag{2}\]
Under the condition of (table) causality (see Dimitriou-Fakalou (2019)), it is secured that is independent of , l > 0, so that (1) implies that
\[\mathbb {P} (X _ {t} = x \mid X _ {t - n} = x _ {n}, n \in \mathbb {N}) \equiv \mathbb {P} (X _ {t} = x \mid X _ {t - n} = x _ {n}, n = 1, \ldots , p) = \pi_ {x} ^ {(x _ {1}, \ldots , x _ {n})}\tag{3}\]
for , and hence exhibits the th order (homogeneous) Markovian structure.
Model (1) uses the latent equalizer predictor to define and this is explained here further. For the linear series, we write the best (linear) predictor of based on, say, for some decided values of the : this is a linear function of the with deterministic coefficients, i.e. a random variable; knowledge of the leads to a realization of the predictor. The difference of minus the predictor is defined as the prediction error, so that what singles out the best linear predictor as unique amongst other linear functions, is that the best linear prediction error is uncorrelated with the relevant .
For the discretized series, we write the latent equalizer predictor of based on , which is a random variable that is set equal to , using the information according to as it is their function; for each one of their combined value an variable is assigned and, hence, knowledge of the leads to a random variable, say, the conditional predictor with known distribution. What has characterized the latent equalizer predictor of based on , is that the distribution of the conditional predictor remains unchanged under the presence of the relevant condition or not, implying some form of independence with it: for instance, in this section verifies that is independent of .
Prediction examples: The best linear prediction of based on some , is a linear function of the ; the intercept and coefficients are determined according to the first and second moments of the variables (parameters which are known or to be estimated). The latent equalizer prediction (of based on some ) is the realization of a random variable; its (discrete) distribution are the parameters which are known or to be estimated. In this section's notation, when the latent equalizer prediction of can be the realization of a random variable with identical distribution as that of .
III. SPATIAL MODELING
Next suppose that instead of the time axis, the strictly stationary process of interest, say takes place in the spatial line transect (see Whittle, 1954); this can still be expressed as in (1), yielding a unilateral representation based on the variables of one side of the transect , but it is explored here what is the spatial analogue. Subject to the condition of causality that links (1) to an adequate strictly stationary process , it is now written that
\[X_{s} := Y_{s}^{[(X_{s-p},\dots,X_{s-1});(X_{s+1},\dots,X_{s+p})]}\]
where for (and still a row vector), , it holds that (set any , )
\[m_{s}^{[\mathbf{i}_P;\mathbf{i}_F]} \cdot Y_{s}^{[\mathbf{i}_P;\mathbf{i}_F]} := M_{s}^{[\mathbf{i}_P;\mathbf{i}_F]},\]
which must be accompanied by
\[m_{s}^{[\mathbf{i}_{P}; \mathbf{i}_{F}]} := \sum_{w = 0, v_{1}, \ldots , v_{k}} f_{w}(I_{s}^{(\mathbf{i}_{P}^{{\tau}})}) \cdot f_{i_{F,1}}(I_{s+1}^{(w, i_{P,1}, \ldots , i_{P,p-1})}) \dots f_{i_{F,p}}(I_{s+p}^{(i_{F,p-1}, \ldots , i_{F,1}, w)})\]
and
\[M_{s}^{[\mathbf{i}_{P}; \mathbf{i}_{F}]} := I_{s}^{(\mathbf{i}_{P}^\tau)} \cdot f_{i_{F,1}} (I_{s+1}^{(I_{s}^{(\mathbf{i}_{P}^\tau)}, i_{P,1}, \ldots , i_{P,p-1})}) \dots f_{i_{F,p}} (I_{s+p}^{(i_{F,p-1}, \ldots , i_{F,1}, I_{s}^{(\mathbf{i}_{P}^\tau)})});\]
the last equation results in
\[\begin{array}{r c l} {f _ {w} (M _ {s} ^ {[ \mathbf {i} _ {P}; \mathbf {i} _ {F} ]})} & = & {f _ {w} (I _ {s} ^ {(\mathbf {i} _ {P} ^ {\tau})}) \cdot f _ {i _ {F, 1}} (I _ {s + 1} ^ {(w, i _ {P, 1}, \ldots , i _ {P, p - 1})}) \ldots f _ {i _ {F, p}} (I _ {s + p} ^ {(i _ {F, p - 1}, \ldots , i _ {F, 1}, w)}), w = v _ {1}, \ldots , v _ {k},} \\& & \\{f _ {0} (M _ {s} ^ {[ \mathbf {i} _ {P}; \mathbf {i} _ {F} ]})} & = & {1 - \sum_ {w = v _ {1}, \ldots , v _ {k}} f _ {w} (M _ {s} ^ {[ \mathbf {i} _ {P}; \mathbf {i} _ {F} ]}).} \end{array}\]
Note that if then is equal to zero as well. Hence, it is clarified that if the system (5) is null, then the variable Y cannot be considered at all, i.e. is defined on the sample space . Provided that , (5) can be replaced by .
In addition, (4) is equivalent to
\[f_{y}(X_{s}) = \sum_{\substack{i_{P,p},\ldots ,i_{P,1},i_{F,1},\ldots i_{F,p} = 0,v_{1},\ldots ,v_{k}}}f_{y}(Y_{s}^{([{\bf i}_{P};{\bf i}_{F}])}\cdot f_{i_{P,p}}(X_{s - p})\ldots f_{i_{P,1}}(X_{s - 1})\\f_{i_{F,1}}(X_{s + 1})\ldots f_{i_{F,p}}(X_{s + p})\tag{6}\]
for all . Directly from (6), it can be deduced that for any , it holds that
\[\mathbb {P} (X _ {s} = y \mid X _ {s - p} = i _ {P, p}, \ldots , X _ {s - 1} = i _ {P, 1}, X _ {s + 1} = i _ {F, 1}, \ldots , X _ {s + p} = i _ {F, p}) =\]
\[\mathbb {P} (Y _ {s} ^ {[ \mathbf {i} _ {P}; \mathbf {i} _ {F} ]} = y \mid X _ {s - p} = i _ {P, p}, \ldots , X _ {s - 1} = i _ {P, 1}, X _ {s + 1} = i _ {F, 1}, \ldots , X _ {s + p} = i _ {F, p}) (7)\]
\[\big (\{X _ {s - 1} = i _ {P, 1}, \dots , X _ {s - p} = i _ {P, p}, X _ {s + 1} = i _ {F, 1}, \dots , X _ {s + p} = i _ {F, p} \} \subseteq \{m _ {s} ^ {[ \mathbf {i} _ {P}; \mathbf {i} _ {F} ]} = 1 \} \big).\]
3.1 Interpretations
The variable is Bernoulli and it holds that
\[\mathbb {P} (m _ {s} ^ {[ \mathbf {i} _ {P}; \mathbf {i} _ {F} ]} = 1) = \sum_ {w = 0, v _ {1}, \ldots , v _ {k}} \mathbb {E} \{f _ {w} (I _ {s} ^ {(\mathbf {i} _ {P} ^ {\tau})}) \} \mathbb {E} \{f _ {i _ {F, 1}} (I _ {s + 1} ^ {(w, i _ {P, 1}, \ldots , i _ {P, p - 1})}) \} \ldots \mathbb {E} \{f _ {i _ {F, p}} (I _ {s + p} ^ {(i _ {F, p - 1}, \ldots , i _ {F, 1}, w)}) \}\tag{8}\]
(remember that are assumed to be 'serially' independent on different spots of the line transect). The variable is 0 or , , and it holds that
\[\begin{array}{r l} & {\mathbb {P} (M _ {s} ^ {[ \mathbf {i} _ {P}; \mathbf {i} _ {F} ]} = w, m _ {s} ^ {[ \mathbf {i} _ {P}; \mathbf {i} _ {F} ]} = 1) =} \\& {\mathbb {E} \{f _ {w} (I _ {s} ^ {(\mathbf {i} _ {P} ^ {\tau})}) \} \mathbb {E} \{f _ {i _ {F, 1}} (I _ {s + 1} ^ {(w, i _ {P, 1}, \ldots , i _ {P, p - 1})}) \} \dots \mathbb {E} \{f _ {i _ {F, p}} (I _ {s + p} ^ {(i _ {F, p - 1}, \ldots , i _ {F, 1}, w)}) \}} \end{array}\tag{9}\]
(for , it holds additionally that ). It can be written, in general, then
\[\begin{array}{r c l} \mathbb {P} (Y _ {s} ^ {[ \mathbf {i} _ {P}; \mathbf {i} _ {F} ]} = w) & \equiv & \mathbb {P} (Y _ {s} ^ {[ \mathbf {i} _ {P}; \mathbf {i} _ {F} ]} = w \mid m _ {s} ^ {[ \mathbf {i} _ {P}; \mathbf {i} _ {F} ]} = 1) = \mathbb {P} (M _ {s} ^ {[ \mathbf {i} _ {P}; \mathbf {i} _ {F} ]} = w \mid m _ {s} ^ {[ \mathbf {i} _ {P}; \mathbf {i} _ {F} ]} = 1) = \\& & \frac {\mathbb {P} (M _ {s} ^ {[ \mathbf {i} _ {P} ; \mathbf {i} _ {F} ]} = w , m _ {s} ^ {[ \mathbf {i} _ {P} ; \mathbf {i} _ {F} ]} = 1)}{\mathbb {P} (m _ {s} ^ {[ \mathbf {i} _ {P} ; \mathbf {i} _ {F} ]} = 1)}, w = 0, v _ {1}, \dots , v _ {k}. \end{array} \tag {10}\]
In the other hand, it holds that
\[\begin{array}{r l} & {\mathbb {P} (X _ {s} = w \mid (X _ {s - 1}, \ldots , X _ {s - p}) = \mathbf {i} _ {P} ^ {\tau}, (X _ {s + 1}, \ldots , X _ {s + p}) = \mathbf {i} _ {F}) =} \\& {\qquad \mathbb {P} (X _ {s} = w, X _ {s + 1} = i _ {F, 1}, \ldots , X _ {s + p} = i _ {F, p} \mid X _ {s - 1} = i _ {P, 1}, \ldots , X _ {s - p} = i _ {P, p})} \\& {\overline {{\sum_ {w ^ {*} = 0 , v _ {1} , \ldots , v _ {k}} \mathbb {P} (X _ {s} = w ^ {*} , X _ {s + 1} = i _ {F , 1} , \ldots , X _ {s + p} = i _ {F , p} \mid X _ {s - 1} = i _ {P , 1} , \ldots , X _ {s - p} = i _ {P , p})}}} \\& {\qquad \mathbb {E} \{f _ {w} (I _ {s} ^ {(i _ {P, 1}, \ldots , i _ {P, p})}) f _ {i _ {F, 1}} (I _ {s + 1} ^ {(w, i _ {P, 1}, \ldots , i _ {P, p - 1})}) \ldots f _ {i _ {F, p}} (I _ {s + p} ^ {(i _ {F, p - 1}, \ldots , i _ {F, 1}, w)}) \}} \\& \overline {{\sum_ {w ^ {*} = 0 , v _ {1} , \ldots , v _ {k}} \mathbb {E} \{f _ {w ^ {*}} (I _ {s} ^ {(i _ {P , 1}, \ldots , i _ {P , p})}) f _ {i _ {F , 1}} (I _ {s + 1} ^ {(w ^ {*} , i _ {P , 1}, \ldots , i _ {P , p - 1})}) \ldots f _ {i _ {F , p}} (I _ {s + p} ^ {(i _ {F , p - 1}, \ldots , i _ {F , 1}, w ^ {*})}) \}}} \end{array}\]
based on the representation . By combining the derivation above together with (7) as well as (8), (9) and (10), it can be concluded that
\[\mathbb {P} (Y _ {s} ^ {[ \mathbf {i} _ {P}; \mathbf {i} _ {F} ]} = w) = \mathbb {P} (Y _ {s} ^ {[ \mathbf {i} _ {P}; \mathbf {i} _ {F} ]} = w \mid (X _ {s - 1}, \ldots , X _ {s - p}) = \mathbf {i} _ {P} ^ {\tau}, (X _ {s + 1}, \ldots , X _ {s + p}) = \mathbf {i} _ {F}).\tag{11}\]
In fact using similar arguments, it can be shown (for any , ), that
\[\begin{array}{c} \mathbb {P} (Y _ {s} ^ {[ \mathbf {i} _ {P}; \mathbf {i} _ {F} ]} = w) = \mathbb {P} (Y _ {s} ^ {[ \mathbf {i} _ {P}; \mathbf {i} _ {F} ]} = w \mid (X _ {s - 1}, \ldots , X _ {s - p}) = \mathbf {i} _ {P} ^ {\tau}, \\(X _ {s + 1}, \ldots , X _ {s + p}) = \mathbf {i} _ {F}, X _ {s - l} = x _ {l}, | l | = p + 1, p + 2, \ldots). \end{array}\tag{12}\]
It has been obvious already that (properly defined) is independent of , i > 0, as the former is a function of , and the latter is a function of , : this is no improvement over what was there before defining , i.e. the random variable being independent of , i > 0. The additional element that the equation (11) (together with (12)) brings in, is that it allows the probability of Y to remain unchanged under the condition of interest or not (implying a form of independence of with each Bernoulli variable on the set , ), subject to shrinking the sample space for the variable; this seems to justify being a latent equalizer predictor of based on , .
3.2 Symmetrical Probabilities
For variables with two values only , say , ( ), there is interest whether it holds that
\[\mathbb {P} (Y _ {s} ^ {[ \mathbf {i} _ {P}; \mathbf {i} _ {F} ]} = w) \equiv \mathbb {P} (Y _ {s} ^ {[ \mathbf {i} _ {F} ^ {\tau}; \mathbf {i} _ {P} ^ {\tau} ]} = w), w = v _ {0}, v _ {1}\tag{13}\]
(it is reminded that the two random variables are defined on different spaces and the index 's' for both is used 'loosely'). If (13) takes place, then
\[\frac \frac {\mathbb {E} \{f _ {w} (I _ {s} ^ {(i _ {P , 1} , \ldots , i _ {P , p})}) f _ {i _ {F , 1}} (I _ {s + 1} ^ {(w , i _ {P , 1} , \ldots , i _ {P , p - 1})}) \ldots f _ {i _ {F , p}} (I _ {s + p} ^ {(i _ {F , p - 1} , \ldots , i _ {F , 1}, w)}) \}}{\sum_ {w ^ {*} = v _ {0}, v _ {1}} \mathbb {E} \{f _ {w ^ {*}} (I _ {s} ^ {(i _ {P , 1}, \ldots , i _ {P, p})}) f _ {i _ {F , 1}} (I _ {s + 1} ^ {(w ^ {*}, i _ {P , 1}, \ldots , i _ {P, p - 1})}) \ldots f _ {i _ {F, p}} (I _ {s + p} ^ {(i _ {F, p - 1}, \ldots , i _ {F, 1}, w ^ {*})}) \}} = \frac {\mathbb {E} \{f _ {w} (I _ {s} ^ {(i _ {F , 1}, \ldots , i _ {F , p})}) f _ {i _ {P , 1}} (I _ {s + 1} ^ {(w , i _ {F , 1}, \ldots , i _ {F , p - 1})}) \ldots f _ {i _ {P , p}} (I _ {s + p} ^ {(i _ {P , p - 1}, \ldots , i _ {P , 1}, w)}) \}}{\sum_ {w ^ {*} = v _ {0}, v _ {1}} \mathbb {E} \{f _ {w ^ {*}} (I _ {s} ^ {(i _ {F , 1}, \ldots , i _ {F, p})}) f _ {i _ {P , 1}} (I _ {s + 1} ^ {(w ^ {*}, i _ {F, 1}, \ldots , i _ {F, p - 1})}) \ldots f _ {i _ {P, p}} (I _ {s + p} ^ {(i _ {P, p - 1}, \ldots , i _ {P, 1}, w ^ {*})}) \}}\]
and, consequently, (13) results in the key equality
\[\begin{array}{r l} & {\mathbb {E} \{f _ {v _ {0}} (I _ {s} ^ {(i _ {P, 1}, \ldots , i _ {P, p})}) f _ {i _ {F, 1}} (I _ {s + 1} ^ {(v _ {0}, i _ {P, 1}, \ldots , i _ {P, p - 1})}) \ldots f _ {i _ {F, p}} (I _ {s + p} ^ {(i _ {F, p - 1}, \ldots , i _ {F, 1}, v _ {0})}) \}} \\& {\mathbb {E} \{f _ {v _ {1}} (I _ {s} ^ {(i _ {F, 1}, \ldots , i _ {F, p})}) f _ {i _ {P, 1}} (I _ {s + 1} ^ {(v _ {1}, i _ {F, 1}, \ldots , i _ {F, p - 1})}) \ldots f _ {i _ {P, p}} (I _ {s + p} ^ {(i _ {P, p - 1}, \ldots , i _ {P, 1}, v _ {1})}) \} =} \\& {\mathbb {E} \{f _ {v _ {1}} (I _ {s} ^ {(i _ {P, 1}, \ldots , i _ {P, p})}) f _ {i _ {F, 1}} (I _ {s + 1} ^ {(v _ {1}, i _ {P, 1}, \ldots , i _ {P, p - 1})}) \ldots f _ {i _ {F, p}} (I _ {s + p} ^ {(i _ {F, p - 1}, \ldots , i _ {F, 1}, v _ {1})}) \}} \\& {\mathbb {E} \{f _ {v _ {0}} (I _ {s} ^ {(i _ {F, 1}, \ldots , i _ {F, p})}) f _ {i _ {P, 1}} (I _ {s + 1} ^ {(v _ {0}, i _ {F, 1}, \ldots , i _ {F, p - 1})}) \ldots f _ {i _ {P, p}} (I _ {s + p} ^ {(i _ {P, p - 1}, \ldots , i _ {P, 1}, v _ {0})}) \}.} \end{array}\tag{14}\]
Lemma 1: If the order or then (13) definitely holds.
Proof: For , (14) becomes
\[\begin{array}{r l} & {\mathbb {E} \{f _ {v _ {0}} (I _ {s} ^ {(i _ {P, 1})}) f _ {i _ {F, 1}} (I _ {s + 1} ^ {(v _ {0})}) \} \mathbb {E} \{f _ {v _ {1}} (I _ {s} ^ {(i _ {F, 1})}) f _ {i _ {P, 1}} (I _ {s + 1} ^ {(v _ {1})}) \} =} \\& {\mathbb {E} \{f _ {v _ {1}} (I _ {s} ^ {(i _ {P, 1})}) f _ {i _ {F, 1}} (I _ {s + 1} ^ {(v _ {1})}) \} \mathbb {E} \{f _ {v _ {0}} (I _ {s} ^ {(i _ {F, 1})}) f _ {i _ {P, 1}} (I _ {s + 1} ^ {(v _ {0})}) \}} \end{array}\]
or , which holds when as well as when or .
For p = 2, the equality under question reduces to
\[\begin{array}{r l} & {\pi_ {v _ {0}} ^ {(i _ {P, 1}, i _ {P, 2})} \pi_ {i _ {F, 1}} ^ {(v _ {0}, i _ {P, 1})} \pi_ {i _ {F, 2}} ^ {(i _ {F, 1}, v _ {0})} \cdot \pi_ {v _ {1}} ^ {(i _ {F, 1}, i _ {F, 2})} \pi_ {i _ {P, 1}} ^ {(v _ {1}, i _ {F, 1})} \pi_ {i _ {P, 2}} ^ {(i _ {P, 1}, v _ {1})} =} \\& {\pi_ {v _ {1}} ^ {(i _ {P, 1}, i _ {P, 2})} \pi_ {i _ {F, 1}} ^ {(v _ {1}, i _ {P, 1})} \pi_ {i _ {F, 2}} ^ {(i _ {F, 1}, v _ {1})} \cdot \pi_ {v _ {0}} ^ {(i _ {F, 1}, i _ {F, 2})} \pi_ {i _ {P, 1}} ^ {(v _ {0}, i _ {F, 1})} \pi_ {i _ {P, 2}} ^ {(i _ {P, 1}, v _ {0})}} \end{array}\]
and the reader may try all different values to verify its validity. □
Next it is explained why (13) is investigated whether it holds. If it is true, it may be derived as well that
\[\begin{array}{l} \mathbb {P} (X _ {s} = w \mid (X _ {s - 1}, \ldots , X _ {s - p}) = \mathbf {i} _ {P} ^ {\tau}, (X _ {s + 1}, \ldots , X _ {s + p}) = \mathbf {i} _ {F}) = \\\mathbb {P} (X _ {s} = w \mid (X _ {s - 1}, \ldots , X _ {s - p}) = \mathbf {i} _ {F}, (X _ {s + 1}, \ldots , X _ {s + p}) = \mathbf {i} _ {P} ^ {\tau}). \end{array}\tag{15}\]
In fact, it was Besag (1974) crystallizing that symmetry conditions often take place regarding the deterministic coefficients for spatial models of interest: for the auto-normal schemes (i.e. Gaussian variables that are characterized by an up to second-order dependence), those coefficients would be found in . Later, in the simultaneous Auto-Linear model of Dimitriou-Fakalou (2010), the symmetrical coefficients were demonstrated to be (proportional to) the auto-covariances of the Y latent process, which was a moving-average series with pairs of variables uncorrelated when more than p (i.e. the order of the model) steps away. Hence the short investigation that took place here regarding Y symmetrical probabilities is natural and just.
For Bernoulli variables and p = 1, the dependence is still linear (the conditional expectation of given , is a linear function of , the conditional variance is no constant though), so that an even auto-covariance function might describe it. The material point is that
\[\mathbb {E} \{X _ {s - n} ^ {(v _ {P})} X _ {s} ^ {(w)} X _ {s + n} ^ {(v _ {F})} \} = \mathbb {E} \{X _ {s - n} ^ {(v _ {F})} X _ {s} ^ {(w)} X _ {s + n} ^ {(v _ {P})} \}, n \in \mathbb {N},\tag{16}\]
for any , where the operator , generates either ( ) or ( ), when is in . The general Bernoulli property (16) results in
\[\mathbb {P} (X _ {s} = w \mid X _ {s - n} = v _ {P}, X _ {s + n} = v _ {F}) = \mathbb {P} (X _ {s} = w \mid X _ {s - n} = v _ {F}, X _ {s + n} = v _ {P}),\]
which easily justifies all symmetrical probabilities (15) for p = 1.
Lemma 2: For any fixed order ( and values ), it holds that
\[\mathbb {P} (X _ {s} = v _ {1} \mid X _ {s - n} = v _ {1}, X _ {s + n} = v _ {0}, n = 1, \ldots , p) =\]
\[\mathbb {P} (X _ {s} = v _ {1} \mid X _ {s - n} = v _ {0}, X _ {s + n} = v _ {1}, n = 1, \ldots , p) \equiv \frac {\pi_ {v _ {1}} ^ {(v _ {1} , \ldots , v _ {1})}}{\pi_ {v _ {1}} ^ {(v _ {1} , \ldots , v _ {1})} + \pi_ {v _ {0}} ^ {(v _ {0} , \ldots , v _ {0})}},\tag{17}\]
which can be linked to a natural interpretation.
Proof: Obvious. □
Lemma 3: For any fixed order (k = 1 and values ), it is true that the 'oscillating' conditional probabilities are symmetrical, i.e.
\[\mathbb {P} (X _ {s} = w \mid X _ {s - 1} = v _ {1}, X _ {s + 1} = v _ {0}, X _ {s - 2} = v _ {0}, X _ {s + 2} = v _ {1}, \ldots\]
\[(X _ {s + p - 1} =) X _ {s - p} \neq X _ {s + p} (= X _ {s - p + 1})) =\]
\[\mathbb {P} (X _ {s} = w \mid X _ {s - 1} = v _ {0}, X _ {s + 1} = v _ {1}, X _ {s - 2} = v _ {1}, X _ {s + 2} = v _ {0}, \ldots\]
\[(X_{s+p-1}=)X_{s-p}\neq X_{s+p}(=X_{s-p+1})\]
holds for both .
Proof: If is even, it holds (for either ) that
\[\mathbb {P} (X _ {s} = w, X _ {s + 1} = v _ {0}, X _ {s + 2} = v _ {1}, \dots X _ {s + p} = v _ {1} \mid X _ {s - 1} = v _ {1}, \dots , X _ {s - p} = v _ {0}) =\]
\[\mathbb {P} (X _ {s} = w, X _ {s + 1} = v _ {1}, X _ {s + 2} = v _ {0}, \ldots , X _ {s + p} = v _ {0} \mid X _ {s - 1} = v _ {0}, \ldots , X _ {s - p} = v _ {1});\]
if is odd, it can be shown that
\[\begin{array}{l} \mathbb {P} (X _ {s} = w, X _ {s + 1} = v _ {0}, X _ {s + 2} = v _ {1}, \ldots , X _ {s + p} = v _ {0} \mid X _ {s - 1} = v _ {1}, \ldots , X _ {s - p} = v _ {1}) \\\hline \mathbb {P} (X _ {s} = w, X _ {s + 1} = v _ {1}, X _ {s + 2} = v _ {0}, \ldots , X _ {s + p} = v _ {1} \mid X _ {s - 1} = v _ {0}, \ldots , X _ {s - p} = v _ {0}) \\\qquad = \frac {\pi_ {v _ {0}} ^ {(v _ {1} v _ {0} . . . . v _ {1})}}{\pi_ {v _ {1}} ^ {(v _ {0} v _ {1} . . . . v _ {0})}}, w = v _ {0}, v _ {1}. \quad \square \end{array}\]
In addition to the Y latent series of the auto-linear model (Dimitriou-Fakalou, 2010), for any , the variables Y as defined in Section 2 also follow a moving-average like pattern, in the sense that is independent of , (p-dependence). Due to the discretization within a finite number of categories and the extensive use of index variables, it is possible to compute joint probabilities of the Y variables. It is reminded that symmetry conditions for the original X modelled naturally in space, such as (15) when it holds, translate for the spatial latent equalizer predictors Y, such as (13), respectively. Nevertheless, the varying conditions under which the different Y series are considered could be making an essential difference to the various equalities, so the probabilities of Y or M should be computed with care. Below, there is a couple of examples of unobvious restrictions on the parameter probabilities of the spatial Y or M.
Probability restriction examples: We consider Bernoulli variables and order p = 1. We start with
\[\mathbb {P} (M _ {s} ^ {[ 0; 1 ]} = 1) = \pi_ {1} ^ {(0)} \pi_ {1} ^ {(1)} \neq \pi_ {1} ^ {(1)} (1 - \pi_ {1} ^ {(1)}) \equiv \mathbb {P} (M _ {s} ^ {[ 1; 0 ]} = 1),\tag{18}\]
as opposed to the special case of (13) when we know that
\[\begin{array}{r c l} \mathbb{P}(Y_{s}^{[0;1]} = 1) & = & \frac{\pi_1^{(0)} \pi_1^{(1)}}{\pi_1^{(0)} \pi_1^{(1)} + (1 - \pi_1^{(0)}) \pi_1^{(0)}} = \\& & \frac{\pi_1^{(1)} (1 - \pi_1^{(1)})}{\pi_1^{(1)} (1 - \pi_1^{(1)}) + (1 - \pi_1^{(1)}) (1 - \pi_1^{(0)})} \equiv \mathbb{P}(Y_{s}^{[1;0]} = 1):\end{array}\tag{19}\]
though the left-hand or right-hand side of (18) seems to be 'proportional' to the left-hand or right-hand, respectively, side of (19), an equality does not become apparent from (18); and are defined on different spaces resulting, here, in .
We continue with interest in moments (probabilities) of two random variables, and we try to discover analogues to the even auto-covariance function of the auto-linear Y latent series by Dimitriou-Fakalou (2010). Hence we consider the case and compare it to , in terms of probability: this seems to be the closest possible to reversing the order in the transect of the two M-variables, i.e. and .
An adequate space must be taking place for either case. In the first case, we write (' is one symbol), with probability of occurrence
\[\begin{array}{r c l} \mathbb{P}(mm_s^{[0;1][1;0]} = 1) & = & \pi_1^{(0)} [\mathbb{P}(I_{s+1}^{(1)} = 1) \mathbb{P}(I_{s+2}^{(1)} = 0) + \mathbb{P}(I_{s+1}^{(1)} = 0) \mathbb{P}(I_{s+2}^{(0)} = 0)] \\& = & \pi_1^{(0)} (1 - \pi_1^{(1)}) (\pi_1^{(1)} + 1 - \pi_1^{(0)}), \end{array}\]
and followed by
\[\begin{array}{r c l} \mathbb {P} (M _ {s} ^ {[ 0; 1 ]} = 1, M _ {s + 1} ^ {[ 1; 0 ]} = 1 \mid m m _ {s} ^ {[ 0; 1 ] [ 1; 0 ]} = 1) & \equiv & \frac {\mathbb {P} (I _ {s} ^ {(0)} = 1 , I _ {s + 1} ^ {(1)} = 1 , I _ {s + 2} ^ {(1)} = 0)}{\mathbb {P} (m m _ {s} ^ {[ 0 ; 1 ] [ 1 ; 0 ]} = 1)} \\& = & \frac {\pi_ {1} ^ {(1)}}{1 - \pi_ {1} ^ {(0)} + \pi_ {1} ^ {(1)}}. \end{array}\]
In the other hand, we write , and consider the events and . Observe that
\[\begin{array}{c} \mathbb {P} (M _ {s} ^ {[ 1; 0 ]} = 0, M _ {s + 1} ^ {[ 0; 1 ]} = 0 \mid m m _ {s} ^ {[ 1; 0 ] [ 0; 1 ]} = 1) = \frac {\mathbb {P} (I _ {s} ^ {(1)} = 0 , I _ {s + 1} ^ {(0)} = 0 , I _ {s + 2} ^ {(0)} = 1)}{\mathbb {P} (m m _ {s} ^ {[ 1 ; 0 ] [ 0 ; 1 ]} = 1)} = \\\frac {(1 - \pi_ {1} ^ {(1)}) (1 - \pi_ {1} ^ {(0)}) \pi_ {1} ^ {(0)}}{(1 - \pi_ {1} ^ {(1)}) \pi_ {1} ^ {(0)} [ \pi_ {1} ^ {(1)} + (1 - \pi_ {1} ^ {(0)}) ]} \equiv \frac {(1 - \pi_ {1} ^ {(0)})}{\pi_ {1} ^ {(1)} + 1 - \pi_ {1} ^ {(0)}}. \end{array}\]
Not only is the sum of the two conditional probabilities equal to unity (this might be attributed to the fact that the realizations 0 and 1 have to switch as well), but even more so it is true that
\[\begin{array}{r c l} \mathbb {P} (M _ {s} ^ {[ 0; 1 ]} = 1, M _ {s + 1} ^ {[ 1; 0 ]} = 1 \mid m m _ {s} ^ {[ 0; 1 ] [ 1; 0 ]} = 1) & = & \mathbb {P} (X _ {s} = 1 \mid X _ {s - 1} = 1, X _ {s + 1} = 0) \\& \equiv & \mathbb {P} (X _ {s} = 1 \mid X _ {s - 1} = 0, X _ {s + 1} = 1), \\\mathbb {P} (M _ {s} ^ {[ 1; 0 ]} = 0, M _ {s + 1} ^ {[ 0; 1 ]} = 0 \mid m m _ {s} ^ {[ 1; 0 ] [ 0; 1 ]} = 1) & = & \mathbb {P} (X _ {s} = 0 \mid X _ {s - 1} = 1, X _ {s + 1} = 0) \\& \equiv & \mathbb {P} (X _ {s} = 0 \mid X _ {s - 1} = 0, X _ {s + 1} = 1), \end{array}\]
((17) might be referred to) and the relation between the symmetrical conditional (from past and future) probabilities of X, translates to the joint probabilities of M-adjusted (Y-like variables). This basic result can be considered as one of a plethora of unobvious restrictions that do take place regarding the Y series dependence: those might concern joint probabilities of not necessarily just two variables (especially for ). Thanks to the model (4), with Y defined from (5), following M and m defined from the I series, there should be no danger that the restrictions might be overlooked by the spatial parameterization. The building block of the I series (hiding behind the m, M and Y) should make sure that any probability is computable and any relationship in the probabilities may reveal itself.
3.3 Time Reversal
It is easily demonstrated that once the pth order Markovian dependence resulting from the 'causal' representation in the transect takes place, the same holds for the unilateral representation from the other side: hence, for any , it holds that
\[\begin{array}{r l} & {\mathbb {P} (X _ {s} = x _ {0} \mid X _ {s + 1} = x _ {1}, \ldots , X _ {s + p} = x _ {p}, \ldots , X _ {s + p + n} = x _ {p + n}) =} \\& {\frac {\mathbb {P} (X _ {s} = x _ {0} , X _ {s + 1} = x _ {1} , \ldots , X _ {s + p} = x _ {p} , \ldots , X _ {s + p + n} = x _ {p + n})}{\mathbb {P} (X _ {s + 1} = x _ {1} , \ldots , X _ {s + p} = x _ {p} , \ldots , X _ {s + p + n} = x _ {p + n})} =} \\& {\qquad \frac {\mathbb {P} (X _ {s} = x _ {0} , X _ {s + 1} = x _ {1} , \ldots , X _ {s + p} = x _ {p})}{\mathbb {P} (X _ {s + 1} = x _ {1} , \ldots , X _ {s + p} = x _ {p})}.} \\& {\frac {\mathbb {P} (X _ {s + p + n} = x _ {p + n} , \ldots , X _ {s + p + 1} = x _ {p + 1} \mid X _ {s + p} = x _ {p} , \ldots , X _ {s + 1} = x _ {1} , X _ {s} = x _ {0})}{\mathbb {P} (X _ {s + p + n} = x _ {p + n} , \ldots , X _ {s + p + 1} = x _ {p + 1} \mid X _ {s + p} = x _ {p} , \ldots , X _ {s + 1} = x _ {1})},} \end{array}\]
where straight from the p-variate memoryless property (3), the last ratio is equal to unity, so that one can continue by writing
\[\frac {\mathbb {P} (X _ {s} = x _ {0} , X _ {s + 1} = x _ {1} , \ldots , X _ {s + p} = x _ {p})}{\mathbb {P} (X _ {s + 1} = x _ {1} , \ldots , X _ {s + p} = x _ {p})} = \mathbb {P} (X _ {s} = x _ {0} \mid X _ {s + 1} = x _ {1}, \ldots , X _ {s + p} = x _ {p}),\]
i.e. a causal satisfying (3) also satisfies
\[\mathbb {P} (X _ {s} = x _ {0} \mid X _ {s + 1} = x _ {1}, \ldots , X _ {s + p} = x _ {p}, \ldots , X _ {s + p + n} = x _ {p + n}) =\]
\[\mathbb {P} (X _ {s} = x _ {0} \mid X _ {s + 1} = x _ {1}, \dots , X _ {s + p} = x _ {p}), \text { for any } n \in \mathbb {N} _ {0}.\tag{20}\]
Additionally, a more general statement concerning a series of variables that occur holds as well, i.e.
\[\begin{array}{l} \mathbb{P} (X _ {s - n} = x _ {- n}, \ldots , X _ {s - 1} = x _ {- 1}, X _ {s} = x _ {0} | X _ {s + 1} = x _ {1}, \ldots , X _ {s + p} = x _ {p}) = \\\prod_ {m = 0} ^ {n} \mathbb{P} (X _ {s - m} = x _ {- m} | X _ {s - m + 1} = x _ {- m + 1}, \ldots , X _ {s - m + p} = x _ {- m + p}), \text{for any} n \in \mathbb{N} _ {0}. \end{array}\tag{21}\]
Thanks to (20) and (21), we may continue under the following assumption: there can also be sequences, say , of independent (on different points of the transect and from either sequence) and identically distributed variables, with marginal distributions
\[\rho_ {w} ^ {(j _ {1}, \dots , j _ {p})} := \mathbb {P} (J _ {s} ^ {(j _ {1}, \dots , j _ {p})} = w) \equiv \mathbb {P} (X _ {s} = w \mid X _ {s + 1} = j _ {1}, \dots , X _ {s + p} = j _ {p})\]
for , such that it can be written for any , that
\[X _ {s} \equiv J _ {s} ^ {(X _ {s + 1}, \dots , X _ {s + p})}.\tag{22}\]
Then it must hold that
\[\rho_{w}^{(j_{1},\ldots,j_{p})} \equiv \pi_{j_{p}}^{(j_{p-1},\ldots,j_{1},w)} \cdot \frac{\psi(j_{p-1},\ldots,j_{1},w)}{\psi(j_{p},\ldots,j_{1})}\]
where are the 'stationary' probabilities, which have been obtained as a unique solution to the linear system (i.e. satisfying)
\[\psi_{(i_1,\dots,i_p)} = \sum_w \pi^{(i_2,\dots,i_p,w)}_{i_1} \cdot \psi_{(i_2,\dots,i_p,w)}\]
The linear system
\[\boldsymbol{\psi}_{(i_1,\dots,i_p)}^{*} = \sum_{w} \rho_{i_p}^{(i_{p-1},\dots,i_1,w)} \cdot \boldsymbol{\psi}_{(w,i_1,\dots,i_{p-1})}^{*}\]
must yield the same stationary distribution to be the unique solution: by substitution in the system above of according to (23), it is easy to verify that is a solution indeed. Additionally, must be a 'causal' (from the 'future') process based on (see Dimitriou-Fakalou (2019)).
Following an identical argument as from the I to the Y, it can be written
\[n _ {s} ^ {[ \mathbf {i} _ {F} ^ {\tau}; \mathbf {i} _ {P} ^ {\tau} ]} := \sum_ {w = 0, v _ {1}, \ldots , v _ {k}} f _ {w} (J _ {s} ^ {(\mathbf {i} _ {F})}) \cdot f _ {i _ {P, 1}} (J _ {s - 1} ^ {(w, i _ {F, 1}, \ldots , i _ {F, p - 1})}) \dots f _ {i _ {P, p}} (J _ {s - p} ^ {(i _ {P, p - 1}, \ldots , i _ {P, 1}, w)})\]
and
\[N_{s}^{[\mathbf{i}_{F}^\tau; \mathbf{i}_{P}^\tau]} := J_{s}^{(\mathbf{i}_{F})} \cdot f_{i_{P,1}} (J_{s-1}^{(J_{s}^{(\mathbf{i}_{F})}, i_{F,1}, \dots , i_{F,p-1})}) \dots f_{i_{P,p}} (J_{s-p}^{(i_{P,p-1}, \dots , i_{P,1}, J_{s}^{(\mathbf{i}_{F})})})\]
so that it is defined
\[n _ {s} ^ {[ \mathbf {i} _ {F} ^ {\tau}; \mathbf {i} _ {P} ^ {\tau} ]} \cdot Y _ {s} ^ {* [ \mathbf {i} _ {F} ^ {\tau}; \mathbf {i} _ {P} ^ {\tau} ]} := N _ {s} ^ {[ \mathbf {i} _ {F} ^ {\tau}; \mathbf {i} _ {P} ^ {\tau} ]},\tag{24}\]
i.e. is defined on the sample space . Then it must hold that
\[X_{s} = Y_{s}^{* [ (X_{s+p},\dots,X_{s+1});(X_{s-1},\dots,X_{s-p}) ]} \equiv Y_{s}^{[ (X_{s-p},\dots,X_{s-1});(X_{s+1},\dots,X_{s+p}) ]}.\]
Equation (25) sets but only provided that .
In fact, regardless of the values of , , it could be considered for every and , whether it holds that ; then, provided that
\[\mathbb {P} (Y _ {s} ^ {[ \mathbf {i} _ {P}; \mathbf {i} _ {F} ]} = Y _ {s} ^ {* [ \mathbf {i} _ {F} ^ {\tau}; \mathbf {i} _ {P} ^ {\tau} ]} = w \mid m _ {s} ^ {[ \mathbf {i} _ {P}; \mathbf {i} _ {F} ]} = n _ {s} ^ {[ \mathbf {i} _ {F} ^ {\tau}; \mathbf {i} _ {P} ^ {\tau} ]} \equiv 1) =\]
\[\mathbb {P} (I _ {s} ^ {(\mathbf {i} _ {P} ^ {\tau})} = J _ {s} ^ {(\mathbf {i} _ {F})} = w \mid m _ {s} ^ {[ \mathbf {i} _ {P}; \mathbf {i} _ {F} ]} = n _ {s} ^ {[ \mathbf {i} _ {F} ^ {\tau}; \mathbf {i} _ {P} ^ {\tau} ]} \equiv 1) =\]
\[\mathbb {P} (X _ {s} = w \mid (X _ {s - 1}, \ldots , X _ {s - p}) = \mathbf {i} _ {P} ^ {\tau}, (X _ {s + 1}, \ldots , X _ {s + p}) = \mathbf {i} _ {F})\tag{26}\]
holds for all , it would be fair to conclude that (defined on the intersection of sample spaces) has all the elements, to take the role of the (conditional) latent equalizer predictor of given , , and , : as it has been explained already, (properly defined) is independent of , and, for the same reasons, (properly defined) would be independent of , . It is mandatory for every , that for at least one out of the possible , , , it holds that and and , together with
\[\begin{array}{r l} & {\mathbb {P} (Y _ {s} ^ {[ (X _ {s - p}, \ldots , X _ {s - 1}); (X _ {s + 1}, \ldots , X _ {s + p}) ]} = Y _ {s} ^ {* [ (X _ {s + p}, \ldots , X _ {s + 1}); (X _ {s - 1}, \ldots , X _ {s - p}) ]} = w \mid X _ {s - i}, i \neq 0) =} \\& {\qquad \mathbb {P} (Y _ {s} ^ {[ (X _ {s - p}, \ldots , X _ {s - 1}); (X _ {s + 1}, \ldots , X _ {s + p}) ]} = w \mid X _ {s - i}, i \neq 0) \equiv} \\& {\qquad \mathbb {P} (Y _ {s} ^ {* [ (X _ {s + p}, \ldots , X _ {s + 1}); (X _ {s - 1}, \ldots , X _ {s - p}) ]} = w \mid X _ {s - i}, i \neq 0);} \end{array}\]
it remains a question to be answered though, whether (26) can or should be attempted to hold. Formally, one would be able to consider for every , that the latent equalizer predictor of based on , , is and should proceed conditionally via Y or .
IV. AN INTRODUCTION TO THE TALMA
Write (and ), and consider to be a series of independent and identically distributed -vectors , which will be used as a building block for the series of interest. Write
\[\pi_ {x} ^ {(\mathbf {i} _ {P} ^ {\tau} | \mathbf {j} _ {P} ^ {\tau})} := \mathbb {P} (I _ {s} ^ {(\mathbf {i} _ {P} ^ {\tau} | \mathbf {j} _ {P} ^ {\tau})} = x), x = 0, v _ {1}, \ldots , v _ {k}, \sum_ {x = 0, v _ {1}, \ldots , v _ {k}} \pi_ {x} ^ {(\mathbf {i} _ {P} ^ {\tau} | \mathbf {j} _ {P} ^ {\tau})} = 1,\]
as well as and
Definition 1: will be called a Table Auto-Linear Moving-Average process of order and it will be written , if it holds that
\[X _ {s} := Y _ {s} ^ {\left([ (X _ {s - p}, \dots , X _ {s - 1}); (X _ {s + 1}, \dots , X _ {s + p}) ] | [ (Y _ {s - q}, \dots , Y _ {s - 1}); (Y _ {s + 1}, \dots , Y _ {s + q}) ]\right)}, s \in \mathbb {Z},\tag{27}\]
paired with
\[Y _ {s} := y _ {s} ^ {\left([ (X _ {s - p}, \dots , X _ {s - 1}); (X _ {s + 1}, \dots , X _ {s + p}) ] | [ (Y _ {s - q}, \dots , Y _ {s - 1}); (Y _ {s + 1}, \dots , Y _ {s + q}) ]\right)}, s \in \mathbb {Z},\]
where, for every , , and , it is set
\[\begin{array}{r c l} m _ {s} ^ {([ \mathbf {i} _ {P}; \mathbf {i} _ {F} ] | [ \mathbf {j} _ {P}; \mathbf {j} _ {F} ])} & := & f _ {j _ {F, 1}} (I _ {s + 1}) \cdot \ldots \cdot f _ {j _ {F, q}} (I _ {s + q}) \cdot \\& & f _ {i _ {F, 1}} (I _ {s + 1} ^ {((I _ {s} ^ {((\mathbf {i} _ {P} ^ {\tau} | \mathbf {j} _ {P} ^ {\tau}))}, i _ {P, 1}, \ldots , i _ {P, p - 1}) | (I _ {s}, j _ {P, 1}, \ldots , j _ {P, q - 1}))}) \cdot \ldots \cdot \\& & f _ {i _ {F, p}} (I _ {s + p} ^ {((i _ {F, p - 1}, \ldots , i _ {F, 1}, I _ {s} ^ {(\mathbf {i} _ {P} ^ {\tau} | \mathbf {j} _ {P} ^ {\tau})}) | (I _ {s + p - 1} (\mathrm{?} = j _ {F}), \ldots , j _ {F, 1}, I _ {s}, j _ {P, 1}, \ldots , j _ {P}))}), \end{array}\tag{28}\]
followed by defining on the sample space only, the variables
\[y _ {s} ^ {([ \mathbf {i} _ {P}; \mathbf {i} _ {F} ] | [ \mathbf {j} _ {P}; \mathbf {j} _ {F} ])} := I _ {s}, \quad \text { and } \quad Y _ {s} ^ {([ \mathbf {i} _ {P}; \mathbf {i} _ {F} ] | [ \mathbf {j} _ {P}; \mathbf {j} _ {F} ])} := I _ {s} ^ {(\mathbf {i} _ {P} ^ {\tau} | \mathbf {j} _ {P} ^ {\tau})}.\]
- Definition 1 presents the spatial analogue to the Table ARMA equation (see Dimitriou-Fakalou (2019)), which models the infinite Markovian dependence (see Dimitriou-Fakalou (2022)), in the same way that the standard time series ARMA is an . Indeed it can be verified for the process of interest that , : it is a prerequisite that this is causal, implying strictly stationary, (as in Dimitriou-Fakalou (2019)) and invertible (as in Dimitriou-Fakalou (2022)). Additionally, the main latent process converts into , which shares the same realizations but, each time, the -value is picked from a different probability rule, depending on the fixed values of and from both sides. Note that, if (28) is re-written as
\[\begin{array}{l} f _ {j _ {F, 1}} (Y _ {s + 1}) \cdot \ldots \cdot f _ {j _ {F, q}} (Y _ {s + q}) \cdot \\f _ {i _ {F, 1}} (I _ {s + 1} ^ {((I _ {s} ^ {((\mathbf {i} _ {P} ^ {\tau} | \mathbf {j} _ {P} ^ {\tau}))}, i _ {P, 1}, \dots , i _ {P, p - 1}) | (I _ {s, j _ {P, 1}}, \dots , j _ {P, q - 1}))}) \cdot \ldots \cdot \\f _ {i _ {F, p}} (I _ {s + p} ^ {((i _ {F, p - 1}, \dots , i _ {F, 1}, I _ {s} ^ {(\mathbf {i} _ {P} ^ {\tau} | \mathbf {j} _ {P} ^ {\tau})}) | (I _ {s + p - 1} (\mathrm{?} = j _ {F}), \dots , j _ {F, 1}, I _ {s}, j _ {P, 1}, \dots , j _ {P}))}), \end{array}\]
then the definition of resembles that of a TARMA caused from the future, though the comparisons should be done with care (this is due to merging the different y processes into one Y for the 'AR' part and considering a p-dependence for the 'MA' part). Still, it should be clear that and , , for positive (and valid) l, are independent (in the same way that and , l > 0, are independent).
To exhibit the independence of and , , l > 0, some form of time reversal would be required, i.e. writing as a 'causal' (from the future) function of another sequence of, say J, iid variables, with the same variables from the past only determining the values of y, Y. Nevertheless, not only would time reversal be expected to be an arduous (if not impossible) task, but also it is not really needed, in order to solidify the vital property of the TALMA model (as in (29) and the statement following it, which conclude this point).
So without resorting to any other than the I building blocks, first see that
\[\{X _ {s - 1} = i _ {P, 1}, \ldots , X _ {s - p} = i _ {P, p}, X _ {s + 1} = i _ {F, 1}, \ldots , X _ {s + p} = i _ {F, p}\}\]
\[\begin{array}{c} I _ {s - 1} = j _ {P, 1}, \ldots , I _ {s - q} = j _ {P, q}, I _ {s + 1} = j _ {F, 1}, \ldots , I _ {s + q} = j _ {F, q} \} \subseteq \\ \{m _ {s} ^ {([ \mathbf {i} _ {P}; \mathbf {i} _ {F} ] | [ \mathbf {j} _ {P}; \mathbf {j} _ {F} ])} = 1 \}. \end{array}\]
Secondly, it is easy to spot
\[\mathbb {P} (X _ {s} = x, Y _ {s} = y \mid X _ {s - 1} = i _ {P, 1}, \ldots , X _ {s - p} = i _ {P, p}, X _ {s + 1} = i _ {F, 1}, \ldots , X _ {s + p} = i _ {F, p},\]
\[I _ {s - 1} = j _ {P, 1}, \dots , I _ {s - q} = j _ {P, q}, I _ {s + 1} = j _ {F, 1}, \dots , I _ {s + q} = j _ {F, q}) =\]
\[\mathbb {P} \big (I _ {s} ^ {(\mathfrak {i} _ {P} ^ {\tau} | \mathfrak {j} _ {P} ^ {\tau})} = x, I _ {s} = y \quad | \quad X _ {s - 1} = i _ {P, 1}, \ldots , X _ {s - p} = i _ {P, p}, I _ {s - 1} = j _ {P, 1}, \ldots , I _ {s - q} = j _ {P, q},\]
\[m _ {s} ^ {([ \mathbf {i} _ {P}; \mathbf {i} _ {F} ] | [ \mathbf {j} _ {P}; \mathbf {j} _ {F} ])} = 1) \equiv\]
\[\mathbb {P} (I _ {s} ^ {(\mathbf {i} _ {P} ^ {\tau} | \mathbf {j} _ {P} ^ {\tau})} = x, I _ {s} = y \quad | \quad m _ {s} ^ {([ \mathbf {i} _ {P}; \mathbf {i} _ {F} ] | [ \mathbf {j} _ {P}; \mathbf {j} _ {F} ])} = 1),\]
where the last equality is due to the fact that , , l, n > 0 are jointly independent of , . Identical arguments write, for any , that
\[\begin{array}{r c l} \mathbb {P} (X _ {s} = x, Y _ {s} = y & | & X _ {s - 1} = i _ {P, 1}, \ldots , X _ {s - p} = i _ {P, p}, X _ {s + 1} = i _ {F, 1}, \ldots , X _ {s + p} = i _ {F, p}, \\& & I _ {s - 1} = j _ {P, 1}, \ldots , I _ {s - q} = j _ {P, q}, I _ {s + 1} = j _ {F, 1}, \ldots , I _ {s + q} = j _ {F, q}, \\& & X _ {s - p - l}, l = 1, \ldots , l _ {1}, I _ {s - q - m}, m = 1, \ldots , m _ {1}) = \end{array}\]
\[\mathbb {P} (I _ {s} ^ {(\mathbf {i} _ {P} ^ {\tau} | \mathbf {j} _ {P} ^ {\tau})} = x, I _ {s} = y \quad | \quad m _ {s} ^ {([ \mathbf {i} _ {P}; \mathbf {i} _ {F} ] | [ \mathbf {j} _ {P}; \mathbf {j} _ {F} ])} = 1).\]
Thirdly, set for convenience , , , and slightly shrink the sample space to , where
\[\begin{array}{r c l} m _ {s} ^ {* ([ \mathbf {i} _ {P}; \mathbf {i} _ {F} ^ {*} ] | [ \mathbf {j} _ {P}; \mathbf {j} _ {F} ^ {*} ])} & := & f _ {j _ {F, 1}} (I _ {s + 1}) \cdot \ldots \cdot f _ {j _ {F, o}} (I _ {s + o}) \cdot \\& & f _ {i _ {F, 1}} (I _ {s + 1} ^ {((I _ {s} ^ {((\mathbf {i} _ {P} ^ {\tau} | \mathbf {j} _ {P} ^ {\tau}))}, i _ {P, 1}, \ldots , i _ {P, p - 1}) | (I _ {s}, j _ {P, 1}, \ldots , j _ {P, q - 1}))}) \cdot \ldots \cdot \\& & f _ {i _ {F, p}} (I _ {s + p} ^ {((i _ {F, p - 1}, \ldots , i _ {F, 1}, I _ {s} ^ {(\mathbf {i} _ {P} ^ {\tau} | \mathbf {j} _ {P} ^ {\tau})}) | (j _ {F, p - 1}, \ldots , j _ {F, 1}, I _ {s}, j _ {P, 1}, \ldots , j _ {P}))}) \cdot \ldots \cdot \\& & f _ {i _ {F, o}} (I _ {s + o} ^ {((i _ {F, o - 1}, \ldots) | (j _ {F, o - 1}, \ldots , j _ {F, 1}, I _ {s}))}) \end{array}\]
(the first line fixes the case , so that exists, and the last line takes care of when , so that is the last entry in the MA part), so that it becomes clear why it holds that
\[\begin{array}{r l} \mathbb {P} (I _ {s} ^ {(\mathbf {i} _ {P} ^ {\tau} | \mathbf {j} _ {P} ^ {\tau})} = x, I _ {s} = y \quad | \quad X _ {s - 1} = i _ {P, 1}, \ldots , X _ {s - p} = i _ {P, p}, I _ {s - 1} = j _ {P, 1}, \ldots , I _ {s - q} = j _ {P, q}, \\m _ {s} ^ {* ([ \mathbf {i} _ {P}; \mathbf {i} _ {F} ^ {*} ] | [ \mathbf {j} _ {P}; \mathbf {j} _ {F} ^ {*} ])} = 1) \equiv \end{array}\]
\[\begin{array}{r l} \mathbb {P} (I _ {s} ^ {(\mathbf {i} _ {P} ^ {\tau} | \mathbf {j} _ {P} ^ {\tau})} = x, I _ {s} = y \mid X _ {s - 1} = i _ {P, 1}, \ldots , X _ {s - p} = i _ {P, p}, I _ {s - 1} = j _ {P, 1}, \ldots , I _ {s - q} = j _ {P, q}, \\m _ {s} ^ {* ([ \mathbf {i} _ {P}; \mathbf {i} _ {F} ^ {*} ] | [ \mathbf {j} _ {P}; \mathbf {j} _ {F} ^ {*} ])} = 1, \end{array}\]
\[X _ {s + o + n}, n = 1, \ldots , n _ {1}, I _ {s + o + n ^ {*}}, n ^ {*} = 1, \ldots , n _ {2}),\]
for any : a similar relation can be found in Section 3.1.
To sum it up, the TALMA model for , is such that the probability
\[\mathbb {P} (X _ {s} = x, Y _ {s} = y \mid X _ {s - 1} = i _ {P, 1}, \ldots , X _ {s - p} = i _ {P, p}, X _ {s - p - l}, l = 1, \ldots , l _ {1},\]
\[Y _ {s - 1} = j _ {P, 1}, \ldots , Y _ {s - q} = j _ {P, q}, Y _ {s - q - m}, m = 1, \ldots , m _ {1},\]
\[X _ {s + 1} = i _ {F, 1}, \ldots , X _ {s + o} = i _ {F, o}, X _ {s + o + n}, n = 1, \ldots , n _ {1},\tag{29}\]
\[Y _ {s + 1} = j _ {F, 1}, \ldots , Y _ {s + o} = j _ {F, o}, Y _ {s + o + n ^ {*}}, n ^ {*} = 1, \ldots , n _ {2}),\]
remains unchanged for any .
The adjustment from to (which is not the case for the linear series) might be attributed to the fact that the model is multiplicative on the AL and MA parts, i.e. those two are tangled together. 2. It is clarified next how to employ the special merit of the TALMA equation, which has been attached to the use of the partitioned latent series. When , the process is none other than what was described in Section 3. It is accustomed that the inclusion of the moving-average part with order , offers a parsimonious way to model the infinite Markovian dependence: its spatial version is what the TALMA is all about.
To demonstrate how to approximate it, first for any , write the 'conditions'
\[\mathcal {P} _ {s, (n)} \left(x _ {- \mu}, \mu = (n + 1), \dots , (n + p), y _ {- \nu}, \nu = (n + 1), \dots , (n + q)\right) :=\]
\[\{X _ {s - \mu} = x _ {- \mu}, \mu = (n + 1), \dots , (n + p), Y _ {s - \nu} = y _ {- \nu}, \nu = (n + 1), \dots , (n + q) \},\]
and
\[\mathcal {F} _ {s, (n)} (x _ {m}, y _ {m}, m = (n + 1), \ldots , (n + o)) :=\]
\[\{X _ {s + m} = x _ {m}, Y _ {s + m} = y _ {m}, m = (n + 1), \ldots , (n + o) \}.\]
Then define the probabilities
\[\omega_ {(x _ {0}, y _ {0})} ^ {([ (x _ {- p}, \ldots , x _ {- 1}); (x _ {1}, \ldots , x _ {o}) ] | [ (y _ {- q}, \ldots , y _ {- 1}); (y _ {1}, \ldots , y _ {o}) ])} := \mathbb {P} \big (X _ {s} = x _ {0}, Y _ {s} = y _ {0} \mid \mathcal {P} _ {s, (0)}, \mathcal {F} _ {s, (0)} \big).\]
More specifically, for any , the quantity
\[\begin{array}{l} \mathbb {P} (X _ {s} = x _ {0} \mid X _ {s + m} = x _ {m}, m = \pm 1, \dots , \pm n, X _ {s + \mu} = x _ {\mu}, \mu = - (n + p), \dots \\- (n + 1), (n + 1), \dots , (n + o), Y _ {s + \nu} = y _ {\nu}, \nu = - (n + q), \dots , - (n + 1), \\(n + 1), \dots , (n + o)) \end{array} \tag {30}\]
will be of interest. To compute (30), first re-write it as
\[\{\sum_{y_m, m = 0, \pm 1, \ldots, \pm n} \mathbb{P}(X_s = x_0, Y_s = y_0, X_{s + m} = x_m, Y_{s + m} = y_m, m = \pm 1, \ldots, \pm n)\}\]
\[X _ {s + \mu} = x _ {\mu}, \mu = - (n + p), \ldots , - (n + 1), (n + 1), \ldots , (n + o),\]
\[Y _ {s + \nu} = y _ {\nu}, \nu = - (n + q), \dots , - (n + 1), (n + 1), \dots , (n + o))\]
\[\{\sum \mathbb{P}(X_{s} = x_{0}^{*}, Y_{s} = y_{0}, X_{s + m} = x_{m}, Y_{s + m} = y_{m}, m = \pm 1, \ldots, \pm n \mid \text{some condition})\]
\[X _ {s + \mu} = x _ {\mu}, \mu = - (n + p), \ldots , - (n + 1), (n + 1), \ldots , (n + o),\]
\[Y _ {s + \nu} = y _ {\nu}, \nu = - (n + q), \ldots , - (n + 1), (n + 1), \ldots , (n + o));\]
then observe the important relation
\[\frac {\mathbb {P} (X _ {s + m} = x x _ {m} , Y _ {s + m} = y y _ {m} , m = 0 , \pm 1 , \ldots , \pm n \mid \mathcal {P} _ {s , (n)} , \mathcal {F} _ {s , (n)})}{\mathbb {P} (X _ {s + m} = x x _ {m} ^ {*} , Y _ {s + m} = y y _ {m} ^ {*} , m = 0 , \pm 1 , \ldots , \pm n \mid \mathcal {P} _ {s , (n)} , \mathcal {F} _ {s , (n)})} = \prod_ {i = - n} ^ {n}\]
\[\{\mathbb{P}(X_{s + i} = x x_{i}, Y_{s + i} = y y_{i} \mid \mathcal{P}_{s, (n)}, X_{s + j} = x x_{j}, Y_{s + j} = y y_{j}, j = -n)\]
\[\ldots , i - 1, X _ {s + j ^ {*}} = x x _ {j ^ {*}} ^ {*}, Y _ {s + j ^ {*}} = y y _ {j ^ {*}} ^ {*}, j ^ {*} = i + 1, \ldots , n, \mathcal {F} _ {s, (n)})\]
\[\{\mathbb{P}(X_{s + i} = x x_i^*, Y_{s + i} = y y_i^* \mid \mathcal{P}_{s, (n)}, X_{s + j} = x x_j, Y_{s + j} = y y_j, j = -n)\]
\[\ldots , i - 1, X _ {s + j ^ {*}} = x x _ {j ^ {*}} ^ {*}, Y _ {s + j ^ {*}} = y y _ {j ^ {*}} ^ {*}, j ^ {*} = i + 1, \ldots , n, \mathcal {F} _ {s, (n)}) \}\tag{31}\]
which finds its roots in Besag (1974) together with a sketch of proof. Indeed, by taking its right-hand side, it can be re-written
\[\begin{array}{l} \prod_ {i = - n} ^ {n} \{\mathbb {P} (X _ {s + j} = x x _ {j}, Y _ {s + j} = y y _ {j}, j = - n, \ldots , i, \\X _ {s + j ^ {*}} = x x _ {j ^ {*}} ^ {*}, Y _ {s + j ^ {*}} = y y _ {j ^ {*}} ^ {*}, j ^ {*} = i + 1, \ldots , n | \mathcal {P} _ {s, (n)}, \mathcal {F} _ {s, (n)}) \} / \\\{\mathbb {P} (X _ {s + j} = x x _ {j}, Y _ {s + j} = y y _ {j}, j = - n, \ldots , i - 1, \\X _ {s + j ^ {*}} = x x _ {j ^ {*}} ^ {*}, Y _ {s + j ^ {*}} = y y _ {j ^ {*}} ^ {*}, j ^ {*} = i, \ldots , n | \mathcal {P} _ {s, (n)}, \mathcal {F} _ {s, (n)}) \}; \end{array}\]
for , the factor becomes
\[\{\mathbb{P}(X_{s - n} = x x_{- n}, Y_{s - n} = y y_{- n}, X_{s + j^{*}} = x x_{j^{*}}^{*}, Y_{s + j^{*}} = y y_{j^{*}}^{*})\]
\[j ^ {*} = - n + 1, \dots , n \mid \left( \mathcal {P} _ {s, (n)}, \mathcal {F} _ {s, (n)} \right) \right\}\]
\[\{\mathbb {P} (X _ {s + j ^ {*}} = x x _ {j ^ {*}} ^ {*}, Y _ {s + j ^ {*}} = y y _ {j ^ {*}} ^ {*}, j ^ {*} = - n, \dots , n \mid \mathcal {P} _ {s, (n)}, \mathcal {F} _ {s, (n)}) \}\]
with its denominator being exactly what is wanted and the numerator will cancel with the denominator of the next factor (for ), which is altogether
\[\{\mathbb{P}(X_{s-n} = x x_{-n}, Y_{s-n} = y y_{-n}, X_{s-n+1} = x x_{-n+1}, Y_{s-n+1} = y y_{-n+1})\}\]
\[X _ {s + j ^ {*}} = x x _ {j ^ {*}} ^ {*}, Y _ {s + j ^ {*}} = y y _ {j ^ {*}} ^ {*}, j ^ {*} = - n + 2, \dots , n \mid \mathcal {P} _ {s, (n)}, \mathcal {F} _ {s, (n)})\]
\[\{\mathbb{P}(X_{s - n} = x x_{- n}, Y_{s - n} = y y_{- n}, X_{s + j^{*}} = x x_{j^{*}}^{*}, Y_{s + j^{*}} = y y_{j^{*}}^{*})\]
\[j ^ {*} = - n + 1, \ldots , n \mid \mathcal {P} _ {s, (n)}, \left. \mathcal {F} _ {s, (n)} \right)\]
and so on. Similarly, the one before the end factor
\[\{\mathbb{P}(X_{s+j}=x, Y_{s+j}=y) \mid j=-n,\ldots,n-1\}\]
\[X _ {s + n} = x x _ {n} ^ {*}, Y _ {s + n} = y y _ {n} ^ {*} \mid \mathcal {P} _ {s, (n)}, \mathcal {F} _ {s, (n)}) \}\]
\[\{\mathbb{P}(X_{s+j}=x x_j, Y_{s+j}=y y_j, j=-n,\ldots,n-2, X_{s+n-1}=x x_{n-1}^{*})\]
\[Y _ {s + n - 1} = y y _ {n - 1} ^ {*}, X _ {s + n} = x x _ {n} ^ {*}, Y _ {s + n} = y y _ {n} ^ {*} \mid \mathcal {P} _ {s, (n)}, \mathcal {F} _ {s, (n)})\]
will have its denominator cancelled with the numerator of the previous factor and its numerator cancelled with the denominator of the last one for i = n, which is
\[\{\mathbb {P} (X _ {s + j} = x x _ {j}, Y _ {s + j} = y y _ {j}, j = - n, \ldots , n | \mathcal {P} _ {s, (n)}, \mathcal {F} _ {s, (n)}) \} /\]
\[\{\mathbb{P}(X_{s+j}=x, Y_{s+j}=y) \mid j=-n,\ldots,n-1\}\]
\[X _ {s + n} = x x _ {n} ^ {*}, Y _ {s + n} = y y _ {n} ^ {*} \mid \mathcal {P} _ {s, (n)}, \mathcal {F} _ {s, (n)})\]
as for the numerator of the last part above, together with the denominator of the first factor, those are combined to verify exactly the statement under question.
Once (31) is valid, given the same condition , the distribution
\[\mathbb {P} (X _ {s + m} = x x _ {m}, Y _ {s + m} = y y _ {m}, m = 0, \pm 1, \ldots , \pm n | \mathcal {P} _ {s, (n)}, \mathcal {F} _ {s, (n)}),\]
i.e. for any , , , can be determined: this should be obvious how. As for the right-hand side of (31), thanks to the conclusion regarding the probability (29) remaining unchanged, it suffices to know the probabilities to have at hand all that is required for it, then for (30). Straight from (30), the infinite spatial Markovian dependence is implied as , and the result is always contained within the prespecified TALMA parameters . 3. The parameters relating to Definition 1 are none other than the probabilities , and those should be expressed in terms of the original , i.e.
\[\begin{array}{r l} & {\omega_ {(x, y)} ^ {([ \mathbf {i} _ {P}; \mathbf {i} _ {F} ^ {*} ] | [ \mathbf {j} _ {P}; \mathbf {j} _ {F} ^ {*} ])} \equiv \mathbb {P} (I _ {s} ^ {(\mathbf {i} _ {P} ^ {\tau} | \mathbf {j} _ {P} ^ {\tau})} = x, I _ {s} = y \mid m _ {s} ^ {* ([ \mathbf {i} _ {P}; \mathbf {i} _ {F} ^ {*} ] | [ \mathbf {j} _ {P}; \mathbf {j} _ {F} ^ {*} ])} = 1)} \\& {\qquad = \mathbb {P} (I _ {s} ^ {(\mathbf {i} _ {P} ^ {\tau} | \mathbf {j} _ {P} ^ {\tau})} = x, I _ {s} = y \mid I _ {s + 1} = j _ {F, 1}, \ldots , I _ {s + o} = j _ {F, o},} \\& {I _ {s + 1} ^ {((I _ {s} ^ {(\mathbf {i} _ {P} ^ {\tau} | \mathbf {j} _ {P} ^ {\tau})}, i _ {P, 1}, \ldots , i _ {P, p - 1}) | (I _ {s}, j _ {P, 1}, \ldots , j _ {P, q - 1}))} = i _ {F, 1}, \ldots , I _ {s + o} ^ {((i _ {F, o - 1}, \ldots) | (j _ {F, o - 1}, \ldots , j _ {F, 1}, I _ {s}))} = i _ {F, o}):} \end{array}\]
particular emphasis must be paid on making sure that the condition considered each time has a strictly positive probability of occurring (this can be referred to as positivity condition). For example, when it is set (i.e. ) this has to be accounted for in the condition first. Other cases rely on the interdependence of and , referring to variables at the same point (rather than the independence of I when considered at different points of the transect): nothing has been said about that here, but the fixes for the causality (or invertibility) of the process usually demand that the distributions of the different must be really near (see Dimitriou-Fakalou (2019) and Dimitriou-Fakalou (2022)).
V. FINAL POINTS: WHY TALMA?
Given the correspondence from the causal TARMA to the TALMA equation, as well as the fact that it is the usual tack, from a sample of observed values to derive optimal inference results via step-by-step projections on the past available, one might wonder, why bother with the TALMA model?
The reasons can be counted on three fingers, the first one being that it might be desirable to understand how the (stationary) spatial dependencies extend. Secondly, it might be preferable to naturally clothe the spatial relations using the TALMA not TARMA equation, when the variables of interest are indexed on more than one coordinates, mainly, in order to avoid the conventional (half-plane) unilateral order of Whittle (1954): instead of writing the conditional probability of one value based on its 'previous' ones, all (other than itself) values can contribute to the information. Finally, there is the practical reason of prediction in spatial settings: for example, the equation provides directly with the whole distribution of the variable of interest based on values from a finite number of 'neighbours' from all around, when additional information can be discarded.
In the cases of multi-indexed discretized stationary processes, it is still advised that simultaneous spatial models would better not be attempted directly: the previous care of causal arrangements in the lattice that will, nevertheless, remain hidden, is essential for the validity of the spatial considerations.
