5. Stochastic Reasoning

Nature loves to hide.

Already briefly introduced in Section 1.​5.​3.

What is asserted by implication is that \( \neg ({\chi_{{{{\mathbf{\mathit{p}}}}_i}}} \wedge (\neg {\psi_{{{{\mathbf{\mathit{p}}}}_j}}})) \), i.e., it is not the case that \( {\chi_{{{{\mathbf{\mathit{p}}}}_i}}} \) and not \( {\psi_{{{{\mathbf{\mathit{p}}}}_j}}} \).

Equivalence is a strong logic operator that means the same as \( ({\chi_{{{{\mathbf{\mathit{p}}}}_i}}} \to {\psi_{{{{\mathbf{\mathit{p}}}}_j}}}) \wedge ({\psi_{{{{\mathbf{\mathit{p}}}}_j}}} \to {\chi_{{{{\mathbf{\mathit{p}}}}_i}}}) \).

Otherwise said, these are realizations of the spatiotemporal random field model (Section 5.3 below).

Whereas in induction the conclusion goes beyond, i.e. “amplifies,” the content of premises (see below).

Heuristics are short-cut solutions to a problem, which, while are often quick and easy (or, in some cases, “quick and dirty”) to implement, do not guarantee a correct solution.

Region in Brazil that has been the theater of NASA’s field-data acquisition campaigns.

In mathematical terms, stochastic reasoning views the temperature attribute as a spatiotemporal random field, see Section 5.3 that follows.

“Known” is here associated with human consciousness.

To use a real life scenario, since there are so many conditions that need to be satisfied simultaneously for Tiberius Finamore to be the only survivor of an airplane crash, the fact that he survived makes it tempting to believe that there was another reason that made his survival highly probable a priori (say, God likes Tiberius) rather than to admit that his chances to survive were indeed extremely small (there are many possible realizations that did not favor Tiberius’ survival but are ignored).

Europe’s ruling elites (cliques would be nearer the mark) are engaged in policies that go so radically against the wishes of ordinary citizens that the rift is widening between the people and the governing elite.

On the contrary, determinism describes events as inevitable, effectively depriving humans of a future.

Remarkably, a similar attitude toward human laws was exhibited in President Lyndon Johnson’s blast to a European ambassador in the 1960s (Wittner 1982: 303): “F*** your Parliament and your Constitution … If your Prime Minister gives me talk about Democracy, Parliament and Constitution, he, his Parliament and his Constitution may not last very long.”

Note that since Eq. (5.10) is deterministic, the solution \( {X_{{\mathbf{\mathit{p}}}}} = {\chi_{{\mathbf{\mathit{p}}}}} \) has probability 1. This is not the case, however, with Eq. (5.12).

A detailed discussion of the underlying physics is beyond the scope of the present discussion, but the interested reader is referred to the numerous volumes on quantum mechanics (e.g., Messiah 1999).

In this sense, the multivariate PDF behaves like the Holy Grail of the legends of the questing king Arthur’s knights, which assumed different shapes, forms, origins, and interpretations.

The reader is reminded that stochastic (or probabilistic) independence is different from logical independence (Table 4.​2).

For example, the Gaussian, Gamma, or Poisson univariate PDF is associated with Hermite, Laguerre, or Charlier polynomials.

The situation may worsen if the theorist happens to be Greek.

I am pure of this blood.

While Newton’s laws deal with actual positions and velocities, the Schrödinger law essentially describes the evolution of probabilities.

The core and the specificatory knowledge bases, G-KB and S-KB, respecitvely, were introduced in Section 1.​2.​3, and are discussed in detail in the Section 3.​6.

The drunk looks for the lost keys only under a sidewalk lamp because this is where the best light is.

It is easily seen that the attribute variance, \( \sigma_{X;\,{{{\mathbf{\mathit{p}}}}_i}}^2 \), is obtained from \( {c_{X;\,{{{\mathbf{\mathit{p}}}}_i},\,{{{\mathbf{\mathit{p}}}}_j}}} \) if we let \( {{{\mathbf{\mathit{p}}}}_i} = {{{\mathbf{\mathit{p}}}}_j} \).

A list of space–time covariance and variogram models together with their permissibility conditions can be found in Christakos (1991a, c, 1992, 2005b, 2008a, b), Christakos and Hristopulos (1998), Gneiting (2002), Kolovos et al. (2004), Christakos et al. (2000, 2005), and Porcu et al. (2006, 2008).

This is easily seen in the discrete case, \( {\beta_{X;{{{\mathbf{\mathit{p}}}}_1},{{{\mathbf{\mathit{p}}}}_2}}} = \sum\nolimits_i {\,f_{1;\,i}\log f_{1;\,i}^{ - 1}} - \sum\nolimits_j {f_{2;j}\log f_{1|2}^{ - 1}} \), where only the probability values are needed and not the numerical values of the realizations.

That is \( ({\chi_{{{{\mathbf{\mathit{p}}}}_1}}},{\chi_{{{{\mathbf{\mathit{p}}}}_2}}}) \to ({\chi ^{\prime}_{{{{\mathbf{\mathit{p}}}}_1}}} = \phi ({\chi_{{{{\mathbf{\mathit{p}}}}_1}}}),{\chi ^{\prime}_{{{{\mathbf{\mathit{p}}}}_2}}} = \phi ({\chi_{{{{\mathbf{\mathit{p}}}}_2}}})) \), in which case \( {f_{X;{{{\mathbf{\mathit{p}}}}_1},{{{\mathbf{\mathit{p}}}}_2}}}\,d{\chi_{{{{\mathbf{\mathit{p}}}}_1}}}\,d{\chi_{{{{\mathbf{\mathit{p}}}}_2}}} = {f^{\prime}_{X;{{{\mathbf{\mathit{p}}}}_1},{{{\mathbf{\mathit{p}}}}_2}}}\,d{\chi ^{\prime}_{{{{\mathbf{\mathit{p}}}}_1}}}\,d{\chi ^{\prime}_{{{{\mathbf{\mathit{p}}}}_2}}} \)

Actually, Pearson defined the discrete-valued contingency as \( \varphi = [{\sum\nolimits_i {\sum\nolimits_j {\tfrac{{{{\eta}^2}({\chi_i},{\chi_j})}}{{{\eta}({\chi_i})\eta({\chi_j})}} - 1]} }^{0.5}} \), where η denotes discrete probabilities. Here the idea is extended in a continuous space–time domain with \( \psi = {\varphi^2} \).

That is, \( \log A = (A - 1) - \tfrac{1}{2}{(A - 1)^2} + \tfrac{1}{3}{(A - 1)^3} -... \)

Again, it is preferable that “nonstationary” be associated with time series rather than spatial functions, the latter being linked to the term “nonhomogeneous.”

Heterogeneity may be interpreted, e.g., in terms of complex spatial patterns combined with varying temporal trends (at local or global scales).

\( {\vartheta_{\nu /\mu, {{\mathbf{\mathit{p}}}}}} = \tfrac{1}{{(\nu + 1)!}}\sum\nolimits_{i = 1}^n {\theta_i^2{\mathbf{\mathit{s}}}_i^{\nu + 1}} + \tfrac{1}{{(\mu + 1)!}}\,{t^{\mu + 1}} \), \( \sum\nolimits_{i = 1}^n {\theta_i^2} = 1 \), and the Green’s function satisfies \( {Q_{\nu /\mu }}\,[G_{0,{{\mathbf{\mathit{p}}}},{{p^{\prime}}}}^{(\nu + 1/\mu + 1)}] = {\delta_{{{\mathbf{\mathit{s}}}} - {{s^{\prime}}}}}{\delta_{t - t^{\prime}}} \), where \( {\delta_{{{\mathbf{\mathit{s}}}} - {{s^{\prime}}}}} \) and \( {\delta_{t - t^{\prime}}} \) are delta functions in space and time, respectively.

\( {G_{1,\tau - \tau ^{\prime}}} = \tfrac{{{{( - 1)}^\mu }}}{{(2\mu + 1)!}}{(\tau - \tau ^{\prime})^{2\mu + 1}}{\theta_{\tau - \tau ^{\prime}}} \) (\( T = ( - \infty, \tau ] \)), \( {G_{1,{h} - {h}^{\prime}}} = \tfrac{{{{( - 1)}^\nu }}}{{(2\nu + 1)!}}{({h} - {h}^{\prime})^{2\nu + 1}}{\theta_{{h} - {h}^{\prime}}} \) (\( V = {R^1} \)), \( {G_{2,{{\mathbf{\mathit{h}}}} - {{h^{\prime}}}}} = \tfrac{1}{{{2^{2\nu + 1}}\,\pi \,{{(\nu !)}^2}}}|{{\mathbf{\mathit{h}}}} - {{h^{\prime}}}{|^{2\nu }}\,\log |{{\mathbf{\mathit{h}}}} - {{h^{\prime}}}| \) (\( V = {R^2} \)), \( {G_{3,{{\mathbf{\mathit{h}}}} - {{h^{\prime}}}}} = \tfrac{{{{( - 1)}^{\nu + 1}}\,\Gamma (\tfrac{1}{2} - \nu )}}{{{2^{2\nu + 2}}\,{\pi^{{3 \mathord{\left/{\vphantom {3 2}} \right.} 2}}}\,\nu !}}|{{\mathbf{\mathit{h}}}} - {{h^{\prime}}}{|^{2\nu - 1}} \) (\( V = {R^3} \)); \( {G_{2,{{\mathbf{\mathit{h}}}} - {{h^{\prime}}}}} = {G_{2,{{h^{\prime}}} - {{\mathbf{\mathit{h}}}}}} \) (\( V = {R^2},\;{R^3} \)); \( \theta \) is the unit step function.

A terminology issue emerges here. Due to mathematical associations of Eq. (5.52) with the theory of generalized functions (distributions), and in order to distinguish it from the STHS covariance \( {c_{Y;{{\mathbf{\mathit{h}}}},\tau }} \), it seems natural to call \( {c_{X;\,{{{\mathbf{\mathit{p}}}}_i},{{{\mathbf{\mathit{p}}}}_j}}} \) a generalized spatiotemporal covariance, keeping in mind that the term “generalized covariance” has been used in physics (Joseph 1965) and geostatistics (Matheron 1973).

For example, \( {U_{{Q_{\nu /\mu }}}} = {( - 1)^{\nu + \mu }}\,{{({\partial^{2\mu + 2}}} \mathord{\left/{\vphantom {{({\partial^{2\mu + 2}}} {\partial {\tau^{2\mu + 2}})}}} \right.} {\partial {\tau^{2\mu + 2}})}}\,{(\nabla_{{\mathbf{\mathit{h}}}}^2)^{\nu + 1}} \); Christakos and Hristopulos (1998: 160).

The coefficients \( {c_0}, {a_\zeta }, {b_\rho }, {a_{\rho \zeta }}, a, b, c \) must satisfy certain permissibility criteria; \( {\delta_r} \) and \( {\delta_\tau } \) are delta functions in space and time, respectively; and \( \gamma \) is an incomplete gamma function.

The \( {r_0} < < r < < {r_m} \), \( {\tau_0} < < \tau < < {\tau_m} \) define space-time fractal ranges; −1< z < 0, \( - 0.5\,(n + 1) < \alpha - \beta \,z < 0 \) are permissibility conditions; \( \sigma_X^2 \) is variance; \( {u_c} \), \( {w_c} \) are cutoffs.

The relevant literature includes (but is not limited to) the following: Serre et al. (2001, 2003a, b), Querido et al. (2007), Tuia et al. (2007), Bogaert and Fasbender (2007), Fasbender et al. (2007), Orton and Lark (2007a, b), Vyas et al. (2004), Bogaert (2002, 2004), Bogaert and Wibrin (2004), Wibrin et al. (2006), Yu et al. (2007a, b,c), Douaik et al. (2004, 2005), Serre and Christakos (1999a, b; 2003), Quilfen et al. (2004), Kolovos et al. (2002), Papantonopoulos and Modis (2006), Akita et al. (2007), Lee et al. (2008a, b, 2009), Bogaert and D’Or (2002), D’Or and Bogaert (2003), Coulliette et al. (2009), Yu and Christakos (2005, 2006), Pang et al. (2009), Puangthongthub et al. (2007), LoBuglio et al. (2007), Choi et al. (2003, 2006, 2007), Savelieva et al. (2005), Parkin et al. (2005), Augustinraj (2002), Law et al. (2006), Wang (2005), Gummer (2009), Kolovos et al. (2010), and de Nazelle et al. (2010).