Probabilistic mapping

Penhune VB, Zatorre RJ, MacDonald JD, Evans AC. 1996. Interhemispheric anatomical differences in human primary auditory cortex Probabilistic mapping and volume measurement from magnetic resonance scans. Cereb Cortex 6(5) 661-672. 

Rademacher J, Morosan P, Schormann T, Schleicher A, Werner C, et al. 2001b. Probabilistic mapping and volume measurment of human primary auditory cortex. Neuro-Image 13 669-683. 

The way contextual correspondences are defined, they are deterministic, not allowing a probabilistic interpretation of a correspondence. Therefore, contextual correspondences are meant to resolve an issue of uncertainty by finding a more refined, yet deterministic, correspondence. It is worth noting, however, that an introduction of stochastic analysis already exists in the form of statistical significance, which can be extended to handle probabilistic mappings as well. 

A schema probabilistic mapping pM is a set of probabilistic mappings between relations in S and in T, where every relation in either S or T appears in at most one probabilistic mapping. 

Fig. 4.2 The running example (a) a probabilistic schema mapping between S and 7 (b) a source instance Ds, (c) the answers of Q over Ds with respect to the probabilistic mapping...

We define query answering under both interpretations. The first interpretation is referred to as the by-table semantics, and the second one is referred to as the by-tuple semantics of probabilistic mappings. Note that one cannot argue for one interpretation over the other the needs of the application should dictate the appropriate semantics. Furthermore, the complexity results for query answering, which will show advantages to by-table semantics, should not be taken as an argument in the favor of by-table semantics. 

A probabilistic mediated schema will avoid this problem. Consider a probabilistic mediated schema M that includes M3 and M4, each with probability 0.5. For each of them and each source schema, we generate a probabilistic mapping (Sect. 3). For example, the set of probabilistic mappings pM for Si is shown in Fig. 4.5a, b. 

Possible improvements might include 1) combining the maps by several independent inventory teams to produce empirical probabilistic maps, 2) using widely applicable statistical distributions of landslide size (area or volume) to estimate the numbers of landslides likely to have been missed in an inventory, and 3) using physics-based watershed scale models of landslide occurrence to identify areas that may have been overlooked in inventories or which may be susceptible to sliding under future rare or unprecedented conditions. 

Proper hazard maps can be used to construct uniform response spectra that can be used for seismic design of structures. For instance, in the United States (US), seismic provisirais are based (Ml probabilistic maps developed by USGS after performing elalxM-ate probabilistic seismic hazard analyses of the entire USA (Petersen et al. 2008). Using these maps, which provide spectral values for different spectral periods (e.g., for 0.2 s in the Sa 2/50 map of Fig. 14), it is possible to develop damped response spectra (for a specific structural damping ration, typically... 

The measures of dimensional variability from Conformability Analysis (CA) (as described in Chapters 2 and 3), specifically the Component Manufacturing Variability Risk, q, is useful in the allocation of tolerances and subsequent analysis of their distributions in probabilistic design. The value is determined from process capability maps for the manufacturing process and knowledge of the component s material and geometry compatibility with the process. In the specific case to the th component bilateral tolerance, it was shown in Chapter 3 that the standard deviation estimates were ... 

For small enough values of p so that pf p) < p for all 0 < p < 1, p = 0 will be the only fixed point. As p increases, there will eventually be some density p for which pf p ) > p in this case, we can expect there to be nonzero fixed point densities as well. Qualitatively, the mean-field-predicted behaviors will depend on the shape of the iterative map. If / has a concave downward profile, for example (i.e. if/" < 0 everywhere), then, as p decreases, Poo decreases continuously to zero at some critical value of p = Pc- Note also that the iterative map /jet for the deterministic rule associated with its minimally diluted probabilistic counterpart is given by /jet = //p-... 

Table 7.3 The four rules in the family of two-state/ one-absorbing-state probabilistic PCA defined in the text, along with their corresponding iterative maps (see equation 7.98).

Probabilistic techniques of estimation provide some Insights Into the potential error of estimation. In the case of krlglng, the variable pCic) spread over the site A is first elevated to the status of a random function PC c). An estimator P (2c) is then built to minimize the estimation variance E [P(2c)-P (2c) ], defined as the expected squared error ( ). The krlglng process not only provides the estimated values pCiyc) from which a kriged map can be produced, but also the corresponding minimum estimation variances 0 (39 ) ... 

In the line of probabilistic approaches, Wilson and Liu [69] solved the transport equation using stochastic differential equations backwards in time. Like in the method developed by Bagtzoglou and co-workers, Wilson and Liu kept the dispersion part positive and reversed the advection part. They provided two types of maps, namely travel time probability and location probability. The travel time probability map was obtained from the normalized flux concentration, given by... 

One problem encountered when assessing exposure of human populations to contaminated land is spatial heterogeneity of pollution. To overcome this problem, Gay and Korre (2006) propose the combinations of spatial statistical methods for mapping soil concentrations, and probabilistic human health risk assessment methods. They applied geostatistical methods to map As concentrations in soil. Subsequently, an age-stratified human population was mapped across the contaminated area, and the intake of As by individuals was calculated using a modified version of the Contaminated Land Exposure Assessment (CLEA) model. This approach allowed a... 

One more way to conduct a similarity-based virtual screening is to retrieve the structures containing a user-defined set of pharmacophoric features. In the Dynamic Mapping of Consensus positions (DMC) algorithm those features are selected by finding common positions in bit strings for all active compounds. The potency-scaled DMC algorithm (POT-DMC) " is a modification of DMC in which compounds activities are taken into account. The latter two methods may be considered as intermediate between conventional similarity search and probabilistic SAR approaches. 

Some historically important artificial neural networks are Hopfield Networks, Per-ceptron Networks and Adaline Networks, while the most well-known are Backpropa-gation Artificial Neural Networks (BP-ANN), Kohonen Networks (K-ANN, or Self-Organizing Maps, SOM), Radial Basis Function Networks (RBFN), Probabilistic Neural Networks (PNN), Generalized Regression Neural Networks (GRNN), Learning Vector Quantization Networks (LVQ), and Adaptive Bidirectional Associative Memory (ABAM). 

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