Discretization. To fit such a continuous, infinite state model into our framework, (i.e., finite set of states and actions), we employ sigma point sampling [BH08] and linear interpolation. This is a non-trivial process, but happens transparently for the user. All that is necessary for him is to add an annotated class like in the following. @ D i s c r e t i z e ( model=ACC. c l a s s ) p u b l i c s t a t i c c l a s s DModel extends ACC {} ; DModel is now a discretized version of ACC and describes a MDP as formally defined above. The specific details of discretization (like number of discrete states used) and sampling are configurable. Note that discretized classes like this one are not part of the hidden internals of the framework but are fully accessible to the engineer. Algorithms We support various algorithms for finding optimal controllers. They differ in how they treat the se- quence of rewards the evolution of the process presents. Let r1, r2, r3 . . . be a sequence of vectors. Then the aggregated reward is defined in one of the following ways. • Total Sum: r1 + r2 + r3 + . . .
Discretization. With stability and existence assured by the weak formulation the next step is to build a mesh. The key purpose of this is to replace an infinite dimensional problem with a finite dimensional problem. In the FEM, Vh, is a subspace of V of piecewise polynomial functions. The discretization then comes from the choice of how and where to triangulate this domain. The simplest mesh might entail equispaced nodes, changing this spacing in order to increase or decrease the number of nodes but this need not be the case. As explained earlier, the adaptability of the mesh is a key strength of FEM/FDM. Depending on the boundaries as well as other components inherent in the problem, we may choose to build a mesh that has heavy focus in certain areas. To be more specific, we know that as h → 0, the error u − uh → 0 but the smaller h is, the higher the computational cost. To best utilize this trade-off, we can make h relatively small in areas of the domain where high accuracy is most desired. For example, with Black-Scholes, often higher accuracy is desired around the exercise price because this is the region that many options will be trading.
Discretization. Many capacity theorems for Gaussian channels, in both classical and quantum information theory, are derived by extending the finite-dimension results to the continuous infinite-dimension Gaussian channel [43]. This requires a discretization limiting argument, as e.g., in [76]. This approach is often more convenient than devising a coding scheme and perform the analysis “from scratch”. Yet, such a proof is less transparent and may give less insight for the design of practical error-correction codes (see also discussion in [102]). − {− − − − − } The most common discretization approach is based on the following operational argument. Consider a classical memoryless channel WY |X , with continuous input X and output Y . We can construct a codebook while restricting ourselves to discrete values, in Lδ, (L 1)δ, . . . , δ, 0, δ, . . . , (L 1)δ, Lδ , and the decoder can also discretize the received signal, with an arbitrarily small discretization step δ > 0 and arbitrarily large L > 0. In this manner, we are effectively coding over a finite- dimension channel, with input Xδ and ouput Yδ over finite alphabets. Thus, by the finite-dimension capacity result, for every input distribution pXδ , a rate R < I(Xδ; Yδ) ε is achievable, for arbitrarily small ε > 0. The achievability proof can then be completed by analyzing the limit of the mutual information I(Xδ; Yδ) as δ tends to zero. A similar argument can be applied for the bosonic channel, where the codebook is restricted to coherent states of discretized values and the output dimension is restricted by the decoding measurement. For basic channel networks, the Gaussian capacity result can be obtained directly. However, in adversarial models, such as the wiretap channel, this approach may become tricky. In particular, a straightforward application will force the eavesdropper to discretize her signal. Clearly, this does not make sense operationally. An alternative discretization approach, which can be applied to adversarial models as well, is based on continuity arguments. In particular, we can view the operational capacity C(W ) as an unknown functional of the probability measure WY |X , which may have either finite or infinite dimensions. Then, if WY |X is a Gaussian channel, then we can define a sequence of discretized { } channels W¯ δk Y |X k≥1, where δ1, δ2, . . . converges to zero uniformly. Since the Gaussian distribution is continuous and smooth, δ { the sequence of probability measures W¯ have Y |X }k≥1 converg...
Discretization. For the purposes of proving Theorem 1.2(a), we find it convenient to work with discrete representations of circular societies. Before switching to dis- crete representations, though, we make a few simplifying assumptions. − −
Discretization. You can use the DISCRETIZATION subcommand to discretize fractional-value variables or to recode categorical variables. Initial configuration. You can specify the kind of initial configuration through the INITIAL subcommand. Tuning the algorithm. You can control the values of algorithm-tuning parameters with the XXXXXXX and CRITITER subcommands. Missing data. You can specify the treatment of missing data with the MISSING subcommand. Optional output. You can request optional output through the PRINT subcommand. Transformation plot per variable. You can request a plot per variable of its quantification against the category numbers. Residual plot per variable. You can request an overlay plot per variable of the residuals and the weighted quantification, against the category numbers. Writing external data. You can write the transformed data (category numbers replaced with optimal quantifications) to an outfile for use in further analyses. You can also write the discretized data to an outfile. Saving variables. You can save the transformed variables, the predicted values, and/or the residuals in the working data file. Basic Specification The basic specification is the command CATREG with the VARIABLES and ANALYSIS subcommands. Syntax Rules • The VARIABLES and ANALYSIS subcommands must always appear, and the VARIABLES subcommand must be the first subcommand specified. The other subcommands, if specified, can be in any order. • Variables specified in the ANALYSIS subcommand must be found in the VARIABLES subcommand. • In the ANALYSIS subcommand, exactly one variable must be specified as a dependent variable and at least one variable must be specified as an independent variable after the keyword WITH. • The word WITH is reserved as a keyword in the CATREG procedure. Thus, it may not be a variable name in CATREG. Also, the word TO is a reserved word in SPSS.
