A Unified Framework

All of the above methods can be unified into a single fl amework [33]. The unified framework builds on the observation that all of the incremental learning algorithms generally follow three core steps (1) update structure information, (2) predict low-dimensional co-ordinates of new data, (3) improve previous and predicted coordinates. Each of these steps are described below. 

The low-dimensional embedding can be given in a number of ways, however, since the unified framework allows for multiple data points to be added at once to the embedding the following proposed method is best followed. The new feature matrix is transformed into a block matrix such that 

The low-dimensional embedding sought is Y = [Y y ] where y corresponds to the predicted co-ordinates of the new data point. It is worth noting that in this 

The low-dimensional embedding of both the previously learnt data, and the newly added data, is then improved using an iterative method similar to Ritz acceleration [34], The basic premise is to refine the low-dimensional embedding using the updated structure information found in F ew. The frill algorithm is displayed in Algorithm 1 of [33]. The termination criteria for this refinement step is to either continue for a set number of iterations, or terminate when the error—measured as the Frobenius norm between the previous iteration s estimate and that of the current iteration—falls below a certain tolerance threshold. 

The ability of the unified framework to extend an existing spectral dimensionality reduction technique to be able to handle incremental data was shown by providing an incremental version of Hessian LLE [35, 33]. Unlike the incremental methods previously discussed, the unified framework allows for the incorporation of multiple data points at the same time. This has obvious benefits when dealing with incremental learning as it is unlikely to encounter a situation where the new data comes one data point at a time. 

---

Cao J and Voth G A 1996 A unified framework for quantum activated rate processes I. General theory J. Chem. Phys. 105 6856... 

Historically azeotropic distillation processes were developed on an individual basis using experimentation to guide the design. The use of residue curve maps as a vehicle to explain the behavior of entire sequences of heterogeneous azeotropic distillation columns as weU as the individual columns that make up the sequence provides a unifying framework for design. This process can be appHed rapidly, and produces an exceUent starting point for detailed simulations and experiments. 

N. Noiray, D. Durox, T. Schuller, and S. Candel. A unified framework for nonlinear combustion instability analysis based on the flame describing function. /. Fluid Mech., 2008 (In press). 

Data Analysis presents a unifying framework for discussing and comparing input and input-output methods. 

The use of statistical tests to characterize model performance has increased markedly within the last few years as demonstrated by the references cited above many other examples are in the literature. The current time is appropriate for integration of existing techniques into a unified framework of guidelines, procedures and quantitative measures for model testing and validation. 

Although the importance of a systemic perspective on metabolism has only recently attained widespread attention, a formal frameworks for systemic analysis has already been developed since the late 1960s. Biochemical Systems Theory (BST), put forward by Savageau and others [142, 144 147], seeks to provide a unified framework for the analysis of cellular reaction networks. Predating Metabolic Control Analysis, BST emphasizes three main aspects in the analysis of metabolism [319] (i) the importance of the interconnections, rather than the components, for cellular function (ii) the nonlinearity of biochemical rate equations (iii) the need for a unified mathematical treatment. Similar to MCA, the achievements associated with BST would warrant a more elaborate treatment, here we will focus on BST solely as a tool for the approximation and numerical simulation of complex biochemical reaction networks. 

In this book and in general, the concentration, d, is used to represent the number of mobile species, ions or vacancies, per unit volume whereas is always the fraction of the crystallographically equivalent sites that are occupied by ions. Often if > 0.5 we talk of ion migration and c refers to ion concentration, whereas if < 0.5 we talk of vacancy migration and c refers to the concentration of vacancies. However, this is simply a convenient way of thinking about ion transport in solids by focusing on the minority species. It is equally possible to describe conduction always in terms of ions or of vacancies provided account is taken of the fact that both the concentration of mobile species and sites to which they may migrate are important. The importance of c and (1 — c) is emphasised and placed within a unified framework in Chapter 3. The concentration of ions Ci is related to the occupancy by = cJC, where C is the concentration of the sites. 

Redish AD, Jensen S, Johnson A A unified framework for addiction Vulnerabilities in the decision process. Behav Brain Sci 2008 31 461. 

The above discussion has relied to a considerable extent on the concepts developed in the chemistry of cyclic phosphates. Obviously pentacovalent species are not required on the basis of the available experimental evidence in many of the above reactions and alternate explanations are plausible. Nevertheless, many diverse observations are explained with equal facility by application of these concepts. Although this certainly does not constitute a proof, the approach offers a unifying framework for productive experimental design with both chemical and biological systems. One relevant example s the previously... 

Naaman N. A unified framework for information assurance, 5 International CC Conference, Berlin, September 2004. 

Bonatti, P. and P. Samarati, A unified framework for regulating access and information release on the web, Journal of Computer Security, 10(3) 241-272, 2002. 

Jajodia, S., P. Samarati, V.S. Subrahmanian, and E. Bertino, A unified framework for enforcing multiple access control policies, in Proc. of the 1997 ACM International SIGMOD Conference on Management of Data, Tucson, AZ, May 1997. 

In the present work we will deal with all the above problems and provide a unified framework to deal with the error correction for static or dynamic systems using multicomponent mass and energy balances. The topological character of the complex process is exploited for an easy classification of the measured and unmeasured variable independently of the linearity or nonlinearity of the balance equations. 

The geometrical setting for the Legendre transformation h(x) —> s(y) that has been introduced above is illustrated below on a few examples. A particularly simple illustration (but still containing all the structure) is developed in Section 3.1.1. The other examples presented below in Sections 2.2.1-2.2.5 deal mostly with well-known and well-studied physical systems. The geometrical setting is demonstrated to provide a unified framework for their investigation on all scales. 

Polarization Model a unified framework for hydration and double layer interactions... 

---