Model selection and error estimation

We study model selection strategies based on penalized empirical loss minimization. We point out a tight relationship between error estimation and data-based complexity penalization: any good error estimate may be converted into a data-based penalty function and the performance of the estimate is governed by the quality of the error estimate. We consider several penalty functions, involving error estimates on independent test data, empirical VC dimension, empirical VC entropy, and margin-based quantities. We also consider the maximal difference between the error on the first half of the training data and the second half, and the expected maximal discrepancy, a closely related capacity estimate that can be calculated by Monte Carlo integration. Maximal discrepancy penalty functions are appealing for pattern classification problems, since their computation is equivalent to empirical risk minimization over the training data with some labels flipped.

Multitask learning with expert advice

We consider the problem of prediction with expert advice in the setting where a forecaster is presented with several online prediction tasks. Instead of competing against the best expert separately on each task, we assume the tasks are related, and thus we expect that a few experts will perform well on the entire set of tasks. That is, our forecaster would like, on each task, to compete against the best expert chosen from a small set of experts. While we describe the “ideal” algorithm and its performance bound, we show that the computation required for this algorithm is as hard as computation of a matrix permanent. We present an efficient algorithm based on mixing priors, and prove a bound that is nearly as good for the sequential task presentation case. We also consider a harder case where the task may change arbitrarily from round to round, and we develop an efficient approximate randomized algorithm based on Markov chain Monte Carlo techniques.

Multitask learning with expert advice

We consider the problem of prediction with expert advice in the setting where a forecaster is presented with several online prediction tasks. Instead of competing against the best expert separately on each task, we assume the tasks are related, and thus we expect that a few experts will perform well on the entire set of tasks. That is, our forecaster would like, on each task, to compete against the best expert chosen from a small set of experts. While we describe the "ideal" algorithm and its performance bound, we show that the computation required for this algorithm is as hard as computation of a matrix permanent. We present an efficient algorithm based on mixing priors, and prove a bound that is nearly as good for the sequential task presentation case. We also consider a harder case where the task may change arbitrarily from round to round, and we develop an efficient approximate randomized algorithm based on Markov chain Monte Carlo techniques.

Application of higher order spectra to identify epileptic EEG

Epilepsy is characterized by the spontaneous and seemingly unforeseeable occurrence of seizures, during which the perception or behavior of patients is disturbed. An automatic system that detects seizure onsets would allow patients or the people near them to take appropriate precautions, and could provide more insight into this phenomenon. Various methods have been proposed to predict the onset of seizures based on EEG recordings. The use of nonlinear features motivated by the higher order spectra (HOS) has been reported to be a promising approach to differentiate between normal, background (pre-ictal) and epileptic EEG signals. In this work, we made a comparative study of the performance of Gaussian mixture model (GMM) and Support Vector Machine (SVM) classifiers using the features derived from HOS and from the power spectrum. Results show that the selected HOS based features achieve 93.11% classification accuracy compared to 88.78% with features derived from the power spectrum for a GMM classifier. The SVM classifier achieves an improvement from 86.89% with features based on the power spectrum to 92.56% with features based on the bispectrum.

Conceptualisation of articular cartilage as a giant reverse micelle: A hypothetical mechanism for joint biocushioning and lubrication

Phospholipid (PL) molecules form the main structure of the membrane that prevents the direct contact of opposing articular cartilage layers. In this paper we conceptualise articular cartilage as a giant reverse micelle (GRM) in which the highly hydrated three-dimensional network of phospholipids is electrically charged and able to resist compressive forces during joint movement, and hence loading. Using this hypothetical base, we describe a hydrophilic-hydrophilic (HL-HL) biopair model of joint lubrication by contacting cartilages, whose mechanism is reliant on lamellar cushioning. To demonstrate the viability of our concept, the electrokinetic properties of the membranous layer on the articular surface were determined by measuring via microelectrophoresis, the adsorption of ions H, OH, Na and Cl on phospholipid membrane of liposomes, leading to the calculation of the effective surface charge density. The surface charge density was found to be -0.08 ± 0.002 cm-2 (mean ± S.D.) for phospholipid membranes, in 0.155 M NaCl solution and physiological pH. This value was approximately five times less than that measured in 0.01 M NaCl. The addition of synovial fluid (SF) to the 0.155 M NaCl solution reduced the surface charge density by 30% which was attributed to the binding of synovial fluid macromolecules to the phospholipid membrane. Our experiments show that particles charge and interact strongly with the polar core of RM. We demonstrate that particles can have strong electrostatic interactions when ions and macromolecules are solubilized by reverse micelle (RM). Since ions are solubilized by reverse micelle, the surface entropy influences the change in the charge density of the phospholipid membrane on cartilage surfaces. Reverse micelles stabilize ions maintaining equilibrium, their surface charges contribute to the stability of particles, while providing additional screening for electrostatic processes. © 2008 Elsevier Ireland Ltd. All rights reserved.

The structure of the mineral leogangite Cu10(OH)6(SO4)(AsO4)4•8H2O–implications for arsenic accumulation and removal

The objective of this research is to determine the molecular structure of the mineral leogangite. The formation of the types of arsenosulphate minerals offers a mechanism for arsenate removal from soils and mine dumps. Raman and infrared spectroscopy have been used to characterise the mineral. Observed bands are assigned to the stretching and bending vibrations of (SO4)2- and (AsO4)3- units, stretching and bending vibrations of hydrogen bonded (OH)- ions and Cu2+-(O,OH) units. The approximate range of O-H...O hydrogen bond lengths is inferred from the Raman spectra. Raman spectra of leogangite from different origins differ in that some spectra are more complex, where bands are sharp and the degenerate bands of (SO4)2- and (AsO4)3- are split and more intense. Lower wavenumbers of  H2O bending vibration in the spectrum may indicate the presence of weaker hydrogen bonds compared with those in a different leogangite samples. The formation of leogangite offers a mechanism for the removal of arsenic from the environment.

Look before you leap : a confidence-based method for selecting species criticality while avoiding negative populations in τ-leaping

The stochastic simulation algorithm was introduced by Gillespie and in a different form by Kurtz. There have been many attempts at accelerating the algorithm without deviating from the behavior of the simulated system. The crux of the explicit τ-leaping procedure is the use of Poisson random variables to approximate the number of occurrences of each type of reaction event during a carefully selected time period, τ. This method is acceptable providing the leap condition, that no propensity function changes “significantly” during any time-step, is met. Using this method there is a possibility that species numbers can, artificially, become negative. Several recent papers have demonstrated methods that avoid this situation. One such method classifies, as critical, those reactions in danger of sending species populations negative. At most, one of these critical reactions is allowed to occur in the next time-step. We argue that the criticality of a reactant species and its dependent reaction channels should be related to the probability of the species number becoming negative. This way only reactions that, if fired, produce a high probability of driving a reactant population negative are labeled critical. The number of firings of more reaction channels can be approximated using Poisson random variables thus speeding up the simulation while maintaining the accuracy. In implementing this revised method of criticality selection we make use of the probability distribution from which the random variable describing the change in species number is drawn. We give several numerical examples to demonstrate the effectiveness of our new method.

Identity management : strengthening one-time password authentication through usability

Usability in HCI (Human-Computer Interaction) is normally understood as the simplicity and clarity with which the interaction with a computer program or a web site is designed. Identity management systems need to provide adequate usability and should have a simple and intuitive interface. The system should not only be designed to satisfy service provider requirements but it has to consider user requirements, otherwise it will lead to inconvenience and poor usability for users when managing their identities. With poor usability and a poor user interface with regard to security, it is highly likely that the system will have poor security. The rapid growth in the number of online services leads to an increasing number of different digital identities each user needs to manage. As a result, many people feel overloaded with credentials, which in turn negatively impacts their ability to manage them securely. Passwords are perhaps the most common type of credential used today. To avoid the tedious task of remembering difficult passwords, users often behave less securely by using low entropy and weak passwords. Weak passwords and bad password habits represent security threats to online services. Some solutions have been developed to eliminate the need for users to create and manage passwords. A typical solution is based on generating one-time passwords, i.e. passwords for single session or transaction usage. Unfortunately, most of these solutions do not satisfy scalability and/or usability requirements, or they are simply insecure. In this thesis, the security and usability aspects of contemporary methods for authentication based on one-time passwords (OTP) are examined and analyzed. In addition, more scalable solutions that provide a good user experience while at the same time preserving strong security are proposed.

Novel numerical methods for time-space fractional reaction diffusion equations in two dimensions

We consider time-space fractional reaction diffusion equations in two dimensions. This equation is obtained from the standard reaction diffusion equation by replacing the first order time derivative with the Caputo fractional derivative, and the second order space derivatives with the fractional Laplacian. Using the matrix transfer technique proposed by Ilic, Liu, Turner and Anh [Fract. Calc. Appl. Anal., 9:333--349, 2006] and the numerical solution strategy used by Yang, Turner, Liu, and Ilic [SIAM J. Scientific Computing, 33:1159--1180, 2011], the solution of the time-space fractional reaction diffusion equations in two dimensions can be written in terms of a matrix function vector product $f(A)b$ at each time step, where $A$ is an approximate matrix representation of the standard Laplacian. We use the finite volume method over unstructured triangular meshes to generate the matrix $A$, which is therefore non-symmetric. However, the standard Lanczos method for approximating $f(A)b$ requires that $A$ is symmetric. We propose a simple and novel transformation in which the standard Lanczos method is still applicable to find $f(A)b$, despite the loss of symmetry. Numerical results are presented to verify the accuracy and efficiency of our newly proposed numerical solution strategy.

The molecular structure of the multianion mineral hidalgoite PbAl3(AsO4)(SO4)(OH)6 : implications for arsenic removal from soils

The objective of this research is to determine the molecular structure of the mineral hidalgoite PbAl3(AsO4)(SO4)(OH)6 using vibrational spectroscopy. The mineral is found in old mine sites. Observed bands are assigned to the stretching and bending vibrations of (SO4)2- and (AsO4)3- units, stretching and bending vibrations of hydrogen bonded (OH)- ions and Al3+-(O,OH) units. The approximate range of O-H...O hydrogen bond lengths is inferred from the Raman and infrared spectra. Values of 2.6989 Å, 2.7682 Å, 2.8659 Å were obtained. The formation of hidalgoite may offer a mechanism for the removal of arsenic from the environment.

An evaluation of the sustainability of agreements reached through mediations conducted by the dispute resolution centres of the Department of Justice and Attorney General, Queensland between 1999 to 2003

In 1990 the Dispute Resolution Centres Act, 1990 (Qld) (the Act) was passed by the Queensland Parliament. In the second reading speech for the Dispute Resolution Centres Bill on May 1990 the Hon Dean Wells stated that the proposed legislation would make mediation services available “in a non-coercive, voluntary forum where, with the help of trained mediators, the disputants will be assisted towards their own solutions to their disputes, thereby ensuring that the result is acceptable to the parties” (Hansard, 1990, 1718). It was recognised at that time that a method for resolving disputes was necessary for which “the conventional court system is not always equipped to provide lasting resolution” (Hansard, 1990, 1717). In particular, the lasting resolution of “disputes between people in continuing relationships” was seen as made possible through the new legislation; for example, “domestic disputes, disputes between employees, and neighbourhood disputes relating to such issues as overhanging tree branches, dividing fences, barking dogs, smoke, noise and other nuisances are occurring continually in the community” (Hansard, 1990, 1717). The key features of the proposed form of mediation in the Act were articulated as follows: “attendance of both parties at mediation sessions is voluntary; a party may withdraw at any time; mediation sessions will be conducted with as little formality and technicality as possible; the rules of evidence will not apply; any agreement reached is not enforceable in any court; although it could be made so if the parties chose to proceed that way; and the provisions of the Act do not affect any rights or remedies that a party to a dispute has apart from the Act” (Hansard, 1990, 1718). Since the introduction of the Act, the Alternative Dispute Resolution Branch of the Queensland Department of Justice and Attorney General has offered mediation services through, first the Community Justice Program (CJP), and then the Dispute Resolution Centres (DRCs) for a range of family, neighbourhood, workplace and community disputes. These services have mirrored those available through similar government agencies in other states such as the Community Justice Centres of NSW and the Victorian Dispute Resolution Centres. Since 1990, mediation has become one of the fastest growing forms of alternative dispute resolution (ADR). Sourdin has commented that "In addition to the growth in court-based and community-based dispute resolution schemes, ADR has been institutionalised and has grown within Australia and overseas” (2005, 14). In Australia, in particular, the development of ADR service provision “has been assisted by the creation and growth of professional organisations such as the Leading Edge Alternative Dispute Resolvers (LEADR), the Australian Commercial Dispute Centres (ACDC), Australian Disputes Resolution Association (ADRA), Conflict Resolution Network, and the Institute of Arbitrators and Mediators Australia (IAMA)” (Sourdin, 2005, 14). The increased emphasis on the use of ADR within education contexts (particularly secondary and tertiary contexts) has “also led to an increasing acceptance and understanding of (ADR) processes” (Sourdin, 2005, 14). Proponents of the mediation process, in particular, argue that much of its success derives from the inherent flexibility and creativity of the agreements reached through the mediation process and that it is a relatively low cost option in many cases (Menkel-Meadow, 1997, 417). It is also accepted that one of the main reasons for the success of mediation can be attributed to the high level of participation by the parties involved and thus creating a sense of ownership of, and commitment to, the terms of the agreement (Boulle, 2005, 65). These characteristics are associated with some of the core values of mediation, particularly as practised in community-based models as found at the DRCs. These core values include voluntary participation, party self-determination and party empowerment (Boulle, 2005, 65). For this reason mediation is argued as being an effective approach to resolving disputes, that creates a lasting resolution of the issues. Evaluation of the mediation process, particularly in the context of the growth of ADR, has been an important aspect of the development of the process (Sourdin, 2008). Writing in 2005 for example, Boulle, states that “although there is a constant refrain for more research into mediation practice, there has been a not insignificant amount of mediation measurement, both in Australia and overseas” (Boulle, 2005, 575). The positive claims of mediation have been supported to a significant degree by evaluations of the efficiency and effectiveness of the process. A common indicator of the effectiveness of mediation is the settlement rate achieved. High settlement rates for mediated disputes have been found for Australia (Altobelli, 2003) and internationally (Alexander, 2003). Boulle notes that mediation agreement rates claimed by service providers range from 55% to 92% (Boulle, 2005, 590). The annual reports for the Alternative Dispute Resolution Branch of the Queensland Department of Justice and Attorney-General considered prior to the commencement of this study indicated generally achievement of an approximate settlement figure of 86% by the Queensland Dispute Resolution Centres. More recently, the 2008-2009 annual report states that of the 2291 civil dispute mediated in 2007-2008, 86% reached an agreement. Further, of the 2693 civil disputes mediated in 2008-2009, 73% reached an agreement. These results are noted in the report as indicating “the effectiveness of mediation in resolving disputes” and as reflecting “the high level of agreement achieved for voluntary mediations” (Annual Report, 2008-2009, online). Whilst the settlement rates for the DRCs are strong, parties are rarely contacted for long term follow-up to assess whether agreements reached during mediation lasted to the satisfaction of each party. It has certainly been the case that the Dispute Resolution Centres of Queensland have not been resourced to conduct long-term follow-up assessments of mediation agreements. As Wade notes, "it is very difficult to compare "success" rates” and whilst “politicians want the comparison studies (they) usually do not want the delay and expense of accurate studies" (1998, 114). To date, therefore, it is fair to say that the efficiency of the mediation process has been evaluated but not necessarily its effectiveness. Rather, the practice at the Queensland DRCs has been to evaluate the quality of mediation service provision and of the practice of the mediation process. This has occurred, for example, through follow-up surveys of parties' satisfaction rates with the mediation service. In most other respects it is fair to say that the Centres have relied on the high settlement rates of the mediation process as a sign of the effectiveness of mediation (Annual Reports 1991 - 2010). Research of the mediation literature conducted for the purpose of this thesis has also indicated that there is little evaluative literature that provides an in-depth analysis and assessment of the longevity of mediated agreements. Instead evaluative studies of mediation tend to assess how mediation is conducted, or compare mediation with other conflict resolution options, or assess the agreement rate of mediations, including parties' levels of satisfaction with the service provision of the dispute resolution service provider (Boulle, 2005, Chapter 16).

Aggregate distance based clustering using Fibonacci Series -FIBCLUS

This paper proposes an innovative instance similarity based evaluation metric that reduces the search map for clustering to be performed. An aggregate global score is calculated for each instance using the novel idea of Fibonacci series. The use of Fibonacci numbers is able to separate the instances effectively and, in hence, the intra-cluster similarity is increased and the inter-cluster similarity is decreased during clustering. The proposed FIBCLUS algorithm is able to handle datasets with numerical, categorical and a mix of both types of attributes. Results obtained with FIBCLUS are compared with the results of existing algorithms such as k-means, x-means expected maximization and hierarchical algorithms that are widely used to cluster numeric, categorical and mix data types. Empirical analysis shows that FIBCLUS is able to produce better clustering solutions in terms of entropy, purity and F-score in comparison to the above described existing algorithms.

Rational design of serine protease inhibitors

Proteases regulate a spectrum of diverse physiological processes, and dysregulation of proteolytic activity drives a plethora of pathological conditions. Understanding protease function is essential to appreciating many aspects of normal physiology and progression of disease. Consequently, development of potent and specific inhibitors of proteolytic enzymes is vital to provide tools for the dissection of protease function in biological systems and for the treatment of diseases linked to aberrant proteolytic activity. The studies in this thesis describe the rational design of potent inhibitors of three proteases that are implicated in disease development. Additionally, key features of the interaction of proteases and their cognate inhibitors or substrates are analysed and a series of rational inhibitor design principles are expounded and tested. Rational design of protease inhibitors relies on a comprehensive understanding of protease structure and biochemistry. Analysis of known protease cleavage sites in proteins and peptides is a commonly used source of such information. However, model peptide substrate and protein sequences have widely differing levels of backbone constraint and hence can adopt highly divergent structures when binding to a protease’s active site. This may result in identical sequences in peptides and proteins having different conformations and diverse spatial distribution of amino acid functionalities. Regardless of this, protein and peptide cleavage sites are often regarded as being equivalent. One of the key findings in the following studies is a definitive demonstration of the lack of equivalence between these two classes of substrate and invalidation of the common practice of using the sequences of model peptide substrates to predict cleavage of proteins in vivo. Another important feature for protease substrate recognition is subsite cooperativity. This type of cooperativity is commonly referred to as protease or substrate binding subsite cooperativity and is distinct from allosteric cooperativity, where binding of a molecule distant from the protease active site affects the binding affinity of a substrate. Subsite cooperativity may be intramolecular where neighbouring residues in substrates are interacting, affecting the scissile bond’s susceptibility to protease cleavage. Subsite cooperativity can also be intermolecular where a particular residue’s contribution to binding affinity changes depending on the identity of neighbouring amino acids. Although numerous studies have identified subsite cooperativity effects, these findings are frequently ignored in investigations probing subsite selectivity by screening against diverse combinatorial libraries of peptides (positional scanning synthetic combinatorial library; PS-SCL). This strategy for determining cleavage specificity relies on the averaged rates of hydrolysis for an uncharacterised ensemble of peptide sequences, as opposed to the defined rate of hydrolysis of a known specific substrate. Further, since PS-SCL screens probe the preference of the various protease subsites independently, this method is inherently unable to detect subsite cooperativity. However, mean hydrolysis rates from PS-SCL screens are often interpreted as being comparable to those produced by single peptide cleavages. Before this study no large systematic evaluation had been made to determine the level of correlation between protease selectivity as predicted by screening against a library of combinatorial peptides and cleavage of individual peptides. This subject is specifically explored in the studies described here. In order to establish whether PS-SCL screens could accurately determine the substrate preferences of proteases, a systematic comparison of data from PS-SCLs with libraries containing individually synthesised peptides (sparse matrix library; SML) was carried out. These SML libraries were designed to include all possible sequence combinations of the residues that were suggested to be preferred by a protease using the PS-SCL method. SML screening against the three serine proteases kallikrein 4 (KLK4), kallikrein 14 (KLK14) and plasmin revealed highly preferred peptide substrates that could not have been deduced by PS-SCL screening alone. Comparing protease subsite preference profiles from screens of the two types of peptide libraries showed that the most preferred substrates were not detected by PS SCL screening as a consequence of intermolecular cooperativity being negated by the very nature of PS SCL screening. Sequences that are highly favoured as result of intermolecular cooperativity achieve optimal protease subsite occupancy, and thereby interact with very specific determinants of the protease. Identifying these substrate sequences is important since they may be used to produce potent and selective inhibitors of protolytic enzymes. This study found that highly favoured substrate sequences that relied on intermolecular cooperativity allowed for the production of potent inhibitors of KLK4, KLK14 and plasmin. Peptide aldehydes based on preferred plasmin sequences produced high affinity transition state analogue inhibitors for this protease. The most potent of these maintained specificity over plasma kallikrein (known to have a very similar substrate preference to plasmin). Furthermore, the efficiency of this inhibitor in blocking fibrinolysis in vitro was comparable to aprotinin, which previously saw clinical use to reduce perioperative bleeding. One substrate sequence particularly favoured by KLK4 was substituted into the 14 amino acid, circular sunflower trypsin inhibitor (SFTI). This resulted in a highly potent and selective inhibitor (SFTI-FCQR) which attenuated protease activated receptor signalling by KLK4 in vitro. Moreover, SFTI-FCQR and paclitaxel synergistically reduced growth of ovarian cancer cells in vitro, making this inhibitor a lead compound for further therapeutic development. Similar incorporation of a preferred KLK14 amino acid sequence into the SFTI scaffold produced a potent inhibitor for this protease. However, the conformationally constrained SFTI backbone enforced a different intramolecular cooperativity, which masked a KLK14 specific determinant. As a consequence, the level of selectivity achievable was lower than that found for the KLK4 inhibitor. Standard mechanism inhibitors such as SFTI rely on a stable acyl-enzyme intermediate for high affinity binding. This is achieved by a conformationally constrained canonical binding loop that allows for reformation of the scissile peptide bond after cleavage. Amino acid substitutions within the inhibitor to target a particular protease may compromise structural determinants that support the rigidity of the binding loop and thereby prevent the engineered inhibitor reaching its full potential. An in silico analysis was carried out to examine the potential for further improvements to the potency and selectivity of the SFTI-based KLK4 and KLK14 inhibitors. Molecular dynamics simulations suggested that the substitutions within SFTI required to target KLK4 and KLK14 had compromised the intramolecular hydrogen bond network of the inhibitor and caused a concomitant loss of binding loop stability. Furthermore in silico amino acid substitution revealed a consistent correlation between a higher frequency of formation and the number of internal hydrogen bonds of SFTI-variants and lower inhibition constants. These predictions allowed for the production of second generation inhibitors with enhanced binding affinity toward both targets and highlight the importance of considering intramolecular cooperativity effects when engineering proteins or circular peptides to target proteases. The findings from this study show that although PS-SCLs are a useful tool for high throughput screening of approximate protease preference, later refinement by SML screening is needed to reveal optimal subsite occupancy due to cooperativity in substrate recognition. This investigation has also demonstrated the importance of maintaining structural determinants of backbone constraint and conformation when engineering standard mechanism inhibitors for new targets. Combined these results show that backbone conformation and amino acid cooperativity have more prominent roles than previously appreciated in determining substrate/inhibitor specificity and binding affinity. The three key inhibitors designed during this investigation are now being developed as lead compounds for cancer chemotherapy, control of fibrinolysis and cosmeceutical applications. These compounds form the basis of a portfolio of intellectual property which will be further developed in the coming years.

Using incomplete online metric maps for topological exploration with the Gap Navigation Tree

This paper presents a general, global approach to the problem of robot exploration, utilizing a topological data structure to guide an underlying Simultaneous Localization and Mapping (SLAM) process. A Gap Navigation Tree (GNT) is used to motivate global target selection and occluded regions of the environment (called “gaps”) are tracked probabilistically. The process of map construction and the motion of the vehicle alters both the shape and location of these regions. The use of online mapping is shown to reduce the difficulties in implementing the GNT.

Variational Bayes and the reduced dependence approximation for the autologistic model on an irregular grid with applications

Discrete Markov random field models provide a natural framework for representing images or spatial datasets. They model the spatial association present while providing a convenient Markovian dependency structure and strong edge-preservation properties. However, parameter estimation for discrete Markov random field models is difficult due to the complex form of the associated normalizing constant for the likelihood function. For large lattices, the reduced dependence approximation to the normalizing constant is based on the concept of performing computationally efficient and feasible forward recursions on smaller sublattices which are then suitably combined to estimate the constant for the whole lattice. We present an efficient computational extension of the forward recursion approach for the autologistic model to lattices that have an irregularly shaped boundary and which may contain regions with no data; these lattices are typical in applications. Consequently, we also extend the reduced dependence approximation to these scenarios enabling us to implement a practical and efficient non-simulation based approach for spatial data analysis within the variational Bayesian framework. The methodology is illustrated through application to simulated data and example images. The supplemental materials include our C++ source code for computing the approximate normalizing constant and simulation studies.