Probabilistic Modeling for Decision Support in Integrated Operations

Chapter 5 - Decision Analysis Fundamentals

Influence diagrams are a powerful graphic representation for decision models, complementary to decision trees. Influence diagrams and decision trees are different graphic representations for the same underlying mathematical model and operations. This article describes the elements of an influence diagram, and shows several familiar decision problems represented as decision trees and as influence diagrams. The authors also contrast the information highlighted in each graphic representation, demonstrate how to calculate the expected utilities of decision alternatives modeled with an influence diagram, provide an overview of the conceptual basis of the solution algorithms that have been developed for influence diagrams, discuss the strengths and limitations of influence diagrams relative to decision trees, and describe the mathematical operations that are used to evaluate both decision trees and influence diagrams. They use clinical examples to illustrate the mathematical operations of the influence-diagram-evaluation algorithm; these operations are arc reversal, chance node removal by averaging, and decision node removal by policy determination. Influence diagrams may be helpful when problems have a high degree of conditional independence, when large models are needed, when communication of the probabilistic relationships is important, or when the analysis requires extensive Bayesian updating. The choice of graphic representation should be governed by convenience, and will depend on the problem being analyzed, on the experience of the analyst, and on the background of the consumers of the analysis.

The high-level decision-making is time-consuming in a large, complex circumstance. The decision-maker needs to consider many factors whatever they are in same domain or not. Influence diagram (ID) is an effective tool to help people make a strategic decision. In a large influence diagram, the high-level decision-maker doesn't need to know the details, and he only needs to know the overall effects of the decisions made in each domain. Starting from this real application, it is necessary to aggregate the nodes in a domain into one block in an influence diagram. In this paper, we are to discuss the aggregation of nodes in influence diagrams. In order to aggregate nodes, we have to partition the influence diagram into blocks first. Each block is composed of nodes which contact each other strongly and contact other nodes loosely. Since the influence diagram has different types of nodes, we have to partition these types of nodes respectively. Based on the relationships among nodes, we aggregate each block into a new node. Further, we combine the blocks with different types of nodes and get a new influence diagram. Preliminary experiments show the feasibility of our proposed methods.

TWO ALGORITHMS THAT LEARN DNF BY DISCOVERING RELEVANT FEATURES

Conversion of a colorectal cancer guideline into clinical decision trees with assessment of validity

Motivation and Objectives Neuroblastoma (NB) is the most common pediatric solid tumor characterized by clinical and molecular risk factors. The mortality is about fifty percent and this makes exploration of new and more effective risk factors for improving stratification mandatory. Hypoxia is a condition of low oxygen tension occurring in poorly vascularized areas of the tumor associated with poor prognosis. We had previously defined a robust gene expression signature measuring the hypoxic component of NB tumors (NB-hypo) that is a novel, independent risk factor (Fardin et al., 2010). Integrating classical risk factors with NB-hypo could improve the stratification of NB patients. We wanted to develop a prognostic classifier of NB patients’ outcome blending existing knowledge on clinical and molecular risk factors with the prognostic NB-hypo signature. Furthermore, we were interested in the decision tree classifier that outputs explicit rules easily translated into the clinical setting. Methods A total of 182 NB patients were enrolled on the bases of availability of gene expression profile by Affymetrix GeneChip HG-U133plus2.0, clinical and molecular information. NB tumor stage was defined according to the International NB Staging System (INSS). Age at diagnosis was dichotomized as greater or equal than 1 year and less than one year. MYCN status was amplified or normal. Good and poor outcome were defined as the patient’s status alive or dead 5 years after diagnosis respectively. The risk group was assigned according to the International Neuroblastoma Risk Group (INRG) Consensus Pretreatment Classification Schema. The 182 NB patients cohort was clustered in High and low hypoxia by k-means analysis of the 62 probsets constituting the NB-hypo signature previously described to measure tumor hypoxia (Fardin et al., 2009). We utilized the k-means algorithm implemented in the WEKA software (Hall et al., 2009) setting up number of clusters to 2, 500 iterations, preserving instances order and using Manhattan distance. The classification was performed by induction of decision trees. We utilized the Weka J48 implementation of the popular C4.5 algorithm (Kotsiantis, 2007; Murthy, 1998) and we set up the following options: pruning parameter was 0.25, pruning method was sub-tree raising and minimum number of instances per leaf was 2. Each leaf of the decision tree classifier identifies a non overlapping group of patients and each decision node identifies a branch which splits the dataset. We utilized Fisher’s exact test to measure the statistical significance of groups and branches. Fisher’s test was utilized in the context of decision trees to design a top-down approach to prune out non statistically significant branches (Liu et al., 2010). For each leaf we counted the number of correctly classified (named n) and the number of incorrectly classified instances (named m). We considered the marginal totals y and ¬y which represent poor and good outcome patients respectively. We designed a 2x2 contingency table of the two possible outcomes (Good or Poor) against the number of instances included in a give leaf and the remaining instances. Application of the Fisher exact test to this table generates a p value giving the probability of observing 2x2 table, or more extreme tables, knowing the marginal totals (y, ¬y) and assuming independency among the patients in a specific leaf and those in other leaves. Results and Discussion Patients were stratified in good and poor outcome on the bases of the following risk factors: Age at diagnosis, INSS stage, MYCN status and NB-hypo. The algorithm generated a decision tree classifier composed by 3 decision nodes and 7 leaves covering 87% of non overlapping good outcome patients and 100% of non overlapping poor outcome patients. Each path from the root to a leaf utilizes some, but not all, considered risk factors. Interestingly NB-hypo was included in the decision node that stratified stage 3 tumors demonstrating its usefulness in NB patients’ stratification. The leaves classifying good outcome patients had the very low error of 2% indicating a good performance of the algorithm in predicting this class. In contrast, the classification of poor outcome patients produced the leaf with the highest error of 13%. This leaf includes stage 4 tumors that are traditionally difficult to stratify by any known risk factor. To test statistical significance of the splits performed by the algorithm and the groups of patients, we utilized the Fisher’s exact test with a confidence set at 95%. The results showed that the groups obtained at each split were statistically significant. Furthermore, analysis of single groups of patients identified by leaves demonstrated that some, but no all reached the significance threshold. The significant leaves included the NB-hypo among the utilized risk factors. These results further strengthen the value of NB-hypo in predicting patients’ outcome. Lack of significance was often associated with a rather low number of patients in the leaf. This is a limitation of the divide-and-conquer approach of the algorithm, applied to relatively small patients’ cohorts, that recurrently splits the dataset. We then assessed the concordance of the predictions with INRG risk assessment. High Risk patients were correctly included in the leaves classifying poor outcome patients and Low Risk patients mapped correctly in the good outcome leaves. Interestingly, NB-hypo generated a leaf identifying a new group of poor outcome patients, sharing the high NB-hypo, whose characteristic fell into both High and Intermediate Risk. We collected and analyzed the results of 1000 10-fold cross validations and we observed that most classifiers had 7 leaves and only 5 out of 10^4 deviated from this pattern. The recurrence of 7 leaves demonstrated the high stability of the decision tree classifier that we generated. Analysis of the pruning parameters revealed optimal performance in the range of 0.1-0.3 in line with what used in this study. The path to reach each leaf can be easily transposed into a “if…than…” rule that, in turn provides an easy readout of the classifier, precious for translating the classification into the clinical setting. In conclusion, we demonstrated that the decision tree algorithm C4.5 can derive explicit rules for NB patients stratification if classical risk factors are blended with the NB hypo signature. These rules are statistically significant and quite stable and suitable to be conveyed to the clinic to design new therapies perhaps taking hypoxia into consideration as a potential target. Acknowledgements The work was supported by the Fondazione Italiana per la Lotta al Neuroblastoma, the Associazione Italiana per la Ricerca sul Cancro, the Società Italiana Glicogenosi, the Fondazione Umberto Veronesi and the Ministero della Salute Italiano. Davide Cangelosi and Fabiola Blengio are recipients of a fellowship from the Fondazione Italiana per la Lotta al Neuroblastoma. References Fardin P, Barla A, Mosci S, Rosasco L, Verri A, Varesio L. (2009) The l1-l2 regularization framework unmasks the hypoxia signature hidden in the transcriptome of a set of heterogeneous neuroblastoma cell lines. BMC Genomics, 10:474. doi:10.1186/1471-2164-10-474 Fardin P, Barla A, Mosci S, Rosasco L, Verri A, et al. (2010) A biology-driven approach identifies the hypoxia gene signature as a predictor of the outcome of neuroblastoma patients, Journal of Molecular Cancer 9:1. 185. doi:10.1186/1476-4598-9-185 Hall M, Eibe F, Holmes G, Pfahringer B, Reutemann P, Witten IH. (2009) The WEKA Data Mining Software: An Update: SIGKDD Explorations. doi: 10.1145/1656274.1656278 Kotsiantis S.B. (2007) Supervised Machine Learning: A Review of Classification Techniques. Informatica 31:249-268. Liu W, Chawla S, Cieslak D, Chawla N. (2010) A Robust Decision Tree Algorithm for Imbalanced Data Sets. In: SDM. 766-777. Murthy S.K. (1998) Automatic Construction of Decision Trees from Data: A Multi-Disciplinary Survey, Data Mining Knowledge Discovery 2:4. 345-389. doi: 10.1023/A:1009744630224

Decision Model for Acute Appendicitis Treatment With Decision Tree Technology—A Modification of the Alvarado Scoring System

Warning and evacuation are among the most effective ways for saving human lives and properties from landslide dam hazards. A new warning decision model for landslide dam break is developed using Influence Diagrams to minimize the total losses. An Influence Diagram is a simple visual representation of a decision problem. It analyzes the qualitative (causal) relationships between the variables via a logic diagram and determines the quantitative relationships via conditional probability and Bayes’ theorem. The model is applied for the warning decision-making of the 2008 Tangjiashan landslide dam. The new model unifies the dam failure probability, evacuation, life loss, and flood damage in an Influence Diagram. Besides, a warning criterion is proposed for efficient decision-making. The model is more advanced than the decision tree since the inter-relationships of influence factors are qualitatively analyzed with causality connections and quantitatively analyzed with conditional probabilities. It is more efficient than a dynamic decision-making model (DYDEM) as it can directly calculate the three types of flood loss (i.e., evacuation cost, flood damage, and monetized life loss) and the expected total loss. Moreover, the probabilities of the influence factors leading to known results can be obtained through inversion analysis based on Bayesian theory. The new warning decision model offers an efficient way to save lives from landslide dam breaking and avoid unnecessary expenses from premature warning and evacuation.

Dynamic decision making without expected utility: An operational approach

We investigate the problem of learning Boolean functions with a short DNF representation using decision trees as a concept description language. Unfortunately, Boolean concepts with a short description may not have a small decision tree representation when the tests at the nodes are limited to the primitive attributes. This representational shortcoming may be overcome by using Boolean features at the decision nodes. We present two new methods that adaptively introduce relevant features while learning a decision tree from examples. We show empirically that these methods outperform a standard decision tree algorithm for learning small random DNF functions when the examples are drawn at random from the uniform distribution.

AimsThis study aimed to clarify the different topographical distribution of tau pathology between progressive supranuclear palsy (PSP) and corticobasal degeneration (CBD) and establish a machine learning‐based decision tree classifier.MethodsParaffin‐embedded sections of the temporal cortex, motor cortex, caudate nucleus, globus pallidus, subthalamic nucleus, substantia nigra, red nucleus, and midbrain tectum from 1020 PSP and 199 CBD cases were assessed by phospho‐tau immunohistochemistry. The severity of tau lesions (i.e., neurofibrillary tangle, coiled body, tufted astrocyte or astrocytic plaque, and tau threads) was semi‐quantitatively scored in each region. Hierarchical cluster analysis was performed using tau pathology scores. A decision tree classifier was made with tau pathology scores using 914 cases. Cross‐validation was done using 305 cases. An additional ten cases were used for a validation study.ResultsCluster analysis displayed two distinct clusters; the first cluster included only CBD, and the other cluster included all PSP and six CBD cases. We built a decision tree, which used only seven decision nodes. The scores of tau threads in the caudate nucleus were the most decisive factor for predicting CBD. In a cross‐validation, 302 out of 305 cases were correctly diagnosed. In the pilot validation study, three investigators made a correct diagnosis in all cases using the decision tree.ConclusionRegardless of the morphology of astrocytic tau lesions, semi‐quantitative tau pathology scores in select brain regions are sufficient to distinguish PSP and CBD. The decision tree simplifies neuropathologic differential diagnosis of PSP and CBD.

Multistage decision making is either under certainty or uncertainty data conditions. Expected monetary values’ perfect information and monetary loss or gain at each stage of decision making are the research topics of a constrained decision making under risk. In the multistage decision making process, one stage’s decision interferes with the subsequent stage’s decision. In this paper, a simple study on decision making for creating Avionic infrastructure within the Odisha region is emphasized by using the mathematical logic tool on a case study problem. In this case study, the first stage D1 has two options, either construct or not construct new airports with utilizing the budgeted allot money. Each option’s outcome probability, conditional value and expected outcome value are determined by applying arithmetic and logical calculations to next two stages D2 and D3. The expected value and percentage of returns are mentioned in the decision trees designed in Section 3. The nodes of the decision tree are stages or chance nodes and the arc lines’ labels are mentioned from the appropriate expressions written in the decision tree and chance nodes table mentioned in Section 4.

Discovery of interesting rules describing the behavioural patterns of smokers' quitting intentions is an important task in the determination of an effective tobacco control strategy. In this paper, we investigate a compact and simplified rule discovery process for predicting smokers' quitting behaviour that can provide feedback to build an scientific evidence-based adaptive tobacco control policy. Standard decision tree (SDT) based rule discovery depends on decision boundaries in the feature space which are orthogonal to the axis of the feature of a particular decision node. This may limit the ability of SDT to learn intermediate concepts for high dimensional large datasets such as tobacco control. In this paper, we propose a cluster based rule discovery model (CRDM) for generation of more compact and simplified rules for the enhancement of tobacco control policy. The cluster-based approach builds conceptual groups from which a set of decision trees (a decision forest) are constructed. Experimental results on the tobacco control data set show that decision rules from the decision forest constructed by CRDM are simpler and can predict smokers' quitting intention more accurately than a single decision tree.

We deal with issues concerning negotiation support for group decisions over influence diagrams, when the group members disagree about utility and probability assessments. We base our discussion on a modification of the balanced increment solution, which guarantees a final negotiated Pareto optimal alternative. As in standard decision analysis textbooks, we deal first with negotiation tables, then with negotiation trees, and finally with negotiation influence diagrams. We show through an example that a naive application of the balanced increment method at each joint decision node in a dynamic decision-making problem, and, more generally, of any standard negotiation approach guaranteeing Pareto optimality, may lead to an inferior solution. Therefore, our strategy proposes computing first the set of nondominated alternatives followed by negotiating over that set.

Influence diagrams were developed as a graphical representation for formulating a decision analysis model, facilitating communication between the decision maker and the analyst. We show several approaches for evaluating influence diagrams, determining the optimal strategy for the decision maker and the value or certain equivalent of the decision situation when that optimal strategy is applied. They can be converted into decision trees, evaluated directly as influence diagrams, or converted into another graphical model, a rooted cluster tree, for analysis.