Decision Tree Analysis

Making Decisions

John J. Fay , David Patterson , in Contemporary Security Management (Fourth Edition), 2018

Analyze the Information

At this stage, all the obtainable relevant information will be on hand and ready for analysis. For analysis to be accurate, the information has to be in a common data set. It is confusing to compare apples against bananas. The most common data set is dollar values. With a common data set established, the data are quantified and then systematically evaluated.

The methods of evaluation vary according to the nature of the information and the practices and resources of the decision maker. Analytical tools include linear modeling, sensitivity analysis to highlight uncertainty, value and probability analysis, decision trees to display relationships between alternatives and outcomes, and ranking of probable outcomes. Mistakes at this stage of the process typically include the following:

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Taking mental shortcuts such as giving greater credence to the most recent or most loudly voiced opinions.

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Wasting valuable time by considering irrelevant information.

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Being overly confident about the quality of information.

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Relying on information that confirms predetermined expectations.

Emerging from the results of the analyses are the outlines of alternatives, singly and in combinations, which if carried out correctly can achieve the decision-maker's objectives. Said another way, the problem is described by its solution.

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Making Decisions

John J. Fay , in Contemporary Security Management (Third Edition), 2011

Analyze the Information

At this stage, all the obtainable relevant information will be on hand and ready for analysis. For analysis to be accurate, the information has to be in a common data set. It is confusing to compare apples against bananas. The most common data set is dollar values. With a common data set established, the data are quantified and then systematically evaluated.

The methods of evaluation vary according to the nature of the information and the practices and resources of the decision maker. Analytical tools include linear modeling, sensitivity analysis to highlight uncertainty, value and probability analysis, decision trees to display relationships between alternatives and outcomes, and ranking of probable outcomes. Mistakes at this stage of the process typically include the following:

•

Taking mental shortcuts such as giving greater credence to the most recent or most loudly voiced opinions.

•

Wasting valuable time by considering irrelevant information.

•

Being overly confident about the quality of information.

•

Relying on information that confirms predetermined expectations.

Emerging from the results of the analyses are the outlines of alternatives, singly and in combinations, which if carried out correctly can achieve the decision-maker's objectives. Said another way, the problem is described by its solution.

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Customer Data Analytics

David Loshin , Abie Reifer , in Using Information to Develop a Culture of Customer Centricity, 2013

Decision Trees

A decision tree is a decision-support model that encapsulates the questions and the possible answers and guides the analyst toward the appropriate result, and can be used for prediction models coupled with classification. Decision tree analysis looks at a collection of data instances and given outcomes, evaluates the frequency and distribution of values across the set of variables, and constructs a decision model in the form of a tree.

The nodes at each level of this tree represent a question, and each possible answer to the question is represented as a branch that points to another node at the next level. For example, we can have a binary decision tree where one internal node asks whether the customer's annual income is between $0 and $25,000, between $25,000 and $65,000, or greater than $65,000. If the answer is the first, the left-hand path is taken; if the second choice is selected, then the middle path is traversed; if the answer is the third, the right-hand path is taken. At each step along the path from the root of the tree to the leaves, the set of records that conform to the answers along the way continues to grow smaller.

Essentially, each node in the tree represents a way of subsetting the customer base in a way that conforms to the answers to the questions along the path one traverses to the specified node. Since each answer divides the set at that point by the number of possible answers, each node reflects a smaller community of individuals with a greater number of similarities among that subset. The predictive models can be integrated with the decision tree to drive specific actions at each customer touch point in reaction to the narrowed customer classes at each level of the tree.

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23rd European Symposium on Computer Aided Process Engineering

Yang Yang , ... Nina F. Thornhill , in Computer Aided Chemical Engineering, 2013

4 Results

The Monte Carlo simulation dataset generated by the discrete-event simulation tool has 1000 data records. Each data record represents one manufacturing batch. Before using the Monte Carlo simulation dataset as training dataset for decision tree analysis, each data record should be allocated a class label since decision tree classification is a supervised learning method ( Grajski, Breiman et al. 1986). The expected purification mass load from the 10,000L mAb upstream process was 20kg. Mass loss per batch exceeding 5% (1kg) of the expected mass load was considered a heavy mass loss. According to the quantity of product loss, each data record in the Monte Carlo simulation dataset was classified into one of three groups: no mass loss, light mass loss and heavy mass loss. The summary of the training dataset is shown in Table 2.

Table 2. Summary of training dataset for CART classification

Classes Description Number of records
No Batches with no mass loss at all 814
Light Batches with mass loss less than 1kg 105
Heavy Batches with mass loss equal to or more than 1kg 71

According to the 10-fold cross validation method, the training dataset was randomly divided into 10 disjoint subsets. Each subset has roughly equal size and roughly the same class proportions as in the training set. Using nine of the subsets, all possible combinations of trees were developed and these were then tested on the 10th subset. This result provides a cross-validation error rate, which gives an equitable evaluation of the predictive precision of tree models of different sizes. Figure 1 shows the resubstitution error rate and cross-validation error rate versus size of decision tree.

Figure 1. Selection of optimal decision tree based on resubstitution error rate and cross-validation error rate.

Resubstitution error describes how well the decision tree fits the training dataset while cross-validation error describes the prediction ability of the decision tree. A larger tree has a smaller resubstitution error but can cause overfitting. The optimal tree with the minimum cross-validation error and tolerance to resubstitution error is selected which is shown in Figure 2. The CART tree model reveals that the key process fluctuations leading to mass loss in this process are product titre, Protein A eluate volume and AEX eluate volume. This reinforces observations in Stonier et al, (2011) that found out the mass loss happened due to tank volume limitations in post Protein A UFDF and post AEX UF where the eluate volumes were collected prior to further concentration. Furthermore, from top to bottom along the branch to each leaf node of the tree, the "if-then" rules can be generated to describe and predict the mass loss degree caused by critical combinations of key process fluctuations. For example, the left branch of the CART tree indicates that if the eluate volume in AEX<3.8CV (column volume) and the eluate volume in Protein A chromatography <2.5CV, then there is no mass loss in the process.

Figure 2. The optimal CART decision tree based on a Monte Carlo simulation dataset. Numerical values are the threshold levels of the split points for the corresponding split conditions. Rectangle nodes are branch nodes which represent the process parameters leading to split. Circle nodes are leaves representing subsets with different class labels for mass loss as no, light or heavy mass loss. The number in each leaf represents the percentage of observations in the leaf.

These decision tree prediction results are applicable to the particular manufacturing process described in this simulation work and the number of correctly classified observations in each leaf is an indicator of the confidence level in the predictions. For example, the leaf with heavy mass loss when the AEX eluate volume is above 3.8CV contains 89.2% correctly classified batches. The explanation for misclassification is that the Monte-Carlo simulation includes fluctuations of other process parameters such as step yield and flux rate which also have an impact on product mass loss. However, the impacts of these fluctuations are not dominant compared to titre, Protein A eluate volume and AEX eluate volume, so that they are ignored in order to generate the decision tree.

In order to clearly display the relationship between mass loss distribution and key process parameters based on the decision tree prediction model, windows of operation of titre vs. protein A chromatography eluate volume under different AEX eluate volume are generated in Figure 3. Generally speaking, batches with heavy mass losses occur when the titre, affinity eluate volume and AEX eluate volume values are high, while batches with no mass loss occur at the opposite extremes. However, compared to the Protein A eluate volume, the AEX eluate volume plays a more decisive role in high and light mass loss distribution as indicated in the decision tree in Figure 2. This observation can be attributed to the larger number of column volumes (CVs) used for the eluate volume collection in AEX versus Protein A chromatography (three versus two CVs) combined with the larger bed volume (196L versus 157L). These two parameters determine that fluctuations in the AEX eluate volume are more likely to lead to tank volume limitations and hence mass loss. In Figure 3a, when the AEX eluate volume is below 3.8 CV, mass loss happens only when the titre is greater than 1.97 g/L and the affinity eluate volume is above 2.5 CV while heavy mass loss happens when the affinity eluate volume is larger than 2.7 CV. When the AEX elute volume is above 3.8 CV, the degree of product mass loss was found to be affected by fluctuation in titre only irrespective of the affinity eluate volume fluctuations as illustrated in Figures 3b and 3c. In Figure 3b, when the AEX eluate volume lies within 3.8-4.0 CV, no or light mass loss occurs depending on the titre level being below or above 1.96 g/L respectively. In Figure 3c, when the AEX eluate volume is in the range of 4.0-4.5 CV, heavy mass loss happens when the titre is higher than 1.96 g/L.

Figure 3. Windows of operation indicating critical combinations of titre and protein A chromatography eluate volume under different AEX eluate volumes that drive mass loss levels. Black area represent batches with heavy mass loss, dark gray area represent batches with light mass loss and light gray area represent batches with no mass loss

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Marketing Information Systems

Robert R. Harmon , in Encyclopedia of Information Systems, 2003

II.E.b. Data Mining Tasks

After the data have been collected and reside in the data warehouse, approaches to analyzing the data are considered. Data mining methodologies may encompass a range of approaches from rigorous scientific methodologies and hypothesis testing to qualitative sifting through massive amounts of data in search of relationships. The type of analysis typically is a function of the task the researcher wants the data mining exercise to accomplish. Typical data mining tasks include:

1.

Classification . A predetermined classification code is assigned to a database record. Decision tree analysis techniques are commonly associated with this task.

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Estimation. Input data are used to estimate continuous variables such as age, income, and likely behaviors. Neural networks are often used for estimation.

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Affinity grouping. Rules of association are developed from the data and used to group variables that seem to go together. Market basket analysis is a preferred technique that analyzes the linkages between items consumers buy in a basket of items.

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Description. Summary observations are made about the data that serve to increase the understanding of the phenomena that generated the data. Description often motivates further research and data analysis. Market basket analysis, query tools, and visualization (mapping) techniques are commonly used.

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Clustering. Clustering is used to segment a large heterogeneous population into homogeneous clusters based on measures of similarity. The researcher must determine the meaning of each cluster. Clustering algorithms are used to analyze the data.

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Prediction. Records are classified based on predicted future values or behaviors. Neural networks, market basket analysis, and decision trees are common techniques.

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Psychopharmacotherapy, Costs of

J. Fritze , in International Encyclopedia of the Social & Behavioral Sciences, 2001

3 Pharmacoeconomic Study Designs

There are two principle approaches to pharmacoeconomic analysis: the clinical trial approach (experimental or observational), and the modeling or systems approach. The former means gathering cost data (retrospectively or, preferably, prospectively) in clinical trials. The latter 'in vitro' approach applies decision-tree analysis to data available from published literature and databases.

Current guidelines favor the clinical trial approach as the 'gold standard.' Modeling, however, can overcome limitations implicit in clinical trials. These include the selection bias concerning the patients recruited in randomized, strictly controlled, and therefore artificial efficacy trials, thus limiting generalizability. Furthermore, their observation period rarely exceeds a few months whereas economic consequences may extend far into the future. Efficacy trials necessarily are underpowered for economic analysis because sample size estimation is based on the critical difference and variability of the efficacy parameter. Obviously, the critical difference and variability of economic data will fundamentally differ. Clinical trials using the mirror image design, i.e., comparing costs and benefits in a defined period before and after administering a specific drug, are of limited value because at least in psychiatric disorders there is a spontaneous decrease of the costs per outcome during the course of illness irrespective of the mode of treatment.

The putatively best experimental design is a prospective, randomized, 'naturalistic,' i.e., non-blind, adequately powered clinical trial. Both principle approaches, the clinical trial approach and modeling, have their weaknesses and strengths so that they are not really exclusive alternatives but complementary.

Friedberg et al. (1999) found that studies sponsored by drug companies were less likely to report unfavorable results than studies funded by nonprofit organizations. This finding could be due to various sources of bias which, however, if present, still have to be identified.

The organization of health-care supply, the health-care utilization patterns, and the prices differ considerably between and even within cultures. Therefore, the results of pharmacoeconomic analyses cannot simply be transferred from one health care market to another.

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GIS Applications for Environment and Resources

Federica Lucà , ... Oreste Terranova , in Comprehensive Geographic Information Systems, 2018

2.03.4 Digital Soil Mapping

DSM can be used to predict both categorical variables (e.g., soil taxonomic classes) and quantitative soil properties by applying various mathematical and statistical approaches from the local (i.e., single field) to regional scale. Such approaches can be summarized into three main categories such as classical statistics, geostatistics, and hybrid methods (McBratney et al., 2000). The classical statistical methods deal with deterministic relations between soil properties and auxiliary variables, but they do not account for spatial autocorrelations of data, especially at the local level. To address this issue, geostatistical methods have been developed. Geostatistics (Matheron, 1971) allows to quantify the spatial variability of soil producing continuous maps starting from sparse data. Geostatistical interpolation techniques can be used even without ancillary information, in case of sufficient data of the soil property of interest within the study area. Since soil properties are the result of environmental covariates, it may be beneficial modeling both (i) the deterministic component of soil spatial variation as a function of the environmental covariates, and (ii) any residual stochastic component. Hybrid methods derive from the combination of both classical and geostatistical techniques and therefore account for both deterministic and stochastic components of soil variability.

Fig. 2 summarizes the main features characterizing the three aforementioned groups. Each of them includes a variety of quantitative available techniques. A detailed description of such approaches goes beyond the scope of the article; interested readers may refer to specific papers (McBratney et al., 2003; Scull et al., 2003). Below, we briefly describe the most commonly applied for predicting soil classes or quantitative properties.

Fig. 2. Schematic overview of pedometric approaches for digital soil mapping.

2.03.4.1 Statistical Methods

Among statistical methods, multiple regression analysis has been the most commonly applied for assessing the relationship between a soil property (dependent variable) and several morphometric attributes as independent predictors (Moore et al., 1993; Gessler et al., 2000). The approach assumes a linear relationship between soil and topography but the simplicity of data processing, model structure, and interpretation explain its wide application for predicting several quantitative soil properties. Regression has been used for example to assessing soil horizon thickness (Moore et al., 1993; Odeh et al., 1994; Gessler et al., 2000; Florinsky et al., 2002). The relationships between soil properties and other topographic or biophysical variables are rarely linear in nature. Such consideration has led to the application of more robust methods such as generalized linear models (GLM) and generalized additive models (GAM). GLM are used both for regression and classification purposes. The assumption is that the dependent variable is normally distributed and that the predictors combine additively on the response. Aside from being able to handle multiple distributions, GLM have additional benefits, such as being able to use both categorical and continuous variables as predictors. Thanks to their ability to model complex data structures, GLM models have been widely applied. In GAM, the linear function between soil properties and topographic covariates is replaced by an unspecified nonparametric function (e.g., spline). Artificial neural network (ANN) is a nonparametric modeling technique used to overcome the nonlinearity in the relationships characterizing the soils. The ANN is a form of artificial intelligence that can use both qualitative and quantitative data. It aims autoanalyzing the relationships between multisource inputs by adopting self-learning methods, and works without any hypothesis on the statistical distribution of variables. Zhao et al. (2009) developed an ANN model to predict soil properties based on hydrological attributes (soil–terrain factor, sediment delivery ratio, and vertical slope position) derived from high-resolution DEM. Fuzzy logic (Zadeh, 1965; McBratney and Odeh, 1997) is a method for grouping multivariate data into clusters, defining the membership of an element belonging to a set of classes. Different from hard logic that allows an individual to lie within a mutually exclusive class, fuzzy logic (sometimes called fuzzy k-means) allows an individual to lie as bridges between classes. Since soil landscapes are characterized by continuous nature, fuzzy logic is useful in predictive soil mapping. Fuzzy logic has been used, for example, to cluster topographic attributes (elevation, slope, plan curvature, TWI, SPI, catchment area) derived from a 5   m DEM in order to predict topsoil clay at field scale (de Bruin and Stein, 1998). The method has resulted useful for predicting chemical properties such as soil mineral nitrogen, organic matter, available phosphorus, and soil pH (Lark, 1999) at the field scale and soil taxonomic classes in large-scale soil mapping (Odeh et al., 1992; Lark, 1999; Barringer et al., 2008). Combination of fuzzy logic with discriminant analysis is also reported in the literature (Sorokina and Kozlov, 2009).

Decision trees work by splitting data into homogeneous subsets. Two main types of the decision tree analysis are used in DSM: classification tree analysis (the dependent variable is categorical) and regression tree analysis (the dependent property is a numeric variable). Classification tree has been applied for predicting soil drainage class using digital elevation and remote sensed data ( Cialella et al., 1997) or soil taxonomic classes (Lagacherie and Holmes, 1997; McBratney et al., 2000; Moran and Bui, 2002; Zhou et al., 2004; Scull et al., 2005; Mendonça-Santos et al., 2008). Regression tree has instead been used for predicting soil cation exchange capacity (Bishop and McBratney, 2001), soil profile thickness, total phosphorus (McKenzie and Ryan, 1999).

Discriminant analysis is used to assess the group membership of an individual based on the attributes of the individual itself. This method allows to determine the attributes adequate to discriminate between classes using a multivariate dataset. The approach has been used to map soil texture classes (Hengl et al., 2007), soil drainage classes (Kravchenko et al., 2002), and taxonomic classes (Thomas et al., 1999; Hengl et al., 2007).

Logistic regression is used to predict a categorical variable from a set of both continuous and/or categorical predictors (Kleinbaum et al., 2008). Logistic regression can be binary or multinomial, based on the number of soil categories to be predicted. For example, multinomial logistic regression has been used to predict soil taxonomic classes or soil texture (Hengl et al., 2007; Giasson et al., 2008). Binary logistic regression has instead been used to assess the presence or absence of specific horizon (Gessler et al., 1995), soil salinity risk (Taylor and Odeh, 2007), and gully erosion (Lucà et al., 2011; Conoscenti et al., 2014).

2.03.4.2 Geostatistical Methods

Geostatistical methods (Matheron, 1971) provide a valuable tool to study the spatial structure of soil properties; geostatistics, in fact, allows one modeling the spatial dependence between neighboring observations as a function of their distance. The mathematical model of spatial correlation is expressed by the variogram. The information provided by variograms is used in one of the different techniques of spatial interpolation (known as kriging) to estimate the variable at unsampled locations. The variographic analysis allows to detect anisotropic behavior of soil variables. The advantage of anisotropic modeling lies in its capability of disclosing important changes in spatial dependence with a particular direction which, in turn, is a function of soil-forming processes. Variogram modeling is sensitive to strong departures from normality, because a few exceptionally large values may contribute to very large squared differences. Geostatistical analysis is therefore most efficient when variables have Gaussian distribution and requires an assumption of data stationarity but such condition is not always verified.

Kriging provides the "best," unbiased, linear estimate of a variable where "best" is defined in a least-square sense (Webster and Oliver, 2007; Chilès and Delfiner, 2012). Ordinary kriging is one of the most commonly used kriging method. The estimation involves only the primary soil variable and the method provides a kriging variance or its square root, the kriging error, which can guide to the reliability of the estimate. Ordinary kriging testifies that geostatistical interpolation techniques can be used at the regional level, even without ancillary data, especially if there is sufficient data in some localities within the study area.

Every kriging algorithm essentially leaves the job of spatial pattern reproduction unfinished because the kriging map is unique and smooth (Caers, 2003). Aiming at modeling heterogeneity and assessing uncertainty of soil variable at unsampled locations, the kriging should be replaced by a set of alternative maps, which honor sample measurements and try to reproduce the true spatial variability of soil properties. Stochastic simulation (Journel and Alabert, 1989; Goovaerts, 2001) represents an alternative modeling technique, which is particularly suitable for applications where global statistics are more important than local accuracy. Simulation consists in computing a set of alternative stochastic images of a random process and then carrying out an uncertainty analysis that is inadequate in the classical methods (Castrignanò et al., 2002; Buttafuoco et al., 2012). Each simulation is an equally probable realization of the unknown soil property and the postprocessing of a large set of simulated images allows assessing uncertainty and evaluating the consequences of data uncertainty on decision-making. The probability of exceeding a particular threshold value can be computed from a set of simulations by counting the percentage of stochastic images that exceed the stated threshold. For example, Lucà et al. (2014) used 500 stochastic images to assess the probability that soil thickness value did not exceed a predefined threshold, useful for identifying where surficial landslide could occur. The approach also allowed delineating the areas characterized by greater uncertainty in pyroclastic thickness estimation, suggesting supplementary measurements to further improve the cover thickness distribution model, thus reducing the uncertainty.

Due to high cost of getting accurate and quantitative information, a high value can be relied upon the available data at different scales.

2.03.4.3 Hybrid Methods

The hybrid geostatistical techniques are based on various combinations of classical statistics and geostatistical methods and use ancillary variables such as landscape attributes or sensed data for soil property estimation (Webster and Oliver, 2007). There are several hybrid geostatistical approaches and the choice of the best method depends on the specific case study, data availability, and presence of a spatial trend.

Cokriging considers two or more variables contemporarily but needs that variables be related. Moreover, it is computing demanding because of the modeling of simple, and cross variograms which describe the correlation of pairwise variables.

Kriging with external drift is similar to universal kriging but uses ancillary variable to describe the spatial changes in the relationship between variables (however such relationship must be linear). Regression kriging combines linear regression models with ordinary or simple kriging (Webster and Oliver, 2007) and is in turn used to interpolate the residuals. Both topographic attributes and electromagnetic data have been used as auxiliary variables to improve the estimation of soil texture and other properties related to soil fertility, in the regression kriging (Moral et al., 2010) or kriging with external drift techniques (De Benedetto et al., 2012).

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Dynamic pricing techniques for Intelligent Transportation System in smart cities: A systematic review

Sandeep Saharan , ... Neeraj Kumar , in Computer Communications, 2020

5.1.1 Mathematical analysis based techniques

Fares can be determined optimally by the techniques such as calculus of variations, dispersion and decision tree analysis [29,30,54]. Mathematical analysis plays critical role in the estimation. Calculus of variations has been used to sell airline tickets where continuous competition exists[29]. It find out the maxima or minima of a function using variations, which captures the small change in the function. Probabilistic formulated demands have also been analyzed. Such demands can get affected by the change in the prices due to competition and also with time remaining to sell tickets. Dispersion (variance, standard deviation or range) which is the stretch or squeeze in the distribution, has been used to analyze demand of the air tickets[54]. As per the study, dispersion can be influenced by factors such as population, and income. This analysis has been done on the air fare data from online travel website, Farecast.com. Similarly, in the railways fare system, Decision Tree Analysis (DTA) which uses tree structure of conditions and results to analyzed data, has been used to study conditions where loyal group's demand goes out of their booking limits[30]. In order to generate high revenue, optimization has been done. The effect of surge multipliers have also been explored using linear model with L1 regularization, also called as LASSO which predicted the surge multipliers in the areas of Pittsburgh[55]. This type of study can be useful and affect the rider or driver behavior when they are aware of such predictions. Such affect in behavior in turn can affect the surge multipliers. Evaluation of pricing strategy of cab aggregators in India has also been area of research in this field[56]. In this study researchers evaluated, how surge prices were changed across different facilitators and also in between different time of the day for any particular facilitator. Advance knowledge of flight fares affects traveler's decision and subsequently pricing decision taken by the flight operators[57]. In this analysis flight fares have been predicted by linear model using Bayesian estimation technique, i.e., Kalman filter which gave low average error with high accuracy. Price analysis of Flixbus-a long distance bus market has also been done[58] which shows increase in fares with increase in number of seats sold. Even the lowest available fares have been shown to be increased as the departure date approaches.

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Real Options in Operations Research: A Review

Lenos Trigeorgis , Andrianos E. Tsekrekos , in European Journal of Operational Research, 2018

3.6 Theme F: valuation and other topics

The last theme combines work on valuation and various other topics not covered above (see Table S6 in the Supplement). The link between RO valuation and traditional decision tree analysis (DTA) is examined in Brandão and Dyer (2005), while Reyck, Degraeve, and Vandenborre (2008) integrate DTA and the certainty-equivalent version of NPV in a RO framework that does not rely on the 'replicating portfolio-spanning' assumption. Wallace (2010) further discusses similarities and differences between RO theory and stochastic programming.

In a general project valuation setting, Keswani and Shackleton (2006) show how incremental levels of investment and divestment flexibility enhance the project's NPV, while Borgonovo, Gatti, and Peccati (2010) propose a novel measure for conducting sensitivity analysis in the valuation of large-scale industrial projects. In terms of RO valuation methods, Hahn and Dyer (2008) show how the work of Nelson and Ramaswamy (1990) on recombining binomial lattices can be extended and applied to model dual correlated mean-reverting processes, typically used to evaluate switching options in natural resources/energy. Wahab and Lee (2011) propose a multi-layered, multinomial lattice approach that discretizes the process followed by a regime-switching underlying asset (following a different gBm in each of the n regimes). They apply this to swing options to change the volume of periodic deliveries of a commodity on certain dates in the future. Folta et al. (2010) analyze hybrid entrepreneurship, i.e. an individual's incremental transition (as opposed to full, one-off immersion) to self-employment from a waged job, as a real switching option. Using matched employee–employer data, the authors examine a sample of Swedish wage earners from the knowledge-intensive sector, providing evidence that such hybrid entrepreneurial forms represent a significant proportion of transitions into and out of entrepreneurship, validating RO theory predictions.

More recently, Wang and Dyer (2102) propose a general framework based on copulas for modeling dependent multivariate uncertainties with arbitrary distributions via the combined use of decision trees and real options. Their dependent decision-tree approach approximates the underlying copula with uniform variables and then transforms it into the desired multivariate probability tree which possesses the required marginal distributions and correlation structure.

Zmeškal (2010) combines fuzzy set theory with binomial real options valuation where all lattice inputs are given vaguely in the form of fuzzy numbers, while Ghosh and Troutt (2012) propose a toolkit that operationalizes complex (compound real) option models for practitioner use. Finally, Brandão, Dyer, and Hahn (2012) discuss the difficulties in obtaining unbiased estimates of project volatility (undeniably the most critical input for RO applications). For project cash flows well-approximated by a gBm process, proper accounting for the temporal resolution of uncertainty and updating of conditional expectations can ameliorate known upward biases that volatility estimates obtained through Monte Carlo simulation exhibit. Extensions for leveraged project cash flows (due to fixed operating costs) and non-constant volatility are also addressed.

In other related topics, a few experimental studies analyze the attitudes of individuals towards (investment) options. Shin and Ariely (2004) consider options that might be unavailable unless sufficient effort is invested in them, asking whether the threat of disappearance changes the way people value such options. They find that decision-makers invest more effort and money in keeping options open that might disappear, even when the options themselves seem to be of little value. Their findings suggest that the tendency of keeping one's options 'open' is likely driven by a type of aversion to loss (Kahneman and Tversky, 1979). Du and Budescu (2005) suggest that human aversion to vagueness might explain deviations from the 'expected utility maximization' model. Subjects were asked to price (or choose) among investment options that varied in vagueness (on probabilities and/or outcomes) and domain (gains or losses). Results suggest that individuals' attitudes towards vagueness are influenced by the reference domain, exhibiting "reversals of attitudes" towards vagueness, particularly in the 'gains' domain.

Glasserman and Wang (2011) analyze the Capital Assistance Program (CAP) initiated by the U.S. government in February 2009 to provide backup capital to qualifying financial institutions that could not raise sufficient capital from private investors. Under CAP, participating banks would sell convertible preferred securities and warrants on bank common shares to the U.S. Treasury. The authors value these preferred securities and the embedded options. The interesting aspect of the program design was that it provided the issuer and the buyer options, since conversion (by the issuing bank) and warrant exercise (by the buying Treasury) both cause significant dilution of common shares. The program is evaluated as a dynamic stochastic game between the parties involving a modified binomial model for the issuer's equity to capture dilution and a determination of the optimal buyer/issuer strategies. The authors' estimates suggest that the terms of the CAP provided substantial benefit to the issuer, while no banks participated in the program.

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Longitudinal K-means approaches to clustering and analyzing EHR opioid use trajectories for clinical subtypes

Sarah Mullin , ... Peter L. Elkin , in Journal of Biomedical Informatics, 2021

2.8 Cluster visualization and interpretation using decision trees

Since k-means is an unsupervised method, we would like to summarize the key characteristics of each cluster. To do this quantitatively, we used external clinical variables and drug prescription variables in a decision tree analysis with cluster as the outcome variable [46]. These variables can be found in Table 3 and Supplementary Table S1. Decision trees can provide insight into the clusters by generating interpretable rules and visualizations for how the cluster was formed. Inspired by Leffondré et al. [47], we extracted various summary measures that describe features of the trajectories, e.g., MME mean, MME standard deviation, regressed linear slope, change in MME from first to last prescription, maximum MME, and minimum MME. We then used those extracted features, combined with additional clinical variables and medication features, to produce a decision tree using R rpart for each of the three methods on the entire training set. The decision trees, therefore, created interpretable rules and visualization for each method's resulting clusters on a per patient level. We used the Gini index to determine splits in the decision tree and a minimum number of 20 observations in a node for a split to occur. The tree is described by the number of nodes, which determine its complexity, and the accuracy of the tree, i.e., the ratio of elements not correctly explained by the resulting tree. We then pruned the trees to avoid overfitting to outliers in the data and chose the complexity parameter following typical rpart convention [48]. For kml, the complexity parameter with minimized cross-validated error was selected to be 0.01. The B-spline and VRAE decision trees had a higher complexity with the complexity parameters associated with minimum error found to both be 0.0007.

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