Minimal Extrapolation Research Articles

Data envelopment analysis of two‐stage processes: an alternative (non‐conventional) approach

AbstractNetwork data envelopment analysis (NDEA) is an extension of standard DEA that models the efficiency assessment of decision‐making units (DMUs) by considering their internal structure. While in standard DEA, the DMU is regarded as a single process, in NDEA, the DMU is viewed as a network of interconnected sub‐processes (stages, divisions), where the flow of the intermediate products (measures) is essential in the efficiency assessment. In the prevalent conventional methodological approach to NDEA, the sub‐processes are assumed as distinct entities with distinct inputs and outputs. Thus, each sub‐process has its own production possibility set (PPS), which can be derived axiomatically from a set of assumptions using the minimum extrapolation principle. The PPS of the overall system is defined as the composition of the individual PPSs. The conventional approach comprises all the methods, in which the system and the divisional efficiencies are computed jointly in a single mathematical program. A fundamental property connecting the system with the divisional efficiencies is that a system is overall efficient if and only if its divisions are all efficient. In NDEA, regardless of the method used, there are cases where none of the observed DMUs is rendered overall efficient as corroborated by real‐world case studies. This is the main issue we discuss in this paper and the motivation to propose an alternative, non‐conventional, approach to address it in the frame of two‐stage processes. We consider the two‐stage process as a system that can be viewed from two perspectives depending on the role of the intermediate measures: the system as producer and as consumer of the intermediates. As our approach is based on standard DEA, it satisfies the basic desirable properties. The fundamental NDEA property, that the overall system is efficient if and only if both perspectives are efficient, is met. The efficient frontier of the system is explicitly defined by the overall efficient observed DMUs, and the inefficient DMUs are projected on the efficient frontier of the system. A convergent procedure is presented for this purpose. The proposed models are equivalently expressed in both the multiplier and the envelopment forms due to strict primal‐dual correspondence and can operate under both constant and variable returns‐to‐scale assumptions. We use the case of 22 automotive manufacturers for the fiscal year 2019 as an example to illustrate our approach. Comparison with other NDEA methods is also provided.

How to peel a data envelopment analysis frontier: A cross-validation-based approach

Data Envelopment Analysis (DEA) presents the typical characteristics of a data-driven approach with the specific objective of determining technical efficiency and production frontiers in Engineering and Microeconomics. However, by construction, the frontier estimator generated by DEA suffers from overfitting problems; something that contrasts with currently accepted models in machine learning. In this regard, DEA can be seen as a preliminary stage of a more complex approach, where the aim is to avoid overfitting in order to determine a proper description of the underlying Data Generating Process that is behind the generation of the observations in a production process. In this paper, we introduce a possible solution to overcome the overfitting problem associated with DEA that is based on cross-validation. This process “peels” the standard DEA frontier (removing certain supporting hyperplanes) until a new convex technology, which also satisfies free disposability in inputs and outputs but not the principle of minimal extrapolation, is determined. Our approach is tested by resorting to a computational experience. Additionally, we illustrate how the new method could be used as a complement to the standard DEA technique through an empirical application based on a PISA (Programme for International Student Assessment) dataset.

Combining Data Envelopment Analysis and Machine Learning

Data Envelopment Analysis (DEA) is one of the most used non-parametric techniques for technical efficiency assessment. DEA is exclusively concerned about the minimization of the empirical error, satisfying, at the same time, some shape constraints (convexity and free disposability). Unfortunately, by construction, DEA is a descriptive methodology that is not concerned about preventing overfitting. In this paper, we introduce a new methodology that allows for estimating polyhedral technologies following the Structural Risk Minimization (SRM) principle. This technique is called Data Envelopment Analysis-based Machines (DEAM). Given that the new method controls the generalization error of the model, the corresponding estimate of the technology does not suffer from overfitting. Moreover, the notion of ε-insensitivity is also introduced, generating a new and more robust definition of technical efficiency. Additionally, we show that DEAM can be seen as a machine learning-type extension of DEA, satisfying the same microeconomic postulates except for minimal extrapolation. Finally, the performance of DEAM is evaluated through simulations. We conclude that the frontier estimator derived from DEAM is better than that associated with DEA. The bias and mean squared error obtained for DEAM are smaller in all the scenarios analyzed, regardless of the number of variables and DMUs.

Multi-output Support Vector Frontiers

In this paper, we show that both Free Disposal Hull (FDH) and Data Envelopment Analysis (DEA), which are well-known modern techniques for efficiency measurement, can be seen as particular cases of a more general model based upon Support Vector Regression (SVR) within machine learning. Our approach is based on the adaptation of SVR in a multi-response framework for dealing with standard microeconomic assumptions, such as free disposability and convexity of the underlying technology. This adaptation allows us to introduce a more robust notion of technical efficiency, linked to the concept of ɛ-insensitivity in standard SVR. Due to computational reasons, we also introduce a simplified version of the initial approach, whose validity is checked through simulation. By resorting to a computational experience, we also show that the new approach, called multi-output Support Vector Frontiers, outperforms FDH and DEA with respect to mean squared error and bias, avoiding the overfitting problem associated with the assumption of the principle of minimal extrapolation in the case of FDH and DEA. We finally show how to implement some usual efficiency measures under the new approach and illustrate their performance through an empirical example.

Selective proportionality and integer-valued data in DEA: an application to performance evaluation of high schools

Conventional data envelopment analysis (DEA) models are often extended for constant or variable returns to scale assumptions based on the under-investigated technology. It is assumed that all inputs and outputs are real-valued data. However, in many practical applications, proportionality or convexity axioms require to be modified. This study attempts to further expand upon the hybrid returns to scale DEA models in the presence of integer-valued input and output data. We refine the previous axioms to introduce a new minimal extrapolation technology set. Moreover, we formulate a couple of mixed-integer linear programming models for efficiency evaluation and target setting. An empirical application on 30 high schools in Iran is provided to validate the proposed approach. The data analysis, including efficiency evaluations along with providing benchmark units, is also performed.

Modelling undesirable products in non-parametric performance analysis

Purpose The purpose of this paper is introducing an alternative model to measure the relative efficiency of observations with undesirable products. Describing the reference set and benchmarking. Design/methodology/approach In this paper, an alternative definition of weak disposability assumption is introduced to handle undesirable outputs. Actually, two types of undesirable outputs are addressed and a substitute definition of weak disposability is presented. Findings Using this assumption a linear production technology set along with a performance analysis model is constructed to assess the relative efficiency of the decision-making units. To illustrate the radial application of the proposed approach, a real case on transportation system of USA during 1992-2009 is given. Originality/value To date, data envelopment analysis studies have investigated undesirable outputs by the assumption of weak disposability, defined as the proportional contraction of good and bad products, which leads to the null-joint assumption between good and bad outputs. Therefore, the only way to produce no undesirable outputs is producing zero desirable outputs. So the production process should be stopped while it is not economically cost-effective. However, in some processes there are some undesirable outputs, which are decreased with non-same percentages. So these undesirable outputs can be stopped while the good outputs have a strictly positive value. In this situation, the good outputs are not null-joint with this type of bad outputs. In the current paper, a new definition of the weak disposability of outputs was represented while two groups of undesirable outputs were considered. Hence, desirable outputs and the first kind of undesirable outputs were decreased proportionally. However, the reduction value was different for the second kind of undesirable outputs. Hence, the null-joint assumption is removed from the production technology. Then, a new technology was proposed based on five postulates as inclusion of observations, free disposability of desirable outputs and inputs, new weak disposability, convexity and minimum extrapolation.

Measuring Taiwanese bank performance: A two-system dynamic network data envelopment analysis approach

This paper deals with the heterogeneity problem related to the determination of meta-technology when measuring performance in a dynamic setting. The authors propose a new way of constructing the non-convex meta-framework of the dynamic network data envelopment analysis. This new approach differs from the minimum extrapolation principle on the aggregation of individual group technologies. The proposed model preserves the role of each group's technology in the determination of the proposed non-convex meta-technology. The proposed model also considers the linking activities between the processes and the carry-over activities between two consecutive terms, illustrated by means of a real-world example from banking that simultaneously evaluates: deposit, lending, period, deposit-period and lending-period efficiencies for 22 Taiwanese banks from 2008 to 2016. The empirical results indicate that lending efficiency outperforms deposit efficiency during the sample period, and thus the improvement in deposit activity has a positive effect on banks’ overall performance. The results also imply that non-FHC banks outperform FHC banks on average in terms of both the deposit and lending processes, but the effect is not statistically significant.

A study of the strength of glued laminated timber

ABSTRACTThe paper presents methodologies for determining the bending strength of structural glued laminated timber (glulam) from lamination tension test strength data. The tension tests involve finger jointed bonded lamination pairs sampled to reflect manufacturer’s practices of lamination stock purchases, defect removal and finger jointing while simultaneously taking account of knot, sloping grain and finger joint reinforcement as occurs within a glulam beam. It uses a modified Weibull weakest link theory with minimal extrapolation of the test data to determine beam strengths. The method is largely independent of any strength class system. The predictions are benchmarked against test data provided by Falk and values provided in European Standard EN14080.

Super-resolution by means of Beurling minimal extrapolation

Let $M(\mathbb{T}^d)$ be the space of complex bounded Radon measures defined on the $d$-dimensional torus group $(\mathbb{R}/\mathbb{Z})^d=\mathbb{T}^d$, equipped with the total variation norm $\|\cdot\|$; and let $\hat\mu$ denote the Fourier transform of $\mu\in M(\mathbb{T}^d)$. We address the super-resolution problem: For given spectral (Fourier transform) data defined on a finite set $\Lambda\subset\mathbb{Z}^d$, determine if there is a unique $\mu\in M(\mathbb{T}^d)$ of minimal norm for which $\hat\mu$ equals this data on $\Lambda$. Without additional assumptions on $\mu$ and $\Lambda$, our main theorem shows that the solutions to the super-resolution problem, which we call minimal extrapolations, depend crucially on the set $\Gamma\subset\Lambda$, defined in terms of $\mu$ and $\Lambda$. For example, when $\Gamma=0$, the minimal extrapolations are singular measures supported in the zero set of an analytic function, and when $\Gamma\geq 2$, the minimal extrapolations are singular measures supported in the intersection of $\Gamma\choose 2$ hyperplanes. By theory and example, we show that the case $\Gamma=1$ is different from other cases and is deeply connected with the existence of positive minimal extrapolations. This theorem has implications to the possibility and impossibility of uniquely recovering $\mu$ from $\Lambda$. We illustrate how to apply our theory to both directions, by computing pertinent analytical examples. These examples are of interest in both super-resolution and deterministic compressed sensing. Our concept of an admissibility range fundamentally connects Beurling's theory of minimal extrapolation with Candes and Fernandez-Granda's work on super-resolution. This connection is exploited to address situations where current algorithms fail to compute a numerical solution to the super-resolution problem.

On estimating optimal α-returns to scale

From a theoretical point of view, -returns to scale is a relevant alternative to traditional DEA models for estimating production technologies under global returns to scale assumptions such as strictly increasing or strictly decreasing returns to scale. However, from a methodological and empirical point of view, a remaining question is the estimation of . This contribution proposes an effective methodology to estimate an optimal value of based upon a goodness-of-fit strategy. A global method using a grid search is presented first. Second, for generalised FDH technologies, a minimum extrapolation principle is developed to estimate directly the optimal -returns from a linear program. An illustration on 63 US industries over the period 1987–2012 shows the relevancy of our approach.

Are Model Transferability And Complexity Antithetical? Insights From Validation of a Variable‐Complexity Empirical Snow Model in Space and Time

AbstractThe related challenges of predictions in ungauged basins and predictions in ungauged climates point to the need to develop environmental models that are transferable across both space and time. Hydrologic modeling has historically focused on modelling one or only a few basins using highly parameterized conceptual or physically based models. However, model parameters and structures have been shown to change significantly when calibrated to new basins or time periods, suggesting that model complexity and model transferability may be antithetical. Empirical space‐for‐time models provide a framework within which to assess model transferability and any tradeoff with model complexity. Using 497 SNOTEL sites in the western U.S., we develop space‐for‐time models of April 1 SWE and Snow Residence Time based on mean winter temperature and cumulative winter precipitation. The transferability of the models to new conditions (in both space and time) is assessed using non‐random cross‐validation tests with consideration of the influence of model complexity on transferability. As others have noted, the algorithmic empirical models transfer best when minimal extrapolation in input variables is required. Temporal split‐sample validations use pseudoreplicated samples, resulting in the selection of overly complex models, which has implications for the design of hydrologic model validation tests. Finally, we show that low to moderate complexity models transfer most successfully to new conditions in space and time, providing empirical confirmation of the parsimony principal.

The overall Malmquist index: a new approach for measuring productivity changes over time

This paper deals with a special case of the non-homogeneity problem related to the determination of the global benchmark technology when measuring productivity changes over time. The authors propose a new way of constructing the global framework of the Malmquist index which applies the minimum extrapolation principle on the aggregation of the experienced contemporaneous technologies. The proposed index, called overall Malmquist index, preserves the role of each contemporaneous technology in the determination of the newly-proposed best practice technology, whereby an acceptable level of discrimination between non-homogeneous observations is provided. With respect to both computational and test properties, the proposed index possesses the circularity property, generates a single measure of productivity change and is immune to infeasibility under variable returns to scale. Furthermore, unlike in the global form, previously computed results by the overall Malmquist index are more stable and less sensitive to changes in the shape of the best practice technology when a new time period is incorporated. Similar to traditional indices, it can be decomposed into various components such as efficiency change, scale efficiency change, and best practice change. The suggested index will be illustrated by means of a real-world example from banking. In particular, it will be compared to the contemporaneous and global forms of the Malmquist index introduced into the literature by Färe et al. (J Product Anal 3:85–101, 1992) and Pastor and Lovell (Econ Lett 88:266–271, 2005) , respectively.

How does selection of climate variables affect predictions of species distributions? A case study of three new weeds in New Zealand

SummarySpecies distribution models are an important tool to predict potential spread of weeds. While recent progress has improved model performance, there is still concern about the validity of such models, especially when applied to novel geographical regions or climates. This study investigates how different sets of variables influence predicted distributions, considering several measures of model performance and how extrapolation to novel geographical regions may affect results. Potential distributions of three new weeds in New Zealand (Archontophoenix cunninghamiana, Psidium guajava and Schefflera actinophylla) are modelled, by training a model based on global data from native and introduced ranges and projecting it to New Zealand, using Maxent. For each species, four models were calibrated: first with a full set of 19 bioclimatic variables, then with a customised set with selection based on analysis of response curves and finally with two reduced sets of uncorrelated variables. Although AUC across all models was very high (AUC ≥ 0.9), correlations between models ranged between 0.27 and 0.98. Inclusion of all variables predicted larger areas to be suitable in the projected region, with highly unlikely predictions in some areas, especially where bioclimatic variables showed values outside the range of the training data (new environments). Conversely, minimal extrapolation and more realistic predictions of weed distributions were obtained from models including a customised set of variables, and even more so from models including only a reduced set of variables. This study shows that careful selection of variables and investigation into extrapolation are vital in generating more realistic predictions of weed distributions.

Variantes sur un théorème de Candès, Romberg et Tao

The CRT theorem reconstructs a signal from a sparse set of frequencies, a paradigm of Compressed sensing. The signal is assumed to be carried by a small number of points, s , in a large cyclic set, of order N ; the frequencies consist of CslogN points chosen randomly in ℤ/Nℤ ; the reconstruction is based on a minimal extrapolation in the Wiener algebra of ℤ/Nℤ of the restriction of the Fourier transform of the signal to the chosen set of frequencies. The probability of reconstructing the signal is nearly 1 when C is large. The statement should be modified when we want all signals carried by s points to be reconstructed in that way. The CRT approach is based on random matrices, here the approach is classical Fourier analysis.

Exact reconstruction using Beurling minimal extrapolation

We show that measures with finite support on the real line are the unique solution to an algorithm, named generalized minimal extrapolation, involving only a finite number of generalized moments (which encompass the standard moments, the Laplace transform, the Stieltjes transformation, etc.).Generalized minimal extrapolation shares related geometric properties with the basis pursuit approach of Chen et al. (1998) [5]. Indeed we also extend some standard results of compressed sensing (the dual polynomial, the nullspace property) to the signed measure framework.We express exact reconstruction in terms of a simple interpolation problem. We prove that every nonnegative measure, supported by a set containing s points, can be exactly recovered from only 2s+1 generalized moments. This result leads to a new construction of deterministic sensing matrices for compressed sensing.

Modelling weak disposability in data envelopment analysis under relaxed convexity assumptions

The treatment of undesirable (bad) outputs in models of efficiency and productivity analysis often requires replacing the assumption of free disposability of outputs by their weak disposability. In a recent publication the authors showed that the Kuosmanen technology is the only correct representation of the fully convex technology exhibiting weak disposability of bad and good outputs. In this paper we relax the assumption of full convexity and consider two further possibilities: the case in which only the output sets are assumed convex and the case in which no convexity is assumed at all. In the first case we show that, although the traditional Shephard technology of nonparametric production analysis satisfies the assumption of convex output sets, it is larger than necessary. Based on the minimum extrapolation principle, we develop a correct model that is based on the assumed axioms. The second case leads to the development of a weakly disposable analogue of the free disposable hull. To complete our study, we give a full axiomatic definition of the Shephard technology.

Cecil Hougie

Idiopathic pulmonary fibrosis (IPF) is a rare disease, with estimates of prevalence varying considerably across countries due to paucity in data collection. The aim of this study was to investigate...

Weak Disposability in Nonparametric Production Analysis: Reply to Färe and Grosskopf

AbstractFäre and Grosskopf (this issue) claim that a single abatement factor suffices for modeling weak disposability in nonparametric production models, and that the Kuosmanen (2005) technology that uses multiple abatement factors is larger than necessary. This article demonstrates by a numerical example that a single abatement factor does not suffice to capture all feasible production plans, and that its use leads to the violation of convexity, one of the maintained assumptions of the model. We also prove that the Kuosmanen technology is the correct minimum extrapolation technology under the stated axioms.

Theory of integer-valued data envelopment analysis

Abstract Conventional data envelopment analysis (DEA) models assume real-valued inputs and outputs. In many occasions, some inputs and/or outputs can only take integer values. In some cases, rounding the DEA solution to the nearest whole number can lead to misleading efficiency assessments and performance targets. This paper develops the axiomatic foundation for DEA in the case of integer-valued data, introducing new axioms of “natural disposability” and “natural divisibility”. We derive a DEA production possibility set that satisfies the minimum extrapolation principle under our refined set of axioms. We also present a mixed integer linear programming formula for computing efficiency scores. An empirical application to Iranian university departments illustrates the approach.

Assessment of the environmental fate and effects of ivermectin in aquatic mesocosms

Pharmaceuticals in the environment have been subject to increasing public concern and scientific investigation over the past years. More than 100 active pharmaceutical ingredients have been detected in surface waters worldwide at the ng to μg L −1 range. At these low levels it is commonly assumed that only chronic and/or mixture toxic effects will be discernible in aquatic ecosystems and that there are orders of magnitude between exposure and effect concentrations. Assessment of potential ecosystem risk of pharmaceuticals are recommended but rarely performed in mesocosms, so for most risk assessments the final tier to reduce extrapolation uncertainty is missing. This paper describes the fate and effects of the anthelmintic drug ivermectin for a 265-day period following treatment (nominal concentration levels of 0, 30, 100, 1000 ng L −1 (or parts per trillion (ppt)) in fifteen 12,000 L outdoor aquatic mesocosms. Although it is established that ivermectin is highly toxic towards invertebrates, it has been believed that ivermectin does not present notable risks to aquatic systems due to the rapid dissipation of the compound and binding to the sediment. Hence, fate and exchange of ivermectin between water and sediment were evaluated in this study. The ivermectin DT 50aqueous in water was found to be 3–5 days, but concentrations increased and appeared to be stabile in the sediment at 20–30 ng kg −1 with no assessable DT 50sed. Acute effects (first week) following ivermectin exposure were identified and cladocerans were particularly sensitive (nom. 100 ppt). Chronic responses (<day 97) were observed for the ecosystem structure and function (nom. 30 ppt). Long-term effects (>229 days) were identified for more sediment-active organisms (e.g. Chydoriae and Ephemeroptera) (nom. 1000 ppt). This is the first study to demonstrate the potential environmental risk of ivermectin at or below the predicted environmental concentration using a standardized test methodology (mesocosm) with minimal extrapolation uncertainty.