abstract

The evaluation of subjective data is a very demanding task. The classification of the information gathered from human evaluators and the possible high noise levels introduced are ones of the most difficult issues to deal with. This situation leads to adopt individuals who can be considered as experts in the specific application domain. Thus, the development of Expert Systems (ES) that consider the opinion of these individuals have been appeared to mitigate the problem. In this work an original methodology for the selection of subjective sequential data for the training of ES is presented. The system is based on the arrangement of knowledge acquired from a group of human experts. An original similarity measure between the subjective evaluations is proposed. Homogeneous groups of experts are produced using this similarity through a clustering algorithm. The methodology was applied to a practical case of the Intelligent Transportation Systems (ITS) domain for the training of ES for driving risk prediction. The results confirm the relevance of selecting homogeneous information (grouping similar opinions) when generating a ground truth (a reliable signal) for the training of ES. Further, the results show the need of considering subjective sequential data when working with phenomena where a set of rules could not be easily learned from human experts, such as risk assessment.

Introduction

The practice of Knowledge Engineering (Van Do, Le Thi, & Nguyen, 2018) has become a very useful approach to solve complex problems that require a high level of human expertise. This discipline involves integrating knowledge into computer systems which emulates the decision-making ability of a human expert in a specific domain. The systems in charge of achieving these tasks are the Expert Systems (ES) (Agarwal & Goel, 2014). The building, maintaining and development of ES (Djamal et al., 2017) are mainly based on the interaction between the knowledge engineer and the domain expert (Yau & Sattar, 1994). The devel- opment of a reliable ES requires a deep understanding and a good representation of the knowledge of the domain expert. In most of the cases, the knowledge representation is based on a set of rules (a production system) that ease the explanation of the decision-making made by the inference engine (Wick & Slagle, 1989). These rules are build from the knowledge acquired from human experts with the application of Machine Learning techniques (such as Neural Networks (Lin & Zhang, 2012), Deep Learning (Wei, He, Chen, Zhou, & Tang, 2017), Decision Trees (Sriram & Yuan, 2012), Fuzzy Logic (Wang, Lee, & Ho, 2007), Bayesian methods (WenBin, XiaoLing, YiJun, & Yu, 2010), Genetic Algorithms (Daza et al., 2011), among others). Knowledge acquisition is a process which aims to extract knowledge, experience and problem-solving procedures from one or more domain experts. Several techniques have been proposed for a correct knowledge acquisition (see Hua, 2008 for a complete review). Nevertheless, there are several problems that must be considered when acquiring knowledge from human experts (Gaines, 1987): • Experts may not be able to express their knowledge in a structured way • Experts may not be aware of the significance of the knowledge they have used. • The expressed knowledge may be irrelevant, incomplete or not understandable. In some cases, depending on the field of application, it may be easier to extract the knowledge from human experts through a continuous scale. This is the case of the risk assessment, where the knowledge could be acquired in a predefined scale (e.g. from 0, no risk, to 100, maximum risk). Here, the knowledge of the experts is gathered in form of subjective sequential data (Prelec, 2004) and could be treated as time series for its study and integration (see, for instance, de Diego, Crespo, Siordia, Conde, & Cabello, 2011; de Diego, Siordia, Conde, & Cabello, 2011; Siordia, de Diego, Conde, & Cabello, 2011a). However, the integration of several opinions into a unique ground truth (i.e. a reliable signal) is a hard-to-achieve task (Liou & Nunamaker, 1990). Two different scenarios appear. The consideration of knowledge from too few experts could provide a ground truth with insufficient information. In contrast, the consideration of knowledge from too many experts could generate a noisy ground truth due to the appearance of possible contradictions between their evaluations (Turban, 1991). Different statistical approaches have been proposed in the past (see, for instance, meta-analysis methods in Brockwell & Gordon (2001)).

abstract

The moment-based method is a fundamental approach to the extraction of affine invariants. However, only integer-order traditional moments can be used to construct affine invariants. No invariants can be constructed by moments with an order lower than 2. Consequently, the obtained invariants are sensitive to noise. In this paper, the moment order is generalized from integer to non-integer. However, the moment order cannot simply be generalized from integer to non-integer to achieve affine invariance. The difficulty of this generalization lies in the fact that the angular factor owing to shearing in the affine transform can hardly be eliminated for non-integer order moments. In order to address this problem, the Mellin polar coordinate moment (MPCM) is proposed, which is directly defined by a repeated integral. The angular factor can easily be eliminated by appropriately selecting a repeated integral. A method is provided for constructing affine invariants by means of MPCMs. The traditional affine moment invariants (AMIs) can be derived in terms of the proposed MPCM. Furthermore, affine invariants constructed with real-order (lower than 2) MPCMs can be derived using the proposed method. These invariants may be more robust to noise than AMIs. Several experiments were conducted to evaluate the proposed method performance.

 Introduction

Images of an object captured from different viewpoints are often subject to perspective distortions [1–4]. If the object is small compared to the camera-to-scene distance, the perspective effect becomes negligible and the affine model provides a reasonable approximation of the projective model [2,5]. Therefore, the extraction of affine invariant features plays an important role in object recognition and registration [6–10]. This method has been used extensively in numerous fields, such as shape recognition and retrieval [11,12], watermarking [13], aircraft identification [14,15], texture classification [16], and image registration [17,18]. The moment-based method is the most commonly used technique for the extraction of affine invariant features. However, only integer-order moments can be used to construct affine invariants [4]. It has been reported that high-order moments are sensitive to noise [19]. Hence, in practice, a moment of the lowest possible order should be used [20]. For similarity transform (including only translation, scaling, and rotation), Fourier Mellin descriptors [21] can be viewed as the invariants constructed by complex number order moments. However, similarity transform is only a spe cial case of affine transform [4]. Thus far, an order of moments for constructing affine invariants can only be an integer. In this paper, we consider generalizing the moment order from integer to noninteger, and construct affine invariants by means of the proposed moment.

1- Introduction

2- Formulation of calibration with varying window sizes: How to endogenise and make different window sizes comparable

3- Application of the Lagrange regularisation method to a simple linear-regression problem

4- Using the Lagrange regularisation method for detecting the beginning of financial bubbles

5- Conclusion

References

Abstract

Motivated by the question of identifying the start time τ of financial bubbles, we propose an improved calibration approach for time series in which the inception of the latest regime of interest is unknown. By taking into account the tendency of a given model to overfit data, we introduce the Lagrange regularisation of the normalised sum of the squared residuals, χ 2 np(Φ), to endogenously detect the optimal fitting window size := w∗ ∈ [τ : t¯ 2] that should be used for calibration, assuming a fixed pseudo present time t¯ 2. The Lagrange regularisation of χ 2 np(Φ) defines the Lagrange regularised sum of the squared residuals, χ 2 λ (Φ). Its performance is exemplified on a simple Linear Regression problem with a change point and compared against the performances of the Residual Sum of Squares (RSS) := χ 2 (Φ) and RSS/(N-p) := χ 2 np(Φ), where N is the sample size, p is the number of degrees of freedom and Φ is the parameter vector. Applied to synthetic models of financial bubbles with a well-defined transition regime and to a number of financial time series (US S&P500, Brazil IBovespa and China SSEC Indices), χ 2 λ (Φ) is found to provide well-defined reasonable determinations of the starting times for major bubbles such as the bubbles ending with the 1987 Black-Monday, the 2008 Sub-prime crisis and minor speculative bubbles on other Indexes, without any further exogenous information. The application of the method thus allows one to endogenise the determination of the starting time of bubbles, a problem that has yet not received a systematic objective solution. Moreover, the technique appears as a practical solution for comparing goodness-of-fit across unbalanced sample sizes.

Introduction

There is an inverse relationship between the tendency of a model to overfit data and the sample size under consideration. In other words, the smaller the sample size, the larger the flexibility for the model with a fixed number of parameters to overfit [1]. Due this characteristic feature, one cannot directly compare goodness-of-fit metrics of statistical models arbitrarily parametrised by the vector Φ of parameters such as the Residual Sum of Squares, RSS := χ 2 (Φ) or its normalised version RSS (N−p) := χ 2 np(Φ), over unequal sized samples. Here, N denotes the sample size while p is the number of degrees of freedom of a model. This has particular relevance when one is interested in selecting the optimal sub-sample of a data set to calibrate a model, a recurrent issue when estimating time series models in a moving window or/and when the model is only valid in a specific time window, which is unknown a priori.

1. Introduction

2. Inverse Gaussian Mixed-Effects Model

3. Estimating Model Parameters

4. Framingham Cholesterol Data

5. Conclusion

Acknowledgements

References

Following [7], we introduce the nested Inverse Gaussian Mixed-Effects model to analyze right-skewed and continuous longitudinal data. The nested random effects don’t follow a specific parameter distribution and rely only on the first two moments assumptions in our model. We apply the truly orthodox best linear unbiased predictor (BLUP) approach to estimate the nested random effects. We derive an optimal estimating equation for the regression parameters under the case of known BLUP of random effects. A real example for Framingham cholesterol data is presented to illustrate our proposed methodology.

Introduction

Skewed continuous longitudinal data frequently appear in many areas of research. Various skew normal models have been proposed to analyze skewed longitudinal data in recent years. For example, [5] for linear mixed models by substituting the skew-normal assumption of random effects for the normal assumption; [9] for Bayesian partial linear model; [1] for Skew-normal antedependence models; [8] for mixed effects model with the skew-normal and skew-t assumption of the distribution of responses and random effects. However, these approaches are generally computationally intensive. The inverse Gaussian regression model is a powerful method for analyzing right-skewed continuous data. Following [6] and [7], we consider a class of nested Inverse Gaussian Mixed-Effects Model for right-skewed and continuous longitudinal data. [7] introduced the nested Tweedie mixed model based on an orthodox BLUP approach. Similarly, this orthodox BLUP approach to our models is still computationally simple and efficient. In addition, our approach consolidates conditional and marginal modeling interpretations.

Mt/G/Infinity queueing systems have been widely used to analyse complex systems, such as telephone call centres, software testing systems, and telecommunication systems. Statistical inferences of performance measures, such as the expected cumulative numbers of arrivals and departures, are indispensable for decision makers in analysing the current scenario, predicting future scenarios, and making cost-effective decisions. In most scenarios, we only obtain interval censored data, namely, counts in fixed time intervals, instead of complete data because we either do not want or are not able to monitor arrivals and departures. We provide a general framework for statistical inference in Mt/G/Infinity queueing systems given interval censored data. A maximum-likelihood estimation (MLE) method is proposed for inferring the arrival rate and service duration. This method is applicable to general forms of the arrival rate functions and general service duration distributions. More importantly, we propose a combination of the bootstrap method and the delta method for inferring the expected cumulative numbers of arrivals and departures. The results of the simulation study demonstrate that the point and interval estimates of the proposed MLE method are satisfactory overall. As the number of intervals increases, the estimates based on the proposed MLE approach the estimates based on MLE with complete data. Our procedure enables estimates to be obtained without the need to keep track of each item, thereby substantially reducing resource consumption for monitoring items and storing data. An application in a software testing system demonstrates that the goodness-of-fit performance of the proposed MLE method is satisfactory.

Introduction

An Mt/G/∞ queueing system is a relatively simple queueing system that has a nonhomogeneous Poisson arrival process with a time-dependent deterministic arrival rate function λa ≡ λa(t), independent and identically distributed (i.i.d.) service durations that are independent of the arrival process, and infinitely many servers. Time-varying queueing models, including Mt/G/∞ models, are standard models for describing the dynamics of large-scale service systems, such as telecommunication systems, call centres, and healthcare systems, e.g., hospitals (Pender, 2016). Researchers have applied Mt/G/∞ models to service systems, such as telemarketing, police patrol, fire fighting, hospitals, copy machine repairs, and automatic teller machine operations. In these applications, an operating policy was to keep customer delays close to zero—a scenario that is consistent with the use of an infinite-server model (Green & Kolesar, 1998). Mt/G/∞ models have been applied to storage systems to assess the day-by-day adequacy of stock (Crawford, 1977), to analyse stock requirements (Hillestad & Carrillo, 1980), and to evaluate war-readiness spare requirements for aircraft (Crawford, 1981). They also have been applied to software testing programs (Vizarreta et al., 2018; Yang, 1996) and internet traffic systems (Fay, Roueff & Soulier, 2007). The Mt/G/∞ model is the offered load model for wireless and packet network systems, which describes the total packet carrying capacity of the channels or links in a packet network (Malhotra, Dey, van Doorn & Koonen, 2001; Palm, 1943; Singhai, Joshi & Bhatt, 2009).

Exponentially weighted moving average (EWMA) control charts are typically used for faster detection of shifts in the process mean, relative to a Shewhart control chart, when the degree of shift is small. Normal guidelines suggest using a small (large) value of the weighting constant, λ, for detecting smaller (larger) shifts in the process mean. Prior research has suggested that the choice of λ should depend on the observed data and have considered the use of a weighting constant, that varies and adapts as monitoring continues and new data are collected. One such adaptive control chart, called the AEWMA chart, utilizes a rather computationally complex scheme to determine the weighting constant λ and it requires knowledge of the size of the shift, to specify whether it is “small” or “large”. A complex twophase optimization scheme is then solved to yield “good solutions”. As an alternative, we propose an adaptive EWMA-type control chart that does not require knowledge of the degree of the shift. Further, the computational scheme is easier and completed in one stage. The performance of the proposed chart is studied using simulations, where the degree of the shift in the process mean is varied over a wide range of values. Based on the average run length (ARL), as a performance measure, the proposed chart is demonstrated to perform uniformly better than the traditional EWMA chart with a constant weight. The proposed chart also performs better than the AEWMA chart for moderate to large shifts.

Introduction

The traditional Shewhart control chart technique, introduced in 1924 by Walter Shewhart, has been used in a variety of manufacturing and service industries. The commonly used control charts are those based on the sample mean (X¯), the range (R), the standard deviation (s), individual data (X) and the moving range (MR). In the first two situations (X¯ and R, X¯ and s), it is assumed that independent and identically distributed observations (X) are selected from a process, in subgroups, where the subgroup size is denoted by n. In the third situation, the subgroup size is 1 and the process is monitored based on individual observations (Mitra, 2016). The popular Shewhart control charts, while easy to implement, have the disadvantage that for small shifts in the process parameters, for example, the mean, it takes a rather long time to detect the shift. Consequently, control charts have been developed with the purpose of faster detection of small changes in the process parameters. These include the cumulative sum (CUSUM) chart and the exponentially-weighted moving average (EWMA) control chart (Albin, Kang, & Shea, 1997; Chen & Elsayed, 2002; Domangue & Patch, 1991; Klein, 1996; Lucas & Saccucci, 1990; Roberts, 1959). The traditional EWMA control chart is based on the monitoring statistic.

We propose a novel approach based on the Marshall–Olkin (MO) copula to estimate the impact of systematic and idiosyncratic components on cross-border systemic risk. To use the data on non-failed banks in the suggested method, we consider the time to bank failure as a censored variable. Therefore, we propose a pseudo-maximum likelihood estimation procedure for the MO copula for a Type I censored sample. We derive the log-likelihood function, the copula parameter estimator and the bootstrap confidence intervals. Empirical data on the banking system of three European countries (Germany, Italy and the UK) shows that the proposed censored model can accurately estimate the systematic component of cross-border systemic risk.

Introduction

The 2007–2008 financial crisis has shown how a shock that originates in one country or asset class can quickly propagate to other markets and across borders. A key aspect of financial contagion is given by the linkages among banks. In the Euro area, the cross-border exposures arose as a prominent issue with the European sovereign debt crisis in 2011 and 2012, where large exposure of many EU banks to stressed sovereigns were revealed by the European Banking Authority (EBA, 2013). In a broader perspective, correlated exposures have recently been shown to be a major source of systemic risk. Given the importance of this research field, this paper is focused on systemic risk in the European banking sector. By definition, systemic risk involves a collection of interconnected institutions that have mutually beneficial business relationships through which insolvency can quickly propagate during periods of financial distress Billio, Getmansky, Lo, and Pelizzon (2012). Systemic risk is mainly due to idiosyncratic and systematic shocks (see De Bandt, Hartmann, & Peydró, 2009 and European Central Bank, 2010). The former affects only the health of a single financial institution, while the latter affects the whole economy, e.g. all financial institutions together at the same time. The component of systemic risk due to idiosyncratic shocks is also known as contagion risk in the literature (De Bandt et al., 2009). One of the the main aims of this paper is to propose a new methodological approach for the analysis of systemic risk to jointly model idiosyncratic and systematic shocks. We propose to apply the copula approach to measure systemic risk between the banking sectors of two countries. To our knowledge, the only papers that previously applied copulae to assess banking system stability are Baglioni and Cherubini (2013a), Baglioni and Cherubini (2013b), Straetmans (2010) and Weiss (2012). In other words, the approach is quite novel to the area of banking and systemic risk. The contributions of this paper are twofold. The first of these is to apply the Marshall and Olkin (MO) copula for modelling systemic risk between two countries. The second innovative aspect is how time to failure is considered for non-failed banks as rightcensored.

In this paper, we study and analyze the regularized least squares for functional linear regression model. The approach is to use the reproducing kernel Hilbert space framework and the integral operators. We show with a more general and realistic assumption on the reproducing kernel and input data statistics that the rate of excess prediction risk by the regularized least squares is minimax optimal.

Introduction

There are increasing cases in practice where the data are collected in the form of random functions or curves. This type of data is becoming more prevalent throughout science, engineering and financial market, as automated on-line data collection facilities are becoming more ubiquitous. Many classical statistical tools and models for multivariate analysis, such as principal components analysis, canonical correlation analysis and linear model are then extended to the infinite-dimensional functional domain. In this paper, we consider the functional linear model where Y is a scalar response, X : I → R is a square integrable functional predictor defined over compact domain I ⊂ R, α0 is the intercept, β0 : I → R is the slope function. and  is the random noise with mean 0 and finite variance σ 2 . Functional linear model was introduced by J.O. Ramsay and C.J. Dalzell [11] and first written in its commonly encountered form (1) by T. Hastie and C. Mallows [7]. Some recent research on the statistical analysis of (1) includes [2, 3, 6, 15, 16]. In this paper, we focus on the random design where X is a path of a square integrable stochastic process defined over I and is independent of . Without loss of much generality, throughout the paper we assume E(X) = 0 and the intercept α0 = 0, since the intercept can be easily estimated. Let L 2 be the Hilbert space of square integrable functions on I (with respect to Lebesgue measure) with standard inner production < u, v >= R I u(s)v(s)ds and norm kuk = R I u 2 (s)ds
1/.

abstract

Constrained approaches to maximum likelihood estimation in the context of finite mixtures of normals have been presented in the literature. A fully data-dependent soft constrained method for maximum likelihood estimation of clusterwise linear regression is proposed, which extends previous work in equivariant data-driven estimation of finite mixtures of normals. The method imposes soft scale bounds based on the homoscedastic variance and a cross-validated tuning parameter c. In our simulation studies and real data examples we show that the selected c will produce an output model with clusterwise linear regressions and clustering as a most-suited-to-the-data solution in between the homoscedastic and the heteroscedastic models.

Introduction

Let {(yi, xi)}n = {(y1, x1),…,(yn, xn)} be a sample of n independent units, where yi is the outcome variable and xi are the J covariates. A clusterwise linear regression model assumes that the density of yi|xi is given by f (yi|xi;ψ) = G g=1 pg f g (yi|xi;σ 2 g ,βg ) = G g=1 pg 1 2πσg 2 exp  − (yi − x i βg )2 2σ 2 g  , (1) where G is the number of clusters, ψ = {(p1,…, pG;β1,…,βG;σ 2 1 ,…,σ 2 G ) ∈ RG( J+2) : p1 +···+ pG = 1, pg ≥ 0,σ 2 g > 0, g = 1,…, G} is the set of model parameters, and pg , βg , and σ 2 g are respectively the mixing proportions, the vector of J regression coefficients, and the variance term for the g-th cluster. The model in Equation (1) is also known under the name of finite mixture of linear regression models, or switching regression model [21,22,15]. The parameters of finite mixtures of linear regression models are identified if some mild regularity conditions are met [10]. The clusterwise linear regression model of Equation (1) can naturally serve as a classification model. Based on the model, one computes the posterior membership probabilities for each observation as follows: and then classify each observation according, for instance, to fuzzy or crisp classification rules. The problem of clustering sample points grouped around linear structures has been receiving a lot of attention in the statistical literature because of its important applications (see, for instance, [16], and references therein. For the robust literature, among the others, see [6,7]). In order to estimate ψ, one has to maximize the following sample likelihood function L(ψ; y) =n i=1  G g=1 pg 1 2πσg 2 exp  − (yi − x i βg )2 2σ 2 g  , (3) which can be done using iterative procedures like the EM algorithm [5], whose clustering can be interpreted as a fuzzy partition [9]. Unfortunately, maximum likelihood (ML) estimation of univariate unconditional or conditional normals suffers from the well-known issue of unboundedness of the likelihood function: whenever a sample point coincides with the group’s centroid and the relative variance approaches zero, the likelihood function increases without bound ([14]; also the multivariate case suffers from the issue of unboundedness. See [4]). Hence a global maximum cannot be found. Yet, ML estimation does not fail: Kiefer [15] showed that there is a sequence of consistent, asymptotically efficient and normally distributed estimators for switching regressions with different group-specific variances (heteroscedastic switching regressions). These estimators correspond, with probability approaching one, to local maxima in the interior of the parameter space. Nonetheless, although there is a local maximum which is also a consistent root, there is no tool for choosing it among the local maxima. Day [4] showed, for multivariate mixtures of normals, that potentially each sample point – or any pair of sample points being sufficiently close together, or co-planar [24] – can generate a singularity in the likelihood function of a mixture with heteroscedastic components. This gives rise, both in univariate and multivariate contexts, to a number of spurious maximizers [18].

مقاله سال 2016

 مجله تحقیقات بازاریابی – Journal of Business Research

The extent to which researchers raise an issue is a subtle indicator of its importance. The benefits and limitations of partial least squares path modeling (PLS) is one such issue that scholars have heatedly debated across a variety of disciplines including marketing (e.g., Fornell & Bookstein, 1982; Hair, Sarstedt, Ringle, & Mena, 2012), strategic management (e.g., Bentler & Huang, 2014; Rigdon, 2012, 2014; Sarstedt, Ringle, Henseler, & Hair, 2014), and management information systems (e.g., Goodhue, Lewis, & Thompson, 2012; Marcoulides & Saunders, 2006; Ringle, Sarstedt, & Straub, 2012). Such scientific debates are important since they serve as a catalyst that sparks further careful examination of a method’s properties. Oftentimes, the result is improved understanding of the advantages and disadvantages of the focal method, but also additional research and methodological advances that stem from such objective and constructive discussions among scholars.

Recently, however, the scholarly community has witnessed a surprising level of acrimony towards PLS. Antonakis, Bendahan, Jacquart, and Lalive (2010, p. 1103) allude that “there is no use for PLS whatsoever […] thus strongly encourage researchers to abandon it.” Other authors similarly suggest that the use of PLS “is very difficult to justify” (Rönkkö & Evermann, 2013, p. 443) or that “PLS should not be adopted as a tool for psychological research.” (Rönkkö, McIntosh, & Antonakis, 2015, p. 82). This new harshness climaxed in an editorial from the editors in chief of the Journal of Operations Management (Guide & Ketokivi, 2015, p. vii) who declared that they were “desk rejecting practically all PLS-based manuscripts.” 1 In a follow-up paper in the very same journal, Rönkkö, McIntosh, Antonakis, and Edwards (2016, p. 16) echo this call by suggesting that “the only logical and reasonable action stemming from objective consideration of these issues is to discontinue the use of PLS.”

Leaving aside the tone of these and similar statements, which aim at shutting down any scholarly debate, the question arises why these authors cannot find even a single positive attribute of PLS despite its acceptance in scholarly research. In an effort to disentangle these opposing views, Rigdon (2016) offers an in-depth discussion of PLS and its origins, concluding that critics just as proponents of the method frequently offer incorrect or incomplete rationale for avoiding as well as using PLS. In addition, Rigdon (2016) concludes that many misconceptions about PLS have their roots in the method’s conceptual underpinnings and particularly the estimation philosophy it relies on (e.g., Rigdon, 2012).

مقاله سال 2016

Unlike soccer, American football and tennis, relatively little work has been published on forecasting in cricket. This seems especially strange given that there are known to be huge betting markets for cricket. The work that has been done on forecasting in cricket has largely been concerned with pre-match forecasting. However, in recent times, the growth in the popularity of in-play betting in all sports, where punters place bets during a game (or match), has meant that models that enable forecasts to be made as the game progresses are in high demand. Cricket is a sport that lends itself particularly to in-play betting: unlike soccer, for example, the discrete nature of the game means that bookmakers and punters alike have ample opportunities tobe active in markets during the game, and as such, cricket attracts extremely large in-play betting volumes. For example, the total amount bet during a typical major OneDay International (ODI) involving Pakistan or India is in the order of $1bn (according to a personal communication from a betting industry insider in 2013). In this paper, we present an in-play forecasting model for One-Day International cricket, and use the model to estimate the probability of victory for a team at any moment during a game.

Of course, betting is not the only use of a forecasting model. A forecasting model like that presented here could be used for several purposes. Team coaches may use in-play forecasting probabilities to assess the merits of various different strategies or to analyse player and team performances. In addition, the media could use the model to identify key moments in a match and enhance the television coverage further.

Previous work in One-day International (ODI) cricket has focussed largely on the problem of resetting targets in limited overs cricket following interruptions to play.

مقاله سال 2016

Consider the case where an analyst has two competing one-step-ahead forecasts for a time series variable yt , namely yˆ1,t and yˆ2,t , for a sample t = 1, 2, . . . , T . The forecasts have the associated forecast errors εˆ1,t and εˆ2,t , respectively. To examine which of the two sets of forecasts provides the best accuracy, the analyst can use criteria based on some average or median of loss functions of the forecast errors. Well-known examples include the root mean squared error (RMSE) and the median absolute error (MAE); see Hyndman and Koehler (2006) for an exhaustive list of criteria, and also Table 1.

t of criteria, and also Table 1. As there is always one set of forecasts that scores lower on some criterion, it seems wise to test whether any observed differences in forecast performances are statistically significant. To test statistically whether the obtained values of these criteria are equal, the analyst can rely on the methodology proposed by Diebold and Mariano (1995) (DM); see also Diebold (2015) for a recent review. This methodology is based on the loss functions li,t = f(yt, yˆi,t) for i = 1, 2. Denoting the sample mean loss differential by d¯ 12, that is, d¯ 12 = 1 T T 1 (l1,t − l2,t), and a consistentestimate of the standard deviation of d¯ 12 by σˆd¯ 12 , the DM test for one-step-ahead forecasts is DM = d¯ 12 σˆd¯ 12 ∼ N(0, 1),

under the null hypothesis of equal forecast accuracy. Even though Diebold and Mariano (1995, p. 254) claim that this result holds for any arbitrary function f , it is quite clear that the function should allow for proper moment conditions in order to yield the asymptotic normality of the test. In fact, as will be argued in Section 2 below, many of the functions that are commonly applied in the forecast literature fail to qualify as useful functions for the DM methodology. This note continues with a brief summary of typical functions in Section 2, along with a concise discussion of which of these functions are useful in the DM framework. It is found that the absolute scaled error (ASE) recommended by Hyndman and Koehler (2006) does have the favorable properties, while various other criteria do not. Section 3 reports on limited simulation experiments which support these insights. The main conclusion of this note is to confirm that the use of the Mean ASE (MASE) criterion is recommended.