On the empirical estimation of integral probability metrics

Abstract Probabilistic modeling methods are increasingly being employed in engineering applications. These approaches make inferences about the distribution for output quantities of interest. A challenge in applying probabilistic computer models (simulators) is validating output distributions against samples from observational data. An ideal validation metric is one that intuitively provides information on key differences between the simulator output and observational distributions, such as statistical distances/divergences. Within the literature, only a small set of statistical distances/divergences have been utilized for this task; often selected based on user experience and without reference to the wider variety available. As a result, this paper offers a unifying framework of statistical distances/divergences, categorizing those implemented within the literature, providing a greater understanding of their benefits, and offering new potential measures as validation metrics. In this paper, two families of measures for quantifying differences between distributions, that encompass the existing statistical distances/divergences within the literature, are analyzed: f-divergence and integral probability metrics (IPMs). Specific measures from these families are highlighted, providing an assessment of current and new validation metrics, with a discussion of their merits in determining simulator adequacy, offering validation metrics with greater sensitivity in quantifying differences across the range of probability mass.

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Integral Probability Metrics and Their Generating Classes of Functions

Advances in Applied Probability ◽

10.2307/1428011 ◽

1997 ◽

Vol 29 (2) ◽

pp. 429-443 ◽

Cited By ~ 56

Author(s):

Alfred Müller

Keyword(s):

Maximal Class ◽

Probability Measures ◽

Probability Metrics ◽

Unified Study ◽

Kolmogorov Metric ◽

Stop Loss ◽

Class Of Functions ◽

Integral Probability

We consider probability metrics of the following type: for a class of functions and probability measures P, Q we define A unified study of such integral probability metrics is given. We characterize the maximal class of functions that generates such a metric. Further, we show how some interesting properties of these probability metrics arise directly from conditions on the generating class of functions. The results are illustrated by several examples, including the Kolmogorov metric, the Dudley metric and the stop-loss metric.

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Integral Probability Metrics and Their Generating Classes of Functions

Advances in Applied Probability ◽

10.1017/s000186780002807x ◽

1997 ◽

Vol 29 (02) ◽

pp. 429-443 ◽

Cited By ~ 8

Author(s):

Alfred Müller

Keyword(s):

Maximal Class ◽

Probability Measures ◽

Probability Metrics ◽

Unified Study ◽

Kolmogorov Metric ◽

Stop Loss ◽

Class Of Functions ◽

Integral Probability

We consider probability metrics of the following type: for a class of functions and probability measures P, Q we define A unified study of such integral probability metrics is given. We characterize the maximal class of functions that generates such a metric. Further, we show how some interesting properties of these probability metrics arise directly from conditions on the generating class of functions. The results are illustrated by several examples, including the Kolmogorov metric, the Dudley metric and the stop-loss metric.

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An Evaluation of Validation Metrics for Probabilistic Model Outputs

ASME 2018 Verification and Validation Symposium ◽

10.1115/vvs2018-9327 ◽

2018 ◽

Cited By ~ 2

Author(s):

Paul Gardner ◽

Charles Lord ◽

Robert J. Barthorpe

Keyword(s):

Statistical Method ◽

Probabilistic Model ◽

Probabilistic Models ◽

Statistical Moments ◽

Probabilistic Modelling ◽

Probability Metrics ◽

Engineering Applications ◽

Validation Metrics ◽

Modelling Methods ◽

Integral Probability

Probabilistic modelling methods are increasingly being employed in engineering applications. These approaches make inferences about the distribution, or summary statistical moments, for output quantities. A challenge in applying probabilistic models is validating output distributions. An ideal validation metric is one that intuitively provides information on key divergences between the output and validation distributions. Furthermore, it should be interpretable across different problems in order to informatively select the appropriate statistical method. In this paper, two families of measures for quantifying differences between distributions are compared: f-divergence and integral probability metrics (IPMs). Discussions and evaluation of these measures as validation metrics are performed with comments on ease of computation, interpretability and quantity of information provided.

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EMPIRICAL ESTIMATION WAITING TIME OF URBAN PUBLIC TRANSPORT WITH THE PURPOSE ENSURING OPTIMAL FUNCTIONING ROUTE NETWORK

Scientific Papers Collection of the Angarsk State Technical University ◽

10.36629/2686-7788-2019-1-1-1179-183 ◽

2019 ◽

Vol 1 (1) ◽

pp. 179-183

Author(s):

O.A Lebedeva ◽

J.O. Poltavskaya

Keyword(s):

Waiting Time ◽

Public Transport ◽

Empirical Estimation ◽

Route Network ◽

Optimal Functioning ◽

Urban Public Transport

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The Economic Effects of COVID-19 on the Producers of Ethanol, Corn, Gasoline, and Oil

Journal of Agricultural & Food Industrial Organization ◽

10.1515/jafio-2020-0025 ◽

2020 ◽

Vol 18 (2) ◽

Author(s):

Andrew Schmitz ◽

Charles B. Moss ◽

Troy G. Schmitz

Keyword(s):

United States ◽

Cost Benefit ◽

Economic Cost ◽

The United States ◽

Welfare Economic ◽

Economic Effects ◽

Economic Losses ◽

Corn Ethanol ◽

Empirical Estimation ◽

Total Oil

AbstractThe COVID-19 crisis created large economic losses for corn, ethanol, gasoline, and oil producers and refineries both in the United States and worldwide. We extend the theory used by Schmitz, A., C. B. Moss, and T. G. Schmitz. 2007. “Ethanol: No Free Lunch.” Journal of Agricultural & Food Industrial Organization 5 (2): 1–28 as a basis for empirical estimation of the effect of COVID-19. We estimate, within a welfare economic cost-benefit framework that, at a minimum, the producer cost in the United States for these four sectors totals $176.8 billion for 2020. For U.S. oil producers alone, the cost was $151 billion. When world oil is added, the costs are much higher, at $1055.8 billion. The total oil producer cost is $1.03 trillion, which is roughly 40 times the effect on U.S. corn, ethanol, and gasoline producers, and refineries. If the assumed unemployment effects from COVID-19 are taken into account, the total effect, including both producers and unemployed workers, is $212.2 billion, bringing the world total to $1266.9 billion.

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Empirical Estimation of Shortest Route Length along U.S. Interstate Highways Based on Great Circle Distance

Transportation Research Record Journal of the Transportation Research Board ◽

10.1177/03611981211015251 ◽

2021 ◽

pp. 036119812110152

Author(s):

Nawei Liu ◽

Fei Xie ◽

Zhenhong Lin ◽

Mingzhou Jin

Keyword(s):

Linear Regression ◽

State Level ◽

Distance Matrix ◽

Great Circle ◽

The State ◽

Empirical Estimation ◽

Interstate Highways ◽

Shortest Route ◽

Route Length ◽

Great Circle Distance

In this study, 98 regression models were specified for easily estimating shortest distances based on great circle distances along the U.S. interstate highways nationwide and for each of the continental 48 states. This allows transportation professionals to quickly generate distance, or even distance matrix, without expending significant efforts on complicated shortest path calculations. For simple usage by all professionals, all models are present in the simple linear regression form. Only one explanatory variable, the great circle distance, is considered to calculate the route distance. For each geographic scope (i.e., the national or one of the states), two different models were considered, with and without the intercept. Based on the adjusted R-squared, it was observed that models without intercepts generally have better fitness. All these models generally have good fitness with the linear regression relationship between the great circle distance and route distance. At the state level, significant variations in the slope coefficients between the state-level models were also observed. Furthermore, a preliminary analysis of the effect of highway density on this variation was conducted.

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