Probabilistic forecasts, calibration and sharpness
Probabilistic forecasts of continuous variables take the form of predictive densities or predictive cumulative distribution functions. We propose a diagnostic approach to the evaluation of predictive performance that is based on the paradigm of maximizing the sharpness of the predictive distribution...
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| Published in | Journal of the Royal Statistical Society. Series B, Statistical methodology Vol. 69; no. 2; pp. 243 - 268 |
|---|---|
| Main Authors | , , |
| Format | Journal Article |
| Language | English |
| Published |
Oxford, UK
Oxford, UK : Blackwell Publishing Ltd
01.04.2007
Blackwell Publishing Ltd Blackwell Publishers Blackwell Royal Statistical Society Oxford University Press |
| Series | Journal of the Royal Statistical Society Series B |
| Subjects | |
| Online Access | Get full text |
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| Abstract | Probabilistic forecasts of continuous variables take the form of predictive densities or predictive cumulative distribution functions. We propose a diagnostic approach to the evaluation of predictive performance that is based on the paradigm of maximizing the sharpness of the predictive distributions subject to calibration. Calibration refers to the statistical consistency between the distributional forecasts and the observations and is a joint property of the predictions and the events that materialize. Sharpness refers to the concentration of the predictive distributions and is a property of the forecasts only. A simple theoretical framework allows us to distinguish between probabilistic calibration, exceedance calibration and marginal calibration. We propose and study tools for checking calibration and sharpness, among them the probability integral transform histogram, marginal calibration plots, the sharpness diagram and proper scoring rules. The diagnostic approach is illustrated by an assessment and ranking of probabilistic forecasts of wind speed at the Stateline wind energy centre in the US Pacific Northwest. In combination with cross-validation or in the time series context, our proposal provides very general, nonparametric alternatives to the use of information criteria for model diagnostics and model selection. |
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| AbstractList | Probabilistic forecasts of continuous variables take the form of predictive densities or predictive cumulative distribution functions. We propose a diagnostic approach to the evaluation of predictive performance that is based on the paradigm of maximizing the sharpness of the predictive distributions subject to calibration . Calibration refers to the statistical consistency between the distributional forecasts and the observations and is a joint property of the predictions and the events that materialize. Sharpness refers to the concentration of the predictive distributions and is a property of the forecasts only. A simple theoretical framework allows us to distinguish between probabilistic calibration, exceedance calibration and marginal calibration. We propose and study tools for checking calibration and sharpness, among them the probability integral transform histogram, marginal calibration plots, the sharpness diagram and proper scoring rules. The diagnostic approach is illustrated by an assessment and ranking of probabilistic forecasts of wind speed at the Stateline wind energy centre in the US Pacific Northwest. In combination with cross-validation or in the time series context, our proposal provides very general, nonparametric alternatives to the use of information criteria for model diagnostics and model selection. Probabilistic forecasts of continuous variables take the form of predictive densities or predictive cumulative distribution functions. We propose a diagnostic approach to the evaluation of predictive performance that is based on the paradigm of maximizing the sharpness of the predictive distributions subject to calibration. Calibration refers to the statistical consistency between the distributional forecasts and the observations and is a joint property of the predictions and the events that materialize. Sharpness refers to the concentration of the predictive distributions and is a property of the forecasts only. A simple theoretical framework allows us to distinguish between probabilistic calibration, exceedance calibration and marginal calibration. We propose and study tools for checking calibration and sharpness, among them the probability integral transform histogram, marginal calibration plots, the sharpness diagram and proper scoring rules. The diagnostic approach is illustrated by an assessment and ranking of probabilistic forecasts of wind speed at the Stateline wind energy centre in the US Pacific Northwest. In combination with cross-validation or in the time series context, our proposal provides very general, nonparametric alternatives to the use of information criteria for model diagnostics and model selection. Reprinted by permission of Blackwell Publishers Probabilistic forecasts of continuous variables take the form of predictive densities or predictive cumulative distribution functions. We propose a diagnostic approach to the evaluation of predictive performance that is based on the paradigm of "maximizing the sharpness of the predictive distributions subject to calibration". Calibration refers to the statistical consistency between the distributional forecasts and the observations and is a joint property of the predictions and the events that materialize. Sharpness refers to the concentration of the predictive distributions and is a property of the forecasts only. A simple theoretical framework allows us to distinguish between probabilistic calibration, exceedance calibration and marginal calibration. We propose and study tools for checking calibration and sharpness, among them the probability integral transform histogram, marginal calibration plots, the sharpness diagram and proper scoring rules. The diagnostic approach is illustrated by an assessment and ranking of probabilistic forecasts of wind speed at the Stateline wind energy centre in the US Pacific Northwest. In combination with cross-validation or in the time series context, our proposal provides very general, nonparametric alternatives to the use of information criteria for model diagnostics and model selection. Copyright 2007 Royal Statistical Society. |
| Author | Balabdaoui, Fadoua Raftery, Adrian E. Gneiting, Tilmann |
| Author_xml | – sequence: 1 fullname: Gneiting, Tilmann – sequence: 2 fullname: Balabdaoui, Fadoua – sequence: 3 fullname: Raftery, Adrian E |
| BackLink | http://pascal-francis.inist.fr/vibad/index.php?action=getRecordDetail&idt=18579548$$DView record in Pascal Francis http://econpapers.repec.org/article/blajorssb/v_3a69_3ay_3a2007_3ai_3a2_3ap_3a243-268.htm$$DView record in RePEc https://hal.science/hal-01575138$$DView record in HAL |
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| References | DeGroot, M. H. and Fienberg, S. E. (1983) The comparison and evaluation of forecasters. Statistician, 12, 12-22. Pearson, K. (1933) On a method of determining whether a sample of size n supposed to have been drawn from a parent population having a known probability integral has probably been drawn at random. Biometrika, 25, 379-410. Dawid, A. P. and Vovk, V. G. (1999) Prequential probability: principles and properties. Bernoulli, 5, 125-162. Murphy, A. H. and Winkler, R. L. (1992) Diagnostic verification of probability forecasts. Int. J. Forecast., 7, 435-455. Dawid, A. P. (1985a) The impossibility of inductive inference. J. Am. Statist. Ass., 80, 340-341. Moyeed, R. A. and Papritz, A. (2002) An empirical comparison of kriging methods for nonlinear spatial point prediction. Math. Geol., 34, 365-386. Shafer, G. and Vovk, V. (2001) Probability and Finance: It's Only a Game! New York: Wiley. Murphy, A. H. (1972) Scalar and vector partitions of the probability score: Part I, Two-state situation. J. Appl. Meteorol., 11, 273-278. Jolliffe, I. T. and Stephenson, D. B. (eds) (2003) Forecast Verification: a Practitioner's Guide in Atmospheric Science. Chichester: Wiley. Seillier-Moiseiwitsch, F. (1993) Sequential probability forecasts and the probability integral transform. Int. Statist. Rev., 61, 395-408. Diebold, F. X. and Mariano, R. S. (1995) Comparing predictive accuracy. J. Bus. Econ. Statist., 13, 253-263. Shephard, N. (1994) Partial non-Gaussian state space. Biometrika, 81, 115-131. Schervish, M. J. (1989) A general method for comparing probability assessors. Ann. Statist., 17, 1856-1879. Bremnes, J. B. (2004) Probabilistic forecasts of precipitation in terms of quantiles using NWP model output. Mnthly Weath Rev., 132, 338-347. Smith, J. Q. (1985) Diagnostic checks of non-standard time series models. J. Forecast., 4, 283-291. Vovk, V. and Shafer, G. (2005) Good randomized sequential probability forecasting is always possible. J. R. Statist. Soc. B, 67, 747-763. Weigend, A. S. and Shi, S. (2000) Predicting daily probability distributions of S&P500 returns. J. Forecast., 19, 375-392. Diebold, F. X., Gunther, T. A. and Tay, A. S. (1998) Evaluating density forecasts with applications to financial risk management. Int. Econ. Rev., 39, 863-883. Bernardo, J. M. (1979) Expected information as expected utility. Ann. Statist., 7, 686-690. Anderson, J. L. (1996) A method for producing and evaluating probabilistic forecasts from ensemble model integrations. J. Clim., 9, 1518-1530. Oakes, D. (1985) Self-calibrating priors do not exist. J. Am. Statist. Ass., 80, 339. Foster, D. P. and Vohra, R. V. (1998) Asymptotic calibration. Biometrika, 85, 379-390. Garratt, A., Lee, K., Pesaran, M. H. and Shin, Y. (2003) Forecast uncertainties in macroeconomic modelling: an application to the UK economy. J. Am. Statist. Ass., 98, 829-838. Brown, B. G., Katz, R. W. and Murphy, A. H. (1984) Time series models to simulate and forecast wind speed and wind power. J. Clim. Appl. Meteorol., 23, 1184-1195. Candille, G. and Talagrand, O. (2005) Evaluation of probabilistic prediction systems for a scalar variable. Q. J. R. Meteorol. Soc., 131, 2131-2150. Duffie, D. and Pan, J. (1997) An overview of value at risk. J. Deriv., 4, 7-49. Palmer, T. N. (2002) The economic value of ensemble forecasts as a tool for risk assessment: from days to decades. Q. J. R. Meteorol. Soc., 128, 747-774. Besag, J., Green, P., Higdon, D. and Mengersen, K. (1995) Bayesian computing and stochastic systems (with discussion). Statist. Sci., 10, 3-66. Rubin, D. B. (1984) Bayesianly justifiable and relevant frequency calculations for the applied statistician. Ann. Statist., 12, 1151-1172. Blum, J. R., Hanson, D. L. and Koopmans, L. H. (1963) On the strong law of large numbers for a class of stochastic processes. Z. Wahrsch. Ver. Geb., 2, 1-11. Selten, R. (1998) Axiomatic characterization of the quadratic scoring rule. Exptl Econ., 1, 43-62. Dawid, A. P. (1984) Statistical theory: the prequential approach (with discussion). J. R. Statist. Soc. A, 147, 278-292. Berkowitz, J. (2001) Testing density forecasts, with applications to risk management. J. Bus. Econ. Statist., 19, 465-474. Frühwirth-Schnatter, S. (1996) Recursive residuals and model diagnostics for normal and non-normal state space models. Environ. Ecol. Statist., 3, 291-309. Gneiting, T. and Raftery, A. E. (2005) Weather forecasting with ensemble methods. Science, 310, 248-249. Gelman, A., Meng, X.-L. and Stern, H. (1996) Posterior predictive assessment of model fitness via realized discrepancies. Statist. Sin., 6, 733-807. Gneiting, T. and Raftery, A. E. (2006) Strictly proper scoring rules, prediction and estimation. J. Am. Statist. Ass., to be published. Gneiting, T., Raftery, A. E., Westveld, A. H. and Goldman, T. (2005) Calibrated probabilistic forecasting using ensemble model output statistics and minimum CRPS estimation. Mnthly Weath. Rev., 133, 1098-1198. Dawid, A. P. (1982) The well-calibrated Bayesian. J. Am. Statist. Ass., 77, 605-610. Good, I. J. (1952) Rational decisions. J. R. Statist. Soc. B, 14, 107-114. Roulston, M. S. and Smith, L. A. (2003) Combining dynamical and statistical ensembles. Tellus A, 55, 16-30. Boero, G. and Marrocu, E. (2004) The performance of SETAR models: a regime conditional evaluation of point, interval and density forecasts. Int. J. Forecast., 20, 305-320. Raftery, A. E., Madigan, D. and Hoeting, J. A. (1997) Bayesian model averaging for linear regression models. J. Am. Statist. Ass., 92, 179-191. Granger, C. W. J. (2006) Some thoughts on the future of forecasting. Oxf. Bull. Econ. Statist., 67S, 707-711. Sandroni, A., Smorodinsky, R. and Vohra, R. V. (2003) Calibration with many checking rules. Math. Oper. Res., 28, 141-153. Roulston, M. S. and Smith, L. A. (2002) Evaluating probabilistic forecasts using information theory. Mnthly Weath. Rev., 130, 1653-1660. Murphy, A. H., Brown, B. G. and Chen, Y.-S. (1989) Diagnostic verification of temperature forecasts. Weath. Forecast., 4, 485-501. Schumacher, M., Graf, E. and Gerds, T. (2003) How to assess prognostic models for survival data: a case study in oncology. Meth. Inform. Med., 42, 564-571. Clements, M. P. and Smith, J. (2000) Evaluating the forecast densities of linear and non-linear models: applications to output growth and unemployment. J. Forecast., 19, 255-276. Rosenblatt, M. (1952) Remarks on a multivariate transformation. Ann. Math. Statist., 23, 470-472. Krzysztofowicz, R. (1999) Bayesian theory of probabilistic forecasting via deterministic hydrologic model. Wat. Resour. Res., 35, 2739-2750. Wallis, K. F. (2003) Chi-squared tests of interval and density forecasts, and the Bank of England's fan charts. Int. J. Forecast., 19, 165-175. Gneiting, T., Larson, K., Westrick, K., Genton, M. G. and Aldrich, E. (2006) Calibrated probabilistic forecasting at the Stateline wind energy centre: the regime-switching space-time (RST) method. J. Am. Statist. Ass., 101, 968-979. Schervish, M. J. (1985) Comment. J. Am. Statist. Ass., 80, 341-342. Brier, G. W. (1950) Verification of forecasts expressed in terms of probability. Mnthly Weath. Rev., 78, 1-3. Murphy, A. H. and Winkler, R. L. (1987) A general framework for forecast verification. Mnthly Weath. Rev., 115, 1330-1338. Hamill, T. M. (2001) Interpretation of rank histograms for verifying ensemble forecasts. Mnthly Weath. Rev., 129, 550-560. Noceti, P., Smith, J. and Hodges, S. (2003) An evaluation of tests of distributional forecasts. J. Forecast., 22, 447-455. Bauwens, L., Giot, P., Grammig, J. and Veredas, D. (2004) A comparison of financial duration models via density forecasts. Int. J. Forecast., 20, 589-609. Christoffersen, P. F. (1998) Evaluating interval forecasts. Int. Econ. Rev., 39, 841-862. Staël von Holstein, C.-A. S. (1970) Assessment and Evaluation of Subjective Probability Distributions. Stockholm: Economics Research Institute. Hamill, T. M. and Colucci, S. J. (1997) Verification of Eta-RSM short-range ensemble forecasts. Mnthly Weath. Rev., 125, 1312-1327. Dawid, A. P. (1985b) Calibration-based empirical probability (with discussion). Ann. Statist., 13, 1251-1285. Krzysztofowicz, R. and Sigrest, A. A. (1999) Calibration of probabilistic quantitative precipitation forecasts. Weath. Forecast., 14, 427-442. Raftery, A. E., Gneiting, T., Balabdaoui, F. and Polakowski, M. (2005) Using Bayesian model averaging to calibrate forecast ensembles. Mnthly Weath. Rev., 133, 1155-1174. 2004; 20 2005; 131 2005; 133 1993; 61 1984; 23 1985b; 13 1970 2003; 19 1998; 85 1997; 4 2003; 98 1933; 25 2005; 67 2003; 55 1983; 12 1977 1992; 7 2004; 132 1952; 23 1950; 78 2000; 19 1985a; 80 1997; 92 2001 1987; 115 1984; 12 1999; 14 2001; 19 1986 1982 1996; 3 1972; 11 2006; 67S 2003; 42 1996; 9 1996; 6 1979; 7 1989; 4 1952; 14 1984; 147 1985; 4 2005; 310 2002; 130 1995; 13 2002; 34 1995; 10 1982; 77 1985; 80 1997 1995 2006 2005 1994 2004 2003 2002 1994; 81 2001; 129 1999; 5 1997; 125 1998; 39 1963; 2 1999; 35 2002; 128 2003; 28 1998; 1 2006; 101 2003; 22 1989; 17 DeGroot (2023040800083686500_) 1982 Moyeed (2023040800083686500_) 2002; 34 Garratt (2023040800083686500_) 2003; 98 Smith (2023040800083686500_) 1985; 4 Pearson (2023040800083686500_) 1933; 25 Bremnes (2023040800083686500_) 2004; 132 Raftery (2023040800083686500_) 1997; 92 Schervish (2023040800083686500_) 1989; 17 Clements (2023040800083686500_) 2000; 19 Besag (2023040800083686500_) 1995; 10 Murphy (2023040800083686500_) 1992; 7 Dawid (2023040800083686500_) 1986 Candille (2023040800083686500_) 2005; 131 Selten (2023040800083686500_) 1998; 1 Raftery (2023040800083686500_) 2005; 133 Hamill (2023040800083686500_) 2001; 129 Bernardo (2023040800083686500_) 1979; 7 Rubin (2023040800083686500_) 1984; 12 Bauwens (2023040800083686500_) 2004; 20 Jolliffe (2023040800083686500_) 2003 Roulston (2023040800083686500_) 2003; 55 Krzysztofowicz (2023040800083686500_) 1999; 14 Gelman (2023040800083686500_) 1996; 6 Gneiting (2023040800083686500_) 2005; 133 Brown (2023040800083686500_) 1984; 23 Dawid (2023040800083686500_) 1984; 147 Gschlöß l (2023040800083686500_) 2005 Boero (2023040800083686500_) 2004; 20 Duffie (2023040800083686500_) 1997; 4 Palmer (2023040800083686500_) 2002; 128 Murphy (2023040800083686500_) 1972; 11 Schumacher (2023040800083686500_) 2003; 42 Talagrand (2023040800083686500_) 1997 Weigend (2023040800083686500_) 2000; 19 Schervish (2023040800083686500_) 1985; 80 Gneiting (2023040800083686500_) 2006; 101 Wallis (2023040800083686500_) 2003; 19 Diebold (2023040800083686500_) 1995; 13 Staël von Holstein (2023040800083686500_) 1970 Winkler (2023040800083686500_) 1977 Shephard (2023040800083686500_) 1994; 81 Granger (2023040800083686500_) 2006; 67S Hamill (2023040800083686500_) 1997; 125 Rosenblatt (2023040800083686500_) 1952; 23 Anderson (2023040800083686500_) 1996; 9 Shafer (2023040800083686500_) 2001 Foster (2023040800083686500_) 1998; 85 Diebold (2023040800083686500_) 1998; 39 Corradi (2023040800083686500_) 2006 Sandroni (2023040800083686500_) 2003; 28 DeGroot (2023040800083686500_) 1983; 12 Berkowitz (2023040800083686500_) 2001; 19 Dawid (2023040800083686500_) 1985; 80 Oakes (2023040800083686500_) 1985; 80 Frühwirth-Schnatter (2023040800083686500_) 1996; 3 Good (2023040800083686500_) 1952; 14 Brier (2023040800083686500_) 1950; 78 Vovk (2023040800083686500_) 2005; 67 Krzysztofowicz (2023040800083686500_) 1999; 35 Noceti (2023040800083686500_) 2003; 22 Gneiting (2023040800083686500_) 2005; 310 Christoffersen (2023040800083686500_) 1998; 39 Roulston (2023040800083686500_) 2002; 130 Hoeting (2023040800083686500_) 1994 Gneiting (2023040800083686500_) 2006 Dawid (2023040800083686500_) 1982; 77 Dawid (2023040800083686500_) 1999; 5 Gneiting (2023040800083686500_) 2004 Murphy (2023040800083686500_) 1989; 4 Murphy (2023040800083686500_) 1987; 115 Gerds (2023040800083686500_) 2002 Blum (2023040800083686500_) 1963; 2 Brocklehurst (2023040800083686500_) 1995 Dawid (2023040800083686500_) 1985; 13 Seillier-Moiseiwitsch (2023040800083686500_) 1993; 61 |
| References_xml | – reference: Shafer, G. and Vovk, V. (2001) Probability and Finance: It's Only a Game! New York: Wiley. – reference: Hamill, T. M. and Colucci, S. J. (1997) Verification of Eta-RSM short-range ensemble forecasts. Mnthly Weath. Rev., 125, 1312-1327. – reference: Murphy, A. H. and Winkler, R. L. (1992) Diagnostic verification of probability forecasts. Int. J. Forecast., 7, 435-455. – reference: DeGroot, M. H. and Fienberg, S. E. (1983) The comparison and evaluation of forecasters. Statistician, 12, 12-22. – reference: Dawid, A. P. (1982) The well-calibrated Bayesian. J. Am. Statist. Ass., 77, 605-610. – reference: Bauwens, L., Giot, P., Grammig, J. and Veredas, D. (2004) A comparison of financial duration models via density forecasts. Int. J. Forecast., 20, 589-609. – reference: Dawid, A. P. (1985a) The impossibility of inductive inference. J. Am. Statist. Ass., 80, 340-341. – reference: Bernardo, J. M. (1979) Expected information as expected utility. Ann. Statist., 7, 686-690. – reference: Berkowitz, J. (2001) Testing density forecasts, with applications to risk management. J. Bus. Econ. Statist., 19, 465-474. – reference: Murphy, A. H. and Winkler, R. L. (1987) A general framework for forecast verification. Mnthly Weath. Rev., 115, 1330-1338. – reference: Rosenblatt, M. (1952) Remarks on a multivariate transformation. Ann. Math. Statist., 23, 470-472. – reference: Smith, J. Q. (1985) Diagnostic checks of non-standard time series models. J. Forecast., 4, 283-291. – reference: Oakes, D. (1985) Self-calibrating priors do not exist. J. Am. Statist. Ass., 80, 339. – reference: Diebold, F. X., Gunther, T. A. and Tay, A. S. (1998) Evaluating density forecasts with applications to financial risk management. Int. Econ. Rev., 39, 863-883. – reference: Vovk, V. and Shafer, G. (2005) Good randomized sequential probability forecasting is always possible. J. R. Statist. Soc. B, 67, 747-763. – reference: Jolliffe, I. T. and Stephenson, D. B. (eds) (2003) Forecast Verification: a Practitioner's Guide in Atmospheric Science. Chichester: Wiley. – reference: Gneiting, T., Raftery, A. E., Westveld, A. H. and Goldman, T. (2005) Calibrated probabilistic forecasting using ensemble model output statistics and minimum CRPS estimation. Mnthly Weath. Rev., 133, 1098-1198. – reference: Duffie, D. and Pan, J. (1997) An overview of value at risk. J. Deriv., 4, 7-49. – reference: Good, I. J. (1952) Rational decisions. J. R. Statist. Soc. B, 14, 107-114. – reference: Moyeed, R. A. and Papritz, A. (2002) An empirical comparison of kriging methods for nonlinear spatial point prediction. Math. Geol., 34, 365-386. – reference: Schervish, M. J. (1989) A general method for comparing probability assessors. Ann. Statist., 17, 1856-1879. – reference: Bremnes, J. B. (2004) Probabilistic forecasts of precipitation in terms of quantiles using NWP model output. Mnthly Weath Rev., 132, 338-347. – reference: Brier, G. W. (1950) Verification of forecasts expressed in terms of probability. Mnthly Weath. Rev., 78, 1-3. – reference: Gneiting, T., Larson, K., Westrick, K., Genton, M. G. and Aldrich, E. (2006) Calibrated probabilistic forecasting at the Stateline wind energy centre: the regime-switching space-time (RST) method. J. Am. Statist. Ass., 101, 968-979. – reference: Selten, R. (1998) Axiomatic characterization of the quadratic scoring rule. Exptl Econ., 1, 43-62. – reference: Gneiting, T. and Raftery, A. E. (2005) Weather forecasting with ensemble methods. Science, 310, 248-249. – reference: Blum, J. R., Hanson, D. L. and Koopmans, L. H. (1963) On the strong law of large numbers for a class of stochastic processes. Z. Wahrsch. Ver. Geb., 2, 1-11. – reference: Candille, G. and Talagrand, O. (2005) Evaluation of probabilistic prediction systems for a scalar variable. Q. J. R. Meteorol. Soc., 131, 2131-2150. – reference: Frühwirth-Schnatter, S. (1996) Recursive residuals and model diagnostics for normal and non-normal state space models. Environ. Ecol. Statist., 3, 291-309. – reference: Sandroni, A., Smorodinsky, R. and Vohra, R. V. (2003) Calibration with many checking rules. Math. Oper. Res., 28, 141-153. – reference: Weigend, A. S. and Shi, S. (2000) Predicting daily probability distributions of S&P500 returns. J. Forecast., 19, 375-392. – reference: Diebold, F. X. and Mariano, R. S. (1995) Comparing predictive accuracy. J. Bus. Econ. Statist., 13, 253-263. – reference: Roulston, M. S. and Smith, L. A. (2003) Combining dynamical and statistical ensembles. Tellus A, 55, 16-30. – reference: Noceti, P., Smith, J. and Hodges, S. (2003) An evaluation of tests of distributional forecasts. J. Forecast., 22, 447-455. – reference: Besag, J., Green, P., Higdon, D. and Mengersen, K. (1995) Bayesian computing and stochastic systems (with discussion). Statist. Sci., 10, 3-66. – reference: Dawid, A. P. (1984) Statistical theory: the prequential approach (with discussion). J. R. Statist. Soc. A, 147, 278-292. – reference: Anderson, J. L. (1996) A method for producing and evaluating probabilistic forecasts from ensemble model integrations. J. Clim., 9, 1518-1530. – reference: Christoffersen, P. F. (1998) Evaluating interval forecasts. Int. Econ. Rev., 39, 841-862. – reference: Krzysztofowicz, R. (1999) Bayesian theory of probabilistic forecasting via deterministic hydrologic model. Wat. Resour. Res., 35, 2739-2750. – reference: Seillier-Moiseiwitsch, F. (1993) Sequential probability forecasts and the probability integral transform. Int. Statist. Rev., 61, 395-408. – reference: Raftery, A. E., Gneiting, T., Balabdaoui, F. and Polakowski, M. (2005) Using Bayesian model averaging to calibrate forecast ensembles. Mnthly Weath. Rev., 133, 1155-1174. – reference: Schumacher, M., Graf, E. and Gerds, T. (2003) How to assess prognostic models for survival data: a case study in oncology. Meth. Inform. Med., 42, 564-571. – reference: Schervish, M. J. (1985) Comment. J. Am. Statist. Ass., 80, 341-342. – reference: Brown, B. G., Katz, R. W. and Murphy, A. H. (1984) Time series models to simulate and forecast wind speed and wind power. J. Clim. Appl. Meteorol., 23, 1184-1195. – reference: Dawid, A. P. (1985b) Calibration-based empirical probability (with discussion). Ann. Statist., 13, 1251-1285. – reference: Boero, G. and Marrocu, E. (2004) The performance of SETAR models: a regime conditional evaluation of point, interval and density forecasts. Int. J. Forecast., 20, 305-320. – reference: Gneiting, T. and Raftery, A. E. (2006) Strictly proper scoring rules, prediction and estimation. J. Am. Statist. Ass., to be published. – reference: Staël von Holstein, C.-A. S. (1970) Assessment and Evaluation of Subjective Probability Distributions. Stockholm: Economics Research Institute. – reference: Dawid, A. P. and Vovk, V. G. (1999) Prequential probability: principles and properties. Bernoulli, 5, 125-162. – reference: Wallis, K. F. (2003) Chi-squared tests of interval and density forecasts, and the Bank of England's fan charts. Int. J. Forecast., 19, 165-175. – reference: Krzysztofowicz, R. and Sigrest, A. A. (1999) Calibration of probabilistic quantitative precipitation forecasts. Weath. Forecast., 14, 427-442. – reference: Murphy, A. H., Brown, B. G. and Chen, Y.-S. (1989) Diagnostic verification of temperature forecasts. Weath. Forecast., 4, 485-501. – reference: Shephard, N. (1994) Partial non-Gaussian state space. Biometrika, 81, 115-131. – reference: Palmer, T. N. (2002) The economic value of ensemble forecasts as a tool for risk assessment: from days to decades. Q. J. R. Meteorol. Soc., 128, 747-774. – reference: Pearson, K. (1933) On a method of determining whether a sample of size n supposed to have been drawn from a parent population having a known probability integral has probably been drawn at random. Biometrika, 25, 379-410. – reference: Clements, M. P. and Smith, J. (2000) Evaluating the forecast densities of linear and non-linear models: applications to output growth and unemployment. J. Forecast., 19, 255-276. – reference: Granger, C. W. J. (2006) Some thoughts on the future of forecasting. Oxf. Bull. Econ. Statist., 67S, 707-711. – reference: Hamill, T. M. (2001) Interpretation of rank histograms for verifying ensemble forecasts. Mnthly Weath. Rev., 129, 550-560. – reference: Rubin, D. B. (1984) Bayesianly justifiable and relevant frequency calculations for the applied statistician. Ann. Statist., 12, 1151-1172. – reference: Gelman, A., Meng, X.-L. and Stern, H. (1996) Posterior predictive assessment of model fitness via realized discrepancies. Statist. Sin., 6, 733-807. – reference: Raftery, A. E., Madigan, D. and Hoeting, J. A. (1997) Bayesian model averaging for linear regression models. J. Am. Statist. Ass., 92, 179-191. – reference: Foster, D. P. and Vohra, R. V. (1998) Asymptotic calibration. Biometrika, 85, 379-390. – reference: Garratt, A., Lee, K., Pesaran, M. H. and Shin, Y. (2003) Forecast uncertainties in macroeconomic modelling: an application to the UK economy. J. Am. Statist. Ass., 98, 829-838. – reference: Murphy, A. H. (1972) Scalar and vector partitions of the probability score: Part I, Two-state situation. J. Appl. Meteorol., 11, 273-278. – reference: Roulston, M. S. and Smith, L. A. (2002) Evaluating probabilistic forecasts using information theory. Mnthly Weath. Rev., 130, 1653-1660. – volume: 80 start-page: 341 year: 1985 end-page: 342 article-title: Comment publication-title: J. Am. Statist. Ass. – volume: 80 start-page: 339 year: 1985 article-title: Self‐calibrating priors do not exist publication-title: J. Am. Statist. Ass. – volume: 19 start-page: 255 year: 2000 end-page: 276 article-title: Evaluating the forecast densities of linear and non‐linear models: applications to output growth and unemployment publication-title: J. 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| SubjectTerms | Analytical forecasting Autocorrelation Calibration Cross-validation cumulative distribution Density forecast diagnostic techniques Distribution Distribution theory Ensemble prediction system Evaluation Ex post evaluation Exact sciences and technology Forecast verification Forecasting Forecasts Histograms Inference from stochastic processes; time series analysis Mathematical methods Mathematics Model diagnostics Model testing Nonparametric inference Posterior predictive assessment prediction Predictive distribution Prequential principle Probabilities Probability Probability and statistics Probability forecasts Probability integral transform Probability theory and stochastic processes Proper scoring rule Sciences and techniques of general use Statistical analysis Statistical methods Statistics Stochastic processes Studies Time series time series analysis Time series forecasting United States Weather forecasting Wind power wind speed Wind velocity |
| Title | Probabilistic forecasts, calibration and sharpness |
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