: This thesis applies neural network feature selection techniques to multivariate time series data to improve prediction of a target time series. Two approaches to feature selection are used. First, a subset enumeration method is used to determine which financial indicators are most useful for aiding in prediction of the S&P 500 futures daily price. The candidate indicators evaluated include RSI, Stochastics and several moving averages. Results indicate that the Stochastics and RSI indicators result in better prediction results than the moving averages. The second approach to feature selection is calculation of individual saliency metrics. A new decision boundary-based individual saliency metric, and a classifier independent saliency metric are developed and tested. Ruck's saliency metric, the decision boundary based saliency metric, and the classifier independent saliency metric are compared for a data set consisting of the RSI and Stochastics indicators as well as delayed closing price values. The decision based metric and the Ruck metric results are similar, but the classifier independent metric agrees with neither of the other metrics. The nine most salient features, determined by the decision boundary based metric, are used to train a neural network and the results are presented and compared to other published results. (AN)

/pdf/nonlinear-time-series-analysis-1j3a35n4y3.pdf

Nonlinear Time Series Analysis.

Stochastic equations in infinite dimensions

5. M. Green, J. Schwarz, and E. Witten, Superstring theory, Cambridge Univ. Press, 1987. 6. J. Isenberg, P. Yasskin, and P. Green, Non-self-dual gauge fields, Phys. Lett. 78B (1978), 462-464. 7. B. Kostant, Graded manifolds, graded Lie theory, and prequantization, Differential Geometric Methods in Mathematicas Physics, Lecture Notes in Math., vol. 570, SpringerVerlag, Berlin and New York, 1977. 8. C. LeBrun, Thickenings and gauge fields, Class. Quantum Grav. 3 (1986), 1039-1059. 9. , Thickenings and conformai gravity, preprint, 1989. 10. C. LeBrun and M. Rothstein, Moduli of super Riemann surfaces, Commun. Math. Phys. 117(1988), 159-176. 11. Y. Manin, Critical dimensions of string theories and the dualizing sheaf on the moduli space of (super) curves, Funct. Anal. Appl. 20 (1987), 244-245. 12. R. Penrose and W. Rindler, Spinors and space-time, V.2, spinor and twistor methods in space-time geometry, Cambridge Univ. Press, 1986. 13. R. Ward, On self-dual gauge fields, Phys. Lett. 61A (1977), 81-82. 14. E. Witten, An interpretation of classical Yang-Mills theory, Phys. Lett. 77NB (1978), 394-398. 15. , Twistor-like transform in ten dimensions, Nucl. Phys. B266 (1986), 245-264. 16. , Physics and geometry, Proc. Internat. Congr. Math., Berkeley, 1986, pp. 267302, Amer. Math. Soc, Providence, R.I., 1987.

Infinite-Dimensional Dynamical Systems in Mechanics and Physics (Roger Temam)

Nonlinear dynamical analysis of EEG and MEG: Review of an emerging field

We describe the construction of a new version of the Europe‐wide E‐OBS temperature (daily minimum, mean and maximum values) and precipitation dataset. This version provides an improved estimation of interpolation uncertainty through the calculation of a 100‐member ensemble of realizations of each daily field. The dataset covers the period back to 1950, and provides gridded fields at a spacing of 0.25° x 0.25° in regular latitude/longitude coordinates. As with the original E‐OBS dataset, the ensemble version is based on the station series collated as part of the ECA&D initiative. Station density varies significantly over the domain, and over time, and a reliable estimation of interpolation uncertainty in the gridded fields is therefore important for users of the dataset. The uncertainty quantified by the ensemble dataset is more realistic than the uncertainty estimates in the original version, although uncertainty is still underestimated in data‐sparse regions. The new dataset is compared against the earlier version of E‐OBS and against regional gridded datasets produced by a selection of National Meteorological Services (NMSs). In terms of both climatological averages and extreme values, the new version of E‐OBS is broadly comparable to the earlier version. Nonetheless, users will notice differences between the two E‐OBS versions, especially for precipitation, which arises from the different gridding method used.

https://agupubs.onlinelibrary.wiley.com/doi/pdf/10.1029/2017JD028200

An Ensemble Version of the E‐OBS Temperature and Precipitation Data Sets

The reliability diagram is a common diagnostic graph used to summarize and evaluate probabilistic forecasts. Its strengths lie in the ease with which it is produced and the transparency of its definition. While visually appealing, major long-noted shortcomings lie in the difficulty of interpreting the graph visually; for the most part, ambiguities arise from variations in the distributions of forecast probabilities and from various binning procedures. A resampling method for assigning consistency bars to the observed frequencies is introduced that allows for immediate visual evaluation as to just how likely the observed relative frequencies are under the assumption that the predicted probabilities are reliable. Further, an alternative presentation of the same information on probability paper eases quantitative evaluation and comparison. Both presentations can easily be employed for any method of binning.

Increasing the Reliability of Reliability Diagrams

Questions remain regarding how the skill of operational probabilistic forecasts is most usefully evaluated or compared, even though probability forecasts have been a long-standing aim in meteorological forecasting. This paper explains the importance of employing proper scores when selecting between the various measures of forecast skill. It is demonstrated that only proper scores provide internally consistent evaluations of probability forecasts, justifying the focus on proper scores independent of any attempt to influence the behavior of a forecaster. Another property of scores (i.e., locality) is discussed. Several scores are examined in this light. There is, effectively, only one proper, local score for probability forecasts of a continuous variable. It is also noted that operational needs of weather forecasts suggest that the current concept of a score may be too narrow; a possible generalization is motivated and discussed in the context of propriety and locality.

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Scoring Probabilistic Forecasts: The Importance of Being Proper

The translation of an ensemble of model runs into a probability distribution is a common task in model-based prediction. Common methods for such ensemble interpretations proceed as if verification and ensemble were draws from the same underlying distribution, an assumption not viable for most, if any, real world ensembles. An alternative is to consider an ensemble as merely a source of information rather than the possible scenarios of reality. This approach, which looks for maps between ensembles and probabilistic distributions, is investigated and extended. Common methods are revisited, and an improvement to standard kernel dressing, called ‘affine kernel dressing’ (AKD), is introduced. AKD assumes an affine mapping between ensemble and verification, typically not acting on individual ensemble members but on the entire ensemble as a whole, the parameters of this mapping are determined in parallel with the other dressing parameters, including a weight assigned to the unconditioned (climatological) distribution. These amendments to standard kernel dressing, albeit simple, can improve performance significantly and are shown to be appropriate for both overdispersive and underdispersive ensembles, unlike standard kernel dressing which exacerbates over dispersion. Studies are presented using operational numerical weather predictions for two locations and data from the Lorenz63 system, demonstrating both effectiveness given operational constraints and statistical significance given a large sample.

/pdf/from-ensemble-forecasts-to-predictive-distribution-functions-4sda4fug8j.pdf

From ensemble forecasts to predictive distribution functions

Scoring rules are an important tool for evaluating the performance of probabilistic forecasting schemes. A scoring rule is called strictly proper if its expectation is optimal if and only if the forecast probability represents the true distribution of the target. In the binary case, strictly proper scoring rules allow for a decomposition into terms related to the resolution and the reliability of a forecast. This fact is particularly well known for the Brier Score. In this article, this result is extended to forecasts for finite-valued targets. Both resolution and reliability are shown to have a positive effect on the score. It is demonstrated that resolution and reliability are directly related to forecast attributes that are desirable on grounds independent of the notion of scores. This finding can be considered an epistemological justification of measuring forecast quality by proper scoring rules. A link is provided to the original work of DeGroot and Fienberg, extending their concepts of sufficiency and refinement. The relation to the conjectured sharpness principle of Gneiting, et al., is elucidated.

/pdf/reliability-sufficiency-and-the-decomposition-of-proper-2ek1lyj203.pdf

Reliability, sufficiency, and the decomposition of proper scores

The continuous ranked probability score (CRPS) is a frequently used scoring rule. In contrast with many other scoring rules, the CRPS evaluates cumulative distribution functions. An ensemble of forecasts can easily be converted into a piecewise constant cumulative distribution function with steps at the ensemble members. This renders the CRPS a convenient scoring rule for the evaluation of ‘raw’ ensembles, obviating the need for sophisticated ensemble model output statistics or dressing methods prior to evaluation. In this article, a relation between the CRPS score and the quantile score is established. The evaluation of ‘raw’ ensembles using the CRPS is discussed in this light. It is shown that latent in this evaluation is an interpretation of the ensemble as quantiles but with non-uniform levels. This needs to be taken into account if the ensemble is evaluated further, for example with rank histograms.

/pdf/evaluating-raw-ensembles-with-the-continuous-ranked-2v1vfunv5w.pdf

Jochen Bröcker

Papers

Increasing the Reliability of Reliability Diagrams

Scoring Probabilistic Forecasts: The Importance of Being Proper

From ensemble forecasts to predictive distribution functions

Reliability, sufficiency, and the decomposition of proper scores

Evaluating raw ensembles with the continuous ranked probability score