Interval Predictor Models Research Articles

Uncertainty Propagation of Missing Data Signals with the Interval Discrete Fourier Transform

The interval discrete Fourier transform (DFT) algorithm can propagate signals carrying interval uncertainty. By addressing the repeated variables problem, the interval DFT algorithm provides exact theoretical bounds on the Fourier amplitude and estimates of the power spectral density (PSD) function while running in polynomial time. Thus, the algorithm can be used to assess the worst-case scenario in terms of maximum or minimum power, and provide insights into the amplitude spectrum bands of the transformed signal. To propagate signals with missing data, an upper and lower value for the missing data present in the signal must be assumed, such that the uncertainty in the spectrum bands can also be interpreted as an indicator of the quality of the reconstructed signal. For missing data reconstruction, there are a number of techniques available that can be used to obtain reliable bounds in the time domain, such as Kriging regressors and interval predictor models. Alternative heuristic strategies based on variable—as opposed to fixed—bounds can also be explored. This work aims to investigate the sensitivity of the algorithm against interval uncertainty in the time signal. The investigation is conducted in different case studies using signals of different lengths generated from the Kanai-Tajimi PSD function, representing earthquakes, and the Joint North Sea Wave Observation Project (JONSWAP) PSD function, representing sea waves as a narrowband PSD model.

Soft-constrained interval predictor models and epistemic reliability intervals: A new tool for uncertainty quantification with limited experimental data

Interval Predictor Models (IPMs) offer a non-probabilistic, interval-valued, characterization of the uncertainty affecting random data generating processes. IPMs are constructed directly from data, with no assumptions on the distributions of the uncertain factors driving the process, and are therefore exempt from the subjectivity induced by such a practice. The reliability of an IPM defines the probability of correct predictions for future samples and, in practice, its true value is always unknown due to finite samples sizes and limited understanding of the process. This paper proposes an overview of scenario optimization programs for the identification of IPMs. Traditional IPM identification methods are compared with a new scheme which softens the scenario constraints and exploits a trade-off between reliability and accuracy. The new methods allows prescribing predictors that achieve higher accuracy for a quantifiable reduction in the reliability. Scenario optimization theory is the mathematical tool used to prescribe formal epistemic bounds on the predictors reliability. A review of relevant theorems and bounds is proposed in this work. Scenario-based reliability bounds hold distribution-free, non asymptotically, and quantify the uncertainty affecting the model’s ability to correctly predict future data. The applicability of the new approach is tested on three examples: i) on the modelling of a trigonometric function affected by a noise term, ii) on the identification of a black-box system-controller dynamic response model and, iii) on the modelling of the vibration response of a car suspension arm crossed by a crack of unknown length. The points of strength and limitations of the new IPM are discussed based on the accuracy, computational cost, and width of the resulting epistemic bounds.

On a class of interval predictor models with universal reliability

An Interval Predictor Model (IPM) is a rule by which some observable variables (system inputs) are mapped into an interval that is used to predict an inaccessible variable (system output). IPMs have been studied in Campi et al. (2009), where the problem of fitting an IPM on a set of observations has been considered. In the same paper, upper-bounds on the probability that a future system output will fall outside the predicted interval (misprediction) have also been derived in a stationary and independent framework. While these bounds have the notable property of being valid independently of the unknown mechanism that has generated the data, in general the actual probability distribution of the misprediction does depend on the data generation mechanism and, hence, these bounds may introduce conservatism when applied to a specific case. In this paper, we study the reliability of an important class of IPMs, called minimax layers, and show that this class exhibits the special property that the probability distribution of the misprediction is known exactly and is universal, i.e., is always the same irrespective of the data generation mechanism. This result carries important consequences on the use of minimax layers in practice.

Robust propagation of probability boxes by interval predictor models

This paper proposes numerical strategies to robustly and efficiently propagate probability boxes through expensive black box models. An interval is obtained for the system failure probability, with a confidence level. The three proposed algorithms are sampling based, and so can be easily parallelised, and make no assumptions about the functional form of the model. In the first two algorithms, the performance function is modelled as a function with unknown noise structure in the aleatory space and supplemented by a modified performance function. In the third algorithm, an Interval Predictor Model is constructed and a re-weighting strategy used to find bounds on the probability of failure. Numerical examples are presented to show the applicability of the approach. The proposed method is flexible and can account for epistemic uncertainty contained inside the limit state function. This is a feature which, to the best of the authors’ knowledge, no existing methods of this type can deal with.

Frequentist history matching with Interval Predictor Models

• History matching is achieved with limited knowledge regarding the measurement error. • Interval Predictor Models are used to robustly quantify the measurement error. • Bounds are obtained on the p -values from frequentist inference . • The method is applied to an industrial oil reservoir test case model. In this paper a novel approach is presented for history matching models without making assumptions about the measurement error. Interval Predictor Models are used to robustly model the observed data and hence a novel figure of merit is proposed to quantify the quality of matches in a frequentist probabilistic framework . The proposed method yields bounds on the p -values from frequentist inference. The method is first applied to a simple example and then to a realistic case study (the Imperial College Fault Model) in order to evaluate its applicability and efficacy. When there is no modelling error the method identifies a feasible region for the matched parameters, which for our test case contained the truth case. When attempting to match one model to data from a different model, a region close to the truth case was identified. The effect of increasing the number of data points on the history matching is also discussed.

Interval Predictor Models With a Linear Parameter Dependency

This paper develops techniques for constructing metamodels that predict the range of an output variable given input–output data. We focus on models depending linearly on the parameters and arbitrarily on the input. This structure enables to rigorously characterize the range of the predicted output and the uncertainty in the model’s parameters. Strategies for calculating optimal interval predictor models (IPMs) that are insensitive to outliers are proposed. The models are optimal in the sense that they yield an interval valued function of minimal spread containing all (or, depending on the formulation, most) of the observations. Outliers are identified as the IPM is calculated by evaluating the extent by which their inclusion into the dataset degrades the tightness of the prediction. When the data generating mechanism (DGM) is stationary, the data are independent, and the optimization program (OP) used for calculating the IPM is convex (or when its solution coincides with the solution to an auxiliary convex program); the model’s reliability, which is the probability that a future observation would fall within the predicted range, is bounded tightly using scenario optimization theory. In contrast to most alternative techniques, this framework does not require making any assumptions on the underlying structure of the DGM.

Interval predictor models: Identification and reliability

This paper addresses the problem of constructing reliable interval predictors directly from observed data. Differently from standard predictor models, interval predictors return a prediction interval as opposed to a single prediction value. We show that, in a stationary and independent observations framework, the reliability of the model (that is, the probability that the future system output falls in the predicted interval) is guaranteed a priori by an explicit and non-asymptotic formula, with no further assumptions on the structure of the unknown mechanism that generates the data. This fact stems from a key result derived in this paper, which relates, at a fundamental level, the reliability of the model to its complexity and to the amount of available information (number of observed data).