Distribution multimodale

Résumé

vignette|Exemple de distribution bimodale de minerais d'or. X : teneur en g/t ; Y : production en tonnes. Le caractère bimodal définit deux groupes de populations statistiques résultant de deux phénomènes différents. En probabilités et statistique, une distribution multimodale est une distribution statistique présentant plusieurs modes. vignette| Histogramme bimodal vignette|Dans ce cas précis, une distribution bimodale un mélange de deux distributions normales avec la même variance mais des moyennes différentes. Si les poids n'étaient pas égaux, la distribution résultante pourrait toujours être bimodale mais avec des pics de hauteurs différentes. Éléments de contexte Les distributions multimodales sont rencontrées dans divers domaines :

les sondages d’opinion,
la production industrielle, qui cherche généralement à éviter les phénomènes multimodaux souvent révélateurs d’un facteur d’influence non maîtrisé dans le processus,
les phénomènes liés à la climatologie, à

Source officielle

https://fr.wikipedia.org/wiki/Distribution_multimodale

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We introduce an online outlier detection algorithm to detect outliers in a sequentially observed data stream. For this purpose, we use a two-stage filtering and hedging approach. In the first stage, we construct a multimodal probability density function to model the normal samples. In the second stage, given a new observation, we label it as an anomaly if the value of aforementioned density function is below a specified threshold at the newly observed point. In order to construct our multimodal density function, we use an incremental decision tree to construct a set of subspaces of the observation space. We train a single component density function of the exponential family using the observations, which fall inside each subspace represented on the tree. These single component density functions are then adaptively combined to produce our multimodal density function, which is shown to achieve the performance of the best convex combination of the density functions defined on the subspaces. As we observe more samples, our tree grows and produces more subspaces. As a result, our modeling power increases in time, while mitigating overfitting issues. In order to choose our threshold level to label the observations, we use an adaptive thresholding scheme. We show that our adaptive threshold level achieves the performance of the optimal prefixed threshold level, which knows the observation labels in hindsight. Our algorithm provides significant performance improvements over the state of the art in our wide set of experiments involving both synthetic as well as real data.

2019

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Validation of likelihood ratio methods for forensic evidence evaluation handling multimodal score distributions

Rudolf Haraksim

This study presents a method for computing likelihood ratios (LRs) from multimodal score distributions, as the ones produced by some commercial off-the-shelf automated fingerprint identification systems (AFISs). The AFIS algorithms used to compare fingermarks and fingerprints were primarily developed for forensic investigation rather than for forensic evaluation purposes. Thus, in some of those algorithms, the computation of discriminating scores is speed-optimised. In the case of the AFIS algorithm used in this work, the speed-optimisation is achieved by performing the comparison in three different stages, each of which outputs scores of different magnitudes. As a consequence, all scores together present a multimodal distribution, even though each fingermark-to-fingerprint comparison generates one single score. This multimodal distribution of scores might be typical for other biometric systems or other algorithms, and the method proposed in this work can be also applied to those cases. As a result, the authors propose a probabilistic model for LR computation that presents more robustness to overfitting and data sparsity than other traditional approaches, like the use of models based on kernel density functions.

Inst Engineering Technology-IET2017

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Safe Motion Planning against Multimodal Distributions Based on a Scenario Approach

Chunhui Chen

We present the design of a motion planning algorithm that ensures safety for an autonomous vehicle. In particular, we consider a multimodal distribution over uncertainties; for example, the uncertain predictions of future trajectories of surrounding vehicles reflect discrete decisions, such as turning or going straight at intersections. We develop a computationally efficient, scenario-based approach that solves the motion planning problem with high confidence given a quantifiable number of samples from the multimodal distribution. Our approach is based on two preprocessing steps, which 1) separate the samples into distinct clusters and 2) compute a bounding polytope for each cluster. Then, we rewrite the motion planning problem approximately as a mixed-integer problem using the polytopes. We demonstrate via simulation on the nuScenes dataset that our approach ensures safety with high probability in the presence of multimodal uncertainties, and is computationally more efficient and less conservative than a conventional scenario approach. © 2017 IEEE.

2022

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