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Linear Algebra
Published in Seyedeh Leili Mirtaheri, Reza Shahbazian, Machine Learning Theory to Applications, 2022
Seyedeh Leili Mirtaheri, Reza Shahbazian
The discrete multimodal distribution k is defined when the distribution has several modes, which can be shown as distinct peaks in the probability density function. The multivariate probability density function of a multimodal distribution can be written as follows: p(k)=k!k1!⋯kN!∏i=1Nyiki,∑i=1Nki=N where k ∈ WD×1 is a random vector which consists of whole numbers (ki ∈ 0,1,2,...) and pi is the probability of i’th element, hence 0 ≤ pi ≤ 1 and ∑i=1Npi=1.
Contextual anomaly detection for high-dimensional data using Dirichlet process variational autoencoder
Published in IISE Transactions, 2023
In this article, we propose a new method for contextual anomaly detection, particularly for complex and high-dimensional response and contextual variables. Specifically, we jointly model the response and contextual variables using separate Variational AutoEncoders (VAEs), which learn low-dimensional representations from high-dimensional data, and we use both the latent response and contextual variables as inputs to the decoder of the VAE to generate the response data, similar to Shulman (2019). However, unlike Shulman (2019) who used a normal prior for the latent space of the VAE for the contextual variables, we impose a Dirichlet process mixture of Gaussians as the prior distribution. This prior allows the latent space of the contextual variables to be represented by several clusters using a Gaussian mixture model, where each cluster represents a different contextual environment. For example, in our case study in Section 6, the contextual variables vary depending on which cord is rubber coated, and each cord may correspond to a specific mode of the multimodal distribution of the latent contextual variables. By contrast, the normal prior in Shulman (2019) only allows a unimodal distribution of the latent space. Moreover, using the proposed prior, the number of clusters in the latent space of the contextual variables can be automatically determined, due to the Dirichlet process.
Evolino Recurrent Neural Network Ensemble for Speculation in Exchange Market in Time of Anomalies
Published in Applied Artificial Intelligence, 2020
Nijolė Maknickienė, Algirdas Maknickas
Third step. Evaluation of the prediction. A multimodal distribution has a shape, mode value, and standard deviation. Calculation of profit and loss probabilities is generated according to the last known value or closing value of previous day. Some methods of evaluation of this combination of distributions were investigated in previous work: high-low strategy (Stankevičienė, Maknickienė, and Maknickas 2014); and UK-New York time zones (Maknickiene and Maknickas 2016). Investigation of distribution parameters before, at moment and after anomaly will be presented on the next section of the paper.
A hierarchical Bayesian logistic regression with a finite mixture for identifying higher-than-expected crash proportions at intersections
Published in Journal of Transportation Safety & Security, 2019
Karen R. Richard, Sungyop Kim, Gudmundur F. Ulfarsson
The largest difference in results occurred for urban four-leg intersections. For rear-end crashes, the threshold proportions vary between methods as well as the number of locations. Like the previous case for rural three-leg intersections, the threshold value for the HB method (0.346) is closer to the median and highest mode than the threshold value for the HSM method , because the HSM method is more skewed by the long tail of the distribution whereas the HB method is better able to capture this tail and a multimodal distribution. Also, 133 locations end up between the two threshold values, because this category of intersections has the largest sample size and the observations are clustered around these measures of central tendency. Angle crashes can be similarly explained. However, it is interesting to note the differences for fixed-object crashes. For that crash type, the HSM method selects 47 locations for all probability levels. However, the number of locations selected by the HB method decreases with increasing probability, which is what would be expected. For this crash type, there are 47 locations with exactly one single-vehicle fixed-object crash, all with a different number of total crashes. Because there are so many locations with zero single-vehicle fixed-object crashes, the threshold proportion for both methods is calculated to be small. For the HSM method, the low threshold proportion leads to a probability of 100% for any of the 47 locations with one crash to exceed the threshold. Consequently, all 47 locations are selected as having a high crash proportion for each threshold level. For the HB method, different probabilities are assigned based on the location's actual calculated proportion as well as variability estimate. Accordingly, this results in a decreasing number of locations selected for each threshold level. The remaining difference in results can be explained through the same issues described above.