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Modeling Images with Undirected Graphical Models
Published in Olivier Lézoray, Leo Grady, Image Processing and Analysis with Graphs, 2012
Similar models have also been used in [15, 13]. This model can be justified in two different ways. The first justification is based on statistical properties of images. The histogram of the response of applying a filter to images has a characteristic shape. Figure 16.7(b) shows the histogram of a derivative filter applied to the image in Figure 16.7. The notable aspect of this shape is that it is sharply peaked around zero, and the tails of the distribution have more mass than would be seen in a Gaussian distribution. This type of distribution is referred to as a heavy-tailed distribution. As shown in Figure 16.7(c), the negative logarithm of this distribution has a similar shape as the Lorentzian penalty. The filters in the Field of Experts model that perform the best are larger than simple derivative filters used to generate the example in Figure 16.7, but the histogram of the response has a similar shape. Thus, the Field of Experts model can be thought of as modeling the marginal statistics of images. Improved versions of the model have also been produced by using Gaussian Scale Mixtures [13, 16].
Robustness assessment of public bus transit system with a response-integrated approach for a resilient public transport system in Hong Kong
Published in Transportmetrica B: Transport Dynamics, 2023
Zizhen Xu, Chuwei Zhang, Shauhrat S. Chopra
Before we present the results of the disruption scenario, we examined the node degree distributions to characterize the FBS network and the road network (shown in Figure 4). A heavy-tailed distribution was observed for the FBS, while the degree distribution of the road network was close to an exponential distribution. It shows that the majority of the bus stops had a very low value of in- and out-degree, and the heavy tail suggests that a small number of stops were considered critical for the entire bus network. In contrast, most nodes in the road network have similar degrees, and such a network does not show significantly different levels of robustness in random failures and targeted attacks. From this, it can be understood that the dependent model was built with these two networks that have different degree distributions and looked into the random infrastructure failures.
A Bayesian Partially Observable Online Change Detection Approach with Thompson Sampling
Published in Technometrics, 2023
Choice of wj and σj: We can set a prior, such as distribution for wj (Ročková and George 2014) or a degenerate point value according to the happening probability of the change (Ishwaran and Rao 2005). In our sparse change scenario, choosing small a and large b will be more effective for targeting at sparse models in high dimensions (Castillo and van der Vaart 2012). For example, in Section 5, since we know that the chance of change on every basis is really small, we set as a reference value. As to σj, it can be set as a hyper-parameter with a heavy-tailed distribution as well, such as Cauchy distribution, or double exponential distribution (Ročková and George 2014). It can also be set as a fixed value following the methods in George and McCulloch (1993, 1997) and Guan and Stephens (2011). As pointed out by Guan and Stephens (2011), the fact that different works using different methods to set σj actually indicates a lack of consensus, while a compelling motivation or interpretation would be enough to support the choice of σj. In Section 5, with prior information about the change magnitude, that is, , we set (within 1-sigma of ) to guarantee that there is a high probability to accommodate all plausible in the slab distribution.