For real-life nonlinear systems, the exact form of nonlinearity is often not known and the known governing equations are often based on certain assumptions and approximations. Such representation introduced model-form error into the system. In this paper, we propose a novel gray-box modeling approach that not only identifies the model-form error but also utilizes it to improve the predictive capability of the known but approximate governing equation. The primary idea is to treat the unknown model-form error as a residual force and estimate it using duel Bayesian filter based joint input-state estimation algorithms. For improving the predictive capability of the underlying physics, we first use machine learning algorithm to learn a mapping between the estimated state and the input (model-form error) and then introduce it into the governing equation as an additional term. This helps in improving the predictive capability of the governing physics and allows the model to generalize to unseen environment. Although in theory, any machine learning algorithm can be used within the proposed framework, we use Gaussian process in this work. To test the performance of proposed framework, case studies discussing four different dynamical systems are discussed; results for which indicate that the framework is applicable to a wide variety of systems and can produce reliable estimates of original system's states.

In this paper, we propose an ensemble learning algorithm called \textit{under-bagging $k$-nearest neighbors} (\textit{under-bagging $k$-NN}) for imbalanced classification problems. On the theoretical side, by developing a new learning theory analysis, we show that with properly chosen parameters, i.e., the number of nearest neighbors $k$, the expected sub-sample size $s$, and the bagging rounds $B$, optimal convergence rates for under-bagging $k$-NN can be achieved under mild assumptions w.r.t.~the arithmetic mean (AM) of recalls. Moreover, we show that with a relatively small $B$, the expected sub-sample size $s$ can be much smaller than the number of training data $n$ at each bagging round, and the number of nearest neighbors $k$ can be reduced simultaneously, especially when the data are highly imbalanced, which leads to substantially lower time complexity and roughly the same space complexity. On the practical side, we conduct numerical experiments to verify the theoretical results on the benefits of the under-bagging technique by the promising AM performance and efficiency of our proposed algorithm.

In the analysis of observational data in social sciences and businesses, it is difficult to obtain a "(quasi) single-source dataset" in which the variables of interest are simultaneously observed. Instead, multiple-source datasets are typically acquired for different individuals or units. Various methods have been proposed to investigate the relationship between the variables in each dataset, e.g., matching and latent variable modeling. It is necessary to utilize these datasets as a single-source dataset with missing variables. Existing methods assume that the datasets to be integrated are acquired from the same population or that the sampling depends on covariates. This assumption is referred to as missing at random (MAR) in terms of missingness. However, as will been shown in application studies, it is likely that this assumption does not hold in actual data analysis and the results obtained may be biased. We propose a data fusion method that does not assume that datasets are homogenous. We use a Gaussian process latent variable model for non-MAR missing data. This model assumes that the variables of concern and the probability of being missing depend on latent variables. A simulation study and real-world data analysis show that the proposed method with a missing-data mechanism and the latent Gaussian process yields valid estimates, whereas an existing method provides severely biased estimates. This is the first study in which non-random assignment to datasets is considered and resolved under resonable assumptions in data fusion problem.

The economy of the European Union (EU) is deeply grounded into Small and Medium Enterprises (SMEs). SMEs represent about 99.8% of the active enterprises in the EU-28 non-financial business sector (NFBS), accounting for almost 60% of value-added within the NFBS and fostering the workforce of the EU with two out of every three jobs (European Commission, 2019a). Thus, there is a wide literature covering various economic aspects of SMEs, with a particular attention to default prediction (for an up-to-date review see Ciampi et al., 2021), which is of interest not only for scholars but also for practitioners such as financial intermediaries and for policy makers in their effort to support SMEs and to ease credit constraints to which they are naturally exposed (Andries et al., 2018; Cornille et al., 2019). Whether it is for private credit-risk assessment or for public funding, independently on the type of data imputed to measure the health status of a firm, prediction of default should success in two aspects: maximise correct classification and clarify the role of the variables involved in the process. Most of the times, the contributions based on Machine Learning (ML) techniques neglect the latter aspect, being rather focused on the former, often with better results with respect to parametric techniques that provide, on the contrary, a clear framework for interpretation.

Here we will see a list of the most known classic and modern types of Data Analysis methods and models. Mathematical and Statistical Methods for Data Analysis Mathematical and statistical sciences have much to give to data mining management and analysis. In fact, most data mining techniques are statistical data analysis tools. Some methods and techniques are well known and very effective. This statistical technique does exactly what the name suggests -"Describe".

Let's say we have the data on consumption statistics of some kind and it has a time stamp on it: In this example, the "Date" column could easily be used to extract additional features and generate powerful insights such as variations of consumption on weekdays or weekends or at a particular time in the year (see yellow highlights below). Feature synthesis is the opposite of feature extraction. In this case, one or more features are combined into creating new features that are more informative than they are individually. Let's say, in a house price dataset you have two columns: floor_space (sqft) and total_house_price (US$). You could use them individually in your analysis but you could also create a new calculated feature called price_per_sqft (US$/sqft). Feature scaling/transformation refers to a variety of methods applied in data preprocessing to rescale or normalize data into a different range.

Dissipative partial differential equations that exhibit chaotic dynamics tend to evolve to attractors that exist on finite-dimensional manifolds. We present a data-driven reduced order modeling method that capitalizes on this fact by finding the coordinates of this manifold and finding an ordinary differential equation (ODE) describing the dynamics in this coordinate system. The manifold coordinates are discovered using an undercomplete autoencoder -- a neural network (NN) that reduces then expands dimension. Then the ODE, in these coordinates, is approximated by a NN using the neural ODE framework. Both of these methods only require snapshots of data to learn a model, and the data can be widely and/or unevenly spaced. We apply this framework to the Kuramoto-Sivashinsky for different domain sizes that exhibit chaotic dynamics. With this system, we find that dimension reduction improves performance relative to predictions in the ambient space, where artifacts arise. Then, with the low-dimensional model, we vary the training data spacing and find excellent short- and long-time statistical recreation of the true dynamics for widely spaced data (spacing of ~0.7 Lyapunov times). We end by comparing performance with various degrees of dimension reduction, and find a "sweet spot" in terms of performance vs. dimension.

Deep learning approaches have produced substantial breakthroughs in fields such as image classification and natural language processing and are making rapid inroads in the area of protein design. Many generative models of proteins have been developed that encompass all known protein sequences, model specific protein families, or extrapolate the dynamics of individual proteins. Those generative models can learn protein representations that are often more informative of protein structure and function than hand-engineered features. Furthermore, they can be used to quickly propose millions of novel proteins that resemble the native counterparts in terms of expression level, stability, or other attributes. The protein design process can further be guided by discriminative oracles to select candidates with the highest probability of having the desired properties. In this review, we discuss five classes of generative models that have been most successful at modeling proteins and provide a framework for model guided protein design.

Softmax working with cross-entropy is widely used in classification, which evaluates the similarity between two discrete distribution columns (predictions and true labels). Inspired by chi-square test, we designed a new loss function called chi-square loss, which is also works for Softmax. Chi-square loss has a statistical background. We proved that it is unbiased in optimization, and clarified its using conditions (its formula determines that it must work with label smoothing). In addition, we studied the sample distribution of this loss function by visualization and found that the distribution is related to the neural network structure, which is distinct compared to cross-entropy. In the past, the influence of structure was often ignored when visualizing. Chi-square loss can notice changes in neural network structure because it is very strict, and we explained the reason for this strictness. We also studied the influence of label smoothing and discussed the relationship between label smoothing and training accuracy and stability. Since the chi-square loss is very strict, the performance will degrade when dealing samples of very many classes.

Evolutionary clustering algorithms have considered as the most popular and widely used evolutionary algorithms for minimising optimisation and practical problems in nearly all fields. In this thesis, a new evolutionary clustering algorithm star (ECA*) is proposed. Additionally, a number of experiments were conducted to evaluate ECA* against five state-of-the-art approaches. For this, 32 heterogeneous and multi-featured datasets were used to examine their performance using internal and external clustering measures, and to measure the sensitivity of their performance towards dataset features in the form of operational framework. The results indicate that ECA* overcomes its competitive techniques in terms of the ability to find the right clusters. Based on its superior performance, exploiting and adapting ECA* on the ontology learning had a vital possibility. In the process of deriving concept hierarchies from corpora, generating formal context may lead to a time-consuming process. Therefore, formal context size reduction results in removing uninterested and erroneous pairs, taking less time to extract the concept lattice and concept hierarchies accordingly. In this premise, this work aims to propose a framework to reduce the ambiguity of the formal context of the existing framework using an adaptive version of ECA*. In turn, an experiment was conducted by applying 385 sample corpora from Wikipedia on the two frameworks to examine the reduction of formal context size, which leads to yield concept lattice and concept hierarchy. The resulting lattice of formal context was evaluated to the original one using concept lattice-invariants. Accordingly, the homomorphic between the two lattices preserves the quality of resulting concept hierarchies by 89% in contrast to the basic ones, and the reduced concept lattice inherits the structural relation of the original one.