You are provided with the abstract of a scientific paper and the abstracts of the papers it references. Your task is to write a Related Work section for the main paper. This section should concisely summarize and integrate the key points from the cited abstracts, reflecting how they relate to the main paper's contributions and context. In your summary, use numerical citation markers like [1], [2], etc., to refer to the corresponding papers.

Main Abstract:
The staggering amount of streaming time series coming from the real world calls for more efficient and effective online modeling solution. For time series modeling, most existing works make some unrealistic assumptions such as the input data is of fixed length or well aligned, which requires extra effort on segmentation or normalization of the raw streaming data. Although some literature claim their approaches to be invariant to data length and misalignment, they are too time-consuming to model a streaming time series in an online manner. We propose a novel and more practical online modeling and classification scheme, DDE-MGM, which does not make any assumptions on the time series while maintaining high efficiency and state-of-the-art performance. The derivative delay embedding (DDE) is developed to incrementally transform time series to the embedding space, where the intrinsic characteristics of data is preserved as recursive patterns regardless of the stream length and misalignment. Then, a non-parametric Markov geographic model (MGM) is proposed to both model and classify the pattern in an online manner. Experimental results demonstrate the effectiveness and superior classification accuracy of the proposed DDE-MGM in an online setting as compared to the state-of-the-art.

Cited Abstract(s):
[1]: 

[2]: In this paper, we present a new framework for large scale online kernel learning, making kernel methods efficient and scalable for large-scale online learning applications. Unlike the regular budget online kernel learning scheme that usually uses some budget maintenance strategies to bound the number of support vectors, our framework explores a completely different approach of kernel functional approximation techniques to make the subsequent online learning task efficient and scalable. Specifically, we present two different online kernel machine learning algorithms: (i) Fourier Online Gradient Descent (FOGD) algorithm that applies the random Fourier features for approximating kernel functions; and (ii) Nystrom Online Gradient Descent (NOGD) algorithm that applies the Nystrom method to approximate large kernel matrices. We explore these two approaches to tackle three online learning tasks: binary classification, multi-class classification, and regression. The encouraging results of our experiments on large-scale datasets validate the effectiveness and efficiency of the proposed algorithms, making them potentially more practical than the family of existing budget online kernel learning approaches.

[3]: We present a discriminative online algorithm with a bounded memory growth, which is based on the kernel-based Perceptron. Generally, the required memory of the kernel-based Perceptron for storing the online hypothesis is not bounded. Previous work has been focused on discarding part of the instances in order to keep the memory bounded. In the proposed algorithm the instances are not discarded, but projected onto the space spanned by the previous online hypothesis. We derive a relative mistake bound and compare our algorithm both analytically and empirically to the state-of-the-art Forgetron algorithm (, 2007). The first variant of our algorithm, called Projectron, outperforms the Forgetron. The second variant, called Projectron++, outperforms even the Perceptron.
Many online learning methods were also proposed in resent years, e.g., [3] proposed the kernel based perceptron with budget, [1] improved the online passive-aggressive algorithm, and [2] extended online gradient descent. However, they require the data to be of the same length or well aligned. In addition, they are more suitable to operate in the feature space rather than on the raw time series. Therefore, we consider these methods as pseudo-online because they need to preprocess (i.e., truncating or aligning) the raw time series.