Modeling multivariate time series is a well-established problem with a wide
range of applications from healthcare to financial markets. Traditional State
Space Models (SSMs) are classical approaches for univariate time series
modeling due to their simplicity and expressive power to represent linear
dependencies. They, however, have fundamentally limited expressive power to
capture non-linear dependencies, are slow in practice, and fail to model the
inter-variate information flow. Despite recent attempts to improve the
expressive power of SSMs by using deep structured SSMs, the existing methods
are either limited to univariate time series, fail to model complex patterns
(e.g., seasonal patterns), fail to dynamically model the dependencies of
variate and time dimensions, and/or are input-independent. We present Chimera
that uses two input-dependent 2-D SSM heads with different discretization
processes to learn long-term progression and seasonal patterns. To improve the
efficiency of complex 2D recurrence, we present a fast training using a new
2-dimensional parallel selective scan. We further present and discuss
2-dimensional Mamba and Mamba-2 as the spacial cases of our 2D SSM. Our
experimental evaluation shows the superior performance of Chimera on extensive
and diverse benchmarks, including ECG and speech time series classification,
long-term and short-term time series forecasting, and time series anomaly
detection.

用两个输入依赖的二维状态空间模型头来学习长期进展和季节模式，提高多元时间序列建模的表达能力和性能。

混合模型：有效地对多变量时间序列进行建模的二维状态空间模型

Chimera: Effectively Modeling Multivariate Time Series with  2-Dimensional State Space Models

Traditional linear methods for forecasting multivariate time series are not
able to satisfactorily model the non-linear dependencies that may exist in
non-Gaussian series. We build on the theory of learning vector-valued functions
in the reproducing kernel Hilbert space and develop a method for learning
prediction functions that accommodate such non-linearities. The method not only
learns the predictive function but also the matrix-valued kernel underlying the
function search space directly from the data. Our approach is based on learning
multiple matrix-valued kernels, each of those composed of a set of input
kernels and a set of output kernels learned in the cone of positive
semi-definite matrices. In addition to superior predictive performance in the
presence of strong non-linearities, our method also recovers the hidden dynamic
relationships between the series and thus is a new alternative to existing
graphical Granger techniques.

使用学习向量值函数的理论，建立一种能处理非高斯序列中的非线性依赖关系预测模型，通过学习多个矩阵值核来提高预测性能，并恢复系列之间的动态关系，该算法是现有图形 Granger 技术的新选择

具有非线性动态依赖关系的预测和 Granger 建模

Forecasting and Granger Modelling with Non-linear Dynamical Dependencies

The proposal of Reshef et al. (2011) is an interesting new approach for
discovering non-linear dependencies among pairs of measurements in exploratory
data mining. However, it has a potentially serious drawback. The authors laud
the fact that MIC has no preference for some alternatives over others, but as
the authors know, there is no free lunch in Statistics: tests which strive to
have high power against all alternatives can have low power in many important
situations. To investigate this, we ran simulations to compare the power of MIC
to that of standard Pearson correlation and distance correlation (dcor). We
simulated pairs of variables with different relationships (most of which were
considered by the Reshef et. al.), but with varying levels of noise added. To
determine proper cutoffs for testing the independence hypothesis, we simulated
independent data with the appropriate marginals. As one can see from the
Figure, MIC has lower power than dcor, in every case except the somewhat
pathological high-frequency sine wave. MIC is sometimes less powerful than
Pearson correlation as well, the linear case being particularly worrisome.

研究比较了 MIC 与 Pearson 相关系数和距离相关系数在不同情况下的功率表现，发现 MIC 在大多数情况下的功率都比 dcro 低，甚至有时候比 Pearson 相关系数还低。