About The Project

The tensor-ssm git repo includes the basic and preconditioned versions of the tensor-train-based (TT) estimation approach in state-space models (SSM).

The tensor-train-based approach solves the filtering, smoothing and parameter estimation problem in state-space models. The approach relies on continuous tensor-train approximation and aims at the exact Bayesian solutions. Consequently, it does not suffer from particle degeneracy which is the common problem in the popular particle-based methods. The sampling methods and the preconditioning technique accompanying TT are also presented.

In our approach, we use the continuous TT package deep-tensor developed by Cui and Dolgov (https://github.com/DeepTransport/deep-tensor) with the latest version published on (https://github.com/DeepTransport/deep-tensor).

The project is built on Matlab 2021a, and proves to work on Matlab 2023a.

Usage

The folders deep-tensor.dev and models are the toolboxes for continuous TT package and model configs. To implement the project, first run the file deep-tensor.dev/load_dir.m to add the paths.

Then, the folders with prefix ''eg'' contain the files for four demonstrating examples in the paper "Tensor-train methods for sequential state and parameter learning in state-space models": Kalman filter, stochastic volatility model, SIR model and predator-prey model. Simply run the script named ''main_script'' in each folder to check the outputs.

The output stats contains:

• stats.samples: the simulated samples obtained from a series of tensor trains.
• stats.ess: the effective sample size.
• stats.time_sample: the computation time to generate samples.

Config

The user can switch the model and change the parameters using the interface in the main script:

name = 'kalman';   % model names: 'kalman','pp', 'sv', 'sir'
T = 50;   % the steps for sequential inference
N = 1e4; % the sample size to estimate precondition in filter

The config for tensor-train-specific commands can also be modified in the main script:

dom = BoundedDomain([-1, 1]);
dom2 = AlgebraicMapping(1);

poly = ApproxBases(Lagrange1(32), dom, d + 2*m); 
poly2 = ApproxBases(Lagrangep(4,8), dom2, d + 2*m); 

opt = TTOption('tt_method', 'random', ...
    'als_tol', 1E-10, 'local_tol', 1E-4, 'max_rank', rank, 'max_als', 6, 'init_rank', 5, 'kick_rank', 5);
lowopt = TTOption('tt_method', 'random', ...
    'als_tol', 1E-10, 'local_tol', 1E-4, 'max_rank', rank, 'max_als', 2, 'init_rank', 5, 'kick_rank', 5);

We set the default domain for the codes to be [-1,1], whereas the user can switch between the BoundedDomain or the AlgebraicMapping as above. The freedom of the polynomials can be modified in ''ApproxBases'' command. The rank of the tensor train, which is a crucial factor for accuracy and efficiency, can be modified in ''opt'' command.

The config for the built-in models can be changed in the script setup in each model folder in [models]. For example, the parameters for the Kalman filter example is stored in models/kalman/setup

model.theta = norminv(([.8; .5]-0.4)/0.6);
model.pre.C = rand(model.n, model.m);

The default parameters are used to generate the results in the paper.

Comments

• Currently, the predator-prey model simply applies Gaussian perturbation. However, this can lead to negative state values. A predator-prey model on the log scale is under development, which solves the above problem.