somata 0.4.1

State-space Oscillator Modeling And Time-series Analysis

somata

Github: https://github.com/mh105/somata

State-space Oscillator Modeling And Time-series Analysis (SOMATA) is a Python library for state-space neural signal processing algorithms developed in the Purdon Lab. Basic state-space models are introduced as class objects for flexible manipulations. Classical exact and approximate inference algorithms are implemented and interfaced as class methods. Advanced neural oscillator modeling techniques are brought together to work synergistically.

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Table of Contents

• Requirements
• Install
• Basic state-space models
  • StateSpaceModel
  • OscillatorModel
  • AutoRegModel
  • GeneralSSModel
• Advanced neural oscillator methods
• Authors
• Citation
• License

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Requirements

somata is built on numpy arrays for computations. joblib is used for multithreading. Additional dependencies include scipy, matplotlib, and spectrum. The source localization module also requires pytorch and MNE-python. Full requirements for each release version will be updated under install_requires in the setup.cfg file. If the environment.yml file is used to create a new conda environment, all and only the required packages will be installed.

Install

$ pip install somata

conda-forge channel

While pip install usually works, an alternative way to install somata is through the conda-forge channel, which utilizes continuous integration (CI) across OS platforms. This means conda-forge packages are more compatible with each other compared to Python Package Index (PyPI) packages installed via pip by default. When somata is installed into an existing conda environment, unmet dependencies are automatically searched, downloaded, and installed from the same repository of packages containing somata. If pip install somata fails to resolve some dependencies, the conda-forge somata feedstock can be used to install:

$ conda install somata -c conda-forge

torch requirement

If the pytorch dependency is not resolved correctly for your OS, first install pytorch manually in a conda environment that you want to install somata in, and then rerun either of the above two lines to install somata.

(For development only)

• Fork this repo to personal git

  How to: GitHub fork

• Clone forked copy to local computer

  How to: GitHub clone

• Install conda

  Recommended conda distribution: Miniforge3

  Apple silicon Mac: choose Miniforge3 native to the ARM64 architecture instead of Intel x86.

• Create a new conda environment

  $ cd <repo root directory with environment.yml>
$ mamba env create -f environment.yml
$ conda activate somata

  You may also install somata in an existing environment by skipping this step.

• Install somata as a package in development mode

  $ cd <repo root directory with setup.py>
$ pip install -e . --config-settings editable_mode=compat

  What is: Editable Installs

• Configure IDEs to use the conda environment

  How to: Configure an existing conda environment

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Basic state-space models

somata, much like a neuron body supported by dendrites, is built on a set of basic state-space models introduced as class objects.

The motivations are to:

• develop a standardized format to store model parameters of state-space equations
• override Python dunder methods so __repr__ and __str__ return something useful
• define arithmetic-like operations such as A + B and A * B
• emulate numpy.array() operations including .append()
• implement inference algorithms like Kalman filtering and parameter update (m-step) equations as callable class methods

At present, and in the near future, somata will be focused on time-invariant Gaussian linear dynamical systems. This limit on models we consider simplifies basic models to avoid nested classes such as transition_model and observation_model, at the cost of restricting somata to classical algorithms with only some extensions to Bayesian inference and learning. This is a deliberate choice to allow easier, faster, and cleaner applications of somata in neural data analysis, instead of to provide a full-fledged statistical inference package.

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class StateSpaceModel

somata.StateSpaceModel(components=None, F=None, Q=None, mu0=None, Q0=None, G=None, R=None, y=None, Fs=None)

StateSpaceModel is the parent class of basic state-space models. The corresponding Gaussian linear dynamical system is:

$$ \mathbf{x}_ {t} = \mathbf{F}\mathbf{x}_{t-1} + \boldsymbol{\eta}_t, \boldsymbol{\eta}_t \sim \mathcal{N}(\mathbf{0}, \mathbf{Q}) $$

$$ \mathbf{y}_ {t} = \mathbf{G}\mathbf{x}_{t} + \boldsymbol{\epsilon}_t, \boldsymbol{\epsilon}_t \sim \mathcal{N}(\mathbf{0}, \mathbf{R}) $$

$$ \mathbf{x}_0 \sim \mathcal{N}(\mathbf{\mu}_0, \mathbf{Q}_0) $$

Most of the constructor input arguments correspond to these model parameters, which are stored as instance attributes. There are two additional arguments: Fs and components.

Fs is the sampling frequency of observed data y.

components is a list of independent components underlying the hidden states $\mathbf{x}$. The independent components are assumed to appear in block-diagonal form in the state equation. For example, $\mathbf{x}_t$ might have two independent autoregressive models (AR) of order 1, and the observation matrix is simply $[1, 1]$ that sums these two components. In this case, components would be a list of two AR1 models. Note that each element of the components list should be an instance of one of basic model class objects. To break the recursion, often the components attribute of a component is set to None, i.e., components[0].components = None.

1. StateSpaceModel.__repr__()

The double-under method __repr__() is overwritten to provide some unique identification info:

>>> s1 = StateSpaceModel()
>>> s1
Ssm(0)<f4c0>

where the number inside parenthesis indicates the number of components (the ncomp attribute) in the model, and the four-digits in <> are the last four digits of the memory address of the object instance.

2. StateSpaceModel.__str__()

The double-under method __str__() is overwritten so print() returns useful info:

>>> print(s1)
<Ssm object at 0x102a8f4c0>
 nstate   = 0     ncomp    = 0
 nchannel = 0     ntime    = 0
 nmodel   = 1
 components = None
 F  .shape = None       Q  .shape = None
 mu0.shape = None       Q0 .shape = None
 G  .shape = None       R  .shape = None
 y  .shape = None       Fs = None

3. Model stacking in StateSpaceModel

In many applications, there are several possible parameter values for a given state-space model structure. Instead of duplicating the same values in multiple instances, somata uses stacking to store multiple model values in the same object instance. Stackable model parameters are F, Q, mu0, Q0, G, R. For example:

>>> s1 = StateSpaceModel(F=1, Q=2)
>>> s2 = StateSpaceModel(F=2, Q=2)
>>> print(s1)
<Ssm object at 0x11fd7bfa0>
 nstate   = 1     ncomp    = 0
 nchannel = 0     ntime    = 0
 nmodel   = 1
 components = None
 F  .shape = (1, 1)     Q  .shape = (1, 1)
 mu0.shape = None       Q0 .shape = None
 G  .shape = None       R  .shape = None
 y  .shape = None       Fs = None

>>> print(s2)
<Ssm object at 0x102acc130>
 nstate   = 1     ncomp    = 0
 nchannel = 0     ntime    = 0
 nmodel   = 1
 components = None
 F  .shape = (1, 1)     Q  .shape = (1, 1)
 mu0.shape = None       Q0 .shape = None
 G  .shape = None       R  .shape = None
 y  .shape = None       Fs = None

>>> s3 = s1+s2
>>> print(s3)
<Ssm object at 0x102acc280>
 nstate   = 1     ncomp    = 0
 nchannel = 0     ntime    = 0
 nmodel   = 2
 components = None
 F  .shape = (1, 1, 2)  Q  .shape = (1, 1)
 mu0.shape = None       Q0 .shape = None
 G  .shape = None       R  .shape = None
 y  .shape = None       Fs = None

Invoking the arithmetic operator + stacks the two instances s1 and s2 into a new instance, where the third dimension of the F attribute is now 2, with the two values from s1 and s2. The nmodel attribute is also updated to 2.

>>> s3.F
array([[[1., 2.]]])

Notice how the third dimension of the Q attribute is still None. This is because the + operator has a built-in duplication check such that the identical model parameters will not be stacked. This behavior of __add__ and __radd__ generalizes to all model parameters, and it is convenient when bootstrapping or testing different parameter values during neural data analysis. Manual stacking of a particular model parameter is also possible with .stack_attr().

4. Model expanding in StateSpaceModel

Similar to stacking, there is a related concept called expanding. Expanding a model is useful when we want to permutate multiple model parameters each with several possible values. For example:

>>> s1 = StateSpaceModel(F=1, Q=3, R=5)
>>> s2 = StateSpaceModel(F=2, Q=4, R=5)
>>> print(s1+s2)
<Ssm object at 0x1059626b0>
 nstate   = 1     ncomp    = 0
 nchannel = 1     ntime    = 0
 nmodel   = 2
 components = None
 F  .shape = (1, 1, 2)  Q  .shape = (1, 1, 2)
 mu0.shape = None       Q0 .shape = None
 G  .shape = None       R  .shape = (1, 1)
 y  .shape = None       Fs = None

>>> s3 = s1*s2
>>> print(s3)
<Ssm object at 0x1059626b0>
 nstate   = 1     ncomp    = 0
 nchannel = 1     ntime    = 0
 nmodel   = 4
 components = None
 F  .shape = (1, 1, 4)  Q  .shape = (1, 1, 4)
 mu0.shape = None       Q0 .shape = None
 G  .shape = None       R  .shape = (1, 1)
 y  .shape = None       Fs = None

>>> s3.F
array([[[1., 1., 2., 2.]]])
>>> s3.Q
array([[[3., 4., 3., 4.]]])

Multiplying two StateSpaceModel instances with more than one differing model parameters results in expanding these parameters into all possible combinations while keeping other identical attributes intact.

5. Arrays of StateSpaceModel

Building on stacking and expanding, we can easily form an array of StateSpaceModel instances using .stack_to_array():

>>> s_array = s3.stack_to_array()
>>> s_array
array([Ssm(0)<4460>, Ssm(0)<4430>, Ssm(0)<4520>, Ssm(0)<4580>],
      dtype=object)

Note that a StateSpaceModel array is duck-typing with a Python list, which means one can also form a valid array with [s1, s2].

6. StateSpaceModel.__len__()

Invoking len() returns the number of stacked models:

>>> len(s2)
1
>>> len(s3)
4

7. StateSpaceModel.append()

Another useful class method on StateSpaceModel is .append(). As one would expect, appending a model to another results in combining them in block-diagonal form in the state equation. Compatibility checks happen in the background to make sure no conflict exists on the respective observation equations and observed data, if any.

>>> s1 = StateSpaceModel(F=1, Q=3, R=5)
>>> s2 = StateSpaceModel(F=2, Q=4, R=5)
>>> s1.append(s2)
>>> print(s1)
<Ssm object at 0x1057cb4c0>
 nstate   = 2     ncomp    = 0
 nchannel = 1     ntime    = 0
 nmodel   = 1
 components = None
 F  .shape = (2, 2)     Q  .shape = (2, 2)
 mu0.shape = None       Q0 .shape = None
 G  .shape = None       R  .shape = (1, 1)
 y  .shape = None       Fs = None

>>> s1.F
array([[1., 0.],
       [0., 2.]])
>>> s1.Q
array([[3., 0.],
       [0., 4.]])

Notice that the nstate attribute is now updated to 2, which is different from the + operator that updates the nmodel attribute to 2.

8. Inference and learning with StateSpaceModel

Two different implementations of Kalman filtering and fixed-interval smoothing are callable class methods:

.kalman_filt_smooth(y=None, R_weights=None, return_dict=False, EM=False, skip_interp=True, seterr=None)

.dejong_filt_smooth(y=None, R_weights=None, return_dict=False, EM=False, skip_interp=True, seterr=None)

With an array of StateSpaceModel, one can easily run Kalman filtering and smoothing on all array elements with multithreading using the static method .par_kalman():

.par_kalman(ssm_array, y=None, method='kalman', R_weights=None, skip_interp=True, return_dict=False)

M-step updates are organized using m_estimate() that will recurse into each element of the components list and use the appropriate m-step update methods associated with different types of state-space models.

Below we explain three kinds of basic state-space models currently supported in somata.

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class OscillatorModel(StateSpaceModel)

somata.OscillatorModel(a=None, freq=None, w=None, sigma2=None, add_dc=False,
                       components='Osc', F=None, Q=None, mu0=None, Q0=None, G=None, R=None, y=None, Fs=None)

OscillatorModel is a child class of StateSpaceModel, which means it inherits all the class methods explained above. It has a particular form of the state-space model:

$$ \begin{bmatrix}x_{t, 1}\newline x_{t, 2}\end{bmatrix} = \mathbf{F}\begin{bmatrix}x_{t-1, 1}\newline x_{t-1, 2}\end{bmatrix} + \mathbf{\eta}_t, \mathbf{\eta}_t \sim \mathcal{N}(\mathbf{0}, \mathbf{Q}) $$

$$ \mathbf{y}_ {t} = \mathbf{G}\begin{bmatrix}x_{t, 1}\newline x_{t, 2}\end{bmatrix} + \mathbf{\epsilon}_t, \mathbf{\epsilon}_t \sim \mathcal{N}(\mathbf{0}, \mathbf{R}) $$

$$ \begin{bmatrix}x_{0, 1}\newline x_{0, 2}\end{bmatrix} \sim \mathcal{N}(\mathbf{\mu}_0, \mathbf{Q}_0) $$

$$ \mathbf{F} = a\begin{bmatrix}\cos\omega & -\sin\omega\newline \sin\omega & \cos\omega\end{bmatrix}, \mathbf{Q} = \begin{bmatrix}\sigma^2 & 0\newline 0 & \sigma^2\end{bmatrix}, \mathbf{G} = \begin{bmatrix}1 & 0 \end{bmatrix} $$

To create a simple oscillator model with rotation frequency $15$ Hz (under $100$ Hz sampling frequency) and damping factor $0.9$:

>>> o1 = OscillatorModel(a=0.9, freq=15, Fs=100)
>>> o1
Osc(1)<81f0>
>>> print(o1)
<Osc object at 0x1058081f0>
 nstate   = 2     ncomp    = 1
 nchannel = 0     ntime    = 0
 nmodel   = 1
 components = [Osc(0)<4b50>]
 F  .shape = (2, 2)     Q  .shape = (2, 2)
 mu0.shape = (2, 1)     Q0 .shape = (2, 2)
 G  .shape = (1, 2)     R  .shape = None
 y  .shape = None       Fs = 100.0 Hz
 damping a = [0.9]
 freq Hz   = [15.]
 sigma2    = [3.]
 obs noise R = None
 dc index  = None

Notice the components attribute auto-populates with a spaceholder OscillatorModel instance, which is different from the o1 instance as can be recognized by different memory addresses. State noise variance $\sigma^2$ defaults to $3$ when not specified and can be changed with the sigma2 argument to the constructor method.

class AutoRegModel(StateSpaceModel)

somata.AutoRegModel(coeff=None, sigma2=None,
                    components='Arn', F=None, Q=None, mu0=None, Q0=None, G=None, R=None, y=None, Fs=None)

AutoRegModel is a child class of StateSpaceModel, which means it inherits all the class methods explained above. It has a particular form of the state-space model. For example, an auto-regressive model of order 3 can be expressed as:

$$ \begin{bmatrix}x_{t}\newline x_{t-1}\newline x_{t-2}\end{bmatrix} = \mathbf{F}\begin{bmatrix}x_{t-1}\newline x_{t-2}\newline x_{t-3}\end{bmatrix} + \mathbf{\eta}_t, \mathbf{\eta}_t \sim \mathcal{N}(\mathbf{0}, \mathbf{Q}) $$

$$ \mathbf{y}_ {t} = \mathbf{G}\begin{bmatrix}x_{t}\newline x_{t-1}\newline x_{t-2}\end{bmatrix} + \mathbf{\epsilon}_t, \mathbf{\epsilon}_t \sim \mathcal{N}(\mathbf{0}, \mathbf{R}) $$

$$ \begin{bmatrix}x_{0}\newline x_{-1}\newline x_{-2}\end{bmatrix} \sim \mathcal{N}(\mathbf{\mu}_0, \mathbf{Q}_0) $$

$$ \mathbf{F} = \begin{bmatrix} a_1 & a_2 & a_3 \newline 1 & 0 & 0 \newline 0 & 1 & 0 \end{bmatrix}, \mathbf{Q} = \begin{bmatrix}\sigma^2 & 0 & 0\newline 0 & 0 & 0\newline 0 & 0 & 0\end{bmatrix}, \mathbf{G} = \begin{bmatrix}1 & 0 & 0\end{bmatrix} $$

To create an AR3 model with parameters $a_1=0.5, a_2=0.3, a_3=0.1$ and $\sigma^2=1$:

>>> a1 = AutoRegModel(coeff=[0.5,0.3,0.1], sigma2=1)
>>> a1
Arn=3<24d0>
>>> print(a1)
<Arn object at 0x1035524d0>
 nstate   = 3     ncomp    = 1
 nchannel = 0     ntime    = 0
 nmodel   = 1
 components = [Arn=3<2680>]
 F  .shape = (3, 3)     Q  .shape = (3, 3)
 mu0.shape = (3, 1)     Q0 .shape = (3, 3)
 G  .shape = (1, 3)     R  .shape = None
 y  .shape = None       Fs = None
 AR order  = [3]
 AR coeff  = ([0.5 0.3 0.1])
 sigma2    = [1.]

Note that __repr__() is slightly different for AutoRegModel, since the key information is not how many components but rather the AR order. We display the order of the auto-regressive model with an = sign as shown above instead of showing the number of components in () as for OscillatorModel and StateSpaceModel.

class GeneralSSModel(StateSpaceModel)

somata.GeneralSSModel(components='Gen', F=None, Q=None, mu0=None, Q0=None, G=None, R=None, y=None, Fs=None)

GeneralSSModel is a child class of StateSpaceModel, which means it inherits all the class methods explained above. The same general Gaussian linear dynamic system as before is followed:

$$ \mathbf{x}_ t = \mathbf{F}\mathbf{x}_{t-1} + \boldsymbol{\eta}_t, \boldsymbol{\eta}_t \sim \mathcal{N}(\mathbf{0}, \mathbf{Q}) $$

$$ \mathbf{y}_ {t} = \mathbf{G}\mathbf{x}_{t} + \boldsymbol{\epsilon}_t, \boldsymbol{\epsilon}_t \sim \mathcal{N}(\mathbf{0}, \mathbf{R}) $$

$$ \mathbf{x}_0 \sim \mathcal{N}(\mathbf{\mu}_0, \mathbf{Q}_0) $$

GeneralSSModel is added to somata so that one can perform the most general Gaussian updates for a state-space model without special structures as specified in OscillatorModel and AutoRegModel. In other words, with non-sparse structures in the model parameters F, Q, Q0, G, R. To create a simple general state-space model:

>>> g1 = GeneralSSModel(F=[[1,2],[3,4]])
>>> g1
Gen(1)<2440>
>>> print(g1)
<Gen object at 0x103552440>
 nstate   = 2     ncomp    = 1
 nchannel = 0     ntime    = 0
 nmodel   = 1
 components = [Gen(0)<2710>]
 F  .shape = (2, 2)     Q  .shape = None
 mu0.shape = None       Q0 .shape = None
 G  .shape = None       R  .shape = None
 y  .shape = None       Fs = None

For more in-depth working examples with the basic models in somata

Look at the demo script basic_models_demo_08302023.py and execute the code line by line to get familiar with class objects and methods of somata basic models.

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Advanced neural oscillator methods

1. Oscillator Model Learning
2. Phase Amplitude Coupling Estimation
3. Iterative Oscillator Algorithm
4. Switching State-Space Inference
5. Multi-channel Oscillator Component Analysis
6. State-Space Event Related Potential
7. Dynamic Source Localization

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1. Oscillator Model Learning

For fitting data with oscillator models, it boils down to three steps:

• Initialize an oscillator model object
• Perform state estimation, i.e., E-step
• Update model parameters, i.e., M-step

Given some time series data, we can fit an oscillator to the data using the expectation-maximization (EM) algorithm.

from somata.basic_models import OscillatorModel as Osc
o1 = Osc(freq=1, Fs=100, y=data)  # create an oscillator object instance
_ = [o1.m_estimate(**o1.kalman_filt_smooth(EM=True))for x in range(50)]  # run 50 steps of EM

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2. Phase Amplitude Coupling Estimation

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3. Iterative Oscillator Algorithm

For a well-commented example script, see IterOsc_example.py.

N.B.: We recommend downsampling to 120 Hz or less, depending on the oscillations present in your data. Highly oversampled data will make it more difficult to identify oscillatory components, increase the computational time, and could also introduce high frequency noise.

One major goal of this method was to produce an algorithm that requires minimal user intervention, if any. This algorithm is designed to fit well automatically in most situations, but there will still be some data sets where it does not fit well without intervention. We recommend starting with the algorithm as is, but in the case of poor fitting, we suggest the following modifications:

1. If the model does not choose the correct number of oscillations, we recommend looking at all fitted models and selecting the best fitting model based on other selection criteria or using your best judgement. You can also choose a subset of well-fitted oscillations and run kalman_filt_smooth() to estimate oscillations using those fitted parameters.

2. This algorithm assumes stationary parameters, and therefore a stationary signal. Although the Kalman smoothing allows the model to work with some time-varying signal, the success of the method depends on the strength and duration of the signal components. The weaker and more brief the time-varying component is, the more poorly the model will capture it, if at all. We recommend decreasing the length of your window until you have a more stationary signal.

When using this module, please cite the following paper:

Beck, A. M., He, M., Gutierrez, R. G., & Purdon, P. L. (2022). An iterative search algorithm to identify oscillatory dynamics in neurophysiological time series. bioRxiv, 2022-10.

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4. Switching State-Space Inference

When using this module, please cite the following paper:

He, M., Das, P., Hotan, G., & Purdon, P. L. (2023). Switching state-space modeling of neural signal dynamics. PLOS Computational Biology, 19(8), e1011395.

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5. Multi-channel Oscillator Component Analysis

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6. State-Space Event Related Potential

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7. Dynamic Source Localization

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Authors

Mingjian He, Proloy Das, Amanda Beck, Patrick Purdon

Citation

Use different citation styles at: https://doi.org/10.5281/zenodo.7242130

License

SOMATA is licensed under the BSD 3-Clause Clear license.
Copyright © 2023. All rights reserved.