Analysis description
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Linear Dynamical Systems (LDS) is used to model statistical properties of sequential or time series data by correlating sequences to a fixed size latent variable vector or a finite-dimensional latent state, whose evolution over the sequential course makes up dynamics of data. In LDS, the state is assumed to be in real domain and the noise terms are assumed to follow the Gaussian distribution. The statistical properties of the model are defined by real-valued vectors which denote the latent variable and observation at time step t, respectively; a transition matrix is a coefficient matrix that controls the evolution of latent states between two successive time steps; an observability matrix which specifies how observations are generated from the present latent state. The initial density is also given as Gaussian distribution. In LDS, the noise terms are assumed to follow zero-mean Gaussian distributions.

How to use this analysis

Inputs:

Sequential or time series data set in a .csv file called “input.csv” under a column called ‘Observations’.

\config: (yaml) a yaml file containing the following parameters:

1) num_measured: size of measurement vector for example if there is only one sensor that measures position x then num_measured = 1.

2) measured_var: this is the mean state variable vector. For example, for a sensor that measures the x position of a car, we can infer the velocity. Therefore the mean state variable vector is both position and velocity for a 2D case, so measured_var: (10,4.5)

3) covar: covariance matrix of all state variables.

4) process_model: transition matrix for process

5) white_noise_var: variance of noise in process

6) dt: time step (in seconds)

7) sensor_covar: variance in sensor measurement

8) measurement_function: converts state to measurement

9) B: control input matrix (optional)

10) U: control input (optional)

See the sample params.yaml for an example file.

Outputs:

LDS_Components: The filtered and smoothed observations and their corresponding covariances in a file called ‘output.csv’ in columns ‘x’ for processed data and ‘c’ processed covariances, respectively.

output.png: A diagram showing a comparison of the smoothed data against the noisy measurements in a file called ‘output.png’.

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