# State Space Models

State space model is an important category of model for sequential data. Through simple assumptions, state space models can achieve quite complicated distributions.

To model a sequence, we can use the joint probability of all the nodes,

$$ p(x_1, x_2, \cdots, x_N), $$

where $x_i$ are the nodes in the sequence.

## Orders

We can introduce different order of dependencies on the past.

The simplest model for the sequence is assuming i.i.d..

To model the dependencies in the sequence, we can assume a node depends on the previous nodes. The first-order model assume that node $x_{i+1}$ only depends on node $x_i$.

To add more complexities, we introduce the second order model.

## Hidden States

For some quantities, such as the GMV of a company, we expect that the dynamics of them are driven by some other factors, which data we may not have.

We introduce a latent space $\{z_i\}$ which is described by first order process. The visible states $\{x_i\}$ are determined using $p(x_i\vert z_i)$ ^{1}.

LM (2022). 'State Space Models', Datumorphism, 02 April. Available at: https://datumorphism.leima.is/wiki/time-series/state-space-models/.