State Space MLM Flashcards
State Space Models (SSM)
State Space Models (SSMs), also known as dynamic linear models (DLM) or system dynamics models, are a class of statistical models used to analyze time series data. They represent the relationship between observed data and latent (unobserved) states of a system, considering the inherent dependencies in the data.
- Introduction
State Space Models are a class of statistical models that use state variables to describe the interactions between observed data and hidden states. They provide a powerful and flexible approach to analyzing a wide range of time series and other data.
- State Variables
These are the underlying, often unobservable, variables that capture the true state of the system at any given time. State variables evolve over time according to certain dynamics, which could be deterministic or stochastic.
- Observation Model
The observation model describes how the observed data are generated from the state variables. The observed data might be a noisy version of the state, or it might be some function of the state.
- Transition Model
This describes how the state evolves from one time point to the next. It often includes some form of randomness, allowing the model to capture a variety of complex behaviors. The transition model is often assumed to be linear, but this is not necessary.
- Kalman Filter
The Kalman Filter is a recursive algorithm used in state space models for estimating the underlying state variable given the observed data. It has two steps, the prediction step and the update step. The prediction step estimates the current state variables based on the previous state. The update step then corrects this prediction based on the current observed data.
- Parameter Estimation
The parameters of the state space model, which might include transition and observation matrices in the linear case, can be estimated using methods such as Maximum Likelihood Estimation (MLE) or Bayesian inference.
- Applications
State Space Models are used in many areas including economics, finance, engineering, physics, and biology. For instance, in economics, these models can be used to analyze and predict economic indicators like GDP, unemployment rates, etc.
- Strengths and Limitations
State Space Models are highly flexible and can capture a wide range of behaviors. However, they can also be complex and computationally intensive, particularly for large-scale problems and when the state or observation models are non-linear.
- Extensions
There are many extensions to the basic state space model, including non-linear and non-Gaussian models, models with multiple observation series, and models with switching or varying coefficients.
State Equation
This models the evolution of the system’s state over time, typically in the form of a linear function with an added Gaussian noise term. The state at a certain time point is often dependent on the state at the previous time point.
Observation (or Measurement) Equation
This models the process generating the observed data. The observed value at a certain time point is usually a linear function of the state at the same time point, again with some added Gaussian noise.
Latent State
The ‘state’ in a state-space model refers to the set of variables that capture the relevant information from the past for predicting the future. These states may not be directly observable, but they can be inferred from the observed data through techniques such as the Kalman filter.
Flexibility
State space models can handle a wide variety of time series patterns and structures, and can easily incorporate external variables.
Applications
State Space Models are used in a wide variety of fields, including economics, engineering, genetics, physics, and signal processing. For instance, they are commonly used in tracking applications (e.g., tracking a plane or missile using radar) and in econometrics for modeling economic indicators.
