The Linear Gaussian State Model: Dynamic Systems

Note: This post is initially composed with deep research and finalized by me.

1. Introduction to State-Space Models

State-space models offer a robust and flexible framework for analyzing and understanding dynamic systems, particularly those evolving over time, such as time series data. This approach provides a powerful lens through which to conceptualize a system’s behavior, distinguishing between its internal, unobservable dynamics and its external, measurable manifestations. The framework is especially adept at managing multivariate data and intricate dynamic relationships, providing a systematic method for modeling processes under uncertainty.

1.1. General Concept of State-Space Modeling

At its core, state-space modeling posits that the true condition of a system at any given time can be encapsulated by a set of unobserved, or “hidden,” state variables. These hidden states are not directly measurable but dictate the system’s evolution and its observable outputs. The fundamental concept involves separating the true, underlying system state—which is often latent or difficult to measure without error—from the noisy, incomplete observations that are actually collected. This separation is a crucial aspect, as it allows for the robust estimation of the true state even when measurements are imperfect or corrupted by noise.

For instance, consider the true position and velocity of an aircraft. While these are the actual states of interest, they cannot be measured perfectly due to sensor limitations and atmospheric disturbances. What is observed are noisy radar readings or GPS signals. State-space models address this challenge by introducing a hidden state variable that represents the aircraft’s true, uncorrupted position and velocity. By modeling how this hidden state evolves and how it relates to the noisy observations, it becomes possible to infer the unobservable truth. This capability to estimate the underlying reality from imperfect data is a significant advantage, addressing a common challenge in real-world data analysis where noise and incompleteness are prevalent.

1.2. The Hidden Markov State Process

A defining characteristic of state-space models is the assumption of a hidden Markov state process for the evolution of the state vector (xₜ). This means that the state of the system at time t depends solely on its state at the previous time step, t−1, along with any exogenous inputs, conditional on all past information. In essence, once the current state is known, all prior states and observations become irrelevant for predicting the future state.

This Markovian assumption, while appearing restrictive, is foundational for the computational efficiency of state estimation algorithms, such as the Kalman filter. Many complex time series problems involve dependencies on a long history of past states and inputs, which can lead to high-dimensional and computationally intractable challenges. By assuming that the future state depends only on the immediate past state, this “memoryless” property (conditional on the previous state) allows for recursive algorithms. These algorithms update the state estimate sequentially, requiring only the previous state estimate and the current observation. Without this pragmatic simplification, exact inference would often be computationally prohibitive for many real-world applications, significantly limiting the practical utility of the state-space framework.

1.3. Observations Independent Given the States

Another core principle of state-space models is that the observations (yₜ) at time t are conditionally independent of all other observations and past states, given the current state xₜ.

This implies that once the true state at time t is known, the observation yₜ provides no additional information about past states or future observations beyond what is already captured by xₜ.

This assumption effectively separates the measurement process from the system’s internal dynamics. The noise inherent in the observation equation is considered to be purely measurement error, uncorrelated with the system’s internal evolution. This separation is crucial for distinguishing between true system changes and inaccuracies arising from the measurement process. The observation at time t is modeled as solely a function of the true state at time t, perturbed by independent measurement noise. It does not carry information about how the state arrived at time t, nor does it directly influence how the state will evolve from time t (that role is reserved for the state equation). This clear decomposition of uncertainty—one part from the system’s evolution (process noise) and another from the measurement process (observation noise)—is vital for accurate state estimation and prediction. It simplifies the likelihood function and facilitates optimal filtering algorithms where the measurement update step precisely accounts for new information from the observation without conflating it with dynamic uncertainties.

2. The Linear Gaussian State Model: Core Formulation

The Linear Gaussian State Model (LGSM) is a specific type of state-space model characterized by linear relationships between states and observations, and by Gaussian (normal) distributions for its noise terms. This combination of linearity and Gaussianity is what makes the model analytically tractable and allows for the development of optimal estimation algorithms like the Kalman filter.

2.1. Fundamental Equations

The LGSM is defined by two fundamental linear equations: the state equation, which describes the evolution of the hidden state, and the observation equation, which relates the hidden state to the observable measurements.

2.1.1. The State Equation (Process Equation)

The state equation describes how the hidden state vector xₜ evolves over time. It is a first-order linear difference equation:

xₜ = Fₜxₜ₋₁ + Bₜuₜ + νₜ

This equation indicates that the state at time t is a linear transformation of the state at time t−1, possibly influenced by exogenous inputs (uₜ), and perturbed by a stochastic term known as process noise (νₜ).

2.1.2. The Observation Equation (Measurement Equation)

The observation equation links the observed data vector yₜ to the hidden state vector xₜ at the same time point:

yₜ = Hₜxₜ + Dₜuₜ + εₜ

This equation shows that the observed data is a linear transformation of the current state, potentially influenced by exogenous inputs (uₜ), and corrupted by observation noise (εₜ).

2.2. Detailed Explanation of Components

Understanding each component of these equations is critical for grasping the LGSM’s functionality.

• State Vector (xₜ): This is an m×1 vector representing the unobserved (hidden) state of the system at time t. It encapsulates the essential information about the system’s internal dynamics. For example, in a tracking application, xₜ might include the position and velocity of an object. In an economic model, it could represent underlying economic trends or latent factors that drive observable indicators. The state vector is the core unobservable variable that evolves according to the system’s internal dynamics.

• Observed Data Vector (yₜ): This is a p×1 vector representing the measurements collected at time t. These are the actual data points that are available, and they are typically noisy or incomplete representations of the true state. The observed data vector serves as the observable output of the system, from which the hidden state is inferred.

• State Transition Matrix (Fₜ): An m×m matrix that defines how the state vector evolves from time t−1 to t. It captures the system’s internal dynamics. If Fₜ is constant over time, it implies that the system’s dynamics are time-invariant. This matrix governs the deterministic evolution of the hidden state over time.

• Measurement Matrix (Hₜ): A p×m matrix that maps the hidden state vector xₜ to the observed data vector yₜ. It describes how the unobserved state variables are linearly transformed into the measurements. This matrix defines the relationship between the unobserved state and the observed data.

• State Noise (νₜ): An m×1 vector representing the process noise or system noise. This stochastic term accounts for unmodeled dynamics, disturbances, or uncertainties in the system’s evolution that are not captured by the deterministic part of the state equation. It captures the inherent stochasticity and inaccuracies in the state evolution model.

• Observation Noise (εₜ): A p×1 vector representing the measurement noise. This term accounts for inaccuracies, errors, or disturbances in the observation process itself, such as sensor noise or environmental interference. It captures the stochasticity and inaccuracies in the measurement process.

• Covariance Matrices (Qₜ, Rₜ): Qₜ is the m×m covariance matrix of the state noise νₜ, and Rₜ is the p×p covariance matrix of the observation noise εₜ. These matrices are crucial as they quantify the uncertainty and correlation within their respective noise terms. They define the statistical properties (variance and covariance) of the noise, which are essential for optimal state estimation algorithms.

The matrices (Fₜ, Hₜ) define the deterministic structure of the system, outlining how the state evolves and how observations are generated. In contrast, the noise terms (νₜ, εₜ) and their covariance matrices (Qₜ, Rₜ) introduce stochasticity and uncertainty into the model. The interplay between these deterministic and stochastic components is critical for accurate modeling and inference. For example, a small Qₜ implies a highly predictable state evolution, suggesting minimal unmodeled disturbances. Conversely, a large Rₜ indicates very noisy measurements, meaning the observations are less reliable. This balance directly impacts how much weight an estimation algorithm, such as the Kalman filter, assigns to the model’s prediction versus the incoming new observations. If process noise (Qₜ) is high, the model’s prediction is less trusted, and the algorithm will rely more heavily on the observation. If measurement noise (Rₜ) is high, the observation is less reliable, and the model will lean more on its own prediction. This dynamic weighting is a fundamental strength of the LGSM for robust estimation.

To summarize the components of the Linear Gaussian State Model, the following table provides a concise overview:

Table 1: Components of the Linear Gaussian State Model Equations

Component Name	Symbol	Typical Dimensions	Description/Role
State Vector	xₜ	m×1	Unobserved (hidden) state of the system
Observed Data Vector	yₜ	p×1	Observed measurements of the system
State Transition Matrix	Fₜ	m×m	Governs the evolution of the hidden state over time
Input Matrix (State)	Bₜ	m×q	Maps exogenous inputs to the state equation
Measurement Matrix	Hₜ	p×m	Defines relationship between state and observations
Input Matrix (Observation)	Dₜ	p×q	Maps exogenous inputs to the observation equation
State Noise	νₜ	m×1	Unmodeled dynamics, disturbances in state evolution
Observation Noise	εₜ	p×1	Inaccuracies, errors in the measurement process
State Noise Covariance	Qₜ	m×m	Quantifies uncertainty in state noise
Observation Noise Cov.	Rₜ	p×p	Quantifies uncertainty in observation noise
Exogenous Input	uₜ	q×1	External factors influencing the system or observations

3. Distributional Assumptions and Properties

The “Gaussian” aspect of the Linear Gaussian State Model refers to critical statistical assumptions about the noise terms and the initial state. These assumptions are not arbitrary; they are the bedrock upon which the analytical tractability and optimality of associated algorithms, particularly the Kalman filter, are built.

3.1. Assumptions for Noise Terms

Both the state noise (νₜ) and observation noise (εₜ) are assumed to follow specific distributional properties:

• Gaussianity: Both noise terms are assumed to be normally (Gaussian) distributed.
  • νₜ ∼ N(0,Qₜ)
  • εₜ ∼ N(0,Rₜ)

• Zero-Mean: Both noise terms are assumed to have a mean of zero. This implies that, in the absence of noise, the model equations accurately capture the central tendency of the system and observation processes.

• White Noise: The noise terms are assumed to be serially uncorrelated over time. This means that νₜ is independent of νₛ for t≠s, and similarly for εₜ. This “memoryless” property for the noise simplifies the statistical analysis.

3.2. Assumptions for the Initial State (x₀)

The initial state x₀ is also assumed to be normally distributed: x₀ ∼ N(x̂₀,P₀)

Here, x̂₀ represents the initial mean estimate of the state, and P₀ is its initial covariance matrix, quantifying the uncertainty in this initial estimate.

3.3. Independence Assumptions

Crucial independence assumptions further simplify the model’s statistical properties:

• Independence between Noise Terms: νₜ and εₛ are assumed to be independent for all t,s. This means that process noise, which affects the system’s internal dynamics, does not directly influence measurement noise, which affects the observation process.

• Independence from Initial State: Both νₜ and εₜ are assumed to be independent of the initial state x₀. This ensures that the initial uncertainty and subsequent noise disturbances are distinct.

These Gaussian and independence assumptions are fundamental because they enable the analytical tractability and optimality of state estimation algorithms. Without them, exact analytical solutions for state estimation, which involve complex conditional probability distributions, would be impossible. Instead, one would need to resort to computationally intensive approximations, such as particle filters for non-Gaussian systems or Extended Kalman Filters for non-linear systems. The mathematical benefit of Gaussianity is that the sum of Gaussian random variables is also Gaussian, and linear transformations of Gaussian variables remain Gaussian. This “closure property” ensures that if the initial state and noise are Gaussian, then all subsequent states and observations will also be Gaussian. The independence assumptions further simplify joint probability distributions into products of marginal distributions, making calculations feasible.

Collectively, these assumptions allow the posterior distribution of the state to be exactly Gaussian at each time step. This means the distribution can be fully characterized by its mean and covariance, leading to the elegant and computationally efficient recursive updates of the Kalman filter. The Kalman filter, under these conditions, provides the Minimum Mean Squared Error (MMSE) estimate, which is the optimal estimate for linear Gaussian systems. Deviations from these assumptions, such as non-Gaussian noise or non-linear dynamics, necessitate more complex, often approximate, filtering techniques, significantly increasing computational burden and theoretical complexity.

The following table summarizes the key assumptions underpinning the Linear Gaussian State Model:

Table 2: Key Assumptions of the Linear Gaussian State Model

Assumption Category	Specific Assumption	Rationale/Implication
Noise Distribution	νₜ ∼ N(0,Qₜ)	Enables analytical solutions like the Kalman filter; assumes unmodeled dynamics are random and centered around zero.
	εₜ ∼ N(0,Rₜ)	Assumes measurement errors are random, centered around zero, and normally distributed.
Initial State	x₀ ∼ N(x̂₀,P₀)	Provides a starting point for the recursive estimation process; initial uncertainty is quantified.
Independence	νₜ ⊥ εₛ for all t,s	Process noise and measurement noise are distinct and do not directly influence each other; simplifies likelihood calculations.
	νₜ ⊥ x₀ for all t	Process disturbances are independent of the initial system state.
	εₜ ⊥ x₀ for all t	Measurement errors are independent of the initial system state.
	νₜ and εₜ are white noise	Noise at one time point is uncorrelated with noise at other time points, simplifying temporal dependencies.

4. Extensions and Enhancements

The basic Linear Gaussian State Model can be extended to handle more complex and realistic scenarios, particularly through the inclusion of exogenous variables and the allowance for time-varying parameters. These enhancements significantly broaden the model’s applicability.

4.1. Incorporating Exogenous Variables (uₜ)

Exogenous variables, also known as control inputs or covariates, are external factors that influence the system but are not themselves part of the state vector. Their incorporation into both the state and observation equations allows the model to account for known external influences.

In the state equation, exogenous inputs are introduced via the Bₜ matrix: xₜ = Fₜxₜ₋₁ + Bₜuₜ + νₜ

Here, Bₜ is an m×q input matrix that maps the q×1 exogenous input vector uₜ to the state space. This allows external control actions (e.g., engine thrust in an aircraft) or known environmental factors (e.g., temperature affecting a chemical process) to directly influence the system’s evolution.

Similarly, in the observation equation, exogenous inputs can be included via the Dₜ matrix: yₜ = Hₜxₜ + Dₜuₜ + εₜ

Here, Dₜ is a p×q feedforward matrix that allows exogenous inputs to directly affect the observations. This is useful when the measurement itself is influenced by external factors independently of their influence on the state. For example, a sensor reading might be directly affected by ambient light conditions (uₜ) in addition to the true state it’s trying to measure.

The ability to incorporate exogenous variables significantly enhances the model’s realism and practical utility, especially in control systems, econometrics, or any domain where external, measurable interventions or influences exist. It transforms the model from a purely descriptive tool into a potentially predictive and prescriptive one. By explicitly accounting for known influences, the model’s predictions become more accurate, as less variance needs to be attributed solely to the noise terms. In control engineering, uₜ represents control signals, allowing the model to be used for designing optimal control strategies. In econometrics, uₜ could be policy variables, enabling “what-if” analyses of different policy interventions. This extension allows for modeling direct causal effects of external factors on both the system’s true state and its observations, making the LGSM a powerful tool for fields requiring active management or prediction under varying external conditions.

4.2. Discussion of Time-Varying Parameters

The notation in the core equations, where matrices like Fₜ,Bₜ,Hₜ,Dₜ and covariance matrices Qₜ,Rₜ are explicitly subscripted with t, signifies that these parameters can change over time. This flexibility allows the model to adapt to evolving system dynamics or measurement characteristics.

Time-varying parameters enable the LGSM to model systems whose underlying dynamics or measurement properties are not static. For instance, a missile’s mass decreases as fuel burns, which changes its dynamic response; an economy’s responsiveness to policy changes might evolve over decades; or a sensor’s precision could degrade over time. If these changes are known or can be estimated, the Kalman filter and other LGSM-based algorithms can still be applied effectively. This adaptability is crucial for modeling long-term or non-stationary processes. It allows the model to capture non-stationarities, regime shifts, or known changes in the system’s physical properties or measurement equipment. This flexibility greatly expands the range of phenomena that can be accurately modeled by the LGSM, moving beyond simple stationary processes to complex, evolving systems.

5. Advantages and Applications

The Linear Gaussian State Model, along with its underlying state-space framework, offers numerous practical benefits that contribute to its widespread adoption across diverse scientific and engineering disciplines.

5.1. Flexibility for Modeling Complex Multivariate Time Series

The state-space formulation naturally handles multiple interacting time series variables by representing them within a single state vector and observation vector. This capability allows for the modeling of complex interdependencies and dynamic relationships among multiple series.

The state-space representation provides a unified and systematic way to manage the inherent complexity of real-world multivariate time series. Instead of developing separate, ad-hoc models for each series, it integrates them into a single, coherent dynamic system. For example, in financial markets, stock prices, interest rates, and inflation are interrelated. In a complex engineering system, multiple sensor readings from different points are interconnected. The LGSM allows for simultaneous estimation and prediction of all these series, capturing lead-lag relationships, common trends, and dynamic dependencies that are often difficult to model with univariate methods. It also offers the potential for dimensionality reduction if the underlying state that drives the observations is lower-dimensional than the observed data itself. This makes the LGSM invaluable for fields like econometrics, signal processing, and control engineering, where systems are inherently multivariate and interconnected.

5.2. Handling Missing Data

A significant practical advantage of the state-space framework is its inherent ability to handle missing observations seamlessly. Many traditional time series analysis methods struggle with or require explicit imputation for missing data, which can introduce bias or complexity into the analysis.

In contrast, the state estimation process within the LGSM, particularly with the Kalman filter, is recursive. If an observation is missing at a particular time step, the filter simply relies on its prediction step (based on the estimated state from the previous time step) to estimate the current state, without performing an observation update. The uncertainty, as quantified by the state covariance matrix, will naturally increase to reflect the lack of new information. This means no explicit imputation is needed; the model naturally propagates uncertainty and provides the best possible estimate given the available data, avoiding the complexities and potential biases of imputation methods. This robustness to incomplete data makes the LGSM highly applicable in scenarios where data collection is intermittent or unreliable, such as in sensor networks, medical monitoring, or financial data analysis where data dropouts are common.

5.3. Representing Various Structures

Many classical time series models, such as Autoregressive Moving Average (ARMA) models, can be reformulated and represented within the state-space framework. This demonstrates the state-space model’s generality and its role as a unifying framework in time series analysis. For instance, a higher-order Autoregressive (AR(p)) model can be transformed into a first-order state-space representation by defining the state vector to include lagged values of the series.

This capability highlights that many seemingly disparate time series models are, in fact, special cases or alternative parameterizations within the broader state-space structure. This provides a common ground for analysis, estimation, and forecasting. The unification means that a single set of powerful algorithms, such as the Kalman filter for estimation and the Kalman smoother for smoothing, can be applied to a wide variety of models. This simplifies the methodological toolkit and allows for easier combination of different model components. The LGSM is thus not just another model, but a meta-framework that encompasses and generalizes many established time series techniques, making it a cornerstone of modern time series analysis.

5.4. Practical Applications

The versatility of the LGSM in modeling dynamic systems under uncertainty has led to its widespread application across numerous scientific and engineering disciplines. While specific examples from the provided material were inaccessible, common application domains illustrate its practical utility.

• Signal Processing & Control Systems: One of the most prominent applications is in tracking objects, such as aircraft, missiles, or vehicles, using noisy sensor data from sources like GPS or radar. The Kalman filter, derived from the LGSM, is fundamental for navigation, guidance, and control systems, providing robust state estimates even with significant measurement noise.

• Econometrics & Finance: In these fields, the LGSM is used for modeling complex economic indicators (e.g., GDP, inflation, interest rates), forecasting stock prices, modeling volatility, and estimating unobserved economic components like permanent income or business cycles. Its ability to handle latent variables and missing data is particularly valuable here.

• Biomedical Engineering: The model is applied to monitor physiological signals (e.g., ECG, EEG) to estimate underlying physiological states, model drug pharmacokinetics, and track patient vital signs. It helps in extracting meaningful information from noisy biological data.

• Environmental Science: LGSMs are used to model climate dynamics, pollutant dispersion, and ecological population dynamics, often leveraging data from noisy sensor networks.

• Robotics: In robotics, the LGSM is crucial for tasks like Simultaneous Localization and Mapping (SLAM), robot navigation, and sensor fusion, where a robot needs to estimate its own state and map its environment simultaneously using imperfect sensor readings.

The following table summarizes common applications of the Linear Gaussian State Model:

Table 3: Common Applications of the Linear Gaussian State Model

Application Domain	Specific Use Case	Benefit of LGSM
Signal Processing & Control Systems	Object Tracking (e.g., aircraft, vehicles)	Robust state estimation in noisy environments, real-time prediction for guidance and control.
Econometrics & Finance	Economic Forecasting, Volatility Modeling	Handles unobserved economic factors, missing data, and complex interdependencies in financial time series.
Biomedical Engineering	Physiological Monitoring, Drug Pharmacokinetics	Infers latent biological states from noisy measurements, tracks dynamic physiological processes.
Environmental Science	Climate Modeling, Pollutant Tracking	Models complex environmental dynamics, handles noisy sensor data from distributed networks.
Robotics	Simultaneous Localization and Mapping (SLAM)	Fuses multiple noisy sensor inputs to estimate robot pose and map environment accurately.

6. Relationship to Other Models

The Linear Gaussian State Model is not an isolated framework but is deeply interconnected with other established time series models, notably ARMA models. This relationship underscores the LGSM’s generality and its role as a unifying paradigm in time series analysis.

6.1. Connection between Stationary ARMA Models and Stationary State-Space Models

A significant theoretical result in time series analysis is that any stationary ARMA (Autoregressive Moving Average) model can be represented in a state-space form. Conversely, under certain conditions, a stationary state-space model can be reduced to an ARMA representation. This equivalence demonstrates the broad applicability and theoretical depth of the state-space framework.

Consider a simple example: an AR(1) process with observational noise.

An AR(1) process is typically defined as: zₜ = ϕzₜ₋₁ + wₜ, where wₜ is white noise.

If we observe yₜ = zₜ + εₜ, where εₜ is observation noise, this system can be perfectly cast into the LGSM form:

State Equation: xₜ = ϕxₜ₋₁ + νₜ Here, the hidden state xₜ is simply zₜ, the true underlying AR(1) process. The state transition matrix Fₜ is the scalar ϕ. The process noise νₜ is wₜ.

Observation Equation: yₜ = 1·xₜ + εₜ The measurement matrix Hₜ is the scalar 1, indicating a direct observation of the state.

For higher-order ARMA models, the state vector would be augmented to include lagged values of the series and/or past noise terms to achieve the first-order Markovian structure required by the state-space representation.

The equivalence between ARMA models and state-space models is more than a theoretical curiosity; it provides a powerful computational advantage. It means that the robust and efficient algorithms developed specifically for state-space models, such as the Kalman filter and Kalman smoother, can be directly applied to estimate and forecast ARMA processes. These algorithms are often more efficient and flexible than traditional ARMA estimation methods, particularly in handling issues like missing data. The Kalman filter, being recursive and linear, avoids the complex iterative optimization often required for maximum likelihood estimation of ARMA models, especially those with moving average components. This unification means that a single, well-understood algorithmic framework can address a wide variety of classical time series problems. This highlights the state-space model not merely as an alternative model, but as a more general framework that encompasses and generalizes many established time series techniques, making it a cornerstone of modern time series analysis.

7. Conclusion

The Linear Gaussian State Model stands as a cornerstone in the field of dynamic system modeling, offering a powerful and flexible framework for understanding and predicting the behavior of systems over time, especially in the presence of uncertainty.

7.1. Summary of Key Concepts

At its core, the LGSM is built upon two fundamental principles: a hidden Markov state process, where the current state encapsulates all necessary information from the past to predict the future, and observations that are conditionally independent given the current state. These principles allow for a clear separation between the unobservable true system dynamics and the noisy, incomplete measurements. The model is mathematically defined by two linear equations—the state equation describing state evolution and the observation equation linking states to measurements—each comprising specific components like state vectors, observation vectors, transition and measurement matrices, and their respective noise terms. The “Gaussian” aspect of the model refers to the critical assumptions that these noise terms and the initial state are normally distributed and mutually independent, which are crucial for the analytical tractability and optimality of associated estimation algorithms like the Kalman filter.

7.2. The Model’s Utility and Versatility

The utility of the LGSM is profound, stemming from its inherent strengths in handling complex multivariate time series, its natural ability to manage missing observations without requiring explicit imputation, and its capacity to represent a wide array of dynamic system structures, including many classical time series models like ARMA processes. Its flexibility allows it to adapt to time-varying system dynamics and measurement characteristics, further extending its applicability to non-stationary real-world phenomena.

The LGSM serves as a fundamental tool across numerous quantitative disciplines, from engineering and signal processing to econometrics, finance, and biomedical research. It enables robust state estimation, accurate prediction, and effective control in environments characterized by inherent uncertainty and noise. Beyond its direct applications, the Linear Gaussian State Model also serves as a foundational building block for more complex and realistic models. Understanding the LGSM is a prerequisite for tackling advanced topics such as non-linear state-space models (e.g., Extended Kalman Filter, Unscented Kalman Filter) and non-Gaussian state-space models (e.g., Particle Filters). The algorithms derived for the LGSM provide the conceptual and algorithmic intuition necessary to develop approximate solutions for these more challenging scenarios. Therefore, mastering the LGSM is not an end in itself but a crucial first step, providing the essential framework for modeling the full spectrum of dynamic systems under uncertainty.

References

• My demo of Gaussian Processes