The Kalman Filter in Sample Survey Methodology: Applications, Theory, and Challenges
I. Introduction: Bridging Kalman Filters and Sample Survey Methodology
The confluence of advanced statistical estimation techniques and the evolving demands of sample survey methodology has led to the adoption of sophisticated tools to enhance the quality, timeliness, and scope of survey-derived information. Among these, the Kalman filter, a powerful algorithm rooted in control theory and time series analysis, has found increasingly diverse applications in the realm of sample surveys. Its ability to recursively estimate the state of dynamic systems from noisy data aligns well with many challenges inherent in survey practice, such as tracking evolving population parameters, handling data imperfections, and improving estimation for domains with limited information.
A. The Kalman Filter: A Primer on Recursive State Estimation and State-Space Models
The Kalman filter is an optimal recursive algorithm designed to estimate the state of a dynamic system from a series of incomplete and noisy measurements.1 Its operation is fundamentally based on the state-space representation of a system, which comprises two key equations. The state equation (or transition equation) describes how the system's state variables evolve over time, often represented as a first-order linear difference equation:
xt​=Axt−1​+But​+wt​
Here, xt​ is the state vector at time t, xt−1​ is the state vector at the previous time point, A is the state transition matrix, ut​ is a vector of control inputs (if any) governed by matrix B, and wt​ is the process noise. The process noise wt​ is typically assumed to be a white noise sequence with zero mean and covariance matrix Q, representing unmodeled dynamics or random shocks to the system.1
The second core equation is the observation equation (or measurement equation), which relates the observed measurements to the state variables:
zt​=Hxt​+vt​
In this equation, zt​ is the vector of observed measurements at time t, H is the observation matrix that maps the true state space into the observed space, and vt​ is the measurement noise. The measurement noise vt​ is also typically assumed to be a white noise sequence with zero mean and covariance matrix R, representing imperfections in the measurement process.1 This state-space structure is powerful because it allows for the modeling of unobserved latent variables (xt​) that are indirectly observed through noisy measurements (zt​).
The Kalman filter operates recursively through a two-stage process: prediction and update.1
1.	Prediction Stage: The filter projects the current state estimate and its error covariance matrix forward in time to obtain an a priori estimate for the next time step. This uses the state equation and knowledge of the system dynamics.
2.	Update Stage: When a new measurement becomes available, the a priori estimate is refined by incorporating the new information. This involves calculating the Kalman gain (Kt​), which determines the optimal weighting between the predicted state estimate and the new measurement based on their respective uncertainties. The innovation, or measurement residual, represents the difference between the actual measurement and the predicted measurement based on the current state estimate; it quantifies the new information brought by the measurement.1 The state estimate and its covariance matrix are then updated.
Under assumptions of linearity in the system and observation models, and Gaussian distributions for the process and measurement noise, the Kalman filter is an optimal estimator in the sense that it minimizes the mean squared error of the state estimate.1
Historically, the Kalman filter has deep connections to classical statistical methods. Its core can be viewed as an extension of the Gauss-Plackett algorithm for recursive least-squares estimation.3 While Gauss demonstrated the possibility of updating least-squares estimates with new equations, Kalman's derivation significantly broadened this concept by embedding it within a state-space model framework capable of handling time-varying parameters and dynamic systems with inherent stochasticity.3 The Kalman filter generalizes recursive least squares by explicitly modeling the evolution of the state (parameters) over time through the state equation and by accommodating a more complex error structure involving both process and measurement noise.3
The explicit modeling and management of uncertainty through covariance matrices is a hallmark of the Kalman filter. Survey data are inherently imperfect reflections of true population parameters, subject to both sampling and non-sampling errors.4 The Kalman filter's architecture, with its distinct process noise covariance (Q) and measurement noise covariance (R), directly addresses this. The recursive updating of the state error covariance matrix (Pt​) provides a dynamic assessment of the uncertainty associated with the state estimates at each time step. This dynamic uncertainty quantification is a considerable advancement over methods that might only offer static error measures. Furthermore, the state-space formulation enables a crucial conceptual separation: it distinguishes the unobservable "true" state of interest (e.g., the actual unemployment rate) from the noisy "observed" survey measurements (e.g., the estimated unemployment rate from a labor force survey). This distinction is foundational for many advanced applications in survey methodology, allowing for more sophisticated modeling of both the underlying phenomenon and the intricacies of the measurement process itself.1
B. Sample Survey Methodology: The Challenge of Dynamic Populations and Data Imperfections
Sample survey methodology is primarily concerned with the efficient estimation of finite population parameters based on data collected from a sample drawn from that population.4 A critical aspect of modern survey practice is the temporal dimension. Many surveys are conducted repeatedly over time (e.g., monthly labor force surveys, quarterly business surveys) or are longitudinal in nature, tracking the same units (individuals, households, or establishments) across multiple time points.7 Such designs are essential for monitoring changes, trends, and dynamic processes within populations. This inherent temporal nature means that survey data often manifest as time series.
The accuracy of survey estimates is affected by two broad categories of errors:
1.	Sampling Error: This error arises because estimates are based on a sample rather than a complete census of the population. It reflects the uncertainty due to observing only a part of the population and can generally be quantified and controlled through sample design and sample size.5
2.	Non-Sampling Errors: These encompass a wide range of errors that can occur at any stage of the survey process, including data collection, processing, and estimation. Major types include coverage errors (e.g., the sampling frame not adequately covering the target population), frame errors (e.g., duplications or omissions in the frame), non-response errors (when data are not obtained from some sample units or for some items), measurement errors (inaccuracies in recorded responses due to questionnaire design, interviewer effects, respondent behavior, or mode of data collection), and processing errors (mistakes during data entry, coding, or editing).4 These errors can introduce both bias and increased variability into survey estimates and are often more challenging to measure and control than sampling error.
The combination of dynamic population characteristics and the pervasive nature of survey errors means that traditional estimation methods, which might treat data from each survey occasion independently or use simplistic adjustments, can be inefficient or even inadequate. This creates a compelling need for advanced statistical techniques, such as the Kalman filter, that can explicitly model temporal dependencies, account for various sources of error, and integrate information in a statistically optimal manner.
The increasing demand from policymakers and researchers for timely and granular data from surveys—often for specific subpopulations or at high frequencies—further underscores the need for sophisticated modeling. However, producing such detailed estimates can be challenging, especially when sample sizes for specific domains or time periods are limited, leading to estimates with high sampling variability.4 Methods like Kalman filtering, which can "borrow strength" across time or integrate auxiliary information through the state-space model, become particularly attractive in this context. Moreover, the very nature of repeated or longitudinal data collection generates observations that are often autocorrelated; for instance, the unemployment rate in one month is typically related to the rate in the previous month.7 Ignoring such autocorrelation, as simpler methods might, can lead to suboptimal inferences and an unrealistic portrayal of the underlying processes. The Kalman filter, through its state equation, explicitly models this temporal dependence, offering a more faithful representation of the data-generating mechanisms.
II. Kalman Filters in the Realm of Repeated and Longitudinal Surveys
Repeated surveys, conducted at regular intervals, and longitudinal surveys, which track the same units over time, are cornerstones of official statistics and social science research. They provide invaluable data for understanding societal trends, economic dynamics, and individual-level changes. The Kalman filter, with its inherent ability to model time-varying systems, offers a powerful framework for enhancing the analysis of data from such surveys.
A. Modeling Evolving Parameters: Capturing Temporal Dynamics in Survey Estimates
A common challenge in analyzing data from repeated surveys is that the parameters of interest (e.g., population means, proportions, unemployment rates) are not static but evolve over time. Traditional approaches often treat survey estimates from different time points as independent or apply simple smoothing techniques. Such methods can be inefficient as they may not fully leverage the information contained in the entire sequence of survey data, especially if the underlying parameters are assumed to change smoothly.7
The state-space framework, operationalized by the Kalman filter, provides a more formal and efficient way to model these evolving parameters. In this context, the state vector (xt​) in the state-space model can be defined to represent the true, unobserved population parameters at each time point t. The state equation then describes how these parameters evolve. For instance, a simple model might assume that the true parameter follows a random walk, implying that the change from one period to the next is random noise. More complex models could involve autoregressive processes or incorporate explanatory variables influencing the parameter's evolution.7 The survey estimates obtained at each time point then serve as the noisy measurements (zt​) in the observation equation, where the measurement noise (vt​) primarily captures the sampling error associated with the survey estimate.
This approach fundamentally shifts the perspective from viewing parameters as fixed within a given survey period to treating them as realizations of a dynamic, stochastic process. This dynamic view often aligns more closely with the reality of many social, economic, and epidemiological phenomena measured by surveys. For example, characteristics like unemployment rates or public opinion do not remain static; they fluctuate due to a multitude of underlying, often unobserved, factors. A state-space model that permits the parameters themselves to evolve (e.g., via a state equation like αt​=Fαt−1​+ξt​, where αt​ is the state vector of parameters 7) is inherently more flexible and realistic than one assuming parameters are constant between survey waves or change only deterministically. The process noise term (ξt​) in such a model is crucial as it allows for unexpected shifts or changes in the trajectory of the true parameters.
A significant benefit of this dynamic modeling is that it implicitly "borrows strength" across time. The estimate of the parameter at time t is informed not only by the direct survey data collected at time t but also by the filtered estimate from time t−1 (which summarizes all historical information) and the model of how parameters are expected to evolve. The recursive nature of the Kalman filter ensures this linkage: the updated state estimate at time t (denoted at∣t​) is a function of the predicted state from the previous period (at∣t−1​) and the new observation zt​.1 This structured, model-based integration of past information often leads to smoother and more stable time series of estimates compared to relying solely on period-specific data.
B. Enhancing Estimation Efficiency and Precision over Time
One of the primary motivations for applying Kalman filters to repeated survey data is the potential for improved statistical properties of the estimates, particularly in terms of efficiency and precision. By optimally combining information from past estimates (encapsulated in the predicted state) with current survey data, Kalman filtering can lead to estimates with lower variance compared to those derived solely from the current data. This is especially true when the underlying process evolves relatively smoothly (i.e., small process noise variance Q) compared to the variability in the survey estimates themselves (i.e., measurement noise variance R).7
Survey estimates, particularly for smaller samples or specific domains within a larger survey, can exhibit considerable variability from one period to the next. The Kalman filtering process can effectively smooth out this noise, making it easier to discern the underlying trend or signal from the random fluctuations inherent in sampling.7 This "signal extraction" capability is a key feature; the Kalman filter acts as a sophisticated tool to separate the "signal" (the true, evolving parameter) from the "noise" (sampling error and other survey imperfections).7 Each survey estimate zt​ can be thought of as a composite of the true parameter xt​ and an error term vt​. The filter attempts to optimally estimate xt​ by modeling both its temporal evolution and the statistical characteristics of vt​. This provides a more formal and powerful approach than many ad-hoc smoothing techniques.
The improved precision achieved through Kalman filtering can have significant practical implications. For instance, it may allow statistical agencies to produce reliable estimates at a higher frequency (e.g., monthly instead of quarterly, or quarterly instead of annually) than would be feasible if relying solely on direct estimates, which might be too unstable at shorter intervals.7 Furthermore, the efficiency gains are not merely statistical; they can translate into resource efficiencies. If more precise estimates can be derived from existing survey data structures, or potentially from surveys with somewhat smaller periodic sample sizes when combined effectively with time series modeling, this could lead to cost savings or allow for the reallocation of resources to other statistical priorities.
To better understand the advantages of the Kalman filter/state-space approach in the context of repeated surveys, Table 1 provides a comparative overview with other common estimation strategies.
Table 1: Comparative Overview of Estimation Approaches in Repeated Surveys

Feature	Traditional Cross-Sectional Estimates	Simple Time Series Smoothing of Estimates	Kalman Filter/State-Space Approach	
Use of historical data	Ignores historical data (each period estimated independently)	Uses historical estimates (e.g., moving average)	Formally incorporates historical data via state prediction and recursive updates	
Modeling of parameter evolution	Assumes parameters are constant within the estimation period	Implicitly assumes smooth evolution (e.g., local linearity)	Explicitly models parameter evolution via the state equation (e.g., random walk, AR process)	
Efficiency in dynamic settings	Lower efficiency if true parameters evolve smoothly or systematically	Moderate efficiency improvement, can adapt to trends	Higher efficiency by optimally combining past information and current data based on model dynamics and noise characteristics	
Handling of survey noise	Directly reflects all survey noise (sampling and non-sampling)	Reduces some noise through averaging, but can lag behind true changes	Formally models and filters survey noise (measurement error vt​) from the underlying state signal (xt​)	
Computational complexity	Generally low, depends on survey design	Low to moderate, depending on smoothing technique	Can be high for complex models, but efficient algorithms exist for many common structures (e.g.7)	
Need for explicit model assumptions	Fewer explicit model assumptions beyond the sampling design	Fewer formal model assumptions, often data-driven	Relies on explicit state-space model assumptions (e.g., linearity, Gaussianity for standard KF, structure of A,H,Q,R)	
Adaptability to structural changes	Poor, treats each period anew	Can be slow to adapt, may smooth over true breaks	Can adapt if process noise Q is appropriately specified or if models for structural breaks are incorporated (e.g., via state augmentation)	
Provision of uncertainty measures	Typically design-based variance for current estimate	Uncertainty often not formally derived or more complex to assess	Provides updated state covariance matrix (Pt​) reflecting uncertainty in state estimates (though hyperparameter uncertainty is a challenge)	
This table highlights that while simpler methods are less demanding in terms of modeling effort, the Kalman filter/state-space approach offers a more principled and potentially more efficient way to analyze time series of survey estimates by explicitly accounting for dynamics and error structures.
C. Computational Strategies and Algorithmic Adaptations for Survey Data
While the theoretical advantages of Kalman filters are compelling, their practical application to survey data, especially large-scale national surveys, can present significant computational challenges. A direct application of the standard Kalman filter equations can involve the manipulation and inversion of large matrices, particularly if the state vector is high-dimensional (e.g., including parameters for many small areas or demographic groups) or if the filter attempts to process raw microdata directly.7 The computational cost of matrix inversion, for example, is typically proportional to the cube of the matrix dimension.9 Such costs can render the approach computationally prohibitive or lead to numerical instability.
Recognizing these challenges, researchers have developed modified algorithms and computational strategies to make Kalman filtering feasible for survey applications. A notable contribution in this area involves reformulating the Kalman filter recursions to reduce their dependence on the sample size of each survey (Nt​). For instance, work detailed in 7 (and further analyzed in 7) demonstrates how, by using properties like the matrix inversion lemma or by working with summary statistics (such as the first and second moments of the data for each period), the computational burden of the filter can be made constant in, or at least less critically dependent on, Nt​. Instead, the complexity becomes primarily dependent on the dimension of the state vector, which is often much smaller than Nt​. This makes the approach viable even for surveys with large periodic samples.
The development of such computationally efficient Kalman filter variants is a critical enabler for their adoption in large-scale official statistics. Without these algorithmic innovations, the theoretical benefits of the Kalman filter would remain largely academic for many practical survey applications conducted by national statistical offices, which often involve processing data from tens or even hundreds of thousands of respondents. The research focusing on algorithms whose complexity scales with the state dimension rather than the sample size has therefore been pivotal for the operationalization of these methods.7
There is also a consideration of whether to process elements of an observation vector sequentially (a univariate approach) or jointly (a multivariate approach). The univariate approach processes each component of the observation vector one at a time, avoiding large matrix inversions but potentially requiring longer sequences of scalar operations. The multivariate approach processes the entire observation vector at once, which can be more statistically efficient if observation errors are correlated but involves larger matrix operations.7 The choice often depends on the specific structure of the problem and the available computational resources.
It is also important to acknowledge a potential trade-off between the desire to utilize full microdata within the Kalman filter framework (which might be seen as theoretically optimal for capturing all available information) and the practical need for computational feasibility. Many practical implementations, such as the one proposed in 7, rely on aggregated data (like means and covariances per period) to manage complexity. While this approach is computationally advantageous, it might involve some loss of information compared to a hypothetical filter operating on all microdata simultaneously, though the latter is often intractable for large surveys. The challenge lies in finding an appropriate balance that retains most of the statistical benefits while ensuring computational tractability. Similar concerns about managing high-dimensional state spaces arise in other fields employing Kalman filters, leading to techniques like reduced-rank approximations or partitioned filters, which aim to simplify the problem by focusing on the most significant components of variation or by breaking down a large problem into smaller, more manageable parts.9
III. The State-Space Framework: Unifying Theory for Survey Applications
The power and versatility of the Kalman filter in survey methodology stem largely from its foundation in the state-space framework. This framework provides a general and coherent way to model dynamic systems and is particularly well-suited to many of the complex data structures and analytical goals encountered in survey research.
A. Conceptualizing Surveys: Distinguishing Latent Processes from Observation Models
A fundamental strength of the state-space framework is its explicit separation of the underlying, often unobservable, dynamic process from the imperfect observation or measurement process.1 This conceptual distinction is highly valuable in the survey context.
1.	Latent Process Model (State Equation): This component describes the evolution of the true underlying quantities of interest over time, independent of how they are measured or observed.6 For example, in an economic survey, the latent process might be the true monthly employment level in a region, which changes due to economic factors. In an epidemiological survey, it might be the actual prevalence of a disease, evolving based on transmission dynamics and interventions. This corresponds to the 'state equation' in the Kalman filter formulation.
2.	Observation Model (Measurement Equation): This component describes how the collected survey data (the observations) are related to the latent process. Crucially, it explicitly incorporates the idea that survey measurements are imperfect and contain "noise," which can encompass sampling error, non-sampling errors, or other sources of discrepancy between the observed value and the true latent state.6 This corresponds to the 'measurement equation'.
This separation provides a powerful paradigm for survey methodologists, allowing them to simultaneously model the phenomenon of interest and the intricacies of the data collection mechanism itself. For instance, changes in survey methodology (e.g., a redesigned questionnaire, a shift in the mode of data collection from face-to-face to web) could be explicitly modeled within the observation equation by adjusting its parameters or error structure, while the state equation continues to represent the underlying trend in the phenomenon being measured. This allows for a more nuanced understanding of how changes in survey design might affect the resulting estimates, distinguishing such effects from genuine changes in the population.
The ability to incorporate covariates into both the process and observation models, as highlighted in the mvgam package for ecological data 10, offers significant flexibility for explaining variability and improving model fit in complex survey scenarios. Many factors can influence both the true value of a parameter and how it is measured. For example, the implementation of new economic policies (a covariate) might directly affect the true unemployment rate (modeled in the process equation), while the level of interviewer experience or the quality of the sampling frame (covariates) might affect the accuracy or bias of the survey's measurement of unemployment (modeled in the observation equation). State-space models that allow for such distinct covariate effects provide a much richer and more realistic representation of the survey environment.
The application of state-space models in ecology, such as analyzing plankton counts from Lake Washington 10, provides a clear illustration. In this example, covariates like water temperature might influence the underlying plankton abundance (the latent process), while other factors, such as weather conditions during sampling (e.g., a thunderstorm), could affect the accuracy of the plankton counts (the observation process). The state-space model allows researchers to disentangle these effects, leading to more robust inferences about the ecological dynamics. This distinction is directly analogous to many situations in social, economic, and health surveys.
To further clarify how the general components of state-space models translate to survey applications, Table 2 provides a mapping.
Table 2: Core Components of State-Space Models in Sample Survey Applications

Component	General Definition	Interpretation/Examples in Survey Context	
State Equation	Describes the evolution of the unobserved state vector over time, e.g., xt​=Axt−1​+But​+wt​.1	Models the true underlying population characteristic (e.g., true monthly unemployment rate, actual disease prevalence). xt​ could be the true rate, A governs its persistence/autocorrelation, wt​ is the process noise representing real unpredictable shocks to the system.	
Observation Equation	Relates the noisy measurements to the unobserved state vector, e.g., zt​=Hxt​+vt​.1	Models the survey process. zt​ is the direct survey estimate (e.g., estimated unemployment rate from a labor force survey). H links the true state to the measurement. vt​ is the measurement noise, representing sampling error and non-sampling errors specific to the survey at time t.	
State Vector (xt​)	A set of variables that completely characterizes the system at time t.	True (unobserved) values of interest, e.g., vector of true employment rates for several regions, parameters of a regression model that are themselves evolving, latent public opinion, or components of a time series like trend and seasonality.	
Process Noise (wt​)	Unpredictable random disturbances or inaccuracies in the process model. Assumed to have a certain covariance Q.	Real-world shocks or unmodelled influences affecting the true characteristic (e.g., sudden economic event impacting employment, unobserved factors influencing disease spread). Its variance Q reflects the volatility or unpredictability of the true underlying process.	
Measurement Noise (vt​)	Errors in the observed measurements. Assumed to have a certain covariance R.	Primarily sampling error associated with the survey estimate. Can also include components of non-sampling error if quantifiable and modeled. Its variance R is often related to the design-based variance of the survey estimate provided by survey practitioners.	
Understanding these components and their interplay is crucial for effectively applying Kalman filters and state-space models to diverse problems in survey methodology.
B. Modeling Complex Dependencies and Hierarchies in Survey Data
Survey data often exhibit complex dependencies and hierarchical structures that need to be accounted for in statistical modeling. The state-space framework, particularly with its Kalman filter engine, offers versatile tools for addressing these complexities.
Multivariate State-Space Models: Many survey applications involve tracking multiple related time series simultaneously. For example, a national statistical office might be interested in unemployment rates for different demographic groups (age, sex, education) or for various geographical regions. These series are often correlated due to common underlying economic or social factors. Multivariate state-space models extend the univariate framework by defining the state vector xt​ to be multidimensional. The transition matrix A can then model not only the evolution of each series individually (diagonal elements) but also the dynamic interdependencies between the series (off-diagonal elements).10 Similarly, the process noise covariance matrix Q can allow for contemporaneous correlations in the random shocks affecting different components of the state vector. This capacity to model interconnected phenomena is vital for policy analysis, where interventions or changes affecting one indicator can have ripple effects on others. A multivariate state-space model provides a more holistic and realistic view than analyzing each time series in isolation.
Hierarchical Structures: Survey data frequently possess a hierarchical or multilevel structure. For instance, in longitudinal household surveys, individuals are nested within households, and repeated measurements are taken on individuals over time. Changes might occur at the individual level, influenced by evolving household-level characteristics. State-space models can be combined with multilevel modeling principles to capture such dependencies. For example, the work by Pfeffermann and Nathan on imputing wave nonresponse in panel surveys uses models that combine standard multi-level (mixed linear) models operating at given time points with time-series state-space models for the random group effects (e.g., household effects) and the individual measurements.8 This allows for modeling how household-specific effects evolve over time and influence the responses of individuals within those households.
Correlated Errors: A critical aspect, especially in survey methodology, is the handling of correlated errors.
●	Correlated Process Errors: As mentioned, multivariate state-space models can accommodate correlations in the process noise wt​ through the Q matrix.10 This means that random shocks affecting one component of the true state (e.g., true youth unemployment) can be correlated with shocks affecting another component (e.g., true adult unemployment).
●	Correlated Measurement Errors: Perhaps more uniquely challenging for survey applications is the issue of correlated measurement errors vt​. Standard Kalman filter theory often assumes that measurement errors are uncorrelated over time and across different components of the observation vector. However, complex survey designs frequently violate this assumption. For example:
○	Rotating panel designs, common in labor force surveys, where a portion of the sample is retained for several consecutive periods, induce correlations in sampling errors for estimates made at different time points due to the overlap in respondents.
○	Cluster sampling, where groups of units (e.g., households within a geographic area) are selected, can lead to correlated sampling errors among units within the same cluster. Ignoring such correlations can lead to inefficient state estimates and, importantly, misstatement of their precision (i.e., incorrect variance estimates). Research has therefore focused on developing modified filtering algorithms that can explicitly handle correlated measurement errors. For instance11 (and 11) discusses the development of such an algorithm in the context of small area estimation, where direct survey estimates (the measurements) for different states can have sampling errors that are correlated due to the survey design or modeling choices. The classical Kalman filter is often invalidated when measurement errors are correlated in a way not captured by the standard model assumptions. Addressing this is a key research frontier for enhancing the accuracy and reliability of survey-based state-space models.
IV. Tackling Survey Errors and Data Deficiencies with Kalman Filters
Survey data are invariably affected by various errors and deficiencies, such as missing data due to nonresponse and inaccuracies arising from measurement error. The Kalman filter and its underlying state-space framework provide sophisticated tools for addressing some of these issues in a model-based manner, aiming to improve the overall quality of survey estimates.
A. Advanced Imputation Techniques for Wave Nonresponse in Panel Surveys
Wave nonresponse is a common and problematic form of missing data in longitudinal or panel surveys. It occurs when a unit (e.g., an individual or household) that participated in earlier waves of the survey fails to provide data for one or more subsequent waves, though they may return in later waves.8 Simple imputation methods, such as mean imputation or last observation carried forward, often fail to adequately utilize the rich longitudinal information available for responding units or the partial information for nonresponding units, and may introduce bias, particularly if the nonresponse is informative (i.e., related to the missing values themselves).
Time-series approaches, leveraging the state-space model and Kalman filter/smoother, offer a more principled way to impute for wave nonresponse. The core idea is to use the observed data from other waves for the same unit, along with a model of how the unit's characteristics evolve over time, to predict the missing values.8
Pfeffermann and Nathan 8 (elaborated in 8) propose a sophisticated approach that combines multi-level (mixed linear) models with time-series state-space models. This recognizes that longitudinal data often have hierarchical structures (e.g., individuals within households) and that changes occur at different levels of this hierarchy over time. Their model might include:
●	Fixed effects of covariates.
●	Random household-level effects that evolve over time according to an autoregressive (AR) process.
●	Random individual-level error terms that also follow an AR process.
The state vector in this formulation includes these evolving random components. The Kalman filter and related smoothing algorithms are then used for prediction and imputation:
●	Kalman Filter: If imputation is performed using only past and current available data for a unit, the Kalman filter is applied to predict the state (and thus the missing components) at the time of nonresponse.
●	Kalman Smoother: If a unit misses a wave but returns to the panel in a subsequent wave (intermittent nonresponse), a smoothing algorithm can be used. Smoothing algorithms utilize all available data for the unit—past, current, and future relative to the missing point—to provide more accurate predictions of the state during the period of nonresponse.1 This "looking back" capability, informed by the unit's subsequent behavior or measurements, is a significant advantage of smoothing over purely forward-predictive filtering for imputation in panel surveys.
The primary benefit of such model-based imputation methods is their potential to reduce imputation bias, even when the nonresponse mechanism is informative (not missing completely at random), by explicitly leveraging the interrelationships among observations for the same unit or group over time and across different levels of the hierarchy.8
B. Addressing Measurement Error: Model-Based Adjustments and Correlated Errors
Measurement error refers to the discrepancy between the true value of a characteristic and the value recorded in a survey. It can arise from various sources, including questionnaire design, interviewer effects, respondent errors (e.g., recall bias, social desirability bias), and data processing mistakes.4
In the standard state-space formulation, the observation noise term (vt​) in the measurement equation (zt​=Hxt​+vt​) inherently accounts for the combined effect of sampling error and non-sampling errors, including measurement error.1 The Kalman filter then optimally separates this composite noise from the estimate of the underlying state (xt​). The magnitude of the measurement noise variance (R) influences how much weight the filter gives to new observations; highly noisy observations (large R) will be down-weighted.
While the basic Kalman filter can account for the presence of measurement noise by incorporating its variance, it does not inherently reduce measurement error at its source or correct for systematic biases unless these are explicitly modeled. However, by modeling the noise, the filter aims to prevent it from unduly distorting the estimate of the underlying state. More complex state-space models might attempt to estimate and adjust for specific biases arising from measurement error if auxiliary information or validated error models are available. For example, if a known bias term affects the measurements, it could potentially be incorporated into the observation equation, perhaps as zt​=Hxt​+biast​+vt​, and this bias term (biast​) might itself be modeled as a state variable, possibly using state augmentation techniques similar to those used for other unobserved components.13 Furthermore, as noted in 10, covariates can be added to the observation model to potentially explain some systematic components of observation error.
A particularly significant development for survey methodology is the extension of Kalman filtering techniques to handle correlated measurement errors.11 As discussed earlier, complex survey designs, such as rotating panels or multi-stage cluster sampling, often lead to sampling errors (a major component of the overall measurement error term vt​ in this context) that are correlated across time or across different units or estimates. For example, estimates for different states produced from a national survey might have sampling errors that are correlated due to the survey's stratification or sample allocation. The classical Kalman filter, which assumes uncorrelated measurement errors (i.e., a diagonal R matrix, or R being independent over time), is invalidated under such conditions and can lead to suboptimal estimates and incorrect assessments of precision.
The work by Pfeffermann, Tiller, and Zimmerman 11 addresses this by developing a filtering algorithm specifically for state-space models with correlated measurement errors. This is crucial, for instance, when benchmarking small area estimates to direct survey estimates from larger areas, as the benchmarking constraints involve these direct estimates and their associated (potentially correlated) sampling errors. The development of such algorithms represents a substantial advancement, making state-space models more robustly applicable to the realities of complex survey data that rarely satisfy the assumption of independent measurement errors. This allows for more accurate modeling of survey error structures, which in turn can lead to more reliable state estimates and more valid measures of their uncertainty.
V. Kalman Filters in Small Area Estimation: Producing Reliable Estimates from Survey Data
One of the most prominent and impactful applications of Kalman filters and state-space models in survey methodology is in the field of Small Area Estimation (SAE). SAE refers to statistical techniques used to produce reliable estimates of characteristics of interest (e.g., poverty rates, unemployment levels, crop yields) for small geographical areas (e.g., counties, districts) or specific demographic sub-populations (e.g., age-sex-race groups), for which direct survey estimates are either not available or are too unreliable due to small sample sizes in those domains.11
A. Structural Time Series Models for Small Domain Estimation
The core challenge in SAE is that the sample sizes allocated to individual small areas are often insufficient to yield direct survey estimates with acceptable precision. To overcome this, SAE methods "borrow strength" by linking the sparse direct information for a small area with data from related areas, from previous time periods, or through statistical models that leverage auxiliary information (e.g., administrative records, census data) correlated with the variable of interest.14
When repeated surveys provide data over time for small areas, even if these data are noisy, Structural Time Series (STS) models offer a powerful framework for SAE. STS models decompose a time series for a small area into several unobserved components, such as a trend (capturing long-term movement), a seasonal component (capturing regular periodic fluctuations), and an irregular component (representing random noise).14 These components are typically modeled as stochastic processes. For example, the trend might be modeled as a local linear trend, where both the level and the slope of the trend can change stochastically over time.
Crucially, STS models can be cast into a state-space form. The unobserved components (level, slope, seasonal effects, etc.) for a small area at a given time t constitute the state vector xt​. The state equation then describes how these components evolve (e.g., the level changes based on the slope and a random shock; the slope itself might be a random walk). The direct survey estimate for the small area at time t serves as the observation zt​ in the measurement equation, with the measurement noise vt​ representing the sampling error of this direct estimate. The Kalman filter is then the primary inferential tool for estimating the evolving state vector (i.e., the underlying trend, seasonal, and other components) over time, thereby producing a model-based estimate for the small area characteristic that is typically more stable and precise than the direct estimate alone.14
The application of Kalman filters in SAE via STS models is a prime example of model-based official statistics. It enables statistical agencies to produce vital estimates for local planning, policy evaluation, and resource allocation that would otherwise be impossible or too unreliable to derive from direct survey data alone.15 The choice of components included in the STS model (e.g., a fixed slope versus a stochastic slope for the trend, the nature of the seasonal component) and the assumptions about their evolution (specifically, the variances of the disturbances in the state equation, which form the Q matrix) are critical modeling decisions. These choices directly influence the smoothness and responsiveness of the resulting small area estimates and require careful consideration, often involving diagnostic checks and expert judgment.
B. Practical Considerations: Hyperparameter Estimation, MSE Calculation, and Benchmarking
While STS models fitted via the Kalman filter offer a robust approach to SAE, their practical implementation involves several significant challenges.
1.	Hyperparameter Estimation: The state-space models used in SAE depend on hyperparameters, which are typically the variances of the process noise components (elements of the Q matrix) and the variance of the measurement noise (the R matrix, often assumed known from survey design or estimated). These hyperparameters are usually unknown and must be estimated from the available data. Common methods for estimating hyperparameters include Maximum Likelihood Estimation (MLE) or Restricted Maximum Likelihood (REML) applied to the time series of direct survey estimates.14 The accuracy of these hyperparameter estimates is crucial, as they determine the relative weight given to the model's predictions versus the incoming survey data in the Kalman filter updates. Inaccurate hyperparameter estimates can lead to suboptimal small area estimates.
2.	Mean Squared Error (MSE) Estimation: A critical requirement for any SAE method is to provide a reliable measure of uncertainty for the model-based estimates. For Kalman filter-based estimates, calculating the Mean Squared Error (MSE) is complex. The MSE should ideally account for three main sources of uncertainty: (a) the uncertainty due to predicting the random effects (the state components), (b) the uncertainty due to estimating the fixed effects in the model (if any), and (c) the uncertainty arising from the estimation of the hyperparameters (the Q and R variances). The standard output from the Kalman filter (the state covariance matrix Pt​) provides a measure of uncertainty for source (a), assuming the model parameters (including hyperparameters) are known. However, if this Pt​ is used directly as the MSE, and the uncertainty from hyperparameter estimation is ignored, the resulting MSE estimates tend to be negatively biased, meaning the published small area estimates might appear more precise than they truly are.14 This underestimation is a serious concern for the credibility of official statistics, spurring research into analytical approximations or resampling methods (like parametric bootstrap) to obtain more accurate MSE estimates that properly account for all significant sources of uncertainty.
3.	Benchmarking: Small area estimates produced by models are often required to be consistent with (i.e., sum up to or align with) reliable direct survey estimates available for larger geographical areas or aggregates (e.g., state-level estimates must sum to a pre-determined national estimate, or county estimates must sum to a state estimate). This process is known as benchmarking. Incorporating such benchmark constraints into the state-space modeling and Kalman filtering framework is non-trivial.11 The constraints often involve the direct survey estimates for the larger areas, which themselves have sampling errors. If these sampling errors are correlated with those of the small areas (or across the larger areas themselves), as is often the case due to complex survey designs, the standard Kalman filter algorithm may be invalidated. Specialized algorithms or adjustments are needed to correctly incorporate these constraints while handling the complex error structures. The U.S. Bureau of Labor Statistics (BLS), for example, uses state-space models to produce monthly employment and unemployment estimates for states and faces these issues, including modeling the complex sampling error structure of the Current Population Survey (CPS) (e.g., using an AR(15) model for the sampling error component) and ensuring consistency with national totals.11 Benchmarking introduces a potential tension between the model-based optimality of estimates for individual small areas and the requirement for consistency with aggregates. Forcing consistency can sometimes degrade the quality of the small area estimates if the benchmark constraint is very stringent or if the underlying models are misspecified.
The development of filtering algorithms that can correctly incorporate benchmark constraints while simultaneously handling issues like correlated measurement errors highlights how practical operational requirements in official statistics drive theoretical and methodological advancements in statistical modeling.
VI. Critical Appraisal: Advantages, Limitations, and Future Horizons
The integration of Kalman filters and state-space models into sample survey methodology has brought significant advancements, but like any sophisticated statistical tool, it comes with its own set of strengths and challenges. A critical appraisal is necessary to understand its appropriate use and potential future directions.
A. Strengths: Efficiency Gains, Dynamic Modeling Capabilities, Coherent Data Integration
The primary advantages of employing Kalman filters in survey methodology are manifold:
●	Improved Efficiency and Precision: Particularly in time series contexts, such as repeated surveys or SAE, Kalman filters can lead to estimates with lower variance and greater stability compared to methods that do not effectively pool information over time or across related domains.7
●	Formal Modeling of Dynamics: The state-space framework allows for the explicit and flexible modeling of how population parameters or underlying processes evolve over time. This is a significant step up from static or ad-hoc approaches to dynamic phenomena.1
●	Principled Handling of Noise: The Kalman filter provides an optimal way (under its assumptions) to separate the underlying signal (the true state) from measurement noise (survey errors), leading to clearer insights into trends and changes.1
●	Framework for Data Integration: While not extensively detailed in the provided materials for survey-specific contexts beyond survey data itself, the Kalman filter is inherently a data fusion algorithm.2 This suggests its potential for coherently combining information from traditional surveys with data from administrative records or other auxiliary sources, provided these can be appropriately modeled within the state-space observation equation.
●	Unified Approach: The state-space framework offers a consistent and coherent structure for tackling a variety of problems in survey methodology, including time series estimation, imputation of missing data, and small area estimation. This unification can lead to more integrated and systematic analytical strategies.
●	Real-Time Recursive Updates: The recursive nature of the Kalman filter, processing measurements one at a time as they become available, is a significant advantage for producing timely survey estimates.1 Many official statistics are released on strict schedules, and a method that can efficiently update estimates with new data without requiring reprocessing of all historical data aligns well with these operational demands.
Table 3 summarizes the key applications and contributions of Kalman filters in different areas of sample survey methodology.
Table 3: Summary of Kalman Filter Applications in Key Sample Survey Methodology Areas

Application Area	Primary Contribution of Kalman Filter/State-Space Model	Key Challenges/Considerations	Illustrative Sources	
Repeated/Longitudinal Survey Estimation	Efficiently models time-varying parameters, improves precision by borrowing strength over time, smooths survey estimates.	Computational cost for large Nt​ (sample size per period), model specification for parameter evolution, hyperparameter estimation.	7	
Wave Nonresponse Imputation in Panel Surveys	Predicts missing values using temporal correlations and hierarchical structure (e.g., individual within household), potentially reducing imputation bias.	Model complexity, estimation of parameters for multi-level models, assumptions about the nonresponse mechanism, choice of filter vs. smoother.	8	
Small Area Estimation (SAE)	Produces reliable estimates for small domains by modeling time series components (trend, seasonality) from repeated survey data.	Hyperparameter estimation, robust MSE calculation (accounting for all sources of uncertainty), benchmarking constraints.	11	
Handling Correlated Measurement/Sampling Errors	Allows for more realistic modeling of complex survey error structures (e.g., due to rotating panels, cluster designs), leading to more accurate estimates and uncertainty assessment.	Developing appropriate filter algorithms for specific correlation structures, defining the correlation structure accurately.	11	
B. Challenges: Model Complexity, Assumption Sensitivity, Computational Demands
Despite their strengths, the application of Kalman filters is not without its challenges and limitations:
●	Model Specification: The performance of the Kalman filter is critically dependent on the correct specification of the state-space model, including the state transition matrix (A), the observation matrix (H), and the covariance matrices of the process noise (Q) and measurement noise (R). Misspecification of these components can lead to biased or inefficient estimates, and potentially to filter divergence where the estimates become unreliable.1
●	Assumption Sensitivity: The standard Kalman filter relies on assumptions of linearity in the system dynamics and observation processes, and on the Gaussianity of the process and measurement noise distributions.1 While extensions such as the Extended Kalman Filter (EKF) and Unscented Kalman Filter (UKF) have been developed for non-linear systems 16, these often involve approximations that can affect performance. Similarly, dealing with strongly non-Gaussian noise can be problematic for the standard filter and may require more advanced techniques like particle filters.
●	Estimation of Noise Covariance Matrices (Q and R): Accurately estimating the noise covariance matrices Q (process noise) and R (measurement noise) is often one of the most challenging aspects of applying Kalman filters in practice.1 These matrices, or their constituent hyperparameters, are crucial for the filter's performance, as they dictate the balance between trusting the model's predictions and incorporating new measurements. Poor estimates of Q and R can lead to suboptimal filtering.
●	Computational Demands: While significant algorithmic improvements have been made, particularly for specific structures relevant to surveys 7, fitting complex state-space models to very large datasets can still pose considerable computational challenges.1 The "curse of dimensionality" can become a factor if the state vector is very large (e.g., when modeling many small areas simultaneously with intricate interactions). This necessitates careful attention to model parsimony, or the use of approximation techniques such as partitioned filters or reduced-rank methods.9
●	Interpretability: As state-space models become more complex to capture intricate real-world dynamics, they can sometimes become "black boxes," making it difficult to intuitively understand why the estimates behave as they do or to pinpoint the sources of model misfit.3
●	Complexity vs. Estimability: There is an inherent tension between the desire to build highly complex models that closely mimic reality and the ability to robustly estimate all the model parameters, especially hyperparameters, from finite (and often noisy) survey data. Overly complex models may suffer from parameter identifiability issues or yield unstable hyperparameter estimates, potentially degrading the filter's performance.
C. Emerging Research and Potential Future Trajectories in Survey Methodology
The use of Kalman filters and state-space models in survey methodology is an active area of research, with several promising future trajectories:
●	Non-linear and Non-Gaussian State-Space Models: Continued development and application of more advanced filtering techniques, such as particle filters, ensemble Kalman filters 9, and other sequential Monte Carlo methods, are likely to expand the toolkit for handling non-linear dynamics and non-Gaussian error distributions that are often encountered in real-world survey data. The Extended Kalman Filter (EKF) is already used in some domains for non-linear systems.16
●	Advanced Handling of Measurement Error: Research is likely to focus on developing more sophisticated models for various types of measurement error (e.g., recall bias, interviewer effects, mode effects) within the state-space framework, potentially allowing for their estimation and adjustment.
●	Dynamic Weighting in Longitudinal Surveys: There is potential for adapting Kalman filter principles to update survey weights dynamically over the life of a panel survey. This could help account for attrition, changes in population composition, and evolving representativeness, although this specific application was not detailed in the provided sources for survey weighting directly (related concepts of dynamic weighting appear in other fields 17).
●	Integration with Machine Learning: The synergy between state-space models and machine learning techniques holds considerable promise. Machine learning could be employed for tasks such as automated covariate selection for state or observation equations, detecting non-linearities, or more robustly tuning hyperparameters.
●	Improved MSE Estimation: Given the importance of accurate uncertainty assessment, particularly in official statistics and SAE, research will continue to focus on developing computationally feasible and statistically sound methods for MSE estimation that fully account for all sources of error, including hyperparameter uncertainty, in complex Kalman filter-based survey estimates.
●	Software and Accessibility: The development of user-friendly and well-documented software packages (such as mvgam for certain types of models 10) is crucial for making these advanced methods more accessible to a broader range of survey practitioners and researchers, facilitating their wider adoption and application.
●	Data Fusion and Integration of Diverse Data Sources: The increasing availability of "big data," administrative records, and other non-survey data sources presents both opportunities and challenges. Kalman filters and the state-space framework offer a natural and powerful paradigm for data fusion 2, where different data sources can be treated as distinct "sensors" (i.e., contributing to different observation equations) that inform the same underlying latent state. Developing methodologies to effectively integrate these diverse data sources with traditional survey data, while accounting for differences in quality, coverage, definitions, and error structures, is a major area for future innovation.
●	Robust Diagnostics and Validation: As models become more intricate to better capture reality (e.g., non-linear, non-Gaussian, high-dimensional), the need for robust diagnostic tools and validation techniques becomes even more critical. This is essential to ensure that the models are not merely fitting noise or producing spurious results that are overly dependent on specific model assumptions. Future research will likely emphasize the development of better tools for model checking, sensitivity analysis, and out-of-sample validation in the context of state-space models for survey data.
VII. Conclusion: The Evolving Role of Kalman Filters in Modern Survey Statistics
The Kalman filter, together with its underlying state-space modeling framework, has emerged as a remarkably versatile and powerful tool within sample survey methodology. Its capacity to recursively estimate evolving states from noisy data provides a principled approach to addressing a range of complex challenges inherent in the collection, analysis, and interpretation of survey information, particularly data with a temporal dimension.
Applications span from enhancing the efficiency and precision of estimates from repeated and longitudinal surveys by explicitly modeling time-varying parameters, to sophisticated imputation techniques for wave nonresponse in panel studies that leverage temporal correlations and hierarchical structures. Furthermore, Kalman filters are central to modern small area estimation, enabling the production of reliable statistics for domains where direct survey estimates lack sufficient precision, often through the use of structural time series models. The ongoing development of specialized filtering algorithms to handle intricacies like correlated measurement errors, which are common in complex survey designs, further underscores the adaptability of this framework to the specific needs of survey practitioners.
However, the adoption of these advanced methods is not without its complexities. The performance of Kalman filter-based approaches is contingent upon careful model specification, including the assumptions about system dynamics and error structures. The estimation of unknown model parameters, particularly noise covariances (hyperparameters), and the robust calculation of mean squared errors that fully account for all sources of uncertainty remain significant practical and research challenges. Computational demands, while mitigated by algorithmic advancements, can still be a consideration for very large or complex models.
Despite these challenges, the trajectory of research and application suggests an expanding role for Kalman filters in survey statistics. The push for more granular, timely, and accurate information from surveys, coupled with the increasing availability of diverse data sources, necessitates the use of sophisticated statistical models. The Kalman filter's ability to integrate information over time, handle various error structures, and provide a coherent framework for dynamic modeling positions it as a key methodology for the future. Ongoing research into non-linear and non-Gaussian extensions, improved error modeling, data fusion techniques, and user-friendly software will likely continue to broaden its applicability and robustness.
Ultimately, the increasing use of Kalman filter-based methods reflects a broader evolution in survey methodology towards more integrated, model-dependent approaches. This shift is driven by the imperative to extract maximum value from survey data, address inherent imperfections, and meet the complex information needs of contemporary society. The Kalman filter, by providing a rigorous and flexible means of navigating the interface between dynamic systems and imperfect measurements, is set to remain an important instrument in the toolkit of survey statisticians and methodologists for years to come.
Works cited
1.	State-Space Models & Kalman Filtering | Intro to Time Series Class ..., accessed June 6, 2025, https://library.fiveable.me/intro-time-series/unit-12
2.	Kalman filter - Wikipedia, accessed June 6, 2025, https://en.wikipedia.org/wiki/Kalman_filter
3.	Recursive Estimation and the Kalman Filter, accessed June 6, 2025, https://www.le.ac.uk/users/dsgp1/COURSES/MESOMET/ECMETXT/recurse.pdf
4.	Measurement Error Estimation Methods in Survey Methodology, accessed June 6, 2025, https://www.pvamu.edu/mathematics/wp-content/uploads/sites/49/05_R701_Saba_RS_Posted_061616_97-114.pdf
5.	Errors in Statistical Data - Australian Bureau of Statistics, accessed June 6, 2025, https://www.abs.gov.au/websitedbs/d3310114.nsf/home/Basic+Survey+Design+-+Errors+in+Statistical+Data
6.	nicholasjclark.github.io, accessed June 6, 2025, https://nicholasjclark.github.io/mvgam/articles/trend_formulas.html#:~:text=State%2DSpace%20models%20allow%20us,%2F%20measure%20this%20underlying%20process).
7.	Repeated surveys and the Kalman filter - CiteSeerX, accessed June 6, 2025, https://citeseerx.ist.psu.edu/document?repid=rep1&type=pdf&doi=97aa288dcf4a9b44df6075f112923ae96ada1daa
8.	Survey Nonresponse. (eds. R. Groves, D. Dillman, J. Eltinge, and R ..., accessed June 6, 2025, https://pluto.huji.ac.il/~gad/papers/Pfeffermann%20Nathan%202001.pdf
9.	A Partitioned Kalman Filter and Smoother - American Meteorological Society, accessed June 6, 2025, https://journals.ametsoc.org/view/journals/mwre/130/5/1520-0493_2002_130_1370_apkfas_2.0.co_2.pdf
10.	State-Space models in mvgam - CRAN, accessed June 6, 2025, https://cran.r-project.org/web/packages/mvgam/vignettes/trend_formulas.html
11.	(PDF) State-Space Modeling with Correlated Measurements with ..., accessed June 6, 2025, https://www.researchgate.net/publication/241632417_State-Space_Modeling_with_Correlated_Measurements_with_Application_to_Small_Area_Estimation_Under_Benchmark_Constraints
12.	THE KALMAN FILTER AND SURVEYING APPLICATIONS - myGeodesy, accessed June 6, 2025, http://www.mygeodesy.id.au/documents/Kalman%20Filter%20Mildura%20Conference.pdf
13.	An Ensemble Kalman Filter and Smoother for Satellite Data Assimilation - ResearchGate, accessed June 6, 2025, https://www.researchgate.net/publication/227369284_An_Ensemble_Kalman_Filter_and_Smoother_for_Satellite_Data_Assimilation
14.	State space time series modelling of the Dutch Labour Force Survey ..., accessed June 6, 2025, https://www150.statcan.gc.ca/n1/pub/12-001-x/2017001/article/14819-eng.pdf
15.	STATISTICS IN TRANSITION new series and SURVEY METHODOLOGY, accessed June 6, 2025, https://stat.gov.pl/files/gfx/portalinformacyjny/en/defaultstronaopisowa/3432/1/1/sae_2014_sit_16_4_28_i_2016_ok.pdf
16.	Analysis of the extended Kalman filter's role in oceanic science - SPIE Digital Library, accessed June 6, 2025, https://www.spiedigitallibrary.org/conference-proceedings-of-spie/13061/130610K/Analysis-of-the-extended-Kalman-filters-role-in-oceanic-science/10.1117/12.3013854.full
17.	A Combined Dynamic–Kinematic Extended Kalman Filter for Estimating Vehicle Sideslip Angle - MDPI, accessed June 6, 2025, https://www.mdpi.com/2076-3417/15/3/1365
18.	Applications and Optimizations of Kalman Filter and Their Variants - AWS, accessed June 6, 2025, https://intech-files.s3.amazonaws.com/a043Y000010Jz76QAC/0015325_Authors_Book%20%282024-07-25%2010%3A03%3A39%29.pdf
19.	A Combined Dynamic–Kinematic Extended Kalman Filter for Estimating Vehicle Sideslip Angle - ResearchGate, accessed June 6, 2025, https://www.researchgate.net/publication/388476263_A_Combined_Dynamic-Kinematic_Extended_Kalman_Filter_for_Estimating_Vehicle_Sideslip_Angle