an approach to time series smoothing and forecasting using the em algorithm pdf

An Approach To Time Series Smoothing And Forecasting Using The Em Algorithm Pdf

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Their variable names and dimensions are as follows. This generic form encapsulates many of the most popular linear time series models see below and is very flexible, allowing estimation with missing observations, forecasting, impulse response functions, and much more.

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Some Applications of the EM Algorithm to Analyzing Incomplete Time Series Data

Their variable names and dimensions are as follows. This generic form encapsulates many of the most popular linear time series models see below and is very flexible, allowing estimation with missing observations, forecasting, impulse response functions, and much more.

An autoregressive model is a good introductory example to putting models in state space form. Recall that an AR 2 model is often written as:. The following are the main estimation classes, which can be accessed through statsmodels. SARIMAX can be used very similarly to tsa models, but works on a wider range of models by adding the estimation of additive and multiplicative seasonal effects, as well as arbitrary trend polynomials.

Behind the scenes, the SARIMAX model creates the design and transition matrices and sometimes some of the other matrices based on the model specification. UnobservedComponentsResults …. For examples of the use of this model, see the example notebook or a notebook on using the unobserved components model to decompose a time series into a trend and cycle or the very brief code snippet below:.

Each of these models has strengths, but in general the DynamicFactorMQ class is recommended. This is because it fits parameters using the Expectation-Maximization EM algorithm, which is more robust and can handle including hundreds of observed series. In addition, it allows customization of which variables load on which factors.

However, it does not yet support including exogenous variables, while DynamicFactor does support that feature. DynamicFactorMQResults …. The DynamicFactor class is suitable for models with a smaller number of observed variables. For an example of the use of the DynamicFactor model, see the Dynamic Factor example notebook. The ExponentialSmoothing class is an implementation of linear exponential smoothing models using a state space approach.

Note : this model is available at sm. ExponentialSmoothing ; it is not the same as the model available at sm. See below for details of the differences between these classes. ExponentialSmoothing endog. ExponentialSmoothingResults …. There are several differences between this model class, available at sm. ExponentialSmoothing , and the model class available at sm. This model class only supports linear exponential smoothing models, while sm.

ExponentialSmoothing also supports multiplicative models. This model class puts the exponential smoothing models into state space form and then applies the Kalman filter to estimate the states, while sm.

ExponentialSmoothing is based on exponential smoothing recursions. In some cases, this can mean that estimating parameters with this model class will be somewhat slower than with sm. This model class can produce confidence intervals for forecasts, based on an assumption of Gaussian errors, while sm. ExponentialSmoothing does not support confidence intervals.

This model class supports concentrating initial values out of the objective function, which can improve performance when there are many initial states to estimate for example when the seasonal periodicity is large.

This model class supports many advanced features available to state space models, such as diagnostics and fixed parameters. The true power of the state space model is to allow the creation and estimation of custom models. Usually that is done by extending the following two classes, which bundle all of state space representation, Kalman filtering, and maximum likelihood fitting functionality for estimation and results output.

For a basic example demonstrating creating and estimating a custom state space model, see the Local Linear Trend example notebook.

In simple cases, the model can be constructed entirely using the MLEModel class. For example, the AR 2 model from above could be constructed and estimated using only the following code:. More advanced usage is possible, including specifying parameter transformations, and specifying names for parameters for a more informative output summary. Apply parameters to the model for example, using fit to construct a results instance.

Interact with the results instance to examine the estimated parameters, explore residual diagnostics, and produce forecasts, simulations, or impulse responses. This is the most commonly used method.

Information criteria, including: aic , aicc , bic , and hquc. It can be useful to compute estimates of the unobserved state vector conditional on the observed data. These are available in the results object states , which contains the following elements:. The estimate of the state vector at time t is based on the observed data up to and including time t. The estimate of the state vector at time t is based on all observed data in the sample. Three diagnostic tests are available after estimation of any statespace model, whether built in or custom, to help assess whether the model conforms to the underlying statistical assumptions.

These tests are:. A number of standard plots of regression residuals are available for the same purpose. There are three methods that can be used to apply estimated parameters from a results object to an updated or different dataset:.

This can be conveniently done using either apply or extend. In the example below, we use the extend method. Statespace model results expose a news method that can be used to understand the impact of data revisions — news — on model parameters. An example of this is:. However, residual diagnostics, in-sample non-dynamic prediction, and out-of-sample forecasting are all still available.

While creation of custom models will almost always be done by extending MLEModel and MLEResults , it can be useful to understand the superstructure behind those classes. Maximum likelihood estimation requires evaluating the likelihood function of the model, and for models in state space form the likelihood function is evaluated as a byproduct of running the Kalman filter. The Representation class is the piece where the state space model representation is defined.

See the class documentation for the full list of available attributes. FrozenRepresentation model. The KalmanFilter class is a subclass of Representation that provides filtering capabilities. Once the state space representation matrices have been constructed, the filter method can be called, producing a FilterResults instance; FilterResults is a subclass of FrozenRepresentation.

The FilterResults class not only holds a frozen representation of the state space model the design, transition, etc. It also provides a predict method, which allows in-sample prediction or out-of-sample forecasting. A similar method, predict , provides additional prediction or forecasting results, including confidence intervals. FilterResults model. The KalmanSmoother class is a subclass of KalmanFilter that provides smoothing capabilities.

Once the state space representation matrices have been constructed, the filter method can be called, producing a SmootherResults instance; SmootherResults is a subclass of FilterResults. The SmootherResults class holds all the output from FilterResults , but also includes smoothing output, including the smoothed state and loglikelihood see the class documentation for the full list of available results. SmootherResults model.

The SimulationSmoother class is a subclass of KalmanSmoother that further provides simulation and simulation smoothing capabilities. The SimulationSmoothResults class has a simulate method, that allows performing simulation smoothing to draw from the joint posterior of the state vector.

This is useful for Bayesian estimation of state space models via Gibbs sampling. SimulationSmoother …. State space representation of a time series process, with Kalman filter and smoother, and with simulation smoother. SimulationSmoothResults …. CFASimulationSmoother model. Transform unconstrained parameters used by the optimizer to constrained parameters used in likelihood evaluation. Transform constrained parameters used in likelihood evaluation to unconstrained parameters used by the optimizer.

Transform unconstrained parameters used by the optimizer to constrained parameters used in likelihood evaluation for a vector autoregression. User Guide. Load the statsmodels api import statsmodels. Variable : y No. L1' : 0.

Some Applications of the EM Algorithm to Analyzing Incomplete Time Series Data

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Data assimilation traditionally refers to the process of quantitatively estimating the state of a time-varying system using all appropriate modeled and measured information available. In geophysical applications, such as in meteorology and oceanography, the primary purpose of data assimilation has been to accurately estimate the flows in the atmosphere and the ocean Ghil and Malanotte-Rizzoli ; Bennett ; Wunsch ; Robinson et al. In these systems, the available information essentially consists of the physical laws that govern the flows, and the indirect, noisy measurements gathered by the sensors observing the system Talagrand ; Kalnay ; Daley In practice, the former is usually available through forecasts and predictions from computational models. Probabilistic frameworks for data assimilation Van Leeuwen and Evensen allow us to naturally combine the information arising from noisy measurements with those given by model predictions and obtain a statistically accurate estimate of the variables of interest. Bayesian filtering and smoothing are two classes of data assimilation problems that differ in their estimation timeline.

One may encounter incompletely specified time series data in several distinct forms: 1 observations in time or space may be irregularly observed or 2 the underlying time series model may be incompletely observed, as in the case where one observes only the sum of a signal and a noise process. Maximum likelihood estimators for parameters in these missing data problems can be developed in a simple, heuristically appealing form by utilizing the EM expectation-maximization algorithm proposed by Dempster, et al. Furthermore, the conditional expectations computed as a by-product of applying the algorithm are the empirical Bayes in the sense of Efron and Morris , estimators for the unobserved components. The EM algorithm is reviewed here within the time series context and applied to i the parameter estimation and smoothing problem for missing data state-space models and ii linear estimation deconvolution in a frequency domain regression model. Unable to display preview. Download preview PDF. Skip to main content.

Some Applications of the EM Algorithm to Analyzing Incomplete Time Series Data

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Learning Nonlinear Dynamical Systems Using an EM Algorithm

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Pinabel L.

An approach to smoothing and forecasting for time series with missing observations is proposed. For an underlying state-space model, the EM algorithm is used in conjunction with the conventional Kalman smoothed estimators to derive a simple recursive procedure for estimating the parameters by maximum likelihood.

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Charlotte P.

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