ABSTRACT
The state space model has become a powerful tool for time series modeling and forecasting. Such models, in conjunction with the Kalman filter, have been used in a wide range of applications (see Chapter 3). A nonlinear state space model (NLSS) or equivalently a Hidden Markov Model (HMM), keeps the hierarchical structure of the Gaussian linear state space model, but removes the limitations of linearity and Gaussianity. An HMM is a discrete time process https://www.w3.org/1998/Math/MathML"> X t , Y t , t ∈ N https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429112638/a69a2ff5-a02c-4831-8fae-bdc86f3c20b4/content/eq9280.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> , where https://www.w3.org/1998/Math/MathML"> X t , t ∈ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429112638/a69a2ff5-a02c-4831-8fae-bdc86f3c20b4/content/eq9281.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> https://www.w3.org/1998/Math/MathML"> N } https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429112638/a69a2ff5-a02c-4831-8fae-bdc86f3c20b4/content/eq9282.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> is a Markov chain and, conditional on https://www.w3.org/1998/Math/MathML"> X t , t ∈ N , Y t , t ∈ N https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429112638/a69a2ff5-a02c-4831-8fae-bdc86f3c20b4/content/eq9283.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> is a sequence of independent random variables such that the conditional distribution of https://www.w3.org/1998/Math/MathML"> Y t https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429112638/a69a2ff5-a02c-4831-8fae-bdc86f3c20b4/content/eq9284.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> only depends on https://www.w3.org/1998/Math/MathML"> X t . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429112638/a69a2ff5-a02c-4831-8fae-bdc86f3c20b4/content/eq9285.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> We denote by https://www.w3.org/1998/Math/MathML"> ( X , X ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429112638/a69a2ff5-a02c-4831-8fae-bdc86f3c20b4/content/eq9286.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> the state space of the hidden Markov chain https://www.w3.org/1998/Math/MathML"> X t , t ∈ N https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429112638/a69a2ff5-a02c-4831-8fae-bdc86f3c20b4/content/eq9287.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> and by https://www.w3.org/1998/Math/MathML"> ( Y , Y ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429112638/a69a2ff5-a02c-4831-8fae-bdc86f3c20b4/content/eq9288.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> the state space of the observations.
