Differences between hidden Markov and state-space models
The Hidden Markov Model (HMM) described in “Overview of Hidden Markov Models and Examples of Applications and Implementations” and the State Space Model (SSM) described in “Overview of State Space Models and Examples of Implementations for Analysing Time Series Data Using R and Python” are statistical models used for modelling time-varying and serial data, but with different approaches. Model, SSM) are statistical models used for modelling temporal changes and series data, but with different approaches. The main differences between them are discussed below.
First, in terms of the purpose of each model, hidden Markov models are used to estimate the hidden state behind the observed series data, whereas only the directly observed ‘observed state’ is available from the observed data and the aim is to infer the ‘hidden state’ behind it, whereas state spatial models aim to simultaneously model the observed data and the continuous ongoing process behind it, called the ‘state’, and there shall be a probabilistic relationship between the observed data and the state.
Looking at this from the perspective of a concrete example, HMMs take a sequence of words in text data as observation data and estimate the context (hidden state) to which each word belongs, whereas SSMs take a physical phenomenon and estimate position and velocity (state) simultaneously from sensor data, for example, by considering the relationship between the observed data and the underlying The problem can be thought of as looking at a continuous ongoing process called a ‘state’ behind the observed data, such as the problem of simultaneously estimating position and velocity (state) from sensor data in a physical phenomenon.
Also, with respect to the state space to be estimated, in HMMs, the hidden states take discrete values, the hidden state at each point in time is chosen from some finite state space, the hidden state is Markovian and the current hidden state depends on the previous hidden state, whereas in SSMs, the hidden state has continuous values and is a continuous time and physical quantities (e.g. position, velocity, temperature) that change in space, the state space has a continuous space and changes in time are usually represented by a continuous model (e.g. differential equations).
Also, for the relationship between observed data and hidden states, in HMMs, the observed data are generated according to a probability distribution generated from the hidden states, and the specific hidden states themselves are not directly observed, so parameter estimation is performed using an EM algorithm or similar, whereas in SSMs, the observed data are generated from the state In SSM, the observation data is generated according to a probability distribution generated from the state, and the relationship between the state and the observation data is expressed using an observation equation (measurement equation), and the parameters of the observation equation are estimated at the same time as the state estimation.
In simple terms, HMMs are mainly aimed at estimating hidden states and the states are discrete, while SSMs are mainly aimed at simultaneous modelling of states and observed data and the states are continuous.
The difference between the algorithms used in hidden Markov models and those used in state space models
Due to these differences, different algorithms are mainly used in HHM and SSM. They are described below.
Algorithms used for Hidden Markov Models (HMMs):
- Forward Algorithm: the forward algorithm is used to calculate the probability of obtaining the observed data for a given HMM and observed data. The algorithm is based on Dynamic Programming and can be applied to other HMM-related problems such as posterior probability and hidden state estimation.
- Backward Algorithm: the backward algorithm is used to calculate the probability of obtaining an observation for a given HMM and observation data. Like the forward algorithm, it can also be applied to estimate posterior probabilities and hidden states.
- Viterbi algorithm: the Viterbi algorithm is used to find the best estimate of the hidden state series when the observed data are available. The algorithm is based on dynamic programming and can efficiently find the optimal hidden state series.
Algorithms used for State Space Models (SSM):
- Kalman Filter: the Kalman filter is an algorithm used to simultaneously estimate both the observed data and the state, optimised for linear Gaussian models and applied to estimate the state and the parameters of the observed equations.
- Particle Filter: the particle filter is an algorithm used for non-linear state-space models, which can be applied to models with non-linear dynamics and non-linear observation equations. Particle filters are based on Monte Carlo simulations and allow the estimation of non-linear relationships.
Case studies of the application of hidden Markov models and state-space models
With such differences, HHM and SSM have their own areas of application in which they excel. They are discussed below
Hidden Markov Models (HMMs) application examples:
1. speech recognition: used to decompose the speech signal into phonemes and to estimate the state of each phoneme. For example, it is used to analyse speech sounds and convert them into words or phrases.
2. natural language processing: used to extract information in text, such as part-of-speech tagging and nomenclature recognition. For example, it is used to determine whether ‘I’ is a noun in a sentence such as ‘I go to the park’.
3. financial market analysis: used to model fluctuations in time-series data, such as stock prices and exchange rates, to predict future prices and manage risk. For example, it is used to capture patterns of stock price rises and falls.
4. bioinformatics: used to analyse DNA and protein sequences and used to predict gene structure and function. For example, it can help identify the location of exons and introns in a gene.
Examples of applications of state-space models:
1. robotics: used to model the behaviour and position of robots and to estimate their state based on data from sensors, e.g. to control robots when they become aware of their surroundings and operate autonomously.
2. economic forecasting: used to predict economic indicators and market trends, e.g. to forecast GDP growth and unemployment rates.
3. weather forecasting: used to predict future weather using meteorological data, e.g. for modelling temperature and precipitation variations for use in weather forecasting.
4. system control: used to design and optimise control systems, used to model and control the operation of machines and processes, e.g. used to control cruise control systems in cars and manufacturing processes.
Example implementations of hidden Markov models and state space models
Examples of HMM and SSM implementations are described below.
Example implementation of a Hidden Markov Model (HMM):
This example is an implementation of an HMM using Python’s hmmlearn library. In this example, a simple HMM is used to generate sequences using hidden and observed states.
import numpy as np
from hmmlearn import hmm
# Define the parameters of the model
n_states = 2
n_observations = 3
# Create an instance of HMM (using Gaussian HMM)
model = hmm.MultinomialHMM(n_components=n_states, random_state=42)
# state-transition matrix
model.transmat_ = np.array([[0.7, 0.3], [0.4, 0.6]])
# Output probability of each state.
model.emissionprob_ = np.array([[0.2, 0.4, 0.4], [0.5, 0.4, 0.1]])
# initial state probability
model.startprob_ = np.array([0.6, 0.4])
# Generate data.
X, Z = model.sample(100)
print("observed data:")
print(X)
print("hidden state:")
print(Z)
# Calculate data logging probabilities.
logprob = model.score(X)
print("log probability:", logprob)Example implementation of a state space model:
This is an example of implementing a state-space model using Python’s pydlm library. In this example, the state-space model is used to fit time series data.
import numpy as np
import matplotlib.pyplot as plt
from pydlm import dlm, trend, season
# Generate time series data.
np.random.seed(42)
n = 100
time = np.arange(n)
data = np.sin(2 * np.pi * time / 12) + np.random.normal(size=n)
# Setting up a state-space model.
model = dlm(data) + trend(degree=2, discount=0.98) + season(period=12)
# Fitting models.
model.fit()
# Get fitting results.
fitted_values = model.predictN(N=n)
# Plotting the results
plt.figure(figsize=(10, 6))
plt.plot(time, data, 'k.', label='observed data')
plt.plot(time, fitted_values, 'r-', label='Fitting results')
plt.legend()
plt.xlabel('Hours.')
plt.ylabel('value')
plt.title('Fitting of state-space models.')
plt.show()The respective libraries and tools are as follows
HMM: a Python library for working with Hidden Markov Models (HMMs)
- hmmlearn: Python library for working with hidden Markov models.
- pomegranate: a more sophisticated HMM library.
State space models:
- pydlm: a Python library for easy handling of state-space models.
- statsmodels: provides a wider range of statistical models and also supports state-space models.
Challenges and remedies for hidden Markov models and challenges and remedies for state space models
HMMs and SSMs have their own specific challenges. The following section describes those challenges and countermeasures.
Hidden Markov Models (HMMs) challenges and remedies:
1. difficulty in parameter estimation:
– Challenges: the EM algorithm (Expectation-Maximisation) is usually used to estimate the parameters of HMMs (transition probabilities, output probabilities and initial state probabilities), but it can lead to locally optimal solutions.
– Solution: to improve parameter estimation, the EM algorithm can be run with several initial conditions and the best results selected. Alternatively, Bayesian estimation or meta-heuristic algorithms (e.g. genetic algorithms described in “Overview of genetic algorithms, application examples, and implementation examples“) can be considered for parameter optimisation.
2. model complexity:
– Challenges: as the number of hidden states increases, the model becomes more complex, which can lead to over-training and increased computational costs.
– Solution: use cross-validation and model selection criteria (e.g. AIC, BIC) to select an appropriate number of hidden states. Regularisation techniques to improve the quantity and quality of data and prevent over-learning are also useful approaches.
3. modelling long-term dependencies:
– Challenge: HMMs basically assume Markovianity, making it difficult to capture long-term dependencies.
– Solution: to capture long-term dependencies, extended models using conditional probabilities or recurrent neural networks (RNNs) such as LSTM (Long Short-Term Memory) can be used.
Challenges and remedies for state-space models:
1. difficulties in specifying models:
– Challenges: state-space models involve the selection of state transitions and observation equations, but it can be difficult to specify the appropriate model for the actual data.
– Solution: use Bayesian approaches and model selection criteria (e.g. cross-validation) to assist in model selection. Prior knowledge and domain knowledge can also help in model specification.
2. computational complexity:
– Challenge: state-space models are computationally expensive, especially for large data sets and complex models.
– Solution: approximation algorithms (e.g. Kalman filter, particle filter) can be used to reduce the computational burden. Efficient numerical libraries and parallel processing techniques can also be used.
3. non-linearity and non-Gaussianity:
– Challenge: state-space models often assume linearity and Gaussianity, but real-world data may contain non-linearity and non-Gaussianity.
– Solution: non-linear state-space models, Gaussian processes and generalised state-space models (e.g. extended Kalman filters, particle filters) can be used to handle non-linearity and non-Gaussianity.
Reference Information and Reference Books
Time series data analysis is described in “Time Series Data Analysis.
Reference book is “
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Stochastic Graphical Models: Principles and Algorithms
Pattern Recognition and Machine Learning
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