the data mining homework solution

the data mining homework solution

Exploring Hidden Markov Models: A Comprehensive Guide

1. Introduction to Hidden Markov Models

The world is becoming more and more obsessed with data. Not a day goes by without new data being created or released for public consumption. However, regardless of the increase in data generation or availability, the fundamental systems around us pay no heed to the data. As such, the role of a data scientist is as important as ever. “Data science means transforming business data…”. Companies invest time and resources in the collection and storage of data because they believe it will help them solve a problem or gain insight. This means in order to maximize the value of data, there is a need to “transform data into more useful resources – e.g., ventures, leagues, and other financial opportunities.” Unlike ever before, more and more companies now look upon data scientists to deal with their data, analyze it, and help predict what comes next. Doing all these means building models.

One type of model quite often used in the area of Probabilistic Graphical Models (PGM) for solving sequence-related problems is the Hidden Markov Model (HMM). The Hidden Markov Model is an essential factor for the domain and is used in robotics, econometrics, medicine, bioinformatics, music instrument recognition, speech recognition and segmentation, natural language processing, part-of-speech tagging, spell checking, machine translation, gene prediction, video and image analysis, financial engineering, environment monitoring, time series prediction, and many other areas with lots of complexity in its model. Simply put, the Hidden Markov Model is a flexible and well-established practical tool. However, a strong full understanding of the HMM is often rare and hard to find. Thus, many just know about the Viterbi or the forward algorithms, despite them being the basics. Furthermore, the explosion of deep learning approaches in AI has made the situation worse and has also led to many researchers knowing only a few HMM algorithms. This is a bit of a problem because there are still several complex solving HMM paradigms and applications that require a fuller understanding. This is why in this work, we will explain and deconstruct the Hidden Markov Model by discussing its origins, explaining its structure, model parameters and formulations, explaining re-estimation of the model parameters using the Baum-Welch algorithm, applying the observed sequence training procedures, explaining the three Base-optimality criteria, discussing the fundamental HMM algorithms, taking a closer look at the problems with their computation algorithms and parallelization, and discussing several Hidden Markov Model disciplines.

2. Fundamental Concepts and Components

2.1 Different types of models

While there are numerous models that could be used (with GARCH being one of the most famous), many of these are effectively ad hoc. That’s not to say that stock market returns display time-dependent properties. It is just that many statistical tools offer inefficient utilization of degrees of freedom present in the data we have on returns.

In our modeling of returns, we use both autoregression and a hidden Markov model to capture these different temporal properties. At a high level, various types of models are listed below:

(A) Autoregression models: These models are the most effective in capturing the autocorrelation of returns. By and large, the predictive power of year-on-year returns of stocks is best modeled using a single lag. The success of using autoregression models in picking up existing autocorrelation is one of the most widely documented phenomena in finance. In general, the predictability of returns tails off quite quickly. A moving average model imparts flexibility to our model, helping us to capture autocorrelation.

(B) Models with intrinsic structure: Utilization of knowledge about the stock market data generating process for return predictions. Economists sometimes claim they can do this, but it’s hard to detect. To our knowledge, the only configuration-based model that is empirically supported is the dividend discount model. Indeed, living through the effects that dividend payments have on returns is often the best way to believe that this structural intuition has predictive power.

3. Learning and Inference Algorithms

In principle, there are several types of learning and inference algorithms involved with Hidden Markov Models, among which the most commonly used are the Baum-Welch algorithm (EM algorithms) and the Viterbi algorithm. In this section, we will introduce the forward-backward algorithm first, which will provide us with a useful tool to build the Baum-Welch algorithm and then the Viterbi algorithm. After learning these algorithms, we will continue with another important step involved with HMM – comparing different HMMs – based on their algorithm products.

3.1 Forward-Backward Algorithm Now that we know that the underlying process of observation data is a forward jumping very special Markov Process, it is essential to learn an algorithm that can backward fit the model parameter back to the observations. The forward-backward algorithm provides exactly such a capability. Based on the chain rule, it is relatively straightforward to calculate the mode, i.e. selecting the observation sequence which is most likely being generated by the model given the model parameters can be translated into a modeling fitting procedure as model fitting is complementary to model parameter estimation. In addition to that, the indirect calculation requires significantly less time and space than the direct calculation, because we introduced the term of dynamic programming, the field in which both the forward-backward and the Viterbi algorithms are recognized as important methodologies. On the other hand, the forward-backward algorithm helps answer another important question raised by the Baum-Welch algorithm.

4. Applications in Various Fields

Similar to all other statistical modelling and probabilistic time series models, the capability of HMMs can be extended into a wide range of problems and domains. Here we provide some examples of HMMs used in various fields such as speech recognition, computer vision, computational biology, and finance.

Speech recognition is a problem of identifying a sequence of words that are being spoken from a source of discrete sound patterns. In the underlying process, there is no one-to-one correspondence between the output and the underlying sequences of words (latent states), and the number of possible words a listener uses is extremely large. Using all possible letter combinations or syllables in a language has led to the use of languages that can be designed specifically for the problem.

Speaker-independent isolated-word recognition is formulated as a Hidden Markov Model.

Computer Vision: The most well-known application of the Hidden Markov model (HMM) in computer vision problems should be object posture recognition, where the spatial and temporal variability of the observation of an object is considered.

Computational Biology: The most well-known application of the Hidden Markov model in computational molecular biology should be gene finding, where the gene structure is modelled probabilistically and the problems of finding a model that fits observed data well or of deciding whether the observed data are consistent with a hypothesized model is formulated.

Implications of HMMs in physical systems are unknown to us, but we can imagine their implications as the traditional continuous-time signals processing techniques being extended to allow for, model and exploit statistical and functional relationships in these time-sequential signals.

Finance: The HMM has been applied to finance in various ways such as stock-price adjustments, currency exchange adjustments, job search process models of job finding, and so forth.

5. Advanced Topics and Future Directions

In this chapter, I explore advanced topics in HMMs, their applications, and still up-and-coming areas. The first section concerns continuous HMMs; these come with a jagged history and are not archived in a state-of-the-art fashion. The next part addresses the dreaded predators, problems associated with HMMs. Since slight variations have their own names, the word Spell Checker, our spell checker, will undo its usual task. Alphabet translation is a subfield of bioinformatics. From understanding genes to deducing protein sequences to finally finding gene function – all of these tasks require some sort of alphabet translation. Sometimes a direct translation is possible. However, the result may not match our intuition – meaning the result may not satisfy the constraints due to the biological context. In addition, translation tables must be learned. The results of these learning procedures must validate the translation. All these steps may require HMMs.