# How Is the Optimal Number of States in an HMM Determined for Financial Data? ▴ Question

**Published:** 2025-08-05
**Categories:** Question

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## Concept

The determination of the optimal number of states in a [Hidden Markov Model](https://prime.greeks.live/area/hidden-markov-model/?utm_source=options.greeks.live&utm_medium=referral&utm_campaign=satellite) (HMM) for financial data is an architectural decision of the first order. It directly governs the model’s ability to interpret the underlying, unobservable dynamics of the market. This process moves far beyond a simple statistical exercise; it is the foundational act of translating the chaotic surface of price movements into a coherent map of [market regimes](https://prime.greeks.live/area/market-regimes/?utm_source=options.greeks.live&utm_medium=referral&utm_campaign=satellite). The core task is to build a model that accurately reflects the recurring, semi-predictable patterns of market behavior ▴ such as high-volatility, low-volatility, trending, or range-bound periods ▴ without succumbing to the modeler’s delusion of perfect foresight.

An HMM operates on the principle that the [financial data](https://prime.greeks.live/area/financial-data/?utm_source=options.greeks.live&utm_medium=referral&utm_campaign=satellite) we observe ▴ stock returns, volatility measures, trading volumes ▴ are “emissions” from a hidden, underlying state. The market, at any given moment, is in a specific regime (a state), and that regime dictates the statistical properties of what we can see. The challenge lies in the fact that [these states](https://prime.greeks.live/question/what-regulatory-frameworks-govern-counterparty-pre-hedging-in-the-united-states-and-europe/?utm_source=options.greeks.live&utm_medium=referral&utm_campaign=satellite) are unobservable. We cannot simply look at a chart and know with certainty if the market is in a “fear” state or a “complacency” state.

Instead, we must infer the existence and properties of these states from the data they generate. The number of states chosen for the model defines the granularity of this interpretation.

> Choosing the number of HMM states is the critical trade-off between creating a model that is too simple to be useful and one that is too complex to be reliable.

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## The Overfitting and Underfitting Dilemma

The central conflict in this process is a classic [quantitative finance](https://prime.greeks.live/area/quantitative-finance/?utm_source=options.greeks.live&utm_medium=referral&utm_campaign=satellite) problem ▴ the balance between model fit and [model complexity](https://prime.greeks.live/area/model-complexity/?utm_source=options.greeks.live&utm_medium=referral&utm_campaign=satellite). This is a trade-off with significant practical consequences for any trading system built upon the model’s output.

- **Underfitting** ▴ A model with too few states is a blunt instrument. It may, for instance, collapse distinct “bearish consolidation” and “high-risk distribution” phases into a single “negative” state. While simple, this model lacks the resolution to provide a meaningful edge. It fails to capture the subtle but critical shifts in market character that often precede major price movements. The common two-state ‘bull/bear’ model often falls into this category, proving insufficient for capturing the nuanced dynamics of modern markets. 

- **Overfitting** ▴ Conversely, a model with too many states becomes a high-strung, nervous system. It begins to model random noise instead of the underlying signal. A state might be “discovered” that corresponds to a single, anomalous news event from the historical data. This model will have exceptional explanatory power on the data it was trained on, but it will fail spectacularly when deployed on new, live data. Its predictive capacity is compromised because it has memorized the past instead of learning its structure. 

Therefore, the objective is to identify the most parsimonious model ▴ the simplest model that still adequately explains the data’s structure. This is the model with the highest likelihood of generalizing to future, unseen market conditions, providing a stable and reliable foundation for strategic decision-making.

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## Strategy

Strategically selecting the number of states for a financial HMM requires a multi-faceted approach that combines statistical rigor with economic intuition. It is a process of disciplined experimentation, where different model architectures are proposed, tested, and compared on their ability to provide a stable and interpretable map of market regimes. The goal is to move beyond a purely mathematical solution and find a model whose states correspond to real-world, actionable market dynamics.

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## What Are the Dominant Methodologies for State Selection?

Three primary strategic pillars support the determination of the optimal number of states. Relying on a single method is insufficient; a robust conclusion is drawn from the confluence of their results. Each method provides a different lens through which to view the trade-off between model complexity and explanatory power.

- **Information Criteria** ▴ This is the most common quantitative approach. Methods like the [Akaike Information Criterion](https://prime.greeks.live/area/akaike-information-criterion/?utm_source=options.greeks.live&utm_medium=referral&utm_campaign=satellite) (AIC) and the [Bayesian Information Criterion](https://prime.greeks.live/area/bayesian-information-criterion/?utm_source=options.greeks.live&utm_medium=referral&utm_campaign=satellite) (BIC) are used to score how well the model fits the data, while applying a penalty for each additional parameter the model uses. The number of parameters grows rapidly with each new state added. The state with the lowest AIC or BIC score is considered the most effective. BIC’s penalty for complexity is harsher than AIC’s, making it tend to favor simpler models. 

- **Out-of-Sample Validation** ▴ This strategy directly tests the model’s predictive power. A common technique is k-fold cross-validation, where the historical data is partitioned into several segments. The model is trained on some segments and tested on the remaining one, and this process is repeated until each segment has served as the test set. The model complexity that performs best on average across all the out-of-sample tests is preferred. This provides a strong defense against overfitting. 

- **Economic Interpretability and State Stability** ▴ A model that is statistically optimal but economically nonsensical is useless. After fitting models with different numbers of states, a qualitative analysis is required. Do the identified states possess consistent and interpretable characteristics? For example, in a three-state model for an equity index, one might expect to find a low-volatility/positive-return state, a high-volatility/negative-return state, and a low-volatility/neutral-return (ranging) state. If adding a fourth state creates two nearly identical states or a state that appears only fleetingly and randomly, it suggests the simpler three-state model is superior. 

> The optimal model is found at the intersection of statistical evidence, predictive performance, and coherent economic interpretation.

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## Comparative Analysis of Selection Criteria

The choice between AIC and BIC is a strategic one. AIC aims to select the model that best approximates the unknown, true data-generating process. BIC, on the other hand, aims to find the true model among the set of candidates.

For financial data, where a “true” model is unlikely to exist, AIC can sometimes be preferred, but its weaker penalty can lead to overly complex models. BIC’s strong penalty helps to enforce parsimony, which is a desirable trait in models intended for forecasting.

### Model Selection Criteria Comparison

| Criterion | Penalty for Complexity | Tendency | Primary Use Case |
| --- | --- | --- | --- |
| Akaike Information Criterion (AIC) | Lower (2k) | Can select more complex models. | Predictive accuracy and approximating an unknown reality. |
| Bayesian Information Criterion (BIC) | Higher (k ln(n)) | Favors more parsimonious models. | Identifying the most likely “true” model from a set of candidates. |

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## Execution

The execution phase of determining the optimal HMM state count is a systematic, data-driven procedure. It translates the strategic principles of model selection into a concrete, repeatable workflow. This operational playbook ensures that the final model architecture is robust, validated, and fit for purpose within a broader quantitative analysis or trading system.

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## A Step-By-Step Implementation Protocol

Executing this analysis involves a clear, sequential process, moving from raw data to a final, validated model. Each step builds upon the last, ensuring a rigorous and defensible outcome.

- **Data Acquisition and Preparation** ▴ The first step is to define and prepare the observational data. For financial applications, this is typically a time series of daily or intraday returns. It is also common to use volatility proxies, such as the daily high-low range, as a second observable. The data must be cleaned, with missing values handled appropriately, and tested for stationarity, a prerequisite for many time-series models. 

- **Iterative Model Fitting** ▴ The core of the execution is a loop. An HMM is fitted to the prepared data for a range of possible state numbers, for instance, from K=2 to K=8. For each value of K, the [Baum-Welch algorithm](https://prime.greeks.live/area/baum-welch-algorithm/?utm_source=options.greeks.live&utm_medium=referral&utm_campaign=satellite) is typically used to estimate the model parameters ▴ the initial state probabilities, the transition matrix, and the emission probabilities (e.g. the mean and variance of returns for each state). 

- **Quantitative Evaluation** ▴ For each fitted model in the loop, key metrics are computed and stored. The primary metrics are the [log-likelihood](https://prime.greeks.live/area/log-likelihood/?utm_source=options.greeks.live&utm_medium=referral&utm_campaign=satellite) of the model, the AIC, and the BIC. The log-likelihood measures how well the model fits the data (higher is better), while AIC and BIC add penalties for complexity. 

- **Selection via Information Criteria** ▴ The calculated AIC and BIC values are plotted against the number of states (K). The optimal K is typically found at the “elbow” of the plot ▴ the point where adding another state yields a diminishing improvement in the criterion score. This provides the first quantitative indication of the best model. 

- **Qualitative State Analysis and Validation** ▴ With a candidate number of states selected (e.g. K=3), the characteristics of these states must be analyzed. This involves examining the estimated parameters for each state. For example, what is the mean return and volatility associated with State 1, State 2, and State 3? Do these correspond to identifiable market regimes? The stability of these states over time is also assessed. If the model is re-trained on different periods of data, do similar states consistently emerge? 

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## How Is the Final Decision Validated in Practice?

Validation moves beyond simple metrics. A key technique is analyzing the [Viterbi path](https://prime.greeks.live/area/viterbi-path/?utm_source=options.greeks.live&utm_medium=referral&utm_campaign=satellite), which is the most likely sequence of hidden states given the observed data. By plotting this path alongside the original price series, a practitioner can visually inspect whether the model’s regime assignments make intuitive sense. For example, does the model consistently switch to a “high-volatility” state during known periods of market turmoil?

> A statistically optimal model is only valuable if its inferred states provide a coherent and stable narrative of market behavior.

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## Hypothetical Model Selection Output

The following table illustrates the kind of data that would be generated during the execution phase. It provides a clear example of how a quantitative analyst would compare models to arrive at an optimal number of states.

### HMM State Selection Analysis for S&P 500 Daily Returns

| Number of States (K) | Log-Likelihood | AIC | BIC | Qualitative State Interpretation |
| --- | --- | --- | --- | --- |
| 2 | -1250.5 | 2513.0 | 2540.1 | State 1 ▴ Bull (Positive return, low vol). State 2 ▴ Bear (Negative return, high vol). Too simplistic. |
| 3 | -1180.2 | 2384.4 | 2425.7 | State 1 ▴ Bull (Positive return, low vol). State 2 ▴ Bear (Slightly negative return, high vol). State 3 ▴ Ranging (Near-zero return, very low vol). Economically intuitive. |
| 4 | -1175.8 | 2387.6 | 2443.0 | States 3 and 4 are very similar (both low vol, near-zero return). Suggests overfitting. |
| 5 | -1172.1 | 2392.2 | 2461.8 | Creates an unstable, fleeting state that captures only a few outlier days. Clearly overfitting noise. |

In this hypothetical analysis, the three-state model is the clear choice. Both AIC and BIC are minimized at K=3. Furthermore, moving to four states offers minimal improvement in log-likelihood at the cost of increased complexity and less distinct states, a classic sign of overfitting. The three identified states provide a much richer and more actionable description of market dynamics than the simple two-state model. 

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## References

- Gatumel, M. and F. Ielpo. “The Number of Regimes Across Asset Returns ▴ Identification and Economic Value.” 2011.

- Scott, S.L. “Bayesian methods for hidden Markov models ▴ Recursive computing in the 21st century.” Journal of the American Statistical Association, vol. 97, no. 457, 2002, pp. 337-351.

- Hamilton, James D. Time Series Analysis. Princeton University Press, 1994.

- Baum, Leonard E. et al. “A Maximization Technique Occurring in the Statistical Analysis of Probabilistic Functions of Markov Chains.” The Annals of Mathematical Statistics, vol. 41, no. 1, 1970, pp. 164-71.

- Leroux, Brian G. “Maximum-Likelihood Estimation for Hidden Markov Models.” Stochastic Processes and their Applications, vol. 40, no. 1, 1992, pp. 127-43.

- Chen, Z. et al. “Determine the Number of States in Hidden Markov Models via Marginal Likelihood.” arXiv preprint arXiv:1705.08883, 2017.

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## Reflection

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## From Model to Mechanism

The selection of an optimal state count for a [Hidden Markov](https://prime.greeks.live/question/how-can-hidden-markov-models-be-calibrated-for-illiquid-assets/?utm_source=options.greeks.live&utm_medium=referral&utm_campaign=satellite) Model is an act of system design. It imposes a structure on the apparent randomness of financial markets, creating a lens through which to interpret flow and volatility. The process, grounded in statistical criteria and validated by economic intuition, yields more than a predictive tool. It delivers a foundational component of a larger analytical architecture.

With this component in place, the operative question evolves. How does this map of market regimes integrate with execution protocols? How can knowledge of the current, hidden state inform decisions about liquidity sourcing or risk exposure? The HMM itself does not provide answers.

Its value is realized when its output ▴ a probabilistic assessment of the market’s underlying character ▴ becomes an input for a more sophisticated decision-making engine. The true edge is found not in the model itself, but in the design of the complete system that surrounds it.

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## Glossary

<a href="https://prime.greeks.live/area/hidden-markov-model/?utm_source=options.greeks.live&utm_medium=referral&utm_campaign=satellite" class="glossary-item-box">
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## Hidden Markov Model

Meaning ▴ A Hidden Markov Model (HMM) is a statistical framework inferring unobservable system states from observable event sequences. </a><a href="https://prime.greeks.live/area/market-regimes/?utm_source=options.greeks.live&utm_medium=referral&utm_campaign=satellite" class="glossary-item-box">
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## Market Regimes

Meaning ▴ Market Regimes denote distinct periods of market behavior characterized by specific statistical properties of price movements, volatility, correlation, and liquidity, which fundamentally influence optimal trading strategies and risk parameters. </a><a href="https://prime.greeks.live/area/financial-data/?utm_source=options.greeks.live&utm_medium=referral&utm_campaign=satellite" class="glossary-item-box">
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## Financial Data

Meaning ▴ Financial data constitutes structured quantitative and qualitative information reflecting economic activities, market events, and financial instrument attributes, serving as the foundational input for analytical models, algorithmic execution, and comprehensive risk management within institutional digital asset derivatives operations. </a><a href="https://prime.greeks.live/question/what-regulatory-frameworks-govern-counterparty-pre-hedging-in-the-united-states-and-europe/?utm_source=options.greeks.live&utm_medium=referral&utm_campaign=satellite" class="glossary-item-box">
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## These States

US and EU frameworks govern pre-hedging via anti-abuse rules, demanding firms manage information and conflicts systemically. </a><a href="https://prime.greeks.live/area/quantitative-finance/?utm_source=options.greeks.live&utm_medium=referral&utm_campaign=satellite" class="glossary-item-box">
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## Quantitative Finance

Meaning ▴ Quantitative Finance applies advanced mathematical, statistical, and computational methods to financial problems. </a><a href="https://prime.greeks.live/area/model-complexity/?utm_source=options.greeks.live&utm_medium=referral&utm_campaign=satellite" class="glossary-item-box">
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## Model Complexity

Meaning ▴ Model Complexity refers to the number of parameters, the degree of non-linearity, and the overall structural intricacy within a quantitative model, directly influencing its capacity to capture patterns in data versus its propensity to overfit, a critical consideration for robust prediction and valuation in dynamic digital asset markets. </a><a href="https://prime.greeks.live/area/bayesian-information-criterion/?utm_source=options.greeks.live&utm_medium=referral&utm_campaign=satellite" class="glossary-item-box">
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## Bayesian Information Criterion

Meaning ▴ The Bayesian Information Criterion, or BIC, is a statistical metric employed for model selection among a finite set of models. </a><a href="https://prime.greeks.live/area/akaike-information-criterion/?utm_source=options.greeks.live&utm_medium=referral&utm_campaign=satellite" class="glossary-item-box">
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## Akaike Information Criterion

Meaning ▴ The Akaike Information Criterion (AIC) functions as a quantitative measure designed for statistical model selection, offering an estimate of the relative quality of multiple statistical models for a given dataset. </a><a href="https://prime.greeks.live/area/baum-welch-algorithm/?utm_source=options.greeks.live&utm_medium=referral&utm_campaign=satellite" class="glossary-item-box">
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## Baum-Welch Algorithm

Meaning ▴ The Baum-Welch Algorithm is an iterative expectation-maximization procedure designed to compute the maximum likelihood estimates of the parameters of a Hidden Markov Model, or HMM, when the underlying states are unobservable. </a><a href="https://prime.greeks.live/area/log-likelihood/?utm_source=options.greeks.live&utm_medium=referral&utm_campaign=satellite" class="glossary-item-box">
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## Log-Likelihood

Meaning ▴ Log-Likelihood represents a fundamental statistical measure that quantifies the probability of observing a given dataset under the assumptions of a specific statistical model. </a><a href="https://prime.greeks.live/area/viterbi-path/?utm_source=options.greeks.live&utm_medium=referral&utm_campaign=satellite" class="glossary-item-box">
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## Viterbi Path

Meaning ▴ The Viterbi Path identifies the single most probable sequence of hidden states within a Hidden Markov Model (HMM), given a series of observed events. </a><a href="https://prime.greeks.live/question/how-can-hidden-markov-models-be-calibrated-for-illiquid-assets/?utm_source=options.greeks.live&utm_medium=referral&utm_campaign=satellite" class="glossary-item-box">
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## Hidden Markov

Calibrating an HMM for illiquid assets decodes sparse data into a map of hidden liquidity regimes, providing a decisive microstructural edge. </a>

**Tags:** [Market Regime Switching](https://options.greeks.live/area/market-regime-switching/), [Viterbi Path](https://options.greeks.live/area/viterbi-path/), [Financial Data](https://options.greeks.live/area/financial-data/), [Hidden Markov Model](https://options.greeks.live/area/hidden-markov-model/), [Bayesian Information Criterion](https://options.greeks.live/area/bayesian-information-criterion/), [Akaike Information Criterion](https://options.greeks.live/area/akaike-information-criterion/)	

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### [How Can a Smart Trading System Use Volatility Analysis to Enhance Risk Management Protocols?](https://options.greeks.live/question/how-can-a-smart-trading-system-use-volatility-analysis-to-enhance-risk-management-protocols/)
![Abstractly depicting an institutional digital asset derivatives trading system. Intersecting beams symbolize cross-asset strategies and high-fidelity execution pathways, integrating a central, translucent disc representing deep liquidity aggregation. Black modules denote precise RFQ protocols and algorithmic trading components, optimizing market microstructure for principal trading.](https://options.greeks.live/wp-content/uploads/2025/07/institutional-digital-asset-derivatives-ecosystem-unifying-rfq-protocols-for-high-fidelity-execution-and-deep-liquidity-aggregation-across-multi-leg-spreads.jpg)

A smart system uses volatility analysis to dynamically calibrate risk, transforming it from a static threat into a manageable system variable.

- 
### [How Can One Effectively Integrate Domain Knowledge into a Deep Learning Model for Volatility Prediction?](https://options.greeks.live/question/how-can-one-effectively-integrate-domain-knowledge-into-a-deep-learning-model-for-volatility-prediction/)
![A sleek, institutional-grade RFQ engine precisely interfaces with a dark blue sphere, symbolizing a deep latent liquidity pool for digital asset derivatives. This robust connection enables high-fidelity execution and price discovery for Bitcoin Options and multi-leg spread strategies.](https://options.greeks.live/wp-content/uploads/2025/07/institutional-crypto-derivatives-rfq-engine-linking-deep-liquidity-pools.jpg)

Integrating domain knowledge transforms a deep learning model from a pattern recognizer into a market-aware predictive system.

- 
### [What Are the Primary Challenges in Sourcing and Normalizing Data for a Dealer Scoring Model?](https://options.greeks.live/question/what-are-the-primary-challenges-in-sourcing-and-normalizing-data-for-a-dealer-scoring-model/)
![A futuristic, metallic structure with reflective surfaces and a central optical mechanism, symbolizing a robust Prime RFQ for institutional digital asset derivatives. It enables high-fidelity execution of RFQ protocols, optimizing price discovery and liquidity aggregation across diverse liquidity pools with minimal slippage.](https://options.greeks.live/wp-content/uploads/2025/07/precision-engineered-platform-for-institutional-digital-asset-derivatives-rfq-protocol-optimized-execution-management-system-with-robust-liquidity-sourcing.jpg)

A dealer scoring model's efficacy hinges on the systematic transmutation of fragmented, multi-format data into a coherent risk signal.

- 
### [How Can Machine Learning Models Enhance the Effectiveness of an Automated Hedging System?](https://options.greeks.live/question/how-can-machine-learning-models-enhance-the-effectiveness-of-an-automated-hedging-system/)
![A precision-engineered, multi-layered system component, symbolizing the intricate market microstructure of institutional digital asset derivatives. Two distinct probes represent RFQ protocols for price discovery and high-fidelity execution, integrating latent liquidity and pre-trade analytics within a robust Prime RFQ framework, ensuring best execution.](https://options.greeks.live/wp-content/uploads/2025/08/digital-asset-derivatives-rfq-protocols-high-fidelity-execution.jpg)

ML models transform hedging from a static, rule-based process to a dynamic system that learns and adapts to minimize total economic risk.

- 
### [How Do Regime-Switching Models Enhance Algorithmic Trading Performance?](https://options.greeks.live/question/how-do-regime-switching-models-enhance-algorithmic-trading-performance/)
![A precision optical system with a reflective lens embodies the Prime RFQ intelligence layer. Gray and green planes represent divergent RFQ protocols or multi-leg spread strategies for institutional digital asset derivatives, enabling high-fidelity execution and optimal price discovery within complex market microstructure.](https://options.greeks.live/wp-content/uploads/2025/07/prime-os-intelligent-core-driving-digital-asset-derivatives-rfq-protocols-high-fidelity-execution-across-diverse-liquidity-pools-for-optimal-price-discovery.jpg)

Regime-switching models enhance algorithms by identifying hidden market states, enabling dynamic strategy and risk parameter adaptation.

- 
### [What Is the Difference between Calibrating to Stylized Facts versus Transactional Data?](https://options.greeks.live/question/what-is-the-difference-between-calibrating-to-stylized-facts-versus-transactional-data/)
![Abstract visualization of institutional digital asset derivatives. Intersecting planes illustrate 'RFQ protocol' pathways, enabling 'price discovery' within 'market microstructure'. Spheres represent 'digital asset tokens' and 'atomic settlement'. The teal surface signifies 'liquidity pool' and 'volatility surface' supporting 'high-fidelity execution' via 'Prime RFQ'.](https://options.greeks.live/wp-content/uploads/2025/07/digital-asset-rfq-execution-liquidity-pool-market-microstructure-protocol.jpg)

Calibration to stylized facts matches a model to broad statistical patterns, while calibration to transactional data rebuilds market mechanics from raw events.

- 
### [How Does a Leakage Model Adapt to Changing Market Regimes?](https://options.greeks.live/question/how-does-a-leakage-model-adapt-to-changing-market-regimes/)
![Angular, transparent forms in teal, clear, and beige dynamically intersect, embodying a multi-leg spread within an RFQ protocol. This depicts aggregated inquiry for institutional liquidity, enabling precise price discovery and atomic settlement of digital asset derivatives, optimizing market microstructure.](https://options.greeks.live/wp-content/uploads/2025/07/multi-leg-digital-asset-derivatives-rfq-protocol-aggregation-dynamics.jpg)

An adaptive leakage model maintains its detection fidelity by dynamically recalibrating its parameters in response to identified shifts in market behavior.

- 
### [How Can Machine Learning Be Used to Classify Market Regimes for Dynamic Algorithmic Adaptation?](https://options.greeks.live/question/how-can-machine-learning-be-used-to-classify-market-regimes-for-dynamic-algorithmic-adaptation/)
![A sleek, multi-layered digital asset derivatives platform highlights a teal sphere, symbolizing a core liquidity pool or atomic settlement node. The perforated white interface represents an RFQ protocol's aggregated inquiry points for multi-leg spread execution, reflecting precise market microstructure.](https://options.greeks.live/wp-content/uploads/2025/07/institutional-digital-asset-derivatives-rfq-core-execution-system.jpg)

Machine learning classifies market regimes by identifying latent states from data, enabling dynamic algorithmic adaptation.

- 
### [What Are the Primary Quantitative Models Dealers Use to Calibrate Their Daily Risk Exposure?](https://options.greeks.live/question/what-are-the-primary-quantitative-models-dealers-use-to-calibrate-their-daily-risk-exposure/)
![A precision probe, symbolizing Smart Order Routing, penetrates a multi-faceted teal crystal, representing Digital Asset Derivatives multi-leg spreads and volatility surface. Mounted on a Prime RFQ base, it illustrates RFQ protocols for high-fidelity execution within market microstructure.](https://options.greeks.live/wp-content/uploads/2025/07/prime-os-rfq-digital-asset-derivatives-high-fidelity-execution-probe.jpg)

Dealers calibrate daily risk using a system of models like VaR, Expected Shortfall, and Stress Tests to quantify potential losses.

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---

**Original URL:** https://options.greeks.live/question/how-is-the-optimal-number-of-states-in-an-hmm-determined-for-financial-data/
