The US Dollar index (USDX) is an indicator that reflects the price of US Dollar futures contracts against several major world currencies, such as the Euro (EUR), Japanese Yen (JPY), British Pound Sterling (GBP), Canadian Dollar (CAD), Swedish Krona (SEK), and Swiss Franc (CHF). The fluctuations in the value of the US Dollar index can provide clues about the strength of the US economy, Federal Reserve monetary policy, and market sentiment towards the currency. As a crucial variable in international finance, exchange rates have a significant impact on decision-making in international financial markets. For example, the crude oil trade is denominated in the US Dollar. The changes in the US Dollar index have a direct impact on crude oil prices [1]. Initially, futures contracts were commonlyused by hedgers or traders to hedge against the commodities they traded. However,over time, the high volatility in commodity futures markets has been exploited byspeculators, also known as traders, to gain financial profits. Futures contracts in futures exchanges are alternative investment instruments that investors can use togain profits and minimize risks when they receive new information [2]. Informationfrom currency futures markets, including the US Dollar index, can be used to fore-cast exchange rate movements and their implications in economic decision-making [3].

Research on this index is important because exchange rate fluctuations couldinfluence economic activity on global financial markets and international trade ac-tivities [4]. Modelling the US Dollar index is crucial for the financial market, asit provides valuable insights for stakeholders in managing currency risk, makinginformed investment decisions, and planning effective business strategies. By un-derstanding the dynamics and predicting the movement of the US Dollar, marketparticipants can anticipate currency volatility, manage currency risk exposure, andmake smarter asset allocation decisions. The US Dollar index was recently re-searched to see its impact during the Covid-19 crisis [5]. Previous research has alsoshown a relationship between the US Dollar Index and other financial factors, suchas the S&P 500 index [6]. The US Dollar index can be modelled using historicaldata by examining the factors influencing it. One model that can be used is thehidden Markov model (HMM). This model uses hidden states to represent marketconditions. In this model, a sequence of observations is inferred using sequences ofhidden states.

Besides HMM-based approaches, several other statistical and machine learn-ing methods have been applied to model or forecast the US Dollar index. Timeseries models, such as autoregressive integrated moving average (ARIMA) and ge-neralized autoregressive conditional heteroskedasticity (GARCH), have been usedto study future trends formed by the Dollar Index [7]. In addition, various machinelearning methods, such as random forest, support vector machines, and artificialneural networks, have been utilized and compared in predicting the US Dollar Index [8]. Moreover, hybrid models that combine statistical and deep learning methodshave shown improved predictive performance in modelling complex financial indi-cators, such as the US Dollar index [9]. These diverse approaches demonstrate thatvarious modelling frameworks can be utilized to understand and forecast USDXdynamics, depending on the research objective and data characteristics. However,the utilization of trend changes based on probabilistic approaches has not beenextensively explored.

The hidden Markov model has been widely used in the past, such as in mea-suring earning quality [10], sleep monitoring for physical conditions [11], electro-cardiogram (ECG) analysis [12], and seismic analysis [13]. In biochemistry, HMMhas been used to model diploid plant crossing [14]. Another application of HMMincludes researching the progression of dementia [15]. In finance, HMM has beenused to construct portfolios and asset allocation [16]. In [17], HMM has been sys-tematically reviewed in terms of its applications and development in various fields.This approach can also be applied to other historical data, such as US dollar indexdata.

A hidden Markov model is based on a set of unobserved (hidden) states which form a Markov chain and each state is associated with a set of possible observations. Hidden Markov models are categorised into two types: discrete HMMs, which are used for discrete observation sequences, and continuous HMMs, which are used for continuous observation sequences. This study models the data of the US Dollar index using a continuous HMM. This choice is made because the USDX data are continuous-valued time series, making the continuous HMM framework more appropriate for capturing the probabilistic structure of such data. The data used consists of the closing index of the US Dollar index from January 1, 2018, to February 29, 2024.

This paper presents the development of a continuous hidden Markov model (HMM) in detail. The modelling process involves several algorithms whose formulations and implementation steps are also described. The analysis focuses on how the resulting model captures the movement patterns derived from the data and how the best-performing continuous HMM is identified for specific test data ranges. Furthermore, the results and discussion sections provide an in-depth examination of the model's performance and behaviour based on the simulated data. Finally, the paper concludes with a summary of the main findings, an overview of the study's limitations, and potential directions for future research and improvement.

According to [18], a hidden Markov model (HMM) consist of a pair of discrete-time stochastic processes . which has a state space , represents a collection of unobserved states that form a Markov chain. So is hidden behind an observation process . Based on the type of the observations, hidden Markov models are divided into two types, discrete HMMs and continuous HMMs.

For a continuous hidden Markov model, is a continuous random variable and conditional distributions of , for come from a certain family of parametric distributions. A continuous hidden Markov model is characterized by the parameter [14] which has the following properties.

(a) is a transition probability matrix, with , for and , for .

(b) is a probability density function matrix, with is the conditional probability density function of , for .

(c) is an initial probability matrix, with for and .

To estimate the parameters from observations , the forward-backward algorithm, the Viterbi algorithm, and the Baum-Welch algorithm are used [18].

This research uses data obtained from investing.com sites. The data used are daily closing data for the US Dollar index for the period from January 1, 2018, to February 29, 2024. The number of data is 1593. The data is divided into training and testing sets, with 1274 (80%) training data and 319 (20%) testing data. This division is used to compare the accuracy of prediction results over the modelling and testing time range. A graph of the daily closing price can be seen in Figure 1.

Figure 1 shows that daily closing data for the US Dollar index moves fluctuatingly and does not have a tendency to form a particular pattern. The data has a range of 88.170 to 114.047, with a mean of 97.638. The first step in data modelling is to determine the optimal number of hidden states and their corresponding distribution. One method for determining model selection is to utilise the Akaike information criterion (AIC).

The Akaike information criterion can be calculated using the following equation.

where L is the likelihood value of the data and k is called the penalty term [26]. In this research, the penalty term is calculated using the following formula.

with N is the number of states in the Markov chain and p is the number of parameters from the distribution used to create the model. The likelihood of the HMM can be calculated using equation (9). Using equation (17) and (18), the results of the AIC values for several types of continuous distribution and different numbers of states can be seen in Table 1 below.

From Table 1 above, it can be seen that the AIC value get smaller as N increases. We find that minimum AIC value for each state always reached by lognormal distribution. To determine the optimal value for N, we define

where is the AIC value of N-state HMM. The plot of for lognormal distributionscan be seen in Figure 2.

In this research, ∆AICN< 0.03 is set as a benchmark for the optimal valueof N . The Figure 2 shows that ∆AICNtend to decrease as N increase. ∆AICN≤0.03 occurs in state 11 and 12. The AIC value difference in state 11 and 12 is0.033052 or almost zero, so 11 is chosen to be best number of state for HMM withlognormal distribution.

Using the Baum-Welch algorithm for N = 11 and lognormal distribution,the estimated parameters of the hidden Markov model consisting of the transitionprobability matrix between statesˆA are obtained , the probability density function matrixˆθ, and the initial probability vector ˆπ can be calculated. Using Mathematica12.3 software, we obtain the estimated parameterˆλ = (ˆA,ˆθ, ˆπ) as follows.

The matrix shows the transition probability between hidden states. Each state tends to remain in its current state, with a slight chance of moving to a nearby state. Matrix shows the parameter of the lognormal distribution for data generated by each state. The matrix shows that the model started with hidden state 1.

Using the Viterbi algorithm, we can estimate the parameter and determine the sequence of hidden states that generated the data. These hidden states are visualized in 11 different colors. The data and the estimated hidden states are shown in Figure 3 as follows.

Figure 3 shows how hidden state moves from one state to another. This figure also shows range of the values of the data generated by each hidden state. From Figure 3, a certain amount of data is generated a hidden state. To make sure that the data for each hidden state is generated by lognormal distribution, we do the Kolmogorov-Smirnov test with the hypothesis:

: that the data distributed lognormally.

: that the data does not distributed lognormally. For each state, the amount of data, distribution parameters that describes the data, and the p-value result for KS-test can be seen in Table 2 below.

Based on Table 2, it can be seen that the 1274 data is divided into 11 states. The amount of data for each state formed varies. The p-value for each data point for each state is greater than 0.05, so it can be concluded that there is not enough evidence to reject , meaning that all data points for each state come from a lognormal distribution with the parameters that have been obtained. A comparison of the mean and variance between theoretical and actual data is presented in Table 3 below.

Based on Table 3, it can be concluded that the actual data and the estimated distribution have almost the same mean and variance. This also suggests that the distribution can effectively represent the original data. Even though there are differences in mean and variances values in the data range that are too high.

The HMM that has been obtained is then used to generate 1274 simulation data as much as the training data used. Data is generated to find the best seeds to make predictions on daily closing data for the US Dollar index. Simulations to identify the optimal seeds were conducted using Mathematica 12.3 software. To determine the accuracy of the model, this research utilises the mean absolute percentage error (MAPE) to calculate it. The MAPE can be calculated using the formula [27] as follows.

where is actual data at time t, is prediction data at time t, and n is the number of data.

Based on the simulation results, the continuous HMM training yielded a MAPE of 1.39819%. Visualization of the training data and the results of continuous HMM training, with a MAPE of 1.39819% can be seen in Figure 4 below. Figure 4 shows that the model simulation results closely follow the ups and downs of the original data.

Predictions are made on testing data using the HMM that has been estimated from training data. Simulations were carried out using Mathematica 12.3 software. MAPE summary results for various testing data ranges can be seen in Table 4 below.

From Table 4, the MAPE results of simulated test data for data all range prediction times are smaller than . This means that the HMM provides very accurate forecasts. Table 4 also shows that the MAPE gets bigger if the prediction time span longer. Plot of simulated data (prediction) and actual data from continuous HMM with MAPE can be seen in Figure 5 below.

From the Figure 5, it can be seen that the prediction results of the testing data are accurate. There is still error in this prediction, but the difference is not very significant. So we come to the cclusion that the continuous HMM can well predict the up and down of data. In Figure 6, the visualisation of the combined training and testing data is shown, illustrating the movement of the simulated data across both datasets as generated by the continuous HMM model.

Figure 6 illustrates that the continuous Hidden Markov Model effectively captures the primary dynamics of the time series, including both short-term fluctuations and longer-term trends, in both the training and testing datasets. The simulated values closely follow the actual data, indicating good generalisation. However, the model still struggles to detect sudden or extreme price changes caused by rapid market shifts. Overall, these results suggest that the continuous HMM can model complex time-series patterns and provide useful forecasts for financial indicators such as the US Dollar index.

Daily closing data on the US Dollar index from 2018 to 2024 can be modelled using a continuous HMM with 11 hidden states and lognormal distributions with different parameters for each hidden state. MAPE measures the accuracy of the model. The MAPE value for both training and testing data is very low, which is smaller than 4%. This indicates that the continuous HMM can model the data very accurately. The plot shows strong agreement between the simulated and actual data, and the model effectively captures the upward and downward fluctuations of the US Dollar index.

However, this study has several limitations. The model utilises only univariate USDX time-series data and does not incorporate external factors, such as interest rates, inflation, or commodity prices, that may influence index movements. Although the use of 11 hidden states performs well empirically, it may not fully represent actual market regimes and could be improved with a more systematic selection process. Additionally, the assumption of constant parameters within each state may not accurately capture structural changes in global financial conditions.

Future research can address these limitations by incorporating multivariate inputs or relevant macroeconomic indicators. Hybrid models that combine HMM with deep learning methods, such as LSTM, may help capture more complex temporal patterns. Additionally, using dynamic HMMs or regime-switching models with time-varying parameters could better reflect changes in currency market behaviour. These improvements may enhance both the interpretability and predictive accuracy of the model for the US Dollar index and other financial time series.