## 3 Methodology
### HMM Foundations for Proficiency Assessment
#### 3.1.1 The Hidden Markov Model: A Framework for Latent Proficiency States  
An HMM consists of several integral components: a set of hidden states representing unobservable conditions, an observation space of measurable signals, and various probability distributions detailing transitions and emissions. For instance, in our study, the hidden states may reflect proficiency levels such as Novice, Intermediate, and Expert. The observable outputs include multimodal data consisting of TCT, HRV, and the NASA-TLX. Each of these data types has unique statistical properties that influence their representation in the HMM.  
An HMM is formally defined by the following components:  
* Set of Hidden States \((S=\{s_{1},s_{2},\ldots,s_{N}\})\): These represent \(N\) distinct, unobservable conditions of the underlying system. In the present study, these states correspond to discrete levels of proficiency, such as Novice, Intermediate, and Expert.\" The state at time \(t\) is denoted as \(q_{t}\in S\).
* Set of Observation Space (\(Y=\{y_{1},y_{2},...,y_{m}\}\)): These are the \(M\) distinct, measurable outputs or signals that can be observed at each time step. In our research, observations are vectors comprising multimodal data: TCT, HRV, and NASA-TLX scores. The observation at time \(t\) is denoted \(O_{t}\in Y\). For continuous observations, \(Y\) represents the space from which observations are drawn, and probability density functions are used instead of discrete probabilities.
* Initial State Distribution \((\pi=\pi_{i})\): The initial probability of hidden states at \(t=1\). \[\pi_{i}=P(q_{1}=s_{i}),for\ i=1,\ldots,N,\sum_{i=1}^{N}\pi_{i}=1.\] (1)  
For example, \(\pi_{\text{Novice}}\) in Equation (1) represents the probability that an individual will begin in the novice proficiency state.  
* State Transition Probability Matrix \(\big{(}A=a_{ij}\big{)}\): This \(N\times N\) matrix defines the probability of transitioning from hidden state \(s_{i}\) to \(s_{j}\): \[a_{ij}=P\big{(}q_{t+1}=s_{j}|q_{1}=s_{i}\big{)},for\ i,j=1,\ldots,N,\sum_{j=1} ^{N}a_{ij}=1\ for\ all\ i.\] (2)  
For example, \(a_{Novice\to Intermediate}\) in Equation (2) denotes the probability of transitioning from Novice to Intermediate.  
* Emission Probability Distribution \(\big{(}B=b_{j}(O_{t})\big{)}\) : The probability of observing \(O_{t}\) when in hidden state \(s_{j}\): \[b_{j}(O_{t})=P\big{(}O_{t}|q_{t}=s_{j}\big{)}.\] (3)  
For example, \(b_{Novice}(longTCT,lowHRV,highTLX)\) in Equation (3) represent the likelihood of observing this specific combination of performance metrics if the individual is currently in the Novice proficiency state.  
The definition of \"time\" t within the HMM is a critical design parameter. It can represent discrete task trials, segments within a continuous task, or fixed time windows. This choice significantly influences the interpretation of transition probabilities A; for instance, short time steps might reveal rapid intra-task fluctuations, while longer steps or trial-based steps might reflect slower learning curves or inter-task proficiency shifts.  
The practical application of HMMs often relies on two key assumptions to maintain mathematical tractability and computational feasibility:  
* First-Order Markov Assumption: The current hidden state \(q_{t}\) depends only on the previous state \(q_{t-1}\): \[P(q_{t}|q_{t-1},q_{t-2},\ldots,q_{1})=P(q_{t}|q_{t-1}).\]
* Output Independence Assumption: The current observation \(Q_{t}\) is conditionally independent of all other observations and past hidden states, given the current hidden state \(q_{t}\): \[P(O_{t}|q_{t-1},q_{t-2},\ldots,q_{1},O_{t-1},O_{t-2},\ldots,O_{1})=P(o_{t}|q_{ t}).\]