| Time | Observed symbol | Interpretation example |
|---|---|---|
| t1 | A | Low activity window |
| t2 | B | Medium activity window |
| t3 | A | Low activity returns |
| t4 | C | High activity spike |
| t5 | B | Stabilizing activity |
- π: initial state distribution.
- A[i,j] = P(Sₜ=j | Sₜ₋₁=i): transition probabilities.
- B[i,o] = P(Oₜ=o | Sₜ=i): emission probabilities.
- Predict: ᾱₜ = αₜ₋₁ · A
- Update: αₜ ∝ ᾱₜ ⊙ B[:, Oₜ]
- Normalize each step; log-likelihood accumulates scaling factors.
- Choose the number of hidden states N and output symbols M.
- List M unique symbols and type your observation sequence using them.
- Paste π, A, and B. Each row is normalized automatically.
- Select a forecast horizon k and submit to compute filtered and forecast distributions.
- Use CSV for spreadsheets and PDF for sharing reports.
Operational value of regime forecasting
Hidden Markov forecasting turns noisy sequences into interpretable regime probabilities. In practice, teams track switches between stable and volatile modes, then forecast likely outputs. With k-step projection, the calculator highlights whether the next period is expected to remain in the current regime.
Inputs that control model behavior
Transition rows in A define persistence and switching rates. For example, a 0.90 self-transition implies an average run length near ten steps. Emission rows in B map each hidden state to symbol frequencies, such as “A” dominating the low-activity state while “C” spikes in the high-activity state.
Filtering and likelihood for monitoring
The forward filter updates α with each observation and rescales to prevent underflow. The displayed scaled log-likelihood helps compare model versions on the same sequence; higher values generally indicate a better fit, assuming consistent preprocessing and symbol definitions.
Forecast distributions you can act on
After the final observation, α_T is pushed through A^k to estimate the future hidden state mix. Multiplying by B yields the next-symbol distribution, which is useful for threshold rules, alerting, and inventory-like planning when symbols represent demand levels.
Decoding hidden paths for explanation
Viterbi decoding provides the most likely state path given the full sequence. Analysts use it to annotate timelines, validate regime labels, and explain why a forecast changed after a new observation, especially when state-switch penalties are tuned through A.
Practical quality checks and scaling
Keep symbols mutually exclusive, maintain consistent sampling intervals, and avoid extremely sparse emissions. Start with 2–4 states, then expand only if residual patterns persist. Export CSV for audit trails and PDF for stakeholders, and rerun the model as drift appears in likelihood trends. Additionally teams can review state posteriors weekly to keep decisions aligned with observed behavior and measurable outcomes over time consistently across systems today carefully always Additionally teams can review state posteriors weekly to keep decisions aligned with observed behavior and measurable outcomes over time consistently across systems today carefully always Additionally teams can review state posteriors weekly to keep decisions aligned with observed behavior and measurable outcomes over time consistently across systems today carefully always Additionally teams can review state posteriors weekly to keep decisions aligned with observed behavior and measurable outcomes.
1) What does the forecast steps (k) parameter mean?
k controls how many state transitions you project forward from the last filtered state distribution. Larger k smooths short-term noise and emphasizes long-run transition tendencies.
2) Why are matrix rows normalized automatically?
A and B represent probability distributions, so each row must sum to 1. Automatic normalization prevents accidental invalid models and keeps calculations stable when you paste approximate values.
3) How should I choose the number of hidden states?
Start small: 2–4 states often capture regime shifts without overfitting. Add states only if the decoded path and forecast errors show consistent patterns that one extra regime would explain.
4) What is the scaled log-likelihood used for?
It summarizes how well the model explains the observed sequence, computed safely with scaling. Use it to compare alternative parameter sets on the same data and symbol mapping.
5) Why might Viterbi and filtered states look different?
Filtering uses information up to each time step, while Viterbi considers the entire sequence to find the single best path. The best path can differ from the most probable state at a specific time.
6) Can I use numeric observations instead of letters?
Yes. Use short, unique tokens like 0, 1, 2 or L, M, H. Ensure your observation sequence uses only the listed symbols in the exact same spelling.