Estimate Monthly Occurrence-State Markov Transition Probabilities

Estimates month-specific transition probabilities for a three-state precipitation occurrence Markov chain (dry, wet, very wet) from observed daily precipitation data, and maps these probabilities onto a simulated time axis.

Transition probabilities are inferred from observed lag-1 to lag-0 precipitation pairs using month-dependent wet and extreme thresholds. A Dirichlet-type smoothing is applied to avoid zero-probability transitions, with the effective smoothing strength decreasing automatically with the number of observed transitions per month.

The function supports both calendar-year and water-year simulations. The regime is inferred from month_order: if month_order[1] == 1, calendar-year logic is used; otherwise, water-year logic is assumed.

Optional spell-persistence modifiers are applied to dry and wet states to influence transition persistence. After these adjustments, transition probabilities are explicitly normalized to ensure stochastic validity (non-negativity and rows summing to one).

The estimated monthly transition probabilities are assigned to all simulated days that fall within the target simulation year and corresponding calendar month.

Usage

estimate_monthly_markov_probs(
  precip_lag0,
  precip_lag1,
  month_lag0,
  month_lag1,
  year_lag0 = NULL,
  year_lag1 = NULL,
  wet_threshold,
  extreme_threshold,
  month_order,
  sim_month,
  sim_wyear,
  year_idx,
  sim_start_year,
  dry_spell_factor_month,
  wet_spell_factor_month,
  n_days_sim,
  dirichlet_alpha = 1
)

Arguments

precip_lag0

Numeric vector. Observed daily precipitation at lag 0 (current day) used to estimate state transitions.

precip_lag1

Numeric vector. Observed daily precipitation at lag 1 (previous day) used to estimate state transitions.

month_lag0

Integer vector (1-12). Calendar month corresponding to precip_lag0.

month_lag1

Integer vector (1-12). Calendar month corresponding to precip_lag1.

year_lag0

Optional integer vector. Calendar year or water year corresponding to precip_lag0. If provided together with year_lag1, transitions are restricted to same-year pairs.

year_lag1

Optional integer vector. Calendar year or water year corresponding to precip_lag1.

wet_threshold

Numeric vector of length 12. Monthly precipitation thresholds separating dry and wet states, aligned to month_order.

extreme_threshold

Numeric vector of length 12. Monthly precipitation thresholds separating wet and very wet states, aligned to month_order.

month_order

Integer vector of length 12 defining the simulated ordering of months (for example 1:12 for calendar years or c(10:12, 1:9) for October-start water years).

sim_month

Integer vector. Calendar month for each simulated day.

sim_wyear

Integer vector. Simulation year (calendar or water year, depending on regime) for each simulated day.

year_idx

Integer. Index of the current simulated year (1-based).

sim_start_year

Integer. First year of the simulation, interpreted as a calendar year or water year depending on the simulation regime.

dry_spell_factor_month

Numeric vector of length 12. Monthly multiplicative factors controlling dry-state persistence.

wet_spell_factor_month

Numeric vector of length 12. Monthly multiplicative factors controlling wet-state persistence.

n_days_sim

Integer. Total number of simulated days. Determines the length of the returned probability vectors.

dirichlet_alpha

Numeric. Base smoothing strength for transition-probability estimation. The effective smoothing applied in each month is dirichlet_alpha / sqrt(n_transitions_m), where n_transitions_m is the number of observed transitions in that month.

Value

A named list of nine numeric vectors, each of length n_days_sim, representing time-varying transition probabilities:

• p00_final: P(dry to dry)

• p01_final: P(dry to wet)

• p02_final: P(dry to very wet)

• p10_final: P(wet to dry)

• p11_final: P(wet to wet)

• p12_final: P(wet to very wet)

• p20_final: P(very wet to dry)

• p21_final: P(very wet to wet)

• p22_final: P(very wet to very wet)

Details

Precipitation occurrence states are defined as:

Dry

Precipitation less than or equal to the monthly wet threshold

Wet

Precipitation greater than the wet threshold and less than or equal to the extreme threshold

Very wet

Precipitation greater than the extreme threshold

All transition-probability rows are explicitly normalized after smoothing and spell adjustments to ensure valid Markov-chain behavior.