murphet 1.4.1

Bayesian time‑series model with Beta/Gaussian heads, smooth changepoints and seasonality

📈 Murphet — Making Prophet a Beta (Pun Intended!)

A Stan-powered time-series model that never breaks the 0-1 bounds while delivering more accurate forecasts than Prophet for rates and probabilities.

Your conversion rates deserve better than to be treated like stock prices!

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🚀 Prophet, but Better for Bounded Data

Prophet is great, but it wasn't built for churn rates, conversion percentages, or hotel occupancy. Murphet was.

When forecasting rates, the last thing you want is predictions that break the logical 0-1 bounds. But that's exactly what happens with vanilla Prophet when forecasting:

• Conversion rates
• Churn probabilities
• Occupancy percentages
• Click-through rates
• Any metric bounded between 0 and 1

Murphet fixes this with a Beta likelihood that respects these natural boundaries while providing more accurate forecasts. It's Prophet's smarter cousin - specifically designed for the data you're actually working with.

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1 · Why Choose Murphet?

Prophet's Limitation	Murphet's Solution	Your Benefit
❌ Predictions can go <0 or >1	✅ Beta head keeps everything in (0,1)	Never explain impossible forecasts to stakeholders again
❌ One-size-fits-all variance model	✅ Heteroscedastic precision adapts to data level	Better uncertainty intervals, especially near boundaries
❌ Hard changepoints create artificial kinks	✅ Smooth logistic transitions between trends	More realistic forecasts with less overfitting
❌ Seasonality coefficients often explode	✅ Regularized Fourier terms with sensible priors	Stable seasonal patterns even with limited data
❌ Ignores autocorrelation in residuals	✅ Latent AR(1) structure captures persistent patterns	Dramatically improved forecast accuracy
❌ Same error structure for all predictions	✅ Data-adaptive variance via smart link functions	Properly calibrated prediction intervals

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2 · Installation in Seconds

pip install murphet              # wheels include pre-compiled Stan models

Requirements

• Python >= 3.8 & CmdStanPy >= 1.0 (auto-installed)
• A recent CmdStan toolchain (gcc/clang) — cmdstanpy.install_cmdstan() will fetch it.

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3 · Quick Start (Just 10 Lines!)

import pandas as pd, numpy as np
from murphet import fit_churn_model

df = pd.read_csv("churn_data.csv")        # cols: ds, y  (0<y<1)
df["ds"] = pd.to_datetime(df["ds"])
df["t"]  = np.arange(len(df))             # integer index

mod = fit_churn_model(
        t              = df["t"],
        y              = df["y"],
        periods        = 12, num_harmonics = 3,    # yearly seasonality
        n_changepoints = 4,
        likelihood     = "beta",                   # default & safest
        inference      = "nuts",                   # full posterior
      )

future_t = np.arange(len(df), len(df)+6)
fcst     = mod.predict(future_t)

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4 · Model Architecture

Murphet combines the best of structural time series modeling with modern Bayesian methods:

Component	Equation	What It Gives You
Trend	μdet(t) = k·t + m + ∑ δj σ(γ (t − sj))	Smooth transitions between trend regimes
Seasonality	Fourier blocks on raw t (fmod)	Flexible multi-periodic patterns
Link / saturation	μ → logit⁻¹ → p	Automatic boundary adherence
Likelihoods	Beta(p·φᵢ,(1-p)·φᵢ) or Student-tν(μ,σᵢ)	Properly scaled uncertainty
Latent error	y* = μdet + ρ·lag	Capture autocorrelated patterns

Advanced Features

Feature	Implementation	Business Impact
AR(1) latent error	real<lower=-1,upper=1> rho; real mu0; + update in partial_sum_*	Catch slow-moving market trends
Heteroscedastic precision	phi_i = exp(log_phi0 - beta_phi*abs(mu_det)); (Beta) / sigma_i = exp(log_sigma0 + beta_sigma*abs(mu_det)); (Gaussian)	More accurate risk assessment
Heavy-tail option	student_t_lpdf(y \ ν, μ, σᵢ) with ν ~ Exp(1/30)	Robustness to outliers and shocks

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5 · Real-World Performance

Hong Kong Hotel Occupancy Rates (2020-2025)

9-Month Forecast	RMSE	Improvement
Murphet β	0.0916	21% better
Prophet (optimized)	0.1159	Baseline

U.S. Retail Inventories-to-Sales Ratio

24-Month Forecast	RMSE	SMAPE	Improvement
Murphet β	0.0496	5.15%	56% better
Prophet	0.1140	13.21%	Baseline

Residual Diagnostics

Murphet's AR(1) and heteroscedastic components virtually eliminate autocorrelation structures that Prophet leaves behind.

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6 · Choosing the Right Model Head

Likelihood	Best For	Technical Details	Examples
Beta (default)	True proportions, rates, percentages	Logit⁻¹ link + Beta(p·φᵢ,(1-p)·φᵢ) likelihood	Conversion rates, CTR, churn %
Gaussian / Student-t	Approximate ratios or unbounded metrics	Identity link + Normal/Student-t(μ,σᵢ)	Price ratios, normalized KPIs

Simply set likelihood="gaussian" to switch; all other API calls remain identical.

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7 · API Reference

Function	Purpose	Example
fit_churn_model(t, y, **kwargs)	Fit the model using MAP, ADVI, or NUTS	See quickstart
model.predict(t_new)	Generate forecasts	forecast = model.predict(future_t)
model.fit_result	Access raw CmdStanPy object	draws = model.fit_result.stan_variable("rho")
model.summary()	Get parameter summary	print(model.summary())

Key Parameters

# Seasonality configuration
periods            # Length of seasonal periods (e.g., 12 for monthly data with yearly seasonality)
num_harmonics      # Number of Fourier terms per period (higher = more flexible seasonality)
season_scale       # Prior scale for seasonal components (0.3-2.0 recommended)

# Trend configuration
n_changepoints     # Number of potential trend changes (Prophet default heuristic = 0.2 * N)
delta_scale        # Prior scale for changepoint magnitudes (0.01-0.6 range)
gamma_scale        # Steepness of changepoint transitions (1.0-10.0 range)

# Inference options
likelihood         # "beta" (default) or "gaussian"
inference          # "map" (fastest), "advi" (quick uncertainty), or "nuts" (most accurate)

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8 · Coming Soon

• Holiday regressors (Prophet style)
• Automatic plotting functionality
• Performance optimizations for long MCMC chains
• Integration with Prophet ecosystem tools

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9 · For Academic Use

If you use Murphet in academic work, please cite:

Murphy, S. (2025). Murphet: A Bayesian Time-Series Model for Bounded Rates.
https://github.com/halsted312/murphet

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10 · Get Started Today

Don't let your bounded metrics be forecasted with unbounded models. Murphet brings the power of Bayesian modeling to your rates and proportions with an easy-to-use API that feels just like Prophet.

# It's as simple as:
from murphet import fit_churn_model

model = fit_churn_model(t=time_index, y=bounded_values)
forecast = model.predict(future_time_points)

Check out the documentation →