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mcp: Multiple Change Point Regression in R

Source: R/mcp-package.R
mcp-package.Rd

Flexible and informed regression with Multiple Change Points. mcp can infer change points in models on means, variances (and other distributional parameters), autocorrelation structure, and any combination of these, as well as the parameters of the segments in between. All parameters are estimated with uncertainty, and posterior predictive intervals are supported - also near the change points. mcp supports hypothesis testing via Savage-Dickey density ratios, posterior contrasts, and PSIS-LOO/WAIC model comparison.

Details

The mcp model

An mcp model divides a continuous predictor \(x\) into \(K\) segments separated by ordered change points \(\Delta_1 < \dots < \Delta_{K-1}\). In each segment \(k \in \{1, \dots, K\}\), the linear predictor \(\eta_i\) is evaluated directly from the segment-local distance \((x_i - \Delta_{k-1})\):

$$\eta_i = \alpha_k + \beta_{k,1} (x_i - \Delta_{k-1}) \quad (\text{with } \Delta_0 = 0)$$

where the segment-start level \(\alpha_k\) is freely estimated for the first and disjoined segments, as in non-segmented regression, and inherited continuously for joined segments:

$$\alpha_k = \begin{cases} \beta_{k,0}, & \text{Disjoined segments } (\sim \texttt{1 + x}, \text{ including } k = 1) \\ \alpha_{k-1} + \beta_{k-1,1} (\Delta_{k-1} - \Delta_{k-2}), & \text{Joined segments } (k \ge 2, \sim \texttt{0 + x}) \end{cases}$$

In all segments, estimated slope and intercept parameters are absolute values (not changes relative to the preceding segment).

If additional continuous covariates or categorical factors are included (e.g., + z + group), they enter additively on their original scale (\(\dots + \sum \gamma_{k,j} z_{j,i}\)); only the change-point predictor \(x\) is converted to segment-local coordinates.

Distributional parameters (sigma(), shape(), etc.) and autoregressive terms (ar(), ma()) follow this exact same segmented structure on their respective link scales.

Extended formulas and model features

  • Distributional regression: Model residual variance and other distributional parameters across segments, e.g., ~ sigma(1 + x) or ~ shape(1).

  • Time-series residuals: Model serial dependence using ar(p) and ma(q) terms with a generalized link-scale recurrence that spans continuously across change points.

  • Group-level effects: Add hierarchical (random) intercepts, slopes, and change points, e.g., ~ 1 + (1|id) or 1 + (1|id) ~ 0 + x.

  • Model comparison and hypothesis testing: Compare models using PSIS-LOO (loo()) or WAIC (waic()), and test directional or point-null hypotheses (hypothesis()) using Savage-Dickey density ratios.

See the mcp website for worked examples and vignettes.

References

  • Lindeløv, J. K. (2020). mcp: An R Package for Regression With Multiple Change Points. OSF Preprints. doi:10.31219/osf.io/fzqxv Introduces the mcp package, formula syntax, default priors, and workflow for regression with multiple change points across generalized linear and time-series models. Newer-than-2020 versions of the paper may be available at that link. Please cite the newest version.

  • Carlin, B. P., Gelfand, A. E., & Smith, A. F. (1992). Hierarchical Bayesian Analysis of Changepoint Problems. Applied Statistics, 41(2), 389–405. doi:10.2307/2347570 Introduced the Gibbs sampling approach for continuous change points in segmented regression and hierarchical models, providing the computational foundation used by BUGS and JAGS.

See also

mcp

Author

Jonas Kristoffer Lindeløv jonas@lindeloev.dk