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Regression with Multiple Change Points

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# mcp: Regression with Multiple Change Points `mcp` does regression with one or Multiple Change Points (MCP) between Generalized and hierarchical Linear Segments using Bayesian inference. `mcp` aims to provide maximum flexibility for analyses with a priori knowledge about the number of change points and the form of the segments in between.

Change points are also called switch points, break points, broken line regression, broken stick regression, bilinear regression, piecewise linear regression, local linear regression, segmented regression, and (performance) discontinuity models. `mcp` aims to be be useful for all of them. See how `mcp` compares to other R packages.

Under the hood, `mcp` takes a formula-representation of linear segments and turns it into JAGS code. `mcp` leverages the power of `tidybayes`, `bayesplot`, `coda`, and `loo` to make change point analysis easy and powerful.

# Install

1. Install the latest version of JAGS. Linux users can fetch binaries here.

2. Install from CRAN:

`install.packages("mcp")`

or install the development version from GitHub:

```if (!requireNamespace("remotes")) install.packages("remotes")
remotes::install_github("lindeloev/mcp")```

# At a glance

Here are some example `mcp` models. `mcp` takes a list of formulas - one for each segment. The change point(s) are the `x` at which data changes from being better predicted by one formula to the next. The first formula is just `response ~ predictors` and the most common formula for segment 2+ would be ` ~ predictors` (more details here). Scroll down to see brief introductions to each of these, or browse the website articles for more thorough worked examples and discussions.

# Brief worked example

## Fit a model

The following model infers the two change points between three segments.

```library(mcp)

# Define the model
model = list(
response ~ 1,  # plateau (int_1)
~ 0 + time,    # joined slope (time_2) at cp_1
~ 1 + time     # disjoined slope (int_3, time_3) at cp_2
)

# Get example data and fit it
ex = mcp_example("demo")
fit = mcp(model, data = ex\$data)```

## Plot and summary

The default plot includes data, fitted lines drawn randomly from the posterior, and change point(s) posterior density for each chain:

`plot(fit)` Use `summary()` to summarise the posterior distribution as well as sampling diagnostics. They were simulated using `mcp` (see `mcp_example("demo")\$call`) so the summary include the "true" values in the column `sim` and the column `match` show whether this true value is within the interval:

`summary(fit)`
```Family: gaussian(link = 'identity')
Iterations: 9000 from 3 chains.
Segments:
1: response ~ 1
2: response ~ 1 ~ 0 + time
3: response ~ 1 ~ 1 + time

Population-level parameters:
name match  sim  mean lower  upper Rhat n.eff
cp_1    OK 30.0 30.27 23.19 38.760    1   384
cp_2    OK 70.0 69.78 69.27 70.238    1  5792
int_1    OK 10.0 10.26  8.82 11.768    1  1480
int_3    OK  0.0  0.44 -2.49  3.428    1   810
sigma_1    OK  4.0  4.01  3.43  4.591    1  3852
time_2    OK  0.5  0.53  0.40  0.662    1   437
time_3    OK -0.2 -0.22 -0.38 -0.035    1   834```

`rhat` is the Gelman-Rubin convergence diagnostic, `eff` is the effective sample size. You may also want to do a posterior predictive check using `pp_check(fit)`.

`plot_pars(fit)` can be used to inspect the posteriors and convergence of all parameters. See the documentation of `plot_pars()` for many other plotting options. Here, we plot just the (population-level) change points. They often have "strange" posterior distributions, highlighting the need for a computational approach:

`plot_pars(fit, regex_pars = "cp_")` Use `fitted(fit)` and `predict(fit)` to get fits and predictions for in-sample and out-of-sample data.

## Tests and model comparison

We can test (joint) probabilities in the model using `hypothesis()` (see more here). For example, what is the evidence (given priors) that the first change point is later than 25 against it being less than 25?

`hypothesis(fit, "cp_1 > 25")`
```     hypothesis mean lower upper     p   BF
1 cp_1 - 25 > 0 5.27 -1.81 13.76 0.917 11.1```

For model comparisons, we can fit a null model and compare the predictive performance of the two models using (approximate) leave-one-out cross-validation (see more here). Our null model omits the first plateau and change point, essentially testing the credence of that change point:

```# Define the model
model_null = list(
response ~ 1 + time,  # intercept (int_1) and slope (time_1)
~ 1 + time            # disjoined slope (int_2, time_1)
)

# Fit it
fit_null = mcp(model_null, ex\$data)```

Leveraging the power of `loo::loo`, we see that the two-change-points model is preferred (it is on top), but the `elpd_diff / se_diff` ratio ratio indicate that this preference is not very strong.

```fit\$loo = loo(fit)
fit_null\$loo = loo(fit_null)

loo::loo_compare(fit\$loo, fit_null\$loo)```
``````       elpd_diff se_diff
model1  0.0       0.0
model2 -7.6       4.6
``````

# Highlights from in-depth guides

The articles on the mcp website go in-depth with the functionality of `mcp`. Here is an executive summary, to give you a quick sense of what mcp can do.

• Parameter names are `int_i` (intercepts), `cp_i` (change points), `x_i` (slopes), `phi_i` (autocorrelation), and `sigma_*` (variance).
• The change point model is basically an `ifelse` model.
• Use `rel()` to specify that parameters are relative to those corresponding in the previous segments.
• Generate data for all supported models using `fit\$simulate()`. See examples in, e.g., `mcp_examples("demo")\$call`.
• See priors in `fit\$prior`.
• Set priors using `mcp(..., prior = list(cp_1 = "dnorm(0, 1)", cp_1 = "dunif(0, 45)")`.
• The default prior for change points is fast for estimation but is mathematically "messy". The Dirichlet prior (`cp_i = "dirichlet(1)"`) is slow but beautiful.
• Fix parameters to specific values using `cp_1 = 45`.
• Share parameters between segments using `slope_1 = "slope_2"`.
• Truncate priors using `T(lower, upper)`, e.g., `int_1 = "dnorm(0, 1) T(0, )"`. `mcp` applies this automatically to change point priors to enforce order restriction. This is true for varying change points too.
• Do prior predictive checks using `mcp(model, data, sample = "prior")`.
• Get posteriors using `ranef(fit)`.
• Plot using `plot(fit, facet_by = "my_group")` and `plot_pars(fit, pars = "varying", type = "dens_overlay", ncol = 3)`.
• The default priors restrict varying change points to lie between the two adjacent change points.
• `mcp` currently supports specific combinations of families (`gaussian()`, `binomial()`, `bernoulli()`, and `poisson()`) and link functions (`identity`, `logit`, `probit`, and `log`).
• Use informative priors to avoid issues when using non-default priors.
• Use `binomial(link = "logit")` for binomial change points in mcp. Also relevant for `bernoulli(link = "logit")`.
• Use `poisson(link = "log")` for Poisson change points in mcp.
• Get results on the parameter scale rather than the observed scale using `plot(fit, scale = "linear")` or `predict(fit, scale = "linear")`.
• Do Leave-One-Out Cross-Validation using `loo(fit)` and `loo::loo_compare(fit1\$loo, fit2\$loo)`.
• Compute Savage-Dickey density rations using `hypothesis(fit, "cp_1 = 40")`.
• Leverage directional and conditional tests to assess interval hypotheses (`hypothesis(fit, "cp_1 > 30 & cp_1 < 50")`), combined other hypotheses (`hypothesis(fit, "cp_1 > 30 & int_1 > int_2")`), etc.

Modeling variance and autoregression:

• `~ sigma(1)` models an intercept change in variance. `~ sigma(0 + x)` models increasing/decreasing variance.
• `~ ar(N)` models Nth order autoregression on residuals. `~ar(N, 0 + x)` models increasing/decreasing autocorrelation.
• You can model anything for `sigma()` and `ar()`. For example, `~ x + sigma(1 + x + I(x^2))` models polynomial change in variance with `x` on top of a slope on the mean.
• `fitted(fit)` and `predict(fit)` take many arguments to predict in-sample and out-of-sample values and intervals.
• Forecasting with prior knowledge about future change points.

Tips, tricks, and debugging

• Speed up fitting using `mcp(..., cores = 3)` / `options(mcp_cores = 3)`, and/or fewer iterations, `mcp(..., adapt = 500)`.
• Help convergence along using `mcp(..., inits = list(cp_1 = 20, int_2 = -3))`.
• Most errors will be caused by circularly defined priors.

# Some examples

`mcp` aims to support a wide variety of models. Here are some example models for inspiration.

## Means

Find the single change point between two plateaus (simulated using `mcp_example("intercepts")\$call`).

```model = list(
y ~ 1,  # plateau (int_1)
~ 1     # plateau (int_2)
)
ex = mcp_example("intercepts")
fit = mcp(model, ex\$data, par_x = "x")
plot(fit)``` ## Varying change points

Here, we find the single change point between two joined slopes. While the slopes are shared by all participants, the change point varies by `id`. Read more about varying change points in mcp.

```model = list(
y ~ 1 + x,          # intercept + slope
1 + (1|id) ~ 0 + x  # joined slope, varying by id
)
ex = mcp_example("varying")
fit = mcp(model, ex\$data)
plot(fit, facet_by = "id")``` Summarise the varying change points using `ranef()` or plot them using `plot_pars(fit, "varying")`. Again, this data was simulated using `mcp` (see `mcp_example("varying")\$call`) so the columns `match` and `sim` are added to show simulation values and whether they are inside the interval. Set the `width` wider for a more lenient criterion.

`ranef(fit, width = 0.98)`
```           name match   sim  mean   lower   upper Rhat n.eff
cp_1_id[Benny]    OK -17.5 -18.1 -21.970 -14.877    1   895
cp_1_id[Bill]    OK -10.5  -7.6 -10.658  -4.451    1   420
cp_1_id[Cath]    OK  -3.5  -2.8  -5.634   0.027    1   888
cp_1_id[Erin]    OK   3.5   3.1   0.041   5.952    1  3622
cp_1_id[John]    OK  10.5  11.3   7.577  14.989    1  2321
cp_1_id[Rose]    OK  17.5  14.1  10.485  18.079    1  5150```

## Generalized linear models

`mcp` supports Generalized Linear Modeling. See extended examples using `binomial()` and `poisson()`.

Here is a binomial change point model with three segments (see simulation code: `mcp_example("binomial")\$call`). We plot the 95% HDI too:

```model = list(
y | trials(N) ~ 1,  # constant rate
~ 0 + x,            # joined changing rate
~ 1 + x             # disjoined changing rate
)
ex = mcp_example("binomial")
fit = mcp(model, ex\$data, family = binomial())
plot(fit, q_fit = TRUE)``` Use `plot(fit, rate = FALSE)` if you want the points and fit lines on the original scale of `y` rather than divided by `N`.

## Time series

`mcp` allows for flexible time series analysis with autoregressive residuals of arbitrary order. Below, we model a change from a plateau with strong positive AR(2) residuals to a slope with medium AR(1) residuals. These data were simulated with `mcp` (see simulation code: `mcp_example("ar")\$call`) and the generating values are in the `sim` column. You can also do regression on the AR coefficients themselves using e.g., `ar(1, 1 + x)`. Read more here.

```model = list(
price ~ 1 + ar(2),
~ 0 + time + ar(1)
)
ex = mcp_example("ar")
fit = mcp(model, ex\$data)
summary(fit)```

The AR(N) parameters on intercepts are named `ar[order]_[segment]`. All parameters, including the change point, are well recovered:

```Population-level parameters:
name match   sim    mean     lower   upper Rhat n.eff
ar1_1    OK   0.7   0.741  5.86e-01   0.892 1.01   713
ar1_2    OK  -0.4  -0.478 -6.88e-01  -0.255 1.00  2151
ar2_1    OK   0.2   0.145 -6.56e-04   0.284 1.01   798
cp_1       120.0 117.313  1.14e+02 118.963 1.05   241
int_1        20.0  17.558  1.51e+01  19.831 1.02   293
sigma_1    OK   5.0   4.829  4.39e+00   5.334 1.00  3750
time_2    OK   0.5   0.517  4.85e-01   0.553 1.00   661```

The fit plot shows the inferred autocorrelated nature:

`plot(fit_ar)` ## Variance change and prediction intervals

You can model variance by adding a `sigma()` term to the formula. The inside `sigma()` can take everything that the formulas outside do. Read more in the article on variance. The example below models two change points. The first is variance-only: variance abruptly increases and then declines linearly with `x`. The second change point is the stop of the variance-decline and the onset of a slope on the mean.

Effects on variance is best visualized using prediction intervals. See more in the documentation for `plot.mcpfit()`.

```model = list(
y ~ 1,
~ 0 + sigma(1 + x),
~ 0 + x
)
ex = mcp_example("variance")
fit = mcp(model, ex\$data, cores = 3, adapt = 5000, iter = 5000)
plot(fit, q_predict = TRUE)``` Write exponents as `I(x^N)`. E.g., quadratic `I(x^2)`, cubic `I(x^3)`, or some other power function `I(x^1.5)`. The example below detects the onset of linear + quadratic growth. This is often called the BLQ model (Broken Line Quadratic) in nutrition research.

```model = list(
y ~ 1,
~ 0 + x + I(x^2)
)
fit = mcp(model, ex\$data)
plot(fit)``` ## Trigonometric and others

You can use `sin(x)`, `cos(x)`, and `tan(x)` to do trigonometry. This can be useful for seasonal trends and other periodic data. You can also do `exp(x)`, `abs(x)`, `log(x)`, and `sqrt(x)`, but beware that the two latter will currently fail in segment 2+. Raise an issue if you need this.

```model = list(
y ~ 1 + sin(x),
~ 1 + cos(x) + x
)
ex = mcp_example("trigonometric")
fit = mcp(model, ex\$data)
plot(fit)``` ## Using `rel()` and priors

Here we find the two change points between three segments. The slope and intercept of segment 2 are parameterized relative to segment 1, i.e., modeling the change in intercept and slope since segment 1. So too with the second change point (`cp_2`) which is now the distance from `cp_1`.

Some of the default priors are overwritten. The first intercept (`int_1`) is forced to be 10, the slopes are in segment 1 and 3 is shared. It is easy to see these effects in the (simulated) data because they violate it somewhat. The first change point has to be at `x = 20` or later.

```model = list(
y ~ 1 + x,
~ rel(1) + rel(x),
rel(1) ~ 0 + x
)

prior = list(
int_1 = 10,  # Constant, not estimated
x_3 = "x_1",  # shared slope in segment 1 and 3
int_2 = "dnorm(0, 20)",
cp_1 = "dunif(20, 50)"  # has to occur in this interval
)
ex = mcp_example("rel_prior")
fit = mcp(model, ex\$data, prior, iter = 10000)
plot(fit)``` Comparing the summary to the fitted lines in the plot, we can see that `int_2` and `x_2` are relative values. We also see that the "wrong" priors made it harder to recover the parameters used to simulate this code (see `mcp_example("rel_prior")\$simulated` which is represented in the `match` and `sim` columns):

`summary(fit)`
```Population-level parameters:
name match   sim  mean  lower upper Rhat n.eff
cp_1    OK  25.0 23.15  20.00 25.81 1.00   297
cp_2        40.0 51.85  47.06 56.36 1.02   428
int_1        25.0 10.00  10.00 10.00  NaN     0
int_2    OK -10.0 -6.86 -21.57 11.89 1.03   190
sigma_1         7.0  9.70   8.32 11.18 1.00  7516
x_1         1.0  1.58   1.24  1.91 1.07   120
x_2    OK  -3.0 -3.28  -3.61 -2.96 1.04   293
x_3         0.5  1.58   1.24  1.91 1.07   120```

# Do much more with the MCMC samples

Don't be constrained by these simple `mcp` functions. `fit\$samples` is a regular `mcmc.list` object and all methods apply. You can work with the MCMC samples just as you would with `brms`, `rstanarm`, `jags`, or other samplers using the always excellent `tidybayes`:

```library(tidybayes)

# Extract all parameters:
tidy_draws(fit\$samples) %>%
# tidybayes stuff here

# Extract some parameters:
fit\$pars\$model  # check out which parameters are inferred.
spread_draws(fit\$samples, cp_1, cp_2, int_1, year_1) %>%
# tidybayes stuff here```

It may be convenient to use `fitted(fit, summary = FALSE)` or `predict(fit, summary = FALSE)` which return draws in `tidybayes` format, extended with additional columns for fits and predictions. For example:

`head(fitted(fit, summary = FALSE))`

# Citation

This preprint formally introduces `mcp`. Find citation info at the link, call `citation("mcp")` or copy-paste this into your reference manager:

``````  @Article{,
title = {mcp: An R Package for Regression With Multiple Change Points},
author = {Jonas Kristoffer Lindeløv},
journal = {OSF Preprints},
year = {2020},
doi = {10.31219/osf.io/fzqxv},
encoding = {UTF-8},
}
``````

Regression with Multiple Change Points

mcp 0.3.1 Latest
Nov 17, 2021

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