# Function that fits a multivariate correlated product partition change point model `ccp_ppm` is a function that fits a Bayesian product partition change point model, where the set of change point indicators between time series are correlated. ```r ccp_ppm(ydata, model=1, nu0, mu0, sigma0, mltypes, thetas, devs, nburn, nskip, nsave, verbose = FALSE) ``` ## Arguments - `ydata`: An $L \times n$ data matrix, where $L$ is the number of time series and $n$, the number of time points. - `model`: Determines of model fit is such that there are p_its (model=1) or only p_t (model=2) - `nu0`: Degrees of freedom of the multivariate Student's t-distribution (see section Details). - `mu0`: Location vector of dimension $L$ (see section Details). - `sigma0`: Positive definite scale matrix of order $L \times L$ (see section Details). - `mltypes`: Type of marginal likelihood. Currently only available is: * `mltypes = 1`. Observations within a block are conditionally independent $Normal(\mu, \sigma^2)$ variates with mean $\mu$ and variance $\sigma^2$. The desired marginal likelihood is obtained after integrating $(\mu, \sigma^2)$ with respect to a $Normal-Inverse-Gamma(\mu_{0}, \kappa_{0}, \alpha_{0}, \beta_{0})$ prior. - `thetas`: An $L \times q$ matrix containing hyperparameters associated with the marginal likelihood. The number of rows $(L)$ corresponds to the number of series. The number of columns $(q)$ depend on the marginal likelihood: * If `mltypes = 1`, then $q = 4$ and `thetas` equals the hyperparameter $(\mu_{0}, \kappa_{0}, \alpha_{0}, \beta_{0})$ of the Normal-Inverse-Gamma prior. - `devs`: An $L \times (n - 1)$ matrix containing the standard deviations of the candidate density associated with the random walk Metropolis-Hastings steps for updating change point probabilities. - `nburn`: The number of initial MCMC iterates to be discarded as burn-in. - `nskip`: The amount to thinning that should be applied to the MCMC chain. - `nsave`: Then number of MCMC iterates to be stored. - `verbose`: Logical indicating whether to print to screen the MCMC progression. The default value is `verbose = FALSE`. ## Details As described in Quinlan et al. (add cite), for each time series $\boldsymbol{y}_{i} = (y_{i,1}, \ldots , y_{i,n})'$: $$ \boldsymbol{y}_{i} \mid \rho_{i} \sim\prod_{j = 1}^{b_{i}}\mathcal{F}(\boldsymbol{y}_{i,j} \mid\boldsymbol{\theta}_{i}) $$ $$ \rho_{i} \mid (p_{i,1}, \ldots , p_{i,n-1})' \sim\prod_{t \in T_{i}}p_{i,t} \prod_{t \notin T_{i}}(1 - p_{i,t}) :T_{i} = \{\tau_{i,1}, \ldots, \tau_{i,b_{i} - 1}\} $$ $$ (p_{1,t}, \ldots , p_{L,t})' \sim logit-t(\nu_{0}, \boldsymbol{\mu}_{0},\boldsymbol{\Sigma}_{0}). $$ Here, $\rho_{i} = \{S_{i,1}, \ldots , S_{i,b_{i}}\}$ is a partition of the set $\{1, \ldots , n\}$ into $b_{i}$ contiguous blocks, and $\boldsymbol{y}_{i,j} = (y_{i,t} : t \in S_{i,j})'$. Also, $\tau_{i,j} = \max(S_{i,j})$ and $\mathcal{F}( \cdot \mid \boldsymbol{\theta}_{i})$ is a marginal likelihood function which depends on the nature of $\boldsymbol{y}_{i}$, indexed by a hyperparameter $\boldsymbol{\theta}_{i}$. In addition, $logit-t(\nu_{0}, \boldsymbol{\mu}_{0}, \boldsymbol{\Sigma}_{0})$ is the logit of a multivariate Student's t-distribution with degrees of freedom $\nu_{0}$, location vector $\boldsymbol{\mu}_{0}$ and scale matrix $\boldsymbol{\Sigma}_{0}$. ## Returns The function returns a list containing arrays filled with MCMC iterates corresponding to model parameters. In order to provide more detail, in what follows let $M$ be the number of MCMC iterates collected. The output list contains the following: * C. An $M \times \{L(n - 1)\}$ matrix containing MCMC iterates associated with each series indicators of a change point. The $m$th row in `C` is divided into $L$ blocks; the first $(n - 1)$ change point indicators for time series 1, the next $(n - 1)$ change point indicators for time series 2, and so on. * P. An $M \times \{L(n - 1)\}$ matrix containing MCMC iterates associated with each series probability of a change point. The $m$th row in `P` is divided into $L$ blocks; the first $(n - 1)$ change point probabilities for time series 1, the next $(n - 1)$ change point probabilities for time series 2, and so on. ## Examples ```r # Generate data that has two series, each with 100 observations y1 <- replicate(25, rnorm(4, c(-1, 0, 1, 2), c(0.1, 0.25, 0.5, 0.75))) y2 <- replicate(25, rnorm(4, c(2, 1, 0, -2), c(0.1, 0.25, 0.5, 0.75))) y <- rbind(c(t(y1)), c(t(y2))) # Marginal likelihood parameters thetas <- matrix(1, nrow = 2,ncol = 4) thetas[1,] <- c(0, 1, 2, 1) thetas[2,] <- c(0, 1, 2, 1) # M-H candidate density standard deviations devs = matrix(0.1, nrow = 2, ncol = (dim(y)[2] - 1)) # Prior parameters for logit-t distribution L <- nrow(y) pivar <- 10 picorr <- 0.9 pimu <- rep(-6, L) # mean associated with logit of p_i piSigma <- pivar*picorr*(rep(1, L) %*% t(rep(1, L))) + pivar*(1 - picorr)*diag(L) nu0 = 3 mu0 = pimu sigma0 = piSigma # Fit the bayesian ppm change point model fit <- ccp_ppm(nburn = 1000, nskip = 1, nsave = 1000, ydata = y, nu0 = nu0, mu0 = mu0, sigma0 = sigma0, mltypes = c(1, 1), thetas = thetas, devs = devs) ```

ccp_ppm function