---
title: "bssm: Bayesian Inference of Non-linear and Non-Gaussian State Space Models in R"
author: 
  - Jouni Helske and Matti Vihola, University of Jyväskylä, Department of Mathematics and Statistics, Finland
date: "February 16, 2021"
link-citations: true
output: html_document
bibliography: bssm.bib
vignette: |
  %\VignetteIndexEntry{bssm: Bayesian Inference of Non-linear and Non-Gaussian State Space Models in R}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
  %\VignetteDepends{ggplot2, dplyr}
---

```{r, echo = FALSE}
if (!requireNamespace("rmarkdown") ||
    !rmarkdown::pandoc_available("1.12.3")) {
  warning(call. = FALSE, "These vignettes assume rmarkdown and pandoc version 1.12.3. These were not found. Older versions will not work.")
  knitr::knit_exit()
}
```

```{r setup, include=FALSE}
knitr::opts_chunk$set(echo = TRUE)
```

This is a short vignette illustrating the `bssm` package. For more detailed exposition, please see the corresponding R Journal paper:

Jouni Helske and Matti Vihola (2021). "bssm: Bayesian Inference of Non-linear and Non-Gaussian State Space Models in R". Accepted to R Journal. [ArXiv preprint](https://arxiv.org/abs/2101.08492).


# Introduction

State space models (SSM) are latent variable models which are commonly applied in analysing time series data due to their flexible and general framework  [cf. @DK2012]. For `R` [@r-core], there is large number of packages available for state space modelling, especially for the two special cases. First special case is linear-Gaussian SSM (LGSSM) where both the observation and state densities are Gaussian with linear relationships with the states. Another special case is SSM with discrete state space, which are sometimes called hidden Markov models (HMM).  What is special about these two classes of models is that the marginal likelihood function, and the conditional state distributions (conditioned on the observations) of these models are analytically tractable, making inference relatively straightforward. See for example [@Petris2010, @Tusell2010, @KFAS, @seqHMM] for review of some of the `R` packages dealing with these type of models. The `R` package `bssm` is designed for Bayesian inference of general state space models with non-Gaussian and/or non-linear observational and state equations. The package aims to provide easy-to-use and efficient functions for fully Bayesian inference of common time series models such basic structural time series model (BSM) [@Harvey1989] with exogenous covariates, simple stochastic volatility models, and discretized diffusion models, making it straightforward and efficient to make predictions and other inference in a Bayesian setting.

The motivation behind the `bssm` package is in [@vihola-helske-franks] which suggests a new computationally efficient, parallelisable approach for Bayesian inference of state space models. The core idea is to use fast approximate Markov chain Monte Carlo (MCMC) targeting the approximate marginal posterior of the hyperparameters (i.e. unknown variables excluding latent state variables), which is then used in importance sampling type weighting phase which provides asymptotically exact samples from the joint posterior of hyperparameters and the hidden states. In addition to this the two-stage procedure, standard pseudo-marginal MCMC and so called delayed acceptance pseudo-marginal MCMC are also supported. For more details, see [@helske-vihola2021]. There is also separate vignette for nonlinear models as well as for discretized diffusion models.

## State space models with linear-Gaussian dynamics

Denote a sequence of observations $(y_1,\ldots,y_T)$ as $y_{1:T}$, and sequence of latent state variables $(\alpha_1,\ldots, \alpha_T)$ as $\alpha_{1:T}$. A general state space model consists of two parts: observation level densities $g_t(y_t | \alpha_t)$ and latent state transition densities $\mu_t(\alpha_{t+1} | \alpha_t)$. We first focus on the case where the state transitions are linear-Gaussian:
$$
\alpha_{t+1} = c_t + T_t \alpha_t + R_t \eta_t,
$$
where $c_t$ is known input vector (often omitted), and $T_t$ and $R_t$ are a system matrices which can depend on unknown parameters. Also, $\eta_t \sim N(0, I_k)$ and $\alpha_1 \sim N(a_1, P_1)$ independently of each other. For observation level density $g_t$, the `bssm` package currently supports basic stochastic volatility model and general exponential family state space models. 

For exponential family models, the observation equation has a general form

$$
g_t(y_t | d_t + Z_t \alpha_t, x'_t\beta, \phi, u_t),
$$
where $d_t$ is a again known input, $x_t$ contains the exogenous covariate values at time $t$, with $\beta$ corresponding to the regression coefficients. Parameter $\phi$ and the known vector $u_t$ are distribution specific and can be omitted in some cases. Currently, following observational level distributions are supported:

* Gaussian distribution: $g_t(y_t | Z_t \alpha_t, x'_t\beta) = x'_t \beta + Z_t \alpha_t + H_t \epsilon_t$ with $\epsilon_t \sim N(0, 1)$.

* Poisson distribution: $g_t(y_t | Z_t \alpha_t, x'_t \beta, u_t) = \textrm{Poisson}(u_t \exp(x'_t \beta + Z_t \alpha_t))$, where $u_t$ is the known exposure at time $t$.

* Binomial distribution: $g_t(y_t | Z_t \alpha_t, x'_t \beta, u_t) = \textrm{binomial}(u_t, \exp(x'_t \beta + Z_t \alpha_t) / (1 + \exp(x'_t \beta + Z_t \alpha_t)))$, where $u_t$ is the number of trials and $\exp(x_t \beta + Z_t \alpha_t) / (1 + \exp(x'_t \beta + Z_t \alpha_t))$ is the probability of the success.

* Negative binomial distribution: $g_t(y_t | Z_t \alpha_t, x'_t \beta, \phi, u_t) = \textrm{negative binomial}(\exp(x'_t \beta + Z_t \alpha_t), \phi, u_t)$, where $u_t \exp(x'_t \beta + Z_t \alpha_t)$ is the expected value and $\phi$ is the dispersion parameter ($u_t$ is again exposure term).

* Gamma distribution: $g_t(y_t | d_t + Z_t \alpha_t, \phi, u_t) = \textrm{Gamma}(\exp( d_t + Z_t \alpha_t), \phi, u_t)$, where $u_t \exp( d_t + Z_t \alpha_t)$ is the expected value, $\phi$ is the shape parameter, and $u_t$ is a known offset term.

For stochastic volatility model, there are two possible parameterizations available. In general for we have
$$
y_t = \sigma \exp(\alpha_t / 2)\epsilon_t, \quad \epsilon_t \sim N(0, 1),
$$
and
$$
\alpha_{t+1} = \mu + \rho (\alpha_t - \mu) + \sigma_{\eta} \eta_t,
$$
with $\alpha_1 \sim  N(\mu, \sigma^2_{\eta} / (1-\rho^2))$. For identifiability purposes we must either choose $\sigma=1$ or $\mu=0$. Although analytically identical, the parameterization with $\mu$ is often preferable in terms of computational efficiency.

Typically some of the model components such as $\beta$, $T_t$ or $R_t$ depend on unknown parameter vector $\theta$, so $g_t(y_t | \alpha_t)$ and $\mu_t(\alpha_{t+1} | \alpha_t)$ depend implicitly on $\theta$. Our goal is to perform Bayesian inference of the joint posterior of $\alpha_{1:T}$ and $\theta$. 

For multivariate models, these distributions can be combined arbitrarily, except the stochastic volatility model case which is currently handled separately. Also, for fully Gaussian model, the observational level errors can be correlated, i.e. $Cov(\epsilon_t) = H_tH'_t$.

## Non-linear models

The general non-linear Gaussian model in the `bssm` has following form:

$$
y_t = Z(t, \alpha_t, \theta) + H(t, \alpha_t, \theta)\epsilon_t,\\
\alpha_{t+1} = T(t, \alpha_t, \theta) + R(t, \alpha_t, \theta)\eta_t,\\
\alpha_1 \sim N(a_1(\theta), P_1(\theta)),
$$
with $t=1,\ldots, n$, $\epsilon_t \sim N(0,\textrm{I}_p)$, and $\eta \sim N(0,\textrm{I}_k)$. Here vector $\theta$ contains the unknown model parameters. Due to their generality and the need to repeated calls within MCMC, functions $T(\cdot)$, $H(\cdot)$, $T(\cdot)$, $R(\cdot)$,$a_1(\cdot)$, $P_1(\cdot)$, as well as functions defining the Jacobians of $Z(\cdot)$ and $T(\cdot)$ needed by the extended Kalman filter and the prior distribution for $\theta$ must be defined by user as a external pointers to user-defined `C++` functions. All of these functions can also depend on some known parameters, defined as `known_params` (vector) and `known_tv_params` (matrix with $n$ columns) arguments to `ssm_nlg` function. See the growth model vignette^[https://cran.r-project.org/package=bssm/vignettes/growth_model.html] for a template for these functions.


## Time-discretised diffusion models

The `bssm` package also supports models where the state equation is defined as a continuous time diffusion model of form
$$
\textrm{d} \alpha_t =
\mu(\alpha_t,\theta) \textrm{d} t +
\sigma(\alpha_t, \theta) \textrm{d} B_t, \quad t\geq0,
$$
where $B_t$ is a Brownian motion and where $\mu$ and $\sigma$ are scalar-valued functions, with the univariate observation density $g(y_k | \alpha_k)$ defined at integer times $k=1\ldots,n$. As these transition densities are generally unavailable for non-linear diffusions, we use Milstein time-discretisation scheme for approximate simulation with bootstrap particle filter. Fine discretisation mesh gives less bias than the coarser one, with increased computational complexity. These models are also defined via `C++` snippets, see the SDE vignette for details.

## Markov chain Monte Carlo

Given the prior $p(\theta)$, the joint posterior of $\theta$ and $\alpha_{1:T}$ is given as

$$
p(\alpha_{1:T}, \theta | y_{1:T}) \propto p(\theta) p(\alpha_{1:T}, y_{1:T} | \theta) = p(\theta) p(y_{1:T} | \theta)  p( \alpha_{1:T} | y_{1:T}, \theta)
$$

where $p(y_{1:T} | \theta)$ is the marginal likelihood, and $p(\alpha_{1:T} | y_{1:T}, \theta)$ is often referred as a smoothing distribution. However, instead of targeting this joint posterior, it is typically more efficient to target the marginal posterior $p(\theta | y)$, and then given the sample $\{\theta^{i}\}^n_{i=1}$ from this marginal posterior, simulate states $\alpha^i_{1:T}$ from the smoothing distribution $p(\alpha_{1:T} | y_{1:T}, \theta^i)$ for $i=1\ldots,n$.

For Gaussian models given the parameters $\theta$, the marginal likelihood $p(y_{1:T} | \theta)$ can be computed using the well known Kalman filter recursions, and there are several algorithms for simulating the states $\alpha_{1:T}$ from the smoothing distribution $p(\alpha_{1:T} | y_{1:T})$ (see for example @DK2012). Therefore we can straightforwardly apply standard MCMC algorithms. In `bssm`, we use an adaptive random walk Metropolis algorithm based on RAM [@Vihola2012] where we fix the target acceptance rate beforehand. There RAM algorithm is provided by the `ramcmc` package [@helske-ram].

For non-linear/non-Gaussian models, the marginal likelihood $p(y_{1:T} | \theta)$ is typically not available in closed form. Thus we need to resort to simulation methods, which leads to pseudo-marginal MCMC algorithm [@lin-liu-sloan, @beaumont, @andrieu-roberts]. `bssm` also supports more efficient inference algorithms based on (intermediate) approximations, see @helske-vihola2021 and @vihola-helske-franks.

# Package functionality

Main functions of `bssm` is written in `C++`, with help of `Rcpp` [@RcppA] and `RcppArmadillo` [@RcppArmadillo] packages. On the `R`side, package uses S3 methods in order to provide relatively unified workflow independent of the type of the model one is working with. The model building functions such as `bsm_ng` and `svm` are used to construct the actual state models which can be then passed to other methods, such as `logLik` and `run_mcmc` which compute the log-likelihood value and run MCMC algorithm respectively. We will now briefly describe the main functions and methods of `bssm`, for more detailed descriptions of different function arguments and return values, see the corresponding documentation in `R`.

## Model building functions

For linear-Gaussian models, `bssm` offers functions `bsm_lg` for basic univariate structural time series models (BSM), `ar1` for univariate, possibly noisy AR(1) process, as well as general `ssm_ulg` and `ssm_mlg` for arbitrary linear gaussian models. As an example, consider a Gaussian local linear trend model of form

$$
\begin{aligned}
y_t &= \mu_t + \epsilon_t,\\
\mu_{t+1} &= \mu_t + \nu_t + \eta_t,\\
\nu_{t+1} &= \nu_t + \xi_t,
\end{aligned}
$$
with zero-mean Gaussian noise terms $\epsilon_t, \eta_t, \xi_t$ with unknown standard deviations. This model can be built with `bsm_lg` function as 

```{r nhtemp}
library("bssm")
data("nhtemp", package = "datasets")
prior <- halfnormal(1, 10)
bsm_model <- bsm_lg(y = nhtemp, sd_y = prior, sd_level = prior,
  sd_slope = prior)
```

Here we use helper function `halfnormal` which defines half-Normal prior distribution for the standard deviation parameters, with first argument defining the initial value of the parameter, and second defines the scale parameter of the half-Normal distribution. Other prior options are `normal` and `uniform`.

For non-Gaussian models, function `bsm_ng` can be used for constructing an BSM model where the observations are assumed to be distributed according to Poisson, binomial, negative binomial, or Gamma distribution. The syntax is nearly identical as in case of `bsm_lg`, but we now define also the distribution via argument `distribution`, and depending on the model, we can also define parameters `u` and `phi`. For Poisson and negative binomial models, the known parameter `u` corresponds to the offset term, whereas in case of binomial model `u` defines the number of trials. For negative binomial model, argument `phi` defines the dispersion term, which can be given as a fixed value, or as a prior function. For same observational densities, a model where the state equation follows a first order autoregressive process can be defined using the function `ng_ar1`. Finally, a stochastic volatility model can be defined using a function `svm`, and an arbitrary linear-Gaussian state model with Poisson, binomial or negative binomial distributed observations can be defined with `ssm_ung` and `ssm_mng` for univariate and multivariate models respectively.

For models where the state equation is no longer linear-Gaussian, we can use our pointer-based C++ interface with the function `ssm_nlg`. Diffusion models can be defined with the function `ssm_sde`.

## Filtering and smoothing

Filtering refers to estimating the conditional densities of the hidden states at time $t$, given the observations up to that point. For linear-Gaussian models, these densities can be efficiently computed using the Kalman filter recursions. The `bssm` has a method `kfilter` for this task. For models defined with the `ssm_mng` ,`bsm_ng`, `ar1_ng`, and `svm` functions, `kfilter` will first construct an approximating Gaussian model for which the Kalman filter is then used. For details of this approximation, see @durbin-koopman1997 and @vihola-helske-franks. For non-linear models defined by `ssm_nlg` it is possible to perform filtering using extended Kalman filter (EKF) with the function `ekf`, or unscented Kalman filter with the function `ukf`. It is also possible to use iterated EKF (IEKF) by changing the argument `iekf_iter` of the `ekf` function. Compared to EKF, in IEKF the observation equation is linearized iteratively within each time step.

While Kalman filter solves the filtering problem exactly in case of linear-Gaussian models, EKF, UKF, and the filtering based on the approximating Gaussian models produce only approximate, possibly biased filtering estimates for general models. This problem can be solved by the use of particle filters (PF). These sequential Monte Carlo methods are computationally more expensive, but can in principle deal with almost arbitrary state space models. The `bssm` supports general bootstrap particle filter (BSF) for all model classes of the `bssm`. For `ssm_mng` ,`bsm_ng`, `ar1_ng`, and `svm` models we recommend the particle filter called $\psi$-APF [@vihola-helske-franks] (see also another vignette on CRAN) which makes use of the previously mentioned approximating Gaussian model in order to produce more efficient filter. It is also available for `ssm_nlg` models but in case of severe non-linearities, it is not necessarily best option.

Compared to filtering problem, in smoothing problems we are interested in the conditional densities of the hidden states at certain time point $t$ given all the observations $y_1,\ldots,y_t,\ldots,y_n$. Again for linear-Gaussian models we can use so called Kalman smoothing recursions, where as in case of more general models we can rely on approximating methods, or smoothing algorithms based on the output of particle filters. Currently only filter-smoother approach [@kitagawa] for particle smoothing is supported.

## Markov chain Monte Carlo

The main purpose of the `bssm` is to allow efficient MCMC-based inference for various state space models. For this task, a method `run_mcmc` can be used. Here we define a random walk model with a drift and stochastic seasonal component for UK gas consumption dataset and use 40 000 MCMC iteration where first half is discarded by default as a burn-in. Note that the number of iterations is quite low and in practice we should run the chain longer. Here we use less iterations to speed up the package checks on CRAN.

```{r mcmc_bsm}
prior <- halfnormal(0.1, 1)
UKgas_model <- bsm_lg(log10(UKgas), sd_y = prior, sd_level = prior,
  sd_slope = prior, sd_seasonal =  prior)

mcmc_bsm <- run_mcmc(UKgas_model, iter = 4e4, seed = 1)
mcmc_bsm
```
Note that all MCMC algorithms of `bssm` output also state forecasts for the timepoint $n + 1$, the summary statistics of this state is also shown in the output above.

Here we use `ggplot2` [@ggplot2] package for the figures, so we transform the MCMC samples to `data.frame`: 

```{r plot}
suppressMessages(library("ggplot2"))
d <- as.data.frame(mcmc_bsm, variable = "theta")
ggplot(d, aes(x = value)) + 
  geom_density(adjust = 3, fill = "#92f0a8") + 
  facet_wrap(~ variable, scales = "free") + 
  theme_bw()
```

```{r trend, dev.args=list(pointsize = 10), fig.cap="Smoothed trend component with 95% intervals."}
suppressMessages(library("dplyr"))
d <- as.data.frame(mcmc_bsm, variable = "states")
level_fit <- d %>% 
  filter(variable == "level") %>%
  group_by(time) %>%
  summarise(consumption = mean(value), 
    lwr = quantile(value, 0.025), 
    upr = quantile(value, 0.975))

ggplot(level_fit, aes(x = time, y = consumption)) + 
  geom_ribbon(aes(ymin = lwr, ymax = upr), 
    fill = "#92f0a8", alpha = 0.25) +
  geom_line(colour = "#92f0a8") +
  geom_line(data = data.frame(
    consumption = log10(UKgas), 
    time = time(UKgas)), 
    colour = "grey30", linetype = "dashed") +
  theme_bw()
```

## Acknowledgements

This work has been supported by the Academy of Finland research grants 284513, 312605, 311877, and 331817.

# References
