We have presented the tools for creating the structure of a DGLM model, specifically, we have shown how to define the relationship between the latent vector \(\vec{\theta}_t\) and the linear predictors \(\vec{\lambda}_t\), along with the temporal dynamic of \(\vec{\theta}_t\). Now we proceed to define the observational model for \(\vec{Y}_t\) and the relationship between \(\vec{\lambda}_t\) and \(\vec{\eta}_t\), i.e., the highlighted part of the following equations:
\[ \require{color} \begin{equation}\begin{aligned} \color{red}{Y_t|\eta_t }&{\color{red}\sim \mathcal{F}\left(\eta_t\right),}\\ {\color{red}g(\eta_t) }&{\color{red}= \lambda_{t}}=F_t'\theta_t,\\ \theta_t &=G_t\theta_{t-1}+\omega_t,\\ \omega_t &\sim \mathcal{N}_n(h_t,W_t), \end{aligned}\end{equation} \]
In each subsection, we will assume that the linear predictors are already defined, along with all the structure that comes along with them (i.e., we will take for granted the part of the model that is not highlighted), moreover, we also assume that the user has created the necessary amount of linear predictors for each type of outcome and that those linear predictors were named as \(\lambda_1\),…,\(\lambda_k\).
Currently, we offer support for the following observational distributions:
We are currently working to include several distributions. In particular, the following distributions shall be supported very soon: Dirichlet; Geometric; Negative Binomial; Rayleigh; Pareto; Asymmetric Laplace with known mean.
In some sense, we can think of this as the most basic case, at least in a theoretical point of view, since the Kalman Filter was first developed for this specific scenario (Kalman, 1960). Indeed, if we have a static observational variance/covariance matrix (even if unknown), we fall within the DLM class, which has an exact analytical solution for the posterior of the latent states. With some adaptations, one can also have some degree of temporal dynamic for the variance/covariance matrix (see Ameen and Harrison, 1985; West and Harrison, 1997, sec. 10.8). Yet, the kDGLM package goes a step further, offering the possibility for predictive structure for both the mean and the observational variance/covariance matrix, allowing the inclusion of dynamic regressions, seasonal trends, autoregressive components, etc., for both parameters.
We will present this case in two contexts: the first, which is a simple implementation of the Kalman Filter and Smoother, deals with data coming from an Normal distribution (possibly multivariate) with unknown mean and known variance/covariance matrix; the second deals with data coming from a univariate Normal distribution with unknown mean and unknown variance.
Also, at the end of the second subsection, we present an extension to the bivariated Normal distribution with unknown mean and unknown covariance matrix. A study is being conducted to expand this approach to the \(k\)-variated case, for any arbitrary \(k\).
Suppose that we have a sequence of \(k\)-dimensional vectors \(\vec{Y}_t\), such that \(\vec{Y}_t=(Y_{1t},...,Y_{kt})'\). We assume that:
\[ \begin{aligned} \vec{Y}_t|\mu_t,V &\sim \mathcal{N}_k\left(\vec{\mu}_t,V\right),\\ \mu_{it}&=\lambda_{it}, i=1,...,k,\\ \end{aligned} \] where \(\vec{\mu}_t=(\mu_{1,t},...,\mu_{k,t})'\) and \(V\) is a known symmetric, definite positive \(k\times k\) matrix. Also, for this model, we assume that the link function \(g\) is the identity function.
To create the outcome for this model, we can make use of the
Normal
function:
Intuitively, the mu
argument must be a character vector
of size \(k\) containing the names of
the linear predictors associated with each \(\mu_{i.}\). The user must also specify one
(and only one) of V
, Tau
or Sd
.
If the user provides V
, \(V\) is assumed to be that value; if the
user provides Tau
, \(V\)
is assumed to be the inverse of the given matrix (i.e., Tau
is the precision matrix); if the user provides Sd
, \(V\) is assumed to be such that the standard
deviation of the observations is equal to the main diagonal of
Sd
and the correlation between observations is assumed the
be equal to the off-diagonal elements of Sd
.
The data
argument must be a \(T \times k\) matrix containing the values
of \(\vec{Y}_t\) for each observation.
Notice that each line \(t\) must have
the values of all categories in time \(t\) and each column \(i\) must represent the values of a category
\(i\) through time. If a value of the
argument data
is not available (NA
) for a
specific time, it is assumed that there was no observation at that time,
thus the update step of the filtering algorithm will be skipped at that
time. Note that the evolution step will still be performed, such that
the predictive distribution for the missing data and the updated
distribution for the latent states at that time will still be
provided.
Next, we present a brief example for the usage of Normal
function for a univariate outcome (the multivariate case works
similarly). We use some functions described in the previous sections, as
well as some functions that will be presented later on. For now, let us
focus on the usage of the Normal
function.
level <- polynomial_block(mu = 1, D = 0.95, order = 2)
season <- harmonic_block(mu = 1, period = 12, D = 0.975)
outcome <- Normal(
mu = "mu", V = 6e-3,
data = c(log(AirPassengers))
)
fitted.model <- fit_model(level, season, outcome)
plot(fitted.model, plot.pkg = "base")
Notice that, since this is the univariate case, the data
argument can be a vector.
For this type of outcome, we assume that:
\[ \begin{aligned} Y_t|\mu_t,\tau_t &\sim \mathcal{N}\left(\mu_t,\tau_t^{-1}\right),\\ \mu_{t}&=\lambda_{1t},\\ \ln\{\tau_{t}\}&=\lambda_{2t}.\\ \end{aligned} \]
To create an outcome for this model, we also make use of the
Normal
function:
Just as before, the mu
argument must be a character
representing the label of the linear predictor associated with \(\mu_t\). The user must also specify one
(and only one) of V
, Tau
or Sd
,
which must be a character string representing the label of the
associated linear predictor.
Similar to the known variance case, we allow multiple
parametrizations of the observational variance. Specifically, if the
user provides V
, we assume that \(\lambda_{2t}=\ln\{\sigma^2_{t}\}=-\ln\{\tau_t\}\);
if the user provides Sd
, we assume that \(\lambda_{2t}=\ln\{\sigma_{t}\}=-\ln\{\tau_t\}/2\);
if the user provides Tau
, then the default parametrization
is used, i.e., \(\lambda_{2t}=\ln\{\tau_t\}\).
The data
argument usually is a \(T \times 1\) matrix containing the values
of \(Y_t\) for each observation. In
cases where \(\vec{Y}_t\) is
univariated, we also accept data
as a line vector, in which
case we assume that each coordinate of data
represents the
observed value at each time. If a value of data is not available
(NA
) for a specific time, it is assumed that there was no
observation at that time, thus the update step of the filtering
algorithm will be skipped at that time. Note that the evolution step
will still be performed, such that the predictive distribution for the
missing data and the updated distribution for the latent states at that
time will still be provided.
Next, we present a brief example for the usage of this outcome. We
use some functions described in the previous sections, as well as some
functions that will be presented later on. For now, let us focus on the
usage of the Normal
function.
structure <- polynomial_block(mu = 1, D = 0.95) +
polynomial_block(V = 1, D = 0.95)
outcome <- Normal(mu = "mu", V = "V", data = cornWheat$corn.log.return[1:500])
fitted.model <- fit_model(structure, outcome)
plot(fitted.model, plot.pkg = "base")
Currently, we also support models with bivariate Normal outcomes. In this scenario we assume the following model:
\[ \begin{aligned} Y_t|\mu_{t},V_t &\sim \mathcal{N}_2\left(\mu_t,V_t\right),\\ \mu_t&=\begin{bmatrix}\mu_{1,t}\\ \mu_{2t}\end{bmatrix},\\ V_t&=\begin{bmatrix}\tau_{1,t}^{-1} & (\tau_{1,t}\tau_{2,t})^{-1/2}\rho_t\\ (\tau_{1,t}\tau_{2,t})^{-1/2}\rho_t & \tau_2^{-1}\end{bmatrix},\\ \mu_{i,t}&=\lambda_{i,t}, i=1,2\\ \tau_{i,t}&=\ln\{\lambda_{(i+2),t}\}, i=1,2\\ \rho_{t}&=\tanh\{\lambda_{5,t}\}.\\ \end{aligned} \]
Notice that \(\rho_t\) represents the (and the covariance) between the series at time \(t\). To guarantee that \(\rho_t \in (-1,1)\), we use the Inverse Fisher transformation (also known as the hyperbolic tangent function) as link function.
For those models, `mu
must be a character vector,
similarly to the case where \(V\) is
known, and V
, Tau
and Sd
must be
a \(2 \times 2\) character matrix. The
main diagonal elements are interpreted as the linear predictors
associated with the precisions, variances or standard deviations,
depending if the user used Tau
, V
or
Sd
, respectively. The off diagonal elements must be equals
(one of them can be NA
) and will be interpreted as the
linear predictor associated with \(\rho_t\).
Bellow we present an example for the bivariate case:
# Bivariate Normal case
structure <- (polynomial_block(mu = 1, D = 0.95) +
polynomial_block(log.V = 1, D = 0.95)) * 2 +
polynomial_block(atanh.rho = 1, D = 0.95)
outcome <- Normal(
mu = c("mu.1", "mu.2"),
V = matrix(c("log.V.1", "atanh.rho", "atanh.rho", "log.V.2"), 2, 2),
data = cornWheat[1:500, c(4, 5)]
)
fitted.model <- fit_model(structure, outcome)
Notice that, by the second plot, the correlation between the series
(represented by atanh.rho
, i.e., the plot shows \(\tanh^{-1}(\rho)\)) is significant and
changes over time, making the proposed model much more adequate than two
independent Normal models (one for each outcome).
In this case, we assume the following observational model:
\[ \begin{equation}\begin{aligned} Y_t|\eta_t &\sim Poisson\left(\eta_t\right),\\ \ln(\eta_t) &=\lambda_{t}. \end{aligned}\end{equation} \]
In the notation introduced before, we have that our link function \(g\) is the (natural) logarithm function.
To define such observational model, we offer the Poisson
function, whose usage is presented bellow:
As usual in the literature, we refer to the rate parameter of the
Poisson distribution as lambda
(although, in the context of
this document, this might seem confusing) and the user must provide for
this argument the name of the linear predictor associated with this
parameter.
For the argument data
the user must provide a sequence
of numerical values consisting of the observed values of \(Y_t\) at each time. Since the \(Y_t\) is a scalar for all \(t\), the user can pass the outcome as a
vector or as a matrix with a single column. If a value of
data
is not available (NA
) for a specific
time, it is assumed that there was no observation at that time, thus the
update step of the filtering algorithm will be skipped at that time.
Note that the evolution step will still be performed, such that the
predictive distribution for the missing data and the updated
distribution for the latent states at that time will still be
provided.
Lastly, the offset
argument is optional and can be used
to provide a measure of the scale of the data. If the offset is provided
and is equal to \(E_t\), then we will
fit a model assuming that:
\[ \begin{equation}\begin{aligned} Y_t|\theta_t &\sim Poisson\left(\eta_tE_t\right),\\ \ln(\eta_t) &=\lambda_{t}. \end{aligned}\end{equation} \]
Bellow we present an example of the usage of this outcome. We use
some functions described in the previous section, as well as some
functions that will present later on, for now, let us focus only on the
usage of the Poisson
function.
data <- c(AirPassengers)
level <- polynomial_block(rate = 1, order = 2, D = 0.95)
season <- harmonic_block(rate = 1, period = 12, order = 2, D = 0.975)
outcome <- Poisson(lambda = "rate", data = data)
fitted.data <- fit_model(level, season,
AirPassengers = outcome
)
plot(fitted.data, plot.pkg = "base")
Notice that, while creating the structure, we defined a linear
predictor named rate
, whose behavior is being explained by
a second order polynomial trend and seasonal component defined by a
second order harmonic block. Since the value passed to rate
equals \(1\) in both blocks, we have
that these components have a constant effect (and equal to \(1\)) on the linear predictor on all times,
although the components themselves change their values over
time such as to capture the behavior of the series.
Later on, when creating the outcome, we pass the name
'rate'
as the linear predictor associated with
lambda
, the rate (or mean) parameter of the Poisson
distribution.
This is a particularly simply usage of the package, the Poisson kernel being the one with the smallest amount of parameters. Moving forward, we will present outcomes whose specification can be a bit more complex.
In this subsection we will present the Gamma case, in which we assume the following observational model:
\[ \begin{equation}\begin{aligned} Y_t|\alpha_t,\beta_t &\sim \mathcal{G}\left(\alpha_t,\beta_t\right),\\ \ln\{\alpha_t\}&=\lambda_{1t},\\ \ln\{\beta_t\}&=\lambda_{2t} \end{aligned}\end{equation} \]
For this outcome we have a few variations. First, there’s a matter of parametrization. We allow the user to define the model by any non redundant pair of:
\[ \begin{equation}\begin{aligned} \alpha_t&,\\ \beta_t&,\\ \phi_t&=\alpha_t,\\ \mu_t&=\frac{\alpha_t}{\beta_t},\\ \sigma_t&=\frac{1}{\beta_t}. \end{aligned}\end{equation} \]
Naturally, the user CANNOT specify both \(\alpha_t\) AND \(\phi_t\) or \(\beta_t\) AND \(\sigma_t\), as such specification is redundant at best, and incoherent at worst. Outside of those cases, in which the package will raise an error, any combination can be used by the user, allowing for the structure of the model to be defined within the variables that are most convenient (it may be easier or more intuitive to specify the structure in the mean \(\mu_t\) and the scale \(\sigma_t\), than on the shape \(\alpha_t\) and rate \(\beta_t\)).
Another particularity of the Gamma outcome is that the user may set the shape parameter \(\phi_t\) to a known constant. In that case, the user must specify the structure to the mean parameter \(\mu_t\) (he is not allowed to specify neither \(\beta_t\) nor \(\sigma_t\)). In general, we do not expect the shape parameter to be known, still, there are some important applications where it is common the use some particular cases of the Gamma distribution, such as the Exponential Model (\(\phi_t=1\)) or the \(\chi^2\) model (\(\phi_t=0.5\)). The estimation of the shape parameter \(\phi_t\) is still under development, as such, the current version of the package does not have support for a unknown \(\phi_t\) (a version of the package with a proper estimation for \(\phi_t\) will be released very soon).
No matter the parametrization, the link function \(g\) will always be the logarithm function, as such, given a certain parametrization, we can write the linear predictor of any other parametrization as a linear transformation of the original.
In the examples of this section, we will always use the parameters \(\phi_t\) (when applicable) and \(\mu_t\), but the code used can be trivially adapted to other parametrizations.
Similar to the Poisson case, the argument data
must
provide a set of numerical values consisting of the observed values of
\(Y_t\) at each time. Since the \(Y_t\) is a scalar for all \(t\), the user can pass the outcome either
as a vector or as a matrix with a single column. If a value of the
argument data
is not available (NA
) for a
specific time, it is assumed that there was no observation at that time,
thus the update step of the filtering algorithm will be skipped at that
time. Note that the evolution step will still be performed, such that
the predictive distribution for the missing data and the updated
distribution for the latent states at that time will still be
provided.
The offset
argument is optional and can be used to
provide a measure of the scale of the data. If the offset is provided
and is equal to \(E_t\), then we will
fit a model assuming that:
\[ \begin{equation}\begin{aligned} Y_t|\theta_t &\sim \mathcal{G}\left(\alpha_t,\beta_t E_t^{-1}\right). \end{aligned}\end{equation} \]
Note that the above model implies that:
\[ \mathbb{E}[Y_t|\theta_t]=\frac{\alpha_t}{\beta_t}E_t. \]
The arguments phi
, mu
, alpha
,
beta
and sigma
should be character strings
indicating the name of the linear predictor associated with their
respective linear predictor. The user may opt to pass phi
as a positive numerical value, it that case, the shape parameter \(\phi_t\) is considered known and equal to
phi
for all \(t\).
Let us assume that we have a sequence of \(k\)-dimensional non-negative integer vectors \(Y_t\), such that \(Y_t=(Y_{1t},...,Y_{kt})'\) and:
\[ \begin{equation}\begin{aligned} Y_t|N_t,\vec{p}_t &\sim Multinom\left(N_t,\vec{p}_t\right),\\ \ln\left\{\frac{p_{it}}{p_{kt}}\right\}&=\lambda_{it}, i=1,...,k-1,\\ N_t&=\sum_{i=1}^{k}Y_{it}, \end{aligned}\end{equation} \] where \(\vec{p}_t=(p_{1t},...,p_{kt})'\), with \(p_{it} > 0, \forall i\) and \(\sum_{i=1}^k p_{it}=1\).
Notice that \(N_t\) is automatically defined by the values of \(Y_t\), such that \(N_t\) is always considered a known parameter. Also, it is important to point out that this model has only \(k-1\) free parameters (instead of \(k\)), since the restriction \(\sum_{i=1}^k p_{it}=1\) implies that defining \(k-1\) entries of \(\vec{p}_t\) defines the remaining value. Specifically, we will always take the last entry (or category) of \(Y_t\) as the reference value, such that \(p_{kt}\) can be considered as the baseline probability of observing data from a category (i.e., we will model how each \(p_{it}\) relates to the baseline probability \(p_{kt}\)).
To create an outcome for this model, we can make use of the
Multinom
function:
For the Multinomial case, p
must be a character
vector of size \(k-1\)
containing the names of the linear predictors associated with \(\ln\left\{\frac{p_{it}}{p_{kt}}\right\}\)
for each \(i=1,...,k-1\).
The data
argument must be a \(T \times k\) matrix containing the values
of \(Y_t\) for each observation. Notice
that each line \(i\) must represent the
values of all categories in time \(i\)
and each column \(j\) must represent
the values of a category \(j\) through
time. If a value of the argument data
is not available
(NA
) for a specific time, it is assumed that there was no
observation at that time, thus the update step of the filtering
algorithm will be skipped at that time. Note that the evolution step
will still be performed, such that the predictive distribution for the
missing data and the updated distribution for the latent states at that
time will still be provided.
The offset
argument is optional and must have the same
dimensions of data
(its dimensions are interpreted in the
same manner). The argument can be used to provide a measure of the scale
of the data and, if the offset is provided, such that, at each time
\(t\), the offset is equal to \(E_t=(E_{1t},...,E_{kt})'\), then we
will fit a model assuming that:
\[ \begin{equation}\begin{aligned} Y_t|\theta_t &\sim Multinom\left(N_t,\vec{p}^*_t\right),\\ \ln\left\{\frac{p^*_{it}}{p^*_{kt}}\right\}&=\ln\left\{\frac{p_{it}}{p_{kt}}\right\}+\ln\left\{\frac{E_{it}}{E_{kt}}\right\}, i=1,...,k-1. \end{aligned}\end{equation} \]
At the end of this subsection we present a brief discussion about the implications of the inclusion of the offset and how to interpret it, as well as a explanation for the way we chose to include it.
Again, we present a brief example for the usage of this outcome:
# Multinomial case
structure <- (
polynomial_block(p = 1, order = 2, D = 0.95) +
harmonic_block(p = 1, period = 12, D = 0.975) +
noise_block(p = 1, R1 = 0.1) +
regression_block(p = chickenPox$date >= as.Date("2013-09-01"))
# Vaccine was introduced in September of 2013
) * 4
outcome <- Multinom(p = structure$pred.names, data = chickenPox[, c(2, 3, 4, 6, 5)])
fitted.data <- fit_model(structure, chickenPox = outcome)
summary(fitted.data)
plot(fitted.data, plot.pkg = "base")
The model presented in this section is intend to describe a phenomena such that we have \(N_t\) subjects that were distributed randomly (but not necessarily uniformly randomly) among \(k\) categories. In this scenario, \(p_{it}\) represent the probability of one observation to fall within the category \(i\), such that:
\[ p_{it}=\mathbb{P}(Y_{it}=1|N_t=1). \]
In some applications, it might be the case that \(N_t\) represents the counting of some event of interest and we want to model the probability of this event occurring in each category. In this scenario, it is not clear how to use the multinomial model, since we will have that:
\[ p_{it}=\mathbb{P}(\text{Observation belong to category }i|\text{Event occured}), \] but we actually want to known:
\[ p^*_{it}=\mathbb{P}(\text{Event occured}|\text{Observation belong to category }i). \]
Notice that we can write:
\[ \begin{aligned} p^*_{it}&=\mathbb{P}(\text{Event occured}|\text{Observation belong to category }i)\\ &=\frac{\mathbb{P}(\text{Observation belong to category }i|\text{Event occured})\mathbb{P}(\text{Event occured})}{\mathbb{P}(\text{Observation belong to category }i)}\\ &=\frac{p_{it}\mathbb{P}(\text{Event occured})}{\mathbb{P}(\text{Observation belong to category }i)}. \end{aligned} \]
The above relation implies that:
\[ \begin{aligned} \ln\left\{\frac{p^*_{it}}{p^*_{kt}}\right\} &=\ln\left\{\frac{p_{it}}{p_{kt}}\right\}-\ln\left\{\frac{\mathbb{P}(\text{Observation belong to category }i)}{\mathbb{P}(\text{Observation belong to category }k)}\right\}. \end{aligned} \]
If we pass to the offset
argument of the
Multinom
function a set of values \(E_t\), such that \(E_{t} \propto (\mathbb{P}(\text{Observation belong
to category }1),...,\mathbb{P}(\text{Observation belong to category
}k))'\), then, by the specification provided in this section,
we have that:
\[ \ln\left\{\frac{p^*_{it}}{p^*_{kt}}\right\}=\lambda_{it}, \] in other words, the linear predictors (and consequently, the model structure) will describe the probability that an event occurs in a specific class (instead of the probability that an observation belongs to that class, given the occurrence of the event).
To obtain \(p^*_{it}\) itself (i.e. the probability of the event occurring given that the observation belongs to the category \(i\)), one can use Bayes formula, as long \(\mathbb{P}(\text{Event occured})\) is known. Indeed, one can write:
\[ \begin{aligned} p^*_{it}&=p_{it}\frac{\mathbb{P}(\text{Event occured})}{\mathbb{P}(\text{Observation belong to category }i)}\\ &=\frac{\exp\{\lambda_i\}}{1+\sum_j \exp\{\lambda_j\}}\frac{\mathbb{P}(\text{Event occured})}{\mathbb{P}(\text{Observation belong to category }i)} \end{aligned} \]
Lastly, the kDGLM package also allows for the user to jointly fit multiple time series, as long as the marginal distribution of each series is one of the supported distributions AND the series are independent given the latent state vector \(\vec{\theta}_t\). In other words, let \(\{\vec{Y}_{i,t}\}_{t=1}^{T}, i =1,...,r\), be a set of time series such that:
\[ \begin{aligned} \vec{Y}_{i,t}|\vec{\eta}_{i,t} &\sim \mathcal{F}_{i}\left(\vec{\eta}_{i,t}\right),\\ g_i(\vec{\eta}_{i,t})&=\vec{\lambda}_{i,t}=F_{i,t}'\vec{\theta}_{t}, \end{aligned} \] and \(\vec{Y}_{1,t}, ...,\vec{Y}_{r,t}\) are mutually independent given \(\vec{\eta}_{1,t}, ...,\vec{\eta}_{r,t}\). Note that the observational distributions \(\mathcal{F}_i\) does not need to be the same for each outcome, as long as each \(\mathcal{F}_i\) is within the supported marginal distributions. For example, we could have three time series (\(r=3\)), such that \(\mathcal{F}_1\) is a Poisson distribution, \(\mathcal{F}_2\) is Normal distribution with unknown mean and precision and \(\mathcal{F}_3\) is a Gamma distribution with known shape. Also, this specification does not impose any restriction on the model structure, such that each outcome can have its own component, with polynomial, regression and harmonic blocks, besides having shared components with each other. See (dos Santos et al., 2024) for a detailed discussion of the approach used to model multiple time series using kDGLMs.
To fit such model, one must only pass the outcomes to the
fit_model
function. As an example, we present the code for
fitting two Poisson series:
structure <- polynomial_block(mu.1 = 1, mu.2 = 1, order = 2, D = 0.95) + # Common factor
harmonic_block(mu.2 = 1, period = 12, order = 2, D = 0.975) + # Seasonality for Series 2
polynomial_block(mu.2 = 1, order = 1, D = 0.95) + # Local level for Series 2
noise_block(mu = 1) * 2 # Overdispersion for both Series
fitted.model <- fit_model(structure,
Adults = Poisson(lambda = "mu.1", data = chickenPox[, 5]),
Infants = Poisson(lambda = "mu.2", data = chickenPox[, 2])
)
plot(fitted.model)
It is important to note that the Multivariate Normal and the Multinomial cases are multivariated outcomes and are not considered multiple outcomes on their own, but instead, they are treated as one outcome each, such that the outcome itself is a vector (note that we made no restrictions on the dimension of each \(\vec{Y}_{i,t}\)). As such, in those cases, the components of the vector \(\vec{Y}_{i,t}\) do not have to be mutually independent given \(\vec{\eta}_{i,t}\).
Also important to note is that our general approach for modeling multiple time series can not, on its own, be considered a generalization of the Multivariate Normal or Multinomial models. Specifically, if we treat each coordinate of the outcome as a outcome of its own, they would not satisfy the hypotheses of independence given the latent states \(\vec{\theta}_t\). This can be compensated with changes to the model structure, but, in general, it is better to model data using a known joint distribution than to assume conditional independence and model the outcomes dependence by shared structure.
There is a special type of specification for a model with multiple outcomes that does not require the outcomes to be independent given the latent states. Indeed, if the user specifies the conditional distribution of each outcome given the previous ones, then no hypotheses is needed for fitting the data.
For instance, lets say that there are three time series \(Y_{1,t},Y_{2,t}\) and \(Y_{3,t}\), such that each series follows a Poisson distribution with parameter \(\eta_{i,t}, i=1,2,3\). Then, \(Z_t=Y_{1,t}+Y_{2,t}+Y_{3,t}\) follows a Poisson distribution with parameter \(\eta_{1,t}+\eta_{2,t}+\eta_{3,t}\) and \(Y_{1,t},Y_{2,t},Y_{3,t}|Z_t\) jointly follows a Multinomial distribution with parameters \(N_t=Z_t\) and \(\vec{p}_t=\left(\frac{\eta_{1,t}}{\eta_{1,t}+\eta_{2,t}+\eta_{3,t}},\frac{\eta_{2,t}}{\eta_{1,t}+\eta_{2,t}+\eta_{3,t}},\frac{\eta_{3,t}}{\eta_{1,t}+\eta_{2,t}+\eta_{3,t}}\right)'\). Then the user may model \(Z_t\) and \(Y_{1,t},Y_{2,t},Y_{3,t}|Z_t\):
structure <- polynomial_block(mu = 1, order = 2, D = 0.95) +
harmonic_block(mu = 1, period = 12, order = 2, D = 0.975) +
noise_block(mu = 1) + polynomial_block(p = 1, D = 0.95) * 2
outcome1 <- Poisson(lambda = "mu", data = rowSums(chickenPox[, c(2, 3, 5)]))
outcome2 <- Multinom(p = c("p.1", "p.2"), data = chickenPox[, c(2, 3, 5)])
fitted.model <- fit_model(structure, Total = outcome1, Proportions = outcome2)
plot(fitted.model, plot.pkg = "base")
See Schmidt et al. (2022) for a discussion of Multinomial-Poisson models. More applications are presented in the advanced examples section of the vignette.