 Defines a new observational or a new auxiliary channel for the model using standard R formula syntax. Formulas of individual response variables can be joined together via +. See 'Details' and the package vignette for more information. The function obs is a shorthand alias for dynamiteformula, and aux is a shorthand alias for dynamiteformula(formula, family = "deterministic").

## Usage

dynamiteformula(formula, family)

obs(formula, family)

aux(formula)

# S3 method for dynamiteformula
+(e1, e2)

# S3 method for dynamiteformula
print(x, ...)

## Arguments

formula

[formula]
An R formula describing the model.

family

[character(1)]
The family name. See 'Details' for the supported families.

e1

[dynamiteformula]
A model formula specification.

e2

[dynamiteformula]
A model formula specification.

x

[dynamiteformula]
The model formula.

...

Ignored.

## Value

A dynamiteformula object.

## Details

Currently the dynamite package supports the following distributions for the observations:

• Categorical: categorical (with a softmax link using the first category as reference). See the documentation of the categorical_logit_glm in the Stan function reference manual (https://mc-stan.org/users/documentation/).

• Multinomial: multinomial (softmax link, first category is reference).

• Gaussian: gaussian (identity link, parameterized using mean and standard deviation).

• Multivariate Gaussian: mvgaussian (identity link, parameterized using mean vector, standard deviation vector and the Cholesky decomposition of the correlation matrix).

• Poisson: poisson (log-link, with an optional known offset variable).

• Negative-binomial: negbin (log-link, using mean and dispersion parameterization, with an optional known offset variable). See the documentation on NegBinomial2 in the Stan function reference manual.

• Bernoulli: bernoulli (logit-link).

• Binomial: binomial (logit-link).

• Exponential: exponential (log-link).

• Gamma: gamma (log-link, using mean and shape parameterization).

• Beta: beta (logit-link, using mean and precision parameterization).

• Student t: student (identity link, parametrized using degrees of freedon, location and scale)

The models in the dynamite package are defined by combining the channel-specific formulas defined via R formula syntax. Each channel is defined via the obs function, and the channels are combined with +. For example a formula obs(y ~ lag(x), family = "gaussian") + obs(x ~ z, family = "poisson") defines a model with two channels; first we declare that y is a gaussian variable depending on a previous value of x (lag(x)), and then we add a second channel declaring x as Poisson distributed depending on some exogenous variable z (for which we do not define any distribution).

Number of trials for binomial channels should be defined via a trials model component, e.g., obs(y ~ x + trials(n), family = "binomial"), where n is a data variable defining the number of trials. For multinomial channels, the number of trials is automatically defined to be the sum of the observations over the categories, but can also be defined using the trials component, for example for prediction.

Multivariate channels are defined by providing a single formula for all components or by providing component-specific formulas separated by a |. The response variables that correspond to the components should be joined by c(). For instance, the following would define c(y1, y2) as multivariate gaussian with x as a predictor for the mean of the first component and x and z as predictors for the mean of the second component: obs(c(y1, y2) ~ x | x + z, family = "mvgaussian"). A multinomial channel should only have a single formula.

In addition to declaring response variables via obs, we can also use the function aux to define auxiliary channels which are deterministic functions of other variables. The values of auxiliary variables are computed dynamically during prediction, making the use of lagged values and other transformations possible. The function aux also does not use the family argument, which is automatically set to deterministic and is a special channel type of obs. Note that lagged values of deterministic aux channels do not imply fixed time points. Instead they must be given starting values using a special function init that directly initializes the lags to specified values, or by past which computes the initial values based on an R expression. Both init and past should appear on the right hand side of the model formula, separated from the primary defining expression via |.

The formula within obs can also contain an additional special function varying, which defines the time-varying part of the model equation, in which case we could write for example obs(x ~ z + varying(~ -1 + w), family = "poisson"), which defines a model equation with a constant intercept and time-invariant effect of z, and a time-varying effect of w. We also remove the duplicate intercept with -1 in order to avoid identifiability issues in the model estimation (we could also define a time varying intercept, in which case we would write obs(x ~ -1 + z + varying(~ w), family = "poisson)). The part of the formula not wrapped with varying is assumed to correspond to the fixed part of the model, so obs(x ~ z + varying(~ -1 + w), family = "poisson") is actually identical to obs(x ~ -1 + fixed(~ z) + varying(~ -1 + w), family = "poisson") and obs(x ~ fixed(~ z) + varying(~ -1 + w), family = "poisson").

When defining varying effects, we also need to define how the these time-varying regression coefficient behave. For this, a splines component should be added to the model, e.g., obs(x ~ varying(~ -1 + w), family = "poisson) + splines(df = 10) defines a cubic B-spline with 10 degrees of freedom for the time-varying coefficient corresponding to the w. If the model contains multiple time-varying coefficients, same spline basis is used for all coefficients, with unique spline coefficients and their standard deviation.

If the desired model contains lagged predictors of each response in each channel, these can be quickly added to the model as either time-invariant or time-varying predictors via lags() instead of writing them manually for each channel.

It is also possible to define group-specific (random) effects term using the special syntax random() similarly as varying(). For example, random(~1) leads to a model where in addition to the common intercept, each individual/group has their own intercept with zero-mean normal prior and unknown standard deviation analogously with the typical mixed models. An additional model component random_spec() can be used to define whether the random effects are allowed to correlate within and across channels and whether to use centered or noncentered parameterization for the random effects.

Model formula construction dynamite(), lags(), lfactor(), random_spec(), splines()

## Examples

# A single gaussian response channel with a time-varying effect of 'x',
# and a time-varying effect of the lag of 'y' using B-splines with
# 20 degrees of freedom for the coefficients of the time-varying terms.
obs(y ~ -1 + varying(~x), family = "gaussian") +
lags(type = "varying") +
splines(df = 20)
#>   Family   Formula
#> y gaussian y ~ -1 + varying(~x)
#>
#> Lagged responses added as varying predictors with: k = 1

# A two-channel categorical model with time-invariant predictors
# here, lag terms are specified manually
obs(x ~ z + lag(x) + lag(y), family = "categorical") +
obs(y ~ z + lag(x) + lag(y), family = "categorical")
#>   Family      Formula
#> x categorical x ~ z + lag(x) + lag(y)
#> y categorical y ~ z + lag(x) + lag(y)

# The same categorical model as above, but with the lag terms
obs(x ~ z, family = "categorical") +
obs(y ~ z, family = "categorical") +
lags(type = "fixed")
#>   Family      Formula
#> x categorical x ~ z
#> y categorical y ~ z
#>
#> Lagged responses added as fixed predictors with: k = 1

# A multichannel model with a gaussian, Poisson and a Bernoulli response and
# an auxiliary channel for the logarithm of 'p' plus one
obs(g ~ lag(g) + lag(logp), family = "gaussian") +
obs(p ~ lag(g) + lag(logp) + lag(b), family = "poisson") +
obs(b ~ lag(b) * lag(logp) + lag(b) * lag(g), family = "bernoulli") +
aux(numeric(logp) ~ log(p + 1))
#>      Family        Formula
#> g    gaussian      g ~ lag(g) + lag(logp)
#> p    poisson       p ~ lag(g) + lag(logp) + lag(b)
#> b    bernoulli     b ~ lag(b) * lag(logp) + lag(b) * lag(g)
#> logp deterministic numeric(logp) ~ log(p + 1)

obs(y ~ x, family = "gaussian") + obs(z ~ w, family = "exponential")
#>   Family      Formula
#> y gaussian    y ~ x
#> z exponential z ~ w

x <- obs(y ~ x + random(~ 1 + lag(d)), family = "gaussian") +
obs(z ~ varying(~w), family = "exponential") +
aux(numeric(d) ~ log(y) | init(c(0, 1))) +
lags(k = 2) +
splines(df = 5) +
random_spec(correlated = FALSE)
#> Warning: Both time-independent and time-varying intercept specified:
#> ℹ Defaulting to time-varying intercept.
print(x)
#>   Family        Formula
#> y gaussian      y ~ x + random(~1 + lag(d))
#> z exponential   z ~ varying(~w)
#> d deterministic numeric(d) ~ log(y) | init(c(0, 1))
#>
#> Lagged responses added as fixed predictors with: k = 2
#> Random effects added for response(s): y