Bold lowercase for vectors
Bold subscript \(\boldsymbol{s}\) since x, y
Bold uppercase for matrices
\(\phi\) for all ‘dispersion’ parameters for consistency with code
Family titles link to TMB or sdmTMB source code
Attempt to link math symbols to argument options in model and code symbols
\[ \begin{align} \mathbb{E}(y_{\boldsymbol{s},t}) &= \mu_{\boldsymbol{s},t},\\ \mu_{\boldsymbol{s},t} &= g^{-1} \left( \boldsymbol{X} \boldsymbol{\beta} + \omega_s + \epsilon_{\boldsymbol{s},t} \right),\\ \boldsymbol{\omega} &\sim \operatorname{MVNormal} \left( \boldsymbol{0}, \boldsymbol{\Sigma}_\omega \right),\\ \boldsymbol{\epsilon}_t &\sim \operatorname{MVNormal} \left( \boldsymbol{0}, \boldsymbol{\Sigma}_{\epsilon} \right). \end{align} \] where \(g\) is a link function and \(g^{-1}\) is the inverse link.
\(\boldsymbol{X} \boldsymbol{\beta}\) is defined by the formula argument
\(\omega_s\) are included if include_spatial = TRUE
\(\epsilon_{\boldsymbol{s},t}\) are included if there are multiple time elements and spatial_only = FALSE (the default if there are multiple time elements)
\[ \begin{align} \mu_{\boldsymbol{s},t} &= g^{-1} \left( \ldots + \gamma_{t} x_{\boldsymbol{s},t} + \ldots \right),\\ \gamma_{t} &\sim \operatorname{Normal} \left(\gamma_{t-1}, \sigma^2_{\gamma} \right). \end{align} \]
\[ \begin{align} \mu_{\boldsymbol{s},t} &= g^{-1} \left( \ldots + \zeta_s x_t + \ldots \right),\\ \boldsymbol{\zeta} &\sim \operatorname{MVNormal} \left( \boldsymbol{0}, \boldsymbol{\Sigma}_\zeta \right). \end{align} \]
Dropping the optional \(\omega_s\) for simplicity:
\[ \begin{align} \mu_{\boldsymbol{s},t} &= g^{-1} \left( \boldsymbol{X} \boldsymbol{\beta} + \delta_{\boldsymbol{s},t} \right),\\ \boldsymbol{\delta}_{t=1} &\sim \operatorname{MVNormal} (\boldsymbol{0}, \boldsymbol{\Sigma}_{\epsilon}),\\ \boldsymbol{\delta}_{t>1} &= \phi \boldsymbol{\delta}_{t-1} + \sqrt{1 - \phi^2} \boldsymbol{\epsilon}_t, \: \boldsymbol{\epsilon}_t \sim \operatorname{MVNormal} \left(\boldsymbol{0}, \boldsymbol{\Sigma}_{\epsilon} \right). \end{align} \]
Offset terms can be included with the reserved word offset in the formula. E.g., y ~ x + offset.
These are included in the linear predictor as
\[ \begin{align} \mu_{\boldsymbol{s},t} &= g^{-1} \left( \ldots + O_{\boldsymbol{s},t} + \ldots \right), \end{align} \] where \(O_{\boldsymbol{s},t}\) is an offset term—a log transformed variable without a coefficient. Offsets only make sense if the link is log.
TODO
These models can be fit by including + breakpt(x) in the model formula, where x is a covariate.
The form is
\[ s(x)=\tau + \psi\ { \left[ 1+{ e }^{ -\ln\ \left(19\right) \cdot \left( x-s50 \right) / \left(s95 - s50 \right) } \right] }^{-1}, \] where \(\psi\) is a scaling parameter (controlling the height of the y-axis for the response, and is unconstrained), \(\tau\) is an intercept, \(s50\) is a parameter controlling the point at which the function reaches 50% of the maximum (\(\psi\)), and \(s95\) is a parameter controlling the point at which the function reaches 95%. The parameter \(s50\) is unconstrained, and \(s95\) is constrained to be larger than \(s50\).
These models can be fit by including + logistic(x) in the model formula, where x is a covariate.
Internally parameterized as the robust version.
\[ \operatorname{Beta} \left(\mu \phi, 1 - \mu \phi \right) \]
where \(\phi\) is variance.
\[ \operatorname{Binomial} \left( N, \mu \right) \]
\(N = 1\) (i.e., ‘size’ currently fixed) and \(\mu\) is probability
As shape, scale:
\[ \operatorname{Gamma} \left( \phi, \frac{\mu}{\phi} \right) \]
where \(\phi\) represents the shape and \(\frac{\mu}{\phi}\) represents the scale.
\[ \operatorname{Normal} \left( \mu, \phi \right) \] where \(\phi\) is the standard deviation (following Stan convention of SD not variance).
\[ \operatorname{Lognormal} \left( \log \mu - \frac{\phi^2}{2}, \phi \right) \]
Internally parameterized as the robust version
\[ \operatorname{NB2} \left( \mu, \phi \right) \]
Variance scales quadratically with mean \(\mathrm{Var}[y] = \mu + \mu^2 / \phi\).
\[ \operatorname{Poisson} \left( \mu \right) \]
\[ \operatorname{Student-t} \left( \mu, \phi, \nu \right) \]
where \(\nu\), the degrees of freedom, is currently fixed at 3.
Source code as in cplm package. https://link.springer.com/article/10.1007/s11222-012-9343-7
\[ \operatorname{Tweedie} \left(\mu, p, \phi \right), \: 1 < p < 2 \]