## Notation conventions

• 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

## Basic model structure

\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)

## Time-varying regression parameters

\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}

## Spatial regression parameters

\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}

## AR1 spatiotemporal random fields

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

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.

## Threshold models

### Linear breakpoint threshold models

TODO

These models can be fit by including + breakpt(x) in the model formula, where x is a covariate.

### Logistic threshold models

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.

## Observation model families

### Binomial

Internally parameterized as the robust version.

### Beta

$\operatorname{Beta} \left(\mu \phi, 1 - \mu \phi \right)$

where $$\phi$$ is variance.

$\operatorname{Binomial} \left( N, \mu \right)$

$$N = 1$$ (i.e., ‘size’ currently fixed) and $$\mu$$ is probability

### Gamma

As shape, scale:

$\operatorname{Gamma} \left( \phi, \frac{\mu}{\phi} \right)$

where $$\phi$$ represents the shape and $$\frac{\mu}{\phi}$$ represents the scale.

### Gaussian

$\operatorname{Normal} \left( \mu, \phi \right)$ where $$\phi$$ is the standard deviation (following Stan convention of SD not variance).

### Lognormal

$\operatorname{Lognormal} \left( \log \mu - \frac{\phi^2}{2}, \phi \right)$

### Negative Binomial

Internally parameterized as the robust version

$\operatorname{NB2} \left( \mu, \phi \right)$

Variance scales quadratically with mean $$\mathrm{Var}[y] = \mu + \mu^2 / \phi$$.

### Poisson

$\operatorname{Poisson} \left( \mu \right)$

### Student-t

$\operatorname{Student-t} \left( \mu, \phi, \nu \right)$

where $$\nu$$, the degrees of freedom, is currently fixed at 3.

### Tweedie

$\operatorname{Tweedie} \left(\mu, p, \phi \right), \: 1 < p < 2$