## Notation conventions

This appendix uses the following notation conventions, which generally follows the guidance in Edwards & Auger-Méthé (2019):

Greek symbols for parameters,

the Latin/Roman alphabet for data (except \(\boldsymbol{Q}\) and \(\boldsymbol{H}\), which are used by convention),

bold symbols for vectors or matrices (e.g., \(\boldsymbol{\omega}\) is a vector and \(\omega_{\boldsymbol{s}}\) is the value of \(\boldsymbol{\omega}\) at point in space \(\boldsymbol{s}\)),

\(\phi\) for all distribution dispersion parameters for consistency with the code,

\(\mathbb{E}[y]\) to define the expected value (mean) of variable \(y\),

\(\mathrm{Var}[y]\) to define the expected variance of the variable \(y\),

a \(^*\) superscript represents interpolated or projected values as opposed to values at knot locations (e.g., \(\boldsymbol{\omega}\) vs. \(\boldsymbol{\omega}^*\)), and

where possible, notation has been chosen to match VAST (Thorson 2019) to maintain consistency (e.g., \(\boldsymbol{\omega}\) for spatial fields and \(\boldsymbol{\epsilon}_t\) for spatiotemporal fields).

We include tables of all major indices (Table 1) and symbols (Table 2).

Symbol | Description |
---|---|

\(\boldsymbol{s}\) | Index for space; a vector of x and y coordinates |

\(t\) | Index for time |

\(g\) | Group |

Symbol | Code | Description |
---|---|---|

\(y\) | `y_i` |
Observed response data |

\(\mu\) | `mu_i` |
Mean |

\(\phi\) | `phi` |
A dispersion parameter for a distribution |

\(f\) | `fit$family$link` |
Link function |

\(f^{-1}\) | `fit$family$linkinv` |
Inverse link function |

\(\boldsymbol{\beta}\) | `b_j` |
Parameter vector |

\(\boldsymbol{X}\) | `X_ij` |
A predictor model matrix |

\(O_{\boldsymbol{s}, t}\) | `offset` |
An offset variable at point \(\boldsymbol{s}\) and time \(t\) |

\(\omega_{\boldsymbol{s}}\) | `omega_s` |
Spatial random field at point \(\boldsymbol{s}\) (knot) |

\(\omega_{\boldsymbol{s}}^*\) | `omega_s_A` |
Spatial random field at point \(\boldsymbol{s}\) (interpolated) |

\(\zeta_{\boldsymbol{s}}\) | `zeta_s` |
Spatially varying coefficient random field at point \(\boldsymbol{s}\) (knot) |

\(\zeta_{\boldsymbol{s}}^*\) | `zeta_s_A` |
Spatially varying coefficient random field at point \(\boldsymbol{s}\) (interpolated) |

\(\epsilon_{\boldsymbol{s}, t}\) | `epsilon_st` |
Spatiotemporal random field at point \(\boldsymbol{s}\) and time \(t\) (knot) |

\(\epsilon_{\boldsymbol{s}, t}^*\) | `epsilon_st_A` |
Spatiotemporal random field at point \(\boldsymbol{s}\) and time \(t\) (interpolated) |

\(\delta_{\boldsymbol{s},t}\) | `b_t` |
AR(1) or random walk spatiotemporal deviations (knot) |

\(\alpha_g\) | `RE` |
IID random intercept deviation for group \(g\) |

\(\boldsymbol{\Sigma}_\omega\) | `-` |
Spatial random field covariance matrix |

\(\boldsymbol{\Sigma}_\zeta\) | `-` |
Spatially varying coefficient random field covariance matrix |

\(\boldsymbol{\Sigma}_\epsilon\) | `-` |
Spatiotemporal random field covariance matrix |

\(\boldsymbol{Q}_\omega\) | `Q_s` |
Spatial random field precision matrix |

\(\boldsymbol{Q}_\zeta\) | `Q_s` |
Spatially varying coefficient random field precision matrix |

\(\boldsymbol{Q}_\epsilon\) | `Q_st` |
Spatiotemporal random field precision matrix |

\(\sigma_\alpha^2\) | `sigma_G` |
IID random intercept variance |

\(\sigma_\epsilon^2\) | `sigma_E` |
Spatiotemporal random field marginal variance |

\(\sigma_\omega^2\) | `sigma_O` |
Spatial random field marginal variance |

\(\sigma_\zeta^2\) | `sigma_Z` |
Spatially varying coefficient random field marginal variance |

\(\kappa_\omega\) | `kappa(0)` |
Spatial decorrelation rate |

\(\kappa_\epsilon\) | `kappa(1)` |
Spatiotemporal decorrelation rate |

\(\rho\) | `ar1_rho` |
Correlation between random fields in subsequent time steps |

\(\boldsymbol{A}\) | `A` |
Sparse projection matrix to interpolate between knot and data locations |

\(\boldsymbol{H}\) | `H` |
2-parameter rotation matrix used to define anisotropy |

## sdmTMB model structure

The complete sdmTMB model can be written as

\[ \begin{aligned} \mathbb{E}[y_{\boldsymbol{s},t}] &= \mu_{\boldsymbol{s},t},\\ \mu_{\boldsymbol{s},t} &= f^{-1} \left( \boldsymbol{X}^{\mathrm{main}}_{\boldsymbol{s},t} \boldsymbol{\beta} + O_{\boldsymbol{s},t} + \alpha_g + \boldsymbol{X}^{\mathrm{tvc}}_{\boldsymbol{s},t} \boldsymbol{\gamma_t} + \boldsymbol{X}^{\mathrm{svc}}_{\boldsymbol{s},t} \zeta_{\boldsymbol{s}} + \omega_{\boldsymbol{s}} + \epsilon_{\boldsymbol{s},t} \right), \end{aligned} \]

where

- \(y_{\boldsymbol{s},t}\) represents the response data at point \(\boldsymbol{s}\) and time \(t\);
- \(\mu\) represents the mean;
- \(f\) represents a link function (e.g., log or logit) and \(f^{-1}\) represents its inverse;
- \(\boldsymbol{X}^{\mathrm{main}}\), \(\boldsymbol{X}^{\mathrm{tvc}}\), and \(\boldsymbol{X}^{\mathrm{svc}}\) represent design matrices (the superscript identifiers ‘main’ = main effects, ‘tvc’ = time varying coefficients, and ‘svc’ = spatially varying coefficients);
- \(\boldsymbol{\beta}\) represents a vector of fixed-effect coefficients;
- \(O_{\boldsymbol{s},t}\) represents an offset: a covariate (usually log transformed) with a coefficient fixed at one;
- \(\alpha_{g}\) represents random intercepts by group \(g\), \(\alpha_{g}\sim \mathrm{N}(0,\sigma^2_\alpha)\);
- \(\gamma_{t}\) represents time-varying coefficients (a random walk), \(\gamma_{t} \sim \mathrm{N}(\gamma_{t-1},\sigma^2_\gamma)\);
- \(\zeta_{\boldsymbol{s}}\) represents spatially varying coefficients (a random field), \(\zeta_{\boldsymbol{s}} \sim \mathrm{MVN}(\boldsymbol{0},\boldsymbol{\Sigma}_\zeta)\);
- \(\omega_{\boldsymbol{s}}\) represents a spatial component (a random field), \(\omega_{\boldsymbol{s}} \sim \mathrm{MVN}(\boldsymbol{0},\boldsymbol{\Sigma}_\omega)\); and
- \(\epsilon_{\boldsymbol{s},t}\) represents a spatiotemporal component (a random field), \(\epsilon_{\boldsymbol{s},t} \sim \mathrm{MVN}(\boldsymbol{0},\boldsymbol{\Sigma}_{\epsilon})\).

A single sdmTMB model will rarely, if ever, contain all of the above components. Next, we will split the model to describe the various parts in more detail using ‘\(\ldots\)’ to represent the other optional components.

### Main effects

\[ \begin{aligned} \mu_{\boldsymbol{s},t} &= f^{-1} \left( \boldsymbol{X}^{\mathrm{main}}_{\boldsymbol{s},t} \boldsymbol{\beta} \ldots \right) \end{aligned} \]

Within `sdmTMB()`

, \(\boldsymbol{X}^{\mathrm{main}}_{\boldsymbol{s},t}
\boldsymbol{\beta}\) is defined by the `formula`

argument and represents the main-effect model matrix and a corresponding
vector of coefficients. This main effect formula can contain optional
penalized smoothers or non-linear functions as defined below.

#### Smoothers

Smoothers in sdmTMB are implemented with the same formula syntax
familiar to mgcv (Wood 2017) users fitting
GAMs (generalized additive models). Smooths are implemented in the
formula using `+ s(x)`

, which implements a smooth from
`mgcv::s()`

. Within these smooths, the same syntax commonly
used in `mgcv::s()`

can be applied, e.g. 2-dimensional
smooths may be constructed with `+ s(x, y)`

; smooths can be
specific to various factor levels, `+ s(x, by = group)`

;
smooths can vary according to a continuous variable,
`+ s(x, by = x2)`

; the basis function dimensions may be
specified, e.g. `+ s(x, k = 4)`

(see
`?mgcv::choose.k`

); and various types of splines may be
constructed such as cyclic splines to model seasonality,
e.g. `+ s(month, bs = "cc", k = 12)`

.

While mgcv can fit unpenalized (e.g., B-splines) or penalized splines
(P-splines), sdmTMB only implements penalized splines. The penalized
splines are constructed in sdmTMB using the function
`mgcv::smooth2random()`

, which transforms splines into random
effects (and associated design matrices) that are estimable in a
mixed-effects modelling framework. This is the same approach as is
implemented in the R packages gamm4 (Wood &
Scheipl 2020) and brms (Bürkner
2017).

#### Linear break-point threshold models

The linear break-point or “hockey stick” model can be used to describe threshold or asymptotic responses. This function consists of two pieces, so that for \(x < b_{1}\), \(s(x) = x \cdot b_{0}\), and for \(x > b_{1}\), \(s(x) = b_{1} \cdot b_{0}\). In both cases, \(b_{0}\) represents the slope of the function up to a threshold, and the product \(b_{1} \cdot b_{0}\) represents the value at the asymptote. No constraints are placed on parameters \(b_{0}\) or \(b_{1}\).

These models can be fit by including `+ breakpt(x)`

in the
model formula, where `x`

is a covariate. The formula can
contain a single break-point covariate.

#### Logistic threshold models

Models with logistic threshold relationships between a predictor and the response can be fit with the form

\[ s(x)=\tau + \psi\ { \left[ 1+{ e }^{ -\ln \left(19\right) \cdot \left( x-s50 \right) / \left(s95 - s50 \right) } \right] }^{-1}, \]

where \(s\) represents the logistic function, \(\psi\) is a scaling parameter (controlling the height of the y-axis for the response; 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% of the maximum. The parameter \(s50\) is unconstrained but \(s95\) 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. The formula can
contain a single logistic covariate.

### Spatial random fields

Spatial random fields, \(\omega_{\boldsymbol{s}}\), are included if
`spatial = 'on'`

(or `TRUE`

) and omitted if
`spatial = 'off'`

(or `FALSE`

).

\[
\begin{aligned}
\mu_{\boldsymbol{s},t} &= f^{-1} \left( \ldots +
\omega_{\boldsymbol{s}} + \ldots \right),\\
\boldsymbol{\omega} &\sim \operatorname{MVNormal} \left(
\boldsymbol{0}, \boldsymbol{\Sigma}_\omega \right),\\
\end{aligned}
\] The marginal standard deviation of \(\boldsymbol{\omega}\) is indicated by
`Spatial SD`

in the printed model output or as
`sigma_O`

in the output of
`sdmTMB::tidy(fit, "ran_pars")`

. The ‘O’ is for ‘omega’
(\(\omega\)).

Internally, the random fields follow a Gaussian Markov random field (GMRF)

\[ \boldsymbol{\omega} \sim \mathrm{MVNormal}\left(\boldsymbol{0}, \sigma_\omega^2 \boldsymbol{Q}^{-1}_\omega\right), \] where \(\boldsymbol{Q}_\omega\) is a sparse precision matrix and \(\sigma_\omega^2\) is the marginal variance.

### Spatiotemporal random fields

Spatiotemporal random fields are included by default if there are
multiple time elements (`time`

argument is not
`NULL`

) and can be set to IID (independent and identically
distributed, `'iid'`

; default), AR(1) (`'ar1'`

),
random walk (`'rw'`

), or off (`'off'`

) via the
`spatiotemporal`

argument. These text values are case
insensitive.

Spatiotemporal random fields are represented by \(\boldsymbol{\epsilon}_t\) within sdmTMB.
This has been chosen to match the representation in VAST (Thorson 2019). The marginal standard deviation
of \(\boldsymbol{\epsilon}_t\) is
indicated by `Spatiotemporal SD`

in the printed model output
or as `sigma_E`

in the output of
`sdmTMB::tidy(fit, "ran_pars")`

. The ‘E’ is for ‘epsilon’
(\(\epsilon\)).

#### IID spatiotemporal random fields

IID spatiotemporal random fields
(`spatiotemporal = 'iid'`

) can be represented as

\[ \begin{aligned} \mu_{\boldsymbol{s},t} &= f^{-1} \left( \ldots + \epsilon_{\boldsymbol{s},t} + \ldots \right),\\ \boldsymbol{\epsilon_{t}} &\sim \operatorname{MVNormal} \left( \boldsymbol{0}, \boldsymbol{\Sigma}_{\epsilon} \right). \end{aligned} \]

where \(\epsilon_{\boldsymbol{s},t}\) represent random field deviations at point \(\boldsymbol{s}\) and time \(t\). The random fields are assumed independent across time steps.

Similarly to the spatial random fields, these spatiotemporal random fields (including all versions described below) are parameterized internally with a sparse precision matrix (\(\boldsymbol{Q}_\epsilon\))

\[ \boldsymbol{\epsilon_{t}} \sim \mathrm{MVNormal}\left(\boldsymbol{0}, \sigma_\epsilon^2 \boldsymbol{Q}^{-1}_\epsilon\right). \]

#### AR(1) spatiotemporal random fields

First-order auto regressive, AR(1), spatiotemporal random fields
(`spatiotemporal = 'ar1'`

) add a parameter defining the
correlation between random field deviations from one time step to the
next. They are defined as

\[
\begin{aligned}
\mu_{\boldsymbol{s},t} &= f^{-1} \left( \ldots +
\delta_{\boldsymbol{s},t} \ldots \right),\\
\boldsymbol{\delta}_{t=1} &\sim \operatorname{MVNormal}
(\boldsymbol{0}, \boldsymbol{\Sigma}_{\epsilon}),\\
\boldsymbol{\delta}_{t>1} &= \rho \boldsymbol{\delta}_{t-1} +
\sqrt{1 - \rho^2} \boldsymbol{\epsilon_{t}}, \:
\boldsymbol{\epsilon_{t}} \sim \operatorname{MVNormal}
\left(\boldsymbol{0}, \boldsymbol{\Sigma}_{\epsilon} \right),
\end{aligned}
\] where \(\rho\) is the
correlation between subsequent spatiotemporal random fields. The \(\rho \boldsymbol{\delta}_{t-1} + \sqrt{1 -
\rho^2}\) term scales the spatiotemporal variance by the
correlation such that it represents the steady-state marginal variance.
The correlation \(\rho\) allows for
mean-reverting spatiotemporal fields, and is constrained to be \(-1 < \rho < 1\). Internally, the
parameter is estimated as `ar1_phi`

, which is unconstrained.
The parameter `ar1_phi`

is transformed to \(\rho\) with \(\rho = 2 \left(
\mathrm{logit}^{-1}(\texttt{ar1\_phi}) - 1 \right)\).

#### Random walk spatiotemporal random fields (RW)

Random walk spatiotemporal random fields
(`spatiotemporal = 'rw'`

) represent a model where the
difference in spatiotemporal deviations from one time step to the next
are IID. They are defined as

\[ \begin{aligned} \mu_{\boldsymbol{s},t} &= f^{-1} \left( \ldots + \delta_{\boldsymbol{s},t} + \ldots \right),\\ \boldsymbol{\delta}_{t=1} &\sim \operatorname{MVNormal} (\boldsymbol{0}, \boldsymbol{\Sigma}_{\epsilon}),\\ \boldsymbol{\delta}_{t>1} &= \boldsymbol{\delta}_{t-1} + \boldsymbol{\epsilon_{t-1}}, \: \boldsymbol{\epsilon_{t-1}} \sim \operatorname{MVNormal} \left(\boldsymbol{0}, \boldsymbol{\Sigma}_{\epsilon} \right), \end{aligned} \]

where the distribution of the spatiotemporal field in the initial time step is the same as for the AR(1) model, but the absence of the \(\rho\) parameter allows the spatiotemporal field to be non-stationary in time. Note that, in contrast to the AR(1) parametrization, the variance is no longer the steady-state marginal variance.

### Time-varying regression parameters

Parameters can be modelled as time varying (a random walk) according to the form

\[ \begin{aligned} \mu_{\boldsymbol{s},t} &= f^{-1} \left( \ldots + \boldsymbol{X}^{\mathrm{tvc}}_{\boldsymbol{s},t} \boldsymbol{\gamma_{t}} + \ldots \right),\\ \gamma_{t=1} &\sim \operatorname{Uniform} \left(-\infty, \infty \right),\\ \gamma_{t>1} &\sim \operatorname{Normal} \left(\gamma_{t-1}, \sigma^2_{\gamma} \right), \end{aligned} \]

where \(\boldsymbol{\gamma_t}\) is
an optional vector of time-varying regression parameters and \(\boldsymbol{X}^{\mathrm{tvc}}_{\boldsymbol{s},t}\)
is the corresponding model matrix with covariate values. This is defined
via the `time_varying`

argument, assuming that the
`time`

argument is also supplied a column name. For example,
`time_varying ~ 0 + x`

. The first value is given an implicit
uniform prior. I.e., the same variable should not appear in the fixed
effect formula since the initial value is estimated as part of the
time-varying formula. The formula `time_varying = ~ 1`

implicitly represents a time-varying intercept and, this case, the
intercept should be omitted from the main effects
(`formula ~ + 0 + ...`

or
`formula ~ -1 + ...`

).

### Spatially varying coefficients (SVC)

Spatially varying coefficient models are defined as

\[ \begin{aligned} \mu_{\boldsymbol{s},t} &= f^{-1} \left( \ldots + \boldsymbol{X}^{\mathrm{svc}}_{\boldsymbol{s}, t} \zeta_{\boldsymbol{s}} + \ldots \right),\\ \boldsymbol{\zeta} &\sim \operatorname{MVNormal} \left( \boldsymbol{0}, \boldsymbol{\Sigma}_\zeta \right), \end{aligned} \]

where \(\boldsymbol{\zeta}\) is a
random field representing a spatially varying coefficient. Usually,
\(\boldsymbol{X}^{\mathrm{svc}}_{\boldsymbol{s},
t}\) would represent a prediction matrix that is constant
spatially for a given time \(t\) as
defined by a one-sided formula supplied to `spatial_varying`

.
For example `spatial_varying = ~ 0 + x`

, where `0`

omits the intercept.

The random fields are parameterized internally with a sparse precision matrix (\(\boldsymbol{Q}_\zeta\))

\[ \boldsymbol{\zeta} \sim \mathrm{MVNormal}\left(\boldsymbol{0}, \sigma_\zeta^2 \boldsymbol{Q}^{-1}_\zeta\right). \]

### IID random or multi-level intercepts

Multilevel/hierchical intercepts are defined as

\[ \begin{aligned} \mu_{\boldsymbol{s},t} &= f^{-1} \left( \ldots + \alpha_{g} + \ldots \right),\\ \alpha_g &\sim \operatorname{Normal} \left(0, \sigma_\alpha^2 \right),\\ \end{aligned} \]

where \(\alpha_g\) is an example
optional “random” intercept—an intercept with mean zero that varies by
level \(g\) and is constrained by \(\sigma_\alpha\). This is defined by the
`formula`

argument via the `(1 | g)`

syntax as in
lme4 or glmmTMB. There can be multiple random intercepts, despite only
showing one above. E.g., `(1 | g1) + (1 | g2)`

, in which case
they are assumed independent and uncorrelated from each other.

### Offset terms

Offset terms can be included through the `offset`

argument
in `sdmTMB()`

. These are included in the linear predictor
as

\[ \begin{aligned} \mu_{\boldsymbol{s},t} &= f^{-1} \left( \ldots + O_{\boldsymbol{s},t} + \ldots \right), \end{aligned} \]

where \(O_{\boldsymbol{s},t}\) is an
offset term—a **log transformed** variable without a
coefficient (assuming a log link). The offset is *not* included
in the prediction. Therefore, if `offset`

represents a
measure of effort, for example, the prediction is for one unit of effort
(`log(1) = 0`

).

## Observation model families

Here we describe the main observation families that are available in sdmTMB and comment on their parametrization, statistical properties, utility, and code representation in sdmTMB.

### Binomial

\[
\operatorname{Binomial} \left(N, \mu \right)
\] where \(N\) is the size or
number of trials, and \(\mu\) is the
probability of success for each trial. If \(N
= 1\), the distribution becomes the Bernoulli distribution.
Internally, the distribution is parameterized as the robust
version in TMB, which is numerically stable when probabilities
approach 0 or 1. Following the structure of `stats::glm()`

,
lme4, and glmmTMB, a binomial family can be specified in one of 4
ways:

- the response may be a factor (and the model classifies the first level versus all others)
- the response may be binomial (0/1)
- the response can be a matrix of form
`cbind(success, failure)`

, or - the response may be the observed proportions, and the
`weights`

argument is used to specify the Binomial size (\(N\)) parameter (`probabilty ~ ..., weights = N`

).

Code defined within TMB.

Example: `family = binomial(link = "logit")`

### Beta

\[ \operatorname{Beta} \left(\mu \phi, (1 - \mu) \phi \right) \] where \(\mu\) is the mean and \(\phi\) is a precision parameter. This parametrization follows Ferrari & Cribari-Neto (2004) and the betareg R package (Cribari-Neto & Zeileis 2010). The variance is \(\mu (1 - \mu) / (\phi + 1)\).

Code defined within TMB.

Example: `family = Beta(link = "logit")`

### Gamma

\[ \operatorname{Gamma} \left( \phi, \frac{\mu}{\phi} \right) \] where \(\phi\) represents the Gamma shape and \(\mu / \phi\) represents the scale. The mean is \(\mu\) and variance is \(\mu \cdot \phi^2\).

Code defined within TMB.

Example: `family = Gamma(link = "log")`

### Gaussian

\[ \operatorname{Normal} \left( \mu, \phi^2 \right) \] where \(\mu\) is the mean and \(\phi\) is the standard deviation. The variance is \(\phi^2\).

Example: `family = Gaussian(link = "identity")`

Code defined within TMB.

### Lognormal

sdmTMB uses the bias-corrected lognormal distribution where \(\phi\) represents the standard deviation in log-space:

\[ \operatorname{Lognormal} \left( \log \mu - \frac{\phi^2}{2}, \phi^2 \right). \] Because of the bias correction, \(\mathbb{E}[y] = \mu\) and \(\mathrm{Var}[\log y] = \phi^2\).

Code defined within
sdmTMB based on the TMB `dnorm()`

normal density.

Example: `family = lognormal(link = "log")`

### Negative Binomial 1 (NB1)

\[ \operatorname{NB1} \left( \mu, \phi \right) \]

where \(\mu\) is the mean and \(\phi\) is the dispersion parameter. The variance scales linearly with the mean \(\mathrm{Var}[y] = \mu + \mu / \phi\) (Hilbe 2011). Internally, the distribution is parameterized as the robust version in TMB.

Code defined within sdmTMB based on NB2 and borrowed from glmmTMB.

Example: `family = nbinom1(link = "log")`

### Negative Binomial 2 (NB2)

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

where \(\mu\) is the mean and \(\phi\) is the dispersion parameter. The variance scales quadratically with the mean \(\mathrm{Var}[y] = \mu + \mu^2 / \phi\) (Hilbe 2011). The NB2 parametrization is more commonly seen in ecology than the NB1. Internally, the distribution is parameterized as the robust version in TMB.

Code defined within TMB.

Example: `family = nbinom2(link = "log")`

### Poisson

\[ \operatorname{Poisson} \left( \mu \right) \] where \(\mu\) represents the mean and \(\mathrm{Var}[y] = \mu\).

Code defined within TMB.

Example: `family = poisson(link = "log")`

### Student-t

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

where \(\nu\), the degrees of
freedom (`df`

), is a user-supplied fixed parameter. Lower
values of \(\nu\) result in heavier
tails compared to the Gaussian distribution. Above approximately
`df = 20`

, the distribution becomes very similar to the
Gaussian. The Student-t distribution with a low degrees of freedom
(e.g., \(\nu \le 7\)) can be helpful
for modelling data that would otherwise be suitable for Gaussian but
needs an approach that is robust to outliers (e.g., Anderson *et al.* 2017).

Code defined within
sdmTMB based on the `dt()`

distribution in TMB.

Example: `family = student(link = "log", df = 7)`

### Tweedie

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

where \(\mu\) is the mean, \(p\) is the power parameter constrained between 1 and 2, and \(\phi\) is the dispersion parameter. The Tweedie distribution can be helpful for modelling data that are positive and continuous but also contain zeros.

Internally, \(p\) is transformed from \(\mathrm{logit}^{-1} (\texttt{thetaf}) + 1\) to constrain it between 1 and 2 and is estimated as an unconstrained variable.

The source code is implemented as in the cplm package (Zhang 2013) and is based on Dunn & Smyth (2005). The TMB version is defined here.

Example: `family = tweedie(link = "log")`

### Gamma mixture

This is a 2 component mixture that extends the Gamma distribution,

\[ (1 - p) \cdot \operatorname{Gamma} \left( \phi, \frac{\mu_{1}}{\phi} \right) + p \cdot \operatorname{Gamma} \left( \phi, \frac{\mu_{2}}{\phi} \right), \] where \(\phi\) represents the Gamma shape, \(\mu_{1} / \phi\) represents the scale for the first (smaller component) of the distribution, \(\mu_{2} / \phi\) represents the scale for the second (larger component) of the distribution, and \(p\) controls the contribution of each component to the mixture (also interpreted as the probability of larger events).

The mean is \((1-p) \cdot \mu_{1} + p \cdot \mu_{2}\) and the variance is \((1-p) ^ 2 \cdot \mu_{1} \cdot \phi^2 + (p) ^ 2 \cdot \mu_{2} \cdot \phi^2\).

Example: `family = gamma_mix(link = "log")`

. See also
`family = delta_gamma_mix()`

for an extension incorporating
this distribution with delta models.

### Lognormal mixture

This is a 2 component mixture that extends the lognormal distribution,

\[ (1 - p) \cdot \operatorname{Lognormal} \left( \log \mu_{1} - \frac{\phi^2}{2}, \phi^2 \right) + p \cdot \operatorname{Lognormal} \left( \log \mu_{2} - \frac{\phi^2}{2}, \phi^2 \right). \]

Because of the bias correction, \(\mathbb{E}[y] = (1-p) \cdot \mu_{1} + p \cdot \mu_{2}\) and \(\mathrm{Var}[\log y] = (1-p)^2 \cdot \phi^2 + p^2 \cdot \phi^2\).

As with the Gamma mixture, \(p\) controls the contribution of each component to the mixture (also interpreted as the probability of larger events).

Example: `family = lognormal_mix(link = "log")`

. See also
`family = delta_lognormal_mix()`

for an extension
incorporating this distribution with delta models.

## Gaussian random fields

### Matérn parameterization

The Matérn defines the covariance \(\Phi \left( s_j, s_k \right)\) between spatial locations \(s_j\) and \(s_k\) as

\[ \Phi\left( s_j,s_k \right) = \tau^2/\Gamma(\nu)2^{\nu - 1} (\kappa d_{jk})^\nu K_\nu \left( \kappa d_{jk} \right), \]

where \(\tau^2\) controls the
spatial variance, \(\nu\) controls the
smoothness, \(\Gamma\) represents the
Gamma function, \(d_{jk}\) represents
the distance between locations \(s_j\)
and \(s_k\), \(K_\nu\) represents the modified Bessel
function of the second kind, and \(\kappa\) represents the decorrelation rate.
The parameter \(\nu\) is set to 1 to
take advantage of the Stochastic Partial Differential Equation (SPDE)
approximation to the GRF to greatly increase computational efficiency
(Lindgren *et al.* 2011).
Internally, the parameters \(\kappa\)
and \(\tau\) are converted to range and
marginal standard deviation \(\sigma\)
as \(\textrm{range} = \sqrt{8} /
\kappa\) and \(\sigma = 1 / \sqrt{4 \pi
\exp \left(2 \log(\tau) + 2 \log(\kappa) \right) }\).

In the case of a spatiotemporal model with both spatial and
spatiotemporal fields, if `share_range = TRUE`

in
`sdmTMB()`

(the default), then a single \(\kappa\) and range are estimated with
separate \(\sigma_\omega\) and \(\sigma_\epsilon\). This often makes sense
since data are often only weakly informative about \(\kappa\). If
`share_range = FALSE`

, then separate \(\kappa_\omega\) and \(\kappa_\epsilon\) are estimated. The
spatially varying coefficient field always shares \(\kappa\) with the spatial random field.

### Projection \(\boldsymbol{A}\) matrix

The values of the spatial variables at the knots are multiplied by a projection matrix \(\boldsymbol{A}\) that bilinearly interpolates from the knot locations to the values at the locations of the observed or predicted data (Lindgren & Rue 2015)

\[ \boldsymbol{\omega}^* = \boldsymbol{A} \boldsymbol{\omega}, \] where \(\boldsymbol{\omega}^*\) represents the values of the spatial random fields at the observed locations or predicted data locations. The matrix \(\boldsymbol{A}\) has a row for each data point or prediction point and a column for each knot. Three non-zero elements on each row define the weight of the neighbouring 3 knot locations for location \(\boldsymbol{s}\). The same bilinear interpolation happens for any spatiotemporal random fields

\[ \boldsymbol{\epsilon}_t^* = \boldsymbol{A} \boldsymbol{\epsilon}_t. \]

### Anisotropy

TMB allows for anisotropy, where spatial covariance may be asymmetric
with respect to latitude and longitude (full
details). Anisotropy can be turned on or off with the logical
`anisotropy`

argument to `sdmTMB()`

. There are a
number of ways to implement anisotropic covariance (Fuglstad *et al.* 2015), and we adopt a
2-parameter rotation matrix \(\textbf{H}\). The elements of \(\textbf{H}\) are defined by the parameter
vector \(\boldsymbol{x}\) so that \(H_{1,1} = x_{1}\), \(H_{1,2} = H_{2,1} = x_{2}\) and \(H_{2,2} = (1 + x_{2}^2) / x_{1}\).

Once a model is fitted with `sdmTMB()`

, the anisotropy
relationships may be plotted using the `plot_anisotropy()`

function, which takes the fitted object as an argument. If a barrier
mesh is used, anisotropy is disabled.

### Incorporating physical barriers into the SPDE

In some cases the spatial domain of interest may be complex and
bounded by some barrier such as by land or water (e.g., coastlines,
islands, lakes). SPDE models allow for physical barriers to be
incorporated into the modelling (Bakka *et
al.* 2019). With `sdmTMB()`

models, the mesh
construction occurs in two steps: the user (1) constructs a mesh with a
call to `sdmTMB::make_mesh()`

, and (2) passes the mesh to
`sdmTMB::add_barrier_mesh()`

. The barriers must be
constructed as `sf`

objects (Pebesma
2018) with polygons defining the barriers. See
`?sdmTMB::add_barrier_mesh`

for an example.

The barrier implementation requires the user to select a fraction
value (`range_fraction`

argument) that defines the fraction
of the usual spatial range when crossing the barrier (Bakka *et al.* 2019). For example, if
the range was estimated at 10 km, `range_fraction = 0.2`

would assume that the range was 2 km across the barrier. This would let
the spatial correlation decay 5 times faster with distance. From
experimentation, values around 0.1 or 0.2 seem to work well but values
much lower than 0.1 can result in convergence issues.

This website by Francesco Serafini and Haakon Bakka provides an illustration with INLA. The implementation within TMB was borrowed from code written by Olav Nikolai Breivik and Hans Skaug at the TMB Case Studies Github site.

## Optimization

### Optimization details

The sdmTMB model is fit by maximum marginal likelihood. Internally, a
TMB (Kristensen *et al.* 2016)
model template calculates the marginal log likelihood and its gradient,
and the negative log likelihood is minimized via the non-linear
optimization routine `stats::nlminb()`

in R (Gay 1990; R Core Team 2021). Random effects are
estimated at values that maximize the log likelihood conditional on the
estimated fixed effects and are integrated over via the Laplace
approximation (Kristensen *et al.*
2016).

Like AD Model Builder (Fournier *et
al.* 2012), TMB allows for parameters to be fit in phases and
we include the `multiphase`

argument in
`sdmTMB::sdmTMBcontrol()`

to allow this. For high-dimensional
models (many fixed and random effects), phased estimation may be faster
and result in more stable convergence. In sdmTMB, phased estimation
proceeds by first estimating all fixed-effect parameters contributing to
the likelihood (holding random effects constant at initial values). In
the second phase, the random-effect parameters (and their variances) are
also estimated. Fixed-effect parameters are also estimated in the second
phase and are initialized at their estimates from the first phase.

In some cases, a single call to `stats::nlminb()`

may not
be result in convergence (e.g., the maximum gradient of the marginal
likelihood with respect to fixed-effect parameters is not small enough
yet), and the algorithm may need to be run multiple times. In the
`sdmTMB::sdmTMBcontrol()`

function, we include an argument
`nlminb_loops`

that will restart the optimization at the
previous best values. The number of `nlminb_loops`

should
generally be small (e.g., 2 or 3 initially), and defaults to 1. For some
sdmTMB models, the Hessian may also be unstable and need to be
re-evaluated. We do this optionally with the
`stats::optimHess()`

routine after the call to
`stats::nlminb()`

. The `stats::optimHess()`

function implements a Newton optimization routine to find the Hessian,
and we include the argument `newton_loops`

in
`sdmTMB::sdmTMBcontrol()`

to allow for multiple function
evaluations (each starting at the previous best value). By default, this
is not included and `newton_loops`

is set to 0. If a model is
already fit, the function `sdmTMB::run_extra_optimization()`

can run additional optimization loops with either routine to further
reduce the maximum gradient.

### Assessing convergence

Much of the guidance around diagnostics and glmmTMB also applies to
sdmTMB, e.g. the
glmmTMB vignette on troubleshooting. Optimization with
`stats::nlminb()`

involves specifying the number of
iterations and evaluations (`eval.max`

and
`iter.max`

) and the tolerances (`abs.tol`

,
`rel.tol`

, `x.tol`

, `xf.tol`

)—a greater
number of iterations and smaller tolerance thresholds increase the
chance that the optimal solution is found, but more evaluations
translates into longer computation time. Warnings of
non-positive-definite Hessian matrices (accompanied by parameters with
`NA`

s for standard errors) often mean models are improperly
specified given the data. Standard errors can be observed in the output
of `print.sdmTMB()`

or by checking
`fit$sd_report`

. The maximum gradient of the marginal
likelihood with respect to fixed-effect parameters can be checked by
inspecting (`fit$gradients`

). Guidance varies, but the
maximum gradient should likely be at least \(< 0.001\) before assuming the fitting
routine is consistent with convergence. If maximum gradients are already
relatively small, they can often be reduced further with additional
optimization calls beginning at the previous best parameter vector as
described above with `sdmTMB::run_extra_optimization()`

.

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