# Simulating fossils

#### Rachel C. M. Warnock

#### 2025-11-06

This vignette provides information about how the package stores and outputs information about fossil sampling times via the `fossils` object and available options for simulating fossil recovery.

## Contents

- The fossils object
- Constant fossil recovery
  - Simulating fossils for a tree object
  - Simulating fossils for a taxonomy object
- Interval-dependent fossil recovery
  - Simulating fossils for a fixed number of intervals
  - Simulating fossils for a set of non-uniform intervals
  - Simulating fossils using per-interval probabilities
  - Incorporating stratigraphic age uncertainty
- Environment-dependent fossil recovery
- Lineage-dependent fossil recovery
  - The autocorrelated fossil recovery model
  - The independent fossil recovery model
  - Environment and lineage-dependent fossil recovery
- Extant species and tip sampling

### The fossils object

The `fossils` object contains information about fossil sampling times and their relationship to the corresponding `phylo` and `taxonomy` objects. In some simulation conditions we may know the age of fossils precisely. Alternatively, there may be some uncertainty associated with specimen ages, i.e. we only know the age to within some stratigraphic interval, which can be incorporated into the `fossils` object.

The object contains a dataframe with species and age information for each fossil with the following 4 fields:

- `sp` the label of the corresponding species. This label matches the edge labels in the corresponding `phylo` object or the species labels in the corresponding `taxonomy` object if taxonomic information was provided

- `edge` the label of the sampled node or tip in the phylogeny, i.e the node at the end of the edge along which the fossil was sampled

- `hmin` the age of the fossil or the youngest bound of the time interval in which the fossil was sampled

- `hmax` the oldest bound of the time interval in which the fossil was sampled. This is equal to `hmin` if exact sampling times are known

The object also contains a logical variable `from.taxonomy` indicating whether the fossil record was simulated using a `taxonomy` object, in which case `from.taxonomy = TRUE`, or from a `phylo` object in which case `from.taxonomy = FALSE` (the default).

## Constant fossil recovery

The simplest model of fossil recovery is a Poisson sampling process, where fossils are sampled along lineages with constant rate \(\psi\) (function argument `rate`).

Fossils can be simulated using a `phylo` object or an existing `taxonomy` object generated using the function `sim.taxonomy`.

### Simulating fossils for a tree object

If a `phylo` rather than a `taxonomy` object is passed to the function, it will assume that all speciation has occurred via symmetric (i.e. bifurcating, \(\beta = 1\)) speciation, i.e. every edge in the tree represents a unique species. This applies to all functions used to simulate fossils.

    # set the random number generator seed to generate the same results using the same code
    set.seed(1234)

    # simulate a tree using TreeSim conditioned on tip number
    lambda = 1
    mu = 0.2
    tips = 8
    t = TreeSim::sim.bd.taxa(n = tips, numbsim = 1, lambda = lambda, mu = mu)[[1]]
    # t is an object of class phylo
    t
    #>
    #> Phylogenetic tree with 8 tips and 7 internal nodes.
    #>
    #> Tip labels:
    #>   t8, t4, t3, t1, t6, t2, ...
    #>
    #> Rooted; includes branch length(s).
    # use t$edge, t$edge.length, t$root.edge to see the tree attributes

    # Simulate fossils
    rate = 3 # poisson sampling rate
    f = sim.fossils.poisson(rate = rate, tree = t)
    f
    #>    sp edge       hmin       hmax
    #> 1   1    1 0.58684752 0.58684752
    #> 2   1    1 0.17795617 0.17795617
    #> 3   1    1 0.58613520 0.58613520
    #> 4   1    1 0.57453591 0.57453591
    #> 5   2    2 0.14585530 0.14585530
    #> 6   2    2 0.70851792 0.70851792
    #> 7   2    2 0.72224865 0.72224865
    #> 8   6    6 0.82173813 0.82173813
    #> 9   6    6 0.35835122 0.35835122
    #> 10  7    7 0.04035432 0.04035432
    #> ...
    #> Fossil record with 19 occurrences representing 9 species
    #> Fossil record not simulated using taxonomy: all speciation events are assumed to be symmetric

Each entry in the `fossils` dataframe corresponds to a fossil sampling time. `sp` and `edge` are the species and edge labels, respectively, which will be the same here since we assume symmetric speciation.

`hmin` and `hmax` are the youngest and oldest sampling ages associated with each fossil, respectively. Since this function returns exact fossil sampling times `hmin` and `hmax` are equal.

    # plot fossil occurrences
    plot(f, tree = t)

    # plot stratigraphic ranges
    plot(f, tree = t, show.ranges = TRUE)

### Simulating fossils for a taxonomy object

    # simulate taxonomy under mixed speciation
    beta = 0.5 # probability of symmetric speciation
    lambda.a = 1.2 # rate of anagenetic speciation
    s = sim.taxonomy(tree = t, beta = beta, lambda.a = lambda.a)

    # simulate fossils
    f = sim.fossils.poisson(rate = rate, taxonomy = s)
    f
    #>    sp edge      hmin      hmax
    #> 1   1    1 1.1276605 1.1276605
    #> 2   1    1 0.7431218 0.7431218
    #> 3   1    1 0.1493434 0.1493434
    #> 4   4   15 0.8912792 0.8912792
    #> 5   4   15 0.4560055 0.4560055
    #> 6   7    7 0.4670495 0.4670495
    #> 7   7    7 0.4936152 0.4936152
    #> 8   7    7 0.6791209 0.6791209
    #> 9   7    7 0.3928259 0.3928259
    #> 10 16    2 0.3277610 0.3277610
    #> ...
    #> Fossil record with 31 occurrences representing 12 species
    #> Fossils record simulated from or assigned to an existing taxonomy

    # plot the output and color fossils by taxonomy
    plot(f, tree = t, taxonomy = s, show.taxonomy = TRUE)

Note that `sp` and `edge` are not all equal, since some species are assigned to multiple edges, and some edges are associated with multiple species.

## Interval-dependent fossil recovery

Under the interval-dependent model, fossil recovery changes in a piece-wise manner across a set of user defined intervals. Interval ages can be specified using a fixed set of uniform length intervals, by passing the function the maximum interval age (`max.age`) and the number of intervals (`strata`), or a vector of non-uniform intervals (`interval.ages`).

The function `tree.max` can be used to assign a maximum age based on tree age. If the tree has a `root.edge` the function will return the origin time, otherwise the function returns the root age.

### Simulating fossils for a fixed number of intervals

    # define max interval age based on tree age
    max.age = tree.max(t)

    # define the number of intervals
    strata = 4

    # define a vector of sampling rates, where the first entry corresponds the youngest interval
    rates = c(1, 0.1, 1, 0.1)

    # simulate fossils
    f = sim.fossils.intervals(tree = t, rates = rates, max.age = max.age, strata = strata)

    # plot the output
    plot(f, tree = t, show.strata = TRUE, strata = strata)

If no maximum is provided to the plotting function and `strata` > 1, the function `tree.max` is used to define `max.age`.

### Simulating fossils for a set of non-uniform intervals

    # the following will sample a random set of interval ages between 0 and max.age
    times = c(0, sort(runif(3, min = 0, max = max.age)), max.age)

    # define a vector of sampling rates from youngest to oldest
    rates = c(1, 0.1, 1, 0.1)

    # simulate fossils
    f = sim.fossils.intervals(tree = t, rates = rates, interval.ages = times)

    # plot the output and show sampling or proxy data
    plot(f, tree = t, show.strata = TRUE, interval.ages = times, show.proxy = TRUE, proxy = rates)

### Simulating fossils using per-interval probabilities

Fossil recovery can also be specified using a set of per-interval probabilities. At most one fossil per species will be sampled per interval using this approach.

    # define max interval age based on tree age
    max.age = tree.max(t)

    # define the number of intervals
    strata = 4

    # define a vector of sampling probabilities from youngest to oldest
    probabilities = c(1.0, 0.5, 0.2, 0.1)

    # simulate fossils
    f = sim.fossils.intervals(tree = t, probabilities = probabilities, max.age = max.age, strata = strata)

    # plot the output
    plot(f, tree = t, show.strata = TRUE, strata = strata)

### Incorporating stratigraphic age uncertainty

To incorporate age uncertainty into the output using the function `sim.fossils.intervals` set `use.exact.times = FALSE`. In this case `hmin` and `hmax` will represent the youngest and oldest age associated with the fossil, respectively. When we plot the output the function will place the fossils at the mid-point of the corresponding interval. This means in some cases fossils will not appear along their corresponding edge.

    # simulate fossils
    f = sim.fossils.intervals(tree = t, probabilities = probabilities, max.age = max.age, strata = strata, use.exact.times = FALSE)
    f
    #>   sp edge      hmin      hmax
    #> 1  1    1 0.0000000 0.5640402
    #> 2  1    1 0.5640402 1.1280804
    #> 3  2    2 0.0000000 0.5640402
    #> 4  3    3 0.0000000 0.5640402
    #> 5  6    6 0.0000000 0.5640402
    #> 6  7    7 0.0000000 0.5640402
    #> 7  8    8 0.0000000 0.5640402
    #> 8 13   13 0.5640402 1.1280804
    #> Fossil record with 8 occurrences representing 7 species
    #> Fossil record not simulated using taxonomy: all speciation events are assumed to be symmetric

    # plot the output
    plot(f, tree = t, show.strata = TRUE, strata = strata)

Alternatively, if you simulate fossils using a function that only returns exact fossil sampling times, e.g. using the `sim.fossils.poisson` function, you can later assign fossils to different intervals using the `sim.interval.ages` function. You can also generate output using this function that respect the species duration times by setting `use.species.ages = TRUE`, i.e. the function will not return a value for `hmin` or `hmax` that are younger or older than the true species age. To use this option you also need to pass a `phylo` or `taxonomy` object to the function.

    # simulate fossils
    f = sim.fossils.poisson(tree = t, rate = 3)

    # reassign fossils to intervals
    f = sim.interval.ages(fossils = f, tree = t, max.age = max.age, strata = strata, use.species.ages = TRUE)

    # plot the output
    plot(f, tree = t, show.strata = TRUE, strata = strata)

## Environment-dependent fossil recovery

The function `sim.fossils.environment` can be used to simulate fossil recovery that reflects species environmental preferences. The variables that control the distribution and abundance of species also control depositional environment, meaning changes in ecological or environmental gradient across a given area will be reflected in the patterns of deposition. A species response curve, which describes the abundance of a species relative to an environmental gradient, can therefore be used to model distributions of fossil occurrences. See the plot below for examples.

Holland (1995) described a three parameter Gaussian model for simulating environment-dependent fossil recovery. In this model each species has three parameters, species preferred depth (\(PD\)), depth tolerance (\(DT\)) and peak abundance (\(PA\)) (function arguments `PD`, `DT` and `PA`). The probability of fossil recovery during a given interval of time is a function of water depth, given by,

\(Pr_{collection_i}(d) = PA * exp(-(d - PD)^2/ (2*DT^2))\),

where \(d\) is current water depth, \(PD\) and \(DT\) are the mean and the standard deviation of the distribution, respectively, and \(PA\) describes the amplitude of the distribution. Here the model is described in the context of marine taxa, however the model and the function `sim.fossils.environment` can be applied to any environmental or depositional context, in which some measure of environmental gradient can be generated.

The following code snippet illustrate how to explore the role of the model parameters \(PD\), \(DT\) and \(PA\) in determining species response curves.

    # define a set of colors for 4 species
    # color palette chosen using RColorBrewer
    cols = c("#E41A1C", "#377EB8", "#984EA3", "#4DAF4A") # red, blue, purple, green

    # species response curve function
    response.curve = function(x, pd, dt, pa) pa * exp(-(x - pd)^2/ (2*dt^2))

    # define species variables # species 1
    PA = 1 # peak abundance
    PD = 2 # preferred depth
    DT = 1 # depth tolerance

    # plot species resonse curve
    curve(response.curve(x, PD, DT, PA), -2, 6, bty = "n", xlab = "relative depth", ylab = "Pr(collection)", xlim = c(-6,6), lwd = 1.5, col = cols[1])

    # redefine depth tolerance and compare the output # species 2
    DT = 0.5
    curve(response.curve(x, PD, DT, PA), -1, 5, add = TRUE, lwd = 1.5, col = cols[2])

    # redefine preferred depth and compare the output # species 3
    PD = -2
    curve(response.curve(x, PD, DT, PA), -5, 1, add = TRUE, lwd = 1.5, col = cols[3])

    # redefine prferred depth and peak abundance # species 4
    PD = -4
    PA = 0.5
    curve(response.curve(x, PD, DT, PA), -6, -2.2, add = TRUE, lwd = 1.5, col = cols[4])

The user is free to use a vector containing any gradient values to simulate fossil occurrences. In the following example we will use a vector of gradient values generated using the function `sim.gradient`, which return values for the following simple sine wave function, \(y = d*sin(cycles*pi*(x-1/4))\), where \(x\) is the number of intervals or strata. The function output can be used to represent changes in relative water depth, where the number of cycles may be equivalent to the number of transgression/regression events.

Interval ages can be specified in the same way as the function `sim.fossils.intervals` via `max.age` and `strata` or the `intervals` vector.

    # define the interval parameters
    strata = 20
    times = seq(0, tree.max(t), length.out = strata+1)

    # simulate gradient values
    wd = sim.gradient(strata, depth = 6)

    # wd is a vector representing relative water depth
    wd
    #>  [1] -6.0000000 -4.7348431 -1.4729129  2.4101725  5.2768425  5.9181678
    #>  [7]  4.0636894  0.4954761 -3.2816889 -5.6749035 -5.6749035 -3.2816889
    #> [13]  0.4954761  4.0636894  5.9181678  5.2768425  2.4101725 -1.4729129
    #> [19] -4.7348431 -6.0000000

    # define species trait values # species 1
    PD = 1
    DT = 2
    PA = 1

    # simulate fossils
    f = sim.fossils.environment(tree = t, interval.ages = times, proxy.data = wd, PD = PD, DT = DT, PA = PA)

    # plot output and include proxy data & preferred depth
    plot(f, tree = t, interval.ages = times, show.strata = TRUE, show.proxy = TRUE, proxy.data = wd, show.preferred.environ = TRUE, preferred.environ = PD, fossil.col = cols[1])

    # redefine species trait values # species 2
    DT = 0.5

    # simulate fossils
    f = sim.fossils.environment(tree = t, interval.ages = times, proxy.data = wd, PD = PD, DT = DT, PA = PA)

    # plot output
    plot(f, tree = t, interval.ages = times, show.strata = TRUE, show.proxy = TRUE, proxy.data = wd, show.preferred.environ = TRUE, preferred.environ = PD, fossil.col = cols[2])

    # redefine species trait values # species 3
    PD = -2

    # simulate fossils
    f = sim.fossils.environment(tree = t, interval.ages = times, proxy.data = wd, PD = PD, DT = DT, PA = PA)

    # plot output
    plot(f, tree = t, interval.ages = times, show.strata = TRUE, show.proxy = TRUE, proxy.data = wd, show.preferred.environ = TRUE, preferred.environ = PD, fossil.col = cols[3])

    # define species trait values # species 4
    PD = -4
    PA = 0.5

    # simulate fossils
    f = sim.fossils.environment(tree = t, interval.ages = times, proxy.data = wd, PD = PD, DT = DT, PA = PA)

    # plot output
    plot(f, tree = t, interval.ages = times, show.strata = TRUE, show.proxy = TRUE, proxy.data = wd, show.preferred.environ = TRUE, preferred.environ = PD, fossil.col = cols[4])

## Lineage-dependent fossil recovery

Variation in fossil recovery parameters across different lineages or species can be generated using the function `sim.trait.values`. The function output is a vector of simulated trait values that can be used to specify the `rate` parameter in `sim.fossils.poisson` or the `PD`, `DT` and `PA` parameters in `sim.fossils.environment`.

Trait values can be simulated for a given `phylo` or `taxonomy` object. If the function is provided with a `phylo` object, trait values are simulated assuming bifurcating speciation and simulated trait values are output for each edge in the order in which they appear in the `phylo` object dataframe. If the tree also has a root edge (`tree$root.edge`), the first entry in the vector will correspond to the first entry in the vector. If the function is provided with a `taxonomy` object, simulated trait values are output for each species in the order in which they appear in the `taxonomy` object dataframe.

### The autocorrelated fossil recovery model

Under the `autocorrelated` model, traits evolve along lineages according to a Brownian motion process, where the strength of the relationship between ancestor and descendant values is determined by the parameter \(\nu\) (function argument `v`). New trait values are drawn from a lognormal distribution, where the mean is equal to the value of the ancestor and the variance is a function of \(\nu\) and the species or edge duration. If \(\nu\) is small values will be more similar between ancestor and descendants, and if \(\nu\) is zero all trait values will be equal. This is model is described in Heath et al. (2014) and is equivalent to the autocorrelated relaxed clock model described in Kishino et al. (2001).

    # define the initial rate at the root or origin
    rate = 1

    # simulate rates under the autocorrelated trait values model (the default option)
    rates = sim.trait.values(init = rate, tree = t, v = 0.01)
    # simulated rates
    rates
    #>  [1] 1.0599176 1.0883147 1.2007113 1.1898586 1.1380816 1.2943105 1.1031386
    #>  [8] 1.0791394 1.0736070 1.1441912 1.2172428 1.3427452 1.0542301 1.0758923
    #> [15] 0.9792786
    #> attr(,"from.taxonomy")
    #> [1] FALSE

    # simulate fossils
    f = sim.fossils.poisson(tree = t, rate = rates)

    # plot the output
    plot(f, tree = t)

### The independent fossil recovery model

Under the `independent` model a new trait value is drawn for each species from any valid user-specified distribution `dist`. To be valid the function just has to return a single value.

    # define the initial rate at the root or origin
    rate = 1

    # define the distribution used to sample new rates
    # in this case an exponential with mean = 1
    dist = function() { rexp(1) }

    # simulate trait values under the independent trait values model
    rates = sim.trait.values(init = rate, tree = t, model = "independent", dist = dist)

    # simulate fossils
    f = sim.fossils.poisson(tree = t, rate = rates)

    # plot the output
    plot(f, tree = t)

The parameter `change.pr` is the probability that changes in trait values are coincident with speciation events. At each speciation event change occurs with probability `change.pr` and new values are drawn from any valid user-specified distribution `dist`. If `change.pr = 1` a change will occur at every speciation event and the model will be the same as above.

    # define the initial value at the root or origin
    rate = 1

    # define the distribution used to sample new rates
    # in this case an exponential with a mean ~ 3
    dist = function() { rexp(1, 1/4) }

    # define the probability of the trait value changing at each speciation event
    change.pr = 0.5

    # simulate trait values under the independent trait values model
    rates = sim.trait.values(init = rate, tree = t, model = "independent", dist = dist, change.pr = change.pr)

    # simulate fossils
    f = sim.fossils.poisson(tree = t, rate = rates)

    # plot the output
    plot(f, tree = t)

### Environment and lineage-dependent fossil recovery

Lineage specific values generated using `sim.trait.values` can also be passed to the the function `sim.fossils.environment` to define the `PD`, `DT`, and `PA` parameters.

    # define constant values for preferred depth and depth tolerance
    PD = 2
    DT = 0.5

    # simulate lineage variable peak abundance values

    # define the distribution used to sample new PA values
    # in this case a uniform in the interval 0, 1
    dist = function() { runif(1) }

    # simulate trait values under the independent model
    PA = sim.trait.values(init = 0.1, tree = t, model = "independent", dist = dist)

    # simulate fossils
    f = sim.fossils.environment(tree = t, interval.ages = times, proxy.data = wd, PD = PD, DT = DT, PA = PA)

    # plot the output
    plot(f, tree = t, show.strata = TRUE, interval.ages = times)

## Extant species and tip sampling

The functions `sim.extant.samples` and `sim.tip.samples` can be used to simulate incomplete extant species and tip sampling, respectively. The parameter `rho` is the probability that an extant species or a tip will be sampled.

    # simulate fossils
    f = sim.fossils.poisson(tree = t, rate = rates)

    # simulate extant species sampling
    f2 = sim.extant.samples(fossils = f, tree = t, rho = 0.5)

    # plot the output
    plot(f2, tree = t, extant.col = "red")

    # for tip sampling only, create a fossil object with no fossils
    f = fossils()

    # simulate tip sampling
    f2 = sim.tip.samples(fossils = f, tree = t, rho = 0.75)

    # plot the output
    plot(f2, tree = t)

## See also

See the `paleotree` vignette to see how `FossilSim` objects can be converted into `paleotree` objects.

## References

Heath, T.A., Huelsenbeck, J.P. & Stadler, T. (2014). The fossilized birth-death process for coherent calibration of divergence-time estimates. Proceedings of the National Academy of Sciences, 111, E2957–E2966.
Holland, S.M. (1995). The stratigraphic distribution of fossils. Paleobiology, 21, 92–109.
Kishino, H., Thorne, J.L. & Bruno, W.J. (2001). Performance of a divergence time estimation method under a probabilistic model of rate evolution. Molecular Biology and Evolution, 18, 352–361.