# Parameterize hatchR Models

#### Morgan Sparks, Bryan M. Maitland

# Overview

This vignette describes two options for selecting parameterized models for predicting fish early life history phenology using hatchR:

1. model parameterizations included in the package
2. custom parameterizations using your own data (i.e., days to hatch or emerge and average incubation temperature)

    library(hatchR)
    library(tibble)
    library(ggplot2)

# Built-in parameterizations

Published model parameterizations are contained in the `model_table` object. This includes parameterizations for several salmonid species from hatchery studies relating temperature to hatch and emergence timing Sparks et al. (2019), and can be selected using `hatchR::model_select()`.

These models are parameterized to 50% hatch or emergence (see Velsen (1987) for details). It is important to remember that while these models predict a point estimate, hatch and emergence, even within a single spawning family would occur as a distribution of the phenological event.

## `model_table`

`model_table` is loaded with hatchR, and is a tibble with 51 rows and 5 columns:

    model_table
    #> # A tibble: 51 × 5
    #>   author                  species model_id development_type expression
    #>   <chr>                   <chr>   <chr>    <chr>            <chr>
    #> 1 Beacham and Murray 1990 sockeye 2        hatch            1 /exp(6.727 - log(…
    #> 2 Beacham and Murray 1990 sockeye 4        hatch            1/exp(8.734 + (-1.5…
    #> 3 Beacham and Murray 1990 sockeye 5        hatch            1/exp(9.848 + (-1.5…
    #> 4 Beacham and Murray 1990 sockeye 7        hatch            1/exp(5.379 + (-0.0…
    #> 5 Beacham and Murray 1990 sockeye 2        emerge           1 / exp(7.227 - log…
    #> # ℹ 46 more rows

- `author`: author-date key denoting publication containing the model parameterization
- `species`: the species the model is parameterized for
- `model`: the model ID (if multiple model parameterizations were built (e.g., Beacham and Murray (1990))
- `development_type`: the phenological development type (i.e., hatch or emerge)
- `expression`: a character string defining the model parameterization

## `model_select()`

Combinations of `author`, `species`, `model`, and `development_type` are unique and used to select parameterized model expressions. For instance, if you wanted to access the expression for sockeye salmon (Oncorhynchus nerka) hatch phenology using model #2 from Beacham and Murray (1990), you would run:

    sockeye_hatch_mod <- model_select(
      author = "Beacham and Murray 1990",
      species = "sockeye",
      model = 2,
      development_type = "hatch"
      )

    sockeye_hatch_mod
    #> # A tibble: 1 × 5
    #>   author                  species model_id development_type expression
    #>   <chr>                   <chr>   <chr>    <chr>            <chr>
    #> 1 Beacham and Murray 1990 sockeye 2        hatch            1 /exp(6.727 - log(…

Note, that the above arguments are equivalent to the first line and four columns from `model_table`. Your model function object—in this case `sockeye_hatch_mod`—can then be passed to `predict_phenology()`, which we will demonstrate in the Predict Phenology: Basic vignette.

To see all available characterizations use:

    View(model_table)

# Creating custom models

hatchR also includes basic functionality to generate your own model parameterizations for predicting hatching and emergence phenology using your own data. Importantly, this functionality implements model form #2 of Beacham and Murray (1990), which we chose because of its overall simplicity and negligible loss of accuracy. See Beacham and Murray (1990) and Sparks et al. (2019) for more specific discussion regarding model #2 and the development of the effective value approach.

The model follows the general format of:

\[ Effective Value_i = 1/exp(log_ea - log_e(Temperature_i - b)) \]

Where i is the daily value and a fish hatches or emerges when the cumulative sum reaches one: \[\sum_{i =1}^nEffectiveValue_i = 1\]

## `fit_model()`

The function `fit_model()` uses data in which average incubation temperature (°C) and days to phenological event are the inputs and estimates parameter coefficients for logea and b using `stats::nls()`.

Here, we borrow data from Table 8.1 (pg. 183) from Quinn (2018) to generate a custom hatch parameterization for brown trout (Salmo trutta).

You could either create a .csv file with those data and import them with `readr::read_csv()` or alternatively, directly input them as an object in R. We’ll use `tibble::tibble()` to create the data.

    bt_data <- tibble(
      temperature = c(2,5,8,11,14),
      days_to_hatch = c(194,87,54,35,28)
      )
    bt_data
    #> # A tibble: 5 × 2
    #>   temperature days_to_hatch
    #>         <dbl>         <dbl>
    #> 1           2           194
    #> 2           5            87
    #> 3           8            54
    #> 4          11            35
    #> 5          14            28

We can plot our data for a validity check using ggplot2:

    bt_data |>
      ggplot(aes(x = temperature, y = days_to_hatch)) +
      geom_point() +
      theme_classic()

We can now use `fit_model()` to create our custom parameterization from our data. You must specify a species and a development type, information which is carried forward in subsequent functions in hatchR.

    bt_fit <- fit_model(temp = bt_data$temperature,
                        days = bt_data$days_to_hatch,
                        species = "brown_trout",
                        development_type = "hatch")

The output of `fit_model()` is a list with several elements:

- `bt_fit$model`: a model object of class “nls” containing the nonlinear regression model
- `bt_fit$log_a`: a named numeric vector of the estimated coefficient logea
- `bt_fit$b`: a named numeric vector of the estimated coefficient b
- `bt_fit$r_squared`: pseudo R-squared value (1 - (residual sum of squares / total sum of squares))
- `bt_fit$mse`: mean squared error (mean(residuals^2))
- `bt_fit$rmse`: root mean squared error (sqrt(mse)
- `bt_fit$expression`: a tibble with the species, development type, and the parameterized model expression
- `bt_fit$pred_plot`: a ggplot object showing the observed data and predicted values

    bt_fit
    #> $model
    #> Nonlinear regression model
    #>   model: y ~ a/(x - b)
    #>    data: df
    #>        a        b
    #> 439.1634  -0.2582
    #>  residual sum-of-squares: 36.92
    #>
    #> Number of iterations to convergence: 5
    #> Achieved convergence tolerance: 2.812e-07
    #>
    #> $log_a
    #> [1] 6.084872
    #>
    #> $b
    #> [1] -0.258236
    #>
    #> $r_squared
    #> [1] 0.9979987
    #>
    #> $mse
    #> [1] 7.38454
    #>
    #> $rmse
    #> [1] 2.717451
    #>
    #> $expression
    #> # A tibble: 1 × 3
    #>   species     development_type expression
    #>   <chr>       <chr>            <chr>
    #> 1 brown_trout hatch            1 / exp(6.08487164096323 - log(x + 0.25823604410…
    #>
    #> $pred_plot

The vast majority of the time, what you will want is the actual expression with parameter estimates for use in the `model = ...` argument of `predict_phenology()`. This is stored in the `expression` element of the list.

You can either pass this tibble directly with the `$` operator by calling `$expression` element of the list (e.g., `model = bt_fit$expression`) or set as an object to pass, like so:

    bt_hatch_exp <- bt_fit$expression
    bt_hatch_exp
    #> # A tibble: 1 × 3
    #>   species     development_type expression
    #>   <chr>       <chr>            <chr>
    #> 1 brown_trout hatch            1 / exp(6.08487164096323 - log(x + 0.25823604410…

`predict_phenology()` will extract the expression from the object and use it to predict phenology. We will demonstrate this in the Predict phenology: Basic vignette.

## Fitting models for other fishes

We demonstrate how the `fit_model()` function may be used to create custom parameterizations for species beyond the Salmonids in the `model_table` included in the package. We include parameterizations from three warm-water species to demonstrate the `fit_model()` utility for fishes beyond the scope of the original effective value approach. These parameterizations are for commonly cultured sportfishes including Smallmouth Bass (Micropterus dolomieu) (Webster 1948), Channel Catfish (Ictalurus punctatus) (Small and Bates 2001) , and Lake Sturgeon (Acipenser fulvescens) (Smith and King 2005).

We demonstrate the utility of this approach by creating a random thermal regime with an ascending thermograph with a mean temperature of 16 °C, parameterizing models for each species, and demonstrating days to hatch and developmental period for each species with the random thermal regime .

    ###  make temp regime
    set.seed(123)
    # create random temps and corresponding dates
    temps_sim <- sort(rnorm(n =30, mean = 16, sd = 1), decreasing = FALSE)
    dates_sim <-  seq(from = ymd("2000-07-01"),
                 to = ymd("2000-07-31"), length.out = 30)

    data_sim <- matrix(NA, 30, 2) |> data.frame()
    data_sim[,1] <- temps_sim
    data_sim[,2] <- dates_sim

    # change names so they aren't the same as the vector objects
    colnames(data_sim) <- c("temp_sim", "date_sim")

Next we’ll parameterize our models for the three different fishes

    ### smallmouth mod
    smallmouth <- matrix(NA, 10, 2) |> data.frame()
    colnames(smallmouth) <- c("hours", "temp_F")
    smallmouth$hours <- c(52, 54, 70, 78, 90, 98, 150, 167, 238, 234)
    smallmouth$temp_F <- c(77, 75, 71, 70, 67, 65, 60, 59, 55, 55)

    # change °F to °C and hours to days
    smallmouth <- smallmouth |>
      mutate(days = ceiling(hours/24),
             temp_C = (temp_F -32) * (5/9))

    # model object for smallmouth bass
    smb_mod <- fit_model(temp = smallmouth$temp_C,
                         days = smallmouth$days,
                         species = "smb",
                         development_type = "hatch")

    ### catfish mod
    catfish <- matrix(NA, 3, 2) |> data.frame()
    colnames(catfish) <- c("days", "temp_C")
    catfish$days <- c(16,21,26)
    catfish$temp_C <- c(22,10,7)

    cat_mod <- fit_model(temp = catfish$temp_C,
                         days = catfish$days,
                         species = "catfish",
                         development_type = "hatch")

    ### lake sturgeon mod
    sturgeon <-  matrix(NA, 7, 2) |> data.frame()
    colnames(sturgeon) <- c("days", "CTU")
    sturgeon$days <- c(7,5,6,6,5,11,7)
    sturgeon$CTU <- c(58.1, 62.2, 61.1, 57.5, 58.1, 71.4, 54.7)

    sturgeon <- sturgeon |>
      mutate(temp_C = CTU/days) # change CTUs to average temp and add column

    sturgeon_mod <- fit_model(days = sturgeon$days,
                              temp = sturgeon$temp_C,
                              species = "sturgeon",
                              development_type = "hatch")

Note the R2 fit from the models below. You can see they generally all preform well and are in line with values from model 2 of Beacham and Murray (1990).

    #model fits
    smb_mod$r_squared; cat_mod$r_squared; sturgeon_mod$r_squared
    #> [1] 0.9868067
    #> [1] 0.9433598
    #> [1] 0.9217358

After we have our fits for each species we can predict phenology using our `data_sim` datset we created above.

    ### predict_phenology

    #smallmouth bass
    smb_res <- predict_phenology(data = data_sim,
                      dates = date_sim,
                      temperature = temp_sim,
                      spawn.date = "2000-07-01",
                      model = smb_mod$expression)

    # catfish
    catfish_res <- predict_phenology(data = data_sim,
                      dates = date_sim,
                      temperature = temp_sim,
                      spawn.date = "2000-07-01",
                      model = cat_mod$expression)

    # sturgeon
    # note that 16 C is pretty far out of range of temps for model fit, not best practice
    sturgeon_res <- predict_phenology(data = data_sim,
                      dates = date_sim,
                      temperature = temp_sim,
                      spawn.date = "2000-07-01",
                      model = sturgeon_mod$expression)

After we have our predictions we’ll combine our results and do a little data cleaning.

    # summary for all species
    all_res <- data.frame(matrix(NA, 3, 2))
    colnames(all_res) <- c("start", "stop")

    all_res$start <- c(catfish_res$dev_period$start,
                       smb_res$dev_period$start,
                       sturgeon_res$dev_period$start)

    all_res$stop <- c(catfish_res$dev_period$stop,
                      smb_res$dev_period$stop,
                      sturgeon_res$dev_period$stop)

    all_res <- all_res |>
      mutate(days = ceiling(stop-start),
             index = c(17,16.5,16)) # index for our horizontal bars

    all_res$Species <- c("Channel Catfish", "Smallmouth Bass", "Lake Sturgeon")

And then finally we can plot our results.

    ggplot() +
      geom_point(data = data_sim, aes(x = date_sim, y = temp_sim )) +
      geom_line(data = data_sim, aes(x = date_sim, y = temp_sim )) +
      geom_rect(data = all_res, aes(xmin = start, xmax = stop, ymax =index-.35, ymin = index-.5, fill = Species)) +
      geom_label(data = all_res, aes(x = start + (stop - start) / 1.25, y = (index -0.425), label = days)) +
      labs(x = "Date", y = "Temperature (°C)") +
      scale_fill_manual(values = c("deepskyblue4", "grey23", "darkolivegreen4")) +
      theme_classic() +
      theme(legend.position = c(0.75, 0.25))

# Important considerations

Here are some important considerations:

> Your model fits will only be as good as the data they are generated from.

1. We recommend a minimum of four temperature x hatch/emerge data points.

2. Data should be spread across temperatures as much as possible.

   - It’s much better to have a fit derived from data for temperatures such as 3, 7, 10, 14 °C than it is 8, 9, 10, 11 °C.
   - The behavior of the model function around the tails of very cold or warm temperatures (relative to the fish species) drive the fit of the function, so more extreme temperatures are helpful.

3. Think hard about whether the data you are generating your parameterization from match the data from which you are trying to predict or if you are extrapolating beyond what is sensible for the model.

4. Understand your response variable, most models are fit to 50% hatch or emergence for a family group or population. However, your data may be different and you should interpret your results accordingly (e.g. comparisons between 50% hatch from population A to 95% hatch of population B may not be reasonable).

# References

Austin, Catherine S., Timothy E. Essington, and Thomas P. Quinn. 2019. “Spawning and Emergence Phenology of Bull Trout (Salvelinus Confluentus) Under Differing Thermal Regimes.” Journal of Fish Biology 94 (1): 191–95. https://doi.org/10.1111/jfb.13864.
Beacham, Terry D., and Clyde B. Murray. 1990. “Temperature, Egg Size, and Development of Embryos and Alevins of Five Species of Pacific Salmon: A Comparative Analysis.” Transactions of the American Fisheries Society 119 (6): 927–45. https://doi.org/10.1577/1548-8659(1990)119%3C0927:TESADO%3E2.3.CO;2.
Quinn, Thomas P. 2018. The Behavior and Ecology of Pacific Salmon and Trout. University of Washington Press.
Small, Brian C., and Terry D. Bates. 2001. “Effect of Low-Temperature Incubation of Channel Catfish Ictalurus Punctatus Eggs on Development, Survival, and Growth.” Journal of the World Aquaculture Society 32 (2): 189–94. https://doi.org/10.1111/j.1749-7345.2001.tb01094.x.
Smith, K. M., and D. K. King. 2005. “Dynamics and Extent of Larval Lake Sturgeon Acipenser Fulvescens Drift in the Upper Black River, Michigan.” Journal of Applied Ichthyology 21 (3): 161–68. https://doi.org/10.1111/j.1439-0426.2005.00623.x.
Sparks, Morgan M., Jeffrey A. Falke, Thomas P. Quinn, Milo D. Adkison, Daniel E. Schindler, Krista Bartz, Dan Young, and Peter A. H. Westley. 2019. “Influences of Spawning Timing, Water Temperature, and Climatic Warming on Early Life History Phenology in Western Alaska Sockeye Salmon.” Canadian Journal of Fisheries and Aquatic Sciences 76 (1): 123–35. https://doi.org/10.1139/cjfas-2017-0468.
Velsen, FPJ. 1987. “Temperature and Incubation in Pacific Salmon and Rainbow Trout: Compilation of Data on Median Hatching Time, Mortality and Embryonic Staging.” Department of Fisheries; Oceans, Fisheries Research Branch, Pacific ….
Webster, Dwight A. 1948. “Relation of Temperature to Survival and Incubation of the Eggs of Smallmouth Bass (Micropterus Dolomieu).” Transactions of the American Fisheries Society 75 (1): 43–47. https://doi.org/10.1577/1548-8659(1945)75[43:ROTTSA]2.0.CO;2.