# Decay functions

Some functions of the accessibility package, such as `floating_catchment_area()` and `gravity()`, use decay functions to continuously discount the weight of opportunities as travel costs become larger. For convenience, the package ships with some of the most frequently used functions in the accessibility literature, which are discussed below. Additionally, users can pass custom functions to convert travel costs into weights to be applied to the opportunities, which is discussed further down in this the vignette.

# Currently available decay functions

## Binary

Also known as the step decay function, it’s most commonly used in cumulative opportunities measures.

\[ \begin{aligned} f(t_{ij})= \left\{ \begin{array}{ll} 1 & \quad \text{for }t_{ij} \leq T \\ 0 & \quad \text{for }t_{ij} > T \end{array} \right.\\ \end{aligned} \]

Where \(t_{ij}\) is the travel cost between origin i and destination j, and \(T\) is the travel cost cutoff.

## Linear

Weights decay linearly until the travel cost cutoff is reached. From this point onward weights assume the value of 0.

\[ \begin{aligned} f(t_{ij})= \left\{ \begin{array}{ll} (1 - t_{ij}/ T) & \quad \text{for }t_{ij} \leq T \\ 0 & \quad \text{for }t_{ij} > T \end{array} \right.\\ \end{aligned} \]

Where \(t_{ij}\) is the travel cost between origin i and destination j, and \(T\) is the travel cost cutoff.

## Negative exponential

\[ \begin{aligned} f(t_{ij})= e^{(-\beta t_{ij})} \end{aligned} \]

Where \(t_{ij}\) is the travel cost between origin i and destination j, and \(\beta\) is the parameter that tells the speed of decay.

## Inverse power

\[ \begin{aligned} f(t_{ij})= \left\{ \begin{array}{ll} 1 & \quad \text{for } t_{ij}\leq 1 \\ t_{ij}^{-\beta} & \quad \text{for }t_{ij} > 1 \end{array} \right.\\ \end{aligned} \]

Where \(t_{ij}\) is the travel cost between origin i and destination j, and \(\beta\) is the parameter that tells the speed of decay.

## Stepped

Similar to the binary function, but can take an arbitrary number of steps. The current implementation assumes that values changes at each step, instead of right after it.

\[ \begin{aligned} f(t_{ij})= \left\{ \begin{array}{ll} 1 & \quad \text{for } t_{ij} \lt S_{1} \\ v_{1} & \quad \text{for } t_{ij} \lt S_{2} \\ v_{2} & \quad \text{for } t_{ij} \lt S_{3} \\ ... \\ v_{n-1} & \quad \text{for } t_{ij} \lt S_{n} \\ v_{n} & \quad \text{otherwise} \\ \end{array} \right.\\ \end{aligned} \]

Where \(t_{ij}\) is the travel cost between origin i and destination j, \(n\) is the total number of steps, \(S_{k}\) is the travel cost cutoff that delimits the \(k^{th}\) step, and \(v_{k}\) is the value that the decay function assumes at the \(k^{th}\) step.

## Logistic

Weights decay sigmoidally, according to a reversed cumulative logistic curve. Currently, the function implements a logistic distribution parameterized with the cutoff that sets its inflection point and the standard deviation that sets its steepness, according to the logistic decay curve proposed by Bauer and Groneberg (2016). Standard deviations values near 0 result in weighting curves that approximate binary decay, while higher values tend to linearize the curve.

\[ \begin{aligned} f(t_{ij}) = \frac{ 1 + e^\frac{-IP \times \pi}{SD \times \sqrt{3}}}{ 1 + e^\frac{(t_{ij} - IP) \times \pi}{SD \times \sqrt{3}}} \end{aligned} \]

Where \(t_{ij}\) is the travel cost between origin i and destination j, \(IP\) is the distribution inflection point, and \(SD\) is the distribution standard deviation.

### Quick demonstration

All decay functions (`decay_*()`) take decay parameters as input and return a function as output. This output function, in turn, takes a vector of numeric values as input, the travel costs, and returns a list of numeric vectors as output. Since the decay function can take multiple decay parameters as input, each element of the output list refers to the opportunities weights calculated with one of these decay parameters.

Let’s check this behavior with an example. With the code snippet below, we calculate the opportunities weights for the same travel costs, but using different negative exponential decay values (0.2 and 0.3):

    library(accessibility)

    output_fn <- decay_exponential(c(0.2, 0.3))

    output_fn(c(10, 15, 20))
    #> $`0.2`
    #> [1] 0.13533528 0.04978707 0.01831564
    #>
    #> $`0.3`
    #> [1] 0.049787068 0.011108997 0.002478752

`decay_stepped()`, as the only decay function that takes more than one argument (both `steps` and `weights`) names the output elements after the combination of steps and weights:

    stepped_output <- decay_stepped(
      steps = list(c(10, 20, 30), c(10, 20, 30, 40)),
      weights = list(c(0.67, 0.33, 0), c(0.75, 0.5, 0.25, 0))
    )

    stepped_output(c(15, 25, 35, 45))
    #> $`s(10,20,30);w(0.67,0.33,0)`
    #> [1] 0.67 0.33 0.00 0.00
    #>
    #> $`s(10,20,30,40);w(0.75,0.5,0.25,0)`
    #> [1] 0.75 0.50 0.25 0.00

With the code below, we demonstrate each decay function with travel costs ranging from 1 to 100:

    library(data.table)
    library(ggplot2)

    binary <- decay_binary(cutoff = 50)
    linear <- decay_linear(cutoff = 50)
    negative_exp <- decay_exponential(decay_value = 0.2)
    inverse_power <- decay_power(decay_value = 0.2)
    stepped <- decay_stepped(steps = c(30, 60, 90), weights = c(0.67, 0.33, 0))
    logistic <- decay_logistic(cutoff = 50, sd = 10)

    travel_costs <- seq(1, 100, 0.1)

    weights <- data.table(
      minutes = travel_costs,
      binary = as.numeric(binary(travel_costs)[["50"]]),
      linear = linear(travel_costs)[["50"]],
      negative_exp = negative_exp(travel_costs)[["0.2"]],
      inverse_power = inverse_power(travel_costs)[["0.2"]],
      stepped = stepped(travel_costs)[["s(30,60,90);w(0.67,0.33,0)"]],
      logistic = logistic(travel_costs)[["c50;sd10"]]
    )

    # reshape data to long format
    weights <- melt(
      weights,
      id.vars = "minutes",
      variable.name = "decay_function",
      value.name = "weights"
    )

    ggplot(weights) +
      geom_line(
        aes(minutes, weights, color = decay_function),
        show.legend = FALSE
      ) +
      facet_wrap(. ~ decay_function, ncol = 2) +
      theme_minimal()

# Using a custom decay function

accessibility also allows you to use a custom decay function, instead of one of the functions shipped with the package. A valid decay function is one that takes a `numeric` vector of travel costs as input and returns either:

- A `numeric` vector of weights, with the same length of input, or;
- A named `list` of `numeric` vectors to be used as weights, each one with the same length of input.

Let’s check the difference between each case with an example. Suppose we want to use a very simple decay function that defines the weights as the multiplicative inverse of travel cost - i.e. \(travel\_cost^{-1}\). We just have to take care of the case when travel cost is less than 1, in which case the function would return values greater than 1 and which we will replace with 1, but otherwise the implementation is pretty simple:

    my_decay <- function(travel_cost) {
      weights <- 1 / travel_cost
      weights[weights > 1] <- 1
      return(weights)
    }

Given a `numeric` vector of travel costs, the function returns a `numeric` vector of weights:

    my_decay(c(0, 0.5, 1, 2, 5, 10))
    #> [1] 1.0 1.0 1.0 0.5 0.2 0.1

Using this function to calculate accessibility is as easy as any of the built-in decay functions:

    data_dir <- system.file("extdata", package = "accessibility")

    travel_matrix <- readRDS(file.path(data_dir, "travel_matrix.rds"))
    land_use_data <- readRDS(file.path(data_dir, "land_use_data.rds"))

    custom_gravity <- gravity(
      travel_matrix,
      land_use_data,
      opportunity = "jobs",
      travel_cost = "travel_time",
      decay_function = my_decay
    )
    head(custom_gravity)
    #>                 id     jobs
    #> 1: 89a88cdb57bffff 11210.42
    #> 2: 89a88cdb597ffff 10775.77
    #> 3: 89a88cdb5b3ffff 11480.25
    #> 4: 89a88cdb5cfffff 12689.44
    #> 5: 89a88cd909bffff 11361.66
    #> 6: 89a88cd90b7ffff 12563.65

Great! But now suppose we want to change this function a bit. Instead of the simple multiplicative inverse of the travel cost, we want to multiply this inverse by a given decay parameter. Now we face a “problem”: our function would need to take two inputs (the travel cost, as above, and the decay parameter to multiply the travel cost), but `gravity()` can only take functions that receive a single input (the travel cost). In this case, we resort to the same strategy used in the decay functions shipped with the package, we create a function that takes a decay parameter as input and returns a function that takes travel cost input as output:

    my_second_decay <- function(decay_parameter) {
      function(travel_cost) {
        weights <- 1 / (decay_parameter * travel_cost)
        weights[weights > 1] <- 1
        return(weights)
      }
    }

    output_fn <- my_second_decay(2)
    output_fn(c(0, 0.5, 1, 2, 5, 10))
    #> [1] 1.00 1.00 0.50 0.25 0.10 0.05

    # compare to the first custom decay function
    my_decay(c(0, 0.5, 1, 2, 5, 10))
    #> [1] 1.0 1.0 1.0 0.5 0.2 0.1

Great, it works! In fact, we can achieve the exact same `gravity()` result shown above if we use `my_second_decay(1)`, instead of `my_decay`:

    second_custom_gravity <- gravity(
      travel_matrix,
      land_use_data,
      opportunity = "jobs",
      travel_cost = "travel_time",
      decay_function = my_second_decay(1)
    )
    head(second_custom_gravity)
    #>                 id     jobs
    #> 1: 89a88cdb57bffff 11210.42
    #> 2: 89a88cdb597ffff 10775.77
    #> 3: 89a88cdb5b3ffff 11480.25
    #> 4: 89a88cdb5cfffff 12689.44
    #> 5: 89a88cd909bffff 11361.66
    #> 6: 89a88cd90b7ffff 12563.65

A small difference is that in the first example we passed `my_decay` (the function object) to the `decay_function` parameter, whereas in the second we passed the function call `my_second_decay(1)`. That’s because `my_second_decay()` (and the built-in decay functions as well) is actually a function “factory”: it’s a function that returns a function.

    decay_power(1)
    #> function (travel_cost)
    #> {
    #>     weights_list <- lapply(decay_value, function(x) {
    #>         weights <- travel_cost^(-x)
    #>         weights[weights > 1] <- 1
    #>         weights
    #>     })
    #>     names(weights_list) <- decay_value
    #>     return(weights_list)
    #> }
    #> <bytecode: 0x5599abde84a0>
    #> <environment: 0x5599b6f25998>

What if we want our custom function to take many decay parameters as inputs, just like the built-in decay functions? In this case, our output function has to return a `list` of `numeric` weights named after the decay parameters we have set:

    my_third_decay <- function(decay_parameter) {
      function(travel_cost) {
        weighting_list <- lapply(
          decay_parameter,
          function(x) {
            weights <- 1 / (x * travel_cost)
            weights[weights > 1] <- 1
            return(weights)
          }
        )

        names(weighting_list) <- decay_parameter
        weighting_list
      }
    }

    output_fn <- my_third_decay(c(1, 2))
    output_fn(c(0, 0.5, 1, 2, 5, 10))
    #> $`1`
    #> [1] 1.0 1.0 1.0 0.5 0.2 0.1
    #>
    #> $`2`
    #> [1] 1.00 1.00 0.50 0.25 0.10 0.05

    # compare to the first and second custom decay functions

    my_decay(c(0, 0.5, 1, 2, 5, 10))
    #> [1] 1.0 1.0 1.0 0.5 0.2 0.1

    my_second_decay(2)(c(0, 0.5, 1, 2, 5, 10))
    #> [1] 1.00 1.00 0.50 0.25 0.10 0.05

This new custom decay function is a bit more complex than the previous two, yet much more powerful. We can use it in conjunction with `gravity()` (and the other accessibility functions that take decay functions as input as well) to calculate the accessibility with multiple decay parameters in a single call:

    third_custom_gravity <- gravity(
      travel_matrix,
      land_use_data,
      opportunity = "jobs",
      travel_cost = "travel_time",
      decay_function = my_third_decay(c(1, 2))
    )
    third_custom_gravity
    #>                    id decay_function_arg      jobs
    #>    1: 89a88cdb57bffff                  1 11210.421
    #>    2: 89a88cdb597ffff                  1 10775.769
    #>    3: 89a88cdb5b3ffff                  1 11480.254
    #>    4: 89a88cdb5cfffff                  1 12689.441
    #>    5: 89a88cd909bffff                  1 11361.664
    #>   ---
    #> 1792: 89a881acda3ffff                  2  5621.512
    #> 1793: 89a88cdb543ffff                  2  7406.822
    #> 1794: 89a88cda667ffff                  2  5947.782
    #> 1795: 89a88cd900fffff                  2  4565.564
    #> 1796: 89a881aebafffff                  2     0.000

When we create our own custom decay function, we don’t have to think about general cases, just the cases that matter for us. Creating a simple function that converts one vector into another, therefore, might be enough, given our needs. If, however, you’re testing a new decay function and you want to check its sensitivity to different decay parameters, for example, you might be better off developing a decay function that can take many decay values as inputs and returns a list of weight vectors as output.

Bauer, Jan, and David A. Groneberg. 2016. “Measuring Spatial Accessibility of Health Care Providers – Introduction of a Variable Distance Decay Function Within the Floating Catchment Area (FCA) Method.” Edited by Kebede Deribe. PLOS ONE 11 (7): e0159148. https://doi.org/10.1371/journal.pone.0159148.