The difference of two delays

Introduction

A Difference is the signed gap between two events, so its support runs on both sides of zero — unlike Convolved (see Convolving distributions) or product, whose supports only ever run in one direction.

What are we going to do in this exercise

2. Plot the density and cumulative probability of the difference of two delays across zero.

3. Take the difference between a Convolved total delay and a single delay.

What might I need to know before starting

This tutorial builds on the Getting started overview and uses AlgebraOfGraphics.jl and CairoMakie.jl for plotting. No fitting or MCMC is involved; every quantity is a direct evaluation.

Packages used

using ConvolvedDistributions, Distributions
using CairoMakie, AlgebraOfGraphics, DataFramesMeta

CairoMakie.activate!(type = "png", px_per_unit = 2)

The difference of two delays

difference builds Z = X - Y, here a reporting delay minus an incubation period. Reflecting the subtracted component makes the support two-sided, so the density crosses zero.

incubation = Gamma(2.0, 1.0)
reporting = LogNormal(1.0, 0.5)

z_dist = difference(reporting, incubation)
z = -8.0:0.05:12.0
difference_df = vcat(
    DataFrame(z = z, value = pdf.(z_dist, z), Quantity = "Density (pdf)"),
    DataFrame(z = z, value = cdf.(z_dist, z), Quantity = "Cumulative (cdf)")
)
draw(
    data(difference_df) *
        mapping(:z, :value, color = :Quantity) *
        visual(Lines, linewidth = 2);
    axis = (
        xlabel = "Reporting delay - incubation period (days)",
        ylabel = "Density / cumulative probability",
    )
)

The mass below zero is the probability that the reporting delay is shorter than the incubation period.

cdf(z_dist, 0.0)

0.2801512628537786

Differencing a Convolved total delay

A Convolved distribution is itself a UnivariateDistribution, so it can be one side of a difference. Here the two-stage delay incubation + reporting is the minuend, and a single Gamma delay the subtrahend.

d = convolved(incubation, reporting)
gap = difference(d, Gamma(2.5, 1.0))
zg = -8.0:0.25:12.0
gap_df = vcat(
    DataFrame(z = zg, value = pdf.(gap, zg), Quantity = "Density (pdf)"),
    DataFrame(z = zg, value = cdf.(gap, zg), Quantity = "Cumulative (cdf)")
)
draw(
    data(gap_df) *
        mapping(:z, :value, color = :Quantity) *
        visual(Lines, linewidth = 2);
    axis = (
        xlabel = "Two-stage delay - single delay (days)",
        ylabel = "Density / cumulative probability",
    )
)

The mass below zero is the probability that the two-stage delay resolves before the single delay.

cdf(gap, 0.0)

0.14688879093119392

Summary

• difference has two-sided support, and its mass below zero is directly interpretable as an ordering probability.

• A Convolved can be one side of a difference, so a multi-stage delay differences against a single delay in one call.

See also: Convolving distributions, The product of two delays, Convolving a timeseries.