Table 1.
A comparison of mixture density networks and Gaussian processes.
Fig 1.
MDN that emulates a model with three inputs and a one-dimensional output with two mixtures.
The inputs are passed through two hidden layers, which are then passed on to the normalised neurons, which represent the parameters of a distribution and its weights e.g. the mean (shown in blue) and variance (shown in green) of a normal distribution. These parameters are used to construct a mixture of distributions (represented as a dashed line).
Fig 2.
Gamma-MDN output emulating a negative binomial model.
(A) For fixed shape parameter k = 2.5, the distribution of output from MDN is shown in blue (mean = solid line, variance = shaded region), the theoretical values are shown as a black dashed line (mean = bold line, variance = normal line). (B) For fixed mean parameter m = 50, the distribution of output from MDN over a range of k values is shown in blue (mean = solid line, variance = shaded region), the theoretical values are shown as a black dashed line (mean = bold line, variance = normal line). (C) Corresponding two-sample K–S statistic where sample of 100 points are drawn from a negative binomial and the MDN over a range of m values. 100 replicates are used to estimate a mean K–S statistic and a 95% range. The dashed line represents significance at α = 0.05, with values less than this indicating that the two samples do not differ significantly. (D) Example empirical CDFs drawn from 100 samples of MDN with inputs m = 50 and k = 2.5. 1,000 empirical CDFs are shown as black transparent lines and true CDF is shown as a blue solid line.
Fig 3.
Binomial-MDN output emulating the final size distribution of a stochastic SIR model.
(A) For random uniform sampling over β and γ a sample of the output from MDN across values for the basic reproductive number R0 = β/γ are shown in blue and the directly simulated values are shown in red. (B) Corresponding two-sample K–S statistic where sample of 100 points are drawn from a negative binomial and the MDN over a range of R0 values. 100 replicates are used to estimate a mean K–S statistic and a 95% range. Dashed line represent significance at α = 0.05, with values less indicating the two samples do not differ significantly. (C) The percentage of 1,000 realisations of the stochastic SIR model with final size greater than 100 is shown in black with dashed line showing a 95% range. Emulated results are shown by the blue line with a 95% range. (D) Example empirical CDFs drawn from 100 samples of MDN with inputs β = 0.4 and γ = 0.2. 1,000 empirical CDF are shown as black transparent lines and true CDF is shown as a blue solid line.
Fig 4.
Beta-MDN output emulating the infection dynamics with time for a stochastic SIR model.
(A–D) A comparison of simulation results with sampled MDN output for fixed γ = 0.2 and N = 1, 000 and different β values that give the following R0 values: (A) R0 = 0.5, (B) R0 = 1.0, (C) R0 = 2.0, and (D) R0 = 5.0. (E–F) Two-sample K–S statistic where sample of 100 points are drawn from a negative binomial and the MDN over a range of time t values. 100 replicates are used to estimate a mean K–S statistic and a 95% range. Dashed line represent significance at α = 0.05, with values less indicating the two samples do not differ significantly. Tests are for (E) number of susceptible people and (F) number of infected people.
Fig 5.
Beta-MDN output emulating the infection dynamics with time for a stochastic SIR model.
(A–D) A comparison of simulation results with sampled MDN output for fixed γ = 0.2 and different β, δ and N values such that (A) R0 = 2.0, δ = 0.01 and N = 1, 000, (B) R0 = 1.0, δ = 0.01 and N = 1, 000, (C) R0 = 2.0, δ = 0.001 and N = 1, 000, (D) R0 = 2.0, δ = 0.01 and N = 100. (E–F) Two-sample K–S statistic where sample of 100 points are drawn from a negative binomial and the MDN over a range of time t values. 100 replicates are used to estimate a mean K–S statistic and a 95% range. Dashed line represent significance at α = 0.05, with values less indicating the two samples do not differ significantly. Tests are for (E) number of susceptible people and (F) number of infected people.