# Bayesian eruption age estimation example notebook¶

This Jupyter notebook demonstrates the Bayesian eruption age estimation approach of Keller, Schoene, and Samperton (2018)

Hint: shift-enter to run a single cell, or from the menu select Cell > Run All to run the whole file

#### (run this first!)¶

In [1]:
# Required packages
using Plots; gr(); default(fmt = :png);
using KernelDensity: kde

if ~isfile("DistTools.jl")
end

# Functions we'll be using here
include("DistTools.jl");


## (2a) Test bayesian eruption age estimation with a synthetic dataset¶

#### Generate synthetic zircon dataset, drawing from MeltsVolcanicZirconDistribution¶

In [2]:
dt_sigma = 30; # Timescale relative to analytical uncertainty
N = 42; # Number of zircons

# Draw set of pseudorandom ages from MELTS volcanic zircon distribution,
# with true minimum == 0 and analytical sigma == 1
ages = draw_from_distribution(MeltsVolcanicZirconDistribution,N).*dt_sigma + randn(N);
uncert = ones(N)

# Calculate the weighted mean age
(wx, wsigma, mswd) = awmean(ages, uncert)

h = plot(ylabel="Time before eruption (sigma)", xlabel="N", fg_color_legend=:white, legend=:topleft)
plot!(h,1:length(ages),sort(ages),yerror=uncert*2,seriestype=:scatter, markerstrokecolor=:auto, label="Synthetic zircon ages")
plot!(h,collect(xlims()), [0,0], label="Eruption age")
plot!(h,collect(xlims()), [wx,wx], label="Weighted mean, MSWD $(signif(mswd,2))")  Out[2]: #### Calculate bootstrapped$\ \mathcal{\vec{f}}_{xtal}(t_r)$¶ In [3]: # Maximum extent of expected analytical tail (beyond eruption/deposition) maxTailLength = mean(uncert) * norm_quantile(1 - 1/(length(ages) + 1)); included = (ages .- minimum(ages)) .>= maxTailLength; # Bootstrapped crystallization distribution, excluding maximum analytical tail if sum(included) > 5 dist = BootstrapCrystDistributionKDE(ages[included]); else # Avoid edge cases at n = 0 and n = 2; # Default to n = 1 instead, which yields a half-normal distribution dist = BootstrapCrystDistributionKDE([0.]); end dist = dist./mean(dist); # Plot bootstrapped distribution plot(linspace(0,1.3,length(dist)),dist, label="bootstrapped", ylabel="Probability Density", xlabel="Time before eruption (scaled)", legend=:bottomleft, fg_color_legend=:white) plot!(linspace(0,1,100),MeltsVolcanicZirconDistribution,label="original")  Out[3]: #### Run MCMC to estimate eruption/deposition age distribution of synthetic dataset¶ In [4]: # Configure model nsteps = 200000; # Length of Markov chain burnin = 100000; # Number of steps to discard at beginning of Markov chain # Run MCMC tminDist = Array{Float64,1}(nsteps); metropolis_min(nsteps,dist,ages,uncert,tminDist); # Print results AgeEst = mean(tminDist[burnin:end]); AgeEst_sigma = std(tminDist[burnin:end]); print("\nEstimated eruption age of synthetic dataset:\n$AgeEst +/- $(2*AgeEst_sigma) Ma (2σ)\n (True synthetic age 0 Ma)") # Plot results h = histogram(tminDist[burnin:end],nbins=50,label="Posterior distribution",xlabel="Eruption Age (Ma)",ylabel="N") plot!(h,[0,0],collect(ylims()),line=(3),label="True (synthetic) age",fg_color_legend=:white) plot!(h,[wx,wx],collect(ylims()),line=(3),label="Weighted mean age",legend=:topright) display(h) sleep(0.5) # (just to make sure this section is finished running)  Estimated eruption age of synthetic dataset: 0.9480287111256819 +/- 2.0507106780895277 Ma (2σ) (True synthetic age 0 Ma) ## (2b) Estimate eruption age for real zircon data¶ The example dataset here is from Wotzlaw et al., 2013 FCT+MLX #### Input dataset (Try pasting in your own data here!)¶ In [5]: # Age and one-sigma uncertainty. ages = [28.196, 28.206, 28.215, 28.224, 28.232, 28.241, 28.246, 28.289, 28.308, 28.332, 28.341, 28.359, 28.379, 28.383, 28.395, 28.4, 28.405, 28.413, 28.415, 28.418, 28.42, 28.422, 28.428, 28.452, 28.454, 28.454, 28.458, 28.468, 28.471, 28.475, 28.482, 28.485, 28.502, 28.52, 28.551, 28.561, 28.565, 28.582, 28.584, 28.586, 28.611, 28.638, 28.655] uncert = [0.019, 0.0155, 0.019, 0.0215, 0.018, 0.023, 0.013, 0.029, 0.0175, 0.0315, 0.0095, 0.0245, 0.0255, 0.0175, 0.0235, 0.014, 0.021, 0.022, 0.0125, 0.0135, 0.016, 0.0195, 0.0175, 0.0125, 0.01, 0.014, 0.015, 0.0205, 0.0155, 0.011, 0.0115, 0.0185, 0.0255, 0.014, 0.0125, 0.013, 0.015, 0.014, 0.012, 0.016, 0.0215, 0.0125, 0.0215] # Sort by age (just to make rank-order plots prettier) t = sortperm(ages) ages = ages[t]; uncert = uncert[t];  #### Calculate bootstrapped$\ \mathcal{\vec{f}}_{xtal}(t_r)$¶ In [6]: # Maximum extent of expected analytical tail (beyond eruption/deposition) maxTailLength = mean(uncert) * norm_quantile(1-1/(length(ages) + 1)); included = (ages-minimum(ages)) .>= maxTailLength; # Bootstrapped crystallization distribution, excluding maximum analytical tail if sum(included) > 5 dist = BootstrapCrystDistributionKDE(ages[included]); else # Avoid edge cases at n = 0 and n = 2; # Default to n = 1 instead, which yields a half-normal distribution dist = BootstrapCrystDistributionKDE([0.]); end dist = dist./mean(dist); # Plot bootstrapped distribution plot(linspace(0,1,length(dist)),dist, label="bootstrapped", ylabel="Probability Density", xlabel="Time before eruption (unitless)", fg_color_legend=:white)  Out[6]: #### Run MCMC to estimate eruption age¶ In [7]: # Configure model nsteps = 400000; # Length of Markov chain burnin = 150000; # Number of steps to discard at beginning of Markov chain # Run MCMC tminDist = Array{Float64,1}(nsteps); metropolis_min(nsteps,dist,ages,uncert,tminDist); # Consider also: # metropolis_min(nsteps,UniformDistribution,ages,uncert,tminDist); # metropolis_min(nsteps,TriangularDistribution,ages,uncert,tminDist); # metropolis_min(nsteps,HalfNormalDistribution,ages,uncert,tminDist); # metropolis_min(nsteps,MeltsVolcanicZirconDistribution,ages,uncert,tminDist); # Print results AgeEst = mean(tminDist[burnin:end]); AgeEst_sigma = std(tminDist[burnin:end]); print("\nEstimated eruption age:\n$AgeEst +/- \$(2*AgeEst_sigma) Ma (2σ)\n")

# Plot results
h = histogram(tminDist[burnin:end],nbins=100,label="Posterior distribution",xlabel="Eruption Age (Ma)",ylabel="N",legend=:topleft,fg_color_legend=:white)
# plot!(h,[wx,wx],collect(ylims()),line=(3),label="Weighted mean age",legend=:topleft)
display(h)
sleep(0.5)

Estimated eruption age:
28.184153232684807 +/- 0.03696267818756048 Ma (2σ)

In [8]:
# Plot eruption age estimate relative to rank-order plot of raw zircon ages
h = plot(ylabel="Age (Ma)", xlabel="N", legend=:topleft, fg_color_legend=:white)
plot!(h,1:length(ages),ages,yerror=uncert*2,seriestype=:scatter, markerstrokecolor=:auto, label="Observed ages")
plot!(h,[length(ages)],[AgeEst],yerror=2*AgeEst_sigma, markerstrokecolor=:auto, label="Bayesian eruption age estimate",color=:red)
plot!(h,collect(xlims()),[AgeEst,AgeEst],color=:red, label="")

Out[8]:

In [ ]: