Simulations
In addition to inference, the Gaussian process modelling framework allows simulations and predictions of the underlying process. This is done by conditioning the Gaussian process on some observations and then sampling from the conditioned distribution.
Sampling from the Gaussian process
Assuming a SingleBendingPowerLaw we can draw realisations from the Gaussian process with a mean. First, we define the power spectral density function and plot it.
using Random
using Plots
using Pioran
rng = MersenneTwister(1234)
# Define power spectral density function
𝓟 = SingleBendingPowerLaw(0.3,1e-2,2.9)
f_min, f_max = 1e-3, 1e3
f0,fM = f_min/20.,f_max*20.
f = 10 .^ range(log10(f0), log10(fM), length=1000)
plot(f, 𝓟(f), xlabel="Frequency (day^-1)",ylabel="Power Spectral Density",legend=false,framestyle = :box,xscale=:log10,yscale=:log10,lw=2)
Then we approximate it to form a covariance function. We then build the Gaussian process and draw realisations from it using rand.
variance = 15.2
# Approximation of the PSD to form a covariance function
𝓡 = approx(𝓟, f_min, f_max, 20, variance, basis_function="SHO")
μ = 1.3
# Build the GP
f = ScalableGP(μ, 𝓡) # Define the GP
T = 450 # days
t = range(0, stop=T, length=1000)
σ² = ones(length(t))*0.25
fx = f(t, σ²) # Gaussian process
realisations = [rand(rng,fx) for _ in 1:5]
plot(t, realisations, label="Realisations", xlabel="Time [days]", framestyle = :box,ylabel="Value")
Sampling from a Gaussian process built with semi-separable covariance functions is very efficient. The time complexity is O(N) where N is the number of data points.
Conditioning the Gaussian process
We can compute the conditioned or posterior distribution of the Gaussian process given some observations. Let's use a subset of the realisations to condition the Gaussian process and then sample from the conditioned distribution.
using StatsBase
idx = sort(sample(1:length(t), 50, replace = false));
t_obs = t[idx]
y_obs = realisations[1][idx]
yerr = 0.25*ones(length(t_obs))
fx = f(t_obs, yerr.^2) # Gaussian processAbstractGPs.FiniteGP{ScalableGP{AbstractGPs.GP{AbstractGPs.ConstMean{Float64}, Pioran.SumOfCelerite{StructArrays.StructArray{<:Pioran.SemiSeparable}}}, Pioran.SumOfCelerite{StructArrays.StructArray{<:Pioran.SemiSeparable}}, Symbol}, Vector{Float64}, LinearAlgebra.Diagonal{Float64, Vector{Float64}}}(
f: ScalableGP{AbstractGPs.GP{AbstractGPs.ConstMean{Float64}, Pioran.SumOfCelerite{StructArrays.StructArray{<:Pioran.SemiSeparable}}}, Pioran.SumOfCelerite{StructArrays.StructArray{<:Pioran.SemiSeparable}}, Symbol}(AbstractGPs.GP{AbstractGPs.ConstMean{Float64}, Pioran.SumOfCelerite{StructArrays.StructArray{<:Pioran.SemiSeparable}}}(AbstractGPs.ConstMean{Float64}(1.3), Pioran.SumOfCelerite{StructArrays.StructArray{<:Pioran.SemiSeparable}}(Celerite[Celerite(0.21179581122061306, 0.21179581122061306, 0.0002221441469079183, 0.0002221441469079183), Celerite(0.38542244831627853, 0.38542244831627853, 0.0006300497947253256, 0.0006300497947253256), Celerite(0.9038882037020084, 0.9038882037020084, 0.0017869601758986303, 0.0017869601758986303), Celerite(1.6067166095810141, 1.6067166095810141, 0.0050682131745472155, 0.0050682131745472155), Celerite(4.964666291074675, 4.964666291074675, 0.014374570362059993, 0.014374570362059993), Celerite(22.755812872815056, 22.755812872815056, 0.04076945187142277, 0.04076945187142277), Celerite(5.378025805869529, 5.378025805869529, 0.11563115724719714, 0.11563115724719714), Celerite(0.5417886639897328, 0.5417886639897328, 0.32795546451037977, 0.32795546451037977), Celerite(0.08070461417658467, 0.08070461417658467, 0.9301540282286335, 0.9301540282286335), Celerite(0.010848075938726597, 0.010848075938726597, 2.638121970376775, 2.638121970376775), Celerite(0.0015090806302273192, 0.0015090806302273192, 7.482295748198312, 7.482295748198312), Celerite(0.0002076692891940062, 0.0002076692891940062, 21.221440969050757, 21.221440969050757), Celerite(2.8676529647079397e-5, 2.8676529647079397e-5, 60.18868699641365, 60.18868699641365), Celerite(3.955543296634202e-6, 3.955543296634202e-6, 170.70839099171252, 170.70839099171252), Celerite(5.458037196298439e-7, 5.458037196298439e-7, 484.16664674402534, 484.16664674402534), Celerite(7.53042020169437e-8, 7.53042020169437e-8, 1373.203393562152, 1373.203393562152), Celerite(1.0389899331678754e-8, 1.0389899331678754e-8, 3894.70768540476, 3894.70768540476), Celerite(1.4338973791019396e-9, 1.4338973791019396e-9, 11046.249977144691, 11046.249977144691), Celerite(1.9677308815711412e-10, 1.9677308815711412e-10, 31329.601195702533, 31329.601195702533), Celerite(3.012647957419647e-11, 3.012647957419647e-11, 88857.65876316732, 88857.65876316732)], [0.21179581122061306, 0.38542244831627853, 0.9038882037020084, 1.6067166095810141, 4.964666291074675, 22.755812872815056, 5.378025805869529, 0.5417886639897328, 0.08070461417658467, 0.010848075938726597, 0.0015090806302273192, 0.0002076692891940062, 2.8676529647079397e-5, 3.955543296634202e-6, 5.458037196298439e-7, 7.53042020169437e-8, 1.0389899331678754e-8, 1.4338973791019396e-9, 1.9677308815711412e-10, 3.012647957419647e-11], [0.21179581122061306, 0.38542244831627853, 0.9038882037020084, 1.6067166095810141, 4.964666291074675, 22.755812872815056, 5.378025805869529, 0.5417886639897328, 0.08070461417658467, 0.010848075938726597, 0.0015090806302273192, 0.0002076692891940062, 2.8676529647079397e-5, 3.955543296634202e-6, 5.458037196298439e-7, 7.53042020169437e-8, 1.0389899331678754e-8, 1.4338973791019396e-9, 1.9677308815711412e-10, 3.012647957419647e-11], [0.0002221441469079183, 0.0006300497947253256, 0.0017869601758986303, 0.0050682131745472155, 0.014374570362059993, 0.04076945187142277, 0.11563115724719714, 0.32795546451037977, 0.9301540282286335, 2.638121970376775, 7.482295748198312, 21.221440969050757, 60.18868699641365, 170.70839099171252, 484.16664674402534, 1373.203393562152, 3894.70768540476, 11046.249977144691, 31329.601195702533, 88857.65876316732], [0.0002221441469079183, 0.0006300497947253256, 0.0017869601758986303, 0.0050682131745472155, 0.014374570362059993, 0.04076945187142277, 0.11563115724719714, 0.32795546451037977, 0.9301540282286335, 2.638121970376775, 7.482295748198312, 21.221440969050757, 60.18868699641365, 170.70839099171252, 484.16664674402534, 1373.203393562152, 3894.70768540476, 11046.249977144691, 31329.601195702533, 88857.65876316732])), Pioran.SumOfCelerite{StructArrays.StructArray{<:Pioran.SemiSeparable}}(Celerite[Celerite(0.21179581122061306, 0.21179581122061306, 0.0002221441469079183, 0.0002221441469079183), Celerite(0.38542244831627853, 0.38542244831627853, 0.0006300497947253256, 0.0006300497947253256), Celerite(0.9038882037020084, 0.9038882037020084, 0.0017869601758986303, 0.0017869601758986303), Celerite(1.6067166095810141, 1.6067166095810141, 0.0050682131745472155, 0.0050682131745472155), Celerite(4.964666291074675, 4.964666291074675, 0.014374570362059993, 0.014374570362059993), Celerite(22.755812872815056, 22.755812872815056, 0.04076945187142277, 0.04076945187142277), Celerite(5.378025805869529, 5.378025805869529, 0.11563115724719714, 0.11563115724719714), Celerite(0.5417886639897328, 0.5417886639897328, 0.32795546451037977, 0.32795546451037977), Celerite(0.08070461417658467, 0.08070461417658467, 0.9301540282286335, 0.9301540282286335), Celerite(0.010848075938726597, 0.010848075938726597, 2.638121970376775, 2.638121970376775), Celerite(0.0015090806302273192, 0.0015090806302273192, 7.482295748198312, 7.482295748198312), Celerite(0.0002076692891940062, 0.0002076692891940062, 21.221440969050757, 21.221440969050757), Celerite(2.8676529647079397e-5, 2.8676529647079397e-5, 60.18868699641365, 60.18868699641365), Celerite(3.955543296634202e-6, 3.955543296634202e-6, 170.70839099171252, 170.70839099171252), Celerite(5.458037196298439e-7, 5.458037196298439e-7, 484.16664674402534, 484.16664674402534), Celerite(7.53042020169437e-8, 7.53042020169437e-8, 1373.203393562152, 1373.203393562152), Celerite(1.0389899331678754e-8, 1.0389899331678754e-8, 3894.70768540476, 3894.70768540476), Celerite(1.4338973791019396e-9, 1.4338973791019396e-9, 11046.249977144691, 11046.249977144691), Celerite(1.9677308815711412e-10, 1.9677308815711412e-10, 31329.601195702533, 31329.601195702533), Celerite(3.012647957419647e-11, 3.012647957419647e-11, 88857.65876316732, 88857.65876316732)], [0.21179581122061306, 0.38542244831627853, 0.9038882037020084, 1.6067166095810141, 4.964666291074675, 22.755812872815056, 5.378025805869529, 0.5417886639897328, 0.08070461417658467, 0.010848075938726597, 0.0015090806302273192, 0.0002076692891940062, 2.8676529647079397e-5, 3.955543296634202e-6, 5.458037196298439e-7, 7.53042020169437e-8, 1.0389899331678754e-8, 1.4338973791019396e-9, 1.9677308815711412e-10, 3.012647957419647e-11], [0.21179581122061306, 0.38542244831627853, 0.9038882037020084, 1.6067166095810141, 4.964666291074675, 22.755812872815056, 5.378025805869529, 0.5417886639897328, 0.08070461417658467, 0.010848075938726597, 0.0015090806302273192, 0.0002076692891940062, 2.8676529647079397e-5, 3.955543296634202e-6, 5.458037196298439e-7, 7.53042020169437e-8, 1.0389899331678754e-8, 1.4338973791019396e-9, 1.9677308815711412e-10, 3.012647957419647e-11], [0.0002221441469079183, 0.0006300497947253256, 0.0017869601758986303, 0.0050682131745472155, 0.014374570362059993, 0.04076945187142277, 0.11563115724719714, 0.32795546451037977, 0.9301540282286335, 2.638121970376775, 7.482295748198312, 21.221440969050757, 60.18868699641365, 170.70839099171252, 484.16664674402534, 1373.203393562152, 3894.70768540476, 11046.249977144691, 31329.601195702533, 88857.65876316732], [0.0002221441469079183, 0.0006300497947253256, 0.0017869601758986303, 0.0050682131745472155, 0.014374570362059993, 0.04076945187142277, 0.11563115724719714, 0.32795546451037977, 0.9301540282286335, 2.638121970376775, 7.482295748198312, 21.221440969050757, 60.18868699641365, 170.70839099171252, 484.16664674402534, 1373.203393562152, 3894.70768540476, 11046.249977144691, 31329.601195702533, 88857.65876316732]), :celerite)
x: [4.504504504504505, 27.47747747747748, 29.27927927927928, 40.990990990990994, 55.4054054054054, 77.92792792792793, 115.76576576576576, 135.13513513513513, 140.54054054054055, 145.04504504504504 … 380.6306306306306, 386.93693693693695, 391.44144144144144, 400.9009009009009, 420.72072072072075, 422.07207207207205, 425.22522522522524, 435.13513513513516, 437.8378378378378, 447.7477477477477]
Σy: [0.0625 0.0 … 0.0 0.0; 0.0 0.0625 … 0.0 0.0; … ; 0.0 0.0 … 0.0625 0.0; 0.0 0.0 … 0.0 0.0625]
)
We can compute the posterior distribution of the Gaussian process given the observations.
fp = posterior(fx, y_obs) # Posterior distributionPioran.PosteriorGP{AbstractGPs.FiniteGP{ScalableGP{AbstractGPs.GP{AbstractGPs.ConstMean{Float64}, Pioran.SumOfCelerite{StructArrays.StructArray{<:Pioran.SemiSeparable}}}, Pioran.SumOfCelerite{StructArrays.StructArray{<:Pioran.SemiSeparable}}, Symbol}, Vector{Float64}, LinearAlgebra.Diagonal{Float64, Vector{Float64}}}, Vector{Float64}}(AbstractGPs.FiniteGP{ScalableGP{AbstractGPs.GP{AbstractGPs.ConstMean{Float64}, Pioran.SumOfCelerite{StructArrays.StructArray{<:Pioran.SemiSeparable}}}, Pioran.SumOfCelerite{StructArrays.StructArray{<:Pioran.SemiSeparable}}, Symbol}, Vector{Float64}, LinearAlgebra.Diagonal{Float64, Vector{Float64}}}(
f: ScalableGP{AbstractGPs.GP{AbstractGPs.ConstMean{Float64}, Pioran.SumOfCelerite{StructArrays.StructArray{<:Pioran.SemiSeparable}}}, Pioran.SumOfCelerite{StructArrays.StructArray{<:Pioran.SemiSeparable}}, Symbol}(AbstractGPs.GP{AbstractGPs.ConstMean{Float64}, Pioran.SumOfCelerite{StructArrays.StructArray{<:Pioran.SemiSeparable}}}(AbstractGPs.ConstMean{Float64}(1.3), Pioran.SumOfCelerite{StructArrays.StructArray{<:Pioran.SemiSeparable}}(Celerite[Celerite(0.21179581122061306, 0.21179581122061306, 0.0002221441469079183, 0.0002221441469079183), Celerite(0.38542244831627853, 0.38542244831627853, 0.0006300497947253256, 0.0006300497947253256), Celerite(0.9038882037020084, 0.9038882037020084, 0.0017869601758986303, 0.0017869601758986303), Celerite(1.6067166095810141, 1.6067166095810141, 0.0050682131745472155, 0.0050682131745472155), Celerite(4.964666291074675, 4.964666291074675, 0.014374570362059993, 0.014374570362059993), Celerite(22.755812872815056, 22.755812872815056, 0.04076945187142277, 0.04076945187142277), Celerite(5.378025805869529, 5.378025805869529, 0.11563115724719714, 0.11563115724719714), Celerite(0.5417886639897328, 0.5417886639897328, 0.32795546451037977, 0.32795546451037977), Celerite(0.08070461417658467, 0.08070461417658467, 0.9301540282286335, 0.9301540282286335), Celerite(0.010848075938726597, 0.010848075938726597, 2.638121970376775, 2.638121970376775), Celerite(0.0015090806302273192, 0.0015090806302273192, 7.482295748198312, 7.482295748198312), Celerite(0.0002076692891940062, 0.0002076692891940062, 21.221440969050757, 21.221440969050757), Celerite(2.8676529647079397e-5, 2.8676529647079397e-5, 60.18868699641365, 60.18868699641365), Celerite(3.955543296634202e-6, 3.955543296634202e-6, 170.70839099171252, 170.70839099171252), Celerite(5.458037196298439e-7, 5.458037196298439e-7, 484.16664674402534, 484.16664674402534), Celerite(7.53042020169437e-8, 7.53042020169437e-8, 1373.203393562152, 1373.203393562152), Celerite(1.0389899331678754e-8, 1.0389899331678754e-8, 3894.70768540476, 3894.70768540476), Celerite(1.4338973791019396e-9, 1.4338973791019396e-9, 11046.249977144691, 11046.249977144691), Celerite(1.9677308815711412e-10, 1.9677308815711412e-10, 31329.601195702533, 31329.601195702533), Celerite(3.012647957419647e-11, 3.012647957419647e-11, 88857.65876316732, 88857.65876316732)], [0.21179581122061306, 0.38542244831627853, 0.9038882037020084, 1.6067166095810141, 4.964666291074675, 22.755812872815056, 5.378025805869529, 0.5417886639897328, 0.08070461417658467, 0.010848075938726597, 0.0015090806302273192, 0.0002076692891940062, 2.8676529647079397e-5, 3.955543296634202e-6, 5.458037196298439e-7, 7.53042020169437e-8, 1.0389899331678754e-8, 1.4338973791019396e-9, 1.9677308815711412e-10, 3.012647957419647e-11], [0.21179581122061306, 0.38542244831627853, 0.9038882037020084, 1.6067166095810141, 4.964666291074675, 22.755812872815056, 5.378025805869529, 0.5417886639897328, 0.08070461417658467, 0.010848075938726597, 0.0015090806302273192, 0.0002076692891940062, 2.8676529647079397e-5, 3.955543296634202e-6, 5.458037196298439e-7, 7.53042020169437e-8, 1.0389899331678754e-8, 1.4338973791019396e-9, 1.9677308815711412e-10, 3.012647957419647e-11], [0.0002221441469079183, 0.0006300497947253256, 0.0017869601758986303, 0.0050682131745472155, 0.014374570362059993, 0.04076945187142277, 0.11563115724719714, 0.32795546451037977, 0.9301540282286335, 2.638121970376775, 7.482295748198312, 21.221440969050757, 60.18868699641365, 170.70839099171252, 484.16664674402534, 1373.203393562152, 3894.70768540476, 11046.249977144691, 31329.601195702533, 88857.65876316732], [0.0002221441469079183, 0.0006300497947253256, 0.0017869601758986303, 0.0050682131745472155, 0.014374570362059993, 0.04076945187142277, 0.11563115724719714, 0.32795546451037977, 0.9301540282286335, 2.638121970376775, 7.482295748198312, 21.221440969050757, 60.18868699641365, 170.70839099171252, 484.16664674402534, 1373.203393562152, 3894.70768540476, 11046.249977144691, 31329.601195702533, 88857.65876316732])), Pioran.SumOfCelerite{StructArrays.StructArray{<:Pioran.SemiSeparable}}(Celerite[Celerite(0.21179581122061306, 0.21179581122061306, 0.0002221441469079183, 0.0002221441469079183), Celerite(0.38542244831627853, 0.38542244831627853, 0.0006300497947253256, 0.0006300497947253256), Celerite(0.9038882037020084, 0.9038882037020084, 0.0017869601758986303, 0.0017869601758986303), Celerite(1.6067166095810141, 1.6067166095810141, 0.0050682131745472155, 0.0050682131745472155), Celerite(4.964666291074675, 4.964666291074675, 0.014374570362059993, 0.014374570362059993), Celerite(22.755812872815056, 22.755812872815056, 0.04076945187142277, 0.04076945187142277), Celerite(5.378025805869529, 5.378025805869529, 0.11563115724719714, 0.11563115724719714), Celerite(0.5417886639897328, 0.5417886639897328, 0.32795546451037977, 0.32795546451037977), Celerite(0.08070461417658467, 0.08070461417658467, 0.9301540282286335, 0.9301540282286335), Celerite(0.010848075938726597, 0.010848075938726597, 2.638121970376775, 2.638121970376775), Celerite(0.0015090806302273192, 0.0015090806302273192, 7.482295748198312, 7.482295748198312), Celerite(0.0002076692891940062, 0.0002076692891940062, 21.221440969050757, 21.221440969050757), Celerite(2.8676529647079397e-5, 2.8676529647079397e-5, 60.18868699641365, 60.18868699641365), Celerite(3.955543296634202e-6, 3.955543296634202e-6, 170.70839099171252, 170.70839099171252), Celerite(5.458037196298439e-7, 5.458037196298439e-7, 484.16664674402534, 484.16664674402534), Celerite(7.53042020169437e-8, 7.53042020169437e-8, 1373.203393562152, 1373.203393562152), Celerite(1.0389899331678754e-8, 1.0389899331678754e-8, 3894.70768540476, 3894.70768540476), Celerite(1.4338973791019396e-9, 1.4338973791019396e-9, 11046.249977144691, 11046.249977144691), Celerite(1.9677308815711412e-10, 1.9677308815711412e-10, 31329.601195702533, 31329.601195702533), Celerite(3.012647957419647e-11, 3.012647957419647e-11, 88857.65876316732, 88857.65876316732)], [0.21179581122061306, 0.38542244831627853, 0.9038882037020084, 1.6067166095810141, 4.964666291074675, 22.755812872815056, 5.378025805869529, 0.5417886639897328, 0.08070461417658467, 0.010848075938726597, 0.0015090806302273192, 0.0002076692891940062, 2.8676529647079397e-5, 3.955543296634202e-6, 5.458037196298439e-7, 7.53042020169437e-8, 1.0389899331678754e-8, 1.4338973791019396e-9, 1.9677308815711412e-10, 3.012647957419647e-11], [0.21179581122061306, 0.38542244831627853, 0.9038882037020084, 1.6067166095810141, 4.964666291074675, 22.755812872815056, 5.378025805869529, 0.5417886639897328, 0.08070461417658467, 0.010848075938726597, 0.0015090806302273192, 0.0002076692891940062, 2.8676529647079397e-5, 3.955543296634202e-6, 5.458037196298439e-7, 7.53042020169437e-8, 1.0389899331678754e-8, 1.4338973791019396e-9, 1.9677308815711412e-10, 3.012647957419647e-11], [0.0002221441469079183, 0.0006300497947253256, 0.0017869601758986303, 0.0050682131745472155, 0.014374570362059993, 0.04076945187142277, 0.11563115724719714, 0.32795546451037977, 0.9301540282286335, 2.638121970376775, 7.482295748198312, 21.221440969050757, 60.18868699641365, 170.70839099171252, 484.16664674402534, 1373.203393562152, 3894.70768540476, 11046.249977144691, 31329.601195702533, 88857.65876316732], [0.0002221441469079183, 0.0006300497947253256, 0.0017869601758986303, 0.0050682131745472155, 0.014374570362059993, 0.04076945187142277, 0.11563115724719714, 0.32795546451037977, 0.9301540282286335, 2.638121970376775, 7.482295748198312, 21.221440969050757, 60.18868699641365, 170.70839099171252, 484.16664674402534, 1373.203393562152, 3894.70768540476, 11046.249977144691, 31329.601195702533, 88857.65876316732]), :celerite)
x: [4.504504504504505, 27.47747747747748, 29.27927927927928, 40.990990990990994, 55.4054054054054, 77.92792792792793, 115.76576576576576, 135.13513513513513, 140.54054054054055, 145.04504504504504 … 380.6306306306306, 386.93693693693695, 391.44144144144144, 400.9009009009009, 420.72072072072075, 422.07207207207205, 425.22522522522524, 435.13513513513516, 437.8378378378378, 447.7477477477477]
Σy: [0.0625 0.0 … 0.0 0.0; 0.0 0.0625 … 0.0 0.0; … ; 0.0 0.0 … 0.0625 0.0; 0.0 0.0 … 0.0 0.0625]
)
, [6.367156475271415, -4.90772098031265, -4.84926961125822, -6.698980668077906, -2.0282528862929174, -0.7388443359234012, 3.784907384188556, -2.4163801619353142, 0.9778276520269376, -0.2913590028965023 … 0.43407210208699065, -0.2209925866175233, 1.9995680229890689, 0.07640695112513551, -7.265195653174087, -7.975772472276943, -8.714588428544848, -4.2393774141685165, -6.091141501299605, 2.831043350348003])The mean and standard deviation of this distribution, can be computed using the mean and std functions. The posterior covariance matrix can be computed using the cov function.
m = mean(fp,t);
s = std(fp,t);1000-element Vector{Float64}:
2.0626517007872334
1.9110763413854732
1.75276351079283
1.5872472345598196
1.4138595245390857
1.2316590526764528
1.0395214066980454
0.8365777882661751
0.6229416261221012
0.4037699022251107
⋮
0.7273957109958538
0.5629224597506416
0.3811475723442188
0.24930315917571488
0.39735949752571564
0.6061667026522791
0.8108042256354694
1.0062576530612704
1.1921904152760825We can plot the realisations, the observations and the posterior distribution.
plot(t, realisations[1], label="Realisation", xlabel="Time [days]", framestyle = :box,ylabel="Value")
plot!(t_obs, y_obs,yerr=yerr, label="Observations", seriestype=:scatter)
plot!(t,m, ribbon=s,label="Posterior distribution", lw=2)
plot!(t,m,ribbon=2*s, label=nothing)
The computation of the mean of the distribution is very efficient. The time complexity is O(N) where N is the number of data points. However, the computation of the covariance is very inefficient as the posterior covariance is not semi-separable.
Sampling from the conditioned distribution
We can draw realisations from the conditioned distribution using rand.
samples_cond = rand(rng,fp,t,5);1000×5 Matrix{Float64}:
8.42748 7.34992 10.6412 10.2005 5.24453
8.66185 7.05551 10.2608 10.0086 5.33955
8.70568 6.6528 10.0825 9.77261 5.29936
8.83709 6.17403 10.1901 9.65821 5.53369
8.31257 6.10017 10.1221 9.08733 5.38345
7.76123 5.96929 9.4228 8.81385 5.02117
7.32911 6.05282 8.90024 8.0894 4.78844
6.95632 6.0115 8.4788 7.52359 4.99107
6.45547 6.41548 8.13547 6.93222 5.32481
6.0703 6.59369 7.18235 6.61222 5.51607
⋮
1.73311 1.81749 1.56871 1.54545 2.22611
2.37855 1.95991 2.02202 1.92103 2.38687
2.81616 1.9061 2.36727 2.57885 2.67545
2.97269 2.5692 2.85837 2.98925 2.51206
3.41462 2.84287 3.10181 2.8865 1.87264
3.8412 2.86026 3.26103 3.13078 1.64703
4.28134 3.39541 3.48248 3.21353 1.48771
4.86719 3.53674 3.73938 3.42635 1.72988
5.31795 3.79149 3.1899 3.84353 1.6074We can plot the realisations, the observations and the posterior distribution.
plot(t_obs, y_obs,yerr=yerr, label="Observations", seriestype=:scatter, xlabel="Time [days]", framestyle = :box,ylabel="Value",color=:black,lw=2)
plot!(t, samples_cond, label=nothing)