Lotka-Volterra Bayesian Parameter Estimation Benchmarks
Parameter Estimation of Lotka-Volterra Equation using DiffEqBayes.jl
using DiffEqBayes, StanSample, DynamicHMC, Turingusing Distributions, BenchmarkTools, StaticArrays
using OrdinaryDiffEq, RecursiveArrayTools, ParameterizedFunctions
using Plots, LinearAlgebragr(fmt=:png)Plots.GRBackend()Initializing the problem
f = @ode_def LotkaVolterraTest begin
dx = a*x - b*x*y
dy = -c*y + d*x*y
end a b c d(::Main.var"##WeaveSandBox#225".LotkaVolterraTest{Main.var"##WeaveSandBox#2
25".var"###ParameterizedDiffEqFunction#227", Main.var"##WeaveSandBox#225".v
ar"###ParameterizedTGradFunction#228", Main.var"##WeaveSandBox#225".var"###
ParameterizedJacobianFunction#229", Nothing, Nothing, ModelingToolkit.ODESy
stem}) (generic function with 1 method)u0 = [1.0,1.0]
tspan = (0.0,10.0)
p = [1.5,1.0,3.0,1,0]5-element Vector{Float64}:
1.5
1.0
3.0
1.0
0.0prob = ODEProblem(f, u0, tspan, p)
sol = solve(prob,Tsit5())retcode: Success
Interpolation: specialized 4th order "free" interpolation
t: 34-element Vector{Float64}:
0.0
0.0776084743154256
0.23264513699277584
0.4291185174543143
0.6790821987497083
0.9444046158046306
1.2674601546021105
1.6192913303893046
1.9869754428624007
2.2640902393538296
⋮
7.584863345264154
7.978068981329682
8.48316543760351
8.719248247740158
8.949206788834692
9.200185054623292
9.438029017301554
9.711808134779586
10.0
u: 34-element Vector{Vector{Float64}}:
[1.0, 1.0]
[1.0454942346944578, 0.8576684823217128]
[1.1758715885138271, 0.6394595703175443]
[1.419680960717083, 0.4569962601282089]
[1.8767193950080012, 0.3247334292791134]
[2.588250064553348, 0.26336255535952197]
[3.860708909220769, 0.2794458098285261]
[5.750812667710401, 0.522007253793458]
[6.8149789991301635, 1.9177826328390826]
[4.392999292571394, 4.1946707928506015]
⋮
[2.6142539677883248, 0.26416945387526314]
[4.24107612719179, 0.3051236762922018]
[6.791123785297775, 1.1345287797146668]
[6.26537067576476, 2.741693507540315]
[3.780765111887945, 4.431165685863443]
[1.816420140681737, 4.064056625315896]
[1.1465021407690763, 2.791170661621642]
[0.9557986135403417, 1.623562295185047]
[1.0337581256020802, 0.9063703842885995]su0 = SA[1.0,1.0]
sp = SA[1.5,1.0,3.0,1,0]
sprob = ODEProblem{false,SciMLBase.FullSpecialize}(f, su0, tspan, sp)
sol = solve(sprob,Tsit5())retcode: Success
Interpolation: specialized 4th order "free" interpolation
t: 34-element Vector{Float64}:
0.0
0.0776084743154256
0.23264513699277584
0.4291185174543143
0.6790821987497083
0.9444046158046306
1.2674601546021105
1.6192913303893046
1.9869754428624007
2.2640902393538296
⋮
7.584863345264154
7.978068981329682
8.48316543760351
8.719248247740158
8.949206788834692
9.200185054623292
9.438029017301554
9.711808134779586
10.0
u: 34-element Vector{StaticArraysCore.SVector{2, Float64}}:
[1.0, 1.0]
[1.0454942346944578, 0.8576684823217128]
[1.1758715885138271, 0.6394595703175443]
[1.419680960717083, 0.4569962601282089]
[1.8767193950080012, 0.3247334292791134]
[2.588250064553348, 0.26336255535952197]
[3.860708909220769, 0.2794458098285261]
[5.750812667710401, 0.522007253793458]
[6.8149789991301635, 1.9177826328390826]
[4.392999292571394, 4.1946707928506015]
⋮
[2.6142539677883248, 0.26416945387526314]
[4.241076127191789, 0.30512367629220183]
[6.791123785297779, 1.1345287797146653]
[6.265370675764766, 2.7416935075403135]
[3.7807651118879293, 4.431165685863457]
[1.8164201406817235, 4.064056625315901]
[1.146502140769069, 2.791170661621637]
[0.9557986135403385, 1.6235622951850437]
[1.033758125602079, 0.9063703842885992]We take the solution data obtained and add noise to it to obtain data for using in the Bayesian Inference of the parameters
t = collect(range(1,stop=10,length=10))
sig = 0.49
data = convert(Array, VectorOfArray([(sol(t[i]) + sig*randn(2)) for i in 1:length(t)]))2×10 Matrix{Float64}:
3.18084 6.72495 1.47715 1.89996 … 4.20071 3.44035 0.862223
0.937477 1.77722 1.14996 -0.101926 1.13171 4.23348 0.929488Plots of the actual data and generated data
scatter(t, data[1,:], lab="#prey (data)")
scatter!(t, data[2,:], lab="#predator (data)")
plot!(sol)
priors = [truncated(Normal(1.5,0.5),0.5,2.5),truncated(Normal(1.2,0.5),0,2),truncated(Normal(3.0,0.5),1,4),truncated(Normal(1.0,0.5),0,2)]4-element Vector{Distributions.Truncated{Distributions.Normal{Float64}, Dis
tributions.Continuous, Float64, Float64, Float64}}:
Truncated(Distributions.Normal{Float64}(μ=1.5, σ=0.5); lower=0.5, upper=2.
5)
Truncated(Distributions.Normal{Float64}(μ=1.2, σ=0.5); lower=0.0, upper=2.
0)
Truncated(Distributions.Normal{Float64}(μ=3.0, σ=0.5); lower=1.0, upper=4.
0)
Truncated(Distributions.Normal{Float64}(μ=1.0, σ=0.5); lower=0.0, upper=2.
0)Stan.jl backend
The solution converges for tolerance values lower than 1e-3, lower tolerance leads to better accuracy in result but is accompanied by longer warmup and sampling time, truncated normal priors are used for preventing Stan from stepping into negative values.
@btime bayesian_result_stan = stan_inference(prob,t,data,priors,num_samples=10_000,print_summary=false,delta = 0.65, vars = (DiffEqBayes.StanODEData(), InverseGamma(2, 3)))37.269094 seconds (2.14 M allocations: 146.043 MiB, 0.12% gc time, 4.36% c
ompilation time)
38.159040 seconds (674 allocations: 56.625 KiB)
30.567834 seconds (674 allocations: 56.625 KiB)
32.539313 seconds (674 allocations: 56.625 KiB)
51.473 s (260872 allocations: 31.96 MiB)
Chains MCMC chain (10000×6×1 Array{Float64, 3}):
Iterations = 1:1:10000
Number of chains = 1
Samples per chain = 10000
parameters = sigma1.1, sigma1.2, theta_1, theta_2, theta_3, theta_4
internals =
Summary Statistics
parameters mean std mcse ess_bulk ess_tail rha
t ⋯
Symbol Float64 Float64 Float64 Float64 Float64 Float6
4 ⋯
sigma1.1 0.5012 0.1526 0.0026 2906.8051 1702.2976 1.000
4 ⋯
sigma1.2 0.7358 0.1993 0.0037 3395.0329 2966.8413 1.000
6 ⋯
theta_1 1.5339 0.1091 0.0024 1969.9347 2373.7754 0.999
9 ⋯
theta_2 1.0615 0.1343 0.0028 2700.0659 2616.4125 1.000
4 ⋯
theta_3 2.9182 0.2926 0.0066 2002.3218 2277.4273 1.000
0 ⋯
theta_4 0.9775 0.1050 0.0023 2019.2970 2571.7774 0.999
9 ⋯
1 column om
itted
Quantiles
parameters 2.5% 25.0% 50.0% 75.0% 97.5%
Symbol Float64 Float64 Float64 Float64 Float64
sigma1.1 0.2938 0.3917 0.4723 0.5777 0.8721
sigma1.2 0.4626 0.5989 0.6990 0.8298 1.2335
theta_1 1.3419 1.4597 1.5253 1.6026 1.7630
theta_2 0.8467 0.9701 1.0466 1.1342 1.3662
theta_3 2.3936 2.7152 2.9064 3.0986 3.5281
theta_4 0.7866 0.9051 0.9737 1.0421 1.1968Direct Turing.jl
@model function fitlv(data, prob)
# Prior distributions.
σ ~ InverseGamma(2, 3)
α ~ truncated(Normal(1.5, 0.5), 0.5, 2.5)
β ~ truncated(Normal(1.2, 0.5), 0, 2)
γ ~ truncated(Normal(3.0, 0.5), 1, 4)
δ ~ truncated(Normal(1.0, 0.5), 0, 2)
# Simulate Lotka-Volterra model.
p = SA[α, β, γ, δ]
_prob = remake(prob, p = p)
predicted = solve(_prob, Tsit5(); saveat=t)
# Observations.
for i in 1:length(predicted)
data[:, i] ~ MvNormal(predicted[i], σ^2 * I)
end
return nothing
end
model = fitlv(data, sprob)
@time chain = sample(model, Turing.NUTS(0.65), 10000; progress=false)47.397708 seconds (144.00 M allocations: 23.410 GiB, 7.01% gc time, 45.57%
compilation time)
Chains MCMC chain (10000×17×1 Array{Float64, 3}):
Iterations = 1001:1:11000
Number of chains = 1
Samples per chain = 10000
Wall duration = 39.44 seconds
Compute duration = 39.44 seconds
parameters = σ, α, β, γ, δ
internals = lp, n_steps, is_accept, acceptance_rate, log_density, h
amiltonian_energy, hamiltonian_energy_error, max_hamiltonian_energy_error,
tree_depth, numerical_error, step_size, nom_step_size
Summary Statistics
parameters mean std mcse ess_bulk ess_tail rha
t ⋯
Symbol Float64 Float64 Float64 Float64 Float64 Float6
4 ⋯
σ 0.5493 0.1061 0.0020 2677.0180 2511.9880 1.000
6 ⋯
α 1.5230 0.0997 0.0022 2030.6911 2654.5861 1.000
5 ⋯
β 1.0405 0.1054 0.0020 2966.9190 3112.9225 0.999
9 ⋯
γ 2.9346 0.2661 0.0058 2130.7072 2787.3330 1.000
1 ⋯
δ 0.9856 0.0990 0.0022 2045.7468 2845.2720 1.000
2 ⋯
1 column om
itted
Quantiles
parameters 2.5% 25.0% 50.0% 75.0% 97.5%
Symbol Float64 Float64 Float64 Float64 Float64
σ 0.3836 0.4747 0.5348 0.6065 0.8004
α 1.3493 1.4539 1.5147 1.5826 1.7397
β 0.8650 0.9677 1.0312 1.1016 1.2737
γ 2.4279 2.7571 2.9310 3.1072 3.4824
δ 0.7973 0.9190 0.9823 1.0502 1.1892Turing.jl backend
@btime bayesian_result_turing = turing_inference(prob, Tsit5(), t, data, priors, num_samples=10_000)19.781 s (112829776 allocations: 18.20 GiB)
Chains MCMC chain (10000×17×1 Array{Float64, 3}):
Iterations = 1001:1:11000
Number of chains = 1
Samples per chain = 10000
Wall duration = 19.71 seconds
Compute duration = 19.71 seconds
parameters = theta[1], theta[2], theta[3], theta[4], σ[1]
internals = lp, n_steps, is_accept, acceptance_rate, log_density, h
amiltonian_energy, hamiltonian_energy_error, max_hamiltonian_energy_error,
tree_depth, numerical_error, step_size, nom_step_size
Summary Statistics
parameters mean std mcse ess_bulk ess_tail rha
t ⋯
Symbol Float64 Float64 Float64 Float64 Float64 Float6
4 ⋯
theta[1] 1.5229 0.0994 0.0022 1965.0563 2831.2972 1.000
3 ⋯
theta[2] 1.0392 0.1039 0.0020 2721.8731 3510.8975 1.001
9 ⋯
theta[3] 2.9354 0.2667 0.0059 2071.7566 2633.9494 1.000
2 ⋯
theta[4] 0.9856 0.1004 0.0023 2025.2440 2813.9824 1.000
2 ⋯
σ[1] 0.5540 0.1068 0.0019 3063.7138 3763.6247 1.001
5 ⋯
1 column om
itted
Quantiles
parameters 2.5% 25.0% 50.0% 75.0% 97.5%
Symbol Float64 Float64 Float64 Float64 Float64
theta[1] 1.3465 1.4543 1.5158 1.5844 1.7337
theta[2] 0.8653 0.9695 1.0283 1.0984 1.2695
theta[3] 2.4385 2.7566 2.9279 3.1048 3.4962
theta[4] 0.8027 0.9182 0.9820 1.0490 1.1961
σ[1] 0.3908 0.4785 0.5379 0.6118 0.8011DynamicHMC.jl backend
@btime bayesian_result_dynamichmc = dynamichmc_inference(prob,Tsit5(),t,data,priors,num_samples=10_000)30.969 s (261383358 allocations: 19.33 GiB)
(posterior = @NamedTuple{parameters::Vector{Float64}, σ::Vector{Float64}}[(
parameters = [1.3222578549539439, 0.9251465117615063, 3.530848595308553, 1.
1911775078852416], σ = [0.42463764407357013, 0.6497097498548102]), (paramet
ers = [1.5966881409820233, 1.0761608434690144, 2.7659684034914207, 0.904737
8893685866], σ = [0.28741256640072377, 0.5415211251806117]), (parameters =
[1.5267631123553367, 0.998973316508563, 2.852759646683009, 0.97494581601418
5], σ = [0.2738405368321744, 0.3894380261298722]), (parameters = [1.6086267
367792269, 1.0683381604321225, 2.678577599677015, 0.9117976103649813], σ =
[0.25623261151085447, 0.49162523869800956]), (parameters = [1.6218643195356
621, 1.0513888812431291, 2.6743531396570304, 0.8834236913394804], σ = [0.28
38363591038976, 0.6641989289723392]), (parameters = [1.5852623782525286, 0.
9964331064541775, 2.7729762553938007, 0.904012682024255], σ = [0.1870111699
57959, 0.6624770223497735]), (parameters = [1.3522273781151966, 0.864562885
6216209, 3.5159024904237537, 1.1498098176160536], σ = [0.4723687132027934,
1.2751931042424078]), (parameters = [1.3565148520061214, 0.8276867738964817
, 3.3569466159970207, 1.178897632344254], σ = [0.3797778520563567, 1.284349
7099794297]), (parameters = [1.4576821942889808, 1.016374779965666, 3.04032
6021596035, 1.0645748245540418], σ = [0.5377907320508352, 0.367595573978023
17]), (parameters = [1.4493162601414449, 0.9201893983066607, 3.080861224754
194, 1.0399963448922047], σ = [0.46906772710926087, 0.38878752615229123])
… (parameters = [1.554818230931119, 1.0818159083635377, 2.722419764298772,
0.9561997456330118], σ = [0.7175899333914487, 0.6408747368287047]), (param
eters = [1.5595825042032552, 1.1847464304262263, 2.963845191541017, 0.92708
85625585458], σ = [0.6233976544301695, 0.533223123599574]), (parameters = [
1.4459879335021069, 0.8688640220318478, 3.062874071472865, 1.05913128189687
01], σ = [0.5545025070022185, 0.5224107406781667]), (parameters = [1.397101
6867280315, 0.9977073039188887, 3.2991756315485583, 1.118608348517123], σ =
[0.32604306786049514, 0.7257923190294405]), (parameters = [1.4232811932108
87, 0.9720791015473629, 3.1762118740371075, 1.1056047010958414], σ = [0.350
68375513341493, 0.6999171472688328]), (parameters = [1.4545906836751918, 0.
8382384558903104, 3.111677634298614, 1.047260443294073], σ = [0.31581182179
82505, 0.8858784898387406]), (parameters = [1.4551968752612483, 0.902735070
5272859, 3.1262658401889922, 1.034105591163242], σ = [0.39554220927201494,
0.6465101702933306]), (parameters = [1.690719259646092, 1.179307672999047,
2.5247317163721252, 0.8368718139618843], σ = [0.2342228010022602, 0.8176197
634998806]), (parameters = [1.6197435091094246, 1.171632016602915, 2.709060
0246296517, 0.8785780013063488], σ = [0.29152522149515253, 0.79119740611525
99]), (parameters = [1.726477996773983, 1.1407042872473159, 2.4880294880820
6, 0.8040843315924617], σ = [0.3054494157039274, 0.6835413700658756])], pos
terior_matrix = [0.2793407715429747 0.4679315721310914 … 0.482267808998906
0.5460834933554528; -0.07780316291873636 0.07339993334604998 … 0.1583976628
453404 0.13164586748010051; … ; -0.8565190759095201 -1.2468365818646918 … -
1.232628753795041 -1.1859710927907037; -0.4312295545054496 -0.6133732009648
201 … -0.23420777709753762 -0.380468097852621], tree_statistics = DynamicHM
C.TreeStatisticsNUTS[DynamicHMC.TreeStatisticsNUTS(-22.538228215771195, 3,
turning at positions -2:-9, 0.9870704272138824, 15, DynamicHMC.Directions(0
xa939da66)), DynamicHMC.TreeStatisticsNUTS(-21.674421348553736, 6, turning
at positions -13:50, 0.9954429776600223, 63, DynamicHMC.Directions(0xd66097
b2)), DynamicHMC.TreeStatisticsNUTS(-18.47899450445755, 5, turning at posit
ions -1:30, 0.9996197634903131, 31, DynamicHMC.Directions(0xa82be7fe)), Dyn
amicHMC.TreeStatisticsNUTS(-18.73036073428713, 5, turning at positions -54:
-61, 0.9963253285213837, 63, DynamicHMC.Directions(0x8bb63742)), DynamicHMC
.TreeStatisticsNUTS(-17.749653072915812, 5, turning at positions -8:-39, 0.
9861189322769188, 63, DynamicHMC.Directions(0x9aed1cd8)), DynamicHMC.TreeSt
atisticsNUTS(-21.788363313172628, 5, turning at positions -13:-44, 0.949617
0628976832, 63, DynamicHMC.Directions(0xfc124f13)), DynamicHMC.TreeStatisti
csNUTS(-27.251629633969245, 6, turning at positions -63:-126, 0.95846352965
61347, 127, DynamicHMC.Directions(0xecc71381)), DynamicHMC.TreeStatisticsNU
TS(-24.314660110108694, 6, turning at positions -36:27, 0.988700417944279,
63, DynamicHMC.Directions(0x28a22cdb)), DynamicHMC.TreeStatisticsNUTS(-26.4
35695988258118, 6, turning at positions -9:54, 0.7729370492648747, 63, Dyna
micHMC.Directions(0x76e02436)), DynamicHMC.TreeStatisticsNUTS(-23.752519834
84638, 5, turning at positions -5:26, 0.9892064412432646, 31, DynamicHMC.Di
rections(0xf464123a)) … DynamicHMC.TreeStatisticsNUTS(-26.15166573453191,
5, turning at positions -10:-41, 0.6651840816551673, 63, DynamicHMC.Direct
ions(0xc5b8d896)), DynamicHMC.TreeStatisticsNUTS(-23.06691505556487, 5, tur
ning at positions 28:59, 0.9123343951538908, 63, DynamicHMC.Directions(0x7b
fcdffb)), DynamicHMC.TreeStatisticsNUTS(-25.25032233897643, 6, turning at p
ositions 37:40, 0.9815649553671265, 83, DynamicHMC.Directions(0x13ba2fd4)),
DynamicHMC.TreeStatisticsNUTS(-21.425626836184648, 5, turning at positions
28:59, 0.9711058940690602, 63, DynamicHMC.Directions(0x3b38fffb)), Dynamic
HMC.TreeStatisticsNUTS(-22.790295956836765, 5, turning at positions -13:18,
0.7463131360877131, 31, DynamicHMC.Directions(0xf8714032)), DynamicHMC.Tre
eStatisticsNUTS(-21.29793971532498, 5, turning at positions 8:39, 0.9864656
496132187, 63, DynamicHMC.Directions(0x9aa30567)), DynamicHMC.TreeStatistic
sNUTS(-20.013803235785254, 5, turning at positions 28:59, 0.991976611545151
9, 63, DynamicHMC.Directions(0x7ab9f17b)), DynamicHMC.TreeStatisticsNUTS(-2
0.83778069899103, 6, turning at positions 55:86, 0.9612983417086235, 95, Dy
namicHMC.Directions(0x23fd45f6)), DynamicHMC.TreeStatisticsNUTS(-19.1953155
74562887, 6, turning at positions -32:31, 0.939434424544655, 63, DynamicHMC
.Directions(0x29091b5f)), DynamicHMC.TreeStatisticsNUTS(-20.551238667390088
, 6, turning at positions -7:56, 0.7336248798249541, 63, DynamicHMC.Directi
ons(0x5a645e38))], κ = Gaussian kinetic energy (Diagonal), √diag(M⁻¹): [0.0
6756663495222748, 0.12064729178457598, 0.09190133503193172, 0.0985857049994
9916, 0.2814517830972898, 0.2642359285268731], ϵ = 0.05661549801185918)Conclusion
Lotka-Volterra Equation is a "predator-prey" model, it models population of two species in which one is the predator (wolf) and the other is the prey (rabbit). It depicts a cyclic behaviour, which is also seen in its Uncertainty Quantification Plots. This behaviour makes it easy to estimate even at very high tolerance values (1e-3).
Appendix
These benchmarks are a part of the SciMLBenchmarks.jl repository, found at: https://github.com/SciML/SciMLBenchmarks.jl. For more information on high-performance scientific machine learning, check out the SciML Open Source Software Organization https://sciml.ai.
To locally run this benchmark, do the following commands:
using SciMLBenchmarks
SciMLBenchmarks.weave_file("benchmarks/BayesianInference","DiffEqBayesLotkaVolterra.jmd")Computer Information:
Julia Version 1.10.9
Commit 5595d20a287 (2025-03-10 12:51 UTC)
Build Info:
Official https://julialang.org/ release
Platform Info:
OS: Linux (x86_64-linux-gnu)
CPU: 128 × AMD EPYC 7502 32-Core Processor
WORD_SIZE: 64
LIBM: libopenlibm
LLVM: libLLVM-15.0.7 (ORCJIT, znver2)
Threads: 1 default, 0 interactive, 1 GC (on 128 virtual cores)
Environment:
JULIA_CPU_THREADS = 128
JULIA_DEPOT_PATH = /cache/julia-buildkite-plugin/depots/5b300254-1738-4989-ae0a-f4d2d937f953
Package Information:
Status `/cache/build/exclusive-amdci1-0/julialang/scimlbenchmarks-dot-jl/benchmarks/BayesianInference/Project.toml`
⌃ [6e4b80f9] BenchmarkTools v1.3.2
⌃ [ebbdde9d] DiffEqBayes v3.6.0
⌅ [459566f4] DiffEqCallbacks v2.29.1
⌃ [31c24e10] Distributions v0.25.100
⌃ [bbc10e6e] DynamicHMC v3.4.6
⌃ [1dea7af3] OrdinaryDiffEq v6.55.0
⌃ [65888b18] ParameterizedFunctions v5.15.0
⌃ [91a5bcdd] Plots v1.39.0
⌅ [731186ca] RecursiveArrayTools v2.38.7
[31c91b34] SciMLBenchmarks v0.1.3
⌃ [c1514b29] StanSample v7.4.2
⌃ [90137ffa] StaticArrays v1.6.2
⌅ [fce5fe82] Turing v0.28.3
[37e2e46d] LinearAlgebra
Info Packages marked with ⌃ and ⌅ have new versions available. Those with ⌃ may be upgradable, but those with ⌅ are restricted by compatibility constraints from upgrading. To see why use `status --outdated`
Warning The project dependencies or compat requirements have changed since the manifest was last resolved. It is recommended to `Pkg.resolve()` or consider `Pkg.update()` if necessary.And the full manifest:
Status `/cache/build/exclusive-amdci1-0/julialang/scimlbenchmarks-dot-jl/benchmarks/BayesianInference/Manifest.toml`
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⌃ [3161d3a3] Zstd_jll v1.5.5+0
⌅ [214eeab7] fzf_jll v0.29.0+0
⌃ [a4ae2306] libaom_jll v3.4.0+0
⌃ [0ac62f75] libass_jll v0.15.1+0
⌃ [f638f0a6] libfdk_aac_jll v2.0.2+0
⌃ [b53b4c65] libpng_jll v1.6.38+0
⌃ [a9144af2] libsodium_jll v1.0.20+0
⌃ [f27f6e37] libvorbis_jll v1.3.7+1
⌅ [1270edf5] x264_jll v2021.5.5+0
⌅ [dfaa095f] x265_jll v3.5.0+0
⌃ [d8fb68d0] xkbcommon_jll v1.4.1+0
[0dad84c5] ArgTools v1.1.1
[56f22d72] Artifacts
[2a0f44e3] Base64
[ade2ca70] Dates
[8ba89e20] Distributed
[f43a241f] Downloads v1.6.0
[7b1f6079] FileWatching
[9fa8497b] Future
[b77e0a4c] InteractiveUtils
[4af54fe1] LazyArtifacts
[b27032c2] LibCURL v0.6.3
[76f85450] LibGit2
[8f399da3] Libdl
[37e2e46d] LinearAlgebra
[56ddb016] Logging
[d6f4376e] Markdown
[a63ad114] Mmap
[ca575930] NetworkOptions v1.2.0
[44cfe95a] Pkg v1.9.0
[de0858da] Printf
[9abbd945] Profile
[3fa0cd96] REPL
[9a3f8284] Random
[ea8e919c] SHA v0.7.0
[9e88b42a] Serialization
[1a1011a3] SharedArrays
[6462fe0b] Sockets
[2f01184e] SparseArrays
[10745b16] Statistics v1.9.0
[4607b0f0] SuiteSparse
[fa267f1f] TOML v1.0.3
[a4e569a6] Tar v1.10.0
[8dfed614] Test
[cf7118a7] UUIDs
[4ec0a83e] Unicode
[e66e0078] CompilerSupportLibraries_jll v1.0.2+0
[deac9b47] LibCURL_jll v7.84.0+0
[29816b5a] LibSSH2_jll v1.10.2+0
[c8ffd9c3] MbedTLS_jll v2.28.2+0
[14a3606d] MozillaCACerts_jll v2022.10.11
[4536629a] OpenBLAS_jll v0.3.21+4
[05823500] OpenLibm_jll v0.8.1+0
[efcefdf7] PCRE2_jll v10.42.0+0
[bea87d4a] SuiteSparse_jll v5.10.1+6
[83775a58] Zlib_jll v1.2.13+0
[8e850b90] libblastrampoline_jll v5.8.0+0
[8e850ede] nghttp2_jll v1.48.0+0
[3f19e933] p7zip_jll v17.4.0+0
Info Packages marked with ⌃ and ⌅ have new versions available. Those with ⌃ may be upgradable, but those with ⌅ are restricted by compatibility constraints from upgrading. To see why use `status --outdated -m`
Warning The project dependencies or compat requirements have changed since the manifest was last resolved. It is recommended to `Pkg.resolve()` or consider `Pkg.update()` if necessary.