using LongMemory, BenchmarkTools, Random, RCall
Random.seed!(1234)TaskLocalRNG()LongMemory.jl: Generating, Estimating, and Forecasting Long Memory Models in Julia
Benchmarks
Abstract
LongMemory.jl is a package for time series long memory modelling in Julia. The package provides functions for generating long memory, estimating the parameters of the models, and simulation. Generating methods include fractional differencing, stochastic error duration, and cross-sectional aggregation. Estimators include those inspired by the log-periodogram regression and parametric ones. This notebook provides benchmarks for the LongMemory.jl package including against current alternatives. The package is closely related to the R packages LongMemoryTS and fracdiff.
Citation
If you use this package in your research, please cite it as:
Vera-Valdés, J.E. (2024). “LongMemory.jl: Generating, Estimating, and Forecasting Long Memory Models in Julia”. arXiv 2401.14077. https://arxiv.org/abs/2401.14077
@article{VERAVALDES2024a,
title = {LongMemory.jl: Generating, Estimating, and Forecasting Long Memory Models in Julia},
year = {2024},
author = {Vera-Valdés, J.E.},
journal = {arXiv preprint arXiv:2401.14077},
url = {https://arxiv.org/abs/2401.14077}
}Loading packages
Loading the package and setting a seed
Long memory generation
The fi_gen() function, which uses the cumulative method.
@benchmark fi_gen(10000, 0.2)BenchmarkTools.Trial: 4970 samples with 1 evaluation. Range (min … max): 546.300 μs … 8.410 ms ┊ GC (min … max): 0.00% … 68.25% Time (median): 690.000 μs ┊ GC (median): 0.00% Time (mean ± σ): 990.161 μs ± 713.281 μs ┊ GC (mean ± σ): 6.98% ± 7.83% █▆▅▄▄▃▂▂▂▄▅▄▄▄▃▃▁▁▁ ▁ ▂ █████████████████████▇█▆▆▆▅▆▃▁▁▁▁▁▁▃▁▃▃▁▁▁▁▁▁▃▇██████▇▇▇▇▇▅▄▅ █ 546 μs Histogram: log(frequency) by time 3.73 ms < Memory estimate: 2.58 MiB, allocs estimate: 44.
R: benchmarking the fracdiff.sim() function.
Loading the R packages. The R package fracdiff is used for fractional differencing generation using fracdiff::fracdiff.sim(). Benchmarking the fractional differencing generation by the R package fracdiff using RCall.
R"library(fracdiff)"
@benchmark R"fracdiff::fracdiff.sim(10000, d = 0.2)"┌ Warning: RCall.jl: Warning: package 'fracdiff' was built under R version 4.3.3
└ @ RCall C:\Users\eduar\.julia\packages\RCall\FEbLj\src\io.jl:172BenchmarkTools.Trial: 47 samples with 1 evaluation. Range (min … max): 100.973 ms … 129.056 ms ┊ GC (min … max): 0.00% … 0.00% Time (median): 106.095 ms ┊ GC (median): 0.00% Time (mean ± σ): 106.919 ms ± 5.563 ms ┊ GC (mean ± σ): 0.00% ± 0.00% ▂█ ▂▅ ▂ ▅ █████▅██▅▁▁█████▅▁▁▁█▁▅▁▁▅▅▅▁▁▁▁█▁▁▁▁▁▁▁▁▁▁▁▅▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▅ ▁ 101 ms Histogram: frequency by time 129 ms < Memory estimate: 1.02 KiB, allocs estimate: 37.
For comparison purposes, the following benchmark is done in R using the microbenchmark package.
R"library(microbenchmark)"
R"microbenchmark(fracdiff::fracdiff.sim(10000, d = 0.2), times = 33)"RObject{VecSxp}
Unit: milliseconds
expr min lq mean median
fracdiff::fracdiff.sim(10000, d = 0.2) 101.0279 102.8038 105.7101 104.5915
uq max neval
109.0149 115.5849 33Note that the times are similar. Hence, for ease of use, the next benchmarks are done using BenchmarkTools.
Fractional differencing
The fi_gen() function
@benchmark fracdiff(randn(10000,1), 0.2)BenchmarkTools.Trial: 6384 samples with 1 evaluation. Range (min … max): 543.000 μs … 6.538 ms ┊ GC (min … max): 0.00% … 81.09% Time (median): 624.700 μs ┊ GC (median): 0.00% Time (mean ± σ): 776.078 μs ± 402.315 μs ┊ GC (mean ± σ): 6.18% ± 9.54% ▅██▇▆▅▄▃▃▂▂▁▁▁ ▁ ▁ ▁ ▁ ▂ █████████████████▇██▇▇███▇██▇██▇▇▆▇▇█▇█████▇██▇▇▆▆▅▆▆▄▅▅▃▄▆▆▄ █ 543 μs Histogram: log(frequency) by time 2.17 ms < Memory estimate: 2.58 MiB, allocs estimate: 44.
R: fracdiff::diffseries()
The R package fracdiff is used for fractional differencing using fracdiff::diffseries().
@benchmark R"fracdiff::diffseries(rnorm(10000), d = 0.2)"BenchmarkTools.Trial: 2163 samples with 1 evaluation. Range (min … max): 1.751 ms … 42.836 ms ┊ GC (min … max): 0.00% … 0.00% Time (median): 2.037 ms ┊ GC (median): 0.00% Time (mean ± σ): 2.301 ms ± 1.334 ms ┊ GC (mean ± σ): 0.00% ± 0.00% ▅▇█▇▆▆▅▅▅▄▃▃▂▁▁ ▁ ▁ █████████████████▇█▇▆█▇▇▇▅▅▅▄▁▅▁▅▄▄▅▄▆▆▅▄▁▆▅▇▅▄▅▄▅▆▅▁▆▁▁▅▄ █ 1.75 ms Histogram: log(frequency) by time 5.8 ms < Memory estimate: 1.02 KiB, allocs estimate: 37.
R: LongMemoryTS:fdiff()
Loading the R package LongMemoryTS. The LongMemoryTS::fdiff() function is used for fractional differencing.
R"library(LongMemoryTS)"
@benchmark R"LongMemoryTS::fdiff(rnorm(10000), 0.2)"BenchmarkTools.Trial: 1256 samples with 1 evaluation. Range (min … max): 3.113 ms … 34.072 ms ┊ GC (min … max): 0.00% … 0.00% Time (median): 3.713 ms ┊ GC (median): 0.00% Time (mean ± σ): 3.965 ms ± 1.155 ms ┊ GC (mean ± σ): 0.00% ± 0.00% ▄▆▆██▇▃▄▂▁▂▂ ▃▆█████████████▆▅▅▅▅▄▄▃▃▄▃▃▃▃▃▃▃▂▂▁▂▃▃▂▃▂▃▁▃▃▂▂▁▂▂▂▂▂▂▂▃▂▃ ▄ 3.11 ms Histogram: frequency by time 6.92 ms < Memory estimate: 1.02 KiB, allocs estimate: 37.
Comparing accuracy
t = 20
x = randn(t,1)
d = 0.2
fd_LongMemoryjl = fracdiff(x, d)
fd_fracdiff = R"fracdiff::diffseries($x, d = $d)"[1:t]
fd_LongMemoryTS = R"LongMemoryTS::fdiff($x, d = $d)"[1:t]
[fd_LongMemoryjl fd_fracdiff fd_LongMemoryTS]20×3 Matrix{Float64}:
1.2558 0.845984 1.2558
0.628208 0.300358 0.628208
-0.0297641 -0.324829 -0.0297641
0.49543 0.220036 0.49543
1.46452 1.20289 1.46452
-0.891909 -1.14307 -0.891909
0.0139176 -0.22887 0.0139176
-1.11488 -1.35073 -1.11488
-0.342751 -0.572705 -0.342751
1.19236 0.967514 1.19236
-0.201157 -0.421504 -0.201157
1.85728 1.64094 1.85728
-0.79888 -1.01161 -0.79888
-0.162949 -0.372412 -0.162949
-0.369188 -0.575659 -0.369188
-0.00385477 -0.207572 -0.00385477
1.84698 1.64581 1.84698
-0.28165 -0.480454 -0.28165
-0.772948 -0.969543 -0.772948
1.08411 0.889586 1.08411Long memory estimation
Data generation.
x = fi_gen(1000, 0.4);The gph_est() function
@benchmark gph_est(x; m = 0.5)BenchmarkTools.Trial: 10000 samples with 1 evaluation. Range (min … max): 28.200 μs … 5.917 ms ┊ GC (min … max): 0.00% … 35.65% Time (median): 35.200 μs ┊ GC (median): 0.00% Time (mean ± σ): 44.094 μs ± 121.568 μs ┊ GC (mean ± σ): 4.81% ± 1.84% ▂▃▃█▆▁ ▂▃▅██████▆▄▄▄▄▃▃▂▃▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁ ▂ 28.2 μs Histogram: frequency by time 84.1 μs < Memory estimate: 66.48 KiB, allocs estimate: 43.
LongMemoryTS: gph()
@benchmark R"LongMemoryTS::gph($x, m = floor(1000^0.5))"BenchmarkTools.Trial: 10000 samples with 1 evaluation. Range (min … max): 138.100 μs … 46.531 ms ┊ GC (min … max): 0.00% … 0.00% Time (median): 166.900 μs ┊ GC (median): 0.00% Time (mean ± σ): 349.421 μs ± 1.242 ms ┊ GC (mean ± σ): 0.00% ± 0.00% █▅▄▃▃▃▃▃▂▁▁ ▁ ██████████████▇█▇▇▆▇▇▇▆▆▆▅▅▄▆▅▅▅▄▄▆▅▄▆▄▅▁▃▄▅▃▄▄▄▅▄▃▃▄▄▃▅▄▄▃▄ █ 138 μs Histogram: log(frequency) by time 3.04 ms < Memory estimate: 2.66 KiB, allocs estimate: 104.
fracdiff: fdGPH()
@benchmark R"fracdiff::fdGPH($x)"BenchmarkTools.Trial: 705 samples with 1 evaluation. Range (min … max): 5.053 ms … 38.156 ms ┊ GC (min … max): 0.00% … 0.00% Time (median): 6.061 ms ┊ GC (median): 0.00% Time (mean ± σ): 7.083 ms ± 3.266 ms ┊ GC (mean ± σ): 0.00% ± 0.00% ▂█▃ ███▇▄▃▃▂▃▄▅▃▃▃▂▂▁▁▁▁▁▁▂▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▂ ▂ 5.05 ms Histogram: frequency by time 31.1 ms < Memory estimate: 2.66 KiB, allocs estimate: 104.
Comparing accuracy
gph_LongMemoryjl = gph_est(x; m = 0.5)
gph_LongMemoryTS = R"LongMemoryTS::gph(X = $x, m = floor(1000^0.5))"[1]
gph_fracdiff = R"fracdiff::fdGPH($x)$d"[1]
[gph_LongMemoryjl gph_LongMemoryTS gph_fracdiff]1×3 Matrix{Float64}:
0.390362 0.390564 0.390564