using LongMemory, BenchmarkTools, RCall, Random
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: 6135 samples with 1 evaluation. Range (min … max): 548.300 μs … 8.784 ms ┊ GC (min … max): 0.00% … 84.61% Time (median): 609.200 μs ┊ GC (median): 0.00% Time (mean ± σ): 809.529 μs ± 445.738 μs ┊ GC (mean ± σ): 6.60% ± 9.47% ▆█▆▄▃▂▂▁▁▁ ▁▁▁▂▃▂▂ ▁▁▁ ▁ ▁ ██████████▇█████████████████▇▆▇▇▅▅▆▅▄▄▅▅▅▅▄▆▇▇▇█▇▇▇▆▇▆▇▆▆▇▄▆▅ █ 548 μs Histogram: log(frequency) by time 2.33 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)"BenchmarkTools.Trial: 51 samples with 1 evaluation. Range (min … max): 98.229 ms … 105.197 ms ┊ GC (min … max): 0.00% … 0.00% Time (median): 98.957 ms ┊ GC (median): 0.00% Time (mean ± σ): 99.429 ms ± 1.301 ms ┊ GC (mean ± σ): 0.00% ± 0.00% ▁▆▁ ▁ █ ▁ ▁ ▄███▇█▇█▁▇█▄▁▄▄▁▄▄▁▁▁▇▁█▄▄▄▁▄▁▄▁▄▄▁▁▁▁▁▁▁▁▄▄▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▄ ▁ 98.2 ms Histogram: frequency by time 103 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) 97.9438 98.2633 98.50093 98.3475
uq max neval
98.6261 99.6666 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: 7629 samples with 1 evaluation. Range (min … max): 534.700 μs … 4.278 ms ┊ GC (min … max): 0.00% … 82.68% Time (median): 567.200 μs ┊ GC (median): 0.00% Time (mean ± σ): 651.414 μs ± 248.458 μs ┊ GC (mean ± σ): 5.20% ± 9.91% ▄██▅▃▂▁▁▁▁▁ ▁▂▂ ▁ ███████████▇█▇▇▅▅▅▄▄▄▃▅▅▅▆▅▅▆▅▆▄▄▄▃▆███▇▇█████▇███▇▆▇▅▅▅▆▄▅▅▅ █ 535 μs Histogram: log(frequency) by time 1.4 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: 2565 samples with 1 evaluation. Range (min … max): 1.585 ms … 44.979 ms ┊ GC (min … max): 0.00% … 0.00% Time (median): 1.756 ms ┊ GC (median): 0.00% Time (mean ± σ): 1.945 ms ± 1.214 ms ┊ GC (mean ± σ): 0.00% ± 0.00% ▅█▁ ▅███▄▃▄▅▄▃▃▂▂▂▂▂▂▂▁▂▂▂▂▁▁▁▂▂▂▂▂▁▁▁▁▁▁▂▁▁▂▁▂▂▂▂▁▂▂▂▁▂▂▂▂▂▂▂ ▂ 1.58 ms Histogram: frequency by time 5.26 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: 1432 samples with 1 evaluation. Range (min … max): 3.082 ms … 42.873 ms ┊ GC (min … max): 0.00% … 0.00% Time (median): 3.271 ms ┊ GC (median): 0.00% Time (mean ± σ): 3.494 ms ± 1.231 ms ┊ GC (mean ± σ): 0.00% ± 0.00% ▅█▄ ▅███▆▅▄▃▃▃▃▃▃▃▂▂▂▂▂▂▂▂▂▂▂▁▂▁▂▁▁▁▁▁▁▂▂▁▁▁▂▁▁▁▁▂▂▂▂▂▂▂▂▂▂▂▁▂ ▃ 3.08 ms Histogram: frequency by time 6.67 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 = $d)"[1:t]
[fd_LongMemoryjl fd_fracdiff fd_LongMemoryTS]20×3 Matrix{Float64}:
1.56774 1.53168 1.56774
0.232034 0.20318 0.232034
-1.8105 -1.83647 -1.8105
-0.503704 -0.527942 -0.503704
1.33832 1.31529 1.33832
0.882536 0.860431 0.882536
1.14611 1.12474 1.14611
-3.29925 -3.32001 -3.29925
0.0745282 0.0542897 0.0745282
0.441823 0.422035 0.441823
0.410045 0.390652 0.410045
-0.683177 -0.702217 -0.683177
-0.607066 -0.625789 -0.607066
-0.454029 -0.472464 -0.454029
1.1894 1.17123 1.1894
0.187175 0.169246 0.187175
1.75967 1.74196 1.75967
-0.547967 -0.565464 -0.547967
-1.58271 -1.60001 -1.58271
0.616853 0.599732 0.616853Long memory estimation
Data generation.
x = fi_gen(1000, 0.4);The gph_est() function
@benchmark gph_est(x; m = 0.5)BenchmarkTools.Trial: 10000 samples with 1 evaluation. Range (min … max): 28.000 μs … 2.978 ms ┊ GC (min … max): 0.00% … 49.13% Time (median): 33.700 μs ┊ GC (median): 0.00% Time (mean ± σ): 40.695 μs ± 96.307 μs ┊ GC (mean ± σ): 4.35% ± 1.83% ▁▁▁▇█▃ ▂██████▆▅▄▄▄▄▃▂▂▂▂▂▂▂▂▂▁▂▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁ ▂ 28 μs Histogram: frequency by time 85.8 μs < Memory estimate: 66.48 KiB, allocs estimate: 43.
LongMemoryTS: gph()
@benchmark R"LongMemoryTS::gph($x, m = floor(1000^0.5))"BenchmarkTools.Trial: 10000 samples with 1 evaluation. Range (min … max): 135.100 μs … 7.234 ms ┊ GC (min … max): 0.00% … 0.00% Time (median): 149.600 μs ┊ GC (median): 0.00% Time (mean ± σ): 175.848 μs ± 153.573 μs ┊ GC (mean ± σ): 0.00% ± 0.00% ▆█▃ ▆███▇▄▃▃▃▂▃▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁ ▂ 135 μs Histogram: frequency by time 366 μs < Memory estimate: 2.66 KiB, allocs estimate: 104.
fracdiff: fdGPH()
@benchmark R"fracdiff::fdGPH($x)"BenchmarkTools.Trial: 707 samples with 1 evaluation. Range (min … max): 5.054 ms … 34.624 ms ┊ GC (min … max): 0.00% … 0.00% Time (median): 5.979 ms ┊ GC (median): 0.00% Time (mean ± σ): 7.071 ms ± 3.428 ms ┊ GC (mean ± σ): 0.00% ± 0.00% ▄█▆ ███▃▃▃▂▂▃▄▆▄▂▂▂▂▂▂▂▂▁▁▂▁▁▁▁▁▂▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▂ ▂ 5.05 ms Histogram: frequency by time 31.3 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}:
0.502649 0.502884 0.502884