quantile: use nth_element, support Eigen vectors for multi-p inputs

[andrjohns]

Summary

Since we have a c++17 requirement, we can use std::nth_element for the quantile function with a single p instead of sorting the entire input. This PR also replaces the std::min_element/std::max_element with Eigen's minCoeff() & maxCoeff() to take advantage of any vectorisation.

For the multi-p case, this also relaxes the requirement that an std::vector<double> is used for the container of p and that an std::vector<double> is returned. Now the return type is the same as the type of the input vector

Tests

Additional tests added for using VectorXd and RowVectorXd for the multi-p inputs

Side Effects

N/A

Release notes

Used std::nth_element to reduce algorithmic complexity of quantile implementation, support using Eigen containers for multiple p

Checklist

• Copyright holder: Andrew Johnson

  The copyright holder is typically you or your assignee, such as a university or company. By submitting this pull request, the copyright holder is agreeing to the license the submitted work under the following licenses:
  - Code: BSD 3-clause (https://opensource.org/licenses/BSD-3-Clause)
  - Documentation: CC-BY 4.0 (https://creativecommons.org/licenses/by/4.0/)

• the basic tests are passing

  • unit tests pass (to run, use: ./runTests.py test/unit)
  • header checks pass, (make test-headers)
  • dependencies checks pass, (make test-math-dependencies)
  • docs build, (make doxygen)
  • code passes the built in C++ standards checks (make cpplint)

• the code is written in idiomatic C++ and changes are documented in the doxygen

• the new changes are tested

[WardBrian]

Upstream ci failures look legitimate, is that going to be addressed here or is there an upstream PR that has to go first?

[andrjohns]

Upstream ci failures look legitimate, is that going to be addressed here or is there an upstream PR that has to go first?

Ah tricky. It looks like the implementation in mcmc/chainset.hpp copies the Eigen::VectorXd of probabilities into a std::vector<double> and then expects quantile(Eigen::VectorXd, std::vector<double>) to return the quantiles as std::vector<double> which are then copied into an Eigen::VectorXd on the return.

This PR changed that vectorised case so that the same type as the input vec is returned (i.e., Eigen::VectorXd) and the second argument can be any vector type (removing two copies).

Two options I can think of here:

• Open a Stan PR with the 'new' code, merge after this is merged (Stan will be broken until this is merged)
• Stan PR implementing the vectorised quantile as a loop over scalar quantiles, replace with the 'new' vectorised code after this one merged

No real preference either way (both small code changes), but the second one makes sure that upstream is always functional. Any pref on your end?

[SteveBronder]

Thanks!

[SteveBronder]

Oh it looks like upstream we caught an issue with the return type
https://jenkins.flatironinstitute.org/blue/organizations/jenkins/Stan%2FStan/detail/downstream_tests/910/pipeline

./src/stan/mcmc/chainset.hpp: In member function 'Eigen::VectorXd stan::mcmc::chainset::quantiles(int, const VectorXd&) const':
./src/stan/mcmc/chainset.hpp:352:54: error: no match for 'operator=' (operand types are 'std::vector<double>' and 'stan::plain_type_t<Eigen::Map<Eigen::Matrix<double, -1, 1>, 0, Eigen::Stride<0, 0> > >' {aka 'Eigen::Matrix<double, -1, 1>'})

       quantiles = stan::math::quantile(map, probs_vec);

[WardBrian]

I don’t mind doing a linked PR in Stan where there will be a temporary breakage

[andrjohns]

Realised that the simple solution was to just set the return type to probs vec type, no other changes required upstream.

I've also simplified the implementation a fair bit

[stan-buildbot]

---

Name	Old Result	New Result	Ratio	Performance change( 1 - new / old )
gp_regr/gp_regr.stan	0.12	0.11	1.13	11.13% faster
gp_regr/gen_gp_data.stan	0.03	0.02	1.06	5.88% faster
arK/arK.stan	1.66	1.57	1.05	5.19% faster
eight_schools/eight_schools.stan	0.06	0.06	1.05	5.16% faster
low_dim_gauss_mix_collapse/low_dim_gauss_mix_collapse.stan	9.16	8.4	1.09	8.33% faster
pkpd/one_comp_mm_elim_abs.stan	20.67	19.29	1.07	6.68% faster
pkpd/sim_one_comp_mm_elim_abs.stan	0.26	0.25	1.05	4.46% faster
sir/sir.stan	73.07	68.1	1.07	6.8% faster
gp_pois_regr/gp_pois_regr.stan	2.55	2.29	1.11	10.12% faster
low_dim_gauss_mix/low_dim_gauss_mix.stan	3.03	2.72	1.11	10.07% faster
irt_2pl/irt_2pl.stan	5.98	5.02	1.19	15.99% faster
arma/arma.stan	0.34	0.31	1.09	8.46% faster
garch/garch.stan	0.51	0.45	1.15	12.71% faster
low_dim_corr_gauss/low_dim_corr_gauss.stan	0.01	0.01	1.04	4.0% faster
performance.compilation	270.88	281.2	0.96	-3.81% slower
Mean result: 1.0824503029713648

---

Jenkins Console Log
Blue Ocean
Commit hash: 53b05f515233eacd215228b863f6706ec835030e

---

Machine information

No LSB modules are available. Distributor ID: Ubuntu Description: Ubuntu 20.04.3 LTS Release: 20.04 Codename: focal

CPU:
Architecture: x86_64
CPU op-mode(s): 32-bit, 64-bit
Byte Order: Little Endian
Address sizes: 46 bits physical, 48 bits virtual
CPU(s): 80
On-line CPU(s) list: 0-79
Thread(s) per core: 2
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Socket(s): 2
NUMA node(s): 2
Vendor ID: GenuineIntel
CPU family: 6
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Model name: Intel(R) Xeon(R) Gold 6148 CPU @ 2.40GHz
Stepping: 4
CPU MHz: 2400.000
CPU max MHz: 3700.0000
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Virtualization: VT-x
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L3 cache: 55 MiB
NUMA node0 CPU(s): 0,2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,38,40,42,44,46,48,50,52,54,56,58,60,62,64,66,68,70,72,74,76,78
NUMA node1 CPU(s): 1,3,5,7,9,11,13,15,17,19,21,23,25,27,29,31,33,35,37,39,41,43,45,47,49,51,53,55,57,59,61,63,65,67,69,71,73,75,77,79
Vulnerability Gather data sampling: Mitigation; Microcode
Vulnerability Itlb multihit: KVM: Mitigation: Split huge pages
Vulnerability L1tf: Mitigation; PTE Inversion; VMX conditional cache flushes, SMT vulnerable
Vulnerability Mds: Mitigation; Clear CPU buffers; SMT vulnerable
Vulnerability Meltdown: Mitigation; PTI
Vulnerability Mmio stale data: Mitigation; Clear CPU buffers; SMT vulnerable
Vulnerability Reg file data sampling: Not affected
Vulnerability Retbleed: Mitigation; IBRS
Vulnerability Spec rstack overflow: Not affected
Vulnerability Spec store bypass: Mitigation; Speculative Store Bypass disabled via prctl
Vulnerability Spectre v1: Mitigation; usercopy/swapgs barriers and __user pointer sanitization
Vulnerability Spectre v2: Mitigation; IBRS; IBPB conditional; STIBP conditional; RSB filling; PBRSB-eIBRS Not affected; BHI Not affected
Vulnerability Srbds: Not affected
Vulnerability Tsx async abort: Mitigation; Clear CPU buffers; SMT vulnerable
Vulnerability Vmscape: Mitigation; IBPB before exit to userspace
Flags: fpu vme de pse tsc msr pae mce cx8 apic sep mtrr pge mca cmov pat pse36 clflush dts acpi mmx fxsr sse sse2 ss ht tm pbe syscall nx pdpe1gb rdtscp lm constant_tsc art arch_perfmon pebs bts rep_good nopl xtopology nonstop_tsc cpuid aperfmperf pni pclmulqdq dtes64 monitor ds_cpl vmx smx est tm2 ssse3 sdbg fma cx16 xtpr pdcm pcid dca sse4_1 sse4_2 x2apic movbe popcnt tsc_deadline_timer aes xsave avx f16c rdrand lahf_lm abm 3dnowprefetch cpuid_fault epb cat_l3 cdp_l3 invpcid_single pti intel_ppin ssbd mba ibrs ibpb stibp tpr_shadow vnmi flexpriority ept vpid ept_ad fsgsbase tsc_adjust bmi1 hle avx2 smep bmi2 erms invpcid rtm cqm mpx rdt_a avx512f avx512dq rdseed adx smap clflushopt clwb intel_pt avx512cd avx512bw avx512vl xsaveopt xsavec xgetbv1 xsaves cqm_llc cqm_occup_llc cqm_mbm_total cqm_mbm_local dtherm ida arat pln pts hwp hwp_act_window hwp_epp hwp_pkg_req pku ospke md_clear flush_l1d arch_capabilities

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