CFR - Continued Fraction Regression

HEAL.CFR is a C#/.NET implementation of Continued Fraction Regression as described in

Pablo Moscato, Haoyuan Sun, Mohammad Nazmul Haque.
Analytic Continued Fractions for Regression: A Memetic Algorithm Approach.
Expert Systems with Applications 179 (2021): 115018, 10.1016/j.eswa.2021.115018

The preprint is available from arxiv.org

The code depends on MathNET.Numerics for the Nelder-Mead Simplex algorithm for derivative-free optimization of model parameters.

Run demo

To run the demo code which tries to reproduce the result for the Gamma function use the following:

git clone https://github.com/heal-research/HEAL.CFR
cd HEAL.CFR/src
dotnet build -c Release
cd Demo
dotnet run

After 200 generations and approximately 200 seconds the result is the following expression with depth=6 and an MSE of 405.6 (NMSE=36%).

2.7364 + 0.037962 * x^3 + (-0.54806 + 5.6751 * x^3) /
  (2.3982 + 2.1704 * x^3 + (-2.2502 + 0.47104 * x^3) / 
    (3.9529 + 0.51109 * x^3 + (1.8246 + -0.54095 * x^3) / 
      (1.5209 + 2.8155 * x^3 + (-1.4091 + 0.93799 * x^3) / 
        (1.6708 + -0.64989 * x^3 + (0.13653 + -2.1031 * x^3) / 
          (3.5042 + 3.4784 * x^3 + (0.66668 + -0.71069 * x^3) / 
            (-2.7499 + -0.42627 * x^3))))))

The result for the Gamma function is worse than the result reported in the paper.

Results for PennML (without Friedman functions)

The following table shows the median MSE values from 30 runs on a part of the PennML problems. When compared to the results reported by Moscato et al. the implementation produces better results on the training set and worse result on the test set. A cause for the difference could be that HEAL.CFR does not stop early when the Nelder-Mead Simplex optimization stagnates.

nr	Instance	Moscato et al. (train)	Median MSE (train)	Stdev (train)	Moscato et al. (test)	Median MSE (test)	Stdev (test)	count
192	vineyard	4.734	3.439	0.656	7.993	9.493	78.494	30
195	auto_price	6540000	7096209.604	1275695.969	7620000	8035568.986	2071094.757	30
207	autoPrice	6350000	6297275.053	1116406.049	7560000	7827328.823	6343703.564	30
210	cloud	0.095	0.061	0.010	0.165	0.335	0.856	30
228	elusage	86.7	83.154	13.131	119	128.058	11662.035	30
230	machine_cpu	2240	2290.480	704.100	3740	4530.875	20231.100	30
485	a.vehicle	21700	19799.554	6464.678	37300	37382.613	26984.828	30
519	vinnie	2.375	2.293	0.099	2.371	2.514	0.351	30
522	pm10	0.681	0.647	0.042	0.693	0.683	0.107	30
523	a.neavote	0.725	0.737	0.079	0.903	0.755	0.454	30
527	a.elec2000	4090000	2369044.844	18009766.336	14200000	13654235.001	106630949.380	30
542	pollution	1290	1062.995	322.863	2400	2180.441	3703.602	30
547	no2	0.303	0.282	0.018	0.32	0.304	0.070	30
556	a.apnea2	1170000	910053.603	157977.790	1170000	1374304.924	473594.166	30
557	a.apnea1	1160000	964706.233	120008.381	1210000	1057566.636	432611.656	30
561	cpu	637	700.402	966.492	1330	2101.871	100387.980	30
659	sl_ex1714	1110000	1522781.116	1906213.556	1860000	2791205.448	3218940.610	30
663	rabe_266	8.381	5.935	6.275	9.641	8.240	35.170	30
665	sl_case2002	55.722	48.428	5.495	59.521	59.223	95.357	30
666	rmftsa_ladata	3.135	3.389	0.479	3.208	3.774	5440.485	30
678	v.env.	7.38	6.690	0.886	8.412	9.895	765.319	30
687	sl_ex1605	95.07	76.615	8.889	103.057	123.636	66.445	30
690	v.galaxy	800	853.299	67.139	886	930.552	4994.997	30
695	chatfield_4	247	220.129	20.880	296	287.365	1429.303	30
706	sl_case1202	2170	2214.299	352.884	2800	2901.182	1284.255	30
712	chs_geyser1	36.539	32.972	2.298	38.619	38.342	7.085	30
1027	ESL	0.265	0.252	0.018	0.309	0.325	0.203	30
1028	SWD	0.484	0.473	0.028	0.488	0.475	0.043	30
1029	LEV	0.409	0.412	0.034	0.418	0.451	0.054	30
1030	ERA	2.553	2.492	0.099	2.665	2.622	0.190	30
1089	USCrime	202	222.194	89.198	372	621.820	124476.809	30
1096	FacultySalry	1.751	1.860	0.628	3.127	3.870	8.632	30