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Fourier Analysis of BOIL (ProShares Ultra Bloomberg Natural Gas ETF)


BOIL (ProShares Ultra Bloomberg Natural Gas ETF) appears to have interesting cyclic behaviour every 26 weeks (26.7119*sine), 32 weeks (26.1366*sine), and 29 weeks (20.9723*sine).

BOIL (ProShares Ultra Bloomberg Natural Gas ETF) has an average price of 121 (topmost row, frequency = 0).



Click on the checkboxes shown on the right to see how the various frequencies contribute to the graph. Look for large magnitude coefficients (sine or cosine), as these are associated with frequencies which contribute most to the associated stock plot. If you find a large magnitude coefficient which dramatically changes the graph, look at the associated "Period" in weeks, as you may have found a significant recurring cycle for the stock of interest.

Right click on the graph above to see the menu of operations (download, full screen, etc.)

Fourier Analysis

Using data from 10/6/2011 to 10/16/2017 for BOIL (ProShares Ultra Bloomberg Natural Gas ETF), this program was able to calculate the following Fourier Series:
Sequence #Cosine Coefficients Sine Coefficients FrequenciesPeriod
0121.0037   0 
132.12533 100.1438 (1*2π)/316316 weeks
249.08952 23.51778 (2*2π)/316158 weeks
329.03134 41.60735 (3*2π)/316105 weeks
435.5932 19.97762 (4*2π)/31679 weeks
536.92416 41.61211 (5*2π)/31663 weeks
623.24607 31.89985 (6*2π)/31653 weeks
723.88703 30.15559 (7*2π)/31645 weeks
821.15549 40.35633 (8*2π)/31640 weeks
95.48122 30.69444 (9*2π)/31635 weeks
108.90406 26.1366 (10*2π)/31632 weeks
118.48363 20.97234 (11*2π)/31629 weeks
1210.13067 26.71185 (12*2π)/31626 weeks
132.42961 22.63471 (13*2π)/31624 weeks
142.29064 19.44722 (14*2π)/31623 weeks
154.23555 10.71461 (15*2π)/31621 weeks
169.81264 18.14909 (16*2π)/31620 weeks
173.7731 17.16359 (17*2π)/31619 weeks
184.31416 16.33662 (18*2π)/31618 weeks
192.42965 15.62385 (19*2π)/31617 weeks
201.37416 14.59996 (20*2π)/31616 weeks
212.1049 12.44989 (21*2π)/31615 weeks
221.09282 11.67021 (22*2π)/31614 weeks
233.71406 10.28498 (23*2π)/31614 weeks
242.19118 11.04333 (24*2π)/31613 weeks
252.42561 11.17231 (25*2π)/31613 weeks
26-2.25734 9.35381 (26*2π)/31612 weeks
272.51528 5.99174 (27*2π)/31612 weeks
283.15796 5.72369 (28*2π)/31611 weeks
295.67484 7.5345 (29*2π)/31611 weeks
303.60216 7.1186 (30*2π)/31611 weeks
316.68696 8.67648 (31*2π)/31610 weeks
322.53881 10.17077 (32*2π)/31610 weeks
333.92616 8.41418 (33*2π)/31610 weeks
341.56766 11.03799 (34*2π)/3169 weeks
35.16877 6.19124 (35*2π)/3169 weeks
362.9084 6.05035 (36*2π)/3169 weeks
372.67068 7.1725 (37*2π)/3169 weeks
382.89925 5.8647 (38*2π)/3168 weeks
392.41202 8.89203 (39*2π)/3168 weeks
401.20928 5.12802 (40*2π)/3168 weeks
413.6988 6.89737 (41*2π)/3168 weeks
422.21107 6.38517 (42*2π)/3168 weeks
431.42694 7.13671 (43*2π)/3167 weeks
441.34218 5.31906 (44*2π)/3167 weeks
45.91924 6.49851 (45*2π)/3167 weeks
46-.45023 3.42319 (46*2π)/3167 weeks
473.70804 2.8115 (47*2π)/3167 weeks
483.63412 5.24702 (48*2π)/3167 weeks
492.26584 5.43888 (49*2π)/3166 weeks
502.92281 4.18312 (50*2π)/3166 weeks
512.61619 5.57925 (51*2π)/3166 weeks
523.05178 5.09923 (52*2π)/3166 weeks
532.14421 6.47546 (53*2π)/3166 weeks
541.4825 4.77949 (54*2π)/3166 weeks
552.14628 5.66147 (55*2π)/3166 weeks
56-.26462 5.70081 (56*2π)/3166 weeks
57.56424 3.5995 (57*2π)/3166 weeks
58.54689 3.0126 (58*2π)/3165 weeks
592.3819 3.43216 (59*2π)/3165 weeks
601.97656 3.80401 (60*2π)/3165 weeks
611.16586 5.32102 (61*2π)/3165 weeks
62-.94294 2.91089 (62*2π)/3165 weeks
631.34232 2.57678 (63*2π)/3165 weeks
64.72323 2.61953 (64*2π)/3165 weeks
651.03822 1.94786 (65*2π)/3165 weeks
661.77075 2.079 (66*2π)/3165 weeks
67.907 1.80365 (67*2π)/3165 weeks
682.5845 .69132 (68*2π)/3165 weeks
692.18015 2.67334 (69*2π)/3165 weeks
702.36003 .59216 (70*2π)/3165 weeks
713.60427 3.04248 (71*2π)/3164 weeks
721.14287 3.17183 (72*2π)/3164 weeks
73.47149 .66559 (73*2π)/3164 weeks
743.28175 -.20193 (74*2π)/3164 weeks
753.92282 .93535 (75*2π)/3164 weeks
763.42855 2.16889 (76*2π)/3164 weeks
773.84784 1.63984 (77*2π)/3164 weeks
783.47867 2.01022 (78*2π)/3164 weeks
794.16272 2.48516 (79*2π)/3164 weeks
803.10912 2.73989 (80*2π)/3164 weeks
814.08511 2.3291 (81*2π)/3164 weeks
823.16422 4.10021 (82*2π)/3164 weeks
832.43322 2.73016 (83*2π)/3164 weeks
842.96863 2.87427 (84*2π)/3164 weeks
852.02315 3.02645 (85*2π)/3164 weeks
862.45829 2.66599 (86*2π)/3164 weeks
872.26715 3.27087 (87*2π)/3164 weeks
881.3379 2.00016 (88*2π)/3164 weeks
892.24244 1.67163 (89*2π)/3164 weeks
902.30392 2.00713 (90*2π)/3164 weeks
911.6112 1.53404 (91*2π)/3163 weeks
922.95224 1.24056 (92*2π)/3163 weeks
932.65103 1.92428 (93*2π)/3163 weeks
943.08205 1.25952 (94*2π)/3163 weeks
952.91753 2.13242 (95*2π)/3163 weeks
963.12426 1.42263 (96*2π)/3163 weeks
973.72155 2.18382 (97*2π)/3163 weeks
983.3127 2.48383 (98*2π)/3163 weeks
993.0822 2.92378 (99*2π)/3163 weeks
1003.06607 3.24243 (100*2π)/3163 weeks
1011.64977 3.84975 (101*2π)/3163 weeks
1021.86287 2.09756 (102*2π)/3163 weeks
1031.58127 3.44281 (103*2π)/3163 weeks
1041.21933 2.1593 (104*2π)/3163 weeks
1051.18813 2.2391 (105*2π)/3163 weeks
106.85117 1.76964 (106*2π)/3163 weeks
107.91271 1.04112 (107*2π)/3163 weeks
1082.32504 1.285 (108*2π)/3163 weeks
1091.11113 1.95968 (109*2π)/3163 weeks
1101.52819 .98962 (110*2π)/3163 weeks
111.92106 1.15693 (111*2π)/3163 weeks
1121.33512 .10442 (112*2π)/3163 weeks
1131.95796 .08527 (113*2π)/3163 weeks
1142.13646 .49557 (114*2π)/3163 weeks
1152.67471 -.21169 (115*2π)/3163 weeks
1162.81686 .82156 (116*2π)/3163 weeks
1172.75003 -.15784 (117*2π)/3163 weeks
1184.01453 .97491 (118*2π)/3163 weeks
1192.95 1.36539 (119*2π)/3163 weeks
1203.1246 1.91606 (120*2π)/3163 weeks
1212.43447 1.15376 (121*2π)/3163 weeks
1223.27138 1.83236 (122*2π)/3163 weeks
1232.5295 1.69337 (123*2π)/3163 weeks
1242.86375 2.59917 (124*2π)/3163 weeks
1251.41935 1.96142 (125*2π)/3163 weeks
1261.40957 1.85675 (126*2π)/3163 weeks
1271.21258 1.027 (127*2π)/3162 weeks
1281.61976 1.39024 (128*2π)/3162 weeks
1291.48726 .2827 (129*2π)/3162 weeks
1302.27454 1.44684 (130*2π)/3162 weeks
1311.22086 .81279 (131*2π)/3162 weeks
1321.70137 .64976 (132*2π)/3162 weeks
1331.32412 .20006 (133*2π)/3162 weeks
1342.14293 -.30163 (134*2π)/3162 weeks
1352.76405 .56368 (135*2π)/3162 weeks
1361.97301 .96118 (136*2π)/3162 weeks
1372.20962 .40433 (137*2π)/3162 weeks
1382.03138 .67696 (138*2π)/3162 weeks
1392.4583 .15874 (139*2π)/3162 weeks
1402.47817 .73037 (140*2π)/3162 weeks
1412.33931 .66991 (141*2π)/3162 weeks
1422.69703 .94219 (142*2π)/3162 weeks
1432.06468 1.38453 (143*2π)/3162 weeks
1441.80606 .81723 (144*2π)/3162 weeks
1452.31832 .74921 (145*2π)/3162 weeks
1461.93906 1.17884 (146*2π)/3162 weeks
1471.70107 .50802 (147*2π)/3162 weeks
1481.69056 .82106 (148*2π)/3162 weeks
1491.98728 .28906 (149*2π)/3162 weeks
1501.72772 .97978 (150*2π)/3162 weeks
1511.67474 1.09878 (151*2π)/3162 weeks
152.69926 .09426 (152*2π)/3162 weeks
1531.92983 .11071 (153*2π)/3162 weeks
1541.35908 .08589 (154*2π)/3162 weeks
1551.87919 -.02683 (155*2π)/3162 weeks
1562.01373 .19868 (156*2π)/3162 weeks
1571.53886 .53878 (157*2π)/3162 weeks
158.92421   (158*2π)/3162 weeks
1591.53886 -.53878 (159*2π)/3162 weeks
1602.01373 -.19868 (160*2π)/3162 weeks
1611.87919 .02683 (161*2π)/3162 weeks
1621.35908 -.08589 (162*2π)/3162 weeks
1631.92983 -.11071 (163*2π)/3162 weeks
164.69926 -.09426 (164*2π)/3162 weeks
1651.67474 -1.09878 (165*2π)/3162 weeks
1661.72772 -.97978 (166*2π)/3162 weeks
1671.98728 -.28906 (167*2π)/3162 weeks
1681.69056 -.82106 (168*2π)/3162 weeks
1691.70107 -.50802 (169*2π)/3162 weeks
1701.93906 -1.17884 (170*2π)/3162 weeks
1712.31832 -.74921 (171*2π)/3162 weeks
1721.80606 -.81723 (172*2π)/3162 weeks
1732.06468 -1.38453 (173*2π)/3162 weeks
1742.69703 -.94219 (174*2π)/3162 weeks
1752.33931 -.66991 (175*2π)/3162 weeks
1762.47817 -.73037 (176*2π)/3162 weeks
1772.4583 -.15874 (177*2π)/3162 weeks
1782.03138 -.67696 (178*2π)/3162 weeks
1792.20962 -.40433 (179*2π)/3162 weeks
1801.97301 -.96118 (180*2π)/3162 weeks
1812.76405 -.56368 (181*2π)/3162 weeks
1822.14293 .30163 (182*2π)/3162 weeks
1831.32412 -.20006 (183*2π)/3162 weeks
1841.70137 -.64976 (184*2π)/3162 weeks
1851.22086 -.81279 (185*2π)/3162 weeks
1862.27454 -1.44684 (186*2π)/3162 weeks
1871.48726 -.2827 (187*2π)/3162 weeks
1881.61976 -1.39024 (188*2π)/3162 weeks
1891.21258 -1.027 (189*2π)/3162 weeks
1901.40957 -1.85675 (190*2π)/3162 weeks
1911.41935 -1.96142 (191*2π)/3162 weeks
1922.86375 -2.59917 (192*2π)/3162 weeks
1932.5295 -1.69337 (193*2π)/3162 weeks
1943.27138 -1.83236 (194*2π)/3162 weeks
1952.43447 -1.15376 (195*2π)/3162 weeks
1963.1246 -1.91606 (196*2π)/3162 weeks
1972.95 -1.36539 (197*2π)/3162 weeks
1984.01453 -.97491 (198*2π)/3162 weeks
1992.75003 .15784 (199*2π)/3162 weeks
2002.81686 -.82156 (200*2π)/3162 weeks
2012.67471 .21169 (201*2π)/3162 weeks
2022.13646 -.49557 (202*2π)/3162 weeks
2031.95796 -.08527 (203*2π)/3162 weeks
2041.33512 -.10442 (204*2π)/3162 weeks
205.92106 -1.15693 (205*2π)/3162 weeks
2061.52819 -.98962 (206*2π)/3162 weeks
2071.11113 -1.95968 (207*2π)/3162 weeks
2082.32504 -1.285 (208*2π)/3162 weeks
209.91271 -1.04112 (209*2π)/3162 weeks
210.85117 -1.76964 (210*2π)/3162 weeks
2111.18813 -2.2391 (211*2π)/3161 weeks
2121.21933 -2.1593 (212*2π)/3161 weeks
2131.58127 -3.44281 (213*2π)/3161 weeks
2141.86287 -2.09756 (214*2π)/3161 weeks
2151.64977 -3.84975 (215*2π)/3161 weeks
2163.06607 -3.24243 (216*2π)/3161 weeks
2173.0822 -2.92378 (217*2π)/3161 weeks
2183.3127 -2.48383 (218*2π)/3161 weeks
2193.72155 -2.18382 (219*2π)/3161 weeks
2203.12426 -1.42263 (220*2π)/3161 weeks
2212.91753 -2.13242 (221*2π)/3161 weeks
2223.08205 -1.25952 (222*2π)/3161 weeks
2232.65103 -1.92428 (223*2π)/3161 weeks
2242.95224 -1.24056 (224*2π)/3161 weeks
2251.6112 -1.53404 (225*2π)/3161 weeks
2262.30392 -2.00713 (226*2π)/3161 weeks
2272.24244 -1.67163 (227*2π)/3161 weeks
2281.3379 -2.00016 (228*2π)/3161 weeks
2292.26715 -3.27087 (229*2π)/3161 weeks
2302.45829 -2.66599 (230*2π)/3161 weeks
2312.02315 -3.02645 (231*2π)/3161 weeks
2322.96863 -2.87427 (232*2π)/3161 weeks
2332.43322 -2.73016 (233*2π)/3161 weeks
2343.16422 -4.10021 (234*2π)/3161 weeks
2354.08511 -2.3291 (235*2π)/3161 weeks
2363.10912 -2.73989 (236*2π)/3161 weeks
2374.16272 -2.48516 (237*2π)/3161 weeks
2383.47867 -2.01022 (238*2π)/3161 weeks
2393.84784 -1.63984 (239*2π)/3161 weeks
2403.42855 -2.16889 (240*2π)/3161 weeks
2413.92282 -.93535 (241*2π)/3161 weeks
2423.28175 .20193 (242*2π)/3161 weeks
243.47149 -.66559 (243*2π)/3161 weeks
2441.14287 -3.17183 (244*2π)/3161 weeks
2453.60427 -3.04248 (245*2π)/3161 weeks
2462.36003 -.59216 (246*2π)/3161 weeks
2472.18015 -2.67334 (247*2π)/3161 weeks
2482.5845 -.69132 (248*2π)/3161 weeks
249.907 -1.80365 (249*2π)/3161 weeks
2501.77075 -2.079 (250*2π)/3161 weeks
2511.03822 -1.94786 (251*2π)/3161 weeks
252.72323 -2.61953 (252*2π)/3161 weeks
2531.34232 -2.57678 (253*2π)/3161 weeks
254-.94294 -2.91089 (254*2π)/3161 weeks
2551.16586 -5.32102 (255*2π)/3161 weeks
2561.97656 -3.80401 (256*2π)/3161 weeks
2572.3819 -3.43216 (257*2π)/3161 weeks
258.54689 -3.0126 (258*2π)/3161 weeks
259.56424 -3.5995 (259*2π)/3161 weeks
260-.26462 -5.70081 (260*2π)/3161 weeks
2612.14628 -5.66147 (261*2π)/3161 weeks
2621.4825 -4.77949 (262*2π)/3161 weeks
2632.14421 -6.47546 (263*2π)/3161 weeks
2643.05178 -5.09923 (264*2π)/3161 weeks
2652.61619 -5.57925 (265*2π)/3161 weeks
2662.92281 -4.18312 (266*2π)/3161 weeks
2672.26584 -5.43888 (267*2π)/3161 weeks
2683.63412 -5.24702 (268*2π)/3161 weeks
2693.70804 -2.8115 (269*2π)/3161 weeks
270-.45023 -3.42319 (270*2π)/3161 weeks
271.91924 -6.49851 (271*2π)/3161 weeks
2721.34218 -5.31906 (272*2π)/3161 weeks
2731.42694 -7.13671 (273*2π)/3161 weeks
2742.21107 -6.38517 (274*2π)/3161 weeks
2753.6988 -6.89737 (275*2π)/3161 weeks
2761.20928 -5.12802 (276*2π)/3161 weeks
2772.41202 -8.89203 (277*2π)/3161 weeks
2782.89925 -5.8647 (278*2π)/3161 weeks
2792.67068 -7.1725 (279*2π)/3161 weeks
2802.9084 -6.05035 (280*2π)/3161 weeks
281.16877 -6.19124 (281*2π)/3161 weeks
2821.56766 -11.03799 (282*2π)/3161 weeks
2833.92616 -8.41418 (283*2π)/3161 weeks
2842.53881 -10.17077 (284*2π)/3161 weeks
2856.68696 -8.67648 (285*2π)/3161 weeks
2863.60216 -7.1186 (286*2π)/3161 weeks
2875.67484 -7.5345 (287*2π)/3161 weeks
2883.15796 -5.72369 (288*2π)/3161 weeks
2892.51528 -5.99174 (289*2π)/3161 weeks
290-2.25734 -9.35381 (290*2π)/3161 weeks
2912.42561 -11.17231 (291*2π)/3161 weeks
2922.19118 -11.04333 (292*2π)/3161 weeks
2933.71406 -10.28498 (293*2π)/3161 weeks
2941.09282 -11.67021 (294*2π)/3161 weeks
2952.1049 -12.44989 (295*2π)/3161 weeks
2961.37416 -14.59996 (296*2π)/3161 weeks
2972.42965 -15.62385 (297*2π)/3161 weeks
2984.31416 -16.33662 (298*2π)/3161 weeks
2993.7731 -17.16359 (299*2π)/3161 weeks
3009.81264 -18.14909 (300*2π)/3161 weeks
3014.23555 -10.71461 (301*2π)/3161 weeks
3022.29064 -19.44722 (302*2π)/3161 weeks
3032.42961 -22.63471 (303*2π)/3161 weeks
30410.13067 -26.71185 (304*2π)/3161 weeks
3058.48363 -20.97234 (305*2π)/3161 weeks
3068.90406 -26.1366 (306*2π)/3161 weeks
3075.48122 -30.69444 (307*2π)/3161 weeks
30821.15549 -40.35633 (308*2π)/3161 weeks
30923.88703 -30.15559 (309*2π)/3161 weeks
31023.24607 -31.89985 (310*2π)/3161 weeks
31136.92416 -41.61211 (311*2π)/3161 weeks
31235.5932 -19.97762 (312*2π)/3161 weeks
31329.03134 -41.60735 (313*2π)/3161 weeks
31449.08952 -23.51778 (314*2π)/3161 weeks



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