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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 25 weeks (27.3079*sine), 30 weeks (25.8427*sine), and 28 weeks (23.0247*sine).

BOIL (ProShares Ultra Bloomberg Natural Gas ETF) has an average price of 125.82 (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 7/17/2017 for BOIL (ProShares Ultra Bloomberg Natural Gas ETF), this program was able to calculate the following Fourier Series:
Sequence #Cosine Coefficients Sine Coefficients FrequenciesPeriod
0125.8198   0 
128.31519 98.09942 (1*2π)/303303 weeks
254.39211 27.67685 (2*2π)/303152 weeks
327.48202 39.19419 (3*2π)/303101 weeks
441.51001 21.95148 (4*2π)/30376 weeks
531.6772 43.86697 (5*2π)/30361 weeks
624.89411 32.55171 (6*2π)/30351 weeks
726.55156 32.38049 (7*2π)/30343 weeks
813.90078 42.64868 (8*2π)/30338 weeks
97.76445 27.43277 (9*2π)/30334 weeks
107.48857 25.84265 (10*2π)/30330 weeks
1112.3832 23.0247 (11*2π)/30328 weeks
124.87194 27.30788 (12*2π)/30325 weeks
132.97686 20.65293 (13*2π)/30323 weeks
14.33608 15.0828 (14*2π)/30322 weeks
1511.04402 15.22651 (15*2π)/30320 weeks
164.68047 19.70116 (16*2π)/30319 weeks
174.69306 16.97336 (17*2π)/30318 weeks
182.95395 16.90026 (18*2π)/30317 weeks
192.00343 15.61203 (19*2π)/30316 weeks
202.09331 12.92335 (20*2π)/30315 weeks
211.30063 12.44318 (21*2π)/30314 weeks
223.76036 10.49707 (22*2π)/30314 weeks
232.29754 11.52532 (23*2π)/30313 weeks
242.51625 11.71072 (24*2π)/30313 weeks
25-2.39374 9.00048 (25*2π)/30312 weeks
262.77234 6.55031 (26*2π)/30312 weeks
273.77985 5.82183 (27*2π)/30311 weeks
285.53024 8.62782 (28*2π)/30311 weeks
294.55176 6.81127 (29*2π)/30310 weeks
306.3759 10.63421 (30*2π)/30310 weeks
312.4587 9.16613 (31*2π)/30310 weeks
324.3696 9.95214 (32*2π)/3039 weeks
33-.53285 9.71914 (33*2π)/3039 weeks
341.82273 5.83155 (34*2π)/3039 weeks
353.26608 7.09536 (35*2π)/3039 weeks
362.24834 6.9106 (36*2π)/3038 weeks
374.32574 7.90669 (37*2π)/3038 weeks
38.36408 6.81552 (38*2π)/3038 weeks
393.8493 5.7932 (39*2π)/3038 weeks
401.97902 7.03734 (40*2π)/3038 weeks
412.00443 7.51843 (41*2π)/3037 weeks
42.97614 5.78542 (42*2π)/3037 weeks
431.39771 6.77282 (43*2π)/3037 weeks
44-.63698 4.12496 (44*2π)/3037 weeks
453.6691 2.70048 (45*2π)/3037 weeks
463.8185 5.43427 (46*2π)/3037 weeks
472.3367 5.65189 (47*2π)/3036 weeks
483.18192 4.41184 (48*2π)/3036 weeks
492.57124 5.70582 (49*2π)/3036 weeks
503.26002 5.5173 (50*2π)/3036 weeks
511.69955 6.64913 (51*2π)/3036 weeks
521.92199 4.9795 (52*2π)/3036 weeks
531.88233 6.38722 (53*2π)/3036 weeks
54-.54873 4.82958 (54*2π)/3036 weeks
55.73003 3.92619 (55*2π)/3036 weeks
561.44492 2.61758 (56*2π)/3035 weeks
572.05107 4.13033 (57*2π)/3035 weeks
582.27882 4.71256 (58*2π)/3035 weeks
59-.53202 4.95429 (59*2π)/3035 weeks
60.79932 1.94938 (60*2π)/3035 weeks
61.93498 2.96653 (61*2π)/3035 weeks
62.90375 2.17196 (62*2π)/3035 weeks
631.66936 1.87225 (63*2π)/3035 weeks
641.0985 2.26562 (64*2π)/3035 weeks
652.05187 .58931 (65*2π)/3035 weeks
662.72154 2.72292 (66*2π)/3035 weeks
672.02334 .76906 (67*2π)/3035 weeks
683.91668 2.94972 (68*2π)/3034 weeks
691.26769 3.37356 (69*2π)/3034 weeks
70.49185 .68776 (70*2π)/3034 weeks
713.51498 -.1141 (71*2π)/3034 weeks
724.18346 1.06696 (72*2π)/3034 weeks
733.38961 2.1434 (73*2π)/3034 weeks
744.08615 1.93835 (74*2π)/3034 weeks
753.74708 1.95657 (75*2π)/3034 weeks
764.15627 2.95119 (76*2π)/3034 weeks
773.25047 2.46038 (77*2π)/3034 weeks
784.52267 3.16803 (78*2π)/3034 weeks
792.42926 3.80003 (79*2π)/3034 weeks
802.83578 2.78686 (80*2π)/3034 weeks
812.83194 3.48121 (81*2π)/3034 weeks
822.38767 2.62501 (82*2π)/3034 weeks
832.65296 2.91605 (83*2π)/3034 weeks
841.26207 2.85748 (84*2π)/3034 weeks
852.02718 1.69989 (85*2π)/3034 weeks
862.41587 2.04887 (86*2π)/3034 weeks
871.81549 2.08243 (87*2π)/3033 weeks
882.86848 1.06332 (88*2π)/3033 weeks
892.94552 1.99938 (89*2π)/3033 weeks
903.01012 1.26878 (90*2π)/3033 weeks
913.15356 2.23322 (91*2π)/3033 weeks
923.19703 1.48645 (92*2π)/3033 weeks
933.88563 2.26968 (93*2π)/3033 weeks
943.46978 2.59756 (94*2π)/3033 weeks
953.19207 3.04253 (95*2π)/3033 weeks
963.16704 3.50105 (96*2π)/3033 weeks
971.3847 3.75329 (97*2π)/3033 weeks
982.3521 2.35969 (98*2π)/3033 weeks
991.12062 3.32186 (99*2π)/3033 weeks
1001.43046 2.31938 (100*2π)/3033 weeks
1011.13347 2.27308 (101*2π)/3033 weeks
102.9641 1.61317 (102*2π)/3033 weeks
1031.56904 .63016 (103*2π)/3033 weeks
1042.07812 2.06698 (104*2π)/3033 weeks
105.97706 1.17962 (105*2π)/3033 weeks
1061.42707 1.46137 (106*2π)/3033 weeks
107.98579 .41565 (107*2π)/3033 weeks
1081.85988 .10413 (108*2π)/3033 weeks
1092.3978 .36083 (109*2π)/3033 weeks
1102.30279 -.20029 (110*2π)/3033 weeks
1113.14488 .68103 (111*2π)/3033 weeks
1122.53975 .0567 (112*2π)/3033 weeks
1134.16901 .70656 (113*2π)/3033 weeks
1143.08429 1.54461 (114*2π)/3033 weeks
1153.33216 1.94406 (115*2π)/3033 weeks
1162.50776 1.23642 (116*2π)/3033 weeks
1173.38532 1.9453 (117*2π)/3033 weeks
1182.66717 1.73681 (118*2π)/3033 weeks
1192.78482 2.87846 (119*2π)/3033 weeks
1201.57543 1.91457 (120*2π)/3033 weeks
1211.34576 1.84221 (121*2π)/3033 weeks
1221.50399 .91082 (122*2π)/3032 weeks
1231.31301 1.36425 (123*2π)/3032 weeks
1242.16858 .29912 (124*2π)/3032 weeks
1251.65976 1.65603 (125*2π)/3032 weeks
1261.50137 .58965 (126*2π)/3032 weeks
1271.51286 .76415 (127*2π)/3032 weeks
1281.72147 -.12445 (128*2π)/3032 weeks
1292.84367 .07431 (129*2π)/3032 weeks
1302.54652 1.06189 (130*2π)/3032 weeks
1312.05575 .44276 (131*2π)/3032 weeks
1322.33299 .71415 (132*2π)/3032 weeks
1332.24081 .15014 (133*2π)/3032 weeks
1342.68396 .71742 (134*2π)/3032 weeks
1352.45353 .74986 (135*2π)/3032 weeks
1362.81113 .84078 (136*2π)/3032 weeks
1372.28105 1.45017 (137*2π)/3032 weeks
1381.87449 .87264 (138*2π)/3032 weeks
1392.39726 .73708 (139*2π)/3032 weeks
1402.00432 1.22025 (140*2π)/3032 weeks
1411.8111 .51355 (141*2π)/3032 weeks
1421.69783 .81015 (142*2π)/3032 weeks
1432.20756 .38012 (143*2π)/3032 weeks
1441.70053 .89335 (144*2π)/3032 weeks
1451.41804 1.24713 (145*2π)/3032 weeks
1461.07839 -.31322 (146*2π)/3032 weeks
1471.82389 .39684 (147*2π)/3032 weeks
1481.57341 -.14951 (148*2π)/3032 weeks
1491.94404 .07082 (149*2π)/3032 weeks
1501.97635 .51296 (150*2π)/3032 weeks
1511.28329 .50002 (151*2π)/3032 weeks
1521.28329 -.50002 (152*2π)/3032 weeks
1531.97635 -.51296 (153*2π)/3032 weeks
1541.94404 -.07082 (154*2π)/3032 weeks
1551.57341 .14951 (155*2π)/3032 weeks
1561.82389 -.39684 (156*2π)/3032 weeks
1571.07839 .31322 (157*2π)/3032 weeks
1581.41804 -1.24713 (158*2π)/3032 weeks
1591.70053 -.89335 (159*2π)/3032 weeks
1602.20756 -.38012 (160*2π)/3032 weeks
1611.69783 -.81015 (161*2π)/3032 weeks
1621.8111 -.51355 (162*2π)/3032 weeks
1632.00432 -1.22025 (163*2π)/3032 weeks
1642.39726 -.73708 (164*2π)/3032 weeks
1651.87449 -.87264 (165*2π)/3032 weeks
1662.28105 -1.45017 (166*2π)/3032 weeks
1672.81113 -.84078 (167*2π)/3032 weeks
1682.45353 -.74986 (168*2π)/3032 weeks
1692.68396 -.71742 (169*2π)/3032 weeks
1702.24081 -.15014 (170*2π)/3032 weeks
1712.33299 -.71415 (171*2π)/3032 weeks
1722.05575 -.44276 (172*2π)/3032 weeks
1732.54652 -1.06189 (173*2π)/3032 weeks
1742.84367 -.07431 (174*2π)/3032 weeks
1751.72147 .12445 (175*2π)/3032 weeks
1761.51286 -.76415 (176*2π)/3032 weeks
1771.50137 -.58965 (177*2π)/3032 weeks
1781.65976 -1.65603 (178*2π)/3032 weeks
1792.16858 -.29912 (179*2π)/3032 weeks
1801.31301 -1.36425 (180*2π)/3032 weeks
1811.50399 -.91082 (181*2π)/3032 weeks
1821.34576 -1.84221 (182*2π)/3032 weeks
1831.57543 -1.91457 (183*2π)/3032 weeks
1842.78482 -2.87846 (184*2π)/3032 weeks
1852.66717 -1.73681 (185*2π)/3032 weeks
1863.38532 -1.9453 (186*2π)/3032 weeks
1872.50776 -1.23642 (187*2π)/3032 weeks
1883.33216 -1.94406 (188*2π)/3032 weeks
1893.08429 -1.54461 (189*2π)/3032 weeks
1904.16901 -.70656 (190*2π)/3032 weeks
1912.53975 -.0567 (191*2π)/3032 weeks
1923.14488 -.68103 (192*2π)/3032 weeks
1932.30279 .20029 (193*2π)/3032 weeks
1942.3978 -.36083 (194*2π)/3032 weeks
1951.85988 -.10413 (195*2π)/3032 weeks
196.98579 -.41565 (196*2π)/3032 weeks
1971.42707 -1.46137 (197*2π)/3032 weeks
198.97706 -1.17962 (198*2π)/3032 weeks
1992.07812 -2.06698 (199*2π)/3032 weeks
2001.56904 -.63016 (200*2π)/3032 weeks
201.9641 -1.61317 (201*2π)/3032 weeks
2021.13347 -2.27308 (202*2π)/3032 weeks
2031.43046 -2.31938 (203*2π)/3031 weeks
2041.12062 -3.32186 (204*2π)/3031 weeks
2052.3521 -2.35969 (205*2π)/3031 weeks
2061.3847 -3.75329 (206*2π)/3031 weeks
2073.16704 -3.50105 (207*2π)/3031 weeks
2083.19207 -3.04253 (208*2π)/3031 weeks
2093.46978 -2.59756 (209*2π)/3031 weeks
2103.88563 -2.26968 (210*2π)/3031 weeks
2113.19703 -1.48645 (211*2π)/3031 weeks
2123.15356 -2.23322 (212*2π)/3031 weeks
2133.01012 -1.26878 (213*2π)/3031 weeks
2142.94552 -1.99938 (214*2π)/3031 weeks
2152.86848 -1.06332 (215*2π)/3031 weeks
2161.81549 -2.08243 (216*2π)/3031 weeks
2172.41587 -2.04887 (217*2π)/3031 weeks
2182.02718 -1.69989 (218*2π)/3031 weeks
2191.26207 -2.85748 (219*2π)/3031 weeks
2202.65296 -2.91605 (220*2π)/3031 weeks
2212.38767 -2.62501 (221*2π)/3031 weeks
2222.83194 -3.48121 (222*2π)/3031 weeks
2232.83578 -2.78686 (223*2π)/3031 weeks
2242.42926 -3.80003 (224*2π)/3031 weeks
2254.52267 -3.16803 (225*2π)/3031 weeks
2263.25047 -2.46038 (226*2π)/3031 weeks
2274.15627 -2.95119 (227*2π)/3031 weeks
2283.74708 -1.95657 (228*2π)/3031 weeks
2294.08615 -1.93835 (229*2π)/3031 weeks
2303.38961 -2.1434 (230*2π)/3031 weeks
2314.18346 -1.06696 (231*2π)/3031 weeks
2323.51498 .1141 (232*2π)/3031 weeks
233.49185 -.68776 (233*2π)/3031 weeks
2341.26769 -3.37356 (234*2π)/3031 weeks
2353.91668 -2.94972 (235*2π)/3031 weeks
2362.02334 -.76906 (236*2π)/3031 weeks
2372.72154 -2.72292 (237*2π)/3031 weeks
2382.05187 -.58931 (238*2π)/3031 weeks
2391.0985 -2.26562 (239*2π)/3031 weeks
2401.66936 -1.87225 (240*2π)/3031 weeks
241.90375 -2.17196 (241*2π)/3031 weeks
242.93498 -2.96653 (242*2π)/3031 weeks
243.79932 -1.94938 (243*2π)/3031 weeks
244-.53202 -4.95429 (244*2π)/3031 weeks
2452.27882 -4.71256 (245*2π)/3031 weeks
2462.05107 -4.13033 (246*2π)/3031 weeks
2471.44492 -2.61758 (247*2π)/3031 weeks
248.73003 -3.92619 (248*2π)/3031 weeks
249-.54873 -4.82958 (249*2π)/3031 weeks
2501.88233 -6.38722 (250*2π)/3031 weeks
2511.92199 -4.9795 (251*2π)/3031 weeks
2521.69955 -6.64913 (252*2π)/3031 weeks
2533.26002 -5.5173 (253*2π)/3031 weeks
2542.57124 -5.70582 (254*2π)/3031 weeks
2553.18192 -4.41184 (255*2π)/3031 weeks
2562.3367 -5.65189 (256*2π)/3031 weeks
2573.8185 -5.43427 (257*2π)/3031 weeks
2583.6691 -2.70048 (258*2π)/3031 weeks
259-.63698 -4.12496 (259*2π)/3031 weeks
2601.39771 -6.77282 (260*2π)/3031 weeks
261.97614 -5.78542 (261*2π)/3031 weeks
2622.00443 -7.51843 (262*2π)/3031 weeks
2631.97902 -7.03734 (263*2π)/3031 weeks
2643.8493 -5.7932 (264*2π)/3031 weeks
265.36408 -6.81552 (265*2π)/3031 weeks
2664.32574 -7.90669 (266*2π)/3031 weeks
2672.24834 -6.9106 (267*2π)/3031 weeks
2683.26608 -7.09536 (268*2π)/3031 weeks
2691.82273 -5.83155 (269*2π)/3031 weeks
270-.53285 -9.71914 (270*2π)/3031 weeks
2714.3696 -9.95214 (271*2π)/3031 weeks
2722.4587 -9.16613 (272*2π)/3031 weeks
2736.3759 -10.63421 (273*2π)/3031 weeks
2744.55176 -6.81127 (274*2π)/3031 weeks
2755.53024 -8.62782 (275*2π)/3031 weeks
2763.77985 -5.82183 (276*2π)/3031 weeks
2772.77234 -6.55031 (277*2π)/3031 weeks
278-2.39374 -9.00048 (278*2π)/3031 weeks
2792.51625 -11.71072 (279*2π)/3031 weeks
2802.29754 -11.52532 (280*2π)/3031 weeks
2813.76036 -10.49707 (281*2π)/3031 weeks
2821.30063 -12.44318 (282*2π)/3031 weeks
2832.09331 -12.92335 (283*2π)/3031 weeks
2842.00343 -15.61203 (284*2π)/3031 weeks
2852.95395 -16.90026 (285*2π)/3031 weeks
2864.69306 -16.97336 (286*2π)/3031 weeks
2874.68047 -19.70116 (287*2π)/3031 weeks
28811.04402 -15.22651 (288*2π)/3031 weeks
289.33608 -15.0828 (289*2π)/3031 weeks
2902.97686 -20.65293 (290*2π)/3031 weeks
2914.87194 -27.30788 (291*2π)/3031 weeks
29212.3832 -23.0247 (292*2π)/3031 weeks
2937.48857 -25.84265 (293*2π)/3031 weeks
2947.76445 -27.43277 (294*2π)/3031 weeks
29513.90078 -42.64868 (295*2π)/3031 weeks
29626.55156 -32.38049 (296*2π)/3031 weeks
29724.89411 -32.55171 (297*2π)/3031 weeks
29831.6772 -43.86697 (298*2π)/3031 weeks
29941.51001 -21.95148 (299*2π)/3031 weeks
30027.48202 -39.19419 (300*2π)/3031 weeks
30154.39211 -27.67685 (301*2π)/3031 weeks



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