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Fourier Analysis of BOIL (ProShares Ultra DJ-UBS Natural Gas ETF)


BOIL (ProShares Ultra DJ-UBS Natural Gas ETF) appears to have interesting cyclic behaviour every 25 weeks (25.4276*sine), 33 weeks (24.8355*sine), and 30 weeks (22.9926*sine).

BOIL (ProShares Ultra DJ-UBS Natural Gas ETF) has an average price of 116.49 (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 1/16/2018 for BOIL (ProShares Ultra DJ-UBS Natural Gas ETF), this program was able to calculate the following Fourier Series:
Sequence #Cosine Coefficients Sine Coefficients FrequenciesPeriod
0116.4919   0 
135.83334 101.3517 (1*2π)/329329 weeks
243.32992 20.83547 (2*2π)/329165 weeks
331.40435 42.87952 (3*2π)/329110 weeks
430.50083 19.95199 (4*2π)/32982 weeks
540.47665 36.49196 (5*2π)/32966 weeks
621.72861 32.88362 (6*2π)/32955 weeks
722.43216 29.63364 (7*2π)/32947 weeks
824.48309 34.66526 (8*2π)/32941 weeks
97.29225 35.74297 (9*2π)/32937 weeks
108.34728 24.83547 (10*2π)/32933 weeks
116.62725 22.99255 (11*2π)/32930 weeks
1211.56673 21.76058 (12*2π)/32927 weeks
134.71024 25.42756 (13*2π)/32925 weeks
142.56916 19.41645 (14*2π)/32924 weeks
15.25778 16.29272 (15*2π)/32922 weeks
168.61971 11.43581 (16*2π)/32921 weeks
176.97071 19.03721 (17*2π)/32919 weeks
184.06385 15.70519 (18*2π)/32918 weeks
193.9401 15.83512 (19*2π)/32917 weeks
202.35893 14.58669 (20*2π)/32916 weeks
211.04743 13.47247 (21*2π)/32916 weeks
222.05564 12.04523 (22*2π)/32915 weeks
23.99918 10.95006 (23*2π)/32914 weeks
243.66408 10.11378 (24*2π)/32914 weeks
252.13758 10.6273 (25*2π)/32913 weeks
262.37903 10.70891 (26*2π)/32913 weeks
27-1.975 9.619 (27*2π)/32912 weeks
282.17714 5.48532 (28*2π)/32912 weeks
292.75486 5.718 (29*2π)/32911 weeks
305.42351 6.50146 (30*2π)/32911 weeks
313.40566 7.63028 (31*2π)/32911 weeks
326.09709 7.06743 (32*2π)/32910 weeks
333.78946 10.60156 (33*2π)/32910 weeks
343.01133 7.80947 (34*2π)/32910 weeks
353.53393 10.05464 (35*2π)/3299 weeks
36-.67379 7.95492 (36*2π)/3299 weeks
371.91678 5.37817 (37*2π)/3299 weeks
383.02885 6.52014 (38*2π)/3299 weeks
392.11006 6.5048 (39*2π)/3298 weeks
403.98793 6.48016 (40*2π)/3298 weeks
41.57201 7.50143 (41*2π)/3298 weeks
422.46012 4.58955 (42*2π)/3298 weeks
432.62193 7.16977 (43*2π)/3298 weeks
442.39256 6.33848 (44*2π)/3297 weeks
45.97821 6.5743 (45*2π)/3297 weeks
461.65714 5.16841 (46*2π)/3297 weeks
47.49449 6.11044 (47*2π)/3297 weeks
48-.19089 2.83348 (48*2π)/3297 weeks
493.68303 2.90119 (49*2π)/3297 weeks
503.45218 5.06957 (50*2π)/3297 weeks
512.18546 5.24163 (51*2π)/3296 weeks
522.67635 4.00083 (52*2π)/3296 weeks
532.67102 5.41044 (53*2π)/3296 weeks
542.80575 4.77542 (54*2π)/3296 weeks
552.46627 6.12132 (55*2π)/3296 weeks
561.10536 4.86056 (56*2π)/3296 weeks
572.10388 5.08938 (57*2π)/3296 weeks
58.53318 6.08709 (58*2π)/3296 weeks
59-.02931 3.61777 (59*2π)/3296 weeks
60.50402 3.5222 (60*2π)/3295 weeks
611.80491 2.53001 (61*2π)/3295 weeks
621.80226 3.80018 (62*2π)/3295 weeks
632.08964 4.39765 (63*2π)/3295 weeks
64-.31827 4.72686 (64*2π)/3295 weeks
65.29573 1.76393 (65*2π)/3295 weeks
661.02872 2.7816 (66*2π)/3295 weeks
67.69661 2.21918 (67*2π)/3295 weeks
681.1671 1.73954 (68*2π)/3295 weeks
691.54963 2.2527 (69*2π)/3295 weeks
70.90511 1.32327 (70*2π)/3295 weeks
712.90707 1.04466 (71*2π)/3295 weeks
721.72589 2.44814 (72*2π)/3295 weeks
732.66847 .57405 (73*2π)/3295 weeks
743.30114 3.09071 (74*2π)/3294 weeks
751.03244 2.99669 (75*2π)/3294 weeks
76.4486 .65458 (76*2π)/3294 weeks
773.05807 -.25979 (77*2π)/3294 weeks
783.69087 .85696 (78*2π)/3294 weeks
793.49378 2.12044 (79*2π)/3294 weeks
803.5353 1.45468 (80*2π)/3294 weeks
813.38116 2.08028 (81*2π)/3294 weeks
823.96482 2.07419 (82*2π)/3294 weeks
833.25375 2.88156 (83*2π)/3294 weeks
843.44338 2.03501 (84*2π)/3294 weeks
853.90028 3.59663 (85*2π)/3294 weeks
862.13075 3.11529 (86*2π)/3294 weeks
872.68863 2.59242 (87*2π)/3294 weeks
882.56417 3.23852 (88*2π)/3294 weeks
892.19149 2.42765 (89*2π)/3294 weeks
902.43288 2.62134 (90*2π)/3294 weeks
911.41826 3.00223 (91*2π)/3294 weeks
921.65171 1.6434 (92*2π)/3294 weeks
932.29098 1.77163 (93*2π)/3294 weeks
942.15356 2.02904 (94*2π)/3294 weeks
951.78202 1.03618 (95*2π)/3293 weeks
962.92091 1.376 (96*2π)/3293 weeks
972.41717 1.74362 (97*2π)/3293 weeks
983.11274 1.29508 (98*2π)/3293 weeks
992.71966 1.99924 (99*2π)/3293 weeks
1003.05283 1.3567 (100*2π)/3293 weeks
1013.56832 2.09256 (101*2π)/3293 weeks
1023.16513 2.37292 (102*2π)/3293 weeks
1032.98142 2.80499 (103*2π)/3293 weeks
1042.94897 3.02388 (104*2π)/3293 weeks
1051.90956 3.79853 (105*2π)/3293 weeks
1061.4017 2.11053 (106*2π)/3293 weeks
1072.01357 3.20314 (107*2π)/3293 weeks
108.92193 2.24026 (108*2π)/3293 weeks
1091.2268 2.20909 (109*2π)/3293 weeks
110.88839 1.97417 (110*2π)/3293 weeks
111.87539 1.39313 (111*2π)/3293 weeks
1121.75224 .624 (112*2π)/3293 weeks
1131.81253 1.96702 (113*2π)/3293 weeks
114.89343 1.10438 (114*2π)/3293 weeks
1151.40462 1.32291 (115*2π)/3293 weeks
116.81754 .59331 (116*2π)/3293 weeks
1171.58708 .08252 (117*2π)/3293 weeks
1182.09345 .12885 (118*2π)/3293 weeks
1191.85775 .3 (119*2π)/3293 weeks
1202.87623 .0414 (120*2π)/3293 weeks
1212.46267 .79512 (121*2π)/3293 weeks
1222.98951 -.18371 (122*2π)/3293 weeks
1233.79111 1.19994 (123*2π)/3293 weeks
1242.87405 1.23424 (124*2π)/3293 weeks
1252.92983 1.88167 (125*2π)/3293 weeks
1262.36539 1.0887 (126*2π)/3293 weeks
1273.1616 1.73665 (127*2π)/3293 weeks
1282.41099 1.66445 (128*2π)/3293 weeks
1292.86826 2.33191 (129*2π)/3293 weeks
1301.32903 2.07989 (130*2π)/3293 weeks
1311.46975 1.82853 (131*2π)/3293 weeks
1321.08037 1.21705 (132*2π)/3292 weeks
1331.75101 1.14669 (133*2π)/3292 weeks
1341.08542 .64071 (134*2π)/3292 weeks
1352.40192 .79091 (135*2π)/3292 weeks
1361.21059 1.24922 (136*2π)/3292 weeks
1371.50037 .51738 (137*2π)/3292 weeks
1381.33437 .65201 (138*2π)/3292 weeks
1391.61064 -.12012 (139*2π)/3292 weeks
1402.59308 -.00543 (140*2π)/3292 weeks
1412.48795 .88585 (141*2π)/3292 weeks
1421.79382 .55649 (142*2π)/3292 weeks
1432.22622 .51861 (143*2π)/3292 weeks
1441.85374 .42751 (144*2π)/3292 weeks
1452.54286 .33349 (145*2π)/3292 weeks
1462.33685 .69295 (146*2π)/3292 weeks
1472.28653 .58137 (147*2π)/3292 weeks
1482.55342 1.01125 (148*2π)/3292 weeks
1491.87513 1.29074 (149*2π)/3292 weeks
1501.73566 .76461 (150*2π)/3292 weeks
1512.23605 .75984 (151*2π)/3292 weeks
1521.87556 1.14206 (152*2π)/3292 weeks
1531.60261 .51078 (153*2π)/3292 weeks
1541.68583 .81414 (154*2π)/3292 weeks
1551.78517 .26856 (155*2π)/3292 weeks
1561.82711 .98896 (156*2π)/3292 weeks
1571.73913 .87817 (157*2π)/3292 weeks
158.63898 .56498 (158*2π)/3292 weeks
1591.72198 -.22637 (159*2π)/3292 weeks
1601.39414 .32247 (160*2π)/3292 weeks
1611.6681 -.15952 (161*2π)/3292 weeks
1621.82721 .06996 (162*2π)/3292 weeks
1631.75228 .51758 (163*2π)/3292 weeks
1641.16319 .44531 (164*2π)/3292 weeks
1651.16319 -.44531 (165*2π)/3292 weeks
1661.75228 -.51758 (166*2π)/3292 weeks
1671.82721 -.06996 (167*2π)/3292 weeks
1681.6681 .15952 (168*2π)/3292 weeks
1691.39414 -.32247 (169*2π)/3292 weeks
1701.72198 .22637 (170*2π)/3292 weeks
171.63898 -.56498 (171*2π)/3292 weeks
1721.73913 -.87817 (172*2π)/3292 weeks
1731.82711 -.98896 (173*2π)/3292 weeks
1741.78517 -.26856 (174*2π)/3292 weeks
1751.68583 -.81414 (175*2π)/3292 weeks
1761.60261 -.51078 (176*2π)/3292 weeks
1771.87556 -1.14206 (177*2π)/3292 weeks
1782.23605 -.75984 (178*2π)/3292 weeks
1791.73566 -.76461 (179*2π)/3292 weeks
1801.87513 -1.29074 (180*2π)/3292 weeks
1812.55342 -1.01125 (181*2π)/3292 weeks
1822.28653 -.58137 (182*2π)/3292 weeks
1832.33685 -.69295 (183*2π)/3292 weeks
1842.54286 -.33349 (184*2π)/3292 weeks
1851.85374 -.42751 (185*2π)/3292 weeks
1862.22622 -.51861 (186*2π)/3292 weeks
1871.79382 -.55649 (187*2π)/3292 weeks
1882.48795 -.88585 (188*2π)/3292 weeks
1892.59308 .00543 (189*2π)/3292 weeks
1901.61064 .12012 (190*2π)/3292 weeks
1911.33437 -.65201 (191*2π)/3292 weeks
1921.50037 -.51738 (192*2π)/3292 weeks
1931.21059 -1.24922 (193*2π)/3292 weeks
1942.40192 -.79091 (194*2π)/3292 weeks
1951.08542 -.64071 (195*2π)/3292 weeks
1961.75101 -1.14669 (196*2π)/3292 weeks
1971.08037 -1.21705 (197*2π)/3292 weeks
1981.46975 -1.82853 (198*2π)/3292 weeks
1991.32903 -2.07989 (199*2π)/3292 weeks
2002.86826 -2.33191 (200*2π)/3292 weeks
2012.41099 -1.66445 (201*2π)/3292 weeks
2023.1616 -1.73665 (202*2π)/3292 weeks
2032.36539 -1.0887 (203*2π)/3292 weeks
2042.92983 -1.88167 (204*2π)/3292 weeks
2052.87405 -1.23424 (205*2π)/3292 weeks
2063.79111 -1.19994 (206*2π)/3292 weeks
2072.98951 .18371 (207*2π)/3292 weeks
2082.46267 -.79512 (208*2π)/3292 weeks
2092.87623 -.0414 (209*2π)/3292 weeks
2101.85775 -.3 (210*2π)/3292 weeks
2112.09345 -.12885 (211*2π)/3292 weeks
2121.58708 -.08252 (212*2π)/3292 weeks
213.81754 -.59331 (213*2π)/3292 weeks
2141.40462 -1.32291 (214*2π)/3292 weeks
215.89343 -1.10438 (215*2π)/3292 weeks
2161.81253 -1.96702 (216*2π)/3292 weeks
2171.75224 -.624 (217*2π)/3292 weeks
218.87539 -1.39313 (218*2π)/3292 weeks
219.88839 -1.97417 (219*2π)/3292 weeks
2201.2268 -2.20909 (220*2π)/3291 weeks
221.92193 -2.24026 (221*2π)/3291 weeks
2222.01357 -3.20314 (222*2π)/3291 weeks
2231.4017 -2.11053 (223*2π)/3291 weeks
2241.90956 -3.79853 (224*2π)/3291 weeks
2252.94897 -3.02388 (225*2π)/3291 weeks
2262.98142 -2.80499 (226*2π)/3291 weeks
2273.16513 -2.37292 (227*2π)/3291 weeks
2283.56832 -2.09256 (228*2π)/3291 weeks
2293.05283 -1.3567 (229*2π)/3291 weeks
2302.71966 -1.99924 (230*2π)/3291 weeks
2313.11274 -1.29508 (231*2π)/3291 weeks
2322.41717 -1.74362 (232*2π)/3291 weeks
2332.92091 -1.376 (233*2π)/3291 weeks
2341.78202 -1.03618 (234*2π)/3291 weeks
2352.15356 -2.02904 (235*2π)/3291 weeks
2362.29098 -1.77163 (236*2π)/3291 weeks
2371.65171 -1.6434 (237*2π)/3291 weeks
2381.41826 -3.00223 (238*2π)/3291 weeks
2392.43288 -2.62134 (239*2π)/3291 weeks
2402.19149 -2.42765 (240*2π)/3291 weeks
2412.56417 -3.23852 (241*2π)/3291 weeks
2422.68863 -2.59242 (242*2π)/3291 weeks
2432.13075 -3.11529 (243*2π)/3291 weeks
2443.90028 -3.59663 (244*2π)/3291 weeks
2453.44338 -2.03501 (245*2π)/3291 weeks
2463.25375 -2.88156 (246*2π)/3291 weeks
2473.96482 -2.07419 (247*2π)/3291 weeks
2483.38116 -2.08028 (248*2π)/3291 weeks
2493.5353 -1.45468 (249*2π)/3291 weeks
2503.49378 -2.12044 (250*2π)/3291 weeks
2513.69087 -.85696 (251*2π)/3291 weeks
2523.05807 .25979 (252*2π)/3291 weeks
253.4486 -.65458 (253*2π)/3291 weeks
2541.03244 -2.99669 (254*2π)/3291 weeks
2553.30114 -3.09071 (255*2π)/3291 weeks
2562.66847 -.57405 (256*2π)/3291 weeks
2571.72589 -2.44814 (257*2π)/3291 weeks
2582.90707 -1.04466 (258*2π)/3291 weeks
259.90511 -1.32327 (259*2π)/3291 weeks
2601.54963 -2.2527 (260*2π)/3291 weeks
2611.1671 -1.73954 (261*2π)/3291 weeks
262.69661 -2.21918 (262*2π)/3291 weeks
2631.02872 -2.7816 (263*2π)/3291 weeks
264.29573 -1.76393 (264*2π)/3291 weeks
265-.31827 -4.72686 (265*2π)/3291 weeks
2662.08964 -4.39765 (266*2π)/3291 weeks
2671.80226 -3.80018 (267*2π)/3291 weeks
2681.80491 -2.53001 (268*2π)/3291 weeks
269.50402 -3.5222 (269*2π)/3291 weeks
270-.02931 -3.61777 (270*2π)/3291 weeks
271.53318 -6.08709 (271*2π)/3291 weeks
2722.10388 -5.08938 (272*2π)/3291 weeks
2731.10536 -4.86056 (273*2π)/3291 weeks
2742.46627 -6.12132 (274*2π)/3291 weeks
2752.80575 -4.77542 (275*2π)/3291 weeks
2762.67102 -5.41044 (276*2π)/3291 weeks
2772.67635 -4.00083 (277*2π)/3291 weeks
2782.18546 -5.24163 (278*2π)/3291 weeks
2793.45218 -5.06957 (279*2π)/3291 weeks
2803.68303 -2.90119 (280*2π)/3291 weeks
281-.19089 -2.83348 (281*2π)/3291 weeks
282.49449 -6.11044 (282*2π)/3291 weeks
2831.65714 -5.16841 (283*2π)/3291 weeks
284.97821 -6.5743 (284*2π)/3291 weeks
2852.39256 -6.33848 (285*2π)/3291 weeks
2862.62193 -7.16977 (286*2π)/3291 weeks
2872.46012 -4.58955 (287*2π)/3291 weeks
288.57201 -7.50143 (288*2π)/3291 weeks
2893.98793 -6.48016 (289*2π)/3291 weeks
2902.11006 -6.5048 (290*2π)/3291 weeks
2913.02885 -6.52014 (291*2π)/3291 weeks
2921.91678 -5.37817 (292*2π)/3291 weeks
293-.67379 -7.95492 (293*2π)/3291 weeks
2943.53393 -10.05464 (294*2π)/3291 weeks
2953.01133 -7.80947 (295*2π)/3291 weeks
2963.78946 -10.60156 (296*2π)/3291 weeks
2976.09709 -7.06743 (297*2π)/3291 weeks
2983.40566 -7.63028 (298*2π)/3291 weeks
2995.42351 -6.50146 (299*2π)/3291 weeks
3002.75486 -5.718 (300*2π)/3291 weeks
3012.17714 -5.48532 (301*2π)/3291 weeks
302-1.975 -9.619 (302*2π)/3291 weeks
3032.37903 -10.70891 (303*2π)/3291 weeks
3042.13758 -10.6273 (304*2π)/3291 weeks
3053.66408 -10.11378 (305*2π)/3291 weeks
306.99918 -10.95006 (306*2π)/3291 weeks
3072.05564 -12.04523 (307*2π)/3291 weeks
3081.04743 -13.47247 (308*2π)/3291 weeks
3092.35893 -14.58669 (309*2π)/3291 weeks
3103.9401 -15.83512 (310*2π)/3291 weeks
3114.06385 -15.70519 (311*2π)/3291 weeks
3126.97071 -19.03721 (312*2π)/3291 weeks
3138.61971 -11.43581 (313*2π)/3291 weeks
314.25778 -16.29272 (314*2π)/3291 weeks
3152.56916 -19.41645 (315*2π)/3291 weeks
3164.71024 -25.42756 (316*2π)/3291 weeks
31711.56673 -21.76058 (317*2π)/3291 weeks
3186.62725 -22.99255 (318*2π)/3291 weeks
3198.34728 -24.83547 (319*2π)/3291 weeks
3207.29225 -35.74297 (320*2π)/3291 weeks
32124.48309 -34.66526 (321*2π)/3291 weeks
32222.43216 -29.63364 (322*2π)/3291 weeks
32321.72861 -32.88362 (323*2π)/3291 weeks
32440.47665 -36.49196 (324*2π)/3291 weeks
32530.50083 -19.95199 (325*2π)/3291 weeks
32631.40435 -42.87952 (326*2π)/3291 weeks
32743.32992 -20.83547 (327*2π)/3291 weeks



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