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Fourier Analysis of BOIL (ProShares Ultra Bloomberg Natur)


BOIL (ProShares Ultra Bloomberg Natur) appears to have interesting cyclic behaviour every 25 weeks (29.339*sine), 28 weeks (24.9872*sine), and 28 weeks (13.5326*cosine).

BOIL (ProShares Ultra Bloomberg Natur) has an average price of 137.01 (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 1/9/2017 for BOIL (ProShares Ultra Bloomberg Natur), this program was able to calculate the following Fourier Series:
Sequence #Cosine Coefficients Sine Coefficients FrequenciesPeriod
0137.0133   0 
120.9017 90.45882 (1*2π)/276276 weeks
261.57893 40.63945 (2*2π)/276138 weeks
328.92275 31.93248 (3*2π)/27692 weeks
451.34908 35.3774 (4*2π)/27669 weeks
525.91079 39.35578 (5*2π)/27655 weeks
626.85989 34.46796 (6*2π)/27646 weeks
723.87822 46.09138 (7*2π)/27639 weeks
86.88911 32.25522 (8*2π)/27635 weeks
98.78836 29.02784 (9*2π)/27631 weeks
1013.53263 24.98723 (10*2π)/27628 weeks
114.69247 29.33895 (11*2π)/27625 weeks
123.40697 22.44062 (12*2π)/27623 weeks
133.06908 13.01135 (13*2π)/27621 weeks
1410.98652 21.05232 (14*2π)/27620 weeks
154.48862 18.908 (15*2π)/27618 weeks
164.43688 18.83764 (16*2π)/27617 weeks
172.76873 17.10659 (17*2π)/27616 weeks
181.64392 14.2986 (18*2π)/27615 weeks
191.58344 13.91379 (19*2π)/27615 weeks
203.8664 11.27465 (20*2π)/27614 weeks
212.42147 12.48137 (21*2π)/27613 weeks
222.35104 13.24432 (22*2π)/27613 weeks
23-1.66379 7.15493 (23*2π)/27612 weeks
242.83413 7.36175 (24*2π)/27612 weeks
255.86672 6.87863 (25*2π)/27611 weeks
263.96802 9.02349 (26*2π)/27611 weeks
277.5633 9.31329 (27*2π)/27610 weeks
282.70125 11.28909 (28*2π)/27610 weeks
294.74625 10.17968 (29*2π)/27610 weeks
30-.41109 11.08507 (30*2π)/2769 weeks
312.04004 6.35023 (31*2π)/2769 weeks
323.41441 7.91088 (32*2π)/2769 weeks
332.60512 6.94924 (33*2π)/2768 weeks
343.16918 10.02282 (34*2π)/2768 weeks
351.61958 5.58806 (35*2π)/2768 weeks
363.4221 8.4017 (36*2π)/2768 weeks
372.81511 7.65126 (37*2π)/2767 weeks
38.81765 7.04451 (38*2π)/2767 weeks
392.01864 7.10074 (39*2π)/2767 weeks
40-.74441 4.93848 (40*2π)/2767 weeks
414.06278 2.93837 (41*2π)/2767 weeks
424.02635 6.05988 (42*2π)/2767 weeks
432.3206 5.82461 (43*2π)/2766 weeks
443.97762 5.49762 (44*2π)/2766 weeks
452.84859 5.61186 (45*2π)/2766 weeks
463.26167 6.99962 (46*2π)/2766 weeks
471.32723 5.69637 (47*2π)/2766 weeks
482.44437 6.36596 (48*2π)/2766 weeks
49-.53928 6.11369 (49*2π)/2766 weeks
50.78631 4.22208 (50*2π)/2766 weeks
511.50865 2.82659 (51*2π)/2765 weeks
522.11322 4.46414 (52*2π)/2765 weeks
532.19266 5.56586 (53*2π)/2765 weeks
54-1.19511 4.1228 (54*2π)/2765 weeks
551.51794 2.82896 (55*2π)/2765 weeks
56.78896 2.82271 (56*2π)/2765 weeks
571.33959 2.03641 (57*2π)/2765 weeks
581.66617 2.67788 (58*2π)/2765 weeks
591.55853 .86129 (59*2π)/2765 weeks
603.33818 2.73541 (60*2π)/2765 weeks
612.10627 .89752 (61*2π)/2765 weeks
624.146 3.43298 (62*2π)/2764 weeks
631.01884 3.37105 (63*2π)/2764 weeks
641.13871 -.12896 (64*2π)/2764 weeks
654.1969 .62975 (65*2π)/2764 weeks
664.69786 2.10635 (66*2π)/2764 weeks
673.99605 1.65417 (67*2π)/2764 weeks
683.9808 2.40208 (68*2π)/2764 weeks
694.74761 2.8279 (69*2π)/2764 weeks
703.4937 2.91336 (70*2π)/2764 weeks
714.9421 3.30667 (71*2π)/2764 weeks
722.58686 4.05648 (72*2π)/2764 weeks
733.16239 3.064 (73*2π)/2764 weeks
742.63303 3.81169 (74*2π)/2764 weeks
752.78513 2.98905 (75*2π)/2764 weeks
762.53277 3.7372 (76*2π)/2764 weeks
771.68126 2.07848 (77*2π)/2764 weeks
782.67166 2.0811 (78*2π)/2764 weeks
792.34815 2.45041 (79*2π)/2763 weeks
802.80219 1.00814 (80*2π)/2763 weeks
813.27802 2.12812 (81*2π)/2763 weeks
823.31186 1.36795 (82*2π)/2763 weeks
833.27526 2.40776 (83*2π)/2763 weeks
843.76165 1.65818 (84*2π)/2763 weeks
854.05152 2.81041 (85*2π)/2763 weeks
863.84871 3.10169 (86*2π)/2763 weeks
873.50458 3.36933 (87*2π)/2763 weeks
882.53493 4.49181 (88*2π)/2763 weeks
891.87681 2.38934 (89*2π)/2763 weeks
901.68695 3.88731 (90*2π)/2763 weeks
911.5282 2.47614 (91*2π)/2763 weeks
921.22848 2.46968 (92*2π)/2763 weeks
931.04198 1.70014 (93*2π)/2763 weeks
942.17032 .75197 (94*2π)/2763 weeks
951.68625 2.3795 (95*2π)/2763 weeks
961.63083 1.02306 (96*2π)/2763 weeks
97.99594 1.19649 (97*2π)/2763 weeks
981.7357 .03238 (98*2π)/2763 weeks
992.50205 .10521 (99*2π)/2763 weeks
1002.25016 .0208 (100*2π)/2763 weeks
1013.52947 .56406 (101*2π)/2763 weeks
1022.7666 .07741 (102*2π)/2763 weeks
1034.59755 .93638 (103*2π)/2763 weeks
1043.38982 1.47557 (104*2π)/2763 weeks
1053.2049 2.26139 (105*2π)/2763 weeks
1063.34017 1.12696 (106*2π)/2763 weeks
1073.03811 2.342 (107*2π)/2763 weeks
1083.48192 2.24144 (108*2π)/2763 weeks
1091.58926 2.59397 (109*2π)/2763 weeks
1101.65199 2.13355 (110*2π)/2763 weeks
1111.46836 1.07152 (111*2π)/2762 weeks
1121.50729 1.52156 (112*2π)/2762 weeks
1132.49989 .37046 (113*2π)/2762 weeks
1141.55608 1.62583 (114*2π)/2762 weeks
1151.83596 .60275 (115*2π)/2762 weeks
1161.46709 .46263 (116*2π)/2762 weeks
1172.39288 -.34182 (117*2π)/2762 weeks
1183.13265 .75051 (118*2π)/2762 weeks
1192.10031 .83186 (119*2π)/2762 weeks
1202.62457 .64594 (120*2π)/2762 weeks
1212.26968 .27303 (121*2π)/2762 weeks
1222.95318 .73643 (122*2π)/2762 weeks
1232.66349 .78459 (123*2π)/2762 weeks
1243.07129 1.00374 (124*2π)/2762 weeks
1252.22406 1.47252 (125*2π)/2762 weeks
1262.09568 .72737 (126*2π)/2762 weeks
1272.53892 1.13041 (127*2π)/2762 weeks
1281.70081 1.02661 (128*2π)/2762 weeks
1292.15081 .9316 (129*2π)/2762 weeks
1302.06522 .31656 (130*2π)/2762 weeks
1311.98269 1.12644 (131*2π)/2762 weeks
1321.72016 1.34773 (132*2π)/2762 weeks
1331.20645 -.35406 (133*2π)/2762 weeks
1341.88271 .47048 (134*2π)/2762 weeks
1351.90961 -.20112 (135*2π)/2762 weeks
1362.19317 .09625 (136*2π)/2762 weeks
1371.83842 .63373 (137*2π)/2762 weeks
1381.0458   (138*2π)/2762 weeks
1391.83842 -.63373 (139*2π)/2762 weeks
1402.19317 -.09625 (140*2π)/2762 weeks
1411.90961 .20112 (141*2π)/2762 weeks
1421.88271 -.47048 (142*2π)/2762 weeks
1431.20645 .35406 (143*2π)/2762 weeks
1441.72016 -1.34773 (144*2π)/2762 weeks
1451.98269 -1.12644 (145*2π)/2762 weeks
1462.06522 -.31656 (146*2π)/2762 weeks
1472.15081 -.9316 (147*2π)/2762 weeks
1481.70081 -1.02661 (148*2π)/2762 weeks
1492.53892 -1.13041 (149*2π)/2762 weeks
1502.09568 -.72737 (150*2π)/2762 weeks
1512.22406 -1.47252 (151*2π)/2762 weeks
1523.07129 -1.00374 (152*2π)/2762 weeks
1532.66349 -.78459 (153*2π)/2762 weeks
1542.95318 -.73643 (154*2π)/2762 weeks
1552.26968 -.27303 (155*2π)/2762 weeks
1562.62457 -.64594 (156*2π)/2762 weeks
1572.10031 -.83186 (157*2π)/2762 weeks
1583.13265 -.75051 (158*2π)/2762 weeks
1592.39288 .34182 (159*2π)/2762 weeks
1601.46709 -.46263 (160*2π)/2762 weeks
1611.83596 -.60275 (161*2π)/2762 weeks
1621.55608 -1.62583 (162*2π)/2762 weeks
1632.49989 -.37046 (163*2π)/2762 weeks
1641.50729 -1.52156 (164*2π)/2762 weeks
1651.46836 -1.07152 (165*2π)/2762 weeks
1661.65199 -2.13355 (166*2π)/2762 weeks
1671.58926 -2.59397 (167*2π)/2762 weeks
1683.48192 -2.24144 (168*2π)/2762 weeks
1693.03811 -2.342 (169*2π)/2762 weeks
1703.34017 -1.12696 (170*2π)/2762 weeks
1713.2049 -2.26139 (171*2π)/2762 weeks
1723.38982 -1.47557 (172*2π)/2762 weeks
1734.59755 -.93638 (173*2π)/2762 weeks
1742.7666 -.07741 (174*2π)/2762 weeks
1753.52947 -.56406 (175*2π)/2762 weeks
1762.25016 -.0208 (176*2π)/2762 weeks
1772.50205 -.10521 (177*2π)/2762 weeks
1781.7357 -.03238 (178*2π)/2762 weeks
179.99594 -1.19649 (179*2π)/2762 weeks
1801.63083 -1.02306 (180*2π)/2762 weeks
1811.68625 -2.3795 (181*2π)/2762 weeks
1822.17032 -.75197 (182*2π)/2762 weeks
1831.04198 -1.70014 (183*2π)/2762 weeks
1841.22848 -2.46968 (184*2π)/2762 weeks
1851.5282 -2.47614 (185*2π)/2761 weeks
1861.68695 -3.88731 (186*2π)/2761 weeks
1871.87681 -2.38934 (187*2π)/2761 weeks
1882.53493 -4.49181 (188*2π)/2761 weeks
1893.50458 -3.36933 (189*2π)/2761 weeks
1903.84871 -3.10169 (190*2π)/2761 weeks
1914.05152 -2.81041 (191*2π)/2761 weeks
1923.76165 -1.65818 (192*2π)/2761 weeks
1933.27526 -2.40776 (193*2π)/2761 weeks
1943.31186 -1.36795 (194*2π)/2761 weeks
1953.27802 -2.12812 (195*2π)/2761 weeks
1962.80219 -1.00814 (196*2π)/2761 weeks
1972.34815 -2.45041 (197*2π)/2761 weeks
1982.67166 -2.0811 (198*2π)/2761 weeks
1991.68126 -2.07848 (199*2π)/2761 weeks
2002.53277 -3.7372 (200*2π)/2761 weeks
2012.78513 -2.98905 (201*2π)/2761 weeks
2022.63303 -3.81169 (202*2π)/2761 weeks
2033.16239 -3.064 (203*2π)/2761 weeks
2042.58686 -4.05648 (204*2π)/2761 weeks
2054.9421 -3.30667 (205*2π)/2761 weeks
2063.4937 -2.91336 (206*2π)/2761 weeks
2074.74761 -2.8279 (207*2π)/2761 weeks
2083.9808 -2.40208 (208*2π)/2761 weeks
2093.99605 -1.65417 (209*2π)/2761 weeks
2104.69786 -2.10635 (210*2π)/2761 weeks
2114.1969 -.62975 (211*2π)/2761 weeks
2121.13871 .12896 (212*2π)/2761 weeks
2131.01884 -3.37105 (213*2π)/2761 weeks
2144.146 -3.43298 (214*2π)/2761 weeks
2152.10627 -.89752 (215*2π)/2761 weeks
2163.33818 -2.73541 (216*2π)/2761 weeks
2171.55853 -.86129 (217*2π)/2761 weeks
2181.66617 -2.67788 (218*2π)/2761 weeks
2191.33959 -2.03641 (219*2π)/2761 weeks
220.78896 -2.82271 (220*2π)/2761 weeks
2211.51794 -2.82896 (221*2π)/2761 weeks
222-1.19511 -4.1228 (222*2π)/2761 weeks
2232.19266 -5.56586 (223*2π)/2761 weeks
2242.11322 -4.46414 (224*2π)/2761 weeks
2251.50865 -2.82659 (225*2π)/2761 weeks
226.78631 -4.22208 (226*2π)/2761 weeks
227-.53928 -6.11369 (227*2π)/2761 weeks
2282.44437 -6.36596 (228*2π)/2761 weeks
2291.32723 -5.69637 (229*2π)/2761 weeks
2303.26167 -6.99962 (230*2π)/2761 weeks
2312.84859 -5.61186 (231*2π)/2761 weeks
2323.97762 -5.49762 (232*2π)/2761 weeks
2332.3206 -5.82461 (233*2π)/2761 weeks
2344.02635 -6.05988 (234*2π)/2761 weeks
2354.06278 -2.93837 (235*2π)/2761 weeks
236-.74441 -4.93848 (236*2π)/2761 weeks
2372.01864 -7.10074 (237*2π)/2761 weeks
238.81765 -7.04451 (238*2π)/2761 weeks
2392.81511 -7.65126 (239*2π)/2761 weeks
2403.4221 -8.4017 (240*2π)/2761 weeks
2411.61958 -5.58806 (241*2π)/2761 weeks
2423.16918 -10.02282 (242*2π)/2761 weeks
2432.60512 -6.94924 (243*2π)/2761 weeks
2443.41441 -7.91088 (244*2π)/2761 weeks
2452.04004 -6.35023 (245*2π)/2761 weeks
246-.41109 -11.08507 (246*2π)/2761 weeks
2474.74625 -10.17968 (247*2π)/2761 weeks
2482.70125 -11.28909 (248*2π)/2761 weeks
2497.5633 -9.31329 (249*2π)/2761 weeks
2503.96802 -9.02349 (250*2π)/2761 weeks
2515.86672 -6.87863 (251*2π)/2761 weeks
2522.83413 -7.36175 (252*2π)/2761 weeks
253-1.66379 -7.15493 (253*2π)/2761 weeks
2542.35104 -13.24432 (254*2π)/2761 weeks
2552.42147 -12.48137 (255*2π)/2761 weeks
2563.8664 -11.27465 (256*2π)/2761 weeks
2571.58344 -13.91379 (257*2π)/2761 weeks
2581.64392 -14.2986 (258*2π)/2761 weeks
2592.76873 -17.10659 (259*2π)/2761 weeks
2604.43688 -18.83764 (260*2π)/2761 weeks
2614.48862 -18.908 (261*2π)/2761 weeks
26210.98652 -21.05232 (262*2π)/2761 weeks
2633.06908 -13.01135 (263*2π)/2761 weeks
2643.40697 -22.44062 (264*2π)/2761 weeks
2654.69247 -29.33895 (265*2π)/2761 weeks
26613.53263 -24.98723 (266*2π)/2761 weeks
2678.78836 -29.02784 (267*2π)/2761 weeks
2686.88911 -32.25522 (268*2π)/2761 weeks
26923.87822 -46.09138 (269*2π)/2761 weeks
27026.85989 -34.46796 (270*2π)/2761 weeks
27125.91079 -39.35578 (271*2π)/2761 weeks
27251.34908 -35.3774 (272*2π)/2761 weeks
27328.92275 -31.93248 (273*2π)/2761 weeks
27461.57893 -40.63945 (274*2π)/2761 weeks

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