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Fourier Analysis of BOIL (ProShares Ultra Bloomberg Natur)


BOIL (ProShares Ultra Bloomberg Natur) appears to have interesting cyclic behaviour every 26 weeks (30.0696*sine), 29 weeks (22.8242*sine), and 20 weeks (11.2283*cosine).

BOIL (ProShares Ultra Bloomberg Natur) has an average price of 133.07 (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 3/13/2017 for BOIL (ProShares Ultra Bloomberg Natur), this program was able to calculate the following Fourier Series:
Sequence #Cosine Coefficients Sine Coefficients FrequenciesPeriod
0133.0685   0 
123.17307 93.58157 (1*2π)/285285 weeks
259.95779 35.78823 (2*2π)/285143 weeks
327.56149 34.52665 (3*2π)/28595 weeks
449.14518 29.45862 (4*2π)/28571 weeks
526.18402 41.52317 (5*2π)/28557 weeks
625.89957 34.09426 (6*2π)/28548 weeks
727.4557 41.07697 (7*2π)/28541 weeks
86.17493 36.38248 (8*2π)/28536 weeks
99.85061 28.78865 (9*2π)/28532 weeks
1010.07664 22.82424 (10*2π)/28529 weeks
118.76893 30.06959 (11*2π)/28526 weeks
122.68284 22.80015 (12*2π)/28524 weeks
13.24681 18.45927 (13*2π)/28522 weeks
1411.22828 14.59908 (14*2π)/28520 weeks
155.32736 21.11632 (15*2π)/28519 weeks
164.99759 17.9332 (16*2π)/28518 weeks
172.96657 17.66804 (17*2π)/28517 weeks
181.68754 16.22065 (18*2π)/28516 weeks
192.38528 13.74638 (19*2π)/28515 weeks
201.1366 12.15097 (20*2π)/28514 weeks
214.02316 12.49838 (21*2π)/28514 weeks
222.57764 11.86004 (22*2π)/28513 weeks
23.66227 13.42482 (23*2π)/28512 weeks
24.71219 5.79581 (24*2π)/28512 weeks
252.92135 6.78842 (25*2π)/28511 weeks
266.24522 7.50808 (26*2π)/28511 weeks
273.89683 8.10421 (27*2π)/28511 weeks
287.3974 9.87454 (28*2π)/28510 weeks
292.52669 10.5069 (29*2π)/28510 weeks
304.67175 10.12264 (30*2π)/28510 weeks
31-.44202 10.61906 (31*2π)/2859 weeks
321.98572 6.19129 (32*2π)/2859 weeks
333.39227 7.65189 (33*2π)/2859 weeks
342.43769 6.97643 (34*2π)/2858 weeks
353.80751 9.45237 (35*2π)/2858 weeks
36.99248 5.91405 (36*2π)/2858 weeks
373.99922 7.70628 (37*2π)/2858 weeks
382.61739 7.10575 (38*2π)/2858 weeks
391.06376 7.5167 (39*2π)/2857 weeks
402.13179 6.23215 (40*2π)/2857 weeks
41-.29934 6.25124 (41*2π)/2857 weeks
422.16378 2.05607 (42*2π)/2857 weeks
434.38147 5.19577 (43*2π)/2857 weeks
442.8094 6.14148 (44*2π)/2856 weeks
452.94806 4.56242 (45*2π)/2856 weeks
462.8756 6.13666 (46*2π)/2856 weeks
473.44049 5.77035 (47*2π)/2856 weeks
481.7052 6.97855 (48*2π)/2856 weeks
492.16849 5.32268 (49*2π)/2856 weeks
501.55936 6.97919 (50*2π)/2856 weeks
51-.29209 4.36738 (51*2π)/2856 weeks
52.55869 3.9758 (52*2π)/2855 weeks
532.46234 3.19244 (53*2π)/2855 weeks
542.07574 4.17147 (54*2π)/2855 weeks
551.35511 5.85421 (55*2π)/2855 weeks
56-.85814 2.79997 (56*2π)/2855 weeks
571.37516 3.0933 (57*2π)/2855 weeks
58.79573 2.57603 (58*2π)/2855 weeks
591.45201 1.94007 (59*2π)/2855 weeks
601.44887 2.59981 (60*2π)/2855 weeks
611.72587 .73652 (61*2π)/2855 weeks
623.12436 2.76426 (62*2π)/2855 weeks
632.08074 .86004 (63*2π)/2855 weeks
644.07364 3.26956 (64*2π)/2854 weeks
651.11128 3.39514 (65*2π)/2854 weeks
66.85091 .16053 (66*2π)/2854 weeks
674.03149 .3975 (67*2π)/2854 weeks
684.64919 1.68767 (68*2π)/2854 weeks
693.59567 1.73284 (69*2π)/2854 weeks
704.03905 2.42824 (70*2π)/2854 weeks
714.53422 2.31107 (71*2π)/2854 weeks
723.5906 3.21485 (72*2π)/2854 weeks
734.42401 2.48404 (73*2π)/2854 weeks
743.35343 4.52083 (74*2π)/2854 weeks
752.83148 2.95437 (75*2π)/2854 weeks
763.24267 3.48156 (76*2π)/2854 weeks
772.38461 2.83208 (77*2π)/2854 weeks
782.79117 3.03489 (78*2π)/2854 weeks
791.3443 3.05619 (79*2π)/2854 weeks
802.20306 1.7928 (80*2π)/2854 weeks
812.54987 2.18409 (81*2π)/2854 weeks
821.77865 1.87172 (82*2π)/2853 weeks
833.28086 1.40354 (83*2π)/2853 weeks
842.79294 2.05333 (84*2π)/2853 weeks
853.68142 1.66141 (85*2π)/2853 weeks
862.94541 2.07095 (86*2π)/2853 weeks
873.9288 1.83017 (87*2π)/2853 weeks
883.73356 2.82262 (88*2π)/2853 weeks
893.62904 3.18697 (89*2π)/2853 weeks
903.42497 3.4085 (90*2π)/2853 weeks
912.09633 4.35827 (91*2π)/2853 weeks
922.08633 2.32372 (92*2π)/2853 weeks
931.46289 3.73135 (93*2π)/2853 weeks
941.50171 2.44112 (94*2π)/2853 weeks
951.19793 2.42031 (95*2π)/2853 weeks
961.01851 1.69164 (96*2π)/2853 weeks
971.95424 .69994 (97*2π)/2853 weeks
981.84774 2.33381 (98*2π)/2853 weeks
991.37746 1.00364 (99*2π)/2853 weeks
1001.10535 1.408 (100*2π)/2853 weeks
1011.43951 .10671 (101*2π)/2853 weeks
1022.21091 .07503 (102*2π)/2853 weeks
1032.17163 .42644 (103*2π)/2853 weeks
1043.34521 .09472 (104*2π)/2853 weeks
1052.60446 .69572 (105*2π)/2853 weeks
1064.12107 .11473 (106*2π)/2853 weeks
1073.56315 1.84237 (107*2π)/2853 weeks
1083.70453 1.81011 (108*2π)/2853 weeks
1092.58203 1.47776 (109*2π)/2853 weeks
1103.64628 1.97718 (110*2π)/2853 weeks
1112.84237 1.83641 (111*2π)/2853 weeks
1122.7755 3.14402 (112*2π)/2853 weeks
1131.77712 1.99014 (113*2π)/2853 weeks
1141.32196 1.7997 (114*2π)/2853 weeks
1151.93057 1.05106 (115*2π)/2852 weeks
1161.19823 .89657 (116*2π)/2852 weeks
1172.76946 1.04662 (117*2π)/2852 weeks
1181.30333 1.11589 (118*2π)/2852 weeks
1191.87634 .6954 (119*2π)/2852 weeks
1201.48897 .17213 (120*2π)/2852 weeks
1212.60102 -.28962 (121*2π)/2852 weeks
1222.95018 .88718 (122*2π)/2852 weeks
1232.03597 .65758 (123*2π)/2852 weeks
1242.5261 .67741 (124*2π)/2852 weeks
1252.24807 .21811 (125*2π)/2852 weeks
1262.85051 .73244 (126*2π)/2852 weeks
1272.58128 .77545 (127*2π)/2852 weeks
1282.97222 .94752 (128*2π)/2852 weeks
1292.23011 1.48652 (129*2π)/2852 weeks
1301.98655 .79322 (130*2π)/2852 weeks
1312.51549 1.01341 (131*2π)/2852 weeks
1321.76719 1.20234 (132*2π)/2852 weeks
1332.16322 .76177 (133*2π)/2852 weeks
1341.76188 .46692 (134*2π)/2852 weeks
1352.2756 1.07653 (135*2π)/2852 weeks
1361.98455 1.00269 (136*2π)/2852 weeks
137.7033 .31698 (137*2π)/2852 weeks
1382.12576 .1337 (138*2π)/2852 weeks
1391.52688 -.012 (139*2π)/2852 weeks
1402.07206 .05343 (140*2π)/2852 weeks
1412.16115 .48317 (141*2π)/2852 weeks
1421.39101 .54304 (142*2π)/2852 weeks
1431.39101 -.54304 (143*2π)/2852 weeks
1442.16115 -.48317 (144*2π)/2852 weeks
1452.07206 -.05343 (145*2π)/2852 weeks
1461.52688 .012 (146*2π)/2852 weeks
1472.12576 -.1337 (147*2π)/2852 weeks
148.7033 -.31698 (148*2π)/2852 weeks
1491.98455 -1.00269 (149*2π)/2852 weeks
1502.2756 -1.07653 (150*2π)/2852 weeks
1511.76188 -.46692 (151*2π)/2852 weeks
1522.16322 -.76177 (152*2π)/2852 weeks
1531.76719 -1.20234 (153*2π)/2852 weeks
1542.51549 -1.01341 (154*2π)/2852 weeks
1551.98655 -.79322 (155*2π)/2852 weeks
1562.23011 -1.48652 (156*2π)/2852 weeks
1572.97222 -.94752 (157*2π)/2852 weeks
1582.58128 -.77545 (158*2π)/2852 weeks
1592.85051 -.73244 (159*2π)/2852 weeks
1602.24807 -.21811 (160*2π)/2852 weeks
1612.5261 -.67741 (161*2π)/2852 weeks
1622.03597 -.65758 (162*2π)/2852 weeks
1632.95018 -.88718 (163*2π)/2852 weeks
1642.60102 .28962 (164*2π)/2852 weeks
1651.48897 -.17213 (165*2π)/2852 weeks
1661.87634 -.6954 (166*2π)/2852 weeks
1671.30333 -1.11589 (167*2π)/2852 weeks
1682.76946 -1.04662 (168*2π)/2852 weeks
1691.19823 -.89657 (169*2π)/2852 weeks
1701.93057 -1.05106 (170*2π)/2852 weeks
1711.32196 -1.7997 (171*2π)/2852 weeks
1721.77712 -1.99014 (172*2π)/2852 weeks
1732.7755 -3.14402 (173*2π)/2852 weeks
1742.84237 -1.83641 (174*2π)/2852 weeks
1753.64628 -1.97718 (175*2π)/2852 weeks
1762.58203 -1.47776 (176*2π)/2852 weeks
1773.70453 -1.81011 (177*2π)/2852 weeks
1783.56315 -1.84237 (178*2π)/2852 weeks
1794.12107 -.11473 (179*2π)/2852 weeks
1802.60446 -.69572 (180*2π)/2852 weeks
1813.34521 -.09472 (181*2π)/2852 weeks
1822.17163 -.42644 (182*2π)/2852 weeks
1832.21091 -.07503 (183*2π)/2852 weeks
1841.43951 -.10671 (184*2π)/2852 weeks
1851.10535 -1.408 (185*2π)/2852 weeks
1861.37746 -1.00364 (186*2π)/2852 weeks
1871.84774 -2.33381 (187*2π)/2852 weeks
1881.95424 -.69994 (188*2π)/2852 weeks
1891.01851 -1.69164 (189*2π)/2852 weeks
1901.19793 -2.42031 (190*2π)/2852 weeks
1911.50171 -2.44112 (191*2π)/2851 weeks
1921.46289 -3.73135 (192*2π)/2851 weeks
1932.08633 -2.32372 (193*2π)/2851 weeks
1942.09633 -4.35827 (194*2π)/2851 weeks
1953.42497 -3.4085 (195*2π)/2851 weeks
1963.62904 -3.18697 (196*2π)/2851 weeks
1973.73356 -2.82262 (197*2π)/2851 weeks
1983.9288 -1.83017 (198*2π)/2851 weeks
1992.94541 -2.07095 (199*2π)/2851 weeks
2003.68142 -1.66141 (200*2π)/2851 weeks
2012.79294 -2.05333 (201*2π)/2851 weeks
2023.28086 -1.40354 (202*2π)/2851 weeks
2031.77865 -1.87172 (203*2π)/2851 weeks
2042.54987 -2.18409 (204*2π)/2851 weeks
2052.20306 -1.7928 (205*2π)/2851 weeks
2061.3443 -3.05619 (206*2π)/2851 weeks
2072.79117 -3.03489 (207*2π)/2851 weeks
2082.38461 -2.83208 (208*2π)/2851 weeks
2093.24267 -3.48156 (209*2π)/2851 weeks
2102.83148 -2.95437 (210*2π)/2851 weeks
2113.35343 -4.52083 (211*2π)/2851 weeks
2124.42401 -2.48404 (212*2π)/2851 weeks
2133.5906 -3.21485 (213*2π)/2851 weeks
2144.53422 -2.31107 (214*2π)/2851 weeks
2154.03905 -2.42824 (215*2π)/2851 weeks
2163.59567 -1.73284 (216*2π)/2851 weeks
2174.64919 -1.68767 (217*2π)/2851 weeks
2184.03149 -.3975 (218*2π)/2851 weeks
219.85091 -.16053 (219*2π)/2851 weeks
2201.11128 -3.39514 (220*2π)/2851 weeks
2214.07364 -3.26956 (221*2π)/2851 weeks
2222.08074 -.86004 (222*2π)/2851 weeks
2233.12436 -2.76426 (223*2π)/2851 weeks
2241.72587 -.73652 (224*2π)/2851 weeks
2251.44887 -2.59981 (225*2π)/2851 weeks
2261.45201 -1.94007 (226*2π)/2851 weeks
227.79573 -2.57603 (227*2π)/2851 weeks
2281.37516 -3.0933 (228*2π)/2851 weeks
229-.85814 -2.79997 (229*2π)/2851 weeks
2301.35511 -5.85421 (230*2π)/2851 weeks
2312.07574 -4.17147 (231*2π)/2851 weeks
2322.46234 -3.19244 (232*2π)/2851 weeks
233.55869 -3.9758 (233*2π)/2851 weeks
234-.29209 -4.36738 (234*2π)/2851 weeks
2351.55936 -6.97919 (235*2π)/2851 weeks
2362.16849 -5.32268 (236*2π)/2851 weeks
2371.7052 -6.97855 (237*2π)/2851 weeks
2383.44049 -5.77035 (238*2π)/2851 weeks
2392.8756 -6.13666 (239*2π)/2851 weeks
2402.94806 -4.56242 (240*2π)/2851 weeks
2412.8094 -6.14148 (241*2π)/2851 weeks
2424.38147 -5.19577 (242*2π)/2851 weeks
2432.16378 -2.05607 (243*2π)/2851 weeks
244-.29934 -6.25124 (244*2π)/2851 weeks
2452.13179 -6.23215 (245*2π)/2851 weeks
2461.06376 -7.5167 (246*2π)/2851 weeks
2472.61739 -7.10575 (247*2π)/2851 weeks
2483.99922 -7.70628 (248*2π)/2851 weeks
249.99248 -5.91405 (249*2π)/2851 weeks
2503.80751 -9.45237 (250*2π)/2851 weeks
2512.43769 -6.97643 (251*2π)/2851 weeks
2523.39227 -7.65189 (252*2π)/2851 weeks
2531.98572 -6.19129 (253*2π)/2851 weeks
254-.44202 -10.61906 (254*2π)/2851 weeks
2554.67175 -10.12264 (255*2π)/2851 weeks
2562.52669 -10.5069 (256*2π)/2851 weeks
2577.3974 -9.87454 (257*2π)/2851 weeks
2583.89683 -8.10421 (258*2π)/2851 weeks
2596.24522 -7.50808 (259*2π)/2851 weeks
2602.92135 -6.78842 (260*2π)/2851 weeks
261.71219 -5.79581 (261*2π)/2851 weeks
262.66227 -13.42482 (262*2π)/2851 weeks
2632.57764 -11.86004 (263*2π)/2851 weeks
2644.02316 -12.49838 (264*2π)/2851 weeks
2651.1366 -12.15097 (265*2π)/2851 weeks
2662.38528 -13.74638 (266*2π)/2851 weeks
2671.68754 -16.22065 (267*2π)/2851 weeks
2682.96657 -17.66804 (268*2π)/2851 weeks
2694.99759 -17.9332 (269*2π)/2851 weeks
2705.32736 -21.11632 (270*2π)/2851 weeks
27111.22828 -14.59908 (271*2π)/2851 weeks
272.24681 -18.45927 (272*2π)/2851 weeks
2732.68284 -22.80015 (273*2π)/2851 weeks
2748.76893 -30.06959 (274*2π)/2851 weeks
27510.07664 -22.82424 (275*2π)/2851 weeks
2769.85061 -28.78865 (276*2π)/2851 weeks
2776.17493 -36.38248 (277*2π)/2851 weeks
27827.4557 -41.07697 (278*2π)/2851 weeks
27925.89957 -34.09426 (279*2π)/2851 weeks
28026.18402 -41.52317 (280*2π)/2851 weeks
28149.14518 -29.45862 (281*2π)/2851 weeks
28227.56149 -34.52665 (282*2π)/2851 weeks
28359.95779 -35.78823 (283*2π)/2851 weeks

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