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

BOIL (ProShares Ultra Bloomberg Natural Gas) appears to have interesting cyclic behaviour every 34 weeks (128.3759*sine), 26 weeks (122.6945*sine), and 31 weeks (120.4057*sine).

BOIL (ProShares Ultra Bloomberg Natural Gas) has an average price of 559.83 (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 4/23/2018 for BOIL (ProShares Ultra Bloomberg Natural Gas), this program was able to calculate the following Fourier Series:
Sequence #Cosine Coefficients Sine Coefficients FrequenciesPeriod
0559.8308   0
1198.3281 509.461 (1*2π)/343343 weeks
2185.8411 97.31962 (2*2π)/343172 weeks
3171.9211 214.396 (3*2π)/343114 weeks
4131.5422 106.0266 (4*2π)/34386 weeks
5206.4071 149.0765 (5*2π)/34369 weeks
6110.0562 175.8408 (6*2π)/34357 weeks
7109.6396 144.7746 (7*2π)/34349 weeks
8118.9321 146.9363 (8*2π)/34343 weeks
967.16843 190.5703 (9*2π)/34338 weeks
1029.2491 128.3759 (10*2π)/34334 weeks
1138.9017 120.4057 (11*2π)/34331 weeks
1241.20558 96.19067 (12*2π)/34329 weeks
1347.85291 122.6945 (13*2π)/34326 weeks
1412.16947 107.9884 (14*2π)/34325 weeks
1513.21049 90.50465 (15*2π)/34323 weeks
165.3459 58.99908 (16*2π)/34321 weeks
1748.87244 68.57395 (17*2π)/34320 weeks
1823.55895 89.64388 (18*2π)/34319 weeks
1920.6984 75.02709 (19*2π)/34318 weeks
2017.39855 76.39756 (20*2π)/34317 weeks
2111.67496 69.36304 (21*2π)/34316 weeks
224.87192 62.81752 (22*2π)/34316 weeks
239.82433 57.8719 (23*2π)/34315 weeks
244.82799 52.19569 (24*2π)/34314 weeks
2517.56503 48.12883 (25*2π)/34314 weeks
2610.37059 51.3089 (26*2π)/34313 weeks
2711.53849 50.46725 (27*2π)/34313 weeks
28-6.49555 51.08244 (28*2π)/34312 weeks
296.39763 24.36175 (29*2π)/34312 weeks
3011.85239 28.93256 (30*2π)/34311 weeks
3122.95679 27.20251 (31*2π)/34311 weeks
3219.66893 39.6281 (32*2π)/34311 weeks
3323.07041 29.91874 (33*2π)/34310 weeks
3427.30051 47.77944 (34*2π)/34310 weeks
3510.30767 42.09574 (35*2π)/34310 weeks
3619.02531 40.86761 (36*2π)/34310 weeks
374.56598 50.32128 (37*2π)/3439 weeks
38.86627 28.67609 (38*2π)/3439 weeks
3912.94843 27.72829 (39*2π)/3439 weeks
4013.19998 33.15283 (40*2π)/3439 weeks
4110.8735 28.62867 (41*2π)/3438 weeks
4218.20681 37.5159 (42*2π)/3438 weeks
431.78006 30.61616 (43*2π)/3438 weeks
4415.46161 23.65076 (44*2π)/3438 weeks
4510.44576 33.82197 (45*2π)/3438 weeks
4611.60406 31.34203 (46*2π)/3437 weeks
474.31426 30.8889 (47*2π)/3437 weeks
488.37954 25.01583 (48*2π)/3437 weeks
492.46388 29.29839 (49*2π)/3437 weeks
50-1.36022 14.38146 (50*2π)/3437 weeks
5116.92879 12.72834 (51*2π)/3437 weeks
5217.2545 23.6051 (52*2π)/3437 weeks
5311.49492 25.64849 (53*2π)/3436 weeks
5410.88386 19.59491 (54*2π)/3436 weeks
5514.97594 25.22608 (55*2π)/3436 weeks
5611.92043 22.58709 (56*2π)/3436 weeks
5714.05022 27.16289 (57*2π)/3436 weeks
585.23902 27.23643 (58*2π)/3436 weeks
599.16955 22.42708 (59*2π)/3436 weeks
608.28711 28.27718 (60*2π)/3436 weeks
61-2.42102 23.25385 (61*2π)/3436 weeks
623.10206 16.89217 (62*2π)/3436 weeks
632.70018 13.57901 (63*2π)/3435 weeks
6410.92548 15.52168 (64*2π)/3435 weeks
658.61207 17.59041 (65*2π)/3435 weeks
667.77739 23.72811 (66*2π)/3435 weeks
67-4.23933 18.91758 (67*2π)/3435 weeks
684.47496 9.14939 (68*2π)/3435 weeks
694.34062 13.3216 (69*2π)/3435 weeks
703.64823 10.21316 (70*2π)/3435 weeks
716.05225 8.24992 (71*2π)/3435 weeks
727.20591 10.9934 (72*2π)/3435 weeks
734.48151 6.21569 (73*2π)/3435 weeks
7413.87055 4.81472 (74*2π)/3435 weeks
759.00464 12.07813 (75*2π)/3435 weeks
7611.24185 2.71478 (76*2π)/3435 weeks
7717.22049 13.30484 (77*2π)/3434 weeks
786.63219 15.51957 (78*2π)/3434 weeks
791.52684 6.20742 (79*2π)/3434 weeks
8011.21828 -2.56357 (80*2π)/3434 weeks
8117.09867 3.50611 (81*2π)/3434 weeks
8219.14625 8.41111 (82*2π)/3434 weeks
8314.9104 7.42562 (83*2π)/3434 weeks
8417.77261 9.63597 (84*2π)/3434 weeks
8516.99685 8.49764 (85*2π)/3434 weeks
8618.51392 12.85183 (86*2π)/3434 weeks
8714.11326 11.73694 (87*2π)/3434 weeks
8819.20689 11.28494 (88*2π)/3434 weeks
8914.59838 18.94665 (89*2π)/3434 weeks
9010.83507 12.94636 (90*2π)/3434 weeks
9113.37755 12.86148 (91*2π)/3434 weeks
9210.50903 15.4405 (92*2π)/3434 weeks
9311.15917 11.88437 (93*2π)/3434 weeks
9411.86394 13.16573 (94*2π)/3434 weeks
956.05489 13.51659 (95*2π)/3434 weeks
968.29884 7.81562 (96*2π)/3434 weeks
9711.05071 8.6363 (97*2π)/3434 weeks
9810.38316 9.74757 (98*2π)/3434 weeks
998.3186 5.23654 (99*2π)/3433 weeks
10013.92714 6.23721 (100*2π)/3433 weeks
10112.0189 8.70862 (101*2π)/3433 weeks
10213.99899 5.67284 (102*2π)/3433 weeks
10313.94085 9.82563 (103*2π)/3433 weeks
10413.36084 6.73215 (104*2π)/3433 weeks
10516.95892 8.81425 (105*2π)/3433 weeks
10615.31707 11.81511 (106*2π)/3433 weeks
10715.41784 13.11735 (107*2π)/3433 weeks
10814.20629 13.60321 (108*2π)/3433 weeks
10913.07207 17.06377 (109*2π)/3433 weeks
1105.21329 14.71819 (110*2π)/3433 weeks
11110.85036 10.81313 (111*2π)/3433 weeks
1125.52074 15.0813 (112*2π)/3433 weeks
1136.03605 9.92935 (113*2π)/3433 weeks
1145.47064 10.25113 (114*2π)/3433 weeks
1153.92296 8.32906 (115*2π)/3433 weeks
1164.03866 5.59561 (116*2π)/3433 weeks
11710.03834 4.17936 (117*2π)/3433 weeks
1187.35307 9.76406 (118*2π)/3433 weeks
1194.98209 4.55469 (119*2π)/3433 weeks
1206.29837 6.50171 (120*2π)/3433 weeks
1214.06254 2.5209 (121*2π)/3433 weeks
1227.70568 .43271 (122*2π)/3433 weeks
12310.03057 .61157 (123*2π)/3433 weeks
1249.08316 1.75128 (124*2π)/3433 weeks
12513.42122 -.31951 (125*2π)/3433 weeks
12612.64233 3.90264 (126*2π)/3433 weeks
12712.78661 -.73638 (127*2π)/3433 weeks
12818.60639 3.77488 (128*2π)/3433 weeks
12913.83927 6.98139 (129*2π)/3433 weeks
13015.4247 7.62646 (130*2π)/3433 weeks
13110.74083 7.21468 (131*2π)/3433 weeks
13214.98355 5.62538 (132*2π)/3433 weeks
13312.17945 9.45178 (133*2π)/3433 weeks
13413.45994 7.94752 (134*2π)/3433 weeks
1359.50157 13.26449 (135*2π)/3433 weeks
1367.48539 8.28171 (136*2π)/3433 weeks
1375.96879 8.05775 (137*2π)/3433 weeks
1386.20743 4.2151 (138*2π)/3432 weeks
1397.16559 6.46958 (139*2π)/3432 weeks
1406.73056 1.39279 (140*2π)/3432 weeks
14111.1173 5.95593 (141*2π)/3432 weeks
1425.4876 4.72926 (142*2π)/3432 weeks
1437.6019 2.56239 (143*2π)/3432 weeks
1446.10748 2.66452 (144*2π)/3432 weeks
1458.02395 -.72771 (145*2π)/3432 weeks
14612.60269 .18841 (146*2π)/3432 weeks
14711.98222 4.24785 (147*2π)/3432 weeks
1488.61023 2.83102 (148*2π)/3432 weeks
14910.68836 2.3031 (149*2π)/3432 weeks
1509.01257 2.5254 (150*2π)/3432 weeks
15111.85015 1.01612 (151*2π)/3432 weeks
15211.45437 3.34814 (152*2π)/3432 weeks
15310.87647 3.13243 (153*2π)/3432 weeks
15412.58218 3.85924 (154*2π)/3432 weeks
15510.61458 6.31289 (155*2π)/3432 weeks
1568.37508 4.09543 (156*2π)/3432 weeks
1579.66818 2.30637 (157*2π)/3432 weeks
1589.86031 4.72246 (158*2π)/3432 weeks
1596.8794 4.43738 (159*2π)/3432 weeks
1608.91902 2.97137 (160*2π)/3432 weeks
1617.07469 2.9049 (161*2π)/3432 weeks
16210.08559 2.2535 (162*2π)/3432 weeks
1637.49068 4.04677 (163*2π)/3432 weeks
1647.16977 5.4094 (164*2π)/3432 weeks
1653.2428 .40168 (165*2π)/3432 weeks
1668.87453 .1886 (166*2π)/3432 weeks
1676.31823 1.01589 (167*2π)/3432 weeks
1688.37366 -.52611 (168*2π)/3432 weeks
1698.89885 .41279 (169*2π)/3432 weeks
1708.22679 2.56216 (170*2π)/3432 weeks
1715.52621 2.10234 (171*2π)/3432 weeks
1725.52621 -2.10234 (172*2π)/3432 weeks
1738.22679 -2.56216 (173*2π)/3432 weeks
1748.89885 -.41279 (174*2π)/3432 weeks
1758.37366 .52611 (175*2π)/3432 weeks
1766.31823 -1.01589 (176*2π)/3432 weeks
1778.87453 -.1886 (177*2π)/3432 weeks
1783.2428 -.40168 (178*2π)/3432 weeks
1797.16977 -5.4094 (179*2π)/3432 weeks
1807.49068 -4.04677 (180*2π)/3432 weeks
18110.08559 -2.2535 (181*2π)/3432 weeks
1827.07469 -2.9049 (182*2π)/3432 weeks
1838.91902 -2.97137 (183*2π)/3432 weeks
1846.8794 -4.43738 (184*2π)/3432 weeks
1859.86031 -4.72246 (185*2π)/3432 weeks
1869.66818 -2.30637 (186*2π)/3432 weeks
1878.37508 -4.09543 (187*2π)/3432 weeks
18810.61458 -6.31289 (188*2π)/3432 weeks
18912.58218 -3.85924 (189*2π)/3432 weeks
19010.87647 -3.13243 (190*2π)/3432 weeks
19111.45437 -3.34814 (191*2π)/3432 weeks
19211.85015 -1.01612 (192*2π)/3432 weeks
1939.01257 -2.5254 (193*2π)/3432 weeks
19410.68836 -2.3031 (194*2π)/3432 weeks
1958.61023 -2.83102 (195*2π)/3432 weeks
19611.98222 -4.24785 (196*2π)/3432 weeks
19712.60269 -.18841 (197*2π)/3432 weeks
1988.02395 .72771 (198*2π)/3432 weeks
1996.10748 -2.66452 (199*2π)/3432 weeks
2007.6019 -2.56239 (200*2π)/3432 weeks
2015.4876 -4.72926 (201*2π)/3432 weeks
20211.1173 -5.95593 (202*2π)/3432 weeks
2036.73056 -1.39279 (203*2π)/3432 weeks
2047.16559 -6.46958 (204*2π)/3432 weeks
2056.20743 -4.2151 (205*2π)/3432 weeks
2065.96879 -8.05775 (206*2π)/3432 weeks
2077.48539 -8.28171 (207*2π)/3432 weeks
2089.50157 -13.26449 (208*2π)/3432 weeks
20913.45994 -7.94752 (209*2π)/3432 weeks
21012.17945 -9.45178 (210*2π)/3432 weeks
21114.98355 -5.62538 (211*2π)/3432 weeks
21210.74083 -7.21468 (212*2π)/3432 weeks
21315.4247 -7.62646 (213*2π)/3432 weeks
21413.83927 -6.98139 (214*2π)/3432 weeks
21518.60639 -3.77488 (215*2π)/3432 weeks
21612.78661 .73638 (216*2π)/3432 weeks
21712.64233 -3.90264 (217*2π)/3432 weeks
21813.42122 .31951 (218*2π)/3432 weeks
2199.08316 -1.75128 (219*2π)/3432 weeks
22010.03057 -.61157 (220*2π)/3432 weeks
2217.70568 -.43271 (221*2π)/3432 weeks
2224.06254 -2.5209 (222*2π)/3432 weeks
2236.29837 -6.50171 (223*2π)/3432 weeks
2244.98209 -4.55469 (224*2π)/3432 weeks
2257.35307 -9.76406 (225*2π)/3432 weeks
22610.03834 -4.17936 (226*2π)/3432 weeks
2274.03866 -5.59561 (227*2π)/3432 weeks
2283.92296 -8.32906 (228*2π)/3432 weeks
2295.47064 -10.25113 (229*2π)/3431 weeks
2306.03605 -9.92935 (230*2π)/3431 weeks
2315.52074 -15.0813 (231*2π)/3431 weeks
23210.85036 -10.81313 (232*2π)/3431 weeks
2335.21329 -14.71819 (233*2π)/3431 weeks
23413.07207 -17.06377 (234*2π)/3431 weeks
23514.20629 -13.60321 (235*2π)/3431 weeks
23615.41784 -13.11735 (236*2π)/3431 weeks
23715.31707 -11.81511 (237*2π)/3431 weeks
23816.95892 -8.81425 (238*2π)/3431 weeks
23913.36084 -6.73215 (239*2π)/3431 weeks
24013.94085 -9.82563 (240*2π)/3431 weeks
24113.99899 -5.67284 (241*2π)/3431 weeks
24212.0189 -8.70862 (242*2π)/3431 weeks
24313.92714 -6.23721 (243*2π)/3431 weeks
2448.3186 -5.23654 (244*2π)/3431 weeks
24510.38316 -9.74757 (245*2π)/3431 weeks
24611.05071 -8.6363 (246*2π)/3431 weeks
2478.29884 -7.81562 (247*2π)/3431 weeks
2486.05489 -13.51659 (248*2π)/3431 weeks
24911.86394 -13.16573 (249*2π)/3431 weeks
25011.15917 -11.88437 (250*2π)/3431 weeks
25110.50903 -15.4405 (251*2π)/3431 weeks
25213.37755 -12.86148 (252*2π)/3431 weeks
25310.83507 -12.94636 (253*2π)/3431 weeks
25414.59838 -18.94665 (254*2π)/3431 weeks
25519.20689 -11.28494 (255*2π)/3431 weeks
25614.11326 -11.73694 (256*2π)/3431 weeks
25718.51392 -12.85183 (257*2π)/3431 weeks
25816.99685 -8.49764 (258*2π)/3431 weeks
25917.77261 -9.63597 (259*2π)/3431 weeks
26014.9104 -7.42562 (260*2π)/3431 weeks
26119.14625 -8.41111 (261*2π)/3431 weeks
26217.09867 -3.50611 (262*2π)/3431 weeks
26311.21828 2.56357 (263*2π)/3431 weeks
2641.52684 -6.20742 (264*2π)/3431 weeks
2656.63219 -15.51957 (265*2π)/3431 weeks
26617.22049 -13.30484 (266*2π)/3431 weeks
26711.24185 -2.71478 (267*2π)/3431 weeks
2689.00464 -12.07813 (268*2π)/3431 weeks
26913.87055 -4.81472 (269*2π)/3431 weeks
2704.48151 -6.21569 (270*2π)/3431 weeks
2717.20591 -10.9934 (271*2π)/3431 weeks
2726.05225 -8.24992 (272*2π)/3431 weeks
2733.64823 -10.21316 (273*2π)/3431 weeks
2744.34062 -13.3216 (274*2π)/3431 weeks
2754.47496 -9.14939 (275*2π)/3431 weeks
276-4.23933 -18.91758 (276*2π)/3431 weeks
2777.77739 -23.72811 (277*2π)/3431 weeks
2788.61207 -17.59041 (278*2π)/3431 weeks
27910.92548 -15.52168 (279*2π)/3431 weeks
2802.70018 -13.57901 (280*2π)/3431 weeks
2813.10206 -16.89217 (281*2π)/3431 weeks
282-2.42102 -23.25385 (282*2π)/3431 weeks
2838.28711 -28.27718 (283*2π)/3431 weeks
2849.16955 -22.42708 (284*2π)/3431 weeks
2855.23902 -27.23643 (285*2π)/3431 weeks
28614.05022 -27.16289 (286*2π)/3431 weeks
28711.92043 -22.58709 (287*2π)/3431 weeks
28814.97594 -25.22608 (288*2π)/3431 weeks
28910.88386 -19.59491 (289*2π)/3431 weeks
29011.49492 -25.64849 (290*2π)/3431 weeks
29117.2545 -23.6051 (291*2π)/3431 weeks
29216.92879 -12.72834 (292*2π)/3431 weeks
293-1.36022 -14.38146 (293*2π)/3431 weeks
2942.46388 -29.29839 (294*2π)/3431 weeks
2958.37954 -25.01583 (295*2π)/3431 weeks
2964.31426 -30.8889 (296*2π)/3431 weeks
29711.60406 -31.34203 (297*2π)/3431 weeks
29810.44576 -33.82197 (298*2π)/3431 weeks
29915.46161 -23.65076 (299*2π)/3431 weeks
3001.78006 -30.61616 (300*2π)/3431 weeks
30118.20681 -37.5159 (301*2π)/3431 weeks
30210.8735 -28.62867 (302*2π)/3431 weeks
30313.19998 -33.15283 (303*2π)/3431 weeks
30412.94843 -27.72829 (304*2π)/3431 weeks
305.86627 -28.67609 (305*2π)/3431 weeks
3064.56598 -50.32128 (306*2π)/3431 weeks
30719.02531 -40.86761 (307*2π)/3431 weeks
30810.30767 -42.09574 (308*2π)/3431 weeks
30927.30051 -47.77944 (309*2π)/3431 weeks
31023.07041 -29.91874 (310*2π)/3431 weeks
31119.66893 -39.6281 (311*2π)/3431 weeks
31222.95679 -27.20251 (312*2π)/3431 weeks
31311.85239 -28.93256 (313*2π)/3431 weeks
3146.39763 -24.36175 (314*2π)/3431 weeks
315-6.49555 -51.08244 (315*2π)/3431 weeks
31611.53849 -50.46725 (316*2π)/3431 weeks
31710.37059 -51.3089 (317*2π)/3431 weeks
31817.56503 -48.12883 (318*2π)/3431 weeks
3194.82799 -52.19569 (319*2π)/3431 weeks
3209.82433 -57.8719 (320*2π)/3431 weeks
3214.87192 -62.81752 (321*2π)/3431 weeks
32211.67496 -69.36304 (322*2π)/3431 weeks
32317.39855 -76.39756 (323*2π)/3431 weeks
32420.6984 -75.02709 (324*2π)/3431 weeks
32523.55895 -89.64388 (325*2π)/3431 weeks
32648.87244 -68.57395 (326*2π)/3431 weeks
3275.3459 -58.99908 (327*2π)/3431 weeks
32813.21049 -90.50465 (328*2π)/3431 weeks
32912.16947 -107.9884 (329*2π)/3431 weeks
33047.85291 -122.6945 (330*2π)/3431 weeks
33141.20558 -96.19067 (331*2π)/3431 weeks
33238.9017 -120.4057 (332*2π)/3431 weeks
33329.2491 -128.3759 (333*2π)/3431 weeks
33467.16843 -190.5703 (334*2π)/3431 weeks
335118.9321 -146.9363 (335*2π)/3431 weeks
336109.6396 -144.7746 (336*2π)/3431 weeks
337110.0562 -175.8408 (337*2π)/3431 weeks
338206.4071 -149.0765 (338*2π)/3431 weeks
339131.5422 -106.0266 (339*2π)/3431 weeks
340171.9211 -214.396 (340*2π)/3431 weeks
341185.8411 -97.31962 (341*2π)/3431 weeks