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Fourier Analysis of BIS (ProShares UltraShort Nasdaq Biotechnology ETF)


BIS (ProShares UltraShort Nasdaq Biotechnology ETF) appears to have interesting cyclic behaviour every 35 weeks (57.6145*sine), 39 weeks (47.0082*sine), and 32 weeks (18.4902*cosine).

BIS (ProShares UltraShort Nasdaq Biotechnology ETF) has an average price of 344.5 (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 4/9/2010 to 9/5/2017 for BIS (ProShares UltraShort Nasdaq Biotechnology ETF), this program was able to calculate the following Fourier Series:
Sequence #Cosine Coefficients Sine Coefficients FrequenciesPeriod
0344.5022   0 
1276.5512 361.7758 (1*2π)/388388 weeks
257.83673 229.6525 (2*2π)/388194 weeks
327.13809 150.6777 (3*2π)/388129 weeks
423.31245 114.7931 (4*2π)/38897 weeks
534.33118 103.0674 (5*2π)/38878 weeks
66.75847 109.8576 (6*2π)/38865 weeks
7-18.14162 81.41499 (7*2π)/38855 weeks
8-12.41624 53.54145 (8*2π)/38849 weeks
9.55234 46.22927 (9*2π)/38843 weeks
107.68828 47.00822 (10*2π)/38839 weeks
111.24473 57.6145 (11*2π)/38835 weeks
12-18.49023 46.05948 (12*2π)/38832 weeks
13-14.80242 28.27727 (13*2π)/38830 weeks
14-5.2603 23.44637 (14*2π)/38828 weeks
15-2.87716 24.59691 (15*2π)/38826 weeks
16-.56305 19.79347 (16*2π)/38824 weeks
172.15118 24.03587 (17*2π)/38823 weeks
181.13752 25.276 (18*2π)/38822 weeks
19-5.11741 23.91797 (19*2π)/38820 weeks
20-4.31698 14.37282 (20*2π)/38819 weeks
21-1.8319 13.34365 (21*2π)/38818 weeks
224.33333 12.5735 (22*2π)/38818 weeks
236.02132 17.78439 (23*2π)/38817 weeks
241.27761 17.99651 (24*2π)/38816 weeks
25.9887 12.89598 (25*2π)/38816 weeks
264.67144 15.83687 (26*2π)/38815 weeks
271.7404 16.6092 (27*2π)/38814 weeks
281.41898 18.96356 (28*2π)/38814 weeks
29-6.04351 16.47683 (29*2π)/38813 weeks
30-6.36273 8.10416 (30*2π)/38813 weeks
311.42543 4.39746 (31*2π)/38813 weeks
325.04272 8.4953 (32*2π)/38812 weeks
334.63276 11.17804 (33*2π)/38812 weeks
342.40925 11.49928 (34*2π)/38811 weeks
351.42197 11.67066 (35*2π)/38811 weeks
36-1.2501 8.36316 (36*2π)/38811 weeks
374.48595 4.67281 (37*2π)/38810 weeks
387.54699 11.16033 (38*2π)/38810 weeks
393.35467 13.77221 (39*2π)/38810 weeks
40-.3918 14.08237 (40*2π)/38810 weeks
41-2.83428 10.17219 (41*2π)/3889 weeks
42-2.28275 7.2917 (42*2π)/3889 weeks
43.68385 4.20921 (43*2π)/3889 weeks
444.21106 6.9714 (44*2π)/3889 weeks
45.54578 10.26053 (45*2π)/3889 weeks
46-.66448 7.15517 (46*2π)/3888 weeks
47-2.08397 7.5258 (47*2π)/3888 weeks
48-1.8724 2.81134 (48*2π)/3888 weeks
492.25587 2.94329 (49*2π)/3888 weeks
503.73856 3.98088 (50*2π)/3888 weeks
512.54858 5.21947 (51*2π)/3888 weeks
52.2761 5.25453 (52*2π)/3887 weeks
53-2.01486 2.43666 (53*2π)/3887 weeks
542.2091 -2.84495 (54*2π)/3887 weeks
559.04735 -.45762 (55*2π)/3887 weeks
568.70595 5.43914 (56*2π)/3887 weeks
575.09473 6.51642 (57*2π)/3887 weeks
583.43178 5.38433 (58*2π)/3887 weeks
593.75555 2.02696 (59*2π)/3887 weeks
608.79072 3.0183 (60*2π)/3886 weeks
617.90777 6.68825 (61*2π)/3886 weeks
625.88412 7.26702 (62*2π)/3886 weeks
634.02426 8.10672 (63*2π)/3886 weeks
643.20304 6.52454 (64*2π)/3886 weeks
654.24806 6.06687 (65*2π)/3886 weeks
664.60495 7.23533 (66*2π)/3886 weeks
672.05989 7.41398 (67*2π)/3886 weeks
68.85609 5.1561 (68*2π)/3886 weeks
692.93599 3.60797 (69*2π)/3886 weeks
703.75426 5.37501 (70*2π)/3886 weeks
712.06076 5.8543 (71*2π)/3885 weeks
722.13655 3.54974 (72*2π)/3885 weeks
732.51349 3.59913 (73*2π)/3885 weeks
743.63538 2.94353 (74*2π)/3885 weeks
753.82513 4.66315 (75*2π)/3885 weeks
762.83032 4.25472 (76*2π)/3885 weeks
772.90797 2.55575 (77*2π)/3885 weeks
784.68617 2.58002 (78*2π)/3885 weeks
795.72465 4.02254 (79*2π)/3885 weeks
804.81084 5.43522 (80*2π)/3885 weeks
813.14186 5.63436 (81*2π)/3885 weeks
822.79678 4.46349 (82*2π)/3885 weeks
833.47558 4.74056 (83*2π)/3885 weeks
842.83124 5.16872 (84*2π)/3885 weeks
852.31251 4.27057 (85*2π)/3885 weeks
861.17168 4.1278 (86*2π)/3885 weeks
871.71504 1.84498 (87*2π)/3884 weeks
883.63695 2.48556 (88*2π)/3884 weeks
893.42956 3.45054 (89*2π)/3884 weeks
901.97345 2.90252 (90*2π)/3884 weeks
912.67812 2.00797 (91*2π)/3884 weeks
923.63483 1.2902 (92*2π)/3884 weeks
934.58438 2.75722 (93*2π)/3884 weeks
943.68088 2.62811 (94*2π)/3884 weeks
954.62263 1.70029 (95*2π)/3884 weeks
964.57198 3.42975 (96*2π)/3884 weeks
974.14117 3.13448 (97*2π)/3884 weeks
983.17407 2.88927 (98*2π)/3884 weeks
993.98267 1.4025 (99*2π)/3884 weeks
1005.66745 2.20072 (100*2π)/3884 weeks
1016.73791 3.90498 (101*2π)/3884 weeks
1025.82171 6.2326 (102*2π)/3884 weeks
1033.46038 6.19561 (103*2π)/3884 weeks
1042.10513 5.66772 (104*2π)/3884 weeks
1051.47619 3.94576 (105*2π)/3884 weeks
1062.83209 3.25429 (106*2π)/3884 weeks
1072.58412 4.50965 (107*2π)/3884 weeks
108.87158 4.13069 (108*2π)/3884 weeks
109.72069 2.63546 (109*2π)/3884 weeks
1101.62775 .94191 (110*2π)/3884 weeks
1113.95062 1.00167 (111*2π)/3883 weeks
1124.09895 3.76317 (112*2π)/3883 weeks
1131.51439 3.79805 (113*2π)/3883 weeks
1141.27973 2.19534 (114*2π)/3883 weeks
1151.81297 1.45464 (115*2π)/3883 weeks
1161.93276 .54655 (116*2π)/3883 weeks
1173.372 .28566 (117*2π)/3883 weeks
1184.11897 1.44586 (118*2π)/3883 weeks
1193.17409 1.85161 (119*2π)/3883 weeks
1203.60279 1.24573 (120*2π)/3883 weeks
1213.74259 1.54093 (121*2π)/3883 weeks
1223.50563 1.026 (122*2π)/3883 weeks
1234.40266 .68895 (123*2π)/3883 weeks
1245.61417 1.83104 (124*2π)/3883 weeks
1255.37087 3.24217 (125*2π)/3883 weeks
1263.80942 3.89283 (126*2π)/3883 weeks
1272.79507 3.02656 (127*2π)/3883 weeks
1283.20129 2.16116 (128*2π)/3883 weeks
1294.05676 2.54478 (129*2π)/3883 weeks
1303.23639 2.69709 (130*2π)/3883 weeks
1313.66308 2.57898 (131*2π)/3883 weeks
1323.16321 3.0305 (132*2π)/3883 weeks
1333.28343 2.33737 (133*2π)/3883 weeks
1343.58913 3.06663 (134*2π)/3883 weeks
1353.37014 3.07759 (135*2π)/3883 weeks
1363.82708 3.41915 (136*2π)/3883 weeks
1373.18158 4.91489 (137*2π)/3883 weeks
138.35892 4.75112 (138*2π)/3883 weeks
139.21726 1.71514 (139*2π)/3883 weeks
1401.98252 1.22139 (140*2π)/3883 weeks
1413.11771 2.08089 (141*2π)/3883 weeks
1422.18676 4.4701 (142*2π)/3883 weeks
143-.59579 3.4464 (143*2π)/3883 weeks
144-.25374 .98104 (144*2π)/3883 weeks
145.9659 1.01696 (145*2π)/3883 weeks
1461.37231 .56913 (146*2π)/3883 weeks
1471.7323 1.44158 (147*2π)/3883 weeks
148.2638 1.54978 (148*2π)/3883 weeks
149-.84844 -.27231 (149*2π)/3883 weeks
150.74031 -2.16595 (150*2π)/3883 weeks
1513.38072 -2.01573 (151*2π)/3883 weeks
1523.58781 .0864 (152*2π)/3883 weeks
1532.88466 .6609 (153*2π)/3883 weeks
1542.46245 .63893 (154*2π)/3883 weeks
1552.18386 .32178 (155*2π)/3883 weeks
1561.87931 .04173 (156*2π)/3882 weeks
1572.03228 -.48325 (157*2π)/3882 weeks
1582.96113 -.71402 (158*2π)/3882 weeks
1593.20321 .45597 (159*2π)/3882 weeks
1601.99652 .79579 (160*2π)/3882 weeks
1611.50004 -.37031 (161*2π)/3882 weeks
1622.32559 -.8451 (162*2π)/3882 weeks
1632.62294 -.79043 (163*2π)/3882 weeks
1642.40112 -1.03329 (164*2π)/3882 weeks
1653.19755 -1.54146 (165*2π)/3882 weeks
1663.6525 -.98248 (166*2π)/3882 weeks
1673.37053 -.66297 (167*2π)/3882 weeks
1683.55796 -1.12391 (168*2π)/3882 weeks
1695.09904 -.3392 (169*2π)/3882 weeks
1703.93659 1.18807 (170*2π)/3882 weeks
1712.09762 .62611 (171*2π)/3882 weeks
1722.25252 -1.11042 (172*2π)/3882 weeks
1733.82191 -1.81714 (173*2π)/3882 weeks
1744.75239 -.697 (174*2π)/3882 weeks
1753.84904 .45223 (175*2π)/3882 weeks
1763.36607 -.01027 (176*2π)/3882 weeks
1773.27997 -.15734 (177*2π)/3882 weeks
1783.47838 -.84145 (178*2π)/3882 weeks
1794.22466 -.90523 (179*2π)/3882 weeks
1804.84412 .70122 (180*2π)/3882 weeks
1812.68317 1.37867 (181*2π)/3882 weeks
1821.52771 -.61865 (182*2π)/3882 weeks
1833.14228 -2.13433 (183*2π)/3882 weeks
1844.12508 -.8411 (184*2π)/3882 weeks
1853.23157 -1.45123 (185*2π)/3882 weeks
1864.7613 -1.39042 (186*2π)/3882 weeks
1874.54223 -.0367 (187*2π)/3882 weeks
1883.63467 -.74771 (188*2π)/3882 weeks
1894.32311 -1.35451 (189*2π)/3882 weeks
1904.81468 -1.03223 (190*2π)/3882 weeks
1914.98606 -.55306 (191*2π)/3882 weeks
1924.54904 -.63505 (192*2π)/3882 weeks
1935.09412 -.63009 (193*2π)/3882 weeks
1945.50053   (194*2π)/3882 weeks
1955.09412 .63009 (195*2π)/3882 weeks
1964.54904 .63505 (196*2π)/3882 weeks
1974.98606 .55306 (197*2π)/3882 weeks
1984.81468 1.03223 (198*2π)/3882 weeks
1994.32311 1.35451 (199*2π)/3882 weeks
2003.63467 .74771 (200*2π)/3882 weeks
2014.54223 .0367 (201*2π)/3882 weeks
2024.7613 1.39042 (202*2π)/3882 weeks
2033.23157 1.45123 (203*2π)/3882 weeks
2044.12508 .8411 (204*2π)/3882 weeks
2053.14228 2.13433 (205*2π)/3882 weeks
2061.52771 .61865 (206*2π)/3882 weeks
2072.68317 -1.37867 (207*2π)/3882 weeks
2084.84412 -.70122 (208*2π)/3882 weeks
2094.22466 .90523 (209*2π)/3882 weeks
2103.47838 .84145 (210*2π)/3882 weeks
2113.27997 .15734 (211*2π)/3882 weeks
2123.36607 .01027 (212*2π)/3882 weeks
2133.84904 -.45223 (213*2π)/3882 weeks
2144.75239 .697 (214*2π)/3882 weeks
2153.82191 1.81714 (215*2π)/3882 weeks
2162.25252 1.11042 (216*2π)/3882 weeks
2172.09762 -.62611 (217*2π)/3882 weeks
2183.93659 -1.18807 (218*2π)/3882 weeks
2195.09904 .3392 (219*2π)/3882 weeks
2203.55796 1.12391 (220*2π)/3882 weeks
2213.37053 .66297 (221*2π)/3882 weeks
2223.6525 .98248 (222*2π)/3882 weeks
2233.19755 1.54146 (223*2π)/3882 weeks
2242.40112 1.03329 (224*2π)/3882 weeks
2252.62294 .79043 (225*2π)/3882 weeks
2262.32559 .8451 (226*2π)/3882 weeks
2271.50004 .37031 (227*2π)/3882 weeks
2281.99652 -.79579 (228*2π)/3882 weeks
2293.20321 -.45597 (229*2π)/3882 weeks
2302.96113 .71402 (230*2π)/3882 weeks
2312.03228 .48325 (231*2π)/3882 weeks
2321.87931 -.04173 (232*2π)/3882 weeks
2332.18386 -.32178 (233*2π)/3882 weeks
2342.46245 -.63893 (234*2π)/3882 weeks
2352.88466 -.6609 (235*2π)/3882 weeks
2363.58781 -.0864 (236*2π)/3882 weeks
2373.38072 2.01573 (237*2π)/3882 weeks
238.74031 2.16595 (238*2π)/3882 weeks
239-.84844 .27231 (239*2π)/3882 weeks
240.2638 -1.54978 (240*2π)/3882 weeks
2411.7323 -1.44158 (241*2π)/3882 weeks
2421.37231 -.56913 (242*2π)/3882 weeks
243.9659 -1.01696 (243*2π)/3882 weeks
244-.25374 -.98104 (244*2π)/3882 weeks
245-.59579 -3.4464 (245*2π)/3882 weeks
2462.18676 -4.4701 (246*2π)/3882 weeks
2473.11771 -2.08089 (247*2π)/3882 weeks
2481.98252 -1.22139 (248*2π)/3882 weeks
249.21726 -1.71514 (249*2π)/3882 weeks
250.35892 -4.75112 (250*2π)/3882 weeks
2513.18158 -4.91489 (251*2π)/3882 weeks
2523.82708 -3.41915 (252*2π)/3882 weeks
2533.37014 -3.07759 (253*2π)/3882 weeks
2543.58913 -3.06663 (254*2π)/3882 weeks
2553.28343 -2.33737 (255*2π)/3882 weeks
2563.16321 -3.0305 (256*2π)/3882 weeks
2573.66308 -2.57898 (257*2π)/3882 weeks
2583.23639 -2.69709 (258*2π)/3882 weeks
2594.05676 -2.54478 (259*2π)/3881 weeks
2603.20129 -2.16116 (260*2π)/3881 weeks
2612.79507 -3.02656 (261*2π)/3881 weeks
2623.80942 -3.89283 (262*2π)/3881 weeks
2635.37087 -3.24217 (263*2π)/3881 weeks
2645.61417 -1.83104 (264*2π)/3881 weeks
2654.40266 -.68895 (265*2π)/3881 weeks
2663.50563 -1.026 (266*2π)/3881 weeks
2673.74259 -1.54093 (267*2π)/3881 weeks
2683.60279 -1.24573 (268*2π)/3881 weeks
2693.17409 -1.85161 (269*2π)/3881 weeks
2704.11897 -1.44586 (270*2π)/3881 weeks
2713.372 -.28566 (271*2π)/3881 weeks
2721.93276 -.54655 (272*2π)/3881 weeks
2731.81297 -1.45464 (273*2π)/3881 weeks
2741.27973 -2.19534 (274*2π)/3881 weeks
2751.51439 -3.79805 (275*2π)/3881 weeks
2764.09895 -3.76317 (276*2π)/3881 weeks
2773.95062 -1.00167 (277*2π)/3881 weeks
2781.62775 -.94191 (278*2π)/3881 weeks
279.72069 -2.63546 (279*2π)/3881 weeks
280.87158 -4.13069 (280*2π)/3881 weeks
2812.58412 -4.50965 (281*2π)/3881 weeks
2822.83209 -3.25429 (282*2π)/3881 weeks
2831.47619 -3.94576 (283*2π)/3881 weeks
2842.10513 -5.66772 (284*2π)/3881 weeks
2853.46038 -6.19561 (285*2π)/3881 weeks
2865.82171 -6.2326 (286*2π)/3881 weeks
2876.73791 -3.90498 (287*2π)/3881 weeks
2885.66745 -2.20072 (288*2π)/3881 weeks
2893.98267 -1.4025 (289*2π)/3881 weeks
2903.17407 -2.88927 (290*2π)/3881 weeks
2914.14117 -3.13448 (291*2π)/3881 weeks
2924.57198 -3.42975 (292*2π)/3881 weeks
2934.62263 -1.70029 (293*2π)/3881 weeks
2943.68088 -2.62811 (294*2π)/3881 weeks
2954.58438 -2.75722 (295*2π)/3881 weeks
2963.63483 -1.2902 (296*2π)/3881 weeks
2972.67812 -2.00797 (297*2π)/3881 weeks
2981.97345 -2.90252 (298*2π)/3881 weeks
2993.42956 -3.45054 (299*2π)/3881 weeks
3003.63695 -2.48556 (300*2π)/3881 weeks
3011.71504 -1.84498 (301*2π)/3881 weeks
3021.17168 -4.1278 (302*2π)/3881 weeks
3032.31251 -4.27057 (303*2π)/3881 weeks
3042.83124 -5.16872 (304*2π)/3881 weeks
3053.47558 -4.74056 (305*2π)/3881 weeks
3062.79678 -4.46349 (306*2π)/3881 weeks
3073.14186 -5.63436 (307*2π)/3881 weeks
3084.81084 -5.43522 (308*2π)/3881 weeks
3095.72465 -4.02254 (309*2π)/3881 weeks
3104.68617 -2.58002 (310*2π)/3881 weeks
3112.90797 -2.55575 (311*2π)/3881 weeks
3122.83032 -4.25472 (312*2π)/3881 weeks
3133.82513 -4.66315 (313*2π)/3881 weeks
3143.63538 -2.94353 (314*2π)/3881 weeks
3152.51349 -3.59913 (315*2π)/3881 weeks
3162.13655 -3.54974 (316*2π)/3881 weeks
3172.06076 -5.8543 (317*2π)/3881 weeks
3183.75426 -5.37501 (318*2π)/3881 weeks
3192.93599 -3.60797 (319*2π)/3881 weeks
320.85609 -5.1561 (320*2π)/3881 weeks
3212.05989 -7.41398 (321*2π)/3881 weeks
3224.60495 -7.23533 (322*2π)/3881 weeks
3234.24806 -6.06687 (323*2π)/3881 weeks
3243.20304 -6.52454 (324*2π)/3881 weeks
3254.02426 -8.10672 (325*2π)/3881 weeks
3265.88412 -7.26702 (326*2π)/3881 weeks
3277.90777 -6.68825 (327*2π)/3881 weeks
3288.79072 -3.0183 (328*2π)/3881 weeks
3293.75555 -2.02696 (329*2π)/3881 weeks
3303.43178 -5.38433 (330*2π)/3881 weeks
3315.09473 -6.51642 (331*2π)/3881 weeks
3328.70595 -5.43914 (332*2π)/3881 weeks
3339.04735 .45762 (333*2π)/3881 weeks
3342.2091 2.84495 (334*2π)/3881 weeks
335-2.01486 -2.43666 (335*2π)/3881 weeks
336.2761 -5.25453 (336*2π)/3881 weeks
3372.54858 -5.21947 (337*2π)/3881 weeks
3383.73856 -3.98088 (338*2π)/3881 weeks
3392.25587 -2.94329 (339*2π)/3881 weeks
340-1.8724 -2.81134 (340*2π)/3881 weeks
341-2.08397 -7.5258 (341*2π)/3881 weeks
342-.66448 -7.15517 (342*2π)/3881 weeks
343.54578 -10.26053 (343*2π)/3881 weeks
3444.21106 -6.9714 (344*2π)/3881 weeks
345.68385 -4.20921 (345*2π)/3881 weeks
346-2.28275 -7.2917 (346*2π)/3881 weeks
347-2.83428 -10.17219 (347*2π)/3881 weeks
348-.3918 -14.08237 (348*2π)/3881 weeks
3493.35467 -13.77221 (349*2π)/3881 weeks
3507.54699 -11.16033 (350*2π)/3881 weeks
3514.48595 -4.67281 (351*2π)/3881 weeks
352-1.2501 -8.36316 (352*2π)/3881 weeks
3531.42197 -11.67066 (353*2π)/3881 weeks
3542.40925 -11.49928 (354*2π)/3881 weeks
3554.63276 -11.17804 (355*2π)/3881 weeks
3565.04272 -8.4953 (356*2π)/3881 weeks
3571.42543 -4.39746 (357*2π)/3881 weeks
358-6.36273 -8.10416 (358*2π)/3881 weeks
359-6.04351 -16.47683 (359*2π)/3881 weeks
3601.41898 -18.96356 (360*2π)/3881 weeks
3611.7404 -16.6092 (361*2π)/3881 weeks
3624.67144 -15.83687 (362*2π)/3881 weeks
363.9887 -12.89598 (363*2π)/3881 weeks
3641.27761 -17.99651 (364*2π)/3881 weeks
3656.02132 -17.78439 (365*2π)/3881 weeks
3664.33333 -12.5735 (366*2π)/3881 weeks
367-1.8319 -13.34365 (367*2π)/3881 weeks
368-4.31698 -14.37282 (368*2π)/3881 weeks
369-5.11741 -23.91797 (369*2π)/3881 weeks
3701.13752 -25.276 (370*2π)/3881 weeks
3712.15118 -24.03587 (371*2π)/3881 weeks
372-.56305 -19.79347 (372*2π)/3881 weeks
373-2.87716 -24.59691 (373*2π)/3881 weeks
374-5.2603 -23.44637 (374*2π)/3881 weeks
375-14.80242 -28.27727 (375*2π)/3881 weeks
376-18.49023 -46.05948 (376*2π)/3881 weeks
3771.24473 -57.6145 (377*2π)/3881 weeks
3787.68828 -47.00822 (378*2π)/3881 weeks
379.55234 -46.22927 (379*2π)/3881 weeks
380-12.41624 -53.54145 (380*2π)/3881 weeks
381-18.14162 -81.41499 (381*2π)/3881 weeks
3826.75847 -109.8576 (382*2π)/3881 weeks
38334.33118 -103.0674 (383*2π)/3881 weeks
38423.31245 -114.7931 (384*2π)/3881 weeks
38527.13809 -150.6777 (385*2π)/3881 weeks
38657.83673 -229.6525 (386*2π)/3881 weeks



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