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Fourier Analysis of BIS (ProShares UltraShort Nasdaq Bio)


BIS (ProShares UltraShort Nasdaq Bio) appears to have interesting cyclic behaviour every 36 weeks (57.7803*sine), 33 weeks (55.9565*sine), and 30 weeks (17.5279*cosine).

BIS (ProShares UltraShort Nasdaq Bio) has an average price of 365.54 (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 3/20/2017 for BIS (ProShares UltraShort Nasdaq Bio), this program was able to calculate the following Fourier Series:
Sequence #Cosine Coefficients Sine Coefficients FrequenciesPeriod
0365.5391   0 
1266.7599 383.3529 (1*2π)/364364 weeks
255.45683 228.3611 (2*2π)/364182 weeks
328.43256 148.5294 (3*2π)/364121 weeks
425.13576 112.6363 (4*2π)/36491 weeks
531.60558 116.7781 (5*2π)/36473 weeks
6-6.37402 108.4185 (6*2π)/36461 weeks
7-18.62184 70.05592 (7*2π)/36452 weeks
8-4.64248 49.55448 (8*2π)/36446 weeks
94.5394 47.73059 (9*2π)/36440 weeks
106.37348 57.78032 (10*2π)/36436 weeks
11-16.39143 55.9565 (11*2π)/36433 weeks
12-17.52791 33.04338 (12*2π)/36430 weeks
13-6.61493 24.85826 (13*2π)/36428 weeks
14-2.88457 26.07818 (14*2π)/36426 weeks
15-.71307 20.92767 (15*2π)/36424 weeks
162.04271 25.47986 (16*2π)/36423 weeks
17.51893 27.20322 (17*2π)/36421 weeks
18-6.76033 23.9626 (18*2π)/36420 weeks
19-2.46819 15.06243 (19*2π)/36419 weeks
20-.16571 12.20242 (20*2π)/36418 weeks
215.96951 14.63512 (21*2π)/36417 weeks
223.4091 20.13717 (22*2π)/36417 weeks
23-.56709 17.04641 (23*2π)/36416 weeks
244.67552 14.76298 (24*2π)/36415 weeks
252.69482 18.48363 (25*2π)/36415 weeks
262.66146 18.73187 (26*2π)/36414 weeks
27-5.10155 19.27431 (27*2π)/36413 weeks
28-7.25123 10.0677 (28*2π)/36413 weeks
29.81616 4.49238 (29*2π)/36413 weeks
305.39405 8.95496 (30*2π)/36412 weeks
314.87191 12.04345 (31*2π)/36412 weeks
322.53232 12.17435 (32*2π)/36411 weeks
33.98978 12.43124 (33*2π)/36411 weeks
34-.73393 7.23337 (34*2π)/36411 weeks
357.24687 6.46098 (35*2π)/36410 weeks
366.56211 13.86466 (36*2π)/36410 weeks
372.43266 14.87354 (37*2π)/36410 weeks
38-2.42418 12.92517 (38*2π)/36410 weeks
39-2.63509 8.74217 (39*2π)/3649 weeks
40-1.26245 5.08906 (40*2π)/3649 weeks
413.77941 5.97608 (41*2π)/3649 weeks
421.99633 11.04361 (42*2π)/3649 weeks
43-.9934 7.91256 (43*2π)/3648 weeks
44-1.74019 8.23798 (44*2π)/3648 weeks
45-2.09185 3.05721 (45*2π)/3648 weeks
462.40559 3.14304 (46*2π)/3648 weeks
473.99769 4.62465 (47*2π)/3648 weeks
482.6873 5.81886 (48*2π)/3648 weeks
49.08152 5.15722 (49*2π)/3647 weeks
50-1.75415 .19062 (50*2π)/3647 weeks
515.24148 -3.41037 (51*2π)/3647 weeks
5210.45927 2.50281 (52*2π)/3647 weeks
537.30068 7.02902 (53*2π)/3647 weeks
544.49245 6.17794 (54*2π)/3647 weeks
553.16427 3.51637 (55*2π)/3647 weeks
567.96874 1.52308 (56*2π)/3647 weeks
578.74213 6.44117 (57*2π)/3646 weeks
586.31849 7.81642 (58*2π)/3646 weeks
594.65658 8.76039 (59*2π)/3646 weeks
603.44313 6.96576 (60*2π)/3646 weeks
614.53353 6.40549 (61*2π)/3646 weeks
624.75462 7.88996 (62*2π)/3646 weeks
631.92608 7.61978 (63*2π)/3646 weeks
641.18452 4.73417 (64*2π)/3646 weeks
653.84796 4.09556 (65*2π)/3646 weeks
663.5273 6.01174 (66*2π)/3646 weeks
671.35859 5.0963 (67*2π)/3645 weeks
682.80442 3.98237 (68*2π)/3645 weeks
693.08535 3.22907 (69*2π)/3645 weeks
704.66739 4.21968 (70*2π)/3645 weeks
713.23417 4.56989 (71*2π)/3645 weeks
722.64687 3.11439 (72*2π)/3645 weeks
734.66264 2.5444 (73*2π)/3645 weeks
746.01902 3.94535 (74*2π)/3645 weeks
755.14218 5.69092 (75*2π)/3645 weeks
763.29716 5.93635 (76*2π)/3645 weeks
772.99577 4.60748 (77*2π)/3645 weeks
783.55049 5.09152 (78*2π)/3645 weeks
792.58489 5.44305 (79*2π)/3645 weeks
802.459 4.73291 (80*2π)/3645 weeks
81.97878 3.52296 (81*2π)/3644 weeks
822.94085 1.72084 (82*2π)/3644 weeks
833.77388 3.15247 (83*2π)/3644 weeks
842.69842 3.93329 (84*2π)/3644 weeks
852.63603 2.23027 (85*2π)/3644 weeks
863.30066 1.51685 (86*2π)/3644 weeks
875.05407 2.33109 (87*2π)/3644 weeks
883.95941 2.8671 (88*2π)/3644 weeks
894.52896 1.73342 (89*2π)/3644 weeks
904.95657 3.62049 (90*2π)/3644 weeks
914.40379 3.33995 (91*2π)/3644 weeks
923.37336 2.95 (92*2π)/3644 weeks
934.61017 1.36765 (93*2π)/3644 weeks
946.25876 2.40295 (94*2π)/3644 weeks
957.0362 4.71022 (95*2π)/3644 weeks
965.06872 7.02084 (96*2π)/3644 weeks
973.13032 6.4244 (97*2π)/3644 weeks
981.58161 5.2903 (98*2π)/3644 weeks
992.12268 3.4322 (99*2π)/3644 weeks
1003.17141 4.40284 (100*2π)/3644 weeks
1011.55053 4.91672 (101*2π)/3644 weeks
102.79664 3.11403 (102*2π)/3644 weeks
1031.29385 1.2315 (103*2π)/3644 weeks
1043.77463 .80231 (104*2π)/3644 weeks
1054.48151 3.79686 (105*2π)/3643 weeks
1061.58472 4.05141 (106*2π)/3643 weeks
1071.31808 2.26172 (107*2π)/3643 weeks
1081.89594 1.53846 (108*2π)/3643 weeks
1092.29811 .37327 (109*2π)/3643 weeks
1103.93135 .46942 (110*2π)/3643 weeks
1114.15178 1.9061 (111*2π)/3643 weeks
1123.2585 1.47939 (112*2π)/3643 weeks
1133.89073 1.47192 (113*2π)/3643 weeks
1143.62123 1.62216 (114*2π)/3643 weeks
1154.22296 .89258 (115*2π)/3643 weeks
1165.77589 1.28461 (116*2π)/3643 weeks
1175.85045 2.9799 (117*2π)/3643 weeks
1184.40395 4.21319 (118*2π)/3643 weeks
1193.03496 3.45164 (119*2π)/3643 weeks
1203.33957 2.35305 (120*2π)/3643 weeks
1214.31164 2.68381 (121*2π)/3643 weeks
1223.44539 2.8406 (122*2π)/3643 weeks
1233.93997 2.82232 (123*2π)/3643 weeks
1243.24796 3.10257 (124*2π)/3643 weeks
1253.77889 2.50903 (125*2π)/3643 weeks
1263.63035 3.24603 (126*2π)/3643 weeks
1273.61661 3.19443 (127*2π)/3643 weeks
1283.96204 4.28533 (128*2π)/3643 weeks
1291.93446 5.71167 (129*2π)/3643 weeks
130-.44973 3.10394 (130*2π)/3643 weeks
1311.45151 1.2952 (131*2π)/3643 weeks
1322.94481 1.87032 (132*2π)/3643 weeks
1333.02931 4.34853 (133*2π)/3643 weeks
134-.31211 4.117 (134*2π)/3643 weeks
135-.45633 1.20069 (135*2π)/3643 weeks
136.99949 1.05349 (136*2π)/3643 weeks
1371.45224 .62055 (137*2π)/3643 weeks
1381.71544 1.69417 (138*2π)/3643 weeks
139.03491 1.53233 (139*2π)/3643 weeks
140-.70149 -.98112 (140*2π)/3643 weeks
1411.48983 -2.6084 (141*2π)/3643 weeks
1424.13766 -1.17793 (142*2π)/3643 weeks
1433.49915 .4158 (143*2π)/3643 weeks
1442.7583 .63798 (144*2π)/3643 weeks
1452.2692 .52007 (145*2π)/3643 weeks
1462.08342 .33003 (146*2π)/3642 weeks
1472.09444 -.29055 (147*2π)/3642 weeks
1482.85734 -.83238 (148*2π)/3642 weeks
1493.53565 .28469 (149*2π)/3642 weeks
1502.29868 .915 (150*2π)/3642 weeks
1511.59201 -.34435 (151*2π)/3642 weeks
1522.48659 -.90739 (152*2π)/3642 weeks
1532.79446 -.81392 (153*2π)/3642 weeks
1542.59719 -1.23184 (154*2π)/3642 weeks
1553.65276 -1.51603 (155*2π)/3642 weeks
1563.96069 -.87698 (156*2π)/3642 weeks
1573.61933 -.91713 (157*2π)/3642 weeks
1584.44977 -1.4846 (158*2π)/3642 weeks
1594.99256 .69214 (159*2π)/3642 weeks
1603.02342 1.29352 (160*2π)/3642 weeks
1612.07184 -.63306 (161*2π)/3642 weeks
1623.32982 -1.99028 (162*2π)/3642 weeks
1634.86105 -1.07583 (163*2π)/3642 weeks
1644.37841 .5046 (164*2π)/3642 weeks
1653.59086 .04639 (165*2π)/3642 weeks
1663.50473 -.1543 (166*2π)/3642 weeks
1673.69272 -.89593 (167*2π)/3642 weeks
1684.588 -.93186 (168*2π)/3642 weeks
1694.93289 1.07189 (169*2π)/3642 weeks
1702.37995 1.20316 (170*2π)/3642 weeks
1711.84147 -1.33447 (171*2π)/3642 weeks
1724.09424 -2.02007 (172*2π)/3642 weeks
1733.82157 -.64458 (173*2π)/3642 weeks
1744.31086 -2.01706 (174*2π)/3642 weeks
1755.28409 -.74058 (175*2π)/3642 weeks
1763.91061 -.35822 (176*2π)/3642 weeks
1774.20229 -1.39891 (177*2π)/3642 weeks
1784.98407 -1.17802 (178*2π)/3642 weeks
1795.28412 -.7004 (179*2π)/3642 weeks
1804.81334 -.54179 (180*2π)/3642 weeks
1815.40207 -.6676 (181*2π)/3642 weeks
1825.87055   (182*2π)/3642 weeks
1835.40207 .6676 (183*2π)/3642 weeks
1844.81334 .54179 (184*2π)/3642 weeks
1855.28412 .7004 (185*2π)/3642 weeks
1864.98407 1.17802 (186*2π)/3642 weeks
1874.20229 1.39891 (187*2π)/3642 weeks
1883.91061 .35822 (188*2π)/3642 weeks
1895.28409 .74058 (189*2π)/3642 weeks
1904.31086 2.01706 (190*2π)/3642 weeks
1913.82157 .64458 (191*2π)/3642 weeks
1924.09424 2.02007 (192*2π)/3642 weeks
1931.84147 1.33447 (193*2π)/3642 weeks
1942.37995 -1.20316 (194*2π)/3642 weeks
1954.93289 -1.07189 (195*2π)/3642 weeks
1964.588 .93186 (196*2π)/3642 weeks
1973.69272 .89593 (197*2π)/3642 weeks
1983.50473 .1543 (198*2π)/3642 weeks
1993.59086 -.04639 (199*2π)/3642 weeks
2004.37841 -.5046 (200*2π)/3642 weeks
2014.86105 1.07583 (201*2π)/3642 weeks
2023.32982 1.99028 (202*2π)/3642 weeks
2032.07184 .63306 (203*2π)/3642 weeks
2043.02342 -1.29352 (204*2π)/3642 weeks
2054.99256 -.69214 (205*2π)/3642 weeks
2064.44977 1.4846 (206*2π)/3642 weeks
2073.61933 .91713 (207*2π)/3642 weeks
2083.96069 .87698 (208*2π)/3642 weeks
2093.65276 1.51603 (209*2π)/3642 weeks
2102.59719 1.23184 (210*2π)/3642 weeks
2112.79446 .81392 (211*2π)/3642 weeks
2122.48659 .90739 (212*2π)/3642 weeks
2131.59201 .34435 (213*2π)/3642 weeks
2142.29868 -.915 (214*2π)/3642 weeks
2153.53565 -.28469 (215*2π)/3642 weeks
2162.85734 .83238 (216*2π)/3642 weeks
2172.09444 .29055 (217*2π)/3642 weeks
2182.08342 -.33003 (218*2π)/3642 weeks
2192.2692 -.52007 (219*2π)/3642 weeks
2202.7583 -.63798 (220*2π)/3642 weeks
2213.49915 -.4158 (221*2π)/3642 weeks
2224.13766 1.17793 (222*2π)/3642 weeks
2231.48983 2.6084 (223*2π)/3642 weeks
224-.70149 .98112 (224*2π)/3642 weeks
225.03491 -1.53233 (225*2π)/3642 weeks
2261.71544 -1.69417 (226*2π)/3642 weeks
2271.45224 -.62055 (227*2π)/3642 weeks
228.99949 -1.05349 (228*2π)/3642 weeks
229-.45633 -1.20069 (229*2π)/3642 weeks
230-.31211 -4.117 (230*2π)/3642 weeks
2313.02931 -4.34853 (231*2π)/3642 weeks
2322.94481 -1.87032 (232*2π)/3642 weeks
2331.45151 -1.2952 (233*2π)/3642 weeks
234-.44973 -3.10394 (234*2π)/3642 weeks
2351.93446 -5.71167 (235*2π)/3642 weeks
2363.96204 -4.28533 (236*2π)/3642 weeks
2373.61661 -3.19443 (237*2π)/3642 weeks
2383.63035 -3.24603 (238*2π)/3642 weeks
2393.77889 -2.50903 (239*2π)/3642 weeks
2403.24796 -3.10257 (240*2π)/3642 weeks
2413.93997 -2.82232 (241*2π)/3642 weeks
2423.44539 -2.8406 (242*2π)/3642 weeks
2434.31164 -2.68381 (243*2π)/3641 weeks
2443.33957 -2.35305 (244*2π)/3641 weeks
2453.03496 -3.45164 (245*2π)/3641 weeks
2464.40395 -4.21319 (246*2π)/3641 weeks
2475.85045 -2.9799 (247*2π)/3641 weeks
2485.77589 -1.28461 (248*2π)/3641 weeks
2494.22296 -.89258 (249*2π)/3641 weeks
2503.62123 -1.62216 (250*2π)/3641 weeks
2513.89073 -1.47192 (251*2π)/3641 weeks
2523.2585 -1.47939 (252*2π)/3641 weeks
2534.15178 -1.9061 (253*2π)/3641 weeks
2543.93135 -.46942 (254*2π)/3641 weeks
2552.29811 -.37327 (255*2π)/3641 weeks
2561.89594 -1.53846 (256*2π)/3641 weeks
2571.31808 -2.26172 (257*2π)/3641 weeks
2581.58472 -4.05141 (258*2π)/3641 weeks
2594.48151 -3.79686 (259*2π)/3641 weeks
2603.77463 -.80231 (260*2π)/3641 weeks
2611.29385 -1.2315 (261*2π)/3641 weeks
262.79664 -3.11403 (262*2π)/3641 weeks
2631.55053 -4.91672 (263*2π)/3641 weeks
2643.17141 -4.40284 (264*2π)/3641 weeks
2652.12268 -3.4322 (265*2π)/3641 weeks
2661.58161 -5.2903 (266*2π)/3641 weeks
2673.13032 -6.4244 (267*2π)/3641 weeks
2685.06872 -7.02084 (268*2π)/3641 weeks
2697.0362 -4.71022 (269*2π)/3641 weeks
2706.25876 -2.40295 (270*2π)/3641 weeks
2714.61017 -1.36765 (271*2π)/3641 weeks
2723.37336 -2.95 (272*2π)/3641 weeks
2734.40379 -3.33995 (273*2π)/3641 weeks
2744.95657 -3.62049 (274*2π)/3641 weeks
2754.52896 -1.73342 (275*2π)/3641 weeks
2763.95941 -2.8671 (276*2π)/3641 weeks
2775.05407 -2.33109 (277*2π)/3641 weeks
2783.30066 -1.51685 (278*2π)/3641 weeks
2792.63603 -2.23027 (279*2π)/3641 weeks
2802.69842 -3.93329 (280*2π)/3641 weeks
2813.77388 -3.15247 (281*2π)/3641 weeks
2822.94085 -1.72084 (282*2π)/3641 weeks
283.97878 -3.52296 (283*2π)/3641 weeks
2842.459 -4.73291 (284*2π)/3641 weeks
2852.58489 -5.44305 (285*2π)/3641 weeks
2863.55049 -5.09152 (286*2π)/3641 weeks
2872.99577 -4.60748 (287*2π)/3641 weeks
2883.29716 -5.93635 (288*2π)/3641 weeks
2895.14218 -5.69092 (289*2π)/3641 weeks
2906.01902 -3.94535 (290*2π)/3641 weeks
2914.66264 -2.5444 (291*2π)/3641 weeks
2922.64687 -3.11439 (292*2π)/3641 weeks
2933.23417 -4.56989 (293*2π)/3641 weeks
2944.66739 -4.21968 (294*2π)/3641 weeks
2953.08535 -3.22907 (295*2π)/3641 weeks
2962.80442 -3.98237 (296*2π)/3641 weeks
2971.35859 -5.0963 (297*2π)/3641 weeks
2983.5273 -6.01174 (298*2π)/3641 weeks
2993.84796 -4.09556 (299*2π)/3641 weeks
3001.18452 -4.73417 (300*2π)/3641 weeks
3011.92608 -7.61978 (301*2π)/3641 weeks
3024.75462 -7.88996 (302*2π)/3641 weeks
3034.53353 -6.40549 (303*2π)/3641 weeks
3043.44313 -6.96576 (304*2π)/3641 weeks
3054.65658 -8.76039 (305*2π)/3641 weeks
3066.31849 -7.81642 (306*2π)/3641 weeks
3078.74213 -6.44117 (307*2π)/3641 weeks
3087.96874 -1.52308 (308*2π)/3641 weeks
3093.16427 -3.51637 (309*2π)/3641 weeks
3104.49245 -6.17794 (310*2π)/3641 weeks
3117.30068 -7.02902 (311*2π)/3641 weeks
31210.45927 -2.50281 (312*2π)/3641 weeks
3135.24148 3.41037 (313*2π)/3641 weeks
314-1.75415 -.19062 (314*2π)/3641 weeks
315.08152 -5.15722 (315*2π)/3641 weeks
3162.6873 -5.81886 (316*2π)/3641 weeks
3173.99769 -4.62465 (317*2π)/3641 weeks
3182.40559 -3.14304 (318*2π)/3641 weeks
319-2.09185 -3.05721 (319*2π)/3641 weeks
320-1.74019 -8.23798 (320*2π)/3641 weeks
321-.9934 -7.91256 (321*2π)/3641 weeks
3221.99633 -11.04361 (322*2π)/3641 weeks
3233.77941 -5.97608 (323*2π)/3641 weeks
324-1.26245 -5.08906 (324*2π)/3641 weeks
325-2.63509 -8.74217 (325*2π)/3641 weeks
326-2.42418 -12.92517 (326*2π)/3641 weeks
3272.43266 -14.87354 (327*2π)/3641 weeks
3286.56211 -13.86466 (328*2π)/3641 weeks
3297.24687 -6.46098 (329*2π)/3641 weeks
330-.73393 -7.23337 (330*2π)/3641 weeks
331.98978 -12.43124 (331*2π)/3641 weeks
3322.53232 -12.17435 (332*2π)/3641 weeks
3334.87191 -12.04345 (333*2π)/3641 weeks
3345.39405 -8.95496 (334*2π)/3641 weeks
335.81616 -4.49238 (335*2π)/3641 weeks
336-7.25123 -10.0677 (336*2π)/3641 weeks
337-5.10155 -19.27431 (337*2π)/3641 weeks
3382.66146 -18.73187 (338*2π)/3641 weeks
3392.69482 -18.48363 (339*2π)/3641 weeks
3404.67552 -14.76298 (340*2π)/3641 weeks
341-.56709 -17.04641 (341*2π)/3641 weeks
3423.4091 -20.13717 (342*2π)/3641 weeks
3435.96951 -14.63512 (343*2π)/3641 weeks
344-.16571 -12.20242 (344*2π)/3641 weeks
345-2.46819 -15.06243 (345*2π)/3641 weeks
346-6.76033 -23.9626 (346*2π)/3641 weeks
347.51893 -27.20322 (347*2π)/3641 weeks
3482.04271 -25.47986 (348*2π)/3641 weeks
349-.71307 -20.92767 (349*2π)/3641 weeks
350-2.88457 -26.07818 (350*2π)/3641 weeks
351-6.61493 -24.85826 (351*2π)/3641 weeks
352-17.52791 -33.04338 (352*2π)/3641 weeks
353-16.39143 -55.9565 (353*2π)/3641 weeks
3546.37348 -57.78032 (354*2π)/3641 weeks
3554.5394 -47.73059 (355*2π)/3641 weeks
356-4.64248 -49.55448 (356*2π)/3641 weeks
357-18.62184 -70.05592 (357*2π)/3641 weeks
358-6.37402 -108.4185 (358*2π)/3641 weeks
35931.60558 -116.7781 (359*2π)/3641 weeks
36025.13576 -112.6363 (360*2π)/3641 weeks
36128.43256 -148.5294 (361*2π)/3641 weeks
36255.45683 -228.3611 (362*2π)/3641 weeks

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