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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 (59.6597*sine), 38 weeks (50.2466*sine), and 32 weeks (19.1899*cosine).

BIS (ProShares UltraShort Nasdaq Biotechnology ETF) has an average price of 351.27 (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 7/10/2017 for BIS (ProShares UltraShort Nasdaq Biotechnology ETF), this program was able to calculate the following Fourier Series:
Sequence #Cosine Coefficients Sine Coefficients FrequenciesPeriod
0351.2748   0 
1273.7398 368.925 (1*2π)/380380 weeks
257.02165 229.1012 (2*2π)/380190 weeks
327.39117 149.8244 (3*2π)/380127 weeks
423.49244 114.1899 (4*2π)/38095 weeks
534.53827 107.4442 (5*2π)/38076 weeks
62.60374 110.0017 (6*2π)/38063 weeks
7-19.046 77.46059 (7*2π)/38054 weeks
8-10.28604 51.49888 (8*2π)/38048 weeks
91.5386 46.82904 (9*2π)/38042 weeks
108.46801 50.2466 (10*2π)/38038 weeks
11-3.83153 59.65972 (11*2π)/38035 weeks
12-19.18989 41.05581 (12*2π)/38032 weeks
13-12.12566 25.82152 (13*2π)/38029 weeks
14-3.68072 23.94993 (14*2π)/38027 weeks
15-3.58841 23.86831 (15*2π)/38025 weeks
162.20517 20.75588 (16*2π)/38024 weeks
171.32752 24.16996 (17*2π)/38022 weeks
18-1.29341 26.35919 (18*2π)/38021 weeks
19-7.49291 20.78609 (19*2π)/38020 weeks
20-1.75422 14.91819 (20*2π)/38019 weeks
21.84063 11.47696 (21*2π)/38018 weeks
225.96416 14.4435 (22*2π)/38017 weeks
233.03799 19.32087 (23*2π)/38017 weeks
24-.55835 16.47941 (24*2π)/38016 weeks
254.27013 13.94505 (25*2π)/38015 weeks
262.9855 17.80118 (26*2π)/38015 weeks
272.60184 17.31653 (27*2π)/38014 weeks
28-3.40308 19.55882 (28*2π)/38014 weeks
29-7.22875 11.73811 (29*2π)/38013 weeks
30-1.74512 4.35962 (30*2π)/38013 weeks
314.77151 7.11261 (31*2π)/38012 weeks
324.97985 10.48413 (32*2π)/38012 weeks
332.83831 12.10256 (33*2π)/38012 weeks
341.99962 11.79567 (34*2π)/38011 weeks
35-1.05249 10.24839 (35*2π)/38011 weeks
362.51207 4.72522 (36*2π)/38011 weeks
378.1289 9.87181 (37*2π)/38010 weeks
384.20285 14.10719 (38*2π)/38010 weeks
39.65521 14.6283 (39*2π)/38010 weeks
40-2.75277 11.02092 (40*2π)/38010 weeks
41-2.29727 7.78298 (41*2π)/3809 weeks
42.10005 4.34405 (42*2π)/3809 weeks
434.18057 6.62088 (43*2π)/3809 weeks
441.00088 10.58374 (44*2π)/3809 weeks
45-.7401 7.37145 (45*2π)/3808 weeks
46-1.94844 7.76733 (46*2π)/3808 weeks
47-1.90607 2.89021 (47*2π)/3808 weeks
482.33802 3.00861 (48*2π)/3808 weeks
493.86897 4.18015 (49*2π)/3808 weeks
502.64773 5.38459 (50*2π)/3808 weeks
51.24726 5.19252 (51*2π)/3807 weeks
52-2.05033 1.74635 (52*2π)/3807 weeks
533.11775 -3.1456 (53*2π)/3807 weeks
549.70849 .43857 (54*2π)/3807 weeks
558.35693 6.07449 (55*2π)/3807 weeks
564.84644 6.36927 (56*2π)/3807 weeks
573.16698 4.96007 (57*2π)/3807 weeks
584.78384 1.38783 (58*2π)/3807 weeks
599.17272 4.46547 (59*2π)/3806 weeks
607.55239 7.46203 (60*2π)/3806 weeks
616.12526 7.66581 (61*2π)/3806 weeks
623.67082 7.55024 (62*2π)/3806 weeks
633.49669 5.95591 (63*2π)/3806 weeks
644.39075 6.24406 (64*2π)/3806 weeks
653.74124 8.06942 (65*2π)/3806 weeks
661.42926 6.96344 (66*2π)/3806 weeks
671.51449 4.12049 (67*2π)/3806 weeks
683.9372 4.23988 (68*2π)/3806 weeks
693.25047 5.8502 (69*2π)/3806 weeks
701.29559 4.70886 (70*2π)/3805 weeks
712.693 3.83744 (71*2π)/3805 weeks
722.92479 3.13471 (72*2π)/3805 weeks
734.49528 3.85521 (73*2π)/3805 weeks
743.1606 4.45179 (74*2π)/3805 weeks
752.41758 3.38114 (75*2π)/3805 weeks
764.11149 2.35278 (76*2π)/3805 weeks
775.54253 3.30878 (77*2π)/3805 weeks
785.25416 5.23517 (78*2π)/3805 weeks
793.8245 6.01622 (79*2π)/3805 weeks
802.81783 4.9097 (80*2π)/3805 weeks
813.5888 4.47629 (81*2π)/3805 weeks
823.1542 5.05801 (82*2π)/3805 weeks
832.11165 4.44221 (83*2π)/3805 weeks
841.54697 4.57947 (84*2π)/3805 weeks
851.28746 2.273 (85*2π)/3804 weeks
863.57195 2.20193 (86*2π)/3804 weeks
873.55548 3.32218 (87*2π)/3804 weeks
882.05966 3.2846 (88*2π)/3804 weeks
892.67771 2.08726 (89*2π)/3804 weeks
903.49918 1.33134 (90*2π)/3804 weeks
914.75781 2.64955 (91*2π)/3804 weeks
923.75624 2.69837 (92*2π)/3804 weeks
934.58264 1.68948 (93*2π)/3804 weeks
944.67856 3.48602 (94*2π)/3804 weeks
954.21179 3.1939 (95*2π)/3804 weeks
963.22336 2.90302 (96*2π)/3804 weeks
974.1584 1.37445 (97*2π)/3804 weeks
985.82125 2.25514 (98*2π)/3804 weeks
996.82109 4.16062 (99*2π)/3804 weeks
1005.59962 6.54836 (100*2π)/3804 weeks
1013.35471 6.25762 (101*2π)/3804 weeks
1021.90198 5.59918 (102*2π)/3804 weeks
1031.60561 3.73934 (103*2π)/3804 weeks
1043.08824 3.56784 (104*2π)/3804 weeks
1052.33853 4.70456 (105*2π)/3804 weeks
106.69558 3.70158 (106*2π)/3804 weeks
107.66259 2.22709 (107*2π)/3804 weeks
1082.15368 .73166 (108*2π)/3804 weeks
1094.55087 1.75933 (109*2π)/3803 weeks
1103.33435 4.34624 (110*2π)/3803 weeks
1111.17036 3.15973 (111*2π)/3803 weeks
1121.34883 1.84181 (112*2π)/3803 weeks
1131.73916 1.41322 (113*2π)/3803 weeks
1142.54186 .27896 (114*2π)/3803 weeks
1153.97922 .66525 (115*2π)/3803 weeks
1163.86903 1.95302 (116*2π)/3803 weeks
1173.18017 1.36053 (117*2π)/3803 weeks
1183.73779 1.44008 (118*2π)/3803 weeks
1193.48178 1.58821 (119*2π)/3803 weeks
1204.01181 .89252 (120*2π)/3803 weeks
1215.41325 1.0831 (121*2π)/3803 weeks
1225.635 2.67425 (122*2π)/3803 weeks
1234.5701 4.03218 (123*2π)/3803 weeks
1243.1163 3.60759 (124*2π)/3803 weeks
1253.03714 2.4321 (125*2π)/3803 weeks
1263.9126 2.20045 (126*2π)/3803 weeks
1273.46862 3.06359 (127*2π)/3803 weeks
1283.60228 2.5653 (128*2π)/3803 weeks
1293.52644 3.17511 (129*2π)/3803 weeks
1303.15821 2.54969 (130*2π)/3803 weeks
1313.85426 2.9779 (131*2π)/3803 weeks
1323.48838 3.1297 (132*2π)/3803 weeks
1333.82264 3.24603 (133*2π)/3803 weeks
1343.52935 4.71314 (134*2π)/3803 weeks
135.81754 5.17247 (135*2π)/3803 weeks
136-.05266 2.05474 (136*2π)/3803 weeks
1371.82984 1.22226 (137*2π)/3803 weeks
1383.0685 1.99793 (138*2π)/3803 weeks
1392.47154 4.47646 (139*2π)/3803 weeks
140-.51278 3.67102 (140*2π)/3803 weeks
141-.309 1.05287 (141*2π)/3803 weeks
142.98638 1.03537 (142*2π)/3803 weeks
1431.40834 .59303 (143*2π)/3803 weeks
1441.74317 1.53106 (144*2π)/3803 weeks
145.20664 1.55551 (145*2π)/3803 weeks
146-.81934 -.49338 (146*2π)/3803 weeks
147.96474 -2.30767 (147*2π)/3803 weeks
1483.69838 -1.77871 (148*2π)/3803 weeks
1493.56122 .2132 (149*2π)/3803 weeks
1502.82846 .68409 (150*2π)/3803 weeks
1512.4112 .67053 (151*2π)/3803 weeks
1522.23938 .36501 (152*2π)/3803 weeks
1531.97937 -.0832 (153*2π)/3802 weeks
1542.24729 -.66028 (154*2π)/3802 weeks
1553.33025 -.50452 (155*2π)/3802 weeks
1563.04696 .69969 (156*2π)/3802 weeks
1571.72251 .47291 (157*2π)/3802 weeks
1581.75616 -.81346 (158*2π)/3802 weeks
1592.43136 -.8694 (159*2π)/3802 weeks
1602.58412 -.72449 (160*2π)/3802 weeks
1612.59976 -1.36243 (161*2π)/3802 weeks
1623.61472 -1.29876 (162*2π)/3802 weeks
1633.77832 -.73194 (163*2π)/3802 weeks
1643.50479 -.91237 (164*2π)/3802 weeks
1654.39818 -1.40634 (165*2π)/3802 weeks
1664.77262 .6724 (166*2π)/3802 weeks
1672.9713 1.25514 (167*2π)/3802 weeks
1681.94646 -.47295 (168*2π)/3802 weeks
1692.94741 -1.81922 (169*2π)/3802 weeks
1704.55251 -1.20668 (170*2π)/3802 weeks
1714.56308 .32796 (171*2π)/3802 weeks
1723.5238 .1382 (172*2π)/3802 weeks
1733.46092 -.16665 (173*2π)/3802 weeks
1743.26667 -.70049 (174*2π)/3802 weeks
1753.96381 -.92008 (175*2π)/3802 weeks
1765.06182 .02444 (176*2π)/3802 weeks
1773.38103 1.57002 (177*2π)/3802 weeks
1781.56758 .02112 (178*2π)/3802 weeks
1792.65216 -2.06907 (179*2π)/3802 weeks
1804.30572 -1.18118 (180*2π)/3802 weeks
1813.23408 -1.10498 (181*2π)/3802 weeks
1824.71292 -1.61914 (182*2π)/3802 weeks
1834.8626 -.19833 (183*2π)/3802 weeks
1843.69739 -.64309 (184*2π)/3802 weeks
1854.2943 -1.39861 (185*2π)/3802 weeks
1864.86922 -1.08324 (186*2π)/3802 weeks
1875.08493 -.60155 (187*2π)/3802 weeks
1884.62713 -.60631 (188*2π)/3802 weeks
1895.19144 -.63971 (189*2π)/3802 weeks
1905.61716   (190*2π)/3802 weeks
1915.19144 .63971 (191*2π)/3802 weeks
1924.62713 .60631 (192*2π)/3802 weeks
1935.08493 .60155 (193*2π)/3802 weeks
1944.86922 1.08324 (194*2π)/3802 weeks
1954.2943 1.39861 (195*2π)/3802 weeks
1963.69739 .64309 (196*2π)/3802 weeks
1974.8626 .19833 (197*2π)/3802 weeks
1984.71292 1.61914 (198*2π)/3802 weeks
1993.23408 1.10498 (199*2π)/3802 weeks
2004.30572 1.18118 (200*2π)/3802 weeks
2012.65216 2.06907 (201*2π)/3802 weeks
2021.56758 -.02112 (202*2π)/3802 weeks
2033.38103 -1.57002 (203*2π)/3802 weeks
2045.06182 -.02444 (204*2π)/3802 weeks
2053.96381 .92008 (205*2π)/3802 weeks
2063.26667 .70049 (206*2π)/3802 weeks
2073.46092 .16665 (207*2π)/3802 weeks
2083.5238 -.1382 (208*2π)/3802 weeks
2094.56308 -.32796 (209*2π)/3802 weeks
2104.55251 1.20668 (210*2π)/3802 weeks
2112.94741 1.81922 (211*2π)/3802 weeks
2121.94646 .47295 (212*2π)/3802 weeks
2132.9713 -1.25514 (213*2π)/3802 weeks
2144.77262 -.6724 (214*2π)/3802 weeks
2154.39818 1.40634 (215*2π)/3802 weeks
2163.50479 .91237 (216*2π)/3802 weeks
2173.77832 .73194 (217*2π)/3802 weeks
2183.61472 1.29876 (218*2π)/3802 weeks
2192.59976 1.36243 (219*2π)/3802 weeks
2202.58412 .72449 (220*2π)/3802 weeks
2212.43136 .8694 (221*2π)/3802 weeks
2221.75616 .81346 (222*2π)/3802 weeks
2231.72251 -.47291 (223*2π)/3802 weeks
2243.04696 -.69969 (224*2π)/3802 weeks
2253.33025 .50452 (225*2π)/3802 weeks
2262.24729 .66028 (226*2π)/3802 weeks
2271.97937 .0832 (227*2π)/3802 weeks
2282.23938 -.36501 (228*2π)/3802 weeks
2292.4112 -.67053 (229*2π)/3802 weeks
2302.82846 -.68409 (230*2π)/3802 weeks
2313.56122 -.2132 (231*2π)/3802 weeks
2323.69838 1.77871 (232*2π)/3802 weeks
233.96474 2.30767 (233*2π)/3802 weeks
234-.81934 .49338 (234*2π)/3802 weeks
235.20664 -1.55551 (235*2π)/3802 weeks
2361.74317 -1.53106 (236*2π)/3802 weeks
2371.40834 -.59303 (237*2π)/3802 weeks
238.98638 -1.03537 (238*2π)/3802 weeks
239-.309 -1.05287 (239*2π)/3802 weeks
240-.51278 -3.67102 (240*2π)/3802 weeks
2412.47154 -4.47646 (241*2π)/3802 weeks
2423.0685 -1.99793 (242*2π)/3802 weeks
2431.82984 -1.22226 (243*2π)/3802 weeks
244-.05266 -2.05474 (244*2π)/3802 weeks
245.81754 -5.17247 (245*2π)/3802 weeks
2463.52935 -4.71314 (246*2π)/3802 weeks
2473.82264 -3.24603 (247*2π)/3802 weeks
2483.48838 -3.1297 (248*2π)/3802 weeks
2493.85426 -2.9779 (249*2π)/3802 weeks
2503.15821 -2.54969 (250*2π)/3802 weeks
2513.52644 -3.17511 (251*2π)/3802 weeks
2523.60228 -2.5653 (252*2π)/3802 weeks
2533.46862 -3.06359 (253*2π)/3802 weeks
2543.9126 -2.20045 (254*2π)/3801 weeks
2553.03714 -2.4321 (255*2π)/3801 weeks
2563.1163 -3.60759 (256*2π)/3801 weeks
2574.5701 -4.03218 (257*2π)/3801 weeks
2585.635 -2.67425 (258*2π)/3801 weeks
2595.41325 -1.0831 (259*2π)/3801 weeks
2604.01181 -.89252 (260*2π)/3801 weeks
2613.48178 -1.58821 (261*2π)/3801 weeks
2623.73779 -1.44008 (262*2π)/3801 weeks
2633.18017 -1.36053 (263*2π)/3801 weeks
2643.86903 -1.95302 (264*2π)/3801 weeks
2653.97922 -.66525 (265*2π)/3801 weeks
2662.54186 -.27896 (266*2π)/3801 weeks
2671.73916 -1.41322 (267*2π)/3801 weeks
2681.34883 -1.84181 (268*2π)/3801 weeks
2691.17036 -3.15973 (269*2π)/3801 weeks
2703.33435 -4.34624 (270*2π)/3801 weeks
2714.55087 -1.75933 (271*2π)/3801 weeks
2722.15368 -.73166 (272*2π)/3801 weeks
273.66259 -2.22709 (273*2π)/3801 weeks
274.69558 -3.70158 (274*2π)/3801 weeks
2752.33853 -4.70456 (275*2π)/3801 weeks
2763.08824 -3.56784 (276*2π)/3801 weeks
2771.60561 -3.73934 (277*2π)/3801 weeks
2781.90198 -5.59918 (278*2π)/3801 weeks
2793.35471 -6.25762 (279*2π)/3801 weeks
2805.59962 -6.54836 (280*2π)/3801 weeks
2816.82109 -4.16062 (281*2π)/3801 weeks
2825.82125 -2.25514 (282*2π)/3801 weeks
2834.1584 -1.37445 (283*2π)/3801 weeks
2843.22336 -2.90302 (284*2π)/3801 weeks
2854.21179 -3.1939 (285*2π)/3801 weeks
2864.67856 -3.48602 (286*2π)/3801 weeks
2874.58264 -1.68948 (287*2π)/3801 weeks
2883.75624 -2.69837 (288*2π)/3801 weeks
2894.75781 -2.64955 (289*2π)/3801 weeks
2903.49918 -1.33134 (290*2π)/3801 weeks
2912.67771 -2.08726 (291*2π)/3801 weeks
2922.05966 -3.2846 (292*2π)/3801 weeks
2933.55548 -3.32218 (293*2π)/3801 weeks
2943.57195 -2.20193 (294*2π)/3801 weeks
2951.28746 -2.273 (295*2π)/3801 weeks
2961.54697 -4.57947 (296*2π)/3801 weeks
2972.11165 -4.44221 (297*2π)/3801 weeks
2983.1542 -5.05801 (298*2π)/3801 weeks
2993.5888 -4.47629 (299*2π)/3801 weeks
3002.81783 -4.9097 (300*2π)/3801 weeks
3013.8245 -6.01622 (301*2π)/3801 weeks
3025.25416 -5.23517 (302*2π)/3801 weeks
3035.54253 -3.30878 (303*2π)/3801 weeks
3044.11149 -2.35278 (304*2π)/3801 weeks
3052.41758 -3.38114 (305*2π)/3801 weeks
3063.1606 -4.45179 (306*2π)/3801 weeks
3074.49528 -3.85521 (307*2π)/3801 weeks
3082.92479 -3.13471 (308*2π)/3801 weeks
3092.693 -3.83744 (309*2π)/3801 weeks
3101.29559 -4.70886 (310*2π)/3801 weeks
3113.25047 -5.8502 (311*2π)/3801 weeks
3123.9372 -4.23988 (312*2π)/3801 weeks
3131.51449 -4.12049 (313*2π)/3801 weeks
3141.42926 -6.96344 (314*2π)/3801 weeks
3153.74124 -8.06942 (315*2π)/3801 weeks
3164.39075 -6.24406 (316*2π)/3801 weeks
3173.49669 -5.95591 (317*2π)/3801 weeks
3183.67082 -7.55024 (318*2π)/3801 weeks
3196.12526 -7.66581 (319*2π)/3801 weeks
3207.55239 -7.46203 (320*2π)/3801 weeks
3219.17272 -4.46547 (321*2π)/3801 weeks
3224.78384 -1.38783 (322*2π)/3801 weeks
3233.16698 -4.96007 (323*2π)/3801 weeks
3244.84644 -6.36927 (324*2π)/3801 weeks
3258.35693 -6.07449 (325*2π)/3801 weeks
3269.70849 -.43857 (326*2π)/3801 weeks
3273.11775 3.1456 (327*2π)/3801 weeks
328-2.05033 -1.74635 (328*2π)/3801 weeks
329.24726 -5.19252 (329*2π)/3801 weeks
3302.64773 -5.38459 (330*2π)/3801 weeks
3313.86897 -4.18015 (331*2π)/3801 weeks
3322.33802 -3.00861 (332*2π)/3801 weeks
333-1.90607 -2.89021 (333*2π)/3801 weeks
334-1.94844 -7.76733 (334*2π)/3801 weeks
335-.7401 -7.37145 (335*2π)/3801 weeks
3361.00088 -10.58374 (336*2π)/3801 weeks
3374.18057 -6.62088 (337*2π)/3801 weeks
338.10005 -4.34405 (338*2π)/3801 weeks
339-2.29727 -7.78298 (339*2π)/3801 weeks
340-2.75277 -11.02092 (340*2π)/3801 weeks
341.65521 -14.6283 (341*2π)/3801 weeks
3424.20285 -14.10719 (342*2π)/3801 weeks
3438.1289 -9.87181 (343*2π)/3801 weeks
3442.51207 -4.72522 (344*2π)/3801 weeks
345-1.05249 -10.24839 (345*2π)/3801 weeks
3461.99962 -11.79567 (346*2π)/3801 weeks
3472.83831 -12.10256 (347*2π)/3801 weeks
3484.97985 -10.48413 (348*2π)/3801 weeks
3494.77151 -7.11261 (349*2π)/3801 weeks
350-1.74512 -4.35962 (350*2π)/3801 weeks
351-7.22875 -11.73811 (351*2π)/3801 weeks
352-3.40308 -19.55882 (352*2π)/3801 weeks
3532.60184 -17.31653 (353*2π)/3801 weeks
3542.9855 -17.80118 (354*2π)/3801 weeks
3554.27013 -13.94505 (355*2π)/3801 weeks
356-.55835 -16.47941 (356*2π)/3801 weeks
3573.03799 -19.32087 (357*2π)/3801 weeks
3585.96416 -14.4435 (358*2π)/3801 weeks
359.84063 -11.47696 (359*2π)/3801 weeks
360-1.75422 -14.91819 (360*2π)/3801 weeks
361-7.49291 -20.78609 (361*2π)/3801 weeks
362-1.29341 -26.35919 (362*2π)/3801 weeks
3631.32752 -24.16996 (363*2π)/3801 weeks
3642.20517 -20.75588 (364*2π)/3801 weeks
365-3.58841 -23.86831 (365*2π)/3801 weeks
366-3.68072 -23.94993 (366*2π)/3801 weeks
367-12.12566 -25.82152 (367*2π)/3801 weeks
368-19.18989 -41.05581 (368*2π)/3801 weeks
369-3.83153 -59.65972 (369*2π)/3801 weeks
3708.46801 -50.2466 (370*2π)/3801 weeks
3711.5386 -46.82904 (371*2π)/3801 weeks
372-10.28604 -51.49888 (372*2π)/3801 weeks
373-19.046 -77.46059 (373*2π)/3801 weeks
3742.60374 -110.0017 (374*2π)/3801 weeks
37534.53827 -107.4442 (375*2π)/3801 weeks
37623.49244 -114.1899 (376*2π)/3801 weeks
37727.39117 -149.8244 (377*2π)/3801 weeks
37857.02165 -229.1012 (378*2π)/3801 weeks



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