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# Fourier Analysis of UVXY (ProShares Ultra VIX Short-Term Futures)

UVXY (ProShares Ultra VIX Short-Term Futures) appears to have interesting cyclic behaviour every 35 weeks (36592.66*sine), 32 weeks (33428.35*sine), and 21 weeks (9816.036*cosine).

UVXY (ProShares Ultra VIX Short-Term Futures) has an average price of 184,642.6 (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 10/4/2011 to 6/18/2018 for UVXY (ProShares Ultra VIX Short-Term Futures), this program was able to calculate the following Fourier Series:
Sequence #Cosine Coefficients Sine Coefficients FrequenciesPeriod
0184,642.6   0
1267,650.4 188,848.2 (1*2π)/351351 weeks
2111,795.5 229,127.3 (2*2π)/351176 weeks
35,055.124 180,864 (3*2π)/351117 weeks
4-35,617.06 109,814.3 (4*2π)/35188 weeks
5-30,890.9 55,541.54 (5*2π)/35170 weeks
6-7,109.103 29,591.35 (6*2π)/35159 weeks
711,156.82 28,903.3 (7*2π)/35150 weeks
816,983.24 37,079.31 (8*2π)/35144 weeks
99,526.651 41,773.02 (9*2π)/35139 weeks
103,873.155 36,592.66 (10*2π)/35135 weeks
112,685.452 33,428.35 (11*2π)/35132 weeks
12-295.7067 30,785.23 (12*2π)/35129 weeks
13-3,311.123 25,220.73 (13*2π)/35127 weeks
14-2,189.061 18,881.99 (14*2π)/35125 weeks
152,128.903 13,561.75 (15*2π)/35123 weeks
169,053.374 15,035.46 (16*2π)/35122 weeks
179,816.036 20,722.38 (17*2π)/35121 weeks
185,271.993 23,440.44 (18*2π)/35120 weeks
19295.2153 22,175.88 (19*2π)/35118 weeks
20-2,445.606 17,494.67 (20*2π)/35118 weeks
21-2,675.687 12,431.23 (21*2π)/35117 weeks
221,677.745 7,847.404 (22*2π)/35116 weeks
237,389.07 8,919.055 (23*2π)/35115 weeks
249,214.367 14,472.8 (24*2π)/35115 weeks
254,277.364 17,793.94 (25*2π)/35114 weeks
26525.3197 14,906.95 (26*2π)/35114 weeks
27529.1658 12,247.13 (27*2π)/35113 weeks
28599.7543 10,975.47 (28*2π)/35113 weeks
29410.1715 9,243.749 (29*2π)/35112 weeks
301,892.905 6,026.704 (30*2π)/35112 weeks
316,974.298 6,572.218 (31*2π)/35111 weeks
327,961.55 11,191.35 (32*2π)/35111 weeks
334,626.886 14,545.23 (33*2π)/35111 weeks
34-96.74403 13,428.85 (34*2π)/35110 weeks
35-2,374.26 9,267.689 (35*2π)/35110 weeks
36-1,002.085 4,927.28 (36*2π)/35110 weeks
373,301.472 2,279.408 (37*2π)/3519 weeks
388,637.135 5,040.412 (38*2π)/3519 weeks
398,552.533 11,213.04 (39*2π)/3519 weeks
402,912.579 14,358.86 (40*2π)/3519 weeks
41-2,987.691 10,603.91 (41*2π)/3519 weeks
42-2,116.312 3,659.231 (42*2π)/3518 weeks
433,607.889 2,128.561 (43*2π)/3518 weeks
446,509.137 5,253.405 (44*2π)/3518 weeks
455,377.996 8,618.696 (45*2π)/3518 weeks
462,758.171 8,720.845 (46*2π)/3518 weeks
471,339.749 7,541.428 (47*2π)/3517 weeks
481,384.697 5,286.349 (48*2π)/3517 weeks
493,433.45 4,857.568 (49*2π)/3517 weeks
503,929.565 6,481.297 (50*2π)/3517 weeks
512,244.506 7,072.377 (51*2π)/3517 weeks
521,161.214 5,326.871 (52*2π)/3517 weeks
532,391.854 3,619.951 (53*2π)/3517 weeks
544,499.091 3,698.127 (54*2π)/3517 weeks
555,750.693 6,068.181 (55*2π)/3516 weeks
563,477.453 8,430.604 (56*2π)/3516 weeks
57467.7934 7,343.561 (57*2π)/3516 weeks
58-547.7125 4,252.587 (58*2π)/3516 weeks
591,502.157 1,488.882 (59*2π)/3516 weeks
605,171.952 1,716.249 (60*2π)/3516 weeks
616,574.445 5,098.027 (61*2π)/3516 weeks
624,428.274 7,765.119 (62*2π)/3516 weeks
631,285.726 7,366.891 (63*2π)/3516 weeks
64-135.6998 4,924.967 (64*2π)/3515 weeks
65482.4391 2,243.403 (65*2π)/3515 weeks
663,535.02 983.8291 (66*2π)/3515 weeks
675,895.173 3,469.159 (67*2π)/3515 weeks
684,329.716 6,038.878 (68*2π)/3515 weeks
692,296.605 5,656.639 (69*2π)/3515 weeks
701,614.574 4,231.393 (70*2π)/3515 weeks
712,377.714 3,481.149 (71*2π)/3515 weeks
722,569.716 3,570.628 (72*2π)/3515 weeks
732,594.008 3,269.39 (73*2π)/3515 weeks
743,066.514 3,234.495 (74*2π)/3515 weeks
753,007.581 3,229.141 (75*2π)/3515 weeks
763,412.508 3,219.575 (76*2π)/3515 weeks
773,931.804 3,611.551 (77*2π)/3515 weeks
783,388.826 5,015.984 (78*2π)/3515 weeks
791,276.03 4,756.977 (79*2π)/3514 weeks
80913.5878 2,295.06 (80*2π)/3514 weeks
812,704.146 1,322.232 (81*2π)/3514 weeks
824,281.5 2,014.145 (82*2π)/3514 weeks
834,399.653 3,654.217 (83*2π)/3514 weeks
843,277.78 4,383.414 (84*2π)/3514 weeks
852,061.586 4,094.058 (85*2π)/3514 weeks
861,797.899 2,709.005 (86*2π)/3514 weeks
872,601.201 2,436.335 (87*2π)/3514 weeks
882,740.187 2,537.45 (88*2π)/3514 weeks
892,627.783 2,327.676 (89*2π)/3514 weeks
903,286.382 1,930.313 (90*2π)/3514 weeks
914,022.474 2,637.457 (91*2π)/3514 weeks
923,456.151 3,625.321 (92*2π)/3514 weeks
932,608.978 3,739.106 (93*2π)/3514 weeks
941,637.684 3,004.523 (94*2π)/3514 weeks
951,881.524 2,005.357 (95*2π)/3514 weeks
962,325.408 1,348.838 (96*2π)/3514 weeks
973,454.799 1,052.986 (97*2π)/3514 weeks
984,543.053 2,011.237 (98*2π)/3514 weeks
994,214.001 3,644.722 (99*2π)/3514 weeks
1002,456.206 4,241.596 (100*2π)/3514 weeks
1011,077.212 2,990.014 (101*2π)/3513 weeks
1021,157.618 1,344.008 (102*2π)/3513 weeks
1032,399.269 128.899 (103*2π)/3513 weeks
1044,507.738 422.0941 (104*2π)/3513 weeks
1055,239.325 2,823.225 (105*2π)/3513 weeks
1063,283.268 4,347.689 (106*2π)/3513 weeks
1071,282.545 3,489.672 (107*2π)/3513 weeks
108757.4129 1,419.167 (108*2π)/3513 weeks
1092,454.113 -166.9727 (109*2π)/3513 weeks
1104,465.363 876.7538 (110*2π)/3513 weeks
1114,143.462 2,760.439 (111*2π)/3513 weeks
1122,524.41 3,126.248 (112*2π)/3513 weeks
1131,758.776 1,841.655 (113*2π)/3513 weeks
1142,478.617 890.207 (114*2π)/3513 weeks
1153,567.276 1,044.838 (115*2π)/3513 weeks
1163,833.949 2,280.615 (116*2π)/3513 weeks
1172,500.111 2,893.979 (117*2π)/3513 weeks
1181,465.602 1,780.018 (118*2π)/3513 weeks
1192,019.121 281.7216 (119*2π)/3513 weeks
1203,697.103 153.3708 (120*2π)/3513 weeks
1214,478.569 1,581.53 (121*2π)/3513 weeks
1223,682.262 2,901.846 (122*2π)/3513 weeks
1232,066.236 2,924.687 (123*2π)/3513 weeks
1241,204.148 1,611.287 (124*2π)/3513 weeks
1251,733.401 57.21331 (125*2π)/3513 weeks
1263,406.519 -313.2202 (126*2π)/3513 weeks
1274,581.334 1,043.98 (127*2π)/3513 weeks
1283,714.022 2,530.048 (128*2π)/3513 weeks
1292,355.651 2,351.908 (129*2π)/3513 weeks
1301,952.82 1,335.824 (130*2π)/3513 weeks
1312,539.002 855.4633 (131*2π)/3513 weeks
1322,560.369 947.9763 (132*2π)/3513 weeks
1332,794.66 591.5591 (133*2π)/3513 weeks
1343,220.782 676.2943 (134*2π)/3513 weeks
1353,527.995 1,059.968 (135*2π)/3513 weeks
1363,184.75 1,640.499 (136*2π)/3513 weeks
1372,721.318 1,435.427 (137*2π)/3513 weeks
1382,576.686 1,415.61 (138*2π)/3513 weeks
1392,189.951 1,045.602 (139*2π)/3513 weeks
1402,275.187 454.2407 (140*2π)/3513 weeks
1412,904.302 -7.83123 (141*2π)/3512 weeks
1423,657.141 422.3623 (142*2π)/3512 weeks
1433,747.922 1,139.511 (143*2π)/3512 weeks
1443,248.701 1,856.674 (144*2π)/3512 weeks
1452,098.775 1,775.7 (145*2π)/3512 weeks
1461,556.457 615.4887 (146*2π)/3512 weeks
1472,435.568 -522.7712 (147*2π)/3512 weeks
1483,709.63 -41.6166 (148*2π)/3512 weeks
1493,737.296 1,007.708 (149*2π)/3512 weeks
1502,895.599 1,510.376 (150*2π)/3512 weeks
1512,149.019 759.4629 (151*2π)/3512 weeks
1522,795.198 11.08596 (152*2π)/3512 weeks
1533,490.746 552.6299 (153*2π)/3512 weeks
1543,078.55 1,364.987 (154*2π)/3512 weeks
1551,931.394 1,137.468 (155*2π)/3512 weeks
1561,796.375 -130.8816 (156*2π)/3512 weeks
1572,760.299 -996.3721 (157*2π)/3512 weeks
1584,396.222 -463.3013 (158*2π)/3512 weeks
1594,469.343 1,361.871 (159*2π)/3512 weeks
1603,018.017 2,405.633 (160*2π)/3512 weeks
1611,026.3 1,727.466 (161*2π)/3512 weeks
162750.0397 -670.3862 (162*2π)/3512 weeks
1632,566.197 -1,809.467 (163*2π)/3512 weeks
1644,340.382 -1,195.553 (164*2π)/3512 weeks
1654,952.315 535.5976 (165*2π)/3512 weeks
1663,889.208 2,201.325 (166*2π)/3512 weeks
1671,796.755 2,163.859 (167*2π)/3512 weeks
168727.8611 353.7726 (168*2π)/3512 weeks
1691,571.162 -1,307.745 (169*2π)/3512 weeks
1703,293.203 -1,715.941 (170*2π)/3512 weeks
1714,466.051 -459.6311 (171*2π)/3512 weeks
1724,050.734 926.3475 (172*2π)/3512 weeks
1732,923.997 1,189.738 (173*2π)/3512 weeks
1742,340.357 774.8754 (174*2π)/3512 weeks
1752,062.911 267.6823 (175*2π)/3512 weeks
1762,062.911 -267.6823 (176*2π)/3512 weeks
1772,340.357 -774.8754 (177*2π)/3512 weeks
1782,923.997 -1,189.738 (178*2π)/3512 weeks
1794,050.734 -926.3475 (179*2π)/3512 weeks
1804,466.051 459.6311 (180*2π)/3512 weeks
1813,293.203 1,715.941 (181*2π)/3512 weeks
1821,571.162 1,307.745 (182*2π)/3512 weeks
183727.8611 -353.7726 (183*2π)/3512 weeks
1841,796.755 -2,163.859 (184*2π)/3512 weeks
1853,889.208 -2,201.325 (185*2π)/3512 weeks
1864,952.315 -535.5976 (186*2π)/3512 weeks
1874,340.382 1,195.553 (187*2π)/3512 weeks
1882,566.197 1,809.467 (188*2π)/3512 weeks
189750.0397 670.3862 (189*2π)/3512 weeks
1901,026.3 -1,727.466 (190*2π)/3512 weeks
1913,018.017 -2,405.633 (191*2π)/3512 weeks
1924,469.343 -1,361.871 (192*2π)/3512 weeks
1934,396.222 463.3013 (193*2π)/3512 weeks
1942,760.299 996.3721 (194*2π)/3512 weeks
1951,796.375 130.8816 (195*2π)/3512 weeks
1961,931.394 -1,137.468 (196*2π)/3512 weeks
1973,078.55 -1,364.987 (197*2π)/3512 weeks
1983,490.746 -552.6299 (198*2π)/3512 weeks
1992,795.198 -11.08596 (199*2π)/3512 weeks
2002,149.019 -759.4629 (200*2π)/3512 weeks
2012,895.599 -1,510.376 (201*2π)/3512 weeks
2023,737.296 -1,007.708 (202*2π)/3512 weeks
2033,709.63 41.6166 (203*2π)/3512 weeks
2042,435.568 522.7712 (204*2π)/3512 weeks
2051,556.457 -615.4887 (205*2π)/3512 weeks
2062,098.775 -1,775.7 (206*2π)/3512 weeks
2073,248.701 -1,856.674 (207*2π)/3512 weeks
2083,747.922 -1,139.511 (208*2π)/3512 weeks
2093,657.141 -422.3623 (209*2π)/3512 weeks
2102,904.302 7.83123 (210*2π)/3512 weeks
2112,275.187 -454.2407 (211*2π)/3512 weeks
2122,189.951 -1,045.602 (212*2π)/3512 weeks
2132,576.686 -1,415.61 (213*2π)/3512 weeks
2142,721.318 -1,435.427 (214*2π)/3512 weeks
2153,184.75 -1,640.499 (215*2π)/3512 weeks
2163,527.995 -1,059.968 (216*2π)/3512 weeks
2173,220.782 -676.2943 (217*2π)/3512 weeks
2182,794.66 -591.5591 (218*2π)/3512 weeks
2192,560.369 -947.9763 (219*2π)/3512 weeks
2202,539.002 -855.4633 (220*2π)/3512 weeks
2211,952.82 -1,335.824 (221*2π)/3512 weeks
2222,355.651 -2,351.908 (222*2π)/3512 weeks
2233,714.022 -2,530.048 (223*2π)/3512 weeks
2244,581.334 -1,043.98 (224*2π)/3512 weeks
2253,406.519 313.2202 (225*2π)/3512 weeks
2261,733.401 -57.21331 (226*2π)/3512 weeks
2271,204.148 -1,611.287 (227*2π)/3512 weeks
2282,066.236 -2,924.687 (228*2π)/3512 weeks
2293,682.262 -2,901.846 (229*2π)/3512 weeks
2304,478.569 -1,581.53 (230*2π)/3512 weeks
2313,697.103 -153.3708 (231*2π)/3512 weeks
2322,019.121 -281.7216 (232*2π)/3512 weeks
2331,465.602 -1,780.018 (233*2π)/3512 weeks
2342,500.111 -2,893.979 (234*2π)/3512 weeks
2353,833.949 -2,280.615 (235*2π)/3511 weeks
2363,567.276 -1,044.838 (236*2π)/3511 weeks
2372,478.617 -890.207 (237*2π)/3511 weeks
2381,758.776 -1,841.655 (238*2π)/3511 weeks
2392,524.41 -3,126.248 (239*2π)/3511 weeks
2404,143.462 -2,760.439 (240*2π)/3511 weeks
2414,465.363 -876.7538 (241*2π)/3511 weeks
2422,454.113 166.9727 (242*2π)/3511 weeks
243757.4129 -1,419.167 (243*2π)/3511 weeks
2441,282.545 -3,489.672 (244*2π)/3511 weeks
2453,283.268 -4,347.689 (245*2π)/3511 weeks
2465,239.325 -2,823.225 (246*2π)/3511 weeks
2474,507.738 -422.0941 (247*2π)/3511 weeks
2482,399.269 -128.899 (248*2π)/3511 weeks
2491,157.618 -1,344.008 (249*2π)/3511 weeks
2501,077.212 -2,990.014 (250*2π)/3511 weeks
2512,456.206 -4,241.596 (251*2π)/3511 weeks
2524,214.001 -3,644.722 (252*2π)/3511 weeks
2534,543.053 -2,011.237 (253*2π)/3511 weeks
2543,454.799 -1,052.986 (254*2π)/3511 weeks
2552,325.408 -1,348.838 (255*2π)/3511 weeks
2561,881.524 -2,005.357 (256*2π)/3511 weeks
2571,637.684 -3,004.523 (257*2π)/3511 weeks
2582,608.978 -3,739.106 (258*2π)/3511 weeks
2593,456.151 -3,625.321 (259*2π)/3511 weeks
2604,022.474 -2,637.457 (260*2π)/3511 weeks
2613,286.382 -1,930.313 (261*2π)/3511 weeks
2622,627.783 -2,327.676 (262*2π)/3511 weeks
2632,740.187 -2,537.45 (263*2π)/3511 weeks
2642,601.201 -2,436.335 (264*2π)/3511 weeks
2651,797.899 -2,709.005 (265*2π)/3511 weeks
2662,061.586 -4,094.058 (266*2π)/3511 weeks
2673,277.78 -4,383.414 (267*2π)/3511 weeks
2684,399.653 -3,654.217 (268*2π)/3511 weeks
2694,281.5 -2,014.145 (269*2π)/3511 weeks
2702,704.146 -1,322.232 (270*2π)/3511 weeks
271913.5878 -2,295.06 (271*2π)/3511 weeks
2721,276.03 -4,756.977 (272*2π)/3511 weeks
2733,388.826 -5,015.984 (273*2π)/3511 weeks
2743,931.804 -3,611.551 (274*2π)/3511 weeks
2753,412.508 -3,219.575 (275*2π)/3511 weeks
2763,007.581 -3,229.141 (276*2π)/3511 weeks
2773,066.514 -3,234.495 (277*2π)/3511 weeks
2782,594.008 -3,269.39 (278*2π)/3511 weeks
2792,569.716 -3,570.628 (279*2π)/3511 weeks
2802,377.714 -3,481.149 (280*2π)/3511 weeks
2811,614.574 -4,231.393 (281*2π)/3511 weeks
2822,296.605 -5,656.639 (282*2π)/3511 weeks
2834,329.716 -6,038.878 (283*2π)/3511 weeks
2845,895.173 -3,469.159 (284*2π)/3511 weeks
2853,535.02 -983.8291 (285*2π)/3511 weeks
286482.4391 -2,243.403 (286*2π)/3511 weeks
287-135.6998 -4,924.967 (287*2π)/3511 weeks
2881,285.726 -7,366.891 (288*2π)/3511 weeks
2894,428.274 -7,765.119 (289*2π)/3511 weeks
2906,574.445 -5,098.027 (290*2π)/3511 weeks
2915,171.952 -1,716.249 (291*2π)/3511 weeks
2921,502.157 -1,488.882 (292*2π)/3511 weeks
293-547.7125 -4,252.587 (293*2π)/3511 weeks
294467.7934 -7,343.561 (294*2π)/3511 weeks
2953,477.453 -8,430.604 (295*2π)/3511 weeks
2965,750.693 -6,068.181 (296*2π)/3511 weeks
2974,499.091 -3,698.127 (297*2π)/3511 weeks
2982,391.854 -3,619.951 (298*2π)/3511 weeks
2991,161.214 -5,326.871 (299*2π)/3511 weeks
3002,244.506 -7,072.377 (300*2π)/3511 weeks
3013,929.565 -6,481.297 (301*2π)/3511 weeks
3023,433.45 -4,857.568 (302*2π)/3511 weeks
3031,384.697 -5,286.349 (303*2π)/3511 weeks
3041,339.749 -7,541.428 (304*2π)/3511 weeks
3052,758.171 -8,720.845 (305*2π)/3511 weeks
3065,377.996 -8,618.696 (306*2π)/3511 weeks
3076,509.137 -5,253.405 (307*2π)/3511 weeks
3083,607.889 -2,128.561 (308*2π)/3511 weeks
309-2,116.312 -3,659.231 (309*2π)/3511 weeks
310-2,987.691 -10,603.91 (310*2π)/3511 weeks
3112,912.579 -14,358.86 (311*2π)/3511 weeks
3128,552.533 -11,213.04 (312*2π)/3511 weeks
3138,637.135 -5,040.412 (313*2π)/3511 weeks
3143,301.472 -2,279.408 (314*2π)/3511 weeks
315-1,002.085 -4,927.28 (315*2π)/3511 weeks
316-2,374.26 -9,267.689 (316*2π)/3511 weeks
317-96.74403 -13,428.85 (317*2π)/3511 weeks
3184,626.886 -14,545.23 (318*2π)/3511 weeks
3197,961.55 -11,191.35 (319*2π)/3511 weeks
3206,974.298 -6,572.218 (320*2π)/3511 weeks
3211,892.905 -6,026.704 (321*2π)/3511 weeks
322410.1715 -9,243.749 (322*2π)/3511 weeks
323599.7543 -10,975.47 (323*2π)/3511 weeks
324529.1658 -12,247.13 (324*2π)/3511 weeks
325525.3197 -14,906.95 (325*2π)/3511 weeks
3264,277.364 -17,793.94 (326*2π)/3511 weeks
3279,214.367 -14,472.8 (327*2π)/3511 weeks
3287,389.07 -8,919.055 (328*2π)/3511 weeks
3291,677.745 -7,847.404 (329*2π)/3511 weeks
330-2,675.687 -12,431.23 (330*2π)/3511 weeks
331-2,445.606 -17,494.67 (331*2π)/3511 weeks
332295.2153 -22,175.88 (332*2π)/3511 weeks
3335,271.993 -23,440.44 (333*2π)/3511 weeks
3349,816.036 -20,722.38 (334*2π)/3511 weeks
3359,053.374 -15,035.46 (335*2π)/3511 weeks
3362,128.903 -13,561.75 (336*2π)/3511 weeks
337-2,189.061 -18,881.99 (337*2π)/3511 weeks
338-3,311.123 -25,220.73 (338*2π)/3511 weeks
339-295.7067 -30,785.23 (339*2π)/3511 weeks
3402,685.452 -33,428.35 (340*2π)/3511 weeks
3413,873.155 -36,592.66 (341*2π)/3511 weeks
3429,526.651 -41,773.02 (342*2π)/3511 weeks
34316,983.24 -37,079.31 (343*2π)/3511 weeks
34411,156.82 -28,903.3 (344*2π)/3511 weeks
345-7,109.103 -29,591.35 (345*2π)/3511 weeks
346-30,890.9 -55,541.54 (346*2π)/3511 weeks
347-35,617.06 -109,814.3 (347*2π)/3511 weeks
3485,055.124 -180,864 (348*2π)/3511 weeks
349111,795.5 -229,127.3 (349*2π)/3511 weeks