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Fourier Analysis of SQQQ (ProShares UltraPro Short QQQ)


SQQQ (ProShares UltraPro Short QQQ) appears to have interesting cyclic behaviour every 37 weeks (95.1439*sine), 23 weeks (94.6316*sine), and 29 weeks (72.9925*sine).

SQQQ (ProShares UltraPro Short QQQ) has an average price of 856.97 (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 2/11/2010 to 3/13/2017 for SQQQ (ProShares UltraPro Short QQQ), this program was able to calculate the following Fourier Series:
Sequence #Cosine Coefficients Sine Coefficients FrequenciesPeriod
0856.9684   0 
1701.0757 810.6517 (1*2π)/371371 weeks
2296.2574 570.2338 (2*2π)/371186 weeks
3156.5182 448.2925 (3*2π)/371124 weeks
4125.4721 356.1047 (4*2π)/37193 weeks
579.95245 369.2894 (5*2π)/37174 weeks
6-31.08774 308.1628 (6*2π)/37162 weeks
7-60.66243 205.4599 (7*2π)/37153 weeks
8-37.27076 132.2777 (8*2π)/37146 weeks
9-11.09548 108.2282 (9*2π)/37141 weeks
10-1.57072 95.14394 (10*2π)/37137 weeks
11-1.17508 68.81658 (11*2π)/37134 weeks
1236.46508 59.81019 (12*2π)/37131 weeks
1335.99376 72.99248 (13*2π)/37129 weeks
1438.56135 57.76116 (14*2π)/37127 weeks
1563.07954 66.22254 (15*2π)/37125 weeks
1641.25285 94.63158 (16*2π)/37123 weeks
1724.86246 70.77254 (17*2π)/37122 weeks
1848.59052 62.57097 (18*2π)/37121 weeks
1940.26131 84.55756 (19*2π)/37120 weeks
2021.64902 71.11974 (20*2π)/37119 weeks
2118.22716 58.26912 (21*2π)/37118 weeks
2227.463 48.10033 (22*2π)/37117 weeks
2341.32222 53.6122 (23*2π)/37116 weeks
2433.43621 61.62727 (24*2π)/37115 weeks
2531.35695 50.35606 (25*2π)/37115 weeks
2640.2094 52.46003 (26*2π)/37114 weeks
2746.86742 59.23623 (27*2π)/37114 weeks
2835.42001 79.19453 (28*2π)/37113 weeks
2913.17946 75.52209 (29*2π)/37113 weeks
303.68331 51.86296 (30*2π)/37112 weeks
3120.29695 42.45465 (31*2π)/37112 weeks
3228.45206 56.31548 (32*2π)/37112 weeks
3313.10143 55.85513 (33*2π)/37111 weeks
349.22092 51.78078 (34*2π)/37111 weeks
352.61699 41.63805 (35*2π)/37111 weeks
3613.62663 29.78383 (36*2π)/37110 weeks
3725.48783 31.3681 (37*2π)/37110 weeks
3832.14684 45.71505 (38*2π)/37110 weeks
3917.54684 52.79678 (39*2π)/37110 weeks
409.89817 48.29809 (40*2π)/3719 weeks
415.03549 39.9859 (41*2π)/3719 weeks
429.71053 32.87849 (42*2π)/3719 weeks
4313.08334 39.09689 (43*2π)/3719 weeks
446.31853 39.71483 (44*2π)/3718 weeks
45-5.52828 35.506 (45*2π)/3718 weeks
46-2.39528 14.56357 (46*2π)/3718 weeks
4714.56986 14.62505 (47*2π)/3718 weeks
4813.88901 21.89718 (48*2π)/3718 weeks
4910.65485 20.58476 (49*2π)/3718 weeks
508.98509 9.8971 (50*2π)/3717 weeks
5121.20947 -.28293 (51*2π)/3717 weeks
5241.9924 8.82365 (52*2π)/3717 weeks
5342.20237 29.57318 (53*2π)/3717 weeks
5428.16082 34.92573 (54*2π)/3717 weeks
5525.66342 28.20042 (55*2π)/3717 weeks
5631.82993 31.10958 (56*2π)/3717 weeks
5728.46859 42.06391 (57*2π)/3717 weeks
5816.48433 42.23762 (58*2π)/3716 weeks
5912.04728 36.4113 (59*2π)/3716 weeks
609.62269 33.45721 (60*2π)/3716 weeks
617.40169 32.02111 (61*2π)/3716 weeks
625.1388 28.70232 (62*2π)/3716 weeks
632.56165 21.28781 (63*2π)/3716 weeks
6410.7417 13.19048 (64*2π)/3716 weeks
6518.61831 18.56497 (65*2π)/3716 weeks
6615.49647 22.88518 (66*2π)/3716 weeks
6714.33048 21.83098 (67*2π)/3716 weeks
6814.31452 20.01608 (68*2π)/3715 weeks
6915.15622 21.66679 (69*2π)/3715 weeks
7013.94889 23.93015 (70*2π)/3715 weeks
719.18722 18.64767 (71*2π)/3715 weeks
7217.2413 16.89635 (72*2π)/3715 weeks
7317.01043 20.55658 (73*2π)/3715 weeks
7416.56711 21.54267 (74*2π)/3715 weeks
7515.69442 26.04234 (75*2π)/3715 weeks
769.60442 26.94255 (76*2π)/3715 weeks
775.54591 21.03643 (77*2π)/3715 weeks
787.70958 16.99002 (78*2π)/3715 weeks
799.91337 15.16148 (79*2π)/3715 weeks
8011.91597 18.82528 (80*2π)/3715 weeks
815.32904 20.17069 (81*2π)/3715 weeks
825.60882 13.88855 (82*2π)/3715 weeks
836.24993 11.42056 (83*2π)/3714 weeks
8411.38795 9.21786 (84*2π)/3714 weeks
8510.51628 12.43173 (85*2π)/3714 weeks
866.80096 10.31665 (86*2π)/3714 weeks
8710.90839 6.40679 (87*2π)/3714 weeks
8812.81344 5.72372 (88*2π)/3714 weeks
8915.46708 5.81147 (89*2π)/3714 weeks
9019.09933 4.40557 (90*2π)/3714 weeks
9125.90708 8.04073 (91*2π)/3714 weeks
9222.62637 18.04404 (92*2π)/3714 weeks
9315.48969 19.43321 (93*2π)/3714 weeks
9412.01067 12.69249 (94*2π)/3714 weeks
9519.81304 8.60018 (95*2π)/3714 weeks
9622.66713 17.0746 (96*2π)/3714 weeks
9716.2401 20.63277 (97*2π)/3714 weeks
9811.12838 18.50349 (98*2π)/3714 weeks
9912.33071 15.9763 (99*2π)/3714 weeks
10014.15946 17.76723 (100*2π)/3714 weeks
1018.13791 21.57819 (101*2π)/3714 weeks
102.27327 14.75675 (102*2π)/3714 weeks
1035.93131 4.93569 (103*2π)/3714 weeks
10412.31746 8.8871 (104*2π)/3714 weeks
1059.94703 12.48259 (105*2π)/3714 weeks
1065.40921 12.2569 (106*2π)/3714 weeks
1072.72133 3.95478 (107*2π)/3713 weeks
10810.94208 -1.04352 (108*2π)/3713 weeks
10915.30366 2.64168 (109*2π)/3713 weeks
11014.85813 3.9835 (110*2π)/3713 weeks
11120.12354 4.66016 (111*2π)/3713 weeks
11218.94511 11.47991 (112*2π)/3713 weeks
11315.75004 12.95996 (113*2π)/3713 weeks
11411.91324 13.26911 (114*2π)/3713 weeks
1158.77199 9.71633 (115*2π)/3713 weeks
11611.46083 7.13955 (116*2π)/3713 weeks
11711.16907 8.86338 (117*2π)/3713 weeks
1189.4683 8.3653 (118*2π)/3713 weeks
1197.70797 6.14307 (119*2π)/3713 weeks
12010.38275 1.35641 (120*2π)/3713 weeks
12113.40943 2.82812 (121*2π)/3713 weeks
12213.6157 4.20273 (122*2π)/3713 weeks
12312.91584 4.45296 (123*2π)/3713 weeks
12412.80027 5.53973 (124*2π)/3713 weeks
12510.10248 2.21969 (125*2π)/3713 weeks
12615.11374 -1.55479 (126*2π)/3713 weeks
12717.38582 2.44048 (127*2π)/3713 weeks
12815.64521 3.84546 (128*2π)/3713 weeks
12915.24851 .77626 (129*2π)/3713 weeks
13020.13525 -1.71274 (130*2π)/3713 weeks
13126.57758 3.98889 (131*2π)/3713 weeks
13223.75126 13.90215 (132*2π)/3713 weeks
13315.04304 12.88072 (133*2π)/3713 weeks
13414.94975 8.10684 (134*2π)/3713 weeks
13518.69056 7.15171 (135*2π)/3713 weeks
13620.38654 16.55715 (136*2π)/3713 weeks
1378.10356 19.6536 (137*2π)/3713 weeks
1382.40496 10.42427 (138*2π)/3713 weeks
1394.46195 5.37768 (139*2π)/3713 weeks
1408.64921 3.34038 (140*2π)/3713 weeks
1417.56018 6.09477 (141*2π)/3713 weeks
1423.82753 2.58406 (142*2π)/3713 weeks
1437.14269 -3.12533 (143*2π)/3713 weeks
14412.57946 -4.56586 (144*2π)/3713 weeks
14514.19052 1.07856 (145*2π)/3713 weeks
14611.20624 1.39287 (146*2π)/3713 weeks
14710.79641 -1.19136 (147*2π)/3713 weeks
14811.08165 -3.57169 (148*2π)/3713 weeks
14913.75589 -5.5136 (149*2π)/3712 weeks
15018.41138 -5.91596 (150*2π)/3712 weeks
15122.14441 -1.41115 (151*2π)/3712 weeks
15218.61884 3.2684 (152*2π)/3712 weeks
15315.13384 -1.07582 (153*2π)/3712 weeks
15420.48405 -2.8364 (154*2π)/3712 weeks
15523.01118 1.4606 (155*2π)/3712 weeks
15621.8873 6.01492 (156*2π)/3712 weeks
15718.0638 6.28681 (157*2π)/3712 weeks
15817.48923 3.73744 (158*2π)/3712 weeks
15919.74344 4.56628 (159*2π)/3712 weeks
16019.49862 8.81694 (160*2π)/3712 weeks
16113.27694 11.45764 (161*2π)/3712 weeks
1629.40808 3.49853 (162*2π)/3712 weeks
16315.49263 1.36296 (163*2π)/3712 weeks
16415.61861 6.24221 (164*2π)/3712 weeks
16511.39161 6.36675 (165*2π)/3712 weeks
1669.28408 2.35979 (166*2π)/3712 weeks
16712.17313 -2.80587 (167*2π)/3712 weeks
16816.58746 -.27253 (168*2π)/3712 weeks
16917.8441 1.73083 (169*2π)/3712 weeks
17017.30786 4.82859 (170*2π)/3712 weeks
17113.94926 6.20253 (171*2π)/3712 weeks
17213.19487 3.06059 (172*2π)/3712 weeks
17315.02287 5.64658 (173*2π)/3712 weeks
17410.3632 8.00771 (174*2π)/3712 weeks
1755.58156 4.49963 (175*2π)/3712 weeks
1766.21363 .10488 (176*2π)/3712 weeks
1775.49184 -2.35392 (177*2π)/3712 weeks
1789.08364 -5.66753 (178*2π)/3712 weeks
17910.21144 -4.94284 (179*2π)/3712 weeks
1809.16934 -9.04434 (180*2π)/3712 weeks
18114.02043 -12.26413 (181*2π)/3712 weeks
18221.32368 -11.11303 (182*2π)/3712 weeks
18324.2161 -4.8005 (183*2π)/3712 weeks
18422.067 -4.64068 (184*2π)/3712 weeks
18525.60039 -2.88968 (185*2π)/3712 weeks
18625.60039 2.88968 (186*2π)/3712 weeks
18722.067 4.64068 (187*2π)/3712 weeks
18824.2161 4.8005 (188*2π)/3712 weeks
18921.32368 11.11303 (189*2π)/3712 weeks
19014.02043 12.26413 (190*2π)/3712 weeks
1919.16934 9.04434 (191*2π)/3712 weeks
19210.21144 4.94284 (192*2π)/3712 weeks
1939.08364 5.66753 (193*2π)/3712 weeks
1945.49184 2.35392 (194*2π)/3712 weeks
1956.21363 -.10488 (195*2π)/3712 weeks
1965.58156 -4.49963 (196*2π)/3712 weeks
19710.3632 -8.00771 (197*2π)/3712 weeks
19815.02287 -5.64658 (198*2π)/3712 weeks
19913.19487 -3.06059 (199*2π)/3712 weeks
20013.94926 -6.20253 (200*2π)/3712 weeks
20117.30786 -4.82859 (201*2π)/3712 weeks
20217.8441 -1.73083 (202*2π)/3712 weeks
20316.58746 .27253 (203*2π)/3712 weeks
20412.17313 2.80587 (204*2π)/3712 weeks
2059.28408 -2.35979 (205*2π)/3712 weeks
20611.39161 -6.36675 (206*2π)/3712 weeks
20715.61861 -6.24221 (207*2π)/3712 weeks
20815.49263 -1.36296 (208*2π)/3712 weeks
2099.40808 -3.49853 (209*2π)/3712 weeks
21013.27694 -11.45764 (210*2π)/3712 weeks
21119.49862 -8.81694 (211*2π)/3712 weeks
21219.74344 -4.56628 (212*2π)/3712 weeks
21317.48923 -3.73744 (213*2π)/3712 weeks
21418.0638 -6.28681 (214*2π)/3712 weeks
21521.8873 -6.01492 (215*2π)/3712 weeks
21623.01118 -1.4606 (216*2π)/3712 weeks
21720.48405 2.8364 (217*2π)/3712 weeks
21815.13384 1.07582 (218*2π)/3712 weeks
21918.61884 -3.2684 (219*2π)/3712 weeks
22022.14441 1.41115 (220*2π)/3712 weeks
22118.41138 5.91596 (221*2π)/3712 weeks
22213.75589 5.5136 (222*2π)/3712 weeks
22311.08165 3.57169 (223*2π)/3712 weeks
22410.79641 1.19136 (224*2π)/3712 weeks
22511.20624 -1.39287 (225*2π)/3712 weeks
22614.19052 -1.07856 (226*2π)/3712 weeks
22712.57946 4.56586 (227*2π)/3712 weeks
2287.14269 3.12533 (228*2π)/3712 weeks
2293.82753 -2.58406 (229*2π)/3712 weeks
2307.56018 -6.09477 (230*2π)/3712 weeks
2318.64921 -3.34038 (231*2π)/3712 weeks
2324.46195 -5.37768 (232*2π)/3712 weeks
2332.40496 -10.42427 (233*2π)/3712 weeks
2348.10356 -19.6536 (234*2π)/3712 weeks
23520.38654 -16.55715 (235*2π)/3712 weeks
23618.69056 -7.15171 (236*2π)/3712 weeks
23714.94975 -8.10684 (237*2π)/3712 weeks
23815.04304 -12.88072 (238*2π)/3712 weeks
23923.75126 -13.90215 (239*2π)/3712 weeks
24026.57758 -3.98889 (240*2π)/3712 weeks
24120.13525 1.71274 (241*2π)/3712 weeks
24215.24851 -.77626 (242*2π)/3712 weeks
24315.64521 -3.84546 (243*2π)/3712 weeks
24417.38582 -2.44048 (244*2π)/3712 weeks
24515.11374 1.55479 (245*2π)/3712 weeks
24610.10248 -2.21969 (246*2π)/3712 weeks
24712.80027 -5.53973 (247*2π)/3712 weeks
24812.91584 -4.45296 (248*2π)/3711 weeks
24913.6157 -4.20273 (249*2π)/3711 weeks
25013.40943 -2.82812 (250*2π)/3711 weeks
25110.38275 -1.35641 (251*2π)/3711 weeks
2527.70797 -6.14307 (252*2π)/3711 weeks
2539.4683 -8.3653 (253*2π)/3711 weeks
25411.16907 -8.86338 (254*2π)/3711 weeks
25511.46083 -7.13955 (255*2π)/3711 weeks
2568.77199 -9.71633 (256*2π)/3711 weeks
25711.91324 -13.26911 (257*2π)/3711 weeks
25815.75004 -12.95996 (258*2π)/3711 weeks
25918.94511 -11.47991 (259*2π)/3711 weeks
26020.12354 -4.66016 (260*2π)/3711 weeks
26114.85813 -3.9835 (261*2π)/3711 weeks
26215.30366 -2.64168 (262*2π)/3711 weeks
26310.94208 1.04352 (263*2π)/3711 weeks
2642.72133 -3.95478 (264*2π)/3711 weeks
2655.40921 -12.2569 (265*2π)/3711 weeks
2669.94703 -12.48259 (266*2π)/3711 weeks
26712.31746 -8.8871 (267*2π)/3711 weeks
2685.93131 -4.93569 (268*2π)/3711 weeks
269.27327 -14.75675 (269*2π)/3711 weeks
2708.13791 -21.57819 (270*2π)/3711 weeks
27114.15946 -17.76723 (271*2π)/3711 weeks
27212.33071 -15.9763 (272*2π)/3711 weeks
27311.12838 -18.50349 (273*2π)/3711 weeks
27416.2401 -20.63277 (274*2π)/3711 weeks
27522.66713 -17.0746 (275*2π)/3711 weeks
27619.81304 -8.60018 (276*2π)/3711 weeks
27712.01067 -12.69249 (277*2π)/3711 weeks
27815.48969 -19.43321 (278*2π)/3711 weeks
27922.62637 -18.04404 (279*2π)/3711 weeks
28025.90708 -8.04073 (280*2π)/3711 weeks
28119.09933 -4.40557 (281*2π)/3711 weeks
28215.46708 -5.81147 (282*2π)/3711 weeks
28312.81344 -5.72372 (283*2π)/3711 weeks
28410.90839 -6.40679 (284*2π)/3711 weeks
2856.80096 -10.31665 (285*2π)/3711 weeks
28610.51628 -12.43173 (286*2π)/3711 weeks
28711.38795 -9.21786 (287*2π)/3711 weeks
2886.24993 -11.42056 (288*2π)/3711 weeks
2895.60882 -13.88855 (289*2π)/3711 weeks
2905.32904 -20.17069 (290*2π)/3711 weeks
29111.91597 -18.82528 (291*2π)/3711 weeks
2929.91337 -15.16148 (292*2π)/3711 weeks
2937.70958 -16.99002 (293*2π)/3711 weeks
2945.54591 -21.03643 (294*2π)/3711 weeks
2959.60442 -26.94255 (295*2π)/3711 weeks
29615.69442 -26.04234 (296*2π)/3711 weeks
29716.56711 -21.54267 (297*2π)/3711 weeks
29817.01043 -20.55658 (298*2π)/3711 weeks
29917.2413 -16.89635 (299*2π)/3711 weeks
3009.18722 -18.64767 (300*2π)/3711 weeks
30113.94889 -23.93015 (301*2π)/3711 weeks
30215.15622 -21.66679 (302*2π)/3711 weeks
30314.31452 -20.01608 (303*2π)/3711 weeks
30414.33048 -21.83098 (304*2π)/3711 weeks
30515.49647 -22.88518 (305*2π)/3711 weeks
30618.61831 -18.56497 (306*2π)/3711 weeks
30710.7417 -13.19048 (307*2π)/3711 weeks
3082.56165 -21.28781 (308*2π)/3711 weeks
3095.1388 -28.70232 (309*2π)/3711 weeks
3107.40169 -32.02111 (310*2π)/3711 weeks
3119.62269 -33.45721 (311*2π)/3711 weeks
31212.04728 -36.4113 (312*2π)/3711 weeks
31316.48433 -42.23762 (313*2π)/3711 weeks
31428.46859 -42.06391 (314*2π)/3711 weeks
31531.82993 -31.10958 (315*2π)/3711 weeks
31625.66342 -28.20042 (316*2π)/3711 weeks
31728.16082 -34.92573 (317*2π)/3711 weeks
31842.20237 -29.57318 (318*2π)/3711 weeks
31941.9924 -8.82365 (319*2π)/3711 weeks
32021.20947 .28293 (320*2π)/3711 weeks
3218.98509 -9.8971 (321*2π)/3711 weeks
32210.65485 -20.58476 (322*2π)/3711 weeks
32313.88901 -21.89718 (323*2π)/3711 weeks
32414.56986 -14.62505 (324*2π)/3711 weeks
325-2.39528 -14.56357 (325*2π)/3711 weeks
326-5.52828 -35.506 (326*2π)/3711 weeks
3276.31853 -39.71483 (327*2π)/3711 weeks
32813.08334 -39.09689 (328*2π)/3711 weeks
3299.71053 -32.87849 (329*2π)/3711 weeks
3305.03549 -39.9859 (330*2π)/3711 weeks
3319.89817 -48.29809 (331*2π)/3711 weeks
33217.54684 -52.79678 (332*2π)/3711 weeks
33332.14684 -45.71505 (333*2π)/3711 weeks
33425.48783 -31.3681 (334*2π)/3711 weeks
33513.62663 -29.78383 (335*2π)/3711 weeks
3362.61699 -41.63805 (336*2π)/3711 weeks
3379.22092 -51.78078 (337*2π)/3711 weeks
33813.10143 -55.85513 (338*2π)/3711 weeks
33928.45206 -56.31548 (339*2π)/3711 weeks
34020.29695 -42.45465 (340*2π)/3711 weeks
3413.68331 -51.86296 (341*2π)/3711 weeks
34213.17946 -75.52209 (342*2π)/3711 weeks
34335.42001 -79.19453 (343*2π)/3711 weeks
34446.86742 -59.23623 (344*2π)/3711 weeks
34540.2094 -52.46003 (345*2π)/3711 weeks
34631.35695 -50.35606 (346*2π)/3711 weeks
34733.43621 -61.62727 (347*2π)/3711 weeks
34841.32222 -53.6122 (348*2π)/3711 weeks
34927.463 -48.10033 (349*2π)/3711 weeks
35018.22716 -58.26912 (350*2π)/3711 weeks
35121.64902 -71.11974 (351*2π)/3711 weeks
35240.26131 -84.55756 (352*2π)/3711 weeks
35348.59052 -62.57097 (353*2π)/3711 weeks
35424.86246 -70.77254 (354*2π)/3711 weeks
35541.25285 -94.63158 (355*2π)/3711 weeks
35663.07954 -66.22254 (356*2π)/3711 weeks
35738.56135 -57.76116 (357*2π)/3711 weeks
35835.99376 -72.99248 (358*2π)/3711 weeks
35936.46508 -59.81019 (359*2π)/3711 weeks
360-1.17508 -68.81658 (360*2π)/3711 weeks
361-1.57072 -95.14394 (361*2π)/3711 weeks
362-11.09548 -108.2282 (362*2π)/3711 weeks
363-37.27076 -132.2777 (363*2π)/3711 weeks
364-60.66243 -205.4599 (364*2π)/3711 weeks
365-31.08774 -308.1628 (365*2π)/3711 weeks
36679.95245 -369.2894 (366*2π)/3711 weeks
367125.4721 -356.1047 (367*2π)/3711 weeks
368156.5182 -448.2925 (368*2π)/3711 weeks
369296.2574 -570.2338 (369*2π)/3711 weeks

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