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Fourier Analysis of SDOW (ProShares UltraPro Short Dow 30 ETF)


SDOW (ProShares UltraPro Short Dow 30 ETF) appears to have interesting cyclic behaviour every 39 weeks (33.3818*sine), 35 weeks (32.85*sine), and 20 weeks (14.1803*cosine).

SDOW (ProShares UltraPro Short Dow 30 ETF) has an average price of 288.19 (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 7/17/2017 for SDOW (ProShares UltraPro Short Dow 30 ETF), this program was able to calculate the following Fourier Series:
Sequence #Cosine Coefficients Sine Coefficients FrequenciesPeriod
0288.1938   0 
1181.1109 244.2979 (1*2π)/389389 weeks
256.30062 160.2263 (2*2π)/389195 weeks
335.73771 120.2788 (3*2π)/389130 weeks
424.28498 99.86962 (4*2π)/38997 weeks
519.24705 86.94373 (5*2π)/38978 weeks
6-3.92365 83.34012 (6*2π)/38965 weeks
7-21.34705 55.9395 (7*2π)/38956 weeks
8-7.80682 31.60681 (8*2π)/38949 weeks
92.85724 29.04436 (9*2π)/38943 weeks
105.98122 33.38182 (10*2π)/38939 weeks
11-4.2099 32.84999 (11*2π)/38935 weeks
12-5.70914 14.96218 (12*2π)/38932 weeks
136.66027 13.8135 (13*2π)/38930 weeks
146.69421 18.42016 (14*2π)/38928 weeks
158.1484 18.02774 (15*2π)/38926 weeks
169.81883 17.02222 (16*2π)/38924 weeks
173.62054 17.2804 (17*2π)/38923 weeks
187.45022 11.77364 (18*2π)/38922 weeks
1914.18026 17.21351 (19*2π)/38920 weeks
209.96277 19.42823 (20*2π)/38919 weeks
215.53349 15.9619 (21*2π)/38919 weeks
224.88445 12.52598 (22*2π)/38918 weeks
238.54761 13.19632 (23*2π)/38917 weeks
2411.28393 15.24805 (24*2π)/38916 weeks
258.03908 16.97118 (25*2π)/38916 weeks
266.11583 14.04523 (26*2π)/38915 weeks
278.47381 13.58126 (27*2π)/38914 weeks
289.79048 14.32198 (28*2π)/38914 weeks
299.87201 18.97584 (29*2π)/38913 weeks
303.49678 20.59791 (30*2π)/38913 weeks
31-1.54382 14.833 (31*2π)/38913 weeks
322.42537 10.14469 (32*2π)/38912 weeks
335.00398 10.66416 (33*2π)/38912 weeks
342.97122 14.06598 (34*2π)/38911 weeks
352.77364 11.69212 (35*2π)/38911 weeks
361.45235 11.23105 (36*2π)/38911 weeks
37.51052 8.31209 (37*2π)/38911 weeks
383.50165 7.56576 (38*2π)/38910 weeks
395.03182 8.02706 (39*2π)/38910 weeks
406.63384 9.85581 (40*2π)/38910 weeks
414.96128 11.05875 (41*2π)/3899 weeks
422.77064 12.30646 (42*2π)/3899 weeks
43.30685 9.96132 (43*2π)/3899 weeks
441.07683 6.57158 (44*2π)/3899 weeks
453.88941 7.7038 (45*2π)/3899 weeks
463.30047 8.37939 (46*2π)/3898 weeks
47-.04507 10.72662 (47*2π)/3898 weeks
48-1.60133 3.6976 (48*2π)/3898 weeks
494.04756 2.63494 (49*2π)/3898 weeks
504.63538 5.26179 (50*2π)/3898 weeks
514.26976 6.4944 (51*2π)/3898 weeks
521.48705 5.48066 (52*2π)/3897 weeks
532.28707 2.75209 (53*2π)/3897 weeks
546.71041 .00842 (54*2π)/3897 weeks
5510.36206 5.09255 (55*2π)/3897 weeks
567.93451 9.19223 (56*2π)/3897 weeks
574.43483 8.70973 (57*2π)/3897 weeks
584.57955 6.14287 (58*2π)/3897 weeks
596.73289 7.96767 (59*2π)/3897 weeks
604.31583 9.96936 (60*2π)/3896 weeks
611.86355 8.37754 (61*2π)/3896 weeks
622.84256 7.52918 (62*2π)/3896 weeks
631.49006 6.70777 (63*2π)/3896 weeks
641.55657 5.75981 (64*2π)/3896 weeks
651.9439 6.04704 (65*2π)/3896 weeks
66.61733 5.36106 (66*2π)/3896 weeks
671.54224 3.08539 (67*2π)/3896 weeks
683.64447 3.56146 (68*2π)/3896 weeks
693.49863 3.67451 (69*2π)/3896 weeks
703.69955 3.88584 (70*2π)/3896 weeks
713.2447 3.52065 (71*2π)/3895 weeks
723.94771 4.39754 (72*2π)/3895 weeks
734.06812 3.86029 (73*2π)/3895 weeks
743.84564 3.91963 (74*2π)/3895 weeks
755.03917 3.11001 (75*2π)/3895 weeks
765.244 5.16367 (76*2π)/3895 weeks
774.27332 5.92954 (77*2π)/3895 weeks
784.36305 5.13643 (78*2π)/3895 weeks
793.68471 5.91021 (79*2π)/3895 weeks
802.43523 6.31339 (80*2π)/3895 weeks
811.31364 4.669 (81*2π)/3895 weeks
822.31609 3.98023 (82*2π)/3895 weeks
832.90926 3.1073 (83*2π)/3895 weeks
843.2492 3.96791 (84*2π)/3895 weeks
851.67066 4.40263 (85*2π)/3895 weeks
862.33986 3.50306 (86*2π)/3895 weeks
871.96982 3.1341 (87*2π)/3894 weeks
883.0217 2.51888 (88*2π)/3894 weeks
892.32435 3.12136 (89*2π)/3894 weeks
901.97644 2.60972 (90*2π)/3894 weeks
912.5069 1.92464 (91*2π)/3894 weeks
922.6413 1.60287 (92*2π)/3894 weeks
933.3921 1.07554 (93*2π)/3894 weeks
943.71472 .50823 (94*2π)/3894 weeks
954.58254 .6438 (95*2π)/3894 weeks
966.36112 2.10913 (96*2π)/3894 weeks
975.91412 3.38446 (97*2π)/3894 weeks
984.2196 3.30695 (98*2π)/3894 weeks
994.30839 2.19521 (99*2π)/3894 weeks
1006.30692 2.92551 (100*2π)/3894 weeks
1016.07185 5.0819 (101*2π)/3894 weeks
1024.62968 5.65697 (102*2π)/3894 weeks
1032.95729 4.66086 (103*2π)/3894 weeks
1043.01954 4.10918 (104*2π)/3894 weeks
1053.71479 4.65015 (105*2π)/3894 weeks
1062.15168 5.15743 (106*2π)/3894 weeks
107.85407 4.00137 (107*2π)/3894 weeks
1081.42876 2.29371 (108*2π)/3894 weeks
1092.45997 2.73368 (109*2π)/3894 weeks
1101.6713 3.34018 (110*2π)/3894 weeks
1111.08649 2.69764 (111*2π)/3894 weeks
112.28849 .64284 (112*2π)/3893 weeks
1132.62518 -.6719 (113*2π)/3893 weeks
1143.77699 .30544 (114*2π)/3893 weeks
1153.75692 1.14898 (115*2π)/3893 weeks
1164.03539 .38306 (116*2π)/3893 weeks
1175.06553 2.23495 (117*2π)/3893 weeks
1183.16898 2.6827 (118*2π)/3893 weeks
1193.2718 1.8011 (119*2π)/3893 weeks
1203.51339 1.77016 (120*2π)/3893 weeks
1213.25423 1.93562 (121*2π)/3893 weeks
1223.86795 2.07555 (122*2π)/3893 weeks
1232.79944 2.0623 (123*2π)/3893 weeks
1242.82434 1.37186 (124*2π)/3893 weeks
1253.19643 1.2965 (125*2π)/3893 weeks
1263.69829 1.51347 (126*2π)/3893 weeks
1273.1277 1.83096 (127*2π)/3893 weeks
1282.70816 1.21862 (128*2π)/3893 weeks
1293.56894 .53835 (129*2π)/3893 weeks
1303.93344 1.92558 (130*2π)/3893 weeks
1312.60831 1.44676 (131*2π)/3893 weeks
1323.08761 -.08477 (132*2π)/3893 weeks
1334.31917 .55354 (133*2π)/3893 weeks
1344.34771 1.65862 (134*2π)/3893 weeks
1353.19828 1.23453 (135*2π)/3893 weeks
1363.1057 .03245 (136*2π)/3893 weeks
1375.34441 -.42779 (137*2π)/3893 weeks
1386.79255 1.95448 (138*2π)/3893 weeks
1394.20108 3.86834 (139*2π)/3893 weeks
1402.65421 2.39695 (140*2π)/3893 weeks
1413.11751 .92354 (141*2π)/3893 weeks
1425.49732 1.39389 (142*2π)/3893 weeks
1434.8081 4.29953 (143*2π)/3893 weeks
1441.72177 4.01468 (144*2π)/3893 weeks
1451.10349 1.64419 (145*2π)/3893 weeks
1461.67477 .87724 (146*2π)/3893 weeks
1472.67275 .89861 (147*2π)/3893 weeks
1482.49674 1.49077 (148*2π)/3893 weeks
1491.43673 1.35379 (149*2π)/3893 weeks
1501.41891 -.28334 (150*2π)/3893 weeks
1512.80001 -1.46547 (151*2π)/3893 weeks
1523.89924 .07337 (152*2π)/3893 weeks
1533.30433 .63913 (153*2π)/3893 weeks
1542.89407 .0779 (154*2π)/3893 weeks
1552.75811 -.47561 (155*2π)/3893 weeks
1563.34645 -.89617 (156*2π)/3892 weeks
1574.3652 -1.29545 (157*2π)/3892 weeks
1585.28033 -.26594 (158*2π)/3892 weeks
1594.91535 1.15897 (159*2π)/3892 weeks
1603.007 1.01174 (160*2π)/3892 weeks
1613.54414 -.62369 (161*2π)/3892 weeks
1625.50048 -.27679 (162*2π)/3892 weeks
1635.27443 1.11993 (163*2π)/3892 weeks
1643.71421 1.53531 (164*2π)/3892 weeks
1653.55462 .73038 (165*2π)/3892 weeks
1663.89896 .43581 (166*2π)/3892 weeks
1674.23006 .52135 (167*2π)/3892 weeks
1684.35019 1.29148 (168*2π)/3892 weeks
1692.7492 1.5941 (169*2π)/3892 weeks
1703.25394 -.56116 (170*2π)/3892 weeks
1715.10102 .03995 (171*2π)/3892 weeks
1724.34461 1.95571 (172*2π)/3892 weeks
1732.81933 1.59983 (173*2π)/3892 weeks
1742.62307 .19916 (174*2π)/3892 weeks
1753.12458 -1.12738 (175*2π)/3892 weeks
1764.93269 .07412 (176*2π)/3892 weeks
1774.62729 .7675 (177*2π)/3892 weeks
1783.85033 1.04076 (178*2π)/3892 weeks
1794.05598 .978 (179*2π)/3892 weeks
1803.74182 .84329 (180*2π)/3892 weeks
1814.25068 1.92034 (181*2π)/3892 weeks
1822.55593 2.67172 (182*2π)/3892 weeks
1831.00188 1.52547 (183*2π)/3892 weeks
1841.28502 -.1855 (184*2π)/3892 weeks
1851.69531 -.77703 (185*2π)/3892 weeks
1861.94454 -1.89883 (186*2π)/3892 weeks
1872.82163 -1.40695 (187*2π)/3892 weeks
1882.79576 -1.57887 (188*2π)/3892 weeks
1893.61351 -2.89437 (189*2π)/3892 weeks
1904.94134 -2.60289 (190*2π)/3892 weeks
1915.36398 -.95237 (191*2π)/3892 weeks
1925.31071 -.41457 (192*2π)/3892 weeks
1935.02817 -1.12102 (193*2π)/3892 weeks
1946.20672 -1.07778 (194*2π)/3892 weeks
1956.20672 1.07778 (195*2π)/3892 weeks
1965.02817 1.12102 (196*2π)/3892 weeks
1975.31071 .41457 (197*2π)/3892 weeks
1985.36398 .95237 (198*2π)/3892 weeks
1994.94134 2.60289 (199*2π)/3892 weeks
2003.61351 2.89437 (200*2π)/3892 weeks
2012.79576 1.57887 (201*2π)/3892 weeks
2022.82163 1.40695 (202*2π)/3892 weeks
2031.94454 1.89883 (203*2π)/3892 weeks
2041.69531 .77703 (204*2π)/3892 weeks
2051.28502 .1855 (205*2π)/3892 weeks
2061.00188 -1.52547 (206*2π)/3892 weeks
2072.55593 -2.67172 (207*2π)/3892 weeks
2084.25068 -1.92034 (208*2π)/3892 weeks
2093.74182 -.84329 (209*2π)/3892 weeks
2104.05598 -.978 (210*2π)/3892 weeks
2113.85033 -1.04076 (211*2π)/3892 weeks
2124.62729 -.7675 (212*2π)/3892 weeks
2134.93269 -.07412 (213*2π)/3892 weeks
2143.12458 1.12738 (214*2π)/3892 weeks
2152.62307 -.19916 (215*2π)/3892 weeks
2162.81933 -1.59983 (216*2π)/3892 weeks
2174.34461 -1.95571 (217*2π)/3892 weeks
2185.10102 -.03995 (218*2π)/3892 weeks
2193.25394 .56116 (219*2π)/3892 weeks
2202.7492 -1.5941 (220*2π)/3892 weeks
2214.35019 -1.29148 (221*2π)/3892 weeks
2224.23006 -.52135 (222*2π)/3892 weeks
2233.89896 -.43581 (223*2π)/3892 weeks
2243.55462 -.73038 (224*2π)/3892 weeks
2253.71421 -1.53531 (225*2π)/3892 weeks
2265.27443 -1.11993 (226*2π)/3892 weeks
2275.50048 .27679 (227*2π)/3892 weeks
2283.54414 .62369 (228*2π)/3892 weeks
2293.007 -1.01174 (229*2π)/3892 weeks
2304.91535 -1.15897 (230*2π)/3892 weeks
2315.28033 .26594 (231*2π)/3892 weeks
2324.3652 1.29545 (232*2π)/3892 weeks
2333.34645 .89617 (233*2π)/3892 weeks
2342.75811 .47561 (234*2π)/3892 weeks
2352.89407 -.0779 (235*2π)/3892 weeks
2363.30433 -.63913 (236*2π)/3892 weeks
2373.89924 -.07337 (237*2π)/3892 weeks
2382.80001 1.46547 (238*2π)/3892 weeks
2391.41891 .28334 (239*2π)/3892 weeks
2401.43673 -1.35379 (240*2π)/3892 weeks
2412.49674 -1.49077 (241*2π)/3892 weeks
2422.67275 -.89861 (242*2π)/3892 weeks
2431.67477 -.87724 (243*2π)/3892 weeks
2441.10349 -1.64419 (244*2π)/3892 weeks
2451.72177 -4.01468 (245*2π)/3892 weeks
2464.8081 -4.29953 (246*2π)/3892 weeks
2475.49732 -1.39389 (247*2π)/3892 weeks
2483.11751 -.92354 (248*2π)/3892 weeks
2492.65421 -2.39695 (249*2π)/3892 weeks
2504.20108 -3.86834 (250*2π)/3892 weeks
2516.79255 -1.95448 (251*2π)/3892 weeks
2525.34441 .42779 (252*2π)/3892 weeks
2533.1057 -.03245 (253*2π)/3892 weeks
2543.19828 -1.23453 (254*2π)/3892 weeks
2554.34771 -1.65862 (255*2π)/3892 weeks
2564.31917 -.55354 (256*2π)/3892 weeks
2573.08761 .08477 (257*2π)/3892 weeks
2582.60831 -1.44676 (258*2π)/3892 weeks
2593.93344 -1.92558 (259*2π)/3892 weeks
2603.56894 -.53835 (260*2π)/3891 weeks
2612.70816 -1.21862 (261*2π)/3891 weeks
2623.1277 -1.83096 (262*2π)/3891 weeks
2633.69829 -1.51347 (263*2π)/3891 weeks
2643.19643 -1.2965 (264*2π)/3891 weeks
2652.82434 -1.37186 (265*2π)/3891 weeks
2662.79944 -2.0623 (266*2π)/3891 weeks
2673.86795 -2.07555 (267*2π)/3891 weeks
2683.25423 -1.93562 (268*2π)/3891 weeks
2693.51339 -1.77016 (269*2π)/3891 weeks
2703.2718 -1.8011 (270*2π)/3891 weeks
2713.16898 -2.6827 (271*2π)/3891 weeks
2725.06553 -2.23495 (272*2π)/3891 weeks
2734.03539 -.38306 (273*2π)/3891 weeks
2743.75692 -1.14898 (274*2π)/3891 weeks
2753.77699 -.30544 (275*2π)/3891 weeks
2762.62518 .6719 (276*2π)/3891 weeks
277.28849 -.64284 (277*2π)/3891 weeks
2781.08649 -2.69764 (278*2π)/3891 weeks
2791.6713 -3.34018 (279*2π)/3891 weeks
2802.45997 -2.73368 (280*2π)/3891 weeks
2811.42876 -2.29371 (281*2π)/3891 weeks
282.85407 -4.00137 (282*2π)/3891 weeks
2832.15168 -5.15743 (283*2π)/3891 weeks
2843.71479 -4.65015 (284*2π)/3891 weeks
2853.01954 -4.10918 (285*2π)/3891 weeks
2862.95729 -4.66086 (286*2π)/3891 weeks
2874.62968 -5.65697 (287*2π)/3891 weeks
2886.07185 -5.0819 (288*2π)/3891 weeks
2896.30692 -2.92551 (289*2π)/3891 weeks
2904.30839 -2.19521 (290*2π)/3891 weeks
2914.2196 -3.30695 (291*2π)/3891 weeks
2925.91412 -3.38446 (292*2π)/3891 weeks
2936.36112 -2.10913 (293*2π)/3891 weeks
2944.58254 -.6438 (294*2π)/3891 weeks
2953.71472 -.50823 (295*2π)/3891 weeks
2963.3921 -1.07554 (296*2π)/3891 weeks
2972.6413 -1.60287 (297*2π)/3891 weeks
2982.5069 -1.92464 (298*2π)/3891 weeks
2991.97644 -2.60972 (299*2π)/3891 weeks
3002.32435 -3.12136 (300*2π)/3891 weeks
3013.0217 -2.51888 (301*2π)/3891 weeks
3021.96982 -3.1341 (302*2π)/3891 weeks
3032.33986 -3.50306 (303*2π)/3891 weeks
3041.67066 -4.40263 (304*2π)/3891 weeks
3053.2492 -3.96791 (305*2π)/3891 weeks
3062.90926 -3.1073 (306*2π)/3891 weeks
3072.31609 -3.98023 (307*2π)/3891 weeks
3081.31364 -4.669 (308*2π)/3891 weeks
3092.43523 -6.31339 (309*2π)/3891 weeks
3103.68471 -5.91021 (310*2π)/3891 weeks
3114.36305 -5.13643 (311*2π)/3891 weeks
3124.27332 -5.92954 (312*2π)/3891 weeks
3135.244 -5.16367 (313*2π)/3891 weeks
3145.03917 -3.11001 (314*2π)/3891 weeks
3153.84564 -3.91963 (315*2π)/3891 weeks
3164.06812 -3.86029 (316*2π)/3891 weeks
3173.94771 -4.39754 (317*2π)/3891 weeks
3183.2447 -3.52065 (318*2π)/3891 weeks
3193.69955 -3.88584 (319*2π)/3891 weeks
3203.49863 -3.67451 (320*2π)/3891 weeks
3213.64447 -3.56146 (321*2π)/3891 weeks
3221.54224 -3.08539 (322*2π)/3891 weeks
323.61733 -5.36106 (323*2π)/3891 weeks
3241.9439 -6.04704 (324*2π)/3891 weeks
3251.55657 -5.75981 (325*2π)/3891 weeks
3261.49006 -6.70777 (326*2π)/3891 weeks
3272.84256 -7.52918 (327*2π)/3891 weeks
3281.86355 -8.37754 (328*2π)/3891 weeks
3294.31583 -9.96936 (329*2π)/3891 weeks
3306.73289 -7.96767 (330*2π)/3891 weeks
3314.57955 -6.14287 (331*2π)/3891 weeks
3324.43483 -8.70973 (332*2π)/3891 weeks
3337.93451 -9.19223 (333*2π)/3891 weeks
33410.36206 -5.09255 (334*2π)/3891 weeks
3356.71041 -.00842 (335*2π)/3891 weeks
3362.28707 -2.75209 (336*2π)/3891 weeks
3371.48705 -5.48066 (337*2π)/3891 weeks
3384.26976 -6.4944 (338*2π)/3891 weeks
3394.63538 -5.26179 (339*2π)/3891 weeks
3404.04756 -2.63494 (340*2π)/3891 weeks
341-1.60133 -3.6976 (341*2π)/3891 weeks
342-.04507 -10.72662 (342*2π)/3891 weeks
3433.30047 -8.37939 (343*2π)/3891 weeks
3443.88941 -7.7038 (344*2π)/3891 weeks
3451.07683 -6.57158 (345*2π)/3891 weeks
346.30685 -9.96132 (346*2π)/3891 weeks
3472.77064 -12.30646 (347*2π)/3891 weeks
3484.96128 -11.05875 (348*2π)/3891 weeks
3496.63384 -9.85581 (349*2π)/3891 weeks
3505.03182 -8.02706 (350*2π)/3891 weeks
3513.50165 -7.56576 (351*2π)/3891 weeks
352.51052 -8.31209 (352*2π)/3891 weeks
3531.45235 -11.23105 (353*2π)/3891 weeks
3542.77364 -11.69212 (354*2π)/3891 weeks
3552.97122 -14.06598 (355*2π)/3891 weeks
3565.00398 -10.66416 (356*2π)/3891 weeks
3572.42537 -10.14469 (357*2π)/3891 weeks
358-1.54382 -14.833 (358*2π)/3891 weeks
3593.49678 -20.59791 (359*2π)/3891 weeks
3609.87201 -18.97584 (360*2π)/3891 weeks
3619.79048 -14.32198 (361*2π)/3891 weeks
3628.47381 -13.58126 (362*2π)/3891 weeks
3636.11583 -14.04523 (363*2π)/3891 weeks
3648.03908 -16.97118 (364*2π)/3891 weeks
36511.28393 -15.24805 (365*2π)/3891 weeks
3668.54761 -13.19632 (366*2π)/3891 weeks
3674.88445 -12.52598 (367*2π)/3891 weeks
3685.53349 -15.9619 (368*2π)/3891 weeks
3699.96277 -19.42823 (369*2π)/3891 weeks
37014.18026 -17.21351 (370*2π)/3891 weeks
3717.45022 -11.77364 (371*2π)/3891 weeks
3723.62054 -17.2804 (372*2π)/3891 weeks
3739.81883 -17.02222 (373*2π)/3891 weeks
3748.1484 -18.02774 (374*2π)/3891 weeks
3756.69421 -18.42016 (375*2π)/3891 weeks
3766.66027 -13.8135 (376*2π)/3891 weeks
377-5.70914 -14.96218 (377*2π)/3891 weeks
378-4.2099 -32.84999 (378*2π)/3891 weeks
3795.98122 -33.38182 (379*2π)/3891 weeks
3802.85724 -29.04436 (380*2π)/3891 weeks
381-7.80682 -31.60681 (381*2π)/3891 weeks
382-21.34705 -55.9395 (382*2π)/3891 weeks
383-3.92365 -83.34012 (383*2π)/3891 weeks
38419.24705 -86.94373 (384*2π)/3891 weeks
38524.28498 -99.86962 (385*2π)/3891 weeks
38635.73771 -120.2788 (386*2π)/3891 weeks
38756.30062 -160.2263 (387*2π)/3891 weeks



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