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Fourier Analysis of RUSL (Direxion Daily Russia Bull 3x Shares ETF)


RUSL (Direxion Daily Russia Bull 3x Shares ETF) appears to have interesting cyclic behaviour every 32 weeks (88.1061*sine), 29 weeks (81.9967*sine), and 32 weeks (37.2748*cosine).

RUSL (Direxion Daily Russia Bull 3x Shares ETF) has an average price of 258.66 (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 5/25/2011 to 7/17/2017 for RUSL (Direxion Daily Russia Bull 3x Shares ETF), this program was able to calculate the following Fourier Series:
Sequence #Cosine Coefficients Sine Coefficients FrequenciesPeriod
0258.6623   0 
1134.0018 220.7929 (1*2π)/322322 weeks
2109.2789 75.3025 (2*2π)/322161 weeks
383.57384 95.53028 (3*2π)/322107 weeks
476.01588 77.61587 (4*2π)/32281 weeks
554.60715 80.67221 (5*2π)/32264 weeks
648.697 65.4361 (6*2π)/32254 weeks
763.80293 43.15678 (7*2π)/32246 weeks
870.19223 94.59725 (8*2π)/32240 weeks
934.4636 76.82455 (9*2π)/32236 weeks
1037.27481 88.10606 (10*2π)/32232 weeks
1112.09786 81.99674 (11*2π)/32229 weeks
12.94654 64.52985 (12*2π)/32227 weeks
13-1.75685 47.76189 (13*2π)/32225 weeks
1419.96883 40.48994 (14*2π)/32223 weeks
1515.62819 53.83546 (15*2π)/32221 weeks
168.47694 58.73008 (16*2π)/32220 weeks
17-9.60564 54.73848 (17*2π)/32219 weeks
18-4.1475 43.18565 (18*2π)/32218 weeks
19-13.2124 45.91876 (19*2π)/32217 weeks
20-17.5092 31.91852 (20*2π)/32216 weeks
21-15.65241 22.93278 (21*2π)/32215 weeks
22-8.45834 15.97679 (22*2π)/32215 weeks
23-3.64408 16.25777 (23*2π)/32214 weeks
24-5.17423 15.75278 (24*2π)/32213 weeks
25-2.19987 10.47155 (25*2π)/32213 weeks
264.25481 9.99391 (26*2π)/32212 weeks
271.55987 14.84855 (27*2π)/32212 weeks
288.23928 11.71568 (28*2π)/32212 weeks
29.02583 19.24644 (29*2π)/32211 weeks
30-.35502 11.92118 (30*2π)/32211 weeks
31.32924 7.60225 (31*2π)/32210 weeks
324.40978 9.46497 (32*2π)/32210 weeks
331.92895 6.0188 (33*2π)/32210 weeks
348.24714 7.17654 (34*2π)/3229 weeks
354.09231 8.76401 (35*2π)/3229 weeks
365.70887 5.62746 (36*2π)/3229 weeks
377.1678 2.59654 (37*2π)/3229 weeks
3811.55468 4.30011 (38*2π)/3228 weeks
3912.16591 3.13705 (39*2π)/3228 weeks
4017.14382 3.21466 (40*2π)/3228 weeks
4120.58855 10.49881 (41*2π)/3228 weeks
4214.93356 12.07243 (42*2π)/3228 weeks
4317.88596 13.33851 (43*2π)/3227 weeks
4413.35858 16.32071 (44*2π)/3227 weeks
4515.3177 15.17046 (45*2π)/3227 weeks
4613.1335 18.40365 (46*2π)/3227 weeks
4714.27892 18.78481 (47*2π)/3227 weeks
4810.29828 26.71502 (48*2π)/3227 weeks
493.01912 22.20255 (49*2π)/3227 weeks
50-.04161 22.16776 (50*2π)/3226 weeks
51-1.50323 15.24166 (51*2π)/3226 weeks
52-.93735 16.27884 (52*2π)/3226 weeks
53-2.95162 12.39786 (53*2π)/3226 weeks
54-.3681 10.59597 (54*2π)/3226 weeks
55-2.66829 9.43078 (55*2π)/3226 weeks
56.813 6.31338 (56*2π)/3226 weeks
57-.08478 7.53489 (57*2π)/3226 weeks
58.99047 6.62003 (58*2π)/3226 weeks
591.37698 4.80173 (59*2π)/3225 weeks
601.95132 3.62676 (60*2π)/3225 weeks
614.01412 3.41116 (61*2π)/3225 weeks
625.44098 5.07507 (62*2π)/3225 weeks
634.92611 4.67183 (63*2π)/3225 weeks
644.75843 4.51984 (64*2π)/3225 weeks
657.00764 3.74262 (65*2π)/3225 weeks
668.33374 5.70729 (66*2π)/3225 weeks
677.26016 8.29259 (67*2π)/3225 weeks
683.29565 5.55619 (68*2π)/3225 weeks
697.78447 3.65122 (69*2π)/3225 weeks
708.67284 6.21541 (70*2π)/3225 weeks
718.23395 6.06796 (71*2π)/3225 weeks
729.69805 7.56938 (72*2π)/3224 weeks
737.59515 10.89494 (73*2π)/3224 weeks
746.15525 8.22541 (74*2π)/3224 weeks
756.77705 9.96935 (75*2π)/3224 weeks
764.96051 9.96823 (76*2π)/3224 weeks
774.47216 9.91305 (77*2π)/3224 weeks
781.34285 9.36224 (78*2π)/3224 weeks
793.45225 5.35392 (79*2π)/3224 weeks
803.95763 6.72131 (80*2π)/3224 weeks
815.49336 7.44198 (81*2π)/3224 weeks
823.50794 8.75662 (82*2π)/3224 weeks
833.6047 8.50306 (83*2π)/3224 weeks
841.61711 8.98266 (84*2π)/3224 weeks
851.13207 7.94914 (85*2π)/3224 weeks
86.68686 8.065 (86*2π)/3224 weeks
87-2.07245 7.18219 (87*2π)/3224 weeks
88-1.06533 3.41241 (88*2π)/3224 weeks
89.48706 3.42103 (89*2π)/3224 weeks
90.38601 3.71737 (90*2π)/3224 weeks
91-.0196 2.36437 (91*2π)/3224 weeks
92-.79163 2.08103 (92*2π)/3224 weeks
93.46331 -.38493 (93*2π)/3223 weeks
941.57316 -.13408 (94*2π)/3223 weeks
952.93452 -1.19368 (95*2π)/3223 weeks
964.36244 1.35135 (96*2π)/3223 weeks
972.43367 .34812 (97*2π)/3223 weeks
983.7143 -.8951 (98*2π)/3223 weeks
995.84537 -1.33947 (99*2π)/3223 weeks
1006.82296 1.50454 (100*2π)/3223 weeks
1014.56329 1.95873 (101*2π)/3223 weeks
1023.70254 .80267 (102*2π)/3223 weeks
1035.5475 .04398 (103*2π)/3223 weeks
1045.41016 -.14519 (104*2π)/3223 weeks
1056.5976 .90374 (105*2π)/3223 weeks
1065.65248 .62655 (106*2π)/3223 weeks
1078.05727 1.03233 (107*2π)/3223 weeks
1086.19724 2.99013 (108*2π)/3223 weeks
1095.77679 3.15842 (109*2π)/3223 weeks
1104.07773 2.60744 (110*2π)/3223 weeks
1114.49118 1.30684 (111*2π)/3223 weeks
1123.76748 1.47053 (112*2π)/3223 weeks
1134.21264 -.21382 (113*2π)/3223 weeks
1145.39902 -1.36543 (114*2π)/3223 weeks
1157.41769 .16656 (115*2π)/3223 weeks
1166.68375 .10806 (116*2π)/3223 weeks
1178.049 .60382 (117*2π)/3223 weeks
1188.84518 1.51518 (118*2π)/3223 weeks
1197.87214 3.98805 (119*2π)/3223 weeks
1206.44622 3.45307 (120*2π)/3223 weeks
1215.80085 2.95175 (121*2π)/3223 weeks
1226.13198 3.15682 (122*2π)/3223 weeks
1235.19274 2.44438 (123*2π)/3223 weeks
1246.80251 3.58313 (124*2π)/3223 weeks
1252.46772 4.03279 (125*2π)/3223 weeks
1263.4163 .54435 (126*2π)/3223 weeks
1274.21413 -.52584 (127*2π)/3223 weeks
1286.62731 .95543 (128*2π)/3223 weeks
1295.5788 .6779 (129*2π)/3222 weeks
1306.03674 2.14668 (130*2π)/3222 weeks
1315.22294 1.91798 (131*2π)/3222 weeks
1325.21127 2.05241 (132*2π)/3222 weeks
1334.5451 1.80457 (133*2π)/3222 weeks
1344.55746 .7597 (134*2π)/3222 weeks
1354.55783 1.42819 (135*2π)/3222 weeks
1365.18831 .5905 (136*2π)/3222 weeks
1374.21302 1.2024 (137*2π)/3222 weeks
1384.79563 -.27528 (138*2π)/3222 weeks
1394.24604 -.11866 (139*2π)/3222 weeks
1405.06275 -2.66003 (140*2π)/3222 weeks
1417.5042 -1.82973 (141*2π)/3222 weeks
1428.47397 -.21211 (142*2π)/3222 weeks
1438.03997 1.41688 (143*2π)/3222 weeks
1447.14785 1.1142 (144*2π)/3222 weeks
1458.05747 2.41329 (145*2π)/3222 weeks
1466.5452 2.95798 (146*2π)/3222 weeks
1475.94553 3.59754 (147*2π)/3222 weeks
1484.71987 2.42099 (148*2π)/3222 weeks
1494.95325 3.43961 (149*2π)/3222 weeks
1502.58226 1.76271 (150*2π)/3222 weeks
1513.30813 .53044 (151*2π)/3222 weeks
1523.50732 -.35928 (152*2π)/3222 weeks
1534.11821 -.84859 (153*2π)/3222 weeks
1545.70517 -.40874 (154*2π)/3222 weeks
1554.70925 .46136 (155*2π)/3222 weeks
1563.3269 -.94532 (156*2π)/3222 weeks
1575.12606 -1.78605 (157*2π)/3222 weeks
1586.07229 -1.3328 (158*2π)/3222 weeks
1595.64383 -1.54574 (159*2π)/3222 weeks
1607.43576 -1.00567 (160*2π)/3222 weeks
1617.23058   (161*2π)/3222 weeks
1627.43576 1.00567 (162*2π)/3222 weeks
1635.64383 1.54574 (163*2π)/3222 weeks
1646.07229 1.3328 (164*2π)/3222 weeks
1655.12606 1.78605 (165*2π)/3222 weeks
1663.3269 .94532 (166*2π)/3222 weeks
1674.70925 -.46136 (167*2π)/3222 weeks
1685.70517 .40874 (168*2π)/3222 weeks
1694.11821 .84859 (169*2π)/3222 weeks
1703.50732 .35928 (170*2π)/3222 weeks
1713.30813 -.53044 (171*2π)/3222 weeks
1722.58226 -1.76271 (172*2π)/3222 weeks
1734.95325 -3.43961 (173*2π)/3222 weeks
1744.71987 -2.42099 (174*2π)/3222 weeks
1755.94553 -3.59754 (175*2π)/3222 weeks
1766.5452 -2.95798 (176*2π)/3222 weeks
1778.05747 -2.41329 (177*2π)/3222 weeks
1787.14785 -1.1142 (178*2π)/3222 weeks
1798.03997 -1.41688 (179*2π)/3222 weeks
1808.47397 .21211 (180*2π)/3222 weeks
1817.5042 1.82973 (181*2π)/3222 weeks
1825.06275 2.66003 (182*2π)/3222 weeks
1834.24604 .11866 (183*2π)/3222 weeks
1844.79563 .27528 (184*2π)/3222 weeks
1854.21302 -1.2024 (185*2π)/3222 weeks
1865.18831 -.5905 (186*2π)/3222 weeks
1874.55783 -1.42819 (187*2π)/3222 weeks
1884.55746 -.7597 (188*2π)/3222 weeks
1894.5451 -1.80457 (189*2π)/3222 weeks
1905.21127 -2.05241 (190*2π)/3222 weeks
1915.22294 -1.91798 (191*2π)/3222 weeks
1926.03674 -2.14668 (192*2π)/3222 weeks
1935.5788 -.6779 (193*2π)/3222 weeks
1946.62731 -.95543 (194*2π)/3222 weeks
1954.21413 .52584 (195*2π)/3222 weeks
1963.4163 -.54435 (196*2π)/3222 weeks
1972.46772 -4.03279 (197*2π)/3222 weeks
1986.80251 -3.58313 (198*2π)/3222 weeks
1995.19274 -2.44438 (199*2π)/3222 weeks
2006.13198 -3.15682 (200*2π)/3222 weeks
2015.80085 -2.95175 (201*2π)/3222 weeks
2026.44622 -3.45307 (202*2π)/3222 weeks
2037.87214 -3.98805 (203*2π)/3222 weeks
2048.84518 -1.51518 (204*2π)/3222 weeks
2058.049 -.60382 (205*2π)/3222 weeks
2066.68375 -.10806 (206*2π)/3222 weeks
2077.41769 -.16656 (207*2π)/3222 weeks
2085.39902 1.36543 (208*2π)/3222 weeks
2094.21264 .21382 (209*2π)/3222 weeks
2103.76748 -1.47053 (210*2π)/3222 weeks
2114.49118 -1.30684 (211*2π)/3222 weeks
2124.07773 -2.60744 (212*2π)/3222 weeks
2135.77679 -3.15842 (213*2π)/3222 weeks
2146.19724 -2.99013 (214*2π)/3222 weeks
2158.05727 -1.03233 (215*2π)/3221 weeks
2165.65248 -.62655 (216*2π)/3221 weeks
2176.5976 -.90374 (217*2π)/3221 weeks
2185.41016 .14519 (218*2π)/3221 weeks
2195.5475 -.04398 (219*2π)/3221 weeks
2203.70254 -.80267 (220*2π)/3221 weeks
2214.56329 -1.95873 (221*2π)/3221 weeks
2226.82296 -1.50454 (222*2π)/3221 weeks
2235.84537 1.33947 (223*2π)/3221 weeks
2243.7143 .8951 (224*2π)/3221 weeks
2252.43367 -.34812 (225*2π)/3221 weeks
2264.36244 -1.35135 (226*2π)/3221 weeks
2272.93452 1.19368 (227*2π)/3221 weeks
2281.57316 .13408 (228*2π)/3221 weeks
229.46331 .38493 (229*2π)/3221 weeks
230-.79163 -2.08103 (230*2π)/3221 weeks
231-.0196 -2.36437 (231*2π)/3221 weeks
232.38601 -3.71737 (232*2π)/3221 weeks
233.48706 -3.42103 (233*2π)/3221 weeks
234-1.06533 -3.41241 (234*2π)/3221 weeks
235-2.07245 -7.18219 (235*2π)/3221 weeks
236.68686 -8.065 (236*2π)/3221 weeks
2371.13207 -7.94914 (237*2π)/3221 weeks
2381.61711 -8.98266 (238*2π)/3221 weeks
2393.6047 -8.50306 (239*2π)/3221 weeks
2403.50794 -8.75662 (240*2π)/3221 weeks
2415.49336 -7.44198 (241*2π)/3221 weeks
2423.95763 -6.72131 (242*2π)/3221 weeks
2433.45225 -5.35392 (243*2π)/3221 weeks
2441.34285 -9.36224 (244*2π)/3221 weeks
2454.47216 -9.91305 (245*2π)/3221 weeks
2464.96051 -9.96823 (246*2π)/3221 weeks
2476.77705 -9.96935 (247*2π)/3221 weeks
2486.15525 -8.22541 (248*2π)/3221 weeks
2497.59515 -10.89494 (249*2π)/3221 weeks
2509.69805 -7.56938 (250*2π)/3221 weeks
2518.23395 -6.06796 (251*2π)/3221 weeks
2528.67284 -6.21541 (252*2π)/3221 weeks
2537.78447 -3.65122 (253*2π)/3221 weeks
2543.29565 -5.55619 (254*2π)/3221 weeks
2557.26016 -8.29259 (255*2π)/3221 weeks
2568.33374 -5.70729 (256*2π)/3221 weeks
2577.00764 -3.74262 (257*2π)/3221 weeks
2584.75843 -4.51984 (258*2π)/3221 weeks
2594.92611 -4.67183 (259*2π)/3221 weeks
2605.44098 -5.07507 (260*2π)/3221 weeks
2614.01412 -3.41116 (261*2π)/3221 weeks
2621.95132 -3.62676 (262*2π)/3221 weeks
2631.37698 -4.80173 (263*2π)/3221 weeks
264.99047 -6.62003 (264*2π)/3221 weeks
265-.08478 -7.53489 (265*2π)/3221 weeks
266.813 -6.31338 (266*2π)/3221 weeks
267-2.66829 -9.43078 (267*2π)/3221 weeks
268-.3681 -10.59597 (268*2π)/3221 weeks
269-2.95162 -12.39786 (269*2π)/3221 weeks
270-.93735 -16.27884 (270*2π)/3221 weeks
271-1.50323 -15.24166 (271*2π)/3221 weeks
272-.04161 -22.16776 (272*2π)/3221 weeks
2733.01912 -22.20255 (273*2π)/3221 weeks
27410.29828 -26.71502 (274*2π)/3221 weeks
27514.27892 -18.78481 (275*2π)/3221 weeks
27613.1335 -18.40365 (276*2π)/3221 weeks
27715.3177 -15.17046 (277*2π)/3221 weeks
27813.35858 -16.32071 (278*2π)/3221 weeks
27917.88596 -13.33851 (279*2π)/3221 weeks
28014.93356 -12.07243 (280*2π)/3221 weeks
28120.58855 -10.49881 (281*2π)/3221 weeks
28217.14382 -3.21466 (282*2π)/3221 weeks
28312.16591 -3.13705 (283*2π)/3221 weeks
28411.55468 -4.30011 (284*2π)/3221 weeks
2857.1678 -2.59654 (285*2π)/3221 weeks
2865.70887 -5.62746 (286*2π)/3221 weeks
2874.09231 -8.76401 (287*2π)/3221 weeks
2888.24714 -7.17654 (288*2π)/3221 weeks
2891.92895 -6.0188 (289*2π)/3221 weeks
2904.40978 -9.46497 (290*2π)/3221 weeks
291.32924 -7.60225 (291*2π)/3221 weeks
292-.35502 -11.92118 (292*2π)/3221 weeks
293.02583 -19.24644 (293*2π)/3221 weeks
2948.23928 -11.71568 (294*2π)/3221 weeks
2951.55987 -14.84855 (295*2π)/3221 weeks
2964.25481 -9.99391 (296*2π)/3221 weeks
297-2.19987 -10.47155 (297*2π)/3221 weeks
298-5.17423 -15.75278 (298*2π)/3221 weeks
299-3.64408 -16.25777 (299*2π)/3221 weeks
300-8.45834 -15.97679 (300*2π)/3221 weeks
301-15.65241 -22.93278 (301*2π)/3221 weeks
302-17.5092 -31.91852 (302*2π)/3221 weeks
303-13.2124 -45.91876 (303*2π)/3221 weeks
304-4.1475 -43.18565 (304*2π)/3221 weeks
305-9.60564 -54.73848 (305*2π)/3221 weeks
3068.47694 -58.73008 (306*2π)/3221 weeks
30715.62819 -53.83546 (307*2π)/3221 weeks
30819.96883 -40.48994 (308*2π)/3221 weeks
309-1.75685 -47.76189 (309*2π)/3221 weeks
310.94654 -64.52985 (310*2π)/3221 weeks
31112.09786 -81.99674 (311*2π)/3221 weeks
31237.27481 -88.10606 (312*2π)/3221 weeks
31334.4636 -76.82455 (313*2π)/3221 weeks
31470.19223 -94.59725 (314*2π)/3221 weeks
31563.80293 -43.15678 (315*2π)/3221 weeks
31648.697 -65.4361 (316*2π)/3221 weeks
31754.60715 -80.67221 (317*2π)/3221 weeks
31876.01588 -77.61587 (318*2π)/3221 weeks
31983.57384 -95.53028 (319*2π)/3221 weeks
320109.2789 -75.3025 (320*2π)/3221 weeks



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