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Fourier Analysis of BRZU (Direxion Daily Brazil Bull 3x Shares ETF)


BRZU (Direxion Daily Brazil Bull 3x Shares ETF) appears to have interesting cyclic behaviour every 23 weeks (18.1872*sine), 21 weeks (17.1208*sine), and 23 weeks (9.1607*cosine).

BRZU (Direxion Daily Brazil Bull 3x Shares ETF) has an average price of 108.02 (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 4/10/2013 to 9/5/2017 for BRZU (Direxion Daily Brazil Bull 3x Shares ETF), this program was able to calculate the following Fourier Series:
Sequence #Cosine Coefficients Sine Coefficients FrequenciesPeriod
0108.0157   0 
150.5505 104.5204 (1*2π)/231231 weeks
2-11.97173 19.64255 (2*2π)/231116 weeks
321.32211 14.86052 (3*2π)/23177 weeks
416.40277 35.69579 (4*2π)/23158 weeks
5-11.64204 16.9049 (5*2π)/23146 weeks
610.49752 -2.39621 (6*2π)/23139 weeks
718.2929 13.25971 (7*2π)/23133 weeks
815.46157 14.54283 (8*2π)/23129 weeks
914.12477 17.25722 (9*2π)/23126 weeks
109.16073 18.18723 (10*2π)/23123 weeks
11-2.16022 17.12075 (11*2π)/23121 weeks
122.3975 8.5151 (12*2π)/23119 weeks
134.99693 12.41808 (13*2π)/23118 weeks
14-.71938 13.43896 (14*2π)/23117 weeks
15.25776 7.06789 (15*2π)/23115 weeks
162.4032 5.59163 (16*2π)/23114 weeks
172.18628 8.0567 (17*2π)/23114 weeks
18-1.20204 9.96436 (18*2π)/23113 weeks
19.71979 8.15245 (19*2π)/23112 weeks
20-2.01002 10.0207 (20*2π)/23112 weeks
21-3.4604 2.72632 (21*2π)/23111 weeks
221.13431 5.39224 (22*2π)/23111 weeks
23-2.74108 5.87902 (23*2π)/23110 weeks
24-1.55272 1.93256 (24*2π)/23110 weeks
251.16508 .65912 (25*2π)/2319 weeks
262.03097 3.23359 (26*2π)/2319 weeks
27-1.01618 1.84512 (27*2π)/2319 weeks
282.17033 1.04718 (28*2π)/2318 weeks
291.97715 2.76606 (29*2π)/2318 weeks
301.97582 1.54812 (30*2π)/2318 weeks
311.85531 1.7675 (31*2π)/2317 weeks
323.39158 3.48812 (32*2π)/2317 weeks
33.16463 4.79235 (33*2π)/2317 weeks
34-.49335 1.21101 (34*2π)/2317 weeks
353.54571 .683 (35*2π)/2317 weeks
362.01366 4.23205 (36*2π)/2316 weeks
37-.43609 1.47384 (37*2π)/2316 weeks
383.05889 2.15222 (38*2π)/2316 weeks
39-.28957 3.19761 (39*2π)/2316 weeks
40-.06493 -2.02925 (40*2π)/2316 weeks
414.94592 -.36452 (41*2π)/2316 weeks
423.04585 3.21272 (42*2π)/2316 weeks
43.00726 3.1352 (43*2π)/2315 weeks
441.77326 .43678 (44*2π)/2315 weeks
453.21761 2.86362 (45*2π)/2315 weeks
46.3851 3.49137 (46*2π)/2315 weeks
47-.12549 .94462 (47*2π)/2315 weeks
481.62123 -.22675 (48*2π)/2315 weeks
493.65607 1.11402 (49*2π)/2315 weeks
502.08213 1.98324 (50*2π)/2315 weeks
512.26274 .55922 (51*2π)/2315 weeks
523.12573 2.11802 (52*2π)/2314 weeks
531.05847 2.53088 (53*2π)/2314 weeks
541.02061 1.01018 (54*2π)/2314 weeks
552.02666 1.92287 (55*2π)/2314 weeks
56.38451 .88494 (56*2π)/2314 weeks
572.18214 .28211 (57*2π)/2314 weeks
581.60522 1.99815 (58*2π)/2314 weeks
591.24999 .81617 (59*2π)/2314 weeks
602.29028 1.01025 (60*2π)/2314 weeks
611.86267 1.4412 (61*2π)/2314 weeks
621.37344 .92972 (62*2π)/2314 weeks
631.87299 .41894 (63*2π)/2314 weeks
642.43418 .98428 (64*2π)/2314 weeks
652.31932 1.22215 (65*2π)/2314 weeks
661.37022 1.50284 (66*2π)/2314 weeks
671.27742 .41949 (67*2π)/2313 weeks
681.60665 1.22268 (68*2π)/2313 weeks
691.0894 .59292 (69*2π)/2313 weeks
702.27284 .27727 (70*2π)/2313 weeks
711.59314 1.56991 (71*2π)/2313 weeks
72.37617 .28149 (72*2π)/2313 weeks
732.7 -.36671 (73*2π)/2313 weeks
742.41178 1.43264 (74*2π)/2313 weeks
751.09521 .65325 (75*2π)/2313 weeks
763.04574 -.05583 (76*2π)/2313 weeks
772.08727 1.88895 (77*2π)/2313 weeks
78.44242 -.46045 (78*2π)/2313 weeks
792.69638 -1.51829 (79*2π)/2313 weeks
804.06482 1.26045 (80*2π)/2313 weeks
811.17038 1.43271 (81*2π)/2313 weeks
822.68709 -.62228 (82*2π)/2313 weeks
832.82529 2.01831 (83*2π)/2313 weeks
84.78548 .92524 (84*2π)/2313 weeks
852.52067 .24658 (85*2π)/2313 weeks
862.18876 1.33263 (86*2π)/2313 weeks
871.95103 .85929 (87*2π)/2313 weeks
882.3368 .63814 (88*2π)/2313 weeks
892.44914 1.91683 (89*2π)/2313 weeks
90.89844 1.8132 (90*2π)/2313 weeks
911.23787 .08324 (91*2π)/2313 weeks
921.47306 .91171 (92*2π)/2313 weeks
93.9879 .16811 (93*2π)/2312 weeks
941.27127 .52575 (94*2π)/2312 weeks
95.47867 .0061 (95*2π)/2312 weeks
961.77497 -.39952 (96*2π)/2312 weeks
971.47647 .99308 (97*2π)/2312 weeks
981.43242 .30548 (98*2π)/2312 weeks
991.85308 .442 (99*2π)/2312 weeks
1001.5867 .43719 (100*2π)/2312 weeks
1011.05621 -.10033 (101*2π)/2312 weeks
1021.92986 .34161 (102*2π)/2312 weeks
1031.4469 .1952 (103*2π)/2312 weeks
1041.84992 -.80455 (104*2π)/2312 weeks
1052.13585 -.16281 (105*2π)/2312 weeks
1061.67189 .11332 (106*2π)/2312 weeks
1071.97425 -.43864 (107*2π)/2312 weeks
1081.77123 -.14868 (108*2π)/2312 weeks
1092.01193 -.3719 (109*2π)/2312 weeks
1102.64325 -.03882 (110*2π)/2312 weeks
1111.97334 .56438 (111*2π)/2312 weeks
1122.10044 .9254 (112*2π)/2312 weeks
1131.90868 .74284 (113*2π)/2312 weeks
1141.40197 1.49582 (114*2π)/2312 weeks
115.59209 .43929 (115*2π)/2312 weeks
116.59209 -.43929 (116*2π)/2312 weeks
1171.40197 -1.49582 (117*2π)/2312 weeks
1181.90868 -.74284 (118*2π)/2312 weeks
1192.10044 -.9254 (119*2π)/2312 weeks
1201.97334 -.56438 (120*2π)/2312 weeks
1212.64325 .03882 (121*2π)/2312 weeks
1222.01193 .3719 (122*2π)/2312 weeks
1231.77123 .14868 (123*2π)/2312 weeks
1241.97425 .43864 (124*2π)/2312 weeks
1251.67189 -.11332 (125*2π)/2312 weeks
1262.13585 .16281 (126*2π)/2312 weeks
1271.84992 .80455 (127*2π)/2312 weeks
1281.4469 -.1952 (128*2π)/2312 weeks
1291.92986 -.34161 (129*2π)/2312 weeks
1301.05621 .10033 (130*2π)/2312 weeks
1311.5867 -.43719 (131*2π)/2312 weeks
1321.85308 -.442 (132*2π)/2312 weeks
1331.43242 -.30548 (133*2π)/2312 weeks
1341.47647 -.99308 (134*2π)/2312 weeks
1351.77497 .39952 (135*2π)/2312 weeks
136.47867 -.0061 (136*2π)/2312 weeks
1371.27127 -.52575 (137*2π)/2312 weeks
138.9879 -.16811 (138*2π)/2312 weeks
1391.47306 -.91171 (139*2π)/2312 weeks
1401.23787 -.08324 (140*2π)/2312 weeks
141.89844 -1.8132 (141*2π)/2312 weeks
1422.44914 -1.91683 (142*2π)/2312 weeks
1432.3368 -.63814 (143*2π)/2312 weeks
1441.95103 -.85929 (144*2π)/2312 weeks
1452.18876 -1.33263 (145*2π)/2312 weeks
1462.52067 -.24658 (146*2π)/2312 weeks
147.78548 -.92524 (147*2π)/2312 weeks
1482.82529 -2.01831 (148*2π)/2312 weeks
1492.68709 .62228 (149*2π)/2312 weeks
1501.17038 -1.43271 (150*2π)/2312 weeks
1514.06482 -1.26045 (151*2π)/2312 weeks
1522.69638 1.51829 (152*2π)/2312 weeks
153.44242 .46045 (153*2π)/2312 weeks
1542.08727 -1.88895 (154*2π)/2312 weeks
1553.04574 .05583 (155*2π)/2311 weeks
1561.09521 -.65325 (156*2π)/2311 weeks
1572.41178 -1.43264 (157*2π)/2311 weeks
1582.7 .36671 (158*2π)/2311 weeks
159.37617 -.28149 (159*2π)/2311 weeks
1601.59314 -1.56991 (160*2π)/2311 weeks
1612.27284 -.27727 (161*2π)/2311 weeks
1621.0894 -.59292 (162*2π)/2311 weeks
1631.60665 -1.22268 (163*2π)/2311 weeks
1641.27742 -.41949 (164*2π)/2311 weeks
1651.37022 -1.50284 (165*2π)/2311 weeks
1662.31932 -1.22215 (166*2π)/2311 weeks
1672.43418 -.98428 (167*2π)/2311 weeks
1681.87299 -.41894 (168*2π)/2311 weeks
1691.37344 -.92972 (169*2π)/2311 weeks
1701.86267 -1.4412 (170*2π)/2311 weeks
1712.29028 -1.01025 (171*2π)/2311 weeks
1721.24999 -.81617 (172*2π)/2311 weeks
1731.60522 -1.99815 (173*2π)/2311 weeks
1742.18214 -.28211 (174*2π)/2311 weeks
175.38451 -.88494 (175*2π)/2311 weeks
1762.02666 -1.92287 (176*2π)/2311 weeks
1771.02061 -1.01018 (177*2π)/2311 weeks
1781.05847 -2.53088 (178*2π)/2311 weeks
1793.12573 -2.11802 (179*2π)/2311 weeks
1802.26274 -.55922 (180*2π)/2311 weeks
1812.08213 -1.98324 (181*2π)/2311 weeks
1823.65607 -1.11402 (182*2π)/2311 weeks
1831.62123 .22675 (183*2π)/2311 weeks
184-.12549 -.94462 (184*2π)/2311 weeks
185.3851 -3.49137 (185*2π)/2311 weeks
1863.21761 -2.86362 (186*2π)/2311 weeks
1871.77326 -.43678 (187*2π)/2311 weeks
188.00726 -3.1352 (188*2π)/2311 weeks
1893.04585 -3.21272 (189*2π)/2311 weeks
1904.94592 .36452 (190*2π)/2311 weeks
191-.06493 2.02925 (191*2π)/2311 weeks
192-.28957 -3.19761 (192*2π)/2311 weeks
1933.05889 -2.15222 (193*2π)/2311 weeks
194-.43609 -1.47384 (194*2π)/2311 weeks
1952.01366 -4.23205 (195*2π)/2311 weeks
1963.54571 -.683 (196*2π)/2311 weeks
197-.49335 -1.21101 (197*2π)/2311 weeks
198.16463 -4.79235 (198*2π)/2311 weeks
1993.39158 -3.48812 (199*2π)/2311 weeks
2001.85531 -1.7675 (200*2π)/2311 weeks
2011.97582 -1.54812 (201*2π)/2311 weeks
2021.97715 -2.76606 (202*2π)/2311 weeks
2032.17033 -1.04718 (203*2π)/2311 weeks
204-1.01618 -1.84512 (204*2π)/2311 weeks
2052.03097 -3.23359 (205*2π)/2311 weeks
2061.16508 -.65912 (206*2π)/2311 weeks
207-1.55272 -1.93256 (207*2π)/2311 weeks
208-2.74108 -5.87902 (208*2π)/2311 weeks
2091.13431 -5.39224 (209*2π)/2311 weeks
210-3.4604 -2.72632 (210*2π)/2311 weeks
211-2.01002 -10.0207 (211*2π)/2311 weeks
212.71979 -8.15245 (212*2π)/2311 weeks
213-1.20204 -9.96436 (213*2π)/2311 weeks
2142.18628 -8.0567 (214*2π)/2311 weeks
2152.4032 -5.59163 (215*2π)/2311 weeks
216.25776 -7.06789 (216*2π)/2311 weeks
217-.71938 -13.43896 (217*2π)/2311 weeks
2184.99693 -12.41808 (218*2π)/2311 weeks
2192.3975 -8.5151 (219*2π)/2311 weeks
220-2.16022 -17.12075 (220*2π)/2311 weeks
2219.16073 -18.18723 (221*2π)/2311 weeks
22214.12477 -17.25722 (222*2π)/2311 weeks
22315.46157 -14.54283 (223*2π)/2311 weeks
22418.2929 -13.25971 (224*2π)/2311 weeks
22510.49752 2.39621 (225*2π)/2311 weeks
226-11.64204 -16.9049 (226*2π)/2311 weeks
22716.40277 -35.69579 (227*2π)/2311 weeks
22821.32211 -14.86052 (228*2π)/2311 weeks
229-11.97173 -19.64255 (229*2π)/2311 weeks



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