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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 22 weeks (22.6981*sine), 17 weeks (14.8847*sine), and 20 weeks (12.6757*sine).

BRZU (Direxion Daily Brazil Bull 3x Shares ETF) has an average price of 110.5 (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 7/10/2017 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
0110.5041   0 
146.24297 108.0434 (1*2π)/223223 weeks
2-8.72918 15.91358 (2*2π)/223112 weeks
323.02192 16.42589 (3*2π)/22374 weeks
411.34154 38.02789 (4*2π)/22356 weeks
5-11.61267 9.75688 (5*2π)/22345 weeks
615.04421 -.66859 (6*2π)/22337 weeks
715.07408 14.47046 (7*2π)/22332 weeks
813.62103 16.4308 (8*2π)/22328 weeks
911.14081 20.69484 (9*2π)/22325 weeks
106.90678 22.69814 (10*2π)/22322 weeks
11-2.13709 12.67566 (11*2π)/22320 weeks
124.90459 10.43084 (12*2π)/22319 weeks
132.97662 14.88466 (13*2π)/22317 weeks
14-2.73778 10.01367 (14*2π)/22316 weeks
15.19385 7.87933 (15*2π)/22315 weeks
163.19976 9.5936 (16*2π)/22314 weeks
172.37754 11.67633 (17*2π)/22313 weeks
18.73942 7.78863 (18*2π)/22312 weeks
19.1259 10.02077 (19*2π)/22312 weeks
20-5.00921 5.23174 (20*2π)/22311 weeks
211.52673 4.39093 (21*2π)/22311 weeks
22-1.67485 6.81895 (22*2π)/22310 weeks
23-1.96064 2.16886 (23*2π)/22310 weeks
24.58163 .57645 (24*2π)/2239 weeks
252.17364 3.16617 (25*2π)/2239 weeks
26-1.10924 2.46534 (26*2π)/2239 weeks
272.20755 1.199 (27*2π)/2238 weeks
282.06188 3.04854 (28*2π)/2238 weeks
292.1745 1.82052 (29*2π)/2238 weeks
302.1149 1.86072 (30*2π)/2237 weeks
313.40241 4.0877 (31*2π)/2237 weeks
32-.38283 4.61129 (32*2π)/2237 weeks
33-.14044 .77386 (33*2π)/2237 weeks
344.07226 2.00933 (34*2π)/2237 weeks
351.01368 4.51848 (35*2π)/2236 weeks
36.57278 .60488 (36*2π)/2236 weeks
372.27826 3.7391 (37*2π)/2236 weeks
38-1.59561 2.20126 (38*2π)/2236 weeks
392.59274 -1.9582 (39*2π)/2236 weeks
405.36998 2.601 (40*2π)/2236 weeks
412.6601 4.1126 (41*2π)/2235 weeks
42.65059 1.01114 (42*2π)/2235 weeks
433.63087 1.28148 (43*2π)/2235 weeks
441.97306 3.87477 (44*2π)/2235 weeks
45-.19924 1.7912 (45*2π)/2235 weeks
461.09297 -.21872 (46*2π)/2235 weeks
473.42328 .01179 (47*2π)/2235 weeks
482.46461 2.10828 (48*2π)/2235 weeks
491.74293 .86574 (49*2π)/2235 weeks
503.28053 1.85388 (50*2π)/2234 weeks
511.41418 3.00985 (51*2π)/2234 weeks
52.9586 1.34633 (52*2π)/2234 weeks
532.2683 1.87257 (53*2π)/2234 weeks
54.42268 1.17355 (54*2π)/2234 weeks
552.28064 .33567 (55*2π)/2234 weeks
561.71127 2.12451 (56*2π)/2234 weeks
571.37552 .85572 (57*2π)/2234 weeks
582.39168 1.2078 (58*2π)/2234 weeks
591.88049 1.5926 (59*2π)/2234 weeks
601.49641 .9442 (60*2π)/2234 weeks
612.15514 .54098 (61*2π)/2234 weeks
622.45725 1.28916 (62*2π)/2234 weeks
632.41989 1.70695 (63*2π)/2234 weeks
641.36311 1.47151 (64*2π)/2233 weeks
652.06996 .56777 (65*2π)/2233 weeks
661.50299 1.15384 (66*2π)/2233 weeks
671.56169 .15307 (67*2π)/2233 weeks
682.65301 1.0914 (68*2π)/2233 weeks
691.07879 1.32356 (69*2π)/2233 weeks
701.5901 -.97778 (70*2π)/2233 weeks
713.13996 .67029 (71*2π)/2233 weeks
721.54997 1.19031 (72*2π)/2233 weeks
732.03639 -.67478 (73*2π)/2233 weeks
742.8075 1.46141 (74*2π)/2233 weeks
75.12219 .71575 (75*2π)/2233 weeks
761.96804 -1.38169 (76*2π)/2233 weeks
774.45473 .53313 (77*2π)/2233 weeks
781.68965 2.09192 (78*2π)/2233 weeks
792.21797 -.56969 (79*2π)/2233 weeks
803.32271 1.98832 (80*2π)/2233 weeks
81.925 1.33146 (81*2π)/2233 weeks
822.62735 .24759 (82*2π)/2233 weeks
832.36976 1.44417 (83*2π)/2233 weeks
842.09067 .95241 (84*2π)/2233 weeks
852.53847 .74848 (85*2π)/2233 weeks
862.46062 2.16066 (86*2π)/2233 weeks
87.81592 1.82929 (87*2π)/2233 weeks
881.68507 .30637 (88*2π)/2233 weeks
891.49888 1.11845 (89*2π)/2233 weeks
901.51861 .22963 (90*2π)/2232 weeks
911.45964 .67606 (91*2π)/2232 weeks
921.11207 -.53039 (92*2π)/2232 weeks
932.50937 -.14104 (93*2π)/2232 weeks
941.30719 .32769 (94*2π)/2232 weeks
951.6178 -.03455 (95*2π)/2232 weeks
961.60742 .41532 (96*2π)/2232 weeks
971.31242 .46529 (97*2π)/2232 weeks
981.88321 -.44585 (98*2π)/2232 weeks
991.47764 .15116 (99*2π)/2232 weeks
1001.10131 -.50998 (100*2π)/2232 weeks
1012.13793 -.201 (101*2π)/2232 weeks
1021.96139 .27338 (102*2π)/2232 weeks
1031.7306 -.33228 (103*2π)/2232 weeks
1041.96564 .08455 (104*2π)/2232 weeks
1052.07264 -.23251 (105*2π)/2232 weeks
1062.70433 -.12939 (106*2π)/2232 weeks
1072.27843 .77874 (107*2π)/2232 weeks
1082.42659 .90372 (108*2π)/2232 weeks
1092.0642 .74417 (109*2π)/2232 weeks
1101.64749 1.5187 (110*2π)/2232 weeks
111.68388 .45893 (111*2π)/2232 weeks
112.68388 -.45893 (112*2π)/2232 weeks
1131.64749 -1.5187 (113*2π)/2232 weeks
1142.0642 -.74417 (114*2π)/2232 weeks
1152.42659 -.90372 (115*2π)/2232 weeks
1162.27843 -.77874 (116*2π)/2232 weeks
1172.70433 .12939 (117*2π)/2232 weeks
1182.07264 .23251 (118*2π)/2232 weeks
1191.96564 -.08455 (119*2π)/2232 weeks
1201.7306 .33228 (120*2π)/2232 weeks
1211.96139 -.27338 (121*2π)/2232 weeks
1222.13793 .201 (122*2π)/2232 weeks
1231.10131 .50998 (123*2π)/2232 weeks
1241.47764 -.15116 (124*2π)/2232 weeks
1251.88321 .44585 (125*2π)/2232 weeks
1261.31242 -.46529 (126*2π)/2232 weeks
1271.60742 -.41532 (127*2π)/2232 weeks
1281.6178 .03455 (128*2π)/2232 weeks
1291.30719 -.32769 (129*2π)/2232 weeks
1302.50937 .14104 (130*2π)/2232 weeks
1311.11207 .53039 (131*2π)/2232 weeks
1321.45964 -.67606 (132*2π)/2232 weeks
1331.51861 -.22963 (133*2π)/2232 weeks
1341.49888 -1.11845 (134*2π)/2232 weeks
1351.68507 -.30637 (135*2π)/2232 weeks
136.81592 -1.82929 (136*2π)/2232 weeks
1372.46062 -2.16066 (137*2π)/2232 weeks
1382.53847 -.74848 (138*2π)/2232 weeks
1392.09067 -.95241 (139*2π)/2232 weeks
1402.36976 -1.44417 (140*2π)/2232 weeks
1412.62735 -.24759 (141*2π)/2232 weeks
142.925 -1.33146 (142*2π)/2232 weeks
1433.32271 -1.98832 (143*2π)/2232 weeks
1442.21797 .56969 (144*2π)/2232 weeks
1451.68965 -2.09192 (145*2π)/2232 weeks
1464.45473 -.53313 (146*2π)/2232 weeks
1471.96804 1.38169 (147*2π)/2232 weeks
148.12219 -.71575 (148*2π)/2232 weeks
1492.8075 -1.46141 (149*2π)/2231 weeks
1502.03639 .67478 (150*2π)/2231 weeks
1511.54997 -1.19031 (151*2π)/2231 weeks
1523.13996 -.67029 (152*2π)/2231 weeks
1531.5901 .97778 (153*2π)/2231 weeks
1541.07879 -1.32356 (154*2π)/2231 weeks
1552.65301 -1.0914 (155*2π)/2231 weeks
1561.56169 -.15307 (156*2π)/2231 weeks
1571.50299 -1.15384 (157*2π)/2231 weeks
1582.06996 -.56777 (158*2π)/2231 weeks
1591.36311 -1.47151 (159*2π)/2231 weeks
1602.41989 -1.70695 (160*2π)/2231 weeks
1612.45725 -1.28916 (161*2π)/2231 weeks
1622.15514 -.54098 (162*2π)/2231 weeks
1631.49641 -.9442 (163*2π)/2231 weeks
1641.88049 -1.5926 (164*2π)/2231 weeks
1652.39168 -1.2078 (165*2π)/2231 weeks
1661.37552 -.85572 (166*2π)/2231 weeks
1671.71127 -2.12451 (167*2π)/2231 weeks
1682.28064 -.33567 (168*2π)/2231 weeks
169.42268 -1.17355 (169*2π)/2231 weeks
1702.2683 -1.87257 (170*2π)/2231 weeks
171.9586 -1.34633 (171*2π)/2231 weeks
1721.41418 -3.00985 (172*2π)/2231 weeks
1733.28053 -1.85388 (173*2π)/2231 weeks
1741.74293 -.86574 (174*2π)/2231 weeks
1752.46461 -2.10828 (175*2π)/2231 weeks
1763.42328 -.01179 (176*2π)/2231 weeks
1771.09297 .21872 (177*2π)/2231 weeks
178-.19924 -1.7912 (178*2π)/2231 weeks
1791.97306 -3.87477 (179*2π)/2231 weeks
1803.63087 -1.28148 (180*2π)/2231 weeks
181.65059 -1.01114 (181*2π)/2231 weeks
1822.6601 -4.1126 (182*2π)/2231 weeks
1835.36998 -2.601 (183*2π)/2231 weeks
1842.59274 1.9582 (184*2π)/2231 weeks
185-1.59561 -2.20126 (185*2π)/2231 weeks
1862.27826 -3.7391 (186*2π)/2231 weeks
187.57278 -.60488 (187*2π)/2231 weeks
1881.01368 -4.51848 (188*2π)/2231 weeks
1894.07226 -2.00933 (189*2π)/2231 weeks
190-.14044 -.77386 (190*2π)/2231 weeks
191-.38283 -4.61129 (191*2π)/2231 weeks
1923.40241 -4.0877 (192*2π)/2231 weeks
1932.1149 -1.86072 (193*2π)/2231 weeks
1942.1745 -1.82052 (194*2π)/2231 weeks
1952.06188 -3.04854 (195*2π)/2231 weeks
1962.20755 -1.199 (196*2π)/2231 weeks
197-1.10924 -2.46534 (197*2π)/2231 weeks
1982.17364 -3.16617 (198*2π)/2231 weeks
199.58163 -.57645 (199*2π)/2231 weeks
200-1.96064 -2.16886 (200*2π)/2231 weeks
201-1.67485 -6.81895 (201*2π)/2231 weeks
2021.52673 -4.39093 (202*2π)/2231 weeks
203-5.00921 -5.23174 (203*2π)/2231 weeks
204.1259 -10.02077 (204*2π)/2231 weeks
205.73942 -7.78863 (205*2π)/2231 weeks
2062.37754 -11.67633 (206*2π)/2231 weeks
2073.19976 -9.5936 (207*2π)/2231 weeks
208.19385 -7.87933 (208*2π)/2231 weeks
209-2.73778 -10.01367 (209*2π)/2231 weeks
2102.97662 -14.88466 (210*2π)/2231 weeks
2114.90459 -10.43084 (211*2π)/2231 weeks
212-2.13709 -12.67566 (212*2π)/2231 weeks
2136.90678 -22.69814 (213*2π)/2231 weeks
21411.14081 -20.69484 (214*2π)/2231 weeks
21513.62103 -16.4308 (215*2π)/2231 weeks
21615.07408 -14.47046 (216*2π)/2231 weeks
21715.04421 .66859 (217*2π)/2231 weeks
218-11.61267 -9.75688 (218*2π)/2231 weeks
21911.34154 -38.02789 (219*2π)/2231 weeks
22023.02192 -16.42589 (220*2π)/2231 weeks
221-8.72918 -15.91358 (221*2π)/2231 weeks



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