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Fourier Analysis of JNUG (Direxion Daily Junior Gold Miners Index Bull 3x Shares ETF)


JNUG (Direxion Daily Junior Gold Miners Index Bull 3x Shares ETF) appears to have interesting cyclic behaviour every 21 weeks (44.806*cosine), 18 weeks (37.8254*sine), and 19 weeks (35.9768*sine).

JNUG (Direxion Daily Junior Gold Miners Index Bull 3x Shares ETF) has an average price of 144.8 (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 10/3/2013 to 10/9/2017 for JNUG (Direxion Daily Junior Gold Miners Index Bull 3x Shares ETF), this program was able to calculate the following Fourier Series:
Sequence #Cosine Coefficients Sine Coefficients FrequenciesPeriod
0144.8042   0 
1135.2241 129.409 (1*2π)/211211 weeks
2-8.94404 137.7869 (2*2π)/211106 weeks
3-21.71813 55.71451 (3*2π)/21170 weeks
41.29093 -1.69872 (4*2π)/21153 weeks
518.87718 27.16321 (5*2π)/21142 weeks
67.03745 41.23743 (6*2π)/21135 weeks
7-8.73667 7.77359 (7*2π)/21130 weeks
84.78525 -16.05251 (8*2π)/21126 weeks
934.70376 -4.2461 (9*2π)/21123 weeks
1044.80604 24.6203 (10*2π)/21121 weeks
1125.46679 35.97676 (11*2π)/21119 weeks
1211.63644 37.82537 (12*2π)/21118 weeks
131.98787 32.42177 (13*2π)/21116 weeks
14-3.41554 20.2363 (14*2π)/21115 weeks
15-3.09492 18.15674 (15*2π)/21114 weeks
16.29987 17.29408 (16*2π)/21113 weeks
17.36133 12.12834 (17*2π)/21112 weeks
18-.82138 10.72124 (18*2π)/21112 weeks
191.0886 10.68744 (19*2π)/21111 weeks
202.39248 7.54839 (20*2π)/21111 weeks
215.14964 10.8676 (21*2π)/21110 weeks
226.28251 15.9171 (22*2π)/21110 weeks
23-1.90792 15.73488 (23*2π)/2119 weeks
24-6.54807 12.42495 (24*2π)/2119 weeks
25-6.17823 3.12434 (25*2π)/2118 weeks
26-1.96524 -.20926 (26*2π)/2118 weeks
274.04739 1.8394 (27*2π)/2118 weeks
286.01403 5.94894 (28*2π)/2118 weeks
291.31911 7.86825 (29*2π)/2117 weeks
301.70717 5.34902 (30*2π)/2117 weeks
313.60691 3.35229 (31*2π)/2117 weeks
322.95987 6.05261 (32*2π)/2117 weeks
33.12644 7.24791 (33*2π)/2116 weeks
34.57053 2.61219 (34*2π)/2116 weeks
353.53161 3.01778 (35*2π)/2116 weeks
362.83466 6.58022 (36*2π)/2116 weeks
371.01649 6.34223 (37*2π)/2116 weeks
38.20373 3.61926 (38*2π)/2116 weeks
39-2.02082 1.27872 (39*2π)/2115 weeks
40-1.21953 2.05198 (40*2π)/2115 weeks
412.36216 .13239 (41*2π)/2115 weeks
423.16919 -1.53636 (42*2π)/2115 weeks
432.9672 .51042 (43*2π)/2115 weeks
442.99051 2.04036 (44*2π)/2115 weeks
453.91237 .30056 (45*2π)/2115 weeks
465.50035 1.98764 (46*2π)/2115 weeks
475.69865 6.13306 (47*2π)/2114 weeks
48-.11784 6.3929 (48*2π)/2114 weeks
49-4.20053 2.17032 (49*2π)/2114 weeks
50-1.24641 -1.41206 (50*2π)/2114 weeks
51-.14024 -4.1893 (51*2π)/2114 weeks
521.41249 -4.34194 (52*2π)/2114 weeks
535.47219 -4.32289 (53*2π)/2114 weeks
548.05425 -2.59899 (54*2π)/2114 weeks
556.5927 -.12376 (55*2π)/2114 weeks
566.22442 .26427 (56*2π)/2114 weeks
578.39102 .4436 (57*2π)/2114 weeks
587.7087 4.63938 (58*2π)/2114 weeks
591.78302 7.50853 (59*2π)/2114 weeks
60-2.75125 2.96375 (60*2π)/2114 weeks
61-2.34596 -3.98749 (61*2π)/2113 weeks
622.78334 -7.07789 (62*2π)/2113 weeks
638.10227 -6.00077 (63*2π)/2113 weeks
649.6569 -1.9705 (64*2π)/2113 weeks
659.13822 .54215 (65*2π)/2113 weeks
668.25698 .94904 (66*2π)/2113 weeks
676.46794 2.24567 (67*2π)/2113 weeks
685.7945 2.37518 (68*2π)/2113 weeks
696.06934 .7835 (69*2π)/2113 weeks
705.12012 .53396 (70*2π)/2113 weeks
715.35904 1.3689 (71*2π)/2113 weeks
725.90158 .85364 (72*2π)/2113 weeks
735.86468 .252 (73*2π)/2113 weeks
746.30205 .28117 (74*2π)/2113 weeks
757.05591 2.01483 (75*2π)/2113 weeks
766.48318 2.5956 (76*2π)/2113 weeks
774.89534 .71037 (77*2π)/2113 weeks
784.77479 .45117 (78*2π)/2113 weeks
797.53798 1.02595 (79*2π)/2113 weeks
808.73958 1.12076 (80*2π)/2113 weeks
816.99228 4.05623 (81*2π)/2113 weeks
825.9583 4.55784 (82*2π)/2113 weeks
834.98138 3.04414 (83*2π)/2113 weeks
845.06466 2.94113 (84*2π)/2113 weeks
856.17134 4.05875 (85*2π)/2112 weeks
865.00538 4.63512 (86*2π)/2112 weeks
872.54256 4.46445 (87*2π)/2112 weeks
882.99411 3.35599 (88*2π)/2112 weeks
894.13325 2.75652 (89*2π)/2112 weeks
902.66121 3.53652 (90*2π)/2112 weeks
911.36325 3.55954 (91*2π)/2112 weeks
921.50796 2.35854 (92*2π)/2112 weeks
932.30214 1.51542 (93*2π)/2112 weeks
941.64234 .5495 (94*2π)/2112 weeks
951.67734 1.26785 (95*2π)/2112 weeks
961.54788 1.51818 (96*2π)/2112 weeks
971.00838 -.70443 (97*2π)/2112 weeks
981.85014 -1.83568 (98*2π)/2112 weeks
994.28428 -1.73831 (99*2π)/2112 weeks
1005.38447 -.1572 (100*2π)/2112 weeks
1014.18382 1.54035 (101*2π)/2112 weeks
1023.57451 .56143 (102*2π)/2112 weeks
1034.00675 .62958 (103*2π)/2112 weeks
1043.1948 2.79389 (104*2π)/2112 weeks
105.53929 1.82875 (105*2π)/2112 weeks
106.53929 -1.82875 (106*2π)/2112 weeks
1073.1948 -2.79389 (107*2π)/2112 weeks
1084.00675 -.62958 (108*2π)/2112 weeks
1093.57451 -.56143 (109*2π)/2112 weeks
1104.18382 -1.54035 (110*2π)/2112 weeks
1115.38447 .1572 (111*2π)/2112 weeks
1124.28428 1.73831 (112*2π)/2112 weeks
1131.85014 1.83568 (113*2π)/2112 weeks
1141.00838 .70443 (114*2π)/2112 weeks
1151.54788 -1.51818 (115*2π)/2112 weeks
1161.67734 -1.26785 (116*2π)/2112 weeks
1171.64234 -.5495 (117*2π)/2112 weeks
1182.30214 -1.51542 (118*2π)/2112 weeks
1191.50796 -2.35854 (119*2π)/2112 weeks
1201.36325 -3.55954 (120*2π)/2112 weeks
1212.66121 -3.53652 (121*2π)/2112 weeks
1224.13325 -2.75652 (122*2π)/2112 weeks
1232.99411 -3.35599 (123*2π)/2112 weeks
1242.54256 -4.46445 (124*2π)/2112 weeks
1255.00538 -4.63512 (125*2π)/2112 weeks
1266.17134 -4.05875 (126*2π)/2112 weeks
1275.06466 -2.94113 (127*2π)/2112 weeks
1284.98138 -3.04414 (128*2π)/2112 weeks
1295.9583 -4.55784 (129*2π)/2112 weeks
1306.99228 -4.05623 (130*2π)/2112 weeks
1318.73958 -1.12076 (131*2π)/2112 weeks
1327.53798 -1.02595 (132*2π)/2112 weeks
1334.77479 -.45117 (133*2π)/2112 weeks
1344.89534 -.71037 (134*2π)/2112 weeks
1356.48318 -2.5956 (135*2π)/2112 weeks
1367.05591 -2.01483 (136*2π)/2112 weeks
1376.30205 -.28117 (137*2π)/2112 weeks
1385.86468 -.252 (138*2π)/2112 weeks
1395.90158 -.85364 (139*2π)/2112 weeks
1405.35904 -1.3689 (140*2π)/2112 weeks
1415.12012 -.53396 (141*2π)/2111 weeks
1426.06934 -.7835 (142*2π)/2111 weeks
1435.7945 -2.37518 (143*2π)/2111 weeks
1446.46794 -2.24567 (144*2π)/2111 weeks
1458.25698 -.94904 (145*2π)/2111 weeks
1469.13822 -.54215 (146*2π)/2111 weeks
1479.6569 1.9705 (147*2π)/2111 weeks
1488.10227 6.00077 (148*2π)/2111 weeks
1492.78334 7.07789 (149*2π)/2111 weeks
150-2.34596 3.98749 (150*2π)/2111 weeks
151-2.75125 -2.96375 (151*2π)/2111 weeks
1521.78302 -7.50853 (152*2π)/2111 weeks
1537.7087 -4.63938 (153*2π)/2111 weeks
1548.39102 -.4436 (154*2π)/2111 weeks
1556.22442 -.26427 (155*2π)/2111 weeks
1566.5927 .12376 (156*2π)/2111 weeks
1578.05425 2.59899 (157*2π)/2111 weeks
1585.47219 4.32289 (158*2π)/2111 weeks
1591.41249 4.34194 (159*2π)/2111 weeks
160-.14024 4.1893 (160*2π)/2111 weeks
161-1.24641 1.41206 (161*2π)/2111 weeks
162-4.20053 -2.17032 (162*2π)/2111 weeks
163-.11784 -6.3929 (163*2π)/2111 weeks
1645.69865 -6.13306 (164*2π)/2111 weeks
1655.50035 -1.98764 (165*2π)/2111 weeks
1663.91237 -.30056 (166*2π)/2111 weeks
1672.99051 -2.04036 (167*2π)/2111 weeks
1682.9672 -.51042 (168*2π)/2111 weeks
1693.16919 1.53636 (169*2π)/2111 weeks
1702.36216 -.13239 (170*2π)/2111 weeks
171-1.21953 -2.05198 (171*2π)/2111 weeks
172-2.02082 -1.27872 (172*2π)/2111 weeks
173.20373 -3.61926 (173*2π)/2111 weeks
1741.01649 -6.34223 (174*2π)/2111 weeks
1752.83466 -6.58022 (175*2π)/2111 weeks
1763.53161 -3.01778 (176*2π)/2111 weeks
177.57053 -2.61219 (177*2π)/2111 weeks
178.12644 -7.24791 (178*2π)/2111 weeks
1792.95987 -6.05261 (179*2π)/2111 weeks
1803.60691 -3.35229 (180*2π)/2111 weeks
1811.70717 -5.34902 (181*2π)/2111 weeks
1821.31911 -7.86825 (182*2π)/2111 weeks
1836.01403 -5.94894 (183*2π)/2111 weeks
1844.04739 -1.8394 (184*2π)/2111 weeks
185-1.96524 .20926 (185*2π)/2111 weeks
186-6.17823 -3.12434 (186*2π)/2111 weeks
187-6.54807 -12.42495 (187*2π)/2111 weeks
188-1.90792 -15.73488 (188*2π)/2111 weeks
1896.28251 -15.9171 (189*2π)/2111 weeks
1905.14964 -10.8676 (190*2π)/2111 weeks
1912.39248 -7.54839 (191*2π)/2111 weeks
1921.0886 -10.68744 (192*2π)/2111 weeks
193-.82138 -10.72124 (193*2π)/2111 weeks
194.36133 -12.12834 (194*2π)/2111 weeks
195.29987 -17.29408 (195*2π)/2111 weeks
196-3.09492 -18.15674 (196*2π)/2111 weeks
197-3.41554 -20.2363 (197*2π)/2111 weeks
1981.98787 -32.42177 (198*2π)/2111 weeks
19911.63644 -37.82537 (199*2π)/2111 weeks
20025.46679 -35.97676 (200*2π)/2111 weeks
20144.80604 -24.6203 (201*2π)/2111 weeks
20234.70376 4.2461 (202*2π)/2111 weeks
2034.78525 16.05251 (203*2π)/2111 weeks
204-8.73667 -7.77359 (204*2π)/2111 weeks
2057.03745 -41.23743 (205*2π)/2111 weeks
20618.87718 -27.16321 (206*2π)/2111 weeks
2071.29093 1.69872 (207*2π)/2111 weeks
208-21.71813 -55.71451 (208*2π)/2111 weeks
209-8.94404 -137.7869 (209*2π)/2111 weeks



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