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# Fourier Analysis of DGAZ (VelocityShares 3x Inv Natural Gas ETN)

DGAZ (VelocityShares 3x Inv Natural Gas ETN) appears to have interesting cyclic behaviour every 32 weeks (16.3804*sine), 29 weeks (14.7336*sine), and 20 weeks (10.4076*cosine).

DGAZ (VelocityShares 3x Inv Natural Gas ETN) has an average price of 66.96 (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.

## Fourier Analysis

Using data from 2/8/2012 to 4/16/2018 for DGAZ (VelocityShares 3x Inv Natural Gas ETN), this program was able to calculate the following Fourier Series:
Sequence #Cosine Coefficients Sine Coefficients FrequenciesPeriod
066.95528   0
145.78842 16.4601 (1*2π)/324324 weeks
229.12613 46.09011 (2*2π)/324162 weeks
333.95731 20.93737 (3*2π)/324108 weeks
417.32025 22.9803 (4*2π)/32481 weeks
510.75583 39.49272 (5*2π)/32465 weeks
611.61847 22.21848 (6*2π)/32454 weeks
71.52875 23.8174 (7*2π)/32446 weeks
83.17071 30.20398 (8*2π)/32441 weeks
9-5.49993 21.81139 (9*2π)/32436 weeks
10-8.11939 16.38039 (10*2π)/32432 weeks
11-7.80316 14.73355 (11*2π)/32429 weeks
12-4.92682 11.82556 (12*2π)/32427 weeks
13-5.65727 10.42552 (13*2π)/32425 weeks
14-8.73627 11.85221 (14*2π)/32423 weeks
15-8.01949 10.53503 (15*2π)/32422 weeks
16-10.40757 2.73616 (16*2π)/32420 weeks
17-7.97746 2.71412 (17*2π)/32419 weeks
18-5.08477 .35773 (18*2π)/32418 weeks
19-7.12831 -1.36349 (19*2π)/32417 weeks
20-4.49892 1.27177 (20*2π)/32416 weeks
21-1.3841 -1.68336 (21*2π)/32415 weeks
22-3.6286 -.3694 (22*2π)/32415 weeks
23-.80967 -.21228 (23*2π)/32414 weeks
24-2.54004 -1.67986 (24*2π)/32414 weeks
25-2.95763 1.74978 (25*2π)/32413 weeks
26-.12645 1.0694 (26*2π)/32412 weeks
27-4.61806 -4.17184 (27*2π)/32412 weeks
28-2.69993 -.93019 (28*2π)/32412 weeks
291.83371 -3.51826 (29*2π)/32411 weeks
30.19412 -3.70048 (30*2π)/32411 weeks
311.76495 -2.01876 (31*2π)/32410 weeks
324.93389 -3.59052 (32*2π)/32410 weeks
333.56136 -1.36393 (33*2π)/32410 weeks
345.58847 -.18411 (34*2π)/32410 weeks
356.63477 -1.4161 (35*2π)/3249 weeks
365.76807 2.40046 (36*2π)/3249 weeks
375.96218 3.07651 (37*2π)/3249 weeks
383.8015 3.58828 (38*2π)/3249 weeks
394.03324 4.38188 (39*2π)/3248 weeks
401.46173 3.2147 (40*2π)/3248 weeks
411.70331 4.13325 (41*2π)/3248 weeks
422.62746 5.60105 (42*2π)/3248 weeks
43.76051 4.78991 (43*2π)/3248 weeks
44-1.64124 5.47713 (44*2π)/3247 weeks
45-1.11504 4.00923 (45*2π)/3247 weeks
46-2.8379 4.62172 (46*2π)/3247 weeks
47-3.34391 3.6621 (47*2π)/3247 weeks
48-3.54196 .81134 (48*2π)/3247 weeks
49-2.45164 .57623 (49*2π)/3247 weeks
50-2.12778 .27136 (50*2π)/3246 weeks
51-2.72856 .4697 (51*2π)/3246 weeks
52-1.38401 -1.45714 (52*2π)/3246 weeks
53-1.73365 -2.62948 (53*2π)/3246 weeks
54-.52795 -1.18477 (54*2π)/3246 weeks
551.63766 -1.77772 (55*2π)/3246 weeks
56.28577 -2.6468 (56*2π)/3246 weeks
57.98519 -.66828 (57*2π)/3246 weeks
583.41183 -1.94642 (58*2π)/3246 weeks
591.49783 -.57067 (59*2π)/3245 weeks
603.69928 .57501 (60*2π)/3245 weeks
612.9846 .16928 (61*2π)/3245 weeks
621.87056 2.08993 (62*2π)/3245 weeks
632.12433 2.589 (63*2π)/3245 weeks
64.9601 .35438 (64*2π)/3245 weeks
65.33287 2.05709 (65*2π)/3245 weeks
661.09296 .98376 (66*2π)/3245 weeks
67.14976 .8451 (67*2π)/3245 weeks
68.62107 1.51 (68*2π)/3245 weeks
69.34077 .50313 (69*2π)/3245 weeks
70-1.00778 -.07521 (70*2π)/3245 weeks
71.69028 -.10286 (71*2π)/3245 weeks
72-.15721 -.46684 (72*2π)/3245 weeks
73-.00751 -.7696 (73*2π)/3244 weeks
741.46496 -1.5617 (74*2π)/3244 weeks
752.30624 -1.1584 (75*2π)/3244 weeks
762.10409 1.08554 (76*2π)/3244 weeks
772.19917 .37029 (77*2π)/3244 weeks
78.6869 .44886 (78*2π)/3244 weeks
791.97653 .84463 (79*2π)/3244 weeks
801.01569 .86978 (80*2π)/3244 weeks
811.1258 1.44507 (81*2π)/3244 weeks
82.87805 .49709 (82*2π)/3244 weeks
83.39654 1.01695 (83*2π)/3244 weeks
841.4509 1.8056 (84*2π)/3244 weeks
85.23291 .11122 (85*2π)/3244 weeks
86-.32584 .91693 (86*2π)/3244 weeks
871.40732 .2393 (87*2π)/3244 weeks
88.23148 -.68158 (88*2π)/3244 weeks
89.34441 .14312 (89*2π)/3244 weeks
901.59397 -.42544 (90*2π)/3244 weeks
91.6128 -.34026 (91*2π)/3244 weeks
921.2906 -.25216 (92*2π)/3244 weeks
93.88097 -1.37592 (93*2π)/3243 weeks
941.33205 .20049 (94*2π)/3243 weeks
952.27603 -.26328 (95*2π)/3243 weeks
961.48332 -.40383 (96*2π)/3243 weeks
971.79687 1.01509 (97*2π)/3243 weeks
982.42683 .36492 (98*2π)/3243 weeks
991.57835 .544 (99*2π)/3243 weeks
1002.11759 1.64336 (100*2π)/3243 weeks
1012.33628 .95851 (101*2π)/3243 weeks
1021.68238 1.12032 (102*2π)/3243 weeks
1031.71833 1.72616 (103*2π)/3243 weeks
1041.58366 1.81037 (104*2π)/3243 weeks
105.28808 1.31191 (105*2π)/3243 weeks
106.65408 1.92466 (106*2π)/3243 weeks
107.77287 1.25781 (107*2π)/3243 weeks
108.04466 .93102 (108*2π)/3243 weeks
109.10142 1.959 (109*2π)/3243 weeks
110.77254 1.5455 (110*2π)/3243 weeks
111-.66274 1.09126 (111*2π)/3243 weeks
112-.94863 1.20418 (112*2π)/3243 weeks
113-.41808 .71638 (113*2π)/3243 weeks
114-.7914 -.22409 (114*2π)/3243 weeks
115-.41249 .23544 (115*2π)/3243 weeks
116.29828 -.40085 (116*2π)/3243 weeks
117.00623 -.11045 (117*2π)/3243 weeks
118.489 -.11477 (118*2π)/3243 weeks
119.50452 -.32793 (119*2π)/3243 weeks
120.09693 -.68714 (120*2π)/3243 weeks
121.32376 -.48584 (121*2π)/3243 weeks
122.86033 -.32403 (122*2π)/3243 weeks
123.64916 -.5878 (123*2π)/3243 weeks
124.47231 -.18534 (124*2π)/3243 weeks
125.58809 -.53546 (125*2π)/3243 weeks
126.60808 -.70939 (126*2π)/3243 weeks
127.31608 -.51921 (127*2π)/3243 weeks
1281.30685 -.90348 (128*2π)/3243 weeks
1291.20254 -1.23399 (129*2π)/3243 weeks
1301.23643 -.49535 (130*2π)/3242 weeks
1311.71245 -.30834 (131*2π)/3242 weeks
1321.54147 -.59519 (132*2π)/3242 weeks
1331.70265 .24904 (133*2π)/3242 weeks
1341.58799 .14059 (134*2π)/3242 weeks
1351.14109 .02267 (135*2π)/3242 weeks
1361.17661 .46856 (136*2π)/3242 weeks
1371.43249 .06042 (137*2π)/3242 weeks
1381.30282 -.12242 (138*2π)/3242 weeks
1391.80236 .30428 (139*2π)/3242 weeks
1401.62835 .41715 (140*2π)/3242 weeks
1411.06196 .83764 (141*2π)/3242 weeks
1421.01406 1.01122 (142*2π)/3242 weeks
143.66654 .48186 (143*2π)/3242 weeks
144.32514 .75173 (144*2π)/3242 weeks
145.76382 .29865 (145*2π)/3242 weeks
146.21601 -.08779 (146*2π)/3242 weeks
147.86094 -.34046 (147*2π)/3242 weeks
148.74838 -.73411 (148*2π)/3242 weeks
149.47412 -.37654 (149*2π)/3242 weeks
150.77369 -.13862 (150*2π)/3242 weeks
1511.02789 -.59863 (151*2π)/3242 weeks
152.78172 .15245 (152*2π)/3242 weeks
1531.55871 -.23032 (153*2π)/3242 weeks
154.72211 .1769 (154*2π)/3242 weeks
1551.09905 .79418 (155*2π)/3242 weeks
156.56098 .24627 (156*2π)/3242 weeks
157.55285 -.30741 (157*2π)/3242 weeks
158.75456 .14621 (158*2π)/3242 weeks
1591.25834 -.21957 (159*2π)/3242 weeks
160.57368 -.4955 (160*2π)/3242 weeks
1611.34452 -.06891 (161*2π)/3242 weeks
1622.02653   (162*2π)/3242 weeks
1631.34452 .06891 (163*2π)/3242 weeks
164.57368 .4955 (164*2π)/3242 weeks
1651.25834 .21957 (165*2π)/3242 weeks
166.75456 -.14621 (166*2π)/3242 weeks
167.55285 .30741 (167*2π)/3242 weeks
168.56098 -.24627 (168*2π)/3242 weeks
1691.09905 -.79418 (169*2π)/3242 weeks
170.72211 -.1769 (170*2π)/3242 weeks
1711.55871 .23032 (171*2π)/3242 weeks
172.78172 -.15245 (172*2π)/3242 weeks
1731.02789 .59863 (173*2π)/3242 weeks
174.77369 .13862 (174*2π)/3242 weeks
175.47412 .37654 (175*2π)/3242 weeks
176.74838 .73411 (176*2π)/3242 weeks
177.86094 .34046 (177*2π)/3242 weeks
178.21601 .08779 (178*2π)/3242 weeks
179.76382 -.29865 (179*2π)/3242 weeks
180.32514 -.75173 (180*2π)/3242 weeks
181.66654 -.48186 (181*2π)/3242 weeks
1821.01406 -1.01122 (182*2π)/3242 weeks
1831.06196 -.83764 (183*2π)/3242 weeks
1841.62835 -.41715 (184*2π)/3242 weeks
1851.80236 -.30428 (185*2π)/3242 weeks
1861.30282 .12242 (186*2π)/3242 weeks
1871.43249 -.06042 (187*2π)/3242 weeks
1881.17661 -.46856 (188*2π)/3242 weeks
1891.14109 -.02267 (189*2π)/3242 weeks
1901.58799 -.14059 (190*2π)/3242 weeks
1911.70265 -.24904 (191*2π)/3242 weeks
1921.54147 .59519 (192*2π)/3242 weeks
1931.71245 .30834 (193*2π)/3242 weeks
1941.23643 .49535 (194*2π)/3242 weeks
1951.20254 1.23399 (195*2π)/3242 weeks
1961.30685 .90348 (196*2π)/3242 weeks
197.31608 .51921 (197*2π)/3242 weeks
198.60808 .70939 (198*2π)/3242 weeks
199.58809 .53546 (199*2π)/3242 weeks
200.47231 .18534 (200*2π)/3242 weeks
201.64916 .5878 (201*2π)/3242 weeks
202.86033 .32403 (202*2π)/3242 weeks
203.32376 .48584 (203*2π)/3242 weeks
204.09693 .68714 (204*2π)/3242 weeks
205.50452 .32793 (205*2π)/3242 weeks
206.489 .11477 (206*2π)/3242 weeks
207.00623 .11045 (207*2π)/3242 weeks
208.29828 .40085 (208*2π)/3242 weeks
209-.41249 -.23544 (209*2π)/3242 weeks
210-.7914 .22409 (210*2π)/3242 weeks
211-.41808 -.71638 (211*2π)/3242 weeks
212-.94863 -1.20418 (212*2π)/3242 weeks
213-.66274 -1.09126 (213*2π)/3242 weeks
214.77254 -1.5455 (214*2π)/3242 weeks
215.10142 -1.959 (215*2π)/3242 weeks
216.04466 -.93102 (216*2π)/3242 weeks
217.77287 -1.25781 (217*2π)/3241 weeks
218.65408 -1.92466 (218*2π)/3241 weeks
219.28808 -1.31191 (219*2π)/3241 weeks
2201.58366 -1.81037 (220*2π)/3241 weeks
2211.71833 -1.72616 (221*2π)/3241 weeks
2221.68238 -1.12032 (222*2π)/3241 weeks
2232.33628 -.95851 (223*2π)/3241 weeks
2242.11759 -1.64336 (224*2π)/3241 weeks
2251.57835 -.544 (225*2π)/3241 weeks
2262.42683 -.36492 (226*2π)/3241 weeks
2271.79687 -1.01509 (227*2π)/3241 weeks
2281.48332 .40383 (228*2π)/3241 weeks
2292.27603 .26328 (229*2π)/3241 weeks
2301.33205 -.20049 (230*2π)/3241 weeks
231.88097 1.37592 (231*2π)/3241 weeks
2321.2906 .25216 (232*2π)/3241 weeks
233.6128 .34026 (233*2π)/3241 weeks
2341.59397 .42544 (234*2π)/3241 weeks
235.34441 -.14312 (235*2π)/3241 weeks
236.23148 .68158 (236*2π)/3241 weeks
2371.40732 -.2393 (237*2π)/3241 weeks
238-.32584 -.91693 (238*2π)/3241 weeks
239.23291 -.11122 (239*2π)/3241 weeks
2401.4509 -1.8056 (240*2π)/3241 weeks
241.39654 -1.01695 (241*2π)/3241 weeks
242.87805 -.49709 (242*2π)/3241 weeks
2431.1258 -1.44507 (243*2π)/3241 weeks
2441.01569 -.86978 (244*2π)/3241 weeks
2451.97653 -.84463 (245*2π)/3241 weeks
246.6869 -.44886 (246*2π)/3241 weeks
2472.19917 -.37029 (247*2π)/3241 weeks
2482.10409 -1.08554 (248*2π)/3241 weeks
2492.30624 1.1584 (249*2π)/3241 weeks
2501.46496 1.5617 (250*2π)/3241 weeks
251-.00751 .7696 (251*2π)/3241 weeks
252-.15721 .46684 (252*2π)/3241 weeks
253.69028 .10286 (253*2π)/3241 weeks
254-1.00778 .07521 (254*2π)/3241 weeks
255.34077 -.50313 (255*2π)/3241 weeks
256.62107 -1.51 (256*2π)/3241 weeks
257.14976 -.8451 (257*2π)/3241 weeks
2581.09296 -.98376 (258*2π)/3241 weeks
259.33287 -2.05709 (259*2π)/3241 weeks
260.9601 -.35438 (260*2π)/3241 weeks
2612.12433 -2.589 (261*2π)/3241 weeks
2621.87056 -2.08993 (262*2π)/3241 weeks
2632.9846 -.16928 (263*2π)/3241 weeks
2643.69928 -.57501 (264*2π)/3241 weeks
2651.49783 .57067 (265*2π)/3241 weeks
2663.41183 1.94642 (266*2π)/3241 weeks
267.98519 .66828 (267*2π)/3241 weeks
268.28577 2.6468 (268*2π)/3241 weeks
2691.63766 1.77772 (269*2π)/3241 weeks
270-.52795 1.18477 (270*2π)/3241 weeks
271-1.73365 2.62948 (271*2π)/3241 weeks
272-1.38401 1.45714 (272*2π)/3241 weeks
273-2.72856 -.4697 (273*2π)/3241 weeks
274-2.12778 -.27136 (274*2π)/3241 weeks
275-2.45164 -.57623 (275*2π)/3241 weeks
276-3.54196 -.81134 (276*2π)/3241 weeks
277-3.34391 -3.6621 (277*2π)/3241 weeks
278-2.8379 -4.62172 (278*2π)/3241 weeks
279-1.11504 -4.00923 (279*2π)/3241 weeks
280-1.64124 -5.47713 (280*2π)/3241 weeks
281.76051 -4.78991 (281*2π)/3241 weeks
2822.62746 -5.60105 (282*2π)/3241 weeks
2831.70331 -4.13325 (283*2π)/3241 weeks
2841.46173 -3.2147 (284*2π)/3241 weeks
2854.03324 -4.38188 (285*2π)/3241 weeks
2863.8015 -3.58828 (286*2π)/3241 weeks
2875.96218 -3.07651 (287*2π)/3241 weeks
2885.76807 -2.40046 (288*2π)/3241 weeks
2896.63477 1.4161 (289*2π)/3241 weeks
2905.58847 .18411 (290*2π)/3241 weeks
2913.56136 1.36393 (291*2π)/3241 weeks
2924.93389 3.59052 (292*2π)/3241 weeks
2931.76495 2.01876 (293*2π)/3241 weeks
294.19412 3.70048 (294*2π)/3241 weeks
2951.83371 3.51826 (295*2π)/3241 weeks
296-2.69993 .93019 (296*2π)/3241 weeks
297-4.61806 4.17184 (297*2π)/3241 weeks
298-.12645 -1.0694 (298*2π)/3241 weeks
299-2.95763 -1.74978 (299*2π)/3241 weeks
300-2.54004 1.67986 (300*2π)/3241 weeks
301-.80967 .21228 (301*2π)/3241 weeks
302-3.6286 .3694 (302*2π)/3241 weeks
303-1.3841 1.68336 (303*2π)/3241 weeks
304-4.49892 -1.27177 (304*2π)/3241 weeks
305-7.12831 1.36349 (305*2π)/3241 weeks
306-5.08477 -.35773 (306*2π)/3241 weeks
307-7.97746 -2.71412 (307*2π)/3241 weeks
308-10.40757 -2.73616 (308*2π)/3241 weeks
309-8.01949 -10.53503 (309*2π)/3241 weeks
310-8.73627 -11.85221 (310*2π)/3241 weeks
311-5.65727 -10.42552 (311*2π)/3241 weeks
312-4.92682 -11.82556 (312*2π)/3241 weeks
313-7.80316 -14.73355 (313*2π)/3241 weeks
314-8.11939 -16.38039 (314*2π)/3241 weeks
315-5.49993 -21.81139 (315*2π)/3241 weeks
3163.17071 -30.20398 (316*2π)/3241 weeks
3171.52875 -23.8174 (317*2π)/3241 weeks
31811.61847 -22.21848 (318*2π)/3241 weeks
31910.75583 -39.49272 (319*2π)/3241 weeks
32017.32025 -22.9803 (320*2π)/3241 weeks
32133.95731 -20.93737 (321*2π)/3241 weeks
32229.12613 -46.09011 (322*2π)/3241 weeks