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# Fourier Analysis of SKF (ProShares UltraShort Financials)

SKF (ProShares UltraShort Financials) appears to have interesting cyclic behaviour every 46 weeks (74.6721*sine), 18 weeks (58.6217*sine), and 17 weeks (51.5364*cosine).

SKF (ProShares UltraShort Financials) has an average price of 453.84 (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 2/1/2007 to 6/18/2018 for SKF (ProShares UltraShort Financials), this program was able to calculate the following Fourier Series:
Sequence #Cosine Coefficients Sine Coefficients FrequenciesPeriod
0453.844   0
1400.3723 516.299 (1*2π)/595595 weeks
220.99244 465.9141 (2*2π)/595298 weeks
3-199.8882 269.7682 (3*2π)/595198 weeks
4-178.9326 13.55038 (4*2π)/595149 weeks
5-19.08336 -65.85799 (5*2π)/595119 weeks
678.61375 -1.6007 (6*2π)/59599 weeks
760.82984 89.7345 (7*2π)/59585 weeks
8-26.42795 102.3887 (8*2π)/59574 weeks
9-58.08352 21.73615 (9*2π)/59566 weeks
10-3.54204 -18.4837 (10*2π)/59560 weeks
1137.06831 -1.3043 (11*2π)/59554 weeks
1251.00256 43.0845 (12*2π)/59550 weeks
133.67205 74.67207 (13*2π)/59546 weeks
14-34.62764 36.53414 (14*2π)/59543 weeks
15-16.79695 -1.38111 (15*2π)/59540 weeks
1615.449 -3.47882 (16*2π)/59537 weeks
1730.88468 26.01464 (17*2π)/59535 weeks
184.95503 47.94375 (18*2π)/59533 weeks
19-20.03054 33.21508 (19*2π)/59531 weeks
20-24.6237 5.48029 (20*2π)/59530 weeks
21-.87196 -15.6244 (21*2π)/59528 weeks
2226.94094 -3.08344 (22*2π)/59527 weeks
2328.76612 24.84077 (23*2π)/59526 weeks
247.15221 36.97895 (24*2π)/59525 weeks
25-7.85337 21.23827 (25*2π)/59524 weeks
26.92231 8.71756 (26*2π)/59523 weeks
2716.09919 17.33313 (27*2π)/59522 weeks
284.80755 39.12997 (28*2π)/59521 weeks
29-22.77112 29.94016 (29*2π)/59521 weeks
30-29.58113 1.85664 (30*2π)/59520 weeks
31-7.33772 -23.89999 (31*2π)/59519 weeks
3232.63617 -13.77534 (32*2π)/59519 weeks
3342.13499 29.58806 (33*2π)/59518 weeks
346.93663 58.62165 (34*2π)/59518 weeks
35-39.51111 44.74487 (35*2π)/59517 weeks
36-51.53639 -4.72816 (36*2π)/59517 weeks
37-14.09953 -37.58614 (37*2π)/59516 weeks
3825.82796 -23.73131 (38*2π)/59516 weeks
3934.51508 10.60676 (39*2π)/59515 weeks
4012.13535 32.56924 (40*2π)/59515 weeks
41-12.1649 23.32792 (41*2π)/59515 weeks
42-14.46632 2.4467 (42*2π)/59514 weeks
43-.0687 -6.73857 (43*2π)/59514 weeks
4414.10098 .24396 (44*2π)/59514 weeks
4512.1681 18.50735 (45*2π)/59513 weeks
46-6.25007 19.60327 (46*2π)/59513 weeks
47-10.57474 5.17439 (47*2π)/59513 weeks
48-3.5732 -4.39924 (48*2π)/59512 weeks
4910.61161 -3.0455 (49*2π)/59512 weeks
5014.09859 11.02215 (50*2π)/59512 weeks
514.2938 18.85517 (51*2π)/59512 weeks
52-6.168 18.03347 (52*2π)/59511 weeks
53-12.20341 6.6272 (53*2π)/59511 weeks
54-6.48135 -1.76006 (54*2π)/59511 weeks
551.34113 -2.03347 (55*2π)/59511 weeks
563.81017 3.12596 (56*2π)/59511 weeks
571.14633 4.5743 (57*2π)/59510 weeks
58.12637 1.44097 (58*2π)/59510 weeks
593.88821 .19316 (59*2π)/59510 weeks
607.5648 3.94029 (60*2π)/59510 weeks
614.47932 10.25399 (61*2π)/59510 weeks
62-2.52063 6.95508 (62*2π)/59510 weeks
63-.72085 -.55004 (63*2π)/5959 weeks
648.32112 -.36129 (64*2π)/5959 weeks
6511.96214 9.33259 (65*2π)/5959 weeks
663.96111 20.1017 (66*2π)/5959 weeks
67-13.5805 16.61592 (67*2π)/5959 weeks
68-17.49381 -1.82708 (68*2π)/5959 weeks
69-6.36308 -14.57291 (69*2π)/5959 weeks
7011.93102 -13.59333 (70*2π)/5959 weeks
7119.80573 3.01677 (71*2π)/5958 weeks
728.64517 14.61058 (72*2π)/5958 weeks
73-.9226 10.56244 (73*2π)/5958 weeks
74-2.50417 2.10579 (74*2π)/5958 weeks
756.96716 -1.44165 (75*2π)/5958 weeks
7612.43707 9.51881 (76*2π)/5958 weeks
772.55606 18.64573 (77*2π)/5958 weeks
78-10.67866 14.26289 (78*2π)/5958 weeks
79-15.29983 .095 (79*2π)/5958 weeks
80-3.1899 -12.63963 (80*2π)/5957 weeks
8112.35526 -6.44626 (81*2π)/5957 weeks
8214.0921 7.2219 (82*2π)/5957 weeks
835.46473 18.02809 (83*2π)/5957 weeks
84-13.36355 15.11608 (84*2π)/5957 weeks
85-17.06643 -6.54174 (85*2π)/5957 weeks
86.5327 -17.77727 (86*2π)/5957 weeks
8718.19079 -10.40188 (87*2π)/5957 weeks
8822.23682 10.13044 (88*2π)/5957 weeks
895.71781 22.78127 (89*2π)/5957 weeks
90-10.97337 16.13852 (90*2π)/5957 weeks
91-16.78631 -.25443 (91*2π)/5957 weeks
92-4.53691 -15.29681 (92*2π)/5956 weeks
9312.85988 -10.8753 (93*2π)/5956 weeks
9417.15575 2.68206 (94*2π)/5956 weeks
9510.47969 13.01513 (95*2π)/5956 weeks
96.50607 12.69564 (96*2π)/5956 weeks
97-2.9677 6.34871 (97*2π)/5956 weeks
98-.64909 2.85205 (98*2π)/5956 weeks
991.11937 4.04978 (99*2π)/5956 weeks
100-.68801 5.27683 (100*2π)/5956 weeks
101-3.27319 2.51766 (101*2π)/5956 weeks
102-.1685 -2.00626 (102*2π)/5956 weeks
1034.63953 .58205 (103*2π)/5956 weeks
1043.22066 4.28428 (104*2π)/5956 weeks
105-.2934 5.49852 (105*2π)/5956 weeks
106-3.55058 1.48429 (106*2π)/5956 weeks
107-.1463 -3.59809 (107*2π)/5956 weeks
1085.9629 -2.44775 (108*2π)/5956 weeks
1097.84483 3.48833 (109*2π)/5955 weeks
1102.52257 9.02779 (110*2π)/5955 weeks
111-5.33968 5.66573 (111*2π)/5955 weeks
112-6.07014 -4.05917 (112*2π)/5955 weeks
1132.52268 -10.00598 (113*2π)/5955 weeks
11413.70518 -5.93558 (114*2π)/5955 weeks
11515.41037 7.0866 (115*2π)/5955 weeks
1166.37424 15.09641 (116*2π)/5955 weeks
117-6.33321 12.28943 (117*2π)/5955 weeks
118-10.27148 -1.5489 (118*2π)/5955 weeks
119.45571 -10.7452 (119*2π)/5955 weeks
12013.86326 -5.9194 (120*2π)/5955 weeks
12116.40706 8.23097 (121*2π)/5955 weeks
1225.81773 17.69372 (122*2π)/5955 weeks
123-7.85694 15.01551 (123*2π)/5955 weeks
124-14.17134 1.26991 (124*2π)/5955 weeks
125-4.88924 -11.09791 (125*2π)/5955 weeks
1269.3483 -9.18885 (126*2π)/5955 weeks
12714.02876 2.57983 (127*2π)/5955 weeks
1287.75348 12.51126 (128*2π)/5955 weeks
129-4.39109 12.84286 (129*2π)/5955 weeks
130-10.66177 1.68561 (130*2π)/5955 weeks
131-4.59982 -9.35777 (131*2π)/5955 weeks
1327.63566 -9.60652 (132*2π)/5955 weeks
13314.65202 -.22544 (133*2π)/5954 weeks
13410.43999 10.42568 (134*2π)/5954 weeks
135.87013 12.06781 (135*2π)/5954 weeks
136-5.52787 6.25535 (136*2π)/5954 weeks
137-4.05815 -.87278 (137*2π)/5954 weeks
138.91556 -2.69376 (138*2π)/5954 weeks
1394.21951 -1.07594 (139*2π)/5954 weeks
1405.03036 1.78111 (140*2π)/5954 weeks
1413.22877 4.05027 (141*2π)/5954 weeks
142.07456 3.08102 (142*2π)/5954 weeks
143.41347 -.72466 (143*2π)/5954 weeks
1444.70876 -1.85857 (144*2π)/5954 weeks
1458.05075 .89357 (145*2π)/5954 weeks
1468.55386 5.94697 (146*2π)/5954 weeks
1474.39567 11.46525 (147*2π)/5954 weeks
148-3.76876 10.69389 (148*2π)/5954 weeks
149-7.69285 3.98276 (149*2π)/5954 weeks
150-6.08684 -2.79901 (150*2π)/5954 weeks
151.24075 -6.42168 (151*2π)/5954 weeks
1526.67495 -3.0816 (152*2π)/5954 weeks
1537.39376 2.51186 (153*2π)/5954 weeks
1543.53488 7.11501 (154*2π)/5954 weeks
155-3.56904 4.68616 (155*2π)/5954 weeks
156-2.96965 -3.27428 (156*2π)/5954 weeks
1575.21184 -5.0447 (157*2π)/5954 weeks
15810.20363 1.16891 (158*2π)/5954 weeks
1597.32111 9.18281 (159*2π)/5954 weeks
160-2.01689 10.8645 (160*2π)/5954 weeks
161-7.47628 3.38135 (161*2π)/5954 weeks
162-4.86002 -4.40733 (162*2π)/5954 weeks
1633.10277 -6.9037 (163*2π)/5954 weeks
1649.02387 -.72707 (164*2π)/5954 weeks
1655.93124 5.87979 (165*2π)/5954 weeks
166.24412 5.91035 (166*2π)/5954 weeks
167-2.04592 1.43844 (167*2π)/5954 weeks
1681.14879 -2.03576 (168*2π)/5954 weeks
1695.04187 -.29442 (169*2π)/5954 weeks
1705.37337 3.82014 (170*2π)/5954 weeks
1712.34789 6.47551 (171*2π)/5953 weeks
172-2.8356 5.2596 (172*2π)/5953 weeks
173-3.43223 -.31541 (173*2π)/5953 weeks
174-.31591 -2.34797 (174*2π)/5953 weeks
1751.79163 -2.38736 (175*2π)/5953 weeks
1763.77883 -2.07089 (176*2π)/5953 weeks
1775.78146 -.68913 (177*2π)/5953 weeks
1787.00851 2.79202 (178*2π)/5953 weeks
1794.97415 6.31193 (179*2π)/5953 weeks
180.16957 8.04426 (180*2π)/5953 weeks
181-4.95692 4.0094 (181*2π)/5953 weeks
182-4.2827 -2.51607 (182*2π)/5953 weeks
1831.66835 -5.6944 (183*2π)/5953 weeks
1847.73733 -1.78193 (184*2π)/5953 weeks
1856.73668 5.25197 (185*2π)/5953 weeks
186-.0573 7.81024 (186*2π)/5953 weeks
187-5.49077 2.39675 (187*2π)/5953 weeks
188-3.83826 -4.8386 (188*2π)/5953 weeks
1893.95376 -8.1918 (189*2π)/5953 weeks
19011.44282 -2.46223 (190*2π)/5953 weeks
19110.88522 6.70702 (191*2π)/5953 weeks
1923.04448 12.32278 (192*2π)/5953 weeks
193-7.0419 8.74774 (193*2π)/5953 weeks
194-9.10231 -1.95247 (194*2π)/5953 weeks
195-1.88997 -9.01702 (195*2π)/5953 weeks
1967.84046 -7.79043 (196*2π)/5953 weeks
19712.24727 1.40772 (197*2π)/5953 weeks
1986.41522 8.88666 (198*2π)/5953 weeks
199-1.40037 8.01828 (199*2π)/5953 weeks
200-4.44333 1.19414 (200*2π)/5953 weeks
201.49241 -3.72263 (201*2π)/5953 weeks
2025.90085 -1.12413 (202*2π)/5953 weeks
2036.1176 4.70513 (203*2π)/5953 weeks
204.39712 7.9716 (204*2π)/5953 weeks
205-5.75289 3.5052 (205*2π)/5953 weeks
206-4.05992 -3.5383 (206*2π)/5953 weeks
2071.46283 -5.72897 (207*2π)/5953 weeks
2086.44472 -3.14333 (208*2π)/5953 weeks
2096.27475 2.08731 (209*2π)/5953 weeks
2103.5253 3.01673 (210*2π)/5953 weeks
2112.41164 3.16026 (211*2π)/5953 weeks
212.55787 1.78671 (212*2π)/5953 weeks
2132.40293 .385 (213*2π)/5953 weeks
2143.21398 1.89549 (214*2π)/5953 weeks
2152.80862 2.98799 (215*2π)/5953 weeks
2161.41026 3.98054 (216*2π)/5953 weeks
217-.55451 2.8452 (217*2π)/5953 weeks
218.06365 2.46919 (218*2π)/5953 weeks
219-2.52836 2.73556 (219*2π)/5953 weeks
220-3.37049 -1.12757 (220*2π)/5953 weeks
221-1.56402 -3.8094 (221*2π)/5953 weeks
2222.39884 -6.04868 (222*2π)/5953 weeks
2237.90656 -2.812 (223*2π)/5953 weeks
2247.04702 4.16997 (224*2π)/5953 weeks
225.75345 5.76157 (225*2π)/5953 weeks
226-3.28718 .9126 (226*2π)/5953 weeks
227.08914 -5.37873 (227*2π)/5953 weeks
2288.2452 -4.26029 (228*2π)/5953 weeks
22910.5798 4.31968 (229*2π)/5953 weeks
2304.39562 10.9718 (230*2π)/5953 weeks
231-5.28518 9.74896 (231*2π)/5953 weeks
232-9.77568 -.3885 (232*2π)/5953 weeks
233-2.70625 -8.4512 (233*2π)/5953 weeks
2346.75159 -6.21415 (234*2π)/5953 weeks
2358.61709 2.43161 (235*2π)/5953 weeks
2361.59317 6.76069 (236*2π)/5953 weeks
237-4.13009 1.55272 (237*2π)/5953 weeks
238-.70296 -5.4305 (238*2π)/5953 weeks
2397.27487 -4.48309 (239*2π)/5952 weeks
2409.89212 3.40322 (240*2π)/5952 weeks
2414.38597 10.55621 (241*2π)/5952 weeks
242-6.14326 9.21737 (242*2π)/5952 weeks
243-9.87823 -1.67803 (243*2π)/5952 weeks
244-2.28476 -9.38578 (244*2π)/5952 weeks
2457.13565 -7.69533 (245*2π)/5952 weeks
24611.10746 1.44983 (246*2π)/5952 weeks
2474.54506 8.76885 (247*2π)/5952 weeks
248-3.39218 6.27863 (248*2π)/5952 weeks
249-5.48982 -.77606 (249*2π)/5952 weeks
250-.35577 -5.89476 (250*2π)/5952 weeks
2515.97583 -2.96128 (251*2π)/5952 weeks
2525.78992 2.8422 (252*2π)/5952 weeks
2531.09236 5.41847 (253*2π)/5952 weeks
254-3.75152 2.46132 (254*2π)/5952 weeks
255-3.14293 -3.68056 (255*2π)/5952 weeks
2562.78278 -5.6802 (256*2π)/5952 weeks
2576.29909 -1.28145 (257*2π)/5952 weeks
2584.88835 2.27407 (258*2π)/5952 weeks
2591.33562 3.90609 (259*2π)/5952 weeks
260-.83131 .47077 (260*2π)/5952 weeks
2611.29253 -.96524 (261*2π)/5952 weeks
2622.824 -.35427 (262*2π)/5952 weeks
2632.86867 2.25933 (263*2π)/5952 weeks
264-.64557 2.37859 (264*2π)/5952 weeks
265-1.18027 -.7548 (265*2π)/5952 weeks
266.76574 -1.80902 (266*2π)/5952 weeks
2672.15655 -1.06312 (267*2π)/5952 weeks
2681.9907 .44145 (268*2π)/5952 weeks
269-.58854 -.37792 (269*2π)/5952 weeks
270.74779 -3.95108 (270*2π)/5952 weeks
2715.57805 -3.92808 (271*2π)/5952 weeks
2728.28361 .83738 (272*2π)/5952 weeks
2735.69691 7.17945 (273*2π)/5952 weeks
274-2.62384 8.63903 (274*2π)/5952 weeks
275-9.60807 1.24275 (275*2π)/5952 weeks
276-6.38173 -9.64521 (276*2π)/5952 weeks
2775.88256 -13.60612 (277*2π)/5952 weeks
27816.36525 -3.41378 (278*2π)/5952 weeks
27912.0085 10.493 (279*2π)/5952 weeks
280-1.13878 14.12658 (280*2π)/5952 weeks
281-11.9734 5.77734 (281*2π)/5952 weeks
282-10.39538 -8.67673 (282*2π)/5952 weeks
2832.72076 -14.21005 (283*2π)/5952 weeks
28413.30377 -5.77812 (284*2π)/5952 weeks
28510.22497 6.2609 (285*2π)/5952 weeks
286-.06838 8.62132 (286*2π)/5952 weeks
287-6.20811 1.04055 (287*2π)/5952 weeks
288-2.07546 -7.13382 (288*2π)/5952 weeks
2896.1181 -7.18282 (289*2π)/5952 weeks
29010.74691 .15862 (290*2π)/5952 weeks
2915.66026 8.05177 (291*2π)/5952 weeks
292-3.47753 6.07304 (292*2π)/5952 weeks
293-5.37896 -1.71195 (293*2π)/5952 weeks
294-.82337 -6.39908 (294*2π)/5952 weeks
2955.01606 -5.36663 (295*2π)/5952 weeks
2965.92443 -1.50253 (296*2π)/5952 weeks
2975.21041 -.43582 (297*2π)/5952 weeks
2985.21041 .43582 (298*2π)/5952 weeks
2995.92443 1.50253 (299*2π)/5952 weeks
3005.01606 5.36663 (300*2π)/5952 weeks
301-.82337 6.39908 (301*2π)/5952 weeks
302-5.37896 1.71195 (302*2π)/5952 weeks
303-3.47753 -6.07304 (303*2π)/5952 weeks
3045.66026 -8.05177 (304*2π)/5952 weeks
30510.74691 -.15862 (305*2π)/5952 weeks
3066.1181 7.18282 (306*2π)/5952 weeks
307-2.07546 7.13382 (307*2π)/5952 weeks
308-6.20811 -1.04055 (308*2π)/5952 weeks
309-.06838 -8.62132 (309*2π)/5952 weeks
31010.22497 -6.2609 (310*2π)/5952 weeks
31113.30377 5.77812 (311*2π)/5952 weeks
3122.72076 14.21005 (312*2π)/5952 weeks
313-10.39538 8.67673 (313*2π)/5952 weeks
314-11.9734 -5.77734 (314*2π)/5952 weeks
315-1.13878 -14.12658 (315*2π)/5952 weeks
31612.0085 -10.493 (316*2π)/5952 weeks
31716.36525 3.41378 (317*2π)/5952 weeks
3185.88256 13.60612 (318*2π)/5952 weeks
319-6.38173 9.64521 (319*2π)/5952 weeks
320-9.60807 -1.24275 (320*2π)/5952 weeks
321-2.62384 -8.63903 (321*2π)/5952 weeks
3225.69691 -7.17945 (322*2π)/5952 weeks
3238.28361 -.83738 (323*2π)/5952 weeks
3245.57805 3.92808 (324*2π)/5952 weeks
325.74779 3.95108 (325*2π)/5952 weeks
326-.58854 .37792 (326*2π)/5952 weeks
3271.9907 -.44145 (327*2π)/5952 weeks
3282.15655 1.06312 (328*2π)/5952 weeks
329.76574 1.80902 (329*2π)/5952 weeks
330-1.18027 .7548 (330*2π)/5952 weeks
331-.64557 -2.37859 (331*2π)/5952 weeks
3322.86867 -2.25933 (332*2π)/5952 weeks
3332.824 .35427 (333*2π)/5952 weeks
3341.29253 .96524 (334*2π)/5952 weeks
335-.83131 -.47077 (335*2π)/5952 weeks
3361.33562 -3.90609 (336*2π)/5952 weeks
3374.88835 -2.27407 (337*2π)/5952 weeks
3386.29909 1.28145 (338*2π)/5952 weeks
3392.78278 5.6802 (339*2π)/5952 weeks
340-3.14293 3.68056 (340*2π)/5952 weeks
341-3.75152 -2.46132 (341*2π)/5952 weeks
3421.09236 -5.41847 (342*2π)/5952 weeks
3435.78992 -2.8422 (343*2π)/5952 weeks
3445.97583 2.96128 (344*2π)/5952 weeks
345-.35577 5.89476 (345*2π)/5952 weeks
346-5.48982 .77606 (346*2π)/5952 weeks
347-3.39218 -6.27863 (347*2π)/5952 weeks
3484.54506 -8.76885 (348*2π)/5952 weeks
34911.10746 -1.44983 (349*2π)/5952 weeks
3507.13565 7.69533 (350*2π)/5952 weeks
351-2.28476 9.38578 (351*2π)/5952 weeks
352-9.87823 1.67803 (352*2π)/5952 weeks
353-6.14326 -9.21737 (353*2π)/5952 weeks
3544.38597 -10.55621 (354*2π)/5952 weeks
3559.89212 -3.40322 (355*2π)/5952 weeks
3567.27487 4.48309 (356*2π)/5952 weeks
357-.70296 5.4305 (357*2π)/5952 weeks
358-4.13009 -1.55272 (358*2π)/5952 weeks
3591.59317 -6.76069 (359*2π)/5952 weeks
3608.61709 -2.43161 (360*2π)/5952 weeks
3616.75159 6.21415 (361*2π)/5952 weeks
362-2.70625 8.4512 (362*2π)/5952 weeks
363-9.77568 .3885 (363*2π)/5952 weeks
364-5.28518 -9.74896 (364*2π)/5952 weeks
3654.39562 -10.9718 (365*2π)/5952 weeks
36610.5798 -4.31968 (366*2π)/5952 weeks
3678.2452 4.26029 (367*2π)/5952 weeks
368.08914 5.37873 (368*2π)/5952 weeks
369-3.28718 -.9126 (369*2π)/5952 weeks
370.75345 -5.76157 (370*2π)/5952 weeks
3717.04702 -4.16997 (371*2π)/5952 weeks
3727.90656 2.812 (372*2π)/5952 weeks
3732.39884 6.04868 (373*2π)/5952 weeks
374-1.56402 3.8094 (374*2π)/5952 weeks
375-3.37049 1.12757 (375*2π)/5952 weeks
376-2.52836 -2.73556 (376*2π)/5952 weeks
377.06365 -2.46919 (377*2π)/5952 weeks
378-.55451 -2.8452 (378*2π)/5952 weeks
3791.41026 -3.98054 (379*2π)/5952 weeks
3802.80862 -2.98799 (380*2π)/5952 weeks
3813.21398 -1.89549 (381*2π)/5952 weeks
3822.40293 -.385 (382*2π)/5952 weeks
383.55787 -1.78671 (383*2π)/5952 weeks
3842.41164 -3.16026 (384*2π)/5952 weeks
3853.5253 -3.01673 (385*2π)/5952 weeks
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