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Fourier Analysis of SKF (ProShares UltraShort Financials ETF)


SKF (ProShares UltraShort Financials ETF) appears to have interesting cyclic behaviour every 33 weeks (2.3463*cosine), 28 weeks (1.989*sine), and 33 weeks (1.57*sine).

SKF (ProShares UltraShort Financials ETF) has an average price of 36.31 (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 1/4/2010 to 7/10/2017 for SKF (ProShares UltraShort Financials ETF), this program was able to calculate the following Fourier Series:
Sequence #Cosine Coefficients Sine Coefficients FrequenciesPeriod
036.31086   0 
12.85358 2.08934 (1*2π)/393393 weeks
2-18.92535 -3.60377 (2*2π)/393197 weeks
3-1.45983 -3.25782 (3*2π)/393131 weeks
43.92736 -5.68164 (4*2π)/39398 weeks
52.45882 2.43636 (5*2π)/39379 weeks
62.58185 4.40085 (6*2π)/39366 weeks
7-2.80346 -1.1733 (7*2π)/39356 weeks
8-2.15272 -.09217 (8*2π)/39349 weeks
91.70612 -.60368 (9*2π)/39344 weeks
10-.4362 -1.55577 (10*2π)/39339 weeks
11.10086 -.15041 (11*2π)/39336 weeks
122.34627 -1.57005 (12*2π)/39333 weeks
13-.1504 .27757 (13*2π)/39330 weeks
14-.69783 1.98896 (14*2π)/39328 weeks
15-.43809 -.2087 (15*2π)/39326 weeks
16-.91684 -.39413 (16*2π)/39325 weeks
17.34809 -.87621 (17*2π)/39323 weeks
18.80078 -1.37226 (18*2π)/39322 weeks
19.19451 1.87747 (19*2π)/39321 weeks
20.64477 1.55976 (20*2π)/39320 weeks
21-.39731 -.55542 (21*2π)/39319 weeks
22-.75941 -.3722 (22*2π)/39318 weeks
23.08496 -.39717 (23*2π)/39317 weeks
24-.4943 -.11295 (24*2π)/39316 weeks
25.193 .45224 (25*2π)/39316 weeks
26.7137 -.23596 (26*2π)/39315 weeks
27-.43514 .60121 (27*2π)/39315 weeks
28-.03158 1.00831 (28*2π)/39314 weeks
29-.58193 -.11461 (29*2π)/39314 weeks
30-.86638 -.56295 (30*2π)/39313 weeks
31.92775 -.72396 (31*2π)/39313 weeks
32.66364 -.18096 (32*2π)/39312 weeks
33-.17284 .83049 (33*2π)/39312 weeks
34-.24856 .02857 (34*2π)/39312 weeks
35-.58157 -.15018 (35*2π)/39311 weeks
36-.30149 .18371 (36*2π)/39311 weeks
37.41933 -.81578 (37*2π)/39311 weeks
38-.25878 -.48464 (38*2π)/39310 weeks
39.3076 .70038 (39*2π)/39310 weeks
40.11771 .35986 (40*2π)/39310 weeks
41-.53797 .24868 (41*2π)/39310 weeks
42-.09666 .06665 (42*2π)/3939 weeks
43.02771 -.81346 (43*2π)/3939 weeks
44-.03255 .10425 (44*2π)/3939 weeks
45.2139 .22153 (45*2π)/3939 weeks
46-.08199 -.35574 (46*2π)/3939 weeks
47-.05071 .25946 (47*2π)/3938 weeks
48.47535 .3731 (48*2π)/3938 weeks
49-.64476 .03286 (49*2π)/3938 weeks
50-.44979 -.02615 (50*2π)/3938 weeks
51.3456 -.62246 (51*2π)/3938 weeks
52.38401 -.2653 (52*2π)/3938 weeks
53.35195 .70133 (53*2π)/3937 weeks
54-.39304 .2905 (54*2π)/3937 weeks
55-.88789 -.19697 (55*2π)/3937 weeks
56.14422 -.41563 (56*2π)/3937 weeks
57.32115 -.38122 (57*2π)/3937 weeks
58-.17402 .17032 (58*2π)/3937 weeks
59.11248 -.18122 (59*2π)/3937 weeks
60.13914 -.26056 (60*2π)/3937 weeks
61-.0119 .53476 (61*2π)/3936 weeks
62-.21369 .1753 (62*2π)/3936 weeks
63-.27504 -.2147 (63*2π)/3936 weeks
64.23674 -.18366 (64*2π)/3936 weeks
65.30552 -.40753 (65*2π)/3936 weeks
66-.06512 .16608 (66*2π)/3936 weeks
67-.06557 .40108 (67*2π)/3936 weeks
68-.38091 -.16164 (68*2π)/3936 weeks
69-.1784 .10614 (69*2π)/3936 weeks
70.23894 -.05698 (70*2π)/3936 weeks
71-.07105 -.40678 (71*2π)/3936 weeks
72-.03283 -.02368 (72*2π)/3935 weeks
73.31348 .09273 (73*2π)/3935 weeks
74-.10818 .15888 (74*2π)/3935 weeks
75-.1784 .37883 (75*2π)/3935 weeks
76.02343 -.44891 (76*2π)/3935 weeks
77-.21583 -.24647 (77*2π)/3935 weeks
78.066 .18248 (78*2π)/3935 weeks
79.24005 -.1063 (79*2π)/3935 weeks
80.00616 .23443 (80*2π)/3935 weeks
81.01822 .33118 (81*2π)/3935 weeks
82-.36308 -.1705 (82*2π)/3935 weeks
83-.32389 .04175 (83*2π)/3935 weeks
84.332 -.20705 (84*2π)/3935 weeks
85.19512 -.31325 (85*2π)/3935 weeks
86-.01612 .13843 (86*2π)/3935 weeks
87.04896 .16602 (87*2π)/3935 weeks
88-.26047 .03676 (88*2π)/3934 weeks
89.05612 .16652 (89*2π)/3934 weeks
90-.00207 -.28439 (90*2π)/3934 weeks
91-.17772 -.20278 (91*2π)/3934 weeks
92.18493 .05499 (92*2π)/3934 weeks
93.09514 -.06903 (93*2π)/3934 weeks
94-.02075 .22993 (94*2π)/3934 weeks
95.24656 .06526 (95*2π)/3934 weeks
96-.13873 -.2528 (96*2π)/3934 weeks
97-.25765 .13946 (97*2π)/3934 weeks
98.10559 -.08714 (98*2π)/3934 weeks
99.05696 -.256 (99*2π)/3934 weeks
100.21864 .1307 (100*2π)/3934 weeks
101.0372 .20205 (101*2π)/3934 weeks
102-.54483 .12496 (102*2π)/3934 weeks
103-.21198 .10324 (103*2π)/3934 weeks
104.14376 -.35503 (104*2π)/3934 weeks
105.09178 -.10311 (105*2π)/3934 weeks
106.18209 .21531 (106*2π)/3934 weeks
107-.02912 -.06412 (107*2π)/3934 weeks
108-.10012 .02406 (108*2π)/3934 weeks
109.07651 -.00995 (109*2π)/3934 weeks
110-.20953 -.19147 (110*2π)/3934 weeks
111-.13395 -.07335 (111*2π)/3934 weeks
112.16553 -.13534 (112*2π)/3934 weeks
113.17096 .00173 (113*2π)/3933 weeks
114.14495 .39611 (114*2π)/3933 weeks
115-.08979 .06261 (115*2π)/3933 weeks
116-.28578 -.01151 (116*2π)/3933 weeks
117.07173 -.01172 (117*2π)/3933 weeks
118.03668 -.30156 (118*2π)/3933 weeks
119.10537 -.06441 (119*2π)/3933 weeks
120.17897 .13029 (120*2π)/3933 weeks
121-.16053 .02792 (121*2π)/3933 weeks
122-.18421 .13419 (122*2π)/3933 weeks
123-.09784 -.1045 (123*2π)/3933 weeks
124-.07046 -.26689 (124*2π)/3933 weeks
125.16909 .08294 (125*2π)/3933 weeks
126.1138 -.05528 (126*2π)/3933 weeks
127-.14607 .00152 (127*2π)/3933 weeks
128.04649 .12709 (128*2π)/3933 weeks
129-.02389 -.01796 (129*2π)/3933 weeks
130-.26282 .03807 (130*2π)/3933 weeks
131.02768 -.2 (131*2π)/3933 weeks
132.08562 -.2016 (132*2π)/3933 weeks
133.00232 .23884 (133*2π)/3933 weeks
134.06629 .09705 (134*2π)/3933 weeks
135-.21345 -.11084 (135*2π)/3933 weeks
136-.02559 .10088 (136*2π)/3933 weeks
137.0632 -.0441 (137*2π)/3933 weeks
138-.29762 -.18258 (138*2π)/3933 weeks
139.05774 -.11436 (139*2π)/3933 weeks
140.40992 -.09463 (140*2π)/3933 weeks
141-.01714 .24439 (141*2π)/3933 weeks
142-.15212 .16349 (142*2π)/3933 weeks
143-.29031 -.32519 (143*2π)/3933 weeks
144-.02287 -.22341 (144*2π)/3933 weeks
145.30343 .12596 (145*2π)/3933 weeks
146-.02033 .10808 (146*2π)/3933 weeks
147-.07674 .15945 (147*2π)/3933 weeks
148.03439 -.05815 (148*2π)/3933 weeks
149-.11502 -.04998 (149*2π)/3933 weeks
150-.07967 .11357 (150*2π)/3933 weeks
151-.07994 -.10648 (151*2π)/3933 weeks
152-.13763 -.11279 (152*2π)/3933 weeks
153.23238 -.05966 (153*2π)/3933 weeks
154.07606 -.0276 (154*2π)/3933 weeks
155-.20321 .20798 (155*2π)/3933 weeks
156-.02883 .0684 (156*2π)/3933 weeks
157-.18938 -.23008 (157*2π)/3933 weeks
158-.1247 -.06337 (158*2π)/3932 weeks
159.33603 -.09233 (159*2π)/3932 weeks
160.15549 .02261 (160*2π)/3932 weeks
161-.02064 .35551 (161*2π)/3932 weeks
162-.09277 -.08854 (162*2π)/3932 weeks
163-.21475 -.22572 (163*2π)/3932 weeks
164.04332 .14113 (164*2π)/3932 weeks
165.11621 -.03898 (165*2π)/3932 weeks
166-.04994 -.07242 (166*2π)/3932 weeks
167.17368 .08181 (167*2π)/3932 weeks
168.09275 -.03042 (168*2π)/3932 weeks
169-.06238 .11494 (169*2π)/3932 weeks
170.00665 .12804 (170*2π)/3932 weeks
171-.2378 -.33431 (171*2π)/3932 weeks
172.12591 -.11482 (172*2π)/3932 weeks
173.27324 .11709 (173*2π)/3932 weeks
174-.1918 .06504 (174*2π)/3932 weeks
175-.16665 .16573 (175*2π)/3932 weeks
176.04531 -.04598 (176*2π)/3932 weeks
177-.00802 -.06039 (177*2π)/3932 weeks
178-.00629 .05151 (178*2π)/3932 weeks
179.02185 -.10644 (179*2π)/3932 weeks
180-.04049 -.01914 (180*2π)/3932 weeks
181.09587 .13911 (181*2π)/3932 weeks
182.03733 .05423 (182*2π)/3932 weeks
183-.24682 .11781 (183*2π)/3932 weeks
184-.10381 -.21739 (184*2π)/3932 weeks
185-.03496 -.32526 (185*2π)/3932 weeks
186.10604 .08351 (186*2π)/3932 weeks
187.16833 .07919 (187*2π)/3932 weeks
188.05592 .15392 (188*2π)/3932 weeks
189-.0306 .09525 (189*2π)/3932 weeks
190-.0701 -.21317 (190*2π)/3932 weeks
191-.0688 .06794 (191*2π)/3932 weeks
192.18009 .12184 (192*2π)/3932 weeks
193.05569 -.20045 (193*2π)/3932 weeks
194-.08282 -.02133 (194*2π)/3932 weeks
195.09472 .21471 (195*2π)/3932 weeks
196-.23233 .07347 (196*2π)/3932 weeks
197-.23233 -.07347 (197*2π)/3932 weeks
198.09472 -.21471 (198*2π)/3932 weeks
199-.08282 .02133 (199*2π)/3932 weeks
200.05569 .20045 (200*2π)/3932 weeks
201.18009 -.12184 (201*2π)/3932 weeks
202-.0688 -.06794 (202*2π)/3932 weeks
203-.0701 .21317 (203*2π)/3932 weeks
204-.0306 -.09525 (204*2π)/3932 weeks
205.05592 -.15392 (205*2π)/3932 weeks
206.16833 -.07919 (206*2π)/3932 weeks
207.10604 -.08351 (207*2π)/3932 weeks
208-.03496 .32526 (208*2π)/3932 weeks
209-.10381 .21739 (209*2π)/3932 weeks
210-.24682 -.11781 (210*2π)/3932 weeks
211.03733 -.05423 (211*2π)/3932 weeks
212.09587 -.13911 (212*2π)/3932 weeks
213-.04049 .01914 (213*2π)/3932 weeks
214.02185 .10644 (214*2π)/3932 weeks
215-.00629 -.05151 (215*2π)/3932 weeks
216-.00802 .06039 (216*2π)/3932 weeks
217.04531 .04598 (217*2π)/3932 weeks
218-.16665 -.16573 (218*2π)/3932 weeks
219-.1918 -.06504 (219*2π)/3932 weeks
220.27324 -.11709 (220*2π)/3932 weeks
221.12591 .11482 (221*2π)/3932 weeks
222-.2378 .33431 (222*2π)/3932 weeks
223.00665 -.12804 (223*2π)/3932 weeks
224-.06238 -.11494 (224*2π)/3932 weeks
225.09275 .03042 (225*2π)/3932 weeks
226.17368 -.08181 (226*2π)/3932 weeks
227-.04994 .07242 (227*2π)/3932 weeks
228.11621 .03898 (228*2π)/3932 weeks
229.04332 -.14113 (229*2π)/3932 weeks
230-.21475 .22572 (230*2π)/3932 weeks
231-.09277 .08854 (231*2π)/3932 weeks
232-.02064 -.35551 (232*2π)/3932 weeks
233.15549 -.02261 (233*2π)/3932 weeks
234.33603 .09233 (234*2π)/3932 weeks
235-.1247 .06337 (235*2π)/3932 weeks
236-.18938 .23008 (236*2π)/3932 weeks
237-.02883 -.0684 (237*2π)/3932 weeks
238-.20321 -.20798 (238*2π)/3932 weeks
239.07606 .0276 (239*2π)/3932 weeks
240.23238 .05966 (240*2π)/3932 weeks
241-.13763 .11279 (241*2π)/3932 weeks
242-.07994 .10648 (242*2π)/3932 weeks
243-.07967 -.11357 (243*2π)/3932 weeks
244-.11502 .04998 (244*2π)/3932 weeks
245.03439 .05815 (245*2π)/3932 weeks
246-.07674 -.15945 (246*2π)/3932 weeks
247-.02033 -.10808 (247*2π)/3932 weeks
248.30343 -.12596 (248*2π)/3932 weeks
249-.02287 .22341 (249*2π)/3932 weeks
250-.29031 .32519 (250*2π)/3932 weeks
251-.15212 -.16349 (251*2π)/3932 weeks
252-.01714 -.24439 (252*2π)/3932 weeks
253.40992 .09463 (253*2π)/3932 weeks
254.05774 .11436 (254*2π)/3932 weeks
255-.29762 .18258 (255*2π)/3932 weeks
256.0632 .0441 (256*2π)/3932 weeks
257-.02559 -.10088 (257*2π)/3932 weeks
258-.21345 .11084 (258*2π)/3932 weeks
259.06629 -.09705 (259*2π)/3932 weeks
260.00232 -.23884 (260*2π)/3932 weeks
261.08562 .2016 (261*2π)/3932 weeks
262.02768 .2 (262*2π)/3932 weeks
263-.26282 -.03807 (263*2π)/3931 weeks
264-.02389 .01796 (264*2π)/3931 weeks
265.04649 -.12709 (265*2π)/3931 weeks
266-.14607 -.00152 (266*2π)/3931 weeks
267.1138 .05528 (267*2π)/3931 weeks
268.16909 -.08294 (268*2π)/3931 weeks
269-.07046 .26689 (269*2π)/3931 weeks
270-.09784 .1045 (270*2π)/3931 weeks
271-.18421 -.13419 (271*2π)/3931 weeks
272-.16053 -.02792 (272*2π)/3931 weeks
273.17897 -.13029 (273*2π)/3931 weeks
274.10537 .06441 (274*2π)/3931 weeks
275.03668 .30156 (275*2π)/3931 weeks
276.07173 .01172 (276*2π)/3931 weeks
277-.28578 .01151 (277*2π)/3931 weeks
278-.08979 -.06261 (278*2π)/3931 weeks
279.14495 -.39611 (279*2π)/3931 weeks
280.17096 -.00173 (280*2π)/3931 weeks
281.16553 .13534 (281*2π)/3931 weeks
282-.13395 .07335 (282*2π)/3931 weeks
283-.20953 .19147 (283*2π)/3931 weeks
284.07651 .00995 (284*2π)/3931 weeks
285-.10012 -.02406 (285*2π)/3931 weeks
286-.02912 .06412 (286*2π)/3931 weeks
287.18209 -.21531 (287*2π)/3931 weeks
288.09178 .10311 (288*2π)/3931 weeks
289.14376 .35503 (289*2π)/3931 weeks
290-.21198 -.10324 (290*2π)/3931 weeks
291-.54483 -.12496 (291*2π)/3931 weeks
292.0372 -.20205 (292*2π)/3931 weeks
293.21864 -.1307 (293*2π)/3931 weeks
294.05696 .256 (294*2π)/3931 weeks
295.10559 .08714 (295*2π)/3931 weeks
296-.25765 -.13946 (296*2π)/3931 weeks
297-.13873 .2528 (297*2π)/3931 weeks
298.24656 -.06526 (298*2π)/3931 weeks
299-.02075 -.22993 (299*2π)/3931 weeks
300.09514 .06903 (300*2π)/3931 weeks
301.18493 -.05499 (301*2π)/3931 weeks
302-.17772 .20278 (302*2π)/3931 weeks
303-.00207 .28439 (303*2π)/3931 weeks
304.05612 -.16652 (304*2π)/3931 weeks
305-.26047 -.03676 (305*2π)/3931 weeks
306.04896 -.16602 (306*2π)/3931 weeks
307-.01612 -.13843 (307*2π)/3931 weeks
308.19512 .31325 (308*2π)/3931 weeks
309.332 .20705 (309*2π)/3931 weeks
310-.32389 -.04175 (310*2π)/3931 weeks
311-.36308 .1705 (311*2π)/3931 weeks
312.01822 -.33118 (312*2π)/3931 weeks
313.00616 -.23443 (313*2π)/3931 weeks
314.24005 .1063 (314*2π)/3931 weeks
315.066 -.18248 (315*2π)/3931 weeks
316-.21583 .24647 (316*2π)/3931 weeks
317.02343 .44891 (317*2π)/3931 weeks
318-.1784 -.37883 (318*2π)/3931 weeks
319-.10818 -.15888 (319*2π)/3931 weeks
320.31348 -.09273 (320*2π)/3931 weeks
321-.03283 .02368 (321*2π)/3931 weeks
322-.07105 .40678 (322*2π)/3931 weeks
323.23894 .05698 (323*2π)/3931 weeks
324-.1784 -.10614 (324*2π)/3931 weeks
325-.38091 .16164 (325*2π)/3931 weeks
326-.06557 -.40108 (326*2π)/3931 weeks
327-.06512 -.16608 (327*2π)/3931 weeks
328.30552 .40753 (328*2π)/3931 weeks
329.23674 .18366 (329*2π)/3931 weeks
330-.27504 .2147 (330*2π)/3931 weeks
331-.21369 -.1753 (331*2π)/3931 weeks
332-.0119 -.53476 (332*2π)/3931 weeks
333.13914 .26056 (333*2π)/3931 weeks
334.11248 .18122 (334*2π)/3931 weeks
335-.17402 -.17032 (335*2π)/3931 weeks
336.32115 .38122 (336*2π)/3931 weeks
337.14422 .41563 (337*2π)/3931 weeks
338-.88789 .19697 (338*2π)/3931 weeks
339-.39304 -.2905 (339*2π)/3931 weeks
340.35195 -.70133 (340*2π)/3931 weeks
341.38401 .2653 (341*2π)/3931 weeks
342.3456 .62246 (342*2π)/3931 weeks
343-.44979 .02615 (343*2π)/3931 weeks
344-.64476 -.03286 (344*2π)/3931 weeks
345.47535 -.3731 (345*2π)/3931 weeks
346-.05071 -.25946 (346*2π)/3931 weeks
347-.08199 .35574 (347*2π)/3931 weeks
348.2139 -.22153 (348*2π)/3931 weeks
349-.03255 -.10425 (349*2π)/3931 weeks
350.02771 .81346 (350*2π)/3931 weeks
351-.09666 -.06665 (351*2π)/3931 weeks
352-.53797 -.24868 (352*2π)/3931 weeks
353.11771 -.35986 (353*2π)/3931 weeks
354.3076 -.70038 (354*2π)/3931 weeks
355-.25878 .48464 (355*2π)/3931 weeks
356.41933 .81578 (356*2π)/3931 weeks
357-.30149 -.18371 (357*2π)/3931 weeks
358-.58157 .15018 (358*2π)/3931 weeks
359-.24856 -.02857 (359*2π)/3931 weeks
360-.17284 -.83049 (360*2π)/3931 weeks
361.66364 .18096 (361*2π)/3931 weeks
362.92775 .72396 (362*2π)/3931 weeks
363-.86638 .56295 (363*2π)/3931 weeks
364-.58193 .11461 (364*2π)/3931 weeks
365-.03158 -1.00831 (365*2π)/3931 weeks
366-.43514 -.60121 (366*2π)/3931 weeks
367.7137 .23596 (367*2π)/3931 weeks
368.193 -.45224 (368*2π)/3931 weeks
369-.4943 .11295 (369*2π)/3931 weeks
370.08496 .39717 (370*2π)/3931 weeks
371-.75941 .3722 (371*2π)/3931 weeks
372-.39731 .55542 (372*2π)/3931 weeks
373.64477 -1.55976 (373*2π)/3931 weeks
374.19451 -1.87747 (374*2π)/3931 weeks
375.80078 1.37226 (375*2π)/3931 weeks
376.34809 .87621 (376*2π)/3931 weeks
377-.91684 .39413 (377*2π)/3931 weeks
378-.43809 .2087 (378*2π)/3931 weeks
379-.69783 -1.98896 (379*2π)/3931 weeks
380-.1504 -.27757 (380*2π)/3931 weeks
3812.34627 1.57005 (381*2π)/3931 weeks
382.10086 .15041 (382*2π)/3931 weeks
383-.4362 1.55577 (383*2π)/3931 weeks
3841.70612 .60368 (384*2π)/3931 weeks
385-2.15272 .09217 (385*2π)/3931 weeks
386-2.80346 1.1733 (386*2π)/3931 weeks
3872.58185 -4.40085 (387*2π)/3931 weeks
3882.45882 -2.43636 (388*2π)/3931 weeks
3893.92736 5.68164 (389*2π)/3931 weeks
390-1.45983 3.25782 (390*2π)/3931 weeks
391-18.92535 3.60377 (391*2π)/3931 weeks



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