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Fourier Analysis of XAR (SPDR® S&P Aerospace & Defense ETF)


XAR (SPDR® S&P Aerospace & Defense ETF) appears to have interesting cyclic behaviour every 25 weeks (1.6733*sine), 28 weeks (1.618*sine), and 30 weeks (1.3162*sine).

XAR (SPDR® S&P Aerospace & Defense ETF) has an average price of 45.75 (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 9/29/2011 to 7/10/2017 for XAR (SPDR® S&P Aerospace & Defense ETF), this program was able to calculate the following Fourier Series:
Sequence #Cosine Coefficients Sine Coefficients FrequenciesPeriod
045.74897   0 
1-2.59471 -13.19744 (1*2π)/303303 weeks
23.43684 -7.38617 (2*2π)/303152 weeks
31.40393 -5.4701 (3*2π)/303101 weeks
4-1.65451 -3.74765 (4*2π)/30376 weeks
5.71307 -2.46509 (5*2π)/30361 weeks
6-.63999 -2.8444 (6*2π)/30351 weeks
7-.35895 -2.21023 (7*2π)/30343 weeks
8-.42843 -1.88465 (8*2π)/30338 weeks
9-.54342 -1.52226 (9*2π)/30334 weeks
10.32487 -1.31621 (10*2π)/30330 weeks
11.21069 -1.61801 (11*2π)/30328 weeks
12-.1563 -1.67331 (12*2π)/30325 weeks
13.01098 -1.03559 (13*2π)/30323 weeks
14.06168 -.99762 (14*2π)/30322 weeks
15-.20591 -1.28948 (15*2π)/30320 weeks
16-.45056 -.81531 (16*2π)/30319 weeks
17-.02072 -.52692 (17*2π)/30318 weeks
18.14492 -.94994 (18*2π)/30317 weeks
19-.05537 -.96027 (19*2π)/30316 weeks
20-.19795 -.79405 (20*2π)/30315 weeks
21-.35506 -.99139 (21*2π)/30314 weeks
22-.17346 -.69528 (22*2π)/30314 weeks
23-.55854 -.67715 (23*2π)/30313 weeks
24-.21794 -.26263 (24*2π)/30313 weeks
25-.28515 -.47469 (25*2π)/30312 weeks
26-.00209 -.37326 (26*2π)/30312 weeks
27-.18186 -.71972 (27*2π)/30311 weeks
28-.04392 -.53483 (28*2π)/30311 weeks
29-.16679 -.50994 (29*2π)/30310 weeks
30-.11738 -.5062 (30*2π)/30310 weeks
31-.1746 -.75898 (31*2π)/30310 weeks
32-.23431 -.40503 (32*2π)/3039 weeks
33-.24306 -.34877 (33*2π)/3039 weeks
34-.16754 -.33845 (34*2π)/3039 weeks
35-.19881 -.56657 (35*2π)/3039 weeks
36-.09981 -.27195 (36*2π)/3038 weeks
37-.28508 -.52007 (37*2π)/3038 weeks
38-.10481 -.49464 (38*2π)/3038 weeks
39-.16978 -.40135 (39*2π)/3038 weeks
40-.28355 -.30893 (40*2π)/3038 weeks
41-.07773 -.27276 (41*2π)/3037 weeks
42-.25869 -.26248 (42*2π)/3037 weeks
43.0073 -.35724 (43*2π)/3037 weeks
44-.19874 -.37805 (44*2π)/3037 weeks
45-.06573 -.3221 (45*2π)/3037 weeks
46-.10843 -.44834 (46*2π)/3037 weeks
47-.12789 -.44549 (47*2π)/3036 weeks
48-.10769 -.38039 (48*2π)/3036 weeks
49-.25084 -.31438 (49*2π)/3036 weeks
50-.06448 -.22306 (50*2π)/3036 weeks
51-.05296 -.30004 (51*2π)/3036 weeks
52-.10314 -.30686 (52*2π)/3036 weeks
53-.0652 -.38914 (53*2π)/3036 weeks
54-.18282 -.34126 (54*2π)/3036 weeks
55-.22793 -.25878 (55*2π)/3036 weeks
56-.07749 -.30906 (56*2π)/3035 weeks
57-.17671 -.31874 (57*2π)/3035 weeks
58-.19273 -.42148 (58*2π)/3035 weeks
59-.11386 -.20879 (59*2π)/3035 weeks
60-.11805 -.34676 (60*2π)/3035 weeks
61-.26306 -.22385 (61*2π)/3035 weeks
62-.21626 -.23254 (62*2π)/3035 weeks
63-.15555 -.14153 (63*2π)/3035 weeks
64-.15864 -.33423 (64*2π)/3035 weeks
65-.23311 -.15972 (65*2π)/3035 weeks
66-.16583 -.13514 (66*2π)/3035 weeks
67-.14985 -.25826 (67*2π)/3035 weeks
68-.14915 -.23342 (68*2π)/3034 weeks
69-.26434 -.14256 (69*2π)/3034 weeks
70-.13391 -.17741 (70*2π)/3034 weeks
71-.18136 -.14609 (71*2π)/3034 weeks
72-.09733 -.18602 (72*2π)/3034 weeks
73-.24996 -.20703 (73*2π)/3034 weeks
74-.17273 .02054 (74*2π)/3034 weeks
75-.05582 -.27377 (75*2π)/3034 weeks
76-.2702 -.1428 (76*2π)/3034 weeks
77-.15412 -.01099 (77*2π)/3034 weeks
78-.03328 -.15157 (78*2π)/3034 weeks
79-.06416 -.22888 (79*2π)/3034 weeks
80-.20275 -.18767 (80*2π)/3034 weeks
81-.13736 -.0613 (81*2π)/3034 weeks
82-.23414 -.22053 (82*2π)/3034 weeks
83-.22352 -.1172 (83*2π)/3034 weeks
84-.10575 -.15132 (84*2π)/3034 weeks
85-.16158 -.15033 (85*2π)/3034 weeks
86-.11399 -.15116 (86*2π)/3034 weeks
87-.22737 -.16221 (87*2π)/3033 weeks
88-.17147 -.2253 (88*2π)/3033 weeks
89-.21249 -.15325 (89*2π)/3033 weeks
90-.25572 -.15018 (90*2π)/3033 weeks
91-.11566 -.10414 (91*2π)/3033 weeks
92-.17963 -.08807 (92*2π)/3033 weeks
93-.20973 -.13328 (93*2π)/3033 weeks
94-.14755 -.0749 (94*2π)/3033 weeks
95-.11388 -.17792 (95*2π)/3033 weeks
96-.21314 -.12259 (96*2π)/3033 weeks
97-.20102 -.07975 (97*2π)/3033 weeks
98-.16375 -.09053 (98*2π)/3033 weeks
99-.14601 -.0841 (99*2π)/3033 weeks
100-.12138 -.06868 (100*2π)/3033 weeks
101-.08337 -.0466 (101*2π)/3033 weeks
102-.13027 -.13838 (102*2π)/3033 weeks
103-.2427 -.04957 (103*2π)/3033 weeks
104-.08975 -.09541 (104*2π)/3033 weeks
105-.21142 -.10285 (105*2π)/3033 weeks
106-.22953 -.06885 (106*2π)/3033 weeks
107-.16568 -.0259 (107*2π)/3033 weeks
108-.23948 -.03409 (108*2π)/3033 weeks
109-.18512 -.08099 (109*2π)/3033 weeks
110-.15958 -.01526 (110*2π)/3033 weeks
111-.19844 -.0393 (111*2π)/3033 weeks
112-.19921 -.09097 (112*2π)/3033 weeks
113-.23753 -.12082 (113*2π)/3033 weeks
114-.13409 -.05013 (114*2π)/3033 weeks
115-.17322 -.04954 (115*2π)/3033 weeks
116-.17958 -.01016 (116*2π)/3033 weeks
117-.13711 -.00804 (117*2π)/3033 weeks
118-.19742 -.05307 (118*2π)/3033 weeks
119-.12123 -.0443 (119*2π)/3033 weeks
120-.14037 -.11068 (120*2π)/3033 weeks
121-.17797 -.10059 (121*2π)/3033 weeks
122-.15599 -.15528 (122*2π)/3032 weeks
123-.19618 -.04161 (123*2π)/3032 weeks
124-.20834 -.09775 (124*2π)/3032 weeks
125-.20514 -.03457 (125*2π)/3032 weeks
126-.12472 -.04676 (126*2π)/3032 weeks
127-.18244 -.11904 (127*2π)/3032 weeks
128-.19087 -.05851 (128*2π)/3032 weeks
129-.19703 -.03703 (129*2π)/3032 weeks
130-.21412 -.00925 (130*2π)/3032 weeks
131-.26952 -.10205 (131*2π)/3032 weeks
132-.08085 .02806 (132*2π)/3032 weeks
133-.09691 -.01492 (133*2π)/3032 weeks
134-.15111 .00925 (134*2π)/3032 weeks
135-.11579 -.01642 (135*2π)/3032 weeks
136-.16837 -.13827 (136*2π)/3032 weeks
137-.13665 .03426 (137*2π)/3032 weeks
138-.21129 -.07794 (138*2π)/3032 weeks
139-.15768 .02761 (139*2π)/3032 weeks
140-.20221 -.0707 (140*2π)/3032 weeks
141-.15652 .01504 (141*2π)/3032 weeks
142-.18227 .01233 (142*2π)/3032 weeks
143-.06462 -.02363 (143*2π)/3032 weeks
144-.13758 -.00282 (144*2π)/3032 weeks
145-.25863 -.09597 (145*2π)/3032 weeks
146-.22311 .04718 (146*2π)/3032 weeks
147-.14511 -.04988 (147*2π)/3032 weeks
148-.24795 -.0184 (148*2π)/3032 weeks
149-.20975 .12732 (149*2π)/3032 weeks
150-.08918 .02609 (150*2π)/3032 weeks
151-.12541 .00801 (151*2π)/3032 weeks
152-.12541 -.00801 (152*2π)/3032 weeks
153-.08918 -.02609 (153*2π)/3032 weeks
154-.20975 -.12732 (154*2π)/3032 weeks
155-.24795 .0184 (155*2π)/3032 weeks
156-.14511 .04988 (156*2π)/3032 weeks
157-.22311 -.04718 (157*2π)/3032 weeks
158-.25863 .09597 (158*2π)/3032 weeks
159-.13758 .00282 (159*2π)/3032 weeks
160-.06462 .02363 (160*2π)/3032 weeks
161-.18227 -.01233 (161*2π)/3032 weeks
162-.15652 -.01504 (162*2π)/3032 weeks
163-.20221 .0707 (163*2π)/3032 weeks
164-.15768 -.02761 (164*2π)/3032 weeks
165-.21129 .07794 (165*2π)/3032 weeks
166-.13665 -.03426 (166*2π)/3032 weeks
167-.16837 .13827 (167*2π)/3032 weeks
168-.11579 .01642 (168*2π)/3032 weeks
169-.15111 -.00925 (169*2π)/3032 weeks
170-.09691 .01492 (170*2π)/3032 weeks
171-.08085 -.02806 (171*2π)/3032 weeks
172-.26952 .10205 (172*2π)/3032 weeks
173-.21412 .00925 (173*2π)/3032 weeks
174-.19703 .03703 (174*2π)/3032 weeks
175-.19087 .05851 (175*2π)/3032 weeks
176-.18244 .11904 (176*2π)/3032 weeks
177-.12472 .04676 (177*2π)/3032 weeks
178-.20514 .03457 (178*2π)/3032 weeks
179-.20834 .09775 (179*2π)/3032 weeks
180-.19618 .04161 (180*2π)/3032 weeks
181-.15599 .15528 (181*2π)/3032 weeks
182-.17797 .10059 (182*2π)/3032 weeks
183-.14037 .11068 (183*2π)/3032 weeks
184-.12123 .0443 (184*2π)/3032 weeks
185-.19742 .05307 (185*2π)/3032 weeks
186-.13711 .00804 (186*2π)/3032 weeks
187-.17958 .01016 (187*2π)/3032 weeks
188-.17322 .04954 (188*2π)/3032 weeks
189-.13409 .05013 (189*2π)/3032 weeks
190-.23753 .12082 (190*2π)/3032 weeks
191-.19921 .09097 (191*2π)/3032 weeks
192-.19844 .0393 (192*2π)/3032 weeks
193-.15958 .01526 (193*2π)/3032 weeks
194-.18512 .08099 (194*2π)/3032 weeks
195-.23948 .03409 (195*2π)/3032 weeks
196-.16568 .0259 (196*2π)/3032 weeks
197-.22953 .06885 (197*2π)/3032 weeks
198-.21142 .10285 (198*2π)/3032 weeks
199-.08975 .09541 (199*2π)/3032 weeks
200-.2427 .04957 (200*2π)/3032 weeks
201-.13027 .13838 (201*2π)/3032 weeks
202-.08337 .0466 (202*2π)/3032 weeks
203-.12138 .06868 (203*2π)/3031 weeks
204-.14601 .0841 (204*2π)/3031 weeks
205-.16375 .09053 (205*2π)/3031 weeks
206-.20102 .07975 (206*2π)/3031 weeks
207-.21314 .12259 (207*2π)/3031 weeks
208-.11388 .17792 (208*2π)/3031 weeks
209-.14755 .0749 (209*2π)/3031 weeks
210-.20973 .13328 (210*2π)/3031 weeks
211-.17963 .08807 (211*2π)/3031 weeks
212-.11566 .10414 (212*2π)/3031 weeks
213-.25572 .15018 (213*2π)/3031 weeks
214-.21249 .15325 (214*2π)/3031 weeks
215-.17147 .2253 (215*2π)/3031 weeks
216-.22737 .16221 (216*2π)/3031 weeks
217-.11399 .15116 (217*2π)/3031 weeks
218-.16158 .15033 (218*2π)/3031 weeks
219-.10575 .15132 (219*2π)/3031 weeks
220-.22352 .1172 (220*2π)/3031 weeks
221-.23414 .22053 (221*2π)/3031 weeks
222-.13736 .0613 (222*2π)/3031 weeks
223-.20275 .18767 (223*2π)/3031 weeks
224-.06416 .22888 (224*2π)/3031 weeks
225-.03328 .15157 (225*2π)/3031 weeks
226-.15412 .01099 (226*2π)/3031 weeks
227-.2702 .1428 (227*2π)/3031 weeks
228-.05582 .27377 (228*2π)/3031 weeks
229-.17273 -.02054 (229*2π)/3031 weeks
230-.24996 .20703 (230*2π)/3031 weeks
231-.09733 .18602 (231*2π)/3031 weeks
232-.18136 .14609 (232*2π)/3031 weeks
233-.13391 .17741 (233*2π)/3031 weeks
234-.26434 .14256 (234*2π)/3031 weeks
235-.14915 .23342 (235*2π)/3031 weeks
236-.14985 .25826 (236*2π)/3031 weeks
237-.16583 .13514 (237*2π)/3031 weeks
238-.23311 .15972 (238*2π)/3031 weeks
239-.15864 .33423 (239*2π)/3031 weeks
240-.15555 .14153 (240*2π)/3031 weeks
241-.21626 .23254 (241*2π)/3031 weeks
242-.26306 .22385 (242*2π)/3031 weeks
243-.11805 .34676 (243*2π)/3031 weeks
244-.11386 .20879 (244*2π)/3031 weeks
245-.19273 .42148 (245*2π)/3031 weeks
246-.17671 .31874 (246*2π)/3031 weeks
247-.07749 .30906 (247*2π)/3031 weeks
248-.22793 .25878 (248*2π)/3031 weeks
249-.18282 .34126 (249*2π)/3031 weeks
250-.0652 .38914 (250*2π)/3031 weeks
251-.10314 .30686 (251*2π)/3031 weeks
252-.05296 .30004 (252*2π)/3031 weeks
253-.06448 .22306 (253*2π)/3031 weeks
254-.25084 .31438 (254*2π)/3031 weeks
255-.10769 .38039 (255*2π)/3031 weeks
256-.12789 .44549 (256*2π)/3031 weeks
257-.10843 .44834 (257*2π)/3031 weeks
258-.06573 .3221 (258*2π)/3031 weeks
259-.19874 .37805 (259*2π)/3031 weeks
260.0073 .35724 (260*2π)/3031 weeks
261-.25869 .26248 (261*2π)/3031 weeks
262-.07773 .27276 (262*2π)/3031 weeks
263-.28355 .30893 (263*2π)/3031 weeks
264-.16978 .40135 (264*2π)/3031 weeks
265-.10481 .49464 (265*2π)/3031 weeks
266-.28508 .52007 (266*2π)/3031 weeks
267-.09981 .27195 (267*2π)/3031 weeks
268-.19881 .56657 (268*2π)/3031 weeks
269-.16754 .33845 (269*2π)/3031 weeks
270-.24306 .34877 (270*2π)/3031 weeks
271-.23431 .40503 (271*2π)/3031 weeks
272-.1746 .75898 (272*2π)/3031 weeks
273-.11738 .5062 (273*2π)/3031 weeks
274-.16679 .50994 (274*2π)/3031 weeks
275-.04392 .53483 (275*2π)/3031 weeks
276-.18186 .71972 (276*2π)/3031 weeks
277-.00209 .37326 (277*2π)/3031 weeks
278-.28515 .47469 (278*2π)/3031 weeks
279-.21794 .26263 (279*2π)/3031 weeks
280-.55854 .67715 (280*2π)/3031 weeks
281-.17346 .69528 (281*2π)/3031 weeks
282-.35506 .99139 (282*2π)/3031 weeks
283-.19795 .79405 (283*2π)/3031 weeks
284-.05537 .96027 (284*2π)/3031 weeks
285.14492 .94994 (285*2π)/3031 weeks
286-.02072 .52692 (286*2π)/3031 weeks
287-.45056 .81531 (287*2π)/3031 weeks
288-.20591 1.28948 (288*2π)/3031 weeks
289.06168 .99762 (289*2π)/3031 weeks
290.01098 1.03559 (290*2π)/3031 weeks
291-.1563 1.67331 (291*2π)/3031 weeks
292.21069 1.61801 (292*2π)/3031 weeks
293.32487 1.31621 (293*2π)/3031 weeks
294-.54342 1.52226 (294*2π)/3031 weeks
295-.42843 1.88465 (295*2π)/3031 weeks
296-.35895 2.21023 (296*2π)/3031 weeks
297-.63999 2.8444 (297*2π)/3031 weeks
298.71307 2.46509 (298*2π)/3031 weeks
299-1.65451 3.74765 (299*2π)/3031 weeks
3001.40393 5.4701 (300*2π)/3031 weeks
3013.43684 7.38617 (301*2π)/3031 weeks



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