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Fourier Analysis of BMW


BMW appears to have interesting cyclic behaviour every 98 weeks (1234.316*sine), 90 weeks (1169.382*sine), and 108 weeks (860.4767*cosine).

BMW has an average price of 5,394.38 (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 8/8/1996 to 3/13/2017 for BMW, this program was able to calculate the following Fourier Series:
Sequence #Cosine Coefficients Sine Coefficients FrequenciesPeriod
05,394.38   0 
18,011.989 -6,071.438 (1*2π)/10761,076 weeks
22,229.286 -7,729.49 (2*2π)/1076538 weeks
3-1,767.099 -4,992.784 (3*2π)/1076359 weeks
4-1,938.75 -1,585.489 (4*2π)/1076269 weeks
5-280.1259 -448.0644 (5*2π)/1076215 weeks
6425.3148 -1,086.844 (6*2π)/1076179 weeks
7-172.9859 -1,352.504 (7*2π)/1076154 weeks
8-507.5988 -592.1991 (8*2π)/1076135 weeks
9167.9641 -20.73355 (9*2π)/1076120 weeks
10860.4767 -489.2876 (10*2π)/1076108 weeks
11582.1425 -1,234.316 (11*2π)/107698 weeks
12-184.4016 -1,169.382 (12*2π)/107690 weeks
13-336.9159 -480.6763 (13*2π)/107683 weeks
14206.5598 -209.6133 (14*2π)/107677 weeks
15516.4163 -660.9568 (15*2π)/107672 weeks
16108.45 -1,050.639 (16*2π)/107667 weeks
17-407.3423 -756.2045 (17*2π)/107663 weeks
18-318.1609 -187.9731 (18*2π)/107660 weeks
19178.9459 -88.1023 (19*2π)/107657 weeks
20381.2342 -460.1472 (20*2π)/107654 weeks
21129.8801 -698.4484 (21*2π)/107651 weeks
22-90.56544 -554.8903 (22*2π)/107649 weeks
23-.93866 -410.2544 (23*2π)/107647 weeks
2480.11677 -551.7496 (24*2π)/107645 weeks
25-160.6995 -690.4833 (25*2π)/107643 weeks
26-468.9008 -450.9476 (26*2π)/107641 weeks
27-389.8774 -10.64519 (27*2π)/107640 weeks
2827.30401 123.9874 (28*2π)/107638 weeks
29288.4355 -148.3534 (29*2π)/107637 weeks
30167.4236 -411.2385 (30*2π)/107636 weeks
31-28.27939 -370.7659 (31*2π)/107635 weeks
3210.92779 -237.3366 (32*2π)/107634 weeks
33138.6842 -315.5063 (33*2π)/107633 weeks
3446.78042 -499.8934 (34*2π)/107632 weeks
35-202.0985 -463.3527 (35*2π)/107631 weeks
36-261.188 -207.8359 (36*2π)/107630 weeks
37-55.42134 -81.96479 (37*2π)/107629 weeks
3890.65091 -241.622 (38*2π)/107628 weeks
39-52.32667 -409.761 (39*2π)/107628 weeks
40-273.1151 -296.4212 (40*2π)/107627 weeks
41-238.2521 -31.99892 (41*2π)/107626 weeks
4213.50418 30.35339 (42*2π)/107626 weeks
43139.2281 -163.6418 (43*2π)/107625 weeks
443.02389 -310.4534 (44*2π)/107624 weeks
45-151.6507 -213.6009 (45*2π)/107624 weeks
46-91.27885 -46.66418 (46*2π)/107623 weeks
4782.20543 -58.31797 (47*2π)/107623 weeks
48125.4912 -213.9185 (48*2π)/107622 weeks
4918.20877 -293.8443 (49*2π)/107622 weeks
50-53.73283 -243.1877 (50*2π)/107622 weeks
51-21.80979 -218.9684 (51*2π)/107621 weeks
52-30.95563 -295.0461 (52*2π)/107621 weeks
53-171.0033 -324.3259 (53*2π)/107620 weeks
54-296.234 -180.5979 (54*2π)/107620 weeks
55-232.7888 14.7435 (55*2π)/107620 weeks
56-49.6436 50.4726 (56*2π)/107619 weeks
5732.29691 -72.47218 (57*2π)/107619 weeks
58-50.77026 -153.6762 (58*2π)/107619 weeks
59-140.183 -80.48726 (59*2π)/107618 weeks
60-94.3115 41.33145 (60*2π)/107618 weeks
6137.85316 62.7896 (61*2π)/107618 weeks
62124.0904 -18.86978 (62*2π)/107617 weeks
63128.6926 -109.9092 (63*2π)/107617 weeks
6499.31332 -172.6546 (64*2π)/107617 weeks
6545.3597 -223.6378 (65*2π)/107617 weeks
66-51.20669 -231.4948 (66*2π)/107616 weeks
67-131.4982 -143.3283 (67*2π)/107616 weeks
68-91.229 -13.39677 (68*2π)/107616 weeks
6954.85854 10.34613 (69*2π)/107616 weeks
70142.5307 -119.2135 (70*2π)/107615 weeks
7160.87704 -256.9203 (71*2π)/107615 weeks
72-94.51308 -246.1476 (72*2π)/107615 weeks
73-152.2949 -117.3987 (73*2π)/107615 weeks
74-81.98045 -24.11088 (74*2π)/107615 weeks
753.48381 -36.85665 (75*2π)/107614 weeks
7621.86872 -88.5425 (76*2π)/107614 weeks
775.87189 -109.8318 (77*2π)/107614 weeks
78-.57034 -120.5517 (78*2π)/107614 weeks
79-27.15539 -141.4575 (79*2π)/107614 weeks
80-86.52328 -116.4887 (80*2π)/107613 weeks
81-91.36091 -17.54247 (81*2π)/107613 weeks
8227.39277 46.97536 (82*2π)/107613 weeks
83165.6642 -51.14585 (83*2π)/107613 weeks
84135.0665 -247.2919 (84*2π)/107613 weeks
85-72.04825 -314.4174 (85*2π)/107613 weeks
86-230.0474 -156.3143 (86*2π)/107613 weeks
87-163.1473 47.87485 (87*2π)/107612 weeks
8829.04642 74.83673 (88*2π)/107612 weeks
89118.3045 -64.64514 (89*2π)/107612 weeks
9042.02625 -174.3703 (90*2π)/107612 weeks
91-54.85761 -149.3926 (91*2π)/107612 weeks
92-57.87913 -85.21592 (92*2π)/107612 weeks
93-31.01179 -83.70193 (93*2π)/107612 weeks
94-59.13259 -88.97874 (94*2π)/107611 weeks
95-77.52834 -20.00816 (95*2π)/107611 weeks
9614.05396 48.51038 (96*2π)/107611 weeks
97148.1597 -23.27019 (97*2π)/107611 weeks
98139.7623 -204.5993 (98*2π)/107611 weeks
99-46.16298 -280.2539 (99*2π)/107611 weeks
100-196.207 -133.678 (100*2π)/107611 weeks
101-116.9875 61.91935 (101*2π)/107611 weeks
10290.86843 60.57412 (102*2π)/107611 weeks
103165.3097 -126.9299 (103*2π)/107610 weeks
10428.94881 -261.1395 (104*2π)/107610 weeks
105-132.5789 -195.3702 (105*2π)/107610 weeks
106-133.1219 -44.4156 (106*2π)/107610 weeks
107-15.8178 -1.83114 (107*2π)/107610 weeks
10844.68947 -83.75539 (108*2π)/107610 weeks
109-7.43862 -150.4247 (109*2π)/107610 weeks
110-72.17941 -127.0339 (110*2π)/107610 weeks
111-73.1227 -78.63604 (111*2π)/107610 weeks
112-52.11081 -72.6082 (112*2π)/107610 weeks
113-66.0785 -75.87881 (113*2π)/107610 weeks
114-79.85445 -34.02927 (114*2π)/10769 weeks
115-28.66786 14.3928 (115*2π)/10769 weeks
11655.61982 -15.402 (116*2π)/10769 weeks
11769.27019 -114.7427 (117*2π)/10769 weeks
118-15.877 -174.6902 (118*2π)/10769 weeks
119-107.053 -127.7091 (119*2π)/10769 weeks
120-108.5608 -30.42751 (120*2π)/10769 weeks
121-30.45208 14.78656 (121*2π)/10769 weeks
12242.76723 -25.36295 (122*2π)/10769 weeks
12350.92696 -103.7829 (123*2π)/10769 weeks
124-4.08294 -158.6332 (124*2π)/10769 weeks
125-84.87 -154.2654 (125*2π)/10769 weeks
126-139.9296 -86.03444 (126*2π)/10769 weeks
127-120.7933 7.01218 (127*2π)/10768 weeks
128-30.04505 47.70893 (128*2π)/10768 weeks
12952.15345 -8.3284 (129*2π)/10768 weeks
13040.19003 -103.7621 (130*2π)/10768 weeks
131-48.42199 -129.4069 (131*2π)/10768 weeks
132-102.6594 -58.3887 (132*2π)/10768 weeks
133-57.59407 15.29277 (133*2π)/10768 weeks
13421.74861 1.86092 (134*2π)/10768 weeks
13536.24542 -66.25452 (135*2π)/10768 weeks
136-10.07491 -93.85345 (136*2π)/10768 weeks
137-32.9857 -65.78339 (137*2π)/10768 weeks
138-7.40269 -56.19062 (138*2π)/10768 weeks
139-4.06739 -96.09973 (139*2π)/10768 weeks
140-63.62623 -111.286 (140*2π)/10768 weeks
141-107.1451 -42.30307 (141*2π)/10768 weeks
142-47.26368 34.37063 (142*2π)/10768 weeks
14355.88728 .0537 (143*2π)/10768 weeks
14457.60468 -117.4493 (144*2π)/10767 weeks
145-59.47879 -161.025 (145*2π)/10767 weeks
146-144.8394 -62.37561 (146*2π)/10767 weeks
147-82.53811 54.68681 (147*2π)/10767 weeks
14847.04152 38.54959 (148*2π)/10767 weeks
14974.6572 -83.49308 (149*2π)/10767 weeks
150-26.36517 -144.6042 (150*2π)/10767 weeks
151-109.4229 -71.36614 (151*2π)/10767 weeks
152-67.26052 22.4275 (152*2π)/10767 weeks
15325.03401 11.08992 (153*2π)/10767 weeks
15438.06663 -68.26711 (154*2π)/10767 weeks
155-23.02975 -93.69423 (155*2π)/10767 weeks
156-49.36332 -46.88966 (156*2π)/10767 weeks
157-6.6333 -23.81639 (157*2π)/10767 weeks
15818.52973 -71.65481 (158*2π)/10767 weeks
159-36.16508 -110.2043 (159*2π)/10767 weeks
160-98.92831 -56.68767 (160*2π)/10767 weeks
161-62.87544 34.17923 (161*2π)/10767 weeks
16247.52997 34.88096 (162*2π)/10767 weeks
16397.37414 -72.12757 (163*2π)/10767 weeks
16423.74956 -165.1521 (164*2π)/10767 weeks
165-85.72509 -144.9674 (165*2π)/10767 weeks
166-115.0138 -52.47427 (166*2π)/10766 weeks
167-58.75652 2.24893 (167*2π)/10766 weeks
168-3.42145 -20.17651 (168*2π)/10766 weeks
169-4.49699 -64.4892 (169*2π)/10766 weeks
170-35.56801 -75.76476 (170*2π)/10766 weeks
171-53.26999 -60.4433 (171*2π)/10766 weeks
172-55.26787 -46.40442 (172*2π)/10766 weeks
173-56.46805 -34.85123 (173*2π)/10766 weeks
174-51.14826 -16.99936 (174*2π)/10766 weeks
175-28.83274 -5.65142 (175*2π)/10766 weeks
176-4.73464 -17.9841 (176*2π)/10766 weeks
177-2.71408 -42.73421 (177*2π)/10766 weeks
178-19.81413 -53.78249 (178*2π)/10766 weeks
179-33.66481 -45.51053 (179*2π)/10766 weeks
180-33.72076 -31.96577 (180*2π)/10766 weeks
181-24.47633 -22.61783 (181*2π)/10766 weeks
182-7.89927 -19.59372 (182*2π)/10766 weeks
18314.30226 -32.97369 (183*2π)/10766 weeks
18419.94333 -71.23212 (184*2π)/10766 weeks
185-18.84735 -106.3487 (185*2π)/10766 weeks
186-80.13892 -86.73573 (186*2π)/10766 weeks
187-92.37473 -14.26152 (187*2π)/10766 weeks
188-26.53879 31.44277 (188*2π)/10766 weeks
18946.49426 -12.885 (189*2π)/10766 weeks
19034.99326 -101.7784 (190*2π)/10766 weeks
191-52.30201 -128.4703 (191*2π)/10766 weeks
192-110.5374 -62.33119 (192*2π)/10766 weeks
193-75.94365 8.71044 (193*2π)/10766 weeks
194-10.82415 -.39958 (194*2π)/10766 weeks
195-9.17834 -54.25153 (195*2π)/10766 weeks
196-58.81663 -58.66778 (196*2π)/10765 weeks
197-69.72606 -7.30781 (197*2π)/10765 weeks
198-20.24926 15.68927 (198*2π)/10765 weeks
19912.33064 -25.07416 (199*2π)/10765 weeks
200-20.56988 -59.93209 (200*2π)/10765 weeks
201-57.43031 -26.33125 (201*2π)/10765 weeks
202-19.04099 24.08093 (202*2π)/10765 weeks
20358.25435 -7.69456 (203*2π)/10765 weeks
20456.28492 -110.5262 (204*2π)/10765 weeks
205-50.02515 -155.5761 (205*2π)/10765 weeks
206-137.5084 -75.6916 (206*2π)/10765 weeks
207-100.6956 30.55517 (207*2π)/10765 weeks
208.87161 32.04739 (208*2π)/10765 weeks
20926.92018 -56.62812 (209*2π)/10765 weeks
210-51.04674 -101.2352 (210*2π)/10765 weeks
211-116.6652 -36.70035 (211*2π)/10765 weeks
212-77.40804 47.42621 (212*2π)/10765 weeks
21310.1351 42.60091 (213*2π)/10765 weeks
21434.58154 -29.30835 (214*2π)/10765 weeks
215-12.72187 -66.16306 (215*2π)/10765 weeks
216-45.97083 -37.83192 (216*2π)/10765 weeks
217-23.18609 -13.65208 (217*2π)/10765 weeks
218-1.94009 -41.86911 (218*2π)/10765 weeks
219-36.92363 -72.11075 (219*2π)/10765 weeks
220-84.95357 -39.66683 (220*2π)/10765 weeks
221-68.83024 24.99329 (221*2π)/10765 weeks
2222.0865 34.40192 (222*2π)/10765 weeks
22337.74817 -30.25152 (223*2π)/10765 weeks
224-11.02916 -87.69787 (224*2π)/10765 weeks
225-83.74871 -62.66248 (225*2π)/10765 weeks
226-89.59045 14.52309 (226*2π)/10765 weeks
227-24.81974 52.35806 (227*2π)/10765 weeks
22834.02439 16.43131 (228*2π)/10765 weeks
22932.18848 -43.67904 (229*2π)/10765 weeks
230-12.53012 -67.19795 (230*2π)/10765 weeks
231-47.21887 -43.32304 (231*2π)/10765 weeks
232-41.54067 -7.3642 (232*2π)/10765 weeks
233-9.0562 2.28642 (233*2π)/10765 weeks
23414.79722 -21.57763 (234*2π)/10765 weeks
2357.03847 -55.271 (235*2π)/10765 weeks
236-25.77499 -67.95572 (236*2π)/10765 weeks
237-54.19552 -48.12674 (237*2π)/10765 weeks
238-52.28769 -15.75381 (238*2π)/10765 weeks
239-26.03686 -4.58142 (239*2π)/10765 weeks
240-10.2865 -22.69458 (240*2π)/10764 weeks
241-24.17781 -38.43713 (241*2π)/10764 weeks
242-40.39861 -25.28697 (242*2π)/10764 weeks
243-27.30542 -7.16002 (243*2π)/10764 weeks
244-6.70333 -20.85592 (244*2π)/10764 weeks
245-22.65598 -46.85537 (245*2π)/10764 weeks
246-61.09751 -30.87379 (246*2π)/10764 weeks
247-56.14223 25.23632 (247*2π)/10764 weeks
24811.55602 47.14526 (248*2π)/10764 weeks
24968.58742 -13.19048 (249*2π)/10764 weeks
25036.59992 -97.27396 (250*2π)/10764 weeks
251-54.87418 -103.0035 (251*2π)/10764 weeks
252-93.21078 -21.86819 (252*2π)/10764 weeks
253-30.35675 38.67741 (253*2π)/10764 weeks
25445.30803 -3.48215 (254*2π)/10764 weeks
25527.66789 -88.70004 (255*2π)/10764 weeks
256-57.03519 -98.88894 (256*2π)/10764 weeks
257-90.49135 -26.96486 (257*2π)/10764 weeks
258-36.00967 18.10254 (258*2π)/10764 weeks
25913.20927 -24.61475 (259*2π)/10764 weeks
260-22.93167 -81.7205 (260*2π)/10764 weeks
261-94.30963 -58.38394 (261*2π)/10764 weeks
262-96.64464 22.35235 (262*2π)/10764 weeks
263-23.13914 54.32684 (263*2π)/10764 weeks
26427.21148 1.58196 (264*2π)/10764 weeks
265-5.8278 -51.75422 (265*2π)/10764 weeks
266-57.11691 -28.21255 (266*2π)/10764 weeks
267-38.25449 27.11005 (267*2π)/10764 weeks
26826.23238 19.455 (268*2π)/10764 weeks
26939.57869 -48.89237 (269*2π)/10764 weeks
270-20.14033 -85.1228 (270*2π)/10764 weeks
271-71.51373 -43.13749 (271*2π)/10764 weeks
272-47.55507 13.96587 (272*2π)/10764 weeks
27311.4023 4.1645 (273*2π)/10764 weeks
27415.22257 -56.58968 (274*2π)/10764 weeks
275-44.01926 -77.21928 (275*2π)/10764 weeks
276-81.53894 -28.05748 (276*2π)/10764 weeks
277-49.76476 19.23507 (277*2π)/10764 weeks
278-1.87196 4.56635 (278*2π)/10764 weeks
279-4.42433 -40.41267 (279*2π)/10764 weeks
280-48.08432 -48.51221 (280*2π)/10764 weeks
281-70.61994 -7.70866 (281*2π)/10764 weeks
282-41.50313 30.46136 (282*2π)/10764 weeks
2833.7716 22.62617 (283*2π)/10764 weeks
28417.61091 -14.64185 (284*2π)/10764 weeks
285-.4463 -39.02006 (285*2π)/10764 weeks
286-19.9071 -39.7975 (286*2π)/10764 weeks
287-28.38482 -34.96481 (287*2π)/10764 weeks
288-36.11103 -32.85913 (288*2π)/10764 weeks
289-47.32323 -24.76862 (289*2π)/10764 weeks
290-51.77458 -6.16583 (290*2π)/10764 weeks
291-40.13398 13.1716 (291*2π)/10764 weeks
292-16.80674 18.5323 (292*2π)/10764 weeks
2932.27798 5.44573 (293*2π)/10764 weeks
2944.6043 -15.27355 (294*2π)/10764 weeks
295-9.10085 -28.01426 (295*2π)/10764 weeks
296-26.76951 -21.70956 (296*2π)/10764 weeks
297-26.96867 1.14779 (297*2π)/10764 weeks
2981.7987 12.50514 (298*2π)/10764 weeks
29930.6119 -15.59631 (299*2π)/10764 weeks
30015.16304 -60.95687 (300*2π)/10764 weeks
301-37.45992 -66.56658 (301*2π)/10764 weeks
302-64.98386 -17.32549 (302*2π)/10764 weeks
303-26.6738 29.41455 (303*2π)/10764 weeks
30435.8671 8.97141 (304*2π)/10764 weeks
30541.04565 -61.89049 (305*2π)/10764 weeks
306-24.35228 -95.91195 (306*2π)/10764 weeks
307-80.43378 -50.18734 (307*2π)/10764 weeks
308-60.45317 14.26825 (308*2π)/10763 weeks
309.36198 16.53702 (309*2π)/10763 weeks
31017.29243 -34.86717 (310*2π)/10763 weeks
311-23.86128 -59.37517 (311*2π)/10763 weeks
312-50.67237 -26.08272 (312*2π)/10763 weeks
313-20.37359 1.41492 (313*2π)/10763 weeks
31411.889 -29.65195 (314*2π)/10763 weeks
315-17.50502 -73.58423 (315*2π)/10763 weeks
316-76.02426 -54.27594 (316*2π)/10763 weeks
317-75.13721 15.09218 (317*2π)/10763 weeks
318-1.63586 37.28273 (318*2π)/10763 weeks
31946.79469 -32.87432 (319*2π)/10763 weeks
320-8.54935 -109.5219 (320*2π)/10763 weeks
321-107.7459 -85.05697 (321*2π)/10763 weeks
322-125.972 18.66181 (322*2π)/10763 weeks
323-41.23339 76.72602 (323*2π)/10763 weeks
32439.80605 26.70814 (324*2π)/10763 weeks
32526.48743 -57.17835 (325*2π)/10763 weeks
326-45.53523 -73.15607 (326*2π)/10763 weeks
327-79.93185 -21.07963 (327*2π)/10763 weeks
328-49.96821 19.55436 (328*2π)/10763 weeks
329-14.16579 6.92465 (329*2π)/10763 weeks
330-19.73326 -19.10137 (330*2π)/10763 weeks
331-39.60178 -11.22737 (331*2π)/10763 weeks
332-26.85198 10.6107 (332*2π)/10763 weeks
3333.30846 -2.76338 (333*2π)/10763 weeks
334-3.30386 -42.90629 (334*2π)/10763 weeks
335-50.3605 -50.70691 (335*2π)/10763 weeks
336-78.33585 -4.62039 (336*2π)/10763 weeks
337-45.10737 43.90209 (337*2π)/10763 weeks
33815.95381 35.12932 (338*2π)/10763 weeks
33936.65991 -23.47902 (339*2π)/10763 weeks
340-3.65748 -68.51505 (340*2π)/10763 weeks
341-59.55036 -56.1911 (341*2π)/10763 weeks
342-75.22024 -5.25522 (342*2π)/10763 weeks
343-41.12425 30.34237 (343*2π)/10763 weeks
3443.27363 18.4367 (344*2π)/10763 weeks
34514.11494 -24.30787 (345*2π)/10763 weeks
346-17.02846 -55.478 (346*2π)/10763 weeks
347-61.16398 -45.73958 (347*2π)/10763 weeks
348-77.69754 -3.17281 (348*2π)/10763 weeks
349-53.27714 32.54927 (349*2π)/10763 weeks
350-16.34368 33.20742 (350*2π)/10763 weeks
351.4872 11.52533 (351*2π)/10763 weeks
352-3.4239 -4.64945 (352*2π)/10763 weeks
353-8.6412 -7.79613 (353*2π)/10763 weeks
354-6.8131 -11.52412 (354*2π)/10763 weeks
355-11.81997 -22.75896 (355*2π)/10763 weeks
356-31.32797 -21.61835 (356*2π)/10763 weeks
357-36.89082 5.5623 (357*2π)/10763 weeks
358-2.36452 23.73919 (358*2π)/10763 weeks
35937.62073 -11.42259 (359*2π)/10763 weeks
36018.34199 -77.37327 (360*2π)/10763 weeks
361-63.5227 -93.49577 (361*2π)/10763 weeks
362-124.1967 -19.79975 (362*2π)/10763 weeks
363-85.05959 75.33789 (363*2π)/10763 weeks
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