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Fourier Analysis of BMW


BMW appears to have interesting cyclic behaviour every 87 weeks (1118.009*sine), 95 weeks (1050.333*sine), and 105 weeks (737.8372*cosine).

BMW has an average price of 5,320.64 (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 1/9/2017 for BMW, this program was able to calculate the following Fourier Series:
Sequence #Cosine Coefficients Sine Coefficients FrequenciesPeriod
05,320.641   0 
17,972.723 -5,930.748 (1*2π)/10461,046 weeks
22,357.926 -7,651.896 (2*2π)/1046523 weeks
3-1,635.149 -5,090.593 (3*2π)/1046349 weeks
4-1,946.827 -1,755.994 (4*2π)/1046262 weeks
5-390.6899 -551.2259 (5*2π)/1046209 weeks
6321.6366 -1,120.167 (6*2π)/1046174 weeks
7-265.4702 -1,396.424 (7*2π)/1046149 weeks
8-670.3834 -632.6619 (8*2π)/1046131 weeks
9-47.55072 55.0463 (9*2π)/1046116 weeks
10737.8372 -284.0808 (10*2π)/1046105 weeks
11617.918 -1,050.333 (11*2π)/104695 weeks
12-114.6338 -1,118.009 (12*2π)/104687 weeks
13-369.2564 -475.6288 (13*2π)/104680 weeks
14125.6576 -103.4299 (14*2π)/104675 weeks
15546.1146 -478.6515 (15*2π)/104670 weeks
16260.4106 -964.4527 (16*2π)/104665 weeks
17-306.9312 -828.569 (17*2π)/104662 weeks
18-387.7026 -269.1375 (18*2π)/104658 weeks
1936.489 -17.82793 (19*2π)/104655 weeks
20334.528 -267.5113 (20*2π)/104652 weeks
21205.8128 -521.296 (21*2π)/104650 weeks
2221.8122 -436.5414 (22*2π)/104648 weeks
23132.4193 -298.4662 (23*2π)/104645 weeks
24302.5434 -471.4711 (24*2π)/104644 weeks
25123.5347 -768.8638 (25*2π)/104642 weeks
26-302.6871 -723.3945 (26*2π)/104640 weeks
27-477.8904 -306.0551 (27*2π)/104639 weeks
28-224.9489 16.77115 (28*2π)/104637 weeks
2984.38349 -50.25549 (29*2π)/104636 weeks
3096.78377 -257.7129 (30*2π)/104635 weeks
31-45.74594 -248.2589 (31*2π)/104634 weeks
32-3.63738 -90.89086 (32*2π)/104633 weeks
33204.2537 -114.4223 (33*2π)/104632 weeks
34247.8528 -351.7989 (34*2π)/104631 weeks
3526.84793 -475.1078 (35*2π)/104630 weeks
36-153.7444 -306.0699 (36*2π)/104629 weeks
37-41.61645 -100.1629 (37*2π)/104628 weeks
38179.5351 -167.9336 (38*2π)/104628 weeks
39162.6654 -412.7261 (39*2π)/104627 weeks
40-91.53254 -476.8845 (40*2π)/104626 weeks
41-244.6555 -264.645 (41*2π)/104626 weeks
42-115.9775 -65.21642 (42*2π)/104625 weeks
4377.21609 -121.1704 (43*2π)/104624 weeks
4459.20354 -297.8295 (44*2π)/104624 weeks
45-117.2871 -313.9897 (45*2π)/104623 weeks
46-189.2246 -149.3227 (46*2π)/104623 weeks
47-79.10574 -30.22285 (47*2π)/104622 weeks
4832.85207 -69.39035 (48*2π)/104622 weeks
4920.78798 -128.7681 (49*2π)/104621 weeks
50-3.96854 -80.53929 (50*2π)/104621 weeks
5194.35494 -23.7241 (51*2π)/104621 weeks
52230.1102 -117.9975 (52*2π)/104620 weeks
53211.1874 -310.2618 (53*2π)/104620 weeks
5433.96972 -387.5636 (54*2π)/104619 weeks
55-96.19769 -278.0928 (55*2π)/104619 weeks
56-47.92636 -149.1939 (56*2π)/104619 weeks
5768.95847 -169.1802 (57*2π)/104618 weeks
5875.78462 -291.674 (58*2π)/104618 weeks
59-39.60454 -352.7236 (59*2π)/104618 weeks
60-149.532 -296.6502 (60*2π)/104617 weeks
61-180.4406 -202.2132 (61*2π)/104617 weeks
62-166.8448 -130.0865 (62*2π)/104617 weeks
63-134.5698 -65.42424 (63*2π)/104617 weeks
64-54.26373 -16.78387 (64*2π)/104616 weeks
6555.75406 -52.99615 (65*2π)/104616 weeks
6685.93635 -181.81 (66*2π)/104616 weeks
67-25.55814 -276.3784 (67*2π)/104616 weeks
68-170.874 -214.0632 (68*2π)/104615 weeks
69-183.4204 -50.8266 (69*2π)/104615 weeks
70-50.69225 35.38187 (70*2π)/104615 weeks
7173.36002 -31.2814 (71*2π)/104615 weeks
7273.10365 -145.8356 (72*2π)/104615 weeks
73-1.66487 -184.5568 (73*2π)/104614 weeks
74-46.05413 -154.7749 (74*2π)/104614 weeks
75-49.00752 -130.0557 (75*2π)/104614 weeks
76-57.4269 -115.4058 (76*2π)/104614 weeks
77-54.85924 -72.68259 (77*2π)/104614 weeks
7811.89317 -40.80435 (78*2π)/104613 weeks
7996.11204 -109.0934 (79*2π)/104613 weeks
8062.95242 -250.437 (80*2π)/104613 weeks
81-112.0519 -293.9974 (81*2π)/104613 weeks
82-251.9626 -140.118 (82*2π)/104613 weeks
83-179.4554 70.52982 (83*2π)/104613 weeks
8440.01617 106.7897 (84*2π)/104612 weeks
85164.6263 -60.16911 (85*2π)/104612 weeks
8680.02291 -223.8199 (86*2π)/104612 weeks
87-73.78855 -216.0004 (87*2π)/104612 weeks
88-114.9402 -103.3979 (88*2π)/104612 weeks
89-52.62367 -49.43462 (89*2π)/104612 weeks
90-18.41454 -74.8666 (90*2π)/104612 weeks
91-38.0927 -73.04416 (91*2π)/104611 weeks
92-9.93383 -23.64006 (92*2π)/104611 weeks
9389.58121 -43.61034 (93*2π)/104611 weeks
94121.1262 -182.7467 (94*2π)/104611 weeks
95-16.53025 -292.3514 (95*2π)/104611 weeks
96-195.6289 -206.8786 (96*2π)/104611 weeks
97-194.786 2.89112 (97*2π)/104611 weeks
98-.50948 89.02613 (98*2π)/104611 weeks
99153.6759 -54.16226 (99*2π)/104611 weeks
10084.10944 -247.2918 (100*2π)/104610 weeks
101-111.0158 -260.3 (101*2π)/104610 weeks
102-194.4137 -102.2253 (102*2π)/104610 weeks
103-97.10362 18.2726 (103*2π)/104610 weeks
10424.85371 -21.22993 (104*2π)/104610 weeks
10524.99775 -125.7771 (105*2π)/104610 weeks
106-57.36312 -151.0047 (106*2π)/104610 weeks
107-97.59375 -95.32246 (107*2π)/104610 weeks
108-71.59272 -56.72378 (108*2π)/104610 weeks
109-54.28246 -65.47887 (109*2π)/104610 weeks
110-73.15871 -58.64945 (110*2π)/104610 weeks
111-64.84715 -9.74387 (111*2π)/10469 weeks
1127.07316 11.50339 (112*2π)/10469 weeks
11372.30558 -54.16323 (113*2π)/10469 weeks
11443.33721 -151.9442 (114*2π)/10469 weeks
115-58.76702 -173.9937 (115*2π)/10469 weeks
116-125.6332 -97.27318 (116*2π)/10469 weeks
117-93.89515 -6.81284 (117*2π)/10469 weeks
118-7.04971 11.40764 (118*2π)/10469 weeks
11952.0859 -45.79709 (119*2π)/10469 weeks
12044.23342 -123.654 (120*2π)/10469 weeks
121-18.66063 -170.2873 (121*2π)/10469 weeks
122-100.5128 -156.5303 (122*2π)/10469 weeks
123-149.5656 -80.43893 (123*2π)/10469 weeks
124-119.1034 11.95639 (124*2π)/10468 weeks
125-23.02428 40.60583 (125*2π)/10468 weeks
12649.34404 -26.78616 (126*2π)/10468 weeks
12721.77235 -117.8778 (127*2π)/10468 weeks
128-69.40002 -127.7563 (128*2π)/10468 weeks
129-111.4783 -49.57268 (129*2π)/10468 weeks
130-58.57842 15.56334 (130*2π)/10468 weeks
13112.62078 -5.36948 (131*2π)/10468 weeks
13216.59661 -64.16828 (132*2π)/10468 weeks
133-22.77229 -74.75663 (133*2π)/10468 weeks
134-23.88507 -45.18544 (134*2π)/10468 weeks
13514.15317 -55.23988 (135*2π)/10468 weeks
1366.75937 -114.559 (136*2π)/10468 weeks
137-71.86 -129.5523 (137*2π)/10468 weeks
138-122.5118 -47.1403 (138*2π)/10468 weeks
139-56.31126 39.94514 (139*2π)/10468 weeks
14057.90032 6.36627 (140*2π)/10467 weeks
14167.5658 -120.1154 (141*2π)/10467 weeks
142-52.14165 -177.2996 (142*2π)/10467 weeks
143-151.257 -85.55519 (143*2π)/10467 weeks
144-103.3626 38.34434 (144*2π)/10467 weeks
14523.13881 37.56137 (145*2π)/10467 weeks
14661.72094 -74.21644 (146*2π)/10467 weeks
147-25.66801 -141.1094 (147*2π)/10467 weeks
148-109.7321 -83.80922 (148*2π)/10467 weeks
149-84.65788 4.06818 (149*2π)/10467 weeks
150-6.73077 5.42956 (150*2π)/10467 weeks
1517.78263 -56.19682 (151*2π)/10467 weeks
152-42.6204 -71.89825 (152*2π)/10467 weeks
153-57.07888 -21.90178 (153*2π)/10467 weeks
154.19113 -1.52077 (154*2π)/10467 weeks
15538.37064 -65.68257 (155*2π)/10467 weeks
156-22.25824 -132.81 (156*2π)/10467 weeks
157-121.0471 -96.00565 (157*2π)/10467 weeks
158-130.3178 19.96627 (158*2π)/10467 weeks
159-28.1132 83.38461 (159*2π)/10467 weeks
16072.41348 25.91739 (160*2π)/10467 weeks
16170.26244 -77.61846 (161*2π)/10466 weeks
162-6.87856 -116.0663 (162*2π)/10466 weeks
163-60.22997 -74.94397 (163*2π)/10466 weeks
164-48.23833 -26.02335 (164*2π)/10466 weeks
165-12.1449 -19.81091 (165*2π)/10466 weeks
1663.236 -39.74501 (166*2π)/10466 weeks
167.63666 -54.34296 (167*2π)/10466 weeks
168-1.95305 -63.63615 (168*2π)/10466 weeks
169-9.41266 -77.654 (169*2π)/10466 weeks
170-31.55463 -84.86707 (170*2π)/10466 weeks
171-53.81026 -69.79472 (171*2π)/10466 weeks
172-53.84103 -43.90375 (172*2π)/10466 weeks
173-35.22797 -32.80972 (173*2π)/10466 weeks
174-20.93873 -41.56236 (174*2π)/10466 weeks
175-21.04195 -54.84759 (175*2π)/10466 weeks
176-29.4732 -64.01273 (176*2π)/10466 weeks
177-45.7416 -68.87511 (177*2π)/10466 weeks
178-70.85537 -57.60687 (178*2π)/10466 weeks
179-81.89848 -18.07561 (179*2π)/10466 weeks
180-47.01005 23.70539 (180*2π)/10466 weeks
18118.00152 13.37638 (181*2π)/10466 weeks
18242.21181 -57.56139 (182*2π)/10466 weeks
183-16.4259 -115.6299 (183*2π)/10466 weeks
184-99.77982 -85.08042 (184*2π)/10466 weeks
185-108.1525 7.06444 (185*2π)/10466 weeks
186-29.16157 53.85471 (186*2π)/10466 weeks
18740.97549 5.12209 (187*2π)/10466 weeks
18824.26751 -67.077 (188*2π)/10466 weeks
189-36.96893 -70.41227 (189*2π)/10466 weeks
190-47.77407 -18.98431 (190*2π)/10466 weeks
191.9881 -3.25677 (191*2π)/10465 weeks
19226.42209 -50.77903 (192*2π)/10465 weeks
193-14.26844 -87.32621 (193*2π)/10465 weeks
194-56.12605 -56.55447 (194*2π)/10465 weeks
195-29.58886 -13.46885 (195*2π)/10465 weeks
19622.25285 -42.04935 (196*2π)/10465 weeks
197-.55966 -116.3836 (197*2π)/10465 weeks
198-98.54586 -122.2148 (198*2π)/10465 weeks
199-151.9324 -22.45682 (199*2π)/10465 weeks
200-82.09758 73.68741 (200*2π)/10465 weeks
20130.77897 51.02213 (201*2π)/10465 weeks
20249.36927 -52.9864 (202*2π)/10465 weeks
203-36.16392 -97.60018 (203*2π)/10465 weeks
204-98.82851 -29.69611 (204*2π)/10465 weeks
205-51.50072 50.20172 (205*2π)/10465 weeks
20639.86706 32.90317 (206*2π)/10465 weeks
20757.54363 -53.79222 (207*2π)/10465 weeks
208-6.89735 -97.45763 (208*2π)/10465 weeks
209-57.47393 -61.18901 (209*2π)/10465 weeks
210-39.98462 -19.47728 (210*2π)/10465 weeks
211-9.16463 -34.06181 (211*2π)/10465 weeks
212-29.94607 -64.73422 (212*2π)/10465 weeks
213-73.16562 -42.17907 (213*2π)/10465 weeks
214-62.24685 17.09994 (214*2π)/10465 weeks
2154.95708 29.55429 (215*2π)/10465 weeks
21643.93451 -32.05358 (216*2π)/10465 weeks
217-.16564 -93.1 (217*2π)/10465 weeks
218-74.91457 -75.07196 (218*2π)/10465 weeks
219-88.64581 1.5143 (219*2π)/10465 weeks
220-26.29436 46.79074 (220*2π)/10465 weeks
22139.66734 14.40256 (221*2π)/10465 weeks
22243.36154 -52.25331 (222*2π)/10465 weeks
223-4.97588 -83.77119 (223*2π)/10465 weeks
224-47.68126 -61.7123 (224*2π)/10465 weeks
225-48.50628 -21.82806 (225*2π)/10465 weeks
226-18.13197 -4.94736 (226*2π)/10465 weeks
2279.94363 -22.25896 (227*2π)/10465 weeks
22810.28272 -55.9016 (228*2π)/10465 weeks
229-18.50438 -76.09688 (229*2π)/10465 weeks
230-52.20696 -63.66785 (230*2π)/10465 weeks
231-59.76937 -29.87107 (231*2π)/10465 weeks
232-36.57117 -9.43783 (232*2π)/10465 weeks
233-13.61616 -21.12594 (233*2π)/10464 weeks
234-19.29484 -41.88778 (234*2π)/10464 weeks
235-39.21687 -38.17753 (235*2π)/10464 weeks
236-37.06998 -17.08467 (236*2π)/10464 weeks
237-13.84523 -17.82521 (237*2π)/10464 weeks
238-13.30081 -46.38107 (238*2π)/10464 weeks
239-52.29945 -53.40806 (239*2π)/10464 weeks
240-78.15201 -4.93775 (240*2π)/10464 weeks
241-34.06518 50.43935 (241*2π)/10464 weeks
24247.54688 32.02612 (242*2π)/10464 weeks
24368.66094 -56.8164 (243*2π)/10464 weeks
244-4.78059 -114.5202 (244*2π)/10464 weeks
245-84.59738 -69.46618 (245*2π)/10464 weeks
246-69.77568 18.37132 (246*2π)/10464 weeks
24716.90082 32.1173 (247*2π)/10464 weeks
24857.74714 -44.70809 (248*2π)/10464 weeks
249.93658 -106.9625 (249*2π)/10464 weeks
250-70.57055 -75.3903 (250*2π)/10464 weeks
251-59.50477 -5.50651 (251*2π)/10464 weeks
2527.83767 -3.8431 (252*2π)/10464 weeks
25323.03785 -73.43119 (253*2π)/10464 weeks
254-45.82871 -110.975 (254*2π)/10464 weeks
255-108.4003 -57.36267 (255*2π)/10464 weeks
256-83.08838 17.98197 (256*2π)/10464 weeks
257-12.57466 19.0533 (257*2π)/10464 weeks
2582.08995 -39.88542 (258*2π)/10464 weeks
259-51.3857 -60.60904 (259*2π)/10464 weeks
260-83.00809 -7.27913 (260*2π)/10464 weeks
261-35.90979 42.69651 (261*2π)/10464 weeks
26230.05155 14.16207 (262*2π)/10464 weeks
26326.19518 -55.94321 (263*2π)/10464 weeks
264-37.26981 -72.83389 (264*2π)/10464 weeks
265-68.22221 -19.37181 (265*2π)/10464 weeks
266-24.8329 23.17522 (266*2π)/10464 weeks
26728.98403 -6.76083 (267*2π)/10464 weeks
26818.35969 -67.18986 (268*2π)/10464 weeks
269-39.06113 -77.40651 (269*2π)/10464 weeks
270-64.54399 -30.82915 (270*2π)/10464 weeks
271-29.93912 2.89812 (271*2π)/10464 weeks
2729.86581 -21.41467 (272*2π)/10464 weeks
273-1.55606 -67.74077 (273*2π)/10464 weeks
274-49.43461 -75.48898 (274*2π)/10464 weeks
275-77.30334 -38.49059 (275*2π)/10464 weeks
276-62.46006 -1.07476 (276*2π)/10464 weeks
277-32.33723 5.90261 (277*2π)/10464 weeks
278-16.47199 -6.03738 (278*2π)/10464 weeks
279-13.05679 -14.86003 (279*2π)/10464 weeks
280-7.75403 -20.11506 (280*2π)/10464 weeks
281-1.85043 -32.83703 (281*2π)/10464 weeks
282-7.55373 -51.33111 (282*2π)/10464 weeks
283-26.97889 -61.74566 (283*2π)/10464 weeks
284-48.59084 -55.55903 (284*2π)/10464 weeks
285-59.38159 -36.74493 (285*2π)/10464 weeks
286-53.15733 -17.07755 (286*2π)/10464 weeks
287-34.0114 -11.14825 (287*2π)/10464 weeks
288-19.68415 -26.88482 (288*2π)/10464 weeks
289-31.25209 -50.14432 (289*2π)/10464 weeks
290-64.90439 -48.71798 (290*2π)/10464 weeks
291-82.26883 -10.34413 (291*2π)/10464 weeks
292-51.49786 28.17662 (292*2π)/10464 weeks
2931.77303 15.68726 (293*2π)/10464 weeks
29412.23249 -42.72354 (294*2π)/10464 weeks
295-42.06399 -77.29668 (295*2π)/10464 weeks
296-99.31828 -37.72198 (296*2π)/10464 weeks
297-85.62983 34.34058 (297*2π)/10464 weeks
298-13.72794 51.59013 (298*2π)/10464 weeks
29928.84801 -5.11121 (299*2π)/10463 weeks
300-5.29218 -60.53771 (300*2π)/10463 weeks
301-61.7385 -47.6103 (301*2π)/10463 weeks
302-63.80343 2.88302 (302*2π)/10463 weeks
303-20.59447 12.71981 (303*2π)/10463 weeks
304-7.60615 -26.87839 (304*2π)/10463 weeks
305-48.89574 -44.27929 (305*2π)/10463 weeks
306-77.1284 -.11024 (306*2π)/10463 weeks
307-35.60651 44.89379 (307*2π)/10463 weeks
30827.04142 14.94267 (308*2π)/10463 weeks
30916.9861 -61.7494 (309*2π)/10463 weeks
310-67.10514 -79.17841 (310*2π)/10463 weeks
311-118.3802 .41988 (311*2π)/10463 weeks
312-61.43307 84.81171 (312*2π)/10463 weeks
31342.19952 67.87327 (313*2π)/10463 weeks
31470.66148 -31.23549 (314*2π)/10463 weeks
315-2.18351 -94.53928 (315*2π)/10463 weeks
316-80.6916 -57.91423 (316*2π)/10463 weeks
317-77.75521 17.06889 (317*2π)/10463 weeks
318-18.60601 37.68963 (318*2π)/10463 weeks
31912.57431 -1.04847 (319*2π)/10463 weeks
320-8.90536 -30.96742 (320*2π)/10463 weeks
321-31.37613 -17.84948 (321*2π)/10463 weeks
322-17.33021 -.85217 (322*2π)/10463 weeks
3231.92544 -19.60962 (323*2π)/10463 weeks
324-17.80709 -48.73229 (324*2π)/10463 weeks
325-59.41748 -36.63869 (325*2π)/10463 weeks
326-64.14475 13.52959 (326*2π)/10463 weeks
327-15.19253 41.02724 (327*2π)/10463 weeks
32834.40362 6.65689 (328*2π)/10463 weeks
32926.85546 -55.42664 (329*2π)/10463 weeks
330-30.84735 -79.18267 (330*2π)/10463 weeks
331-78.57329 -42.84958 (331*2π)/10463 weeks
332-72.23173 12.69967 (332*2π)/10463 weeks
333-25.398 33.64743 (333*2π)/10463 weeks
33413.14131 5.97077 (334*2π)/10463 weeks
3358.91875 -39.09251 (335*2π)/10463 weeks
336-31.31336 -58.83592 (336*2π)/10463 weeks
337-70.32107 -35.89426 (337*2π)/10463 weeks
338-74.05721 9.14156 (338*2π)/10463 weeks
339-41.48116 37.54175 (339*2π)/10463 weeks
340-3.99518 29.42584 (340*2π)/10463 weeks
3418.42158 1.72266 (341*2π)/10463 weeks
342-1.65369 -15.39464 (342*2π)/10463 weeks
343-9.88834 -15.8849 (343*2π)/10463 weeks
344-8.42585 -18.28762 (344*2π)/10463 weeks
345-14.58114 -28.7172 (345*2π)/10463 weeks
346-34.17039 -25.66094 (346*2π)/10463 weeks
347-38.55369 1.97554 (347*2π)/10463 weeks
348-3.89866 20.02326 (348*2π)/10463 weeks
34937.04107 -14.6164 (349*2π)/10463 weeks
35019.74108 -82.36032 (350*2π)/10463 weeks
351-62.99885 -102.8272 (351*2π)/10463 weeks
352-129.3528 -32.08511 (352*2π)/10463 weeks
353-98.29375 66.55341 (353*2π)/10463 weeks
3543.66767 86.68832 (354*2π)/10463 weeks
35565.40624 8.92146 (355*2π)/10463 weeks
35625.21228 -73.02346 (356*2π)/10463 weeks
357-56.60516 -69.32639 (357*2π)/10463 weeks
358-81.02195 -1.12983 (358*2π)/10463 weeks
359-34.31818 40.59204 (359*2π)/10463 weeks
36011.66746 18.37131 (360*2π)/10463 weeks
3619.04898 -18.02923 (361*2π)/10463 weeks
362-9.85047 -20.9989 (362*2π)/10463 weeks
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71834.40362 -6.65689