找到相應的MATLAB中有95％的置信區間峯值

Question

假設我們有以下數組：

0.196238259763928 
0.0886250228175519 
0.417543614272817 
0.182403230538167 
0.136500793051860 
0.389922187581014 
0.0344012946153299 
0.381603315802419 
0.0997542838649466 
0.274807632628596 
0.601652859233616 
0.209431489000677 
0.396925294300794 
0.0351587496999554 
0.177321874549738 
0.369200511917405 
0.287108838007101 
0.477076452316346 
0.127558716868438 
0.792431584110476 
0.0459982776925879 
0.612598437936600 
0.228340227044324 
0.190267907472804 
0.564751537228850 
0.00269368929400299 
0.940538666131177 
0.101588565140294 
0.426175626669060 
0.600215481734847 
0.127859067121782 
0.985881201195063 
0.0945679498528667 
0.950077461673118 
0.415212985598547 
0.467423473845033 
1.24336273213410 
0.0848695928658021 
1.84522775800633 
0.289288949281834 
1.38792131632743 
1.73186592736729 
0.554254947026916 
3.46075557122590 
0.0872957577705428 
4.93259798197976 
2.03544238985229 
3.71059303259615 
8.47095716918618 
0.422940369071662 
25.2287636895831 
4.14535369056670 
63.7312173032838 
152.080907190007 
1422.19492782494 
832.134744027851 
0.0220089962114756 
60.8238733887811 
7.71053463387430 
10.4151913932115 
11.3141744831953 
0.988978595613829 
8.65598040591953 
0.219820300144944 
3.92785491164888 
2.28370963778411 
1.60232807621444 
2.51086405960291 
0.0181622519984990 
2.27469230188760 
0.487809730727909 
0.961063613990814 
1.90435488292485 
0.515640996120482 
1.25933693517960 
0.0953200831348589 
1.52851575480462 
0.582109930768162 
0.933543409438383 
0.717947488528521 
0.0445235241119612 
1.21157308704582 
0.0942421028083462 
0.536069075206508 
0.821400666720535 
0.308956823975938 
1.28706199713640 
0.0339217632187507 
1.19575886464231 
0.0853733920496230 
0.736744959694641 
0.635218502184121 
0.262305581223588 
0.986899895695809 
0.0398800891449550 
0.758792061180657 
0.134279188964854 
0.442531129290843 
0.542782326712391 
0.377221037448628 
0.704787750202814 
0.224180325609783 
0.998785634315287 
0.408055416702400 
0.329684702125840 
0.522384453408780 
0.154542718256493 
0.602294251721841 
0.240357912028348 
0.359040779285709 
0.525224294805813 
0.427539247203335 
0.624034405807298 
0.298184846094056 
0.498659616687732 
0.0962076792277457 
0.430092706132805 
0.656212420735658 
0.278310520474744 
0.866037361133916 
0.184971060800812 
0.481149730712771 
0.624405636807668 
0.382388147099945 
0.435350646037440 
0.216499523971397 
1.22960953802959 
0.330841706900755 
0.891793067878849 
0.628241046456751 
0.278687691121678 
1.06358076764171 
0.365652714373067 
1.34921178081181 
0.652888708375276 
0.861138633227739 
1.02878577330537 
0.591174450919664 
1.93594290806582 
0.497631035062465 
1.14486512201656 
0.978067581547298 
0.948931658572253 
2.01004088022982 
0.917415940349743 
2.24124811810385 
1.42691656876436 
2.15636037453584 
1.92812357585099 
1.12786835077183 
4.81721425534142 
1.70055431306602 
4.87939454466131 
3.90293284926105 
5.16542230018432 
10.5783535493504 
1.74023535081791 
27.0572221453758 
7.78813114379733 
69.2528169436690 
167.769806437531 
1490.03057130613 
869.247150795648 
3.27543244752518 
62.3527480644562 
9.74192115073051 
13.6074209231800 
10.5686495478844 
7.70239986387120 
9.62850426896699 
9.85304975304259 
7.09026325332085 
12.8782040428502 
16.3163128995995 
7.00070066635845 
74.1532966917877 
4.80506505312457 
1042.52337489620 
1510.37374385290 
118.514435606795 
80.7915675273571 
2.96352221859211 
27.7825124315786 
1.55102367292252 
8.66382951478539 
5.02910503820560 
1.25219344189599 
7.72195587189507 
0.356973215117373 
6.06702456628919 
1.01953617014621 
2.76489896186652 
3.35353608882459 
0.793376336025486 
4.90341095941571 
0.00742857354167949 
5.07665716731356 
1.16863474789604 
4.47635486149688 
4.33050121578669 
2.42974020115261 
9.79494608790444 
0.0568839453395247 
22.9153086380666 
4.48791386399205 
59.6962194708933 
97.8636220152072 
1119.97978883924 
806.144299041605 
7.33252581243942 
57.0699524267842 
0.900104994068117 
15.2791339483160 
3.31266162202546 
3.20809490583211 
5.36617545130941 
0.648122925703121 
3.90480316969632 
0.0338850542128927 
2.58828964019220 
0.543604662856673 
1.16385064506181 
1.01835324272839 
0.172915006573539 
1.55998411282069 
0.00221570175453666 
1.14803074836796 
0.0769335878967426 
0.421762398811163 
0.468260146832541 
0.203765185125597 
0.467641715366303 
0.00142988680149041 
0.698088976126660 
0.0413316717103625 
0.190548157914037 
0.504713663418641 
0.325697764871308 
0.375910057283262 
0.123307135682793 
0.331115262928959 
0.00263961045860704 
0.204555648718379 
0.139008751575803 
0.182936666944843 
0.154943314848474 
0.0840483576044629 
0.293075999812128 
0.00306911699543199 
0.272993318570981 
0.0864711337990886 
0.280495615619829 
0.091
0.148399626645134 
0.141945002415500 
0.0512001531781583 
0.0295283557338525 

在MATLAB是非常容易使用findpeaks找峯，就像這樣：

[pxx_peaks,location] = findpeaks(Pxx); 

如果我們繪製pxx_peaks，我們得到

plot(pxx_peaks) 

當然，除了這些峯值之外，圖中沒有顯示較小的峯值，但我的目標是找到比所有其他峯值高95-96％的所有峯值。

我已經試過這樣：

>> average = mean(pxx_peaks); 
>> stand = std(pxx_peaks); 
>> final_peaks = pxx_peaks(pxx_peaks > average + 3*stand); 

這樣做的結果是

>> final_peaks 

final_peaks = 

    1.0e+03 * 

    1.4222 
    1.4900 
    1.5104 
    1.1200 

但如何返回其對應的位置嗎？我想把它寫成一個m文件，所以請大家幫我

編輯

也請幫我在這個問題：我可以參數化的置信區間？例如，而不是95％，我想找到高於其他峯值60％的峯值，有可能嗎？

Answer 1

注意3σ≈99.73％

關於你提到的第一個問題，這很容易，你只需要跟蹤地點以同樣的方式，你的山峯做：

inds   = pxx_peaks > mean(pxx_peaks) + 3*std(pxx_peaks); 
final_peaks  = pxx_peaks(inds); 
final_locations = location(inds); 

plot(Pxx), hold on 
plot(final_locations, final_peaks, 'r.') 

至於你的第二個問題，這有點複雜。如果你想像你說的那樣制定它，你必須將所需的百分比轉換爲正確的σ數。這涉及到標準正態分佈的整合，求根：

%// Convert confidence interval percentage to number-of-sigmas 
F = @(P) fzero(@(sig) quadgk(@(x) exp(-x.^2/2),-sig,+sig)/sqrt(2*pi) - P/100, 1); 

% // Repeat with the desired percentage 
inds   = pxx_peaks > mean(pxx_peaks) + F(63)*std(pxx_peaks); %// 63% 
final_peaks  = pxx_peaks(inds); 
final_locations = location(inds); 

plot(final_locations, final_peaks, 'r.')