数値積分：悪い積分動作エラー

Question

これは私の行列

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-3.70907277842741, -3.61955988253518, -3.55951762156004, -3.27305364378077, 
-3.15716747633446, -3.08436906473098, -2.94699678899349, -2.67819176248474, 
-2.51950911610342, -2.45801068672049, -2.38080987129559, -2.23660157184075, 
-2.03595688651075, -1.58836441712333, -1.47187389747016, -1.39555037998287, 
-1.27113029247092, -1.15673682064109, -1.0426545765059, -0.943946623196274, 
-0.835837575946843, -0.805837349555374, -0.689648730659341, -0.648094429117022, 
-0.584092303333525, -0.394149224677363, -0.222824659020402, -0.0906541511144734, 
0, 0.0906541511144734, 0.222824659020402, 0.394149224677363, 
0.584092303333525, 0.648094429117022, 0.689648730659341, 0.805837349555374, 
0.835837575946843, 0.943946623196274, 1.0426545765059, 1.15673682064109, 
1.27113029247092, 1.39555037998287, 1.47187389747016, 1.58836441712333, 
2.03595688651075, 2.23660157184075, 2.38080987129559, 2.45801068672049, 
2.51950911610342, 2.67819176248474, 2.94699678899349, 3.08436906473098, 
3.15716747633446, 3.27305364378077, 3.55951762156004, 3.61955988253518, 
3.70907277842741, 3.77843705689023, 3.93194755639431, 4.26474198682156, 
4.37919276705198, 4.60547103236952, 4.78689576531572, 4.95085939235108, 
5.09102141962306, 5.43365721329602, 5.85854042138457, 6.15110072929745, 
6.36994876479492, 6.48897104737938, 7.13162491301438, 7.50817793772083, 
7.93147284818737, 8.30440651819497, 10.1862843945321, 11.1957507602953, 
12.2966556390103, 12.6000848203634, -12.7233949898779, -12.425009683628, 
-11.2494681735225, -10.2166322197219, -8.3755092456272, -8.00338523998922, 
-7.57923229725886, -7.19608684570266, -6.5405512372713, -6.40269304276938, 
-6.17884044516668, -5.8896632980985, -5.47132446137006, -5.13601542532135, 
-4.9932306291456, -4.82484171541241, -4.64182091659811, -4.41774407285591, 
-4.29324182862204, -3.95837714178017, -3.80090594237893, -3.73788575810611, 
-3.6386462286676, -3.57417931862452, -3.29515190899568, -3.18022337150066, 
-3.10777656095811, -2.97137842876953, -2.69668587115743, -2.54143464829308, 
-2.47346988209172, -2.39673461879726, -2.24821478269518, -2.05469079214647, 
-1.61210502089736, -1.50444445385474, -1.42335954097761, -1.29707578786137, 
-1.18226432293022, -1.06498427720905, -0.963826675880699, -0.85422851115692, 
-0.817708608597364, -0.698957479868998, -0.647234505242058, -0.580874151897233, 
-0.392280434624987, -0.221324908501217, -0.0951930012794266, 
0, 0.0951930012794266, 0.221324908501217, 0.392280434624987, 
0.580874151897233, 0.647234505242058, 0.698957479868998, 0.817708608597364, 
0.85422851115692, 0.963826675880699, 1.06498427720905, 1.18226432293022, 
1.29707578786137, 1.42335954097761, 1.50444445385474, 1.61210502089736, 
2.05469079214647, 2.24821478269518, 2.39673461879726, 2.47346988209172, 
2.54143464829308, 2.69668587115743, 2.97137842876953, 3.10777656095811, 
3.18022337150066, 3.29515190899568, 3.57417931862452, 3.6386462286676, 
3.73788575810611, 3.80090594237893, 3.95837714178017, 4.29324182862204, 
4.41774407285591, 4.64182091659811, 4.82484171541241, 4.9932306291456, 
5.13601542532135, 5.47132446137006, 5.8896632980985, 6.17884044516668, 
6.40269304276938, 6.5405512372713, 7.19608684570266, 7.57923229725886, 
8.00338523998922, 8.3755092456272, 10.2166322197219, 11.2494681735225, 
12.425009683628, 12.7233949898779, -12.9701665831908, -12.6818752140478, 
-11.3569688906404, -10.2773650952434, -8.51780191626772, -8.14868819459235, 
-7.72003821070641, -7.32508978114711, -6.64377488616525, -6.46822176338875, 
-6.23435390289731, -5.95194722735932, -5.54670516078474, -5.22605862710606, 
-5.07802507599128, -4.90078016073291, -4.71456527242172, -4.49489397213016, 
-4.35027647059818, -4.01126873151645, -3.85381415345849, -3.78760418054142, 
-3.67684233255883, -3.6035206970325, -3.33937554552141, -3.22636344257365, 
-3.15462026543263, -3.02017161524238, -2.73369677368098, -2.58531260689116, 
-2.50440723533775, -2.42860364735808, -2.27145544935968, -2.09218158273497, 
-1.66032132882236, -1.56891953844307, -1.47901197414181, -1.34899860381407, 
-1.23335063996306, -1.10967106859385, -1.00361116644947, -0.894069184830091, 
-0.838429443534018, -0.717586396553344, -0.645513602695347, -0.574433901587114, 
-0.388540562231589, -0.218323567844169, -0.104276269037671, 0, 
0.104276269037671, 0.218323567844169, 0.388540562231589, 0.574433901587114, 
0.645513602695347, 0.717586396553344, 0.838429443534018, 0.894069184830091, 
1.00361116644947, 1.10967106859385, 1.23335063996306, 1.34899860381407, 
1.47901197414181, 1.56891953844307, 1.66032132882236, 2.09218158273497, 
2.27145544935968, 2.42860364735808, 2.50440723533775, 2.58531260689116, 
2.73369677368098, 3.02017161524238, 3.15462026543263, 3.22636344257365, 
3.33937554552141, 3.6035206970325, 3.67684233255883, 3.78760418054142, 
3.85381415345849, 4.01126873151645, 4.35027647059818, 4.49489397213016, 
4.71456527242172, 4.90078016073291, 5.07802507599128, 5.22605862710606, 
5.54670516078474, 5.95194722735932, 6.23435390289731, 6.46822176338875, 
6.64377488616525, 7.32508978114711, 7.72003821070641, 8.14868819459235, 
8.51780191626772, 10.2773650952434, 11.3569688906404, 12.6818752140478, 
12.9701665831908), .Dim = c(99L, 11L)) 

で各列は、私が統合し、私のintegr行列に値を入れたいことを私のための機能である：

integr=matrix(0,11) 

for (t in 1:11){ 
    integrating = approxfun(thau,func[,t],rule=2) 
    integr[t,1]=integrate(integrating, lower = 0.01, upper = 0.5,subdivisions = 1000)$value 
} 

私は、このエラーメッセージがあります。

Error in integrate(integrating, lower = 0.01, upper = 0.5, subdivisions = 1000) : 
    extremely bad integrand behaviour 

私はこの問題と続きを回避するにはどうすればよいの関数内でintegrateを使用することをお勧めします。私は非線形アプローチをする必要がありますか？もしそうなら、どうしたらいいですか？

多くのありがとうございます。

Answer 1

まず、最近10日間程度の分位関数に関する数値的な統合的な質問を見ると非常に興味深いです。たとえば：

• Building a function by defining X and Y and then Integrating in R
• Standard Normal Quantile Function Integration in R
• Understanding and implementing numerical integration with a quantile function in R

は、この質問が最初のものと類似しているかに注意してください。 thauが何であるかは言及していませんが、私はそれがthau <- seq(0.01, 0.99, 0.01)だと信じています。さんはthauに対するあなたのfunc行列をスケッチ持ってみましょう：

matplot(thau, func, type = "l") 

また、のはfuncのすべての列が単調に増加していることを確認してみましょう：

all(diff(func) > 0) 
# [1] TRUE 

基本的には、あなたの質問には、答えを使用している私は、第1のリンクされた質問（ただし、rule = 2の正当性は第3のリンクされた質問に記載されています）で提供されます。しかし、あなたの質問に感謝します。私は今、いくつかの潜在的な数値的欠陥があることを認識しています。

---

私は分野の専門家ではないよと私は限られた時間の中で適応直交背後にある数学を理解することが洗練されています。しかし、それはいつかそんなに簡単な仕事でいつかは失敗するだろうと私には驚くべきことです。 2番目のリンク先の質問で述べたように、私たちは台形ルールを使うことさえできます。

私がintegrateをテストすると、それは失敗したと報告した5分の1ではなくfuncの第1列になります。

## get interpolation function for all columns in a list 
flst <- lapply(1:ncol(func), function (i) approxfun(thau, func[,i], rule = 2)) 

## all OK excluding the 1st column 
sapply(flst[-1], function (fun) integrate(fun, 0.01, 0.5)$value) 
# [1] -2.010421 -2.088981 -2.114083 -2.000653 -2.015932 -1.986130 -1.912076 
# [8] -1.877459 -1.892291 -1.921983 

## the 1st one fails 
integrate(flst[[1]], 0.01, 0.5) 
# extremely bad integrand behaviour 

前述したように、私はこの問題は単純な性質のために人工的ではないと信じています。実際には、考えてみましょう

integrate(flst[[1]], 0.01 + .Machine$double.eps^0.25, 0.5) 
# -2.13653 with absolute error < 8.7e-05 

integrate(flst[[1]], 0, 0.5) 
# -2.286034 with absolute error < 0.00027 

これらはすべて動作します。

---

私の知る限り探索することができるように、integrate関数は、2つのFortranサブルーチン使用している。ため

• dqagsを定積分、エンドポイントの特異点に対処することができます。
• dqagi不定積分の場合。

integrateのRドキュメントは、これらのルーチンのエラー処理についてはあまり説明していませんが、Fortranページは少しです。残念なことに、「悪い行為」が何であるかは依然として非常に明確ではありません。しかし、エラーコードに関係なく、Fortranのサブルーチンは常に統合結果を返すことが分かります。

integrateのソースコードを確認してください。積分結果を変数wkに格納され、その後、swtich文はwk$ierrに保存されている整数エラーコードを解釈するために使用されます。

res$message <- switch(wk$ierr + 1L, "OK", "maximum number of subdivisions reached", 
    "roundoff error was detected", "extremely bad integrand behaviour", 
    "roundoff error is detected in the extrapolation table", 
    "the integral is probably divergent", "the input is invalid") 
if (wk$ierr == 6L || (wk$ierr > 0L && stop.on.error)) 
    stop(res$message) 

このswitch次if文は、我々はすべてのエラーを無視するかどうかを決定します。 integrateにはstop.on.error引数があります。デフォルトのTRUEではなくFALSEに設定すると、integrateが常に動作します。したがって、しましょう

z <- integrate(flst[[1]], 0.01, 0.5, stop.on.error = FALSE) 

str(z) 
# $ value  : num -2.14 
# $ abs.error : num 0.000446 
# $ subdivisions: int 69 
# $ message  : chr "extremely bad integrand behaviour" 
# $ call  : language integrate(f = flst[[1]], lower = 0.01, upper = 0.5, stop.on.error = FALSE) 
# - attr(*, "class")= chr "integrate" 

z$value 
# [1] -2.138348 

これはすべて私が現在行うことができます。近い将来、私はAdaptive Quadratureの周りを読む機会があると私は信じています。