Integrating columns of an Array using R (Error)

Question

This is my matrix:

funcao=structure(c(-14.7690673280818, -14.5543581356252, -12.1406211639974, 
-10.7200919648493, -9.55507848352318, -9.20790894914246, -8.74647670464071, 
-8.26548763467919, -7.3962484443768, -6.94590909664862, -6.63903257406218, 
-6.40598087538929, -6.09621085755895, -5.88245015332806, -5.69615454359131, 
-5.4543516801245, -5.24485270997044, -5.05729667581875, -4.7660441793448, 
-4.39683466679171, -4.30668996362841, -4.0828498417172, -3.95528199743041, 
-3.81741171570944, -3.66175435590196, -3.56271256245504, -3.49609868983868, 
-3.37586125647396, -3.0034966287813, -2.90517151506904, -2.72993249556762, 
-2.66092057836039, -2.4408738658527, -2.36547969703899, -2.09546783234301, 
-1.95526398213212, -1.88470374007828, -1.72750243012653, -1.60575660667247, 
-1.43542615745848, -1.29362975715702, -1.18449733575892, -0.98947894198681, 
-0.853386354928181, -0.63296867063245, -0.527486151545778, -0.361277864998242, 
-0.257665212366851, -0.109270291541468, 0, 0.109270291541468, 
0.257665212366851, 0.361277864998242, 0.527486151545778, 0.63296867063245, 
0.853386354928181, 0.98947894198681, 1.18449733575892, 1.29362975715702, 
1.43542615745848, 1.60575660667247, 1.72750243012653, 1.88470374007828, 
1.95526398213212, 2.09546783234301, 2.36547969703899, 2.4408738658527, 
2.66092057836039, 2.72993249556762, 2.90517151506904, 3.0034966287813, 
3.37586125647396, 3.49609868983868, 3.56271256245504, 3.66175435590196, 
3.81741171570944, 3.95528199743041, 4.0828498417172, 4.30668996362841, 
4.39683466679171, 4.7660441793448, 5.05729667581875, 5.24485270997044, 
5.4543516801245, 5.69615454359131, 5.88245015332806, 6.09621085755895, 
6.40598087538929, 6.63903257406218, 6.94590909664862, 7.3962484443768, 
8.26548763467919, 8.74647670464071, 9.20790894914246, 9.55507848352318, 
10.7200919648493, 12.1406211639974, 14.5543581356252, 14.7690673280818, 
-13.7079821475458, -13.4498703618915, -11.67838231268, -10.4589486406842, 
-8.9432388382521, -8.58312548137122, -8.14102991846389, -7.71079209128989, 
-6.95240042017031, -6.66414427349685, -6.40033205538623, -6.13816822549321, 
-5.77208382991545, -5.49527630669887, -5.33154964830415, -5.12782641892739, 
-4.93206161073643, -4.72556232300793, -4.52080276980657, -4.1694078333758, 
-4.03956026669583, -3.90869860153294, -3.79104380757601, -3.69124767313376, 
-3.47159857371935, -3.3643163654408, -3.29467696235366, -3.16605701449292, 
-2.84435464789616, -2.71650210221017, -2.59690597248341, -2.52388797476942, 
-2.34094207608544, -2.20427426100811, -1.83879580218285, -1.72737794398069, 
-1.6454056454373, -1.5042411957216, -1.38609220687676, -1.2432788707001, 
-1.1225617148797, -1.01318771357089, -0.90038209504245, -0.773284479514442, 
-0.640368323748707, -0.555178375238413, -0.377358821158776, -0.230347860323025, 
-0.110436162251717, 0, 0.110436162251717, 0.230347860323025, 
0.377358821158776, 0.555178375238413, 0.640368323748707, 0.773284479514442, 
0.90038209504245, 1.01318771357089, 1.1225617148797, 1.2432788707001, 
1.38609220687676, 1.5042411957216, 1.6454056454373, 1.72737794398069, 
1.83879580218285, 2.20427426100811, 2.34094207608544, 2.52388797476942, 
2.59690597248341, 2.71650210221017, 2.84435464789616, 3.16605701449292, 
3.29467696235366, 3.3643163654408, 3.47159857371935, 3.69124767313376, 
3.79104380757601, 3.90869860153294, 4.03956026669583, 4.1694078333758, 
4.52080276980657, 4.72556232300793, 4.93206161073643, 5.12782641892739, 
5.33154964830415, 5.49527630669887, 5.77208382991545, 6.13816822549321, 
6.40033205538623, 6.66414427349685, 6.95240042017031, 7.71079209128989, 
8.14102991846389, 8.58312548137122, 8.9432388382521, 10.4589486406842, 
11.67838231268, 13.4498703618915, 13.7079821475458, -14.358609077878, 
-14.1271105147046, -11.9618138886078, -10.6190742201334, -9.31840132054895, 
-8.96622474353695, -8.51227248333579, -8.05091545299969, -7.22455526958365, 
-6.83691436333096, -6.54669635165921, -6.30238324376159, -5.97082920578328, 
-5.73268017960003, -5.55511488901834, -5.32804232266972, -5.12385612117062, 
-4.92897229198519, -4.67117775232928, -4.30885942992513, -4.20335651783943, 
-4.0154831357938, -3.89174994433905, -3.76860783727992, -3.58819663306074, 
-3.48596721027473, -3.41818297549918, -3.29470294456033, -2.94193593581688, 
-2.83218876260999, -2.67847401240583, -2.60791242809822, -2.40221737689215, 
-2.3031208003073, -1.99617971268181, -1.86711111167336, -1.79213635811389, 
-1.64113856670334, -1.52078410227216, -1.36109806339877, -1.22745572081485, 
-1.11822984936273, -0.955013720375685, -0.822400646878094, -0.635831069171161, 
-0.538198299721656, -0.367498441110635, -0.24709807532094, -0.109721283846771, 
0, 0.109721283846771, 0.24709807532094, 0.367498441110635, 0.538198299721656, 
0.635831069171161, 0.822400646878094, 0.955013720375685, 1.11822984936273, 
1.22745572081485, 1.36109806339877, 1.52078410227216, 1.64113856670334, 
1.79213635811389, 1.86711111167336, 1.99617971268181, 2.3031208003073, 
2.40221737689215, 2.60791242809822, 2.67847401240583, 2.83218876260999, 
2.94193593581688, 3.29470294456033, 3.41818297549918, 3.48596721027473, 
3.58819663306074, 3.76860783727992, 3.89174994433905, 4.0154831357938, 
4.20335651783943, 4.30885942992513, 4.67117775232928, 4.92897229198519, 
5.12385612117062, 5.32804232266972, 5.55511488901834, 5.73268017960003, 
5.97082920578328, 6.30238324376159, 6.54669635165921, 6.83691436333096, 
7.22455526958365, 8.05091545299969, 8.51227248333579, 8.96622474353695, 
9.31840132054895, 10.6190742201334, 11.9618138886078, 14.1271105147046, 
14.358609077878, -14.5672256639001, -14.3442603466748, -12.0526932005212, 
-10.6704167793823, -9.43869316924387, -9.08906143604798, -8.63130745231583, 
-8.15997238138419, -7.31181881941087, -6.89231124997616, -6.59362650222709, 
-6.3550370379524, -6.03455479085867, -5.80880119944129, -5.62679869904323, 
-5.39223941682433, -5.1853529853784, -4.99419352909143, -4.7193938866676, 
-4.35357309705323, -4.255876038818, -4.04972245780601, -3.92404029302206, 
-3.79341254883719, -3.62558255722496, -3.52497325621722, -3.4577838613747, 
-3.33595188997765, -2.97322433504632, -2.86928245642403, -2.70462793445213, 
-2.63485397290467, -2.42186464857093, -2.33481488889686, -2.04664318502277, 
-1.91191506158551, -1.8391839961936, -1.68503324864501, -1.56397162198931, 
-1.39887553095015, -1.26108886508264, -1.15191048988921, -0.972530768371285, 
-0.838149222100249, -0.634376246882453, -0.532753819606308, -0.364336815412981, 
-0.252468853112474, -0.109492065710681, 0, 0.109492065710681, 
0.252468853112474, 0.364336815412981, 0.532753819606308, 0.634376246882453, 
0.838149222100249, 0.972530768371285, 1.15191048988921, 1.26108886508264, 
1.39887553095015, 1.56397162198931, 1.68503324864501, 1.8391839961936, 
1.91191506158551, 2.04664318502277, 2.33481488889686, 2.42186464857093, 
2.63485397290467, 2.70462793445213, 2.86928245642403, 2.97322433504632, 
3.33595188997765, 3.4577838613747, 3.52497325621722, 3.62558255722496, 
3.79341254883719, 3.92404029302206, 4.04972245780601, 4.255876038818, 
4.35357309705323, 4.7193938866676, 4.99419352909143, 5.1853529853784, 
5.39223941682433, 5.62679869904323, 5.80880119944129, 6.03455479085867, 
6.3550370379524, 6.59362650222709, 6.89231124997616, 7.31181881941087, 
8.15997238138419, 8.63130745231583, 9.08906143604798, 9.43869316924387, 
10.6704167793823, 12.0526932005212, 14.3442603466748, 14.5672256639001, 
-13.6245844768687, -13.3630613956461, -11.6420519179073, -10.4384236674326, 
-8.89515033208928, -8.53401963497411, -8.0934438691888, -7.66719491633901, 
-6.91751548158816, -6.64199852086489, -6.38157100998465, -6.11711906602055, 
-5.74660855271358, -5.46484576488352, -5.30289294897202, -5.10216264876024, 
-4.90747729878566, -4.69948913603343, -4.50152763272595, -4.15153286112639, 
-4.01856478499323, -3.89501090892983, -3.77813524686729, -3.68133161034538, 
-3.45665297878437, -3.34872310204811, -3.27884590260074, -3.14956711773407, 
-2.83184663180451, -2.70167333082681, -2.5864505421683, -2.51311768021067, 
-2.33308777876224, -2.19160406439047, -1.81862225736669, -1.70946687980764, 
-1.6265976326061, -1.48669362532537, -1.36882733539455, -1.22817675030816, 
-1.10911635110033, -0.999723362435733, -0.89337938721453, -0.766988745723311, 
-0.640949911207503, -0.557354889428034, -0.37862272938278, -0.228200809764493, 
-0.110527795707098, 0, 0.110527795707098, 0.228200809764493, 
0.37862272938278, 0.557354889428034, 0.640949911207503, 0.766988745723311, 
0.89337938721453, 0.999723362435733, 1.10911635110033, 1.22817675030816, 
1.36882733539455, 1.48669362532537, 1.6265976326061, 1.70946687980764, 
1.81862225736669, 2.19160406439047, 2.33308777876224, 2.51311768021067, 
2.5864505421683, 2.70167333082681, 2.83184663180451, 3.14956711773407, 
3.27884590260074, 3.34872310204811, 3.45665297878437, 3.68133161034538, 
3.77813524686729, 3.89501090892983, 4.01856478499323, 4.15153286112639, 
4.50152763272595, 4.69948913603343, 4.90747729878566, 5.10216264876024, 
5.30289294897202, 5.46484576488352, 5.74660855271358, 6.11711906602055, 
6.38157100998465, 6.64199852086489, 6.91751548158816, 7.66719491633901, 
8.0934438691888, 8.53401963497411, 8.89515033208928, 10.4384236674326, 
11.6420519179073, 13.3630613956461, 13.6245844768687, -13.7513240158161, 
-13.4949850845108, -11.6972632618919, -10.4696154936035, -8.96823049188587, 
-8.60864584793023, -8.16576044465774, -7.73344959477095, -6.97053016408726, 
-6.67565344738059, -6.41008219291174, -6.14910749861269, -5.78532336166338, 
-5.51109109533802, -5.34644256890291, -5.14116391069274, -4.94483810678764, 
-4.73911258917317, -4.53082008138683, -4.17869747763421, -4.05047164341461, 
-3.91581211159757, -3.79775240187192, -3.6964010627146, -3.47936581711638, 
-3.37242020290272, -3.30290438286576, -3.17462683337767, -2.85085507867997, 
-2.72420863226898, -2.60233967200701, -2.52948530961352, -2.34502396371101, 
-2.21085897720345, -1.84928001732781, -1.73668634525457, -1.65518019199163, 
-1.51336068884896, -1.39506478104353, -1.25112746056258, -1.1295492863675, 
-1.02018515280999, -0.904021410598258, -0.776556379825624, -0.640066072057238, 
-0.554047238235075, -0.376701966557133, -0.231463685042374, -0.110388540236378, 
0, 0.110388540236378, 0.231463685042374, 0.376701966557133, 0.554047238235075, 
0.640066072057238, 0.776556379825624, 0.904021410598258, 1.02018515280999, 
1.1295492863675, 1.25112746056258, 1.39506478104353, 1.51336068884896, 
1.65518019199163, 1.73668634525457, 1.84928001732781, 2.21085897720345, 
2.34502396371101, 2.52948530961352, 2.60233967200701, 2.72420863226898, 
2.85085507867997, 3.17462683337767, 3.30290438286576, 3.37242020290272, 
3.47936581711638, 3.6964010627146, 3.79775240187192, 3.91581211159757, 
4.05047164341461, 4.17869747763421, 4.53082008138683, 4.73911258917317, 
4.94483810678764, 5.14116391069274, 5.34644256890291, 5.51109109533802, 
5.78532336166338, 6.14910749861269, 6.41008219291174, 6.67565344738059, 
6.97053016408726, 7.73344959477095, 8.16576044465774, 8.60864584793023, 
8.96823049188587, 10.4696154936035, 11.6972632618919, 13.4949850845108, 
13.7513240158161, -13.5038153563633, -13.2373523390222, -11.5894414578226, 
-10.4087012189772, -8.82551281645418, -8.46290889731311, -8.02453396711809, 
-7.60406134878737, -6.86699820314081, -6.6099290032563, -6.3544029256183, 
-6.08663753750477, -5.70971751250438, -5.42077894932731, -5.26139485499848, 
-5.06499864973269, -4.87187647554418, -4.66173225584577, -4.47361508632088, 
-4.12564790942336, -3.98816098787005, -3.87518960476143, -3.75944221250401, 
-3.66697204643782, -3.43501009186589, -3.32614231903036, -3.2559207639547, 
-3.12568790968636, -2.81373363068119, -2.68019961742924, -2.57130991397796, 
-2.49752109350884, -2.32171388099973, -2.17325620779303, -1.78940871644707, 
-1.68352966297731, -1.59936153416788, -1.46128278798956, -1.34382587758861, 
-1.20630719708568, -1.08964596607244, -0.980225481600834, -0.88323868593752, 
-0.757871821632761, -0.641792114656659, -0.560506724513046, -0.380453009333927, 
-0.225091641300559, -0.110660491161603, 0, 0.110660491161603, 
0.225091641300559, 0.380453009333927, 0.560506724513046, 0.641792114656659, 
0.757871821632761, 0.88323868593752, 0.980225481600834, 1.08964596607244, 
1.20630719708568, 1.34382587758861, 1.46128278798956, 1.59936153416788, 
1.68352966297731, 1.78940871644707, 2.17325620779303, 2.32171388099973, 
2.49752109350884, 2.57130991397796, 2.68019961742924, 2.81373363068119, 
3.12568790968636, 3.2559207639547, 3.32614231903036, 3.43501009186589, 
3.66697204643782, 3.75944221250401, 3.87518960476143, 3.98816098787005, 
4.12564790942336, 4.47361508632088, 4.66173225584577, 4.87187647554418, 
5.06499864973269, 5.26139485499848, 5.42077894932731, 5.70971751250438, 
6.08663753750477, 6.3544029256183, 6.6099290032563, 6.86699820314081, 
7.60406134878737, 8.02453396711809, 8.46290889731311, 8.82551281645418, 
10.4087012189772, 11.5894414578226, 13.2373523390222, 13.5038153563633, 
-12.8877947524831, -12.5961340486959, -11.3210853809037, -10.2570925911286, 
-8.47030492676585, -8.10018637873416, -7.67303749757961, -7.28202887686486, 
-6.60931905266524, -6.44634841642492, -6.21582362929168, -5.93115698481963, 
-5.5215432443443, -5.19600239869575, -5.04972087076752, -4.87543206990054, 
-4.69028336184037, -4.46914150055587, -4.33123842894087, -3.99361363184664, 
-3.833076928421, -3.77408485453713, -3.6640925546724, -3.59372660759739, 
-3.32461379011727, -3.21096198541142, -3.13898393694403, -3.00388455382712, 
-2.72134261347931, -2.57066623804085, -2.49408041291093, -2.41796583370138, 
-2.26369776447253, -2.07966723692233, -1.64375650300888, -1.54786821816088, 
-1.46043531083332, -1.33166687876378, -1.21629813646533, -1.09475471308712, 
-0.990331188483095, -0.880770453063485, -0.831512873094893, -0.711368103958197, 
-0.64608803628934, -0.576583643383317, -0.389788923647592, -0.219325408944189, 
-0.101244293663156, 0, 0.101244293663156, 0.219325408944189, 
0.389788923647592, 0.576583643383317, 0.64608803628934, 0.711368103958197, 
0.831512873094893, 0.880770453063485, 0.990331188483095, 1.09475471308712, 
1.21629813646533, 1.33166687876378, 1.46043531083332, 1.54786821816088, 
1.64375650300888, 2.07966723692233, 2.26369776447253, 2.41796583370138, 
2.49408041291093, 2.57066623804085, 2.72134261347931, 3.00388455382712, 
3.13898393694403, 3.21096198541142, 3.32461379011727, 3.59372660759739, 
3.6640925546724, 3.77408485453713, 3.833076928421, 3.99361363184664, 
4.33123842894087, 4.46914150055587, 4.69028336184037, 4.87543206990054, 
5.04972087076752, 5.19600239869575, 5.5215432443443, 5.93115698481963, 
6.21582362929168, 6.44634841642492, 6.60931905266524, 7.28202887686486, 
7.67303749757961, 8.10018637873416, 8.47030492676585, 10.2570925911286, 
11.3210853809037, 12.5961340486959, 12.8877947524831, -12.6000848203634, 
-12.2966556390103, -11.1957507602953, -10.1862843945321, -8.30440651819497, 
-7.93147284818737, -7.50817793772083, -7.13162491301438, -6.48897104737938, 
-6.36994876479492, -6.15110072929745, -5.85854042138457, -5.43365721329602, 
-5.09102141962306, -4.95085939235108, -4.78689576531572, -4.60547103236952, 
-4.37919276705198, -4.26474198682156, -3.93194755639431, -3.77843705689023, 
-3.70907277842741, -3.61955988253518, -3.55951762156004, -3.27305364378077, 
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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))

In each column I want to integrate using the function integrate R, and place the value of this integral in the matrix integrais. The problem I find is the mistake:

Error in integrate(integrando.funcao.t, lower = 0.01, upper = 0.5, subdivisions = 1000) : 
  extremely bad integrand behaviour

integrais=matrix(0,11)

for (t in 1:11){


integrando.funcao.t = approxfun(thau,funcao[,t],rule=2)

integrais[t,1]=integrate(integrando.funcao.t, lower = 0.01, upper = 0.5,subdivisions = 1000)$value

}

There is a way around this problem with the same function integrate? Or will I need to approach in a non-linear way? If that’s the case how do I?

Thank you

Answer 1

The function integrate does not use to compute integrals this way. To specifically use this function, you need to write the function to be integrated into R. In your case, as the function values are already calculated, what you should do is use something like Simpson’s Formula, as shown in this answer right here at Stackoverflow.

I plotted the contents of the object funcao and I got the following:

matplot(funcao, type="l", col="black", lty=1)

Given your history of questions around here, these 11 functions each have quartile functions of normal distribution. Ignore the x-axis numbers of this graph, as they are only showing the position of the values within the data frame and therefore vary between 1 and 99. Looking at this figure, I would say that integrals are all equal to zero. To test this, I’ve assumed two hypotheses that are not explicit in your original question:

1. The step between each line of the object funcao is constant, that is, the function in question has been calculated at equidistant points

2. As I am suspicious that it is a quantile function, it should not be calculated between 0.01 and 0.50, but between 0.01 and 0.99

My answer is based on these two hypotheses and I may be mistaken. If I am, please correct me.

# pacote para calcular as integrais pela Fórmula de Simpson
library(Bolstad) 

# vetor variando entre 0,01 e 0,99, com tamanho igual ao numero
# de linhas da matriz funcao
x <- seq(from=0.01, to=0.99, length.out=dim(funcao)[1])

# inicializa o vetor para guardar os resultados das integracoes
integrais <- 0

# loop para calcular um numero de integrais igual ao numero de 
# colunas da matriz funcao
# utilizo j como contador porque t eh uma funcao do R e nao eh
# interessante utilizar palavras reservadas da linguagem
# como nome de objetos
for (j in 1:dim(funcao)[2]){
    integrais[j] <- sintegral(x, funcao[, j])$value
}

# resultado
integrais
 [1] -5.149960e-16 -5.657367e-16 -5.128276e-16 -4.902762e-16
 [5] -5.373306e-16 -5.893723e-16 -5.306085e-16 -5.205255e-16
 [9] -5.434021e-16 -5.186823e-16 -5.606409e-16

As I suspected, all integrals were equal to zero.

One important point to be made is the divergence of this function. It is not set to 0 and 1. Therefore, if you are interested in calculating the integral between 0.01 and 0.50, you will get a value. If you calculate between 0.001 and 0.500, you will get an even higher value. If you calculate between 0.0001 and 0.5000, you will get an even higher value. The closer the function gets to its limits at 0 and 1, the more it will diverge.