Sage peut interagir avec R pour faire des statistiques
Chaque copie de sage possède une copie de R, un logiciel de traitements statistiques open-source puissant et reconnu par la communauté mathématique. Il est possible d'utiliser R directement dans une cellule du Notebook:
%r
x <- c(1,3,5,6,4,9,9,1,1,6,7,8,4,5,5)
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Bon, je ne suis pas un statisticien et ne fait pas de la statistique de haut vol ici. On calcule la moyenne, la varience, etc, directement en utilisant R:
%r
mean(x)
[1] 4.933333 [1] 4.933333 |
%r
var(x)
[1] 7.209524 [1] 7.209524 |
%r
summary(x)
Min. 1st Qu. Median Mean 3rd Qu. Max. 1.000 3.500 5.000 4.933 6.500 9.000 Min. 1st Qu. Median Mean 3rd Qu. Max. 1.000 3.500 5.000 4.933 6.500 9.000 |
On fait ci-dessous un test-T disant que la moyenne de la population est de 5:
%r
results = t.test(x, mu=5)
results
One Sample t-test data: x t = -0.0962, df = 14, p-value = 0.9248 alternative hypothesis: true mean is not equal to 5 95 percent confidence interval: 3.446399 6.420268 sample estimates: mean of x 4.933333 One Sample t-test data: x t = -0.0962, df = 14, p-value = 0.9248 alternative hypothesis: true mean is not equal to 5 95 percent confidence interval: 3.446399 6.420268 sample estimates: mean of x 4.933333 |
%r
results['statistic']
$statistic
t
-0.09616147
$statistic
t
-0.09616147
|
%r
results['p.value']
$p.value [1] 0.9247553 $p.value [1] 0.9247553 |
Intégrer R et Sage
Nous avons vu ci-dessus qu'il était possible d'utiliser l'interpréteur de R directement depuis une cellule de Sage. Il est également possible de faire interagir plus étroitement R et Sage:
x = r([1,3,5,6,4,9,9,1,1,6,7,8,4,5,5])
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r.mean(x)
[1] 4.933333 [1] 4.933333 |
x.var()
[1] 7.209524 [1] 7.209524 |
r.summary(x)
Min. 1st Qu. Median Mean 3rd Qu. Max. 1.000 3.500 5.000 4.933 6.500 9.000 Min. 1st Qu. Median Mean 3rd Qu. Max. 1.000 3.500 5.000 4.933 6.500 9.000 |
On fait ci-dessous un test-T disant que la moyenne de la population est de 5:
results = r.t_test(x, mu=5)
results
One Sample t-test data: sage25 t = -0.0962, df = 14, p-value = 0.9248 alternative hypothesis: true mean is not equal to 5 95 percent confidence interval: 3.446399 6.420268 sample estimates: mean of x 4.933333 One Sample t-test data: sage25 t = -0.0962, df = 14, p-value = 0.9248 alternative hypothesis: true mean is not equal to 5 95 percent confidence interval: 3.446399 6.420268 sample estimates: mean of x 4.933333 |
On peut convertir le résultat du test en une structure de données Python (un dictionnaire) pour en explorer l'infornation de manière plus flexible:
la variable testresult est maintenant un dictionnaire Python qu'il est possible d'utiliser comme n'importe quelle structure de donnée de ce langage.
testresult = results._sage_()
testresult
{'_r_class': 'htest', '_Names': ['statistic', 'parameter', 'p.value',
'conf.int', 'estimate', 'null.value', 'alternative', 'method',
'data.name'], 'DATA': {'p_value': 0.92475529872458795, 'alternative':
'two.sided', 'data_name': 'sage25', 'null_value': {'_Names': 'mean',
'DATA': 5}, 'conf_int': {'conf_level': 0.94999999999999996, 'DATA':
[3.4463990798025899, 6.4202675868640702]}, 'statistic': {'_Names': 't',
'DATA': -0.096161467028551301}, 'estimate': {'_Names': 'mean of x',
'DATA': 4.93333333333333}, 'parameter': {'_Names': 'df', 'DATA': 14},
'method': 'One Sample t-test'}}
{'_r_class': 'htest', '_Names': ['statistic', 'parameter', 'p.value', 'conf.int', 'estimate', 'null.value', 'alternative', 'method', 'data.name'], 'DATA': {'p_value': 0.92475529872458795, 'alternative': 'two.sided', 'data_name': 'sage25', 'null_value': {'_Names': 'mean', 'DATA': 5}, 'conf_int': {'conf_level': 0.94999999999999996, 'DATA': [3.4463990798025899, 6.4202675868640702]}, 'statistic': {'_Names': 't', 'DATA': -0.096161467028551301}, 'estimate': {'_Names': 'mean of x', 'DATA': 4.93333333333333}, 'parameter': {'_Names': 'df', 'DATA': 14}, 'method': 'One Sample t-test'}}
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print 'p-value is: ', testresult['DATA']['p_value']
print 'Statistic: %s = %s' % (testresult['DATA']['statistic']['_Names'], testresult['DATA']['statistic']['DATA'])
print '%s%% C.I.: %s' % (round(100*testresult['DATA']['conf_int']['conf_level']),testresult['DATA']['conf_int']['DATA'])
p-value is: 0.924755298725 Statistic: t = -0.0961614670286 95.0% C.I.: [3.4463990798025899, 6.4202675868640702] p-value is: 0.924755298725 Statistic: t = -0.0961614670286 95.0% C.I.: [3.4463990798025899, 6.4202675868640702] |
Il est possible d'utiliser R pour la simulation et Python pour reste:
En R, rnorm(200, 5, 2) retourne un vecteur de 200 valeurs tirées d'une distribution gaussienne de moyenne 5 et d'écart-type 2.
r_norm = r.rnorm(200, 5, 2)
r_norm
[1] 2.932860130850042 7.019686630221097 -1.255683824281048 1.284682617292919 4.687406533398461 [6] 4.332701977430219 6.751191342748030 7.121697101716489 5.926438605074823 6.334666794362983 [11] 4.635204545604861 4.776158262740585 9.773172896202670 6.206230354875271 5.524138794796816 [16] 1.993938158183552 4.484702257245935 7.071366381243096 3.619284759614675 8.077753304874742 [21] 6.218972984020207 5.247320252027164 2.516019119684823 4.033597052001843 4.958893571987001 [26] 3.403126682206814 5.655490158917621 7.271838898286782 3.765106457817799 5.451735091594872 [31] 4.176481608503499 7.877050770942954 2.307803569500903 7.569826944999753 10.223064206614417 [36] 4.901327685507820 1.935225870705978 3.283511904922625 8.360971478418904 3.976855878787579 [41] 5.521794389631239 4.581571444603891 4.467008550497113 4.778494557810026 7.692700581136485 [46] 5.594038459986956 4.519053718942827 5.832301491611340 5.731401564383916 4.967714648649425 [51] 1.811666211753191 7.904763247132804 5.693298633222565 2.768379816491648 8.478469517126006 [56] 8.072145718690999 7.630006618541160 4.429098915431492 6.163584011128266 5.380622971635888 [61] 7.100721326806628 4.966086824247173 8.130760593831180 4.815648504831355 1.416877381503559 [66] 1.513040533976527 6.125241602775927 5.612790766660552 3.558414126656045 9.511485767492793 [71] 5.851324157742610 5.548342694056521 5.134226798241416 5.266050559599313 4.777542568985157 [76] 6.022256387936848 7.154804013798771 4.140771445791568 4.067403256376745 4.328932794482435 [81] 1.823695047721291 8.064729826025323 4.055429211775283 5.236005540210176 3.471364825012341 [86] 6.457393789264209 5.494824775152410 5.381001231764175 5.130996327173515 7.639242191276614 [91] 10.468845614687545 2.375220080031704 4.673200549235761 8.753786085379827 5.087739130539235 [96] 6.062131594853822 5.080768895902694 1.974211096686897 3.756011504462642 3.493085036373910 [101] 4.743451506993039 5.706417100711520 7.387707487817293 6.839882171968608 6.362991001974821 [106] 4.090167877166839 5.692063290995042 6.639155776705820 4.767580327585668 1.662674019718402 [111] 5.068210944505697 7.521537181860581 1.915789731760151 2.831255966622024 6.586507291395144 [116] 3.604936102650397 4.640749546432859 7.029542313314467 5.580951812457584 7.364958194718141 [121] 4.612703264794814 5.411059149403209 2.181587108537966 4.002556772307646 3.071879648808284 [126] 8.058382122859093 5.606853605478570 5.699699081021991 4.887637035308696 7.031255412229200 [131] 5.428258790807652 5.858763520186419 4.250640895750221 0.648944394495954 5.159199420369405 [136] 3.447743920927558 7.677861778304395 7.101381926107556 1.820231162733710 5.430085092929261 [141] 3.341601603761124 5.429044915448719 9.281791436631117 6.352536931571318 5.342454436616697 [146] 2.739368600097829 3.559195797131109 2.256956048268007 1.527824244197628 5.782138357492101 [151] 8.736381337621978 3.968425946949193 3.917634206736804 6.117988411933440 4.286253894093873 [156] 5.204720547740065 3.064516806581130 4.748036574912142 6.934419375508805 6.966185036662761 [161] 5.946017440217970 4.976883109005738 5.783788335010193 5.706222430681183 3.513150539961214 [166] 7.343510303916510 2.689026331281138 6.823415881985600 4.267755832702060 8.376560923943423 [171] 7.948308133473203 3.004540163354743 2.933666812774234 7.543954036235458 4.955392781847880 [176] 4.307077576940843 5.507427923278904 4.804893580967775 5.779556518636309 6.465924338187170 [181] 7.910016166193128 8.749501642241245 9.126545434210815 3.237828819836731 3.572230427006931 [186] 5.663311542622086 5.186940952932177 5.583700151939425 7.188544347397083 3.580407511612389 [191] 3.629980964356285 3.034240838345398 4.680429657849403 3.303505437948030 6.591872452393569 [196] 7.500828912379446 7.046754679519861 6.350760130949962 3.762547820866622 5.413958248299800 [1] 2.932860130850042 7.019686630221097 -1.255683824281048 1.284682617292919 4.687406533398461 [6] 4.332701977430219 6.751191342748030 7.121697101716489 5.926438605074823 6.334666794362983 [11] 4.635204545604861 4.776158262740585 9.773172896202670 6.206230354875271 5.524138794796816 [16] 1.993938158183552 4.484702257245935 7.071366381243096 3.619284759614675 8.077753304874742 [21] 6.218972984020207 5.247320252027164 2.516019119684823 4.033597052001843 4.958893571987001 [26] 3.403126682206814 5.655490158917621 7.271838898286782 3.765106457817799 5.451735091594872 [31] 4.176481608503499 7.877050770942954 2.307803569500903 7.569826944999753 10.223064206614417 [36] 4.901327685507820 1.935225870705978 3.283511904922625 8.360971478418904 3.976855878787579 [41] 5.521794389631239 4.581571444603891 4.467008550497113 4.778494557810026 7.692700581136485 [46] 5.594038459986956 4.519053718942827 5.832301491611340 5.731401564383916 4.967714648649425 [51] 1.811666211753191 7.904763247132804 5.693298633222565 2.768379816491648 8.478469517126006 [56] 8.072145718690999 7.630006618541160 4.429098915431492 6.163584011128266 5.380622971635888 [61] 7.100721326806628 4.966086824247173 8.130760593831180 4.815648504831355 1.416877381503559 [66] 1.513040533976527 6.125241602775927 5.612790766660552 3.558414126656045 9.511485767492793 [71] 5.851324157742610 5.548342694056521 5.134226798241416 5.266050559599313 4.777542568985157 [76] 6.022256387936848 7.154804013798771 4.140771445791568 4.067403256376745 4.328932794482435 [81] 1.823695047721291 8.064729826025323 4.055429211775283 5.236005540210176 3.471364825012341 [86] 6.457393789264209 5.494824775152410 5.381001231764175 5.130996327173515 7.639242191276614 [91] 10.468845614687545 2.375220080031704 4.673200549235761 8.753786085379827 5.087739130539235 [96] 6.062131594853822 5.080768895902694 1.974211096686897 3.756011504462642 3.493085036373910 [101] 4.743451506993039 5.706417100711520 7.387707487817293 6.839882171968608 6.362991001974821 [106] 4.090167877166839 5.692063290995042 6.639155776705820 4.767580327585668 1.662674019718402 [111] 5.068210944505697 7.521537181860581 1.915789731760151 2.831255966622024 6.586507291395144 [116] 3.604936102650397 4.640749546432859 7.029542313314467 5.580951812457584 7.364958194718141 [121] 4.612703264794814 5.411059149403209 2.181587108537966 4.002556772307646 3.071879648808284 [126] 8.058382122859093 5.606853605478570 5.699699081021991 4.887637035308696 7.031255412229200 [131] 5.428258790807652 5.858763520186419 4.250640895750221 0.648944394495954 5.159199420369405 [136] 3.447743920927558 7.677861778304395 7.101381926107556 1.820231162733710 5.430085092929261 [141] 3.341601603761124 5.429044915448719 9.281791436631117 6.352536931571318 5.342454436616697 [146] 2.739368600097829 3.559195797131109 2.256956048268007 1.527824244197628 5.782138357492101 [151] 8.736381337621978 3.968425946949193 3.917634206736804 6.117988411933440 4.286253894093873 [156] 5.204720547740065 3.064516806581130 4.748036574912142 6.934419375508805 6.966185036662761 [161] 5.946017440217970 4.976883109005738 5.783788335010193 5.706222430681183 3.513150539961214 [166] 7.343510303916510 2.689026331281138 6.823415881985600 4.267755832702060 8.376560923943423 [171] 7.948308133473203 3.004540163354743 2.933666812774234 7.543954036235458 4.955392781847880 [176] 4.307077576940843 5.507427923278904 4.804893580967775 5.779556518636309 6.465924338187170 [181] 7.910016166193128 8.749501642241245 9.126545434210815 3.237828819836731 3.572230427006931 [186] 5.663311542622086 5.186940952932177 5.583700151939425 7.188544347397083 3.580407511612389 [191] 3.629980964356285 3.034240838345398 4.680429657849403 3.303505437948030 6.591872452393569 [196] 7.500828912379446 7.046754679519861 6.350760130949962 3.762547820866622 5.413958248299800 |
r_norm.summary()
Min. 1st Qu. Median Mean 3rd Qu. Max. -1.25568382428 3.97474839583 5.25668540581 5.24372022367 6.58784858164 10.46884561470 Min. 1st Qu. Median Mean 3rd Qu. Max. -1.25568382428 3.97474839583 5.25668540581 5.24372022367 6.58784858164 10.46884561470 |
Il est possible de récupérer les nombres aléatoires générés dans une structure de données python et, par exemple, de les trier avec les outils python prévus pour trier une liste
sage_norm = sorted(r_norm._sage_())
sage_norm
[-1.25568382428105, 0.64894439449595398, 1.28468261729292, 1.41687738150356, 1.51304053397653, 1.52782424419763, 1.6626740197184, 1.8116662117531901, 1.82023116273371, 1.8236950477212901, 1.91578973176015, 1.93522587070598, 1.9742110966869, 1.99393815818355, 2.1815871085379701, 2.2569560482680102, 2.3078035695009, 2.3752200800317, 2.5160191196848198, 2.68902633128114, 2.7393686000978299, 2.7683798164916502, 2.8312559666220198, 2.9328601308500399, 2.9336668127742298, 3.0045401633547399, 3.0342408383453998, 3.0645168065811301, 3.07187964880828, 3.23782881983673, 3.2835119049226198, 3.3035054379480302, 3.34160160376112, 3.40312668220681, 3.44774392092756, 3.4713648250123401, 3.4930850363739099, 3.51315053996121, 3.5584141266560398, 3.5591957971311099, 3.5722304270069301, 3.5804075116123899, 3.6049361026504001, 3.6192847596146702, 3.6299809643562901, 3.7560115044626401, 3.7625478208666201, 3.7651064578178, 3.9176342067368002, 3.96842594694919, 3.9768558787875801, 4.0025567723076501, 4.0335970520018396, 4.0554292117752802, 4.0674032563767497, 4.0901678771668397, 4.1407714457915699, 4.1764816085034999, 4.2506408957502204, 4.2677558327020604, 4.2862538940938704, 4.3070775769408396, 4.3289327944824301, 4.3327019774302196, 4.4290989154314904, 4.46700855049711, 4.4847022572459396, 4.5190537189428301, 4.58157144460389, 4.6127032647948099, 4.6352045456048598, 4.6407495464328603, 4.6732005492357596, 4.6804296578494, 4.6874065333984598, 4.7434515069930399, 4.74803657491214, 4.7675803275856703, 4.7761582627405801, 4.7775425689851598, 4.7784945578100304, 4.8048935809677804, 4.8156485048313504, 4.8876370353086998, 4.9013276855078196, 4.9553927818478796, 4.9588935719869998, 4.9660868242471699, 4.9677146486494204, 4.9768831090057404, 5.0682109445057, 5.0807688959026898, 5.0877391305392301, 5.1309963271735102, 5.1342267982414196, 5.1591994203693998, 5.1869409529321802, 5.2047205477400604, 5.2360055402101802, 5.2473202520271602, 5.2660505595993099, 5.3424544366167002, 5.3806229716358898, 5.3810012317641798, 5.4110591494032096, 5.4139582482998003, 5.4282587908076501, 5.4290449154487197, 5.4300850929292599, 5.4517350915948697, 5.4948247751524102, 5.5074279232789003, 5.52179438963124, 5.5241387947968201, 5.5483426940565197, 5.5809518124575801, 5.5837001519394196, 5.5940384599869599, 5.6068536054785696, 5.6127907666605497, 5.6554901589176199, 5.6633115426220897, 5.6920632909950397, 5.6932986332225699, 5.6996990810219899, 5.70622243068118, 5.7064171007115201, 5.73140156438392, 5.7795565186363103, 5.7821383574920997, 5.7837883350102004, 5.8323014916113403, 5.8513241577426101, 5.8587635201864199, 5.9264386050748197, 5.94601744021797, 6.0222563879368503, 6.0621315948538204, 6.1179884119334398, 6.1252416027759304, 6.16358401112827, 6.2062303548752702, 6.2189729840202101, 6.3346667943629802, 6.3507601309499604, 6.3525369315713203, 6.3629910019748204, 6.4573937892642101, 6.4659243381871701, 6.5865072913951401, 6.5918724523935701, 6.6391557767058202, 6.7511913427480303, 6.8234158819855999, 6.8398821719686103, 6.9344193755088002, 6.9661850366627602, 7.0196866302211003, 7.0295423133144697, 7.0312554122291999, 7.0467546795198599, 7.0713663812430996, 7.1007213268066298, 7.1013819261075604, 7.1216971017164896, 7.1548040137987696, 7.1885443473970803, 7.2718388982867799, 7.3435103039165099, 7.3649581947181399, 7.3877074878173001, 7.5008289123794496, 7.5215371818605803, 7.5439540362354602, 7.5698269449997504, 7.6300066185411604, 7.63924219127661, 7.6778617783043996, 7.6927005811364904, 7.87705077094295, 7.9047632471328004, 7.9100161661931301, 7.9483081334732004, 8.0583821228591006, 8.0647298260253208, 8.0721457186910008, 8.0777533048747401, 8.1307605938311802, 8.3609714784189002, 8.3765609239434191, 8.4784695171259994, 8.7363813376219799, 8.7495016422412508, 8.7537860853798293, 9.1265454342108097, 9.2817914366311207, 9.5114857674928004, 9.77317289620267, 10.223064206614399, 10.468845614687501] [-1.25568382428105, 0.64894439449595398, 1.28468261729292, 1.41687738150356, 1.51304053397653, 1.52782424419763, 1.6626740197184, 1.8116662117531901, 1.82023116273371, 1.8236950477212901, 1.91578973176015, 1.93522587070598, 1.9742110966869, 1.99393815818355, 2.1815871085379701, 2.2569560482680102, 2.3078035695009, 2.3752200800317, 2.5160191196848198, 2.68902633128114, 2.7393686000978299, 2.7683798164916502, 2.8312559666220198, 2.9328601308500399, 2.9336668127742298, 3.0045401633547399, 3.0342408383453998, 3.0645168065811301, 3.07187964880828, 3.23782881983673, 3.2835119049226198, 3.3035054379480302, 3.34160160376112, 3.40312668220681, 3.44774392092756, 3.4713648250123401, 3.4930850363739099, 3.51315053996121, 3.5584141266560398, 3.5591957971311099, 3.5722304270069301, 3.5804075116123899, 3.6049361026504001, 3.6192847596146702, 3.6299809643562901, 3.7560115044626401, 3.7625478208666201, 3.7651064578178, 3.9176342067368002, 3.96842594694919, 3.9768558787875801, 4.0025567723076501, 4.0335970520018396, 4.0554292117752802, 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On peut ensuite utiliser les possibilités de visualisation de Sage:
list_plot(sage_norm, xmin=-2)
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