Ungakhetha Kanjani Ngokwezibalo Imigqomo Elungile Ye-Histogram Yakho

Wake wazibuza ukuthi ungayikhetha kanjani imigqomo yakho ku-histogram? Wake wazibuza ukuthi ngabe zikhona yini izizathu ezijulile zokukhetha okudlula lokho okubukeka kukuhle nje? Nakuba ama-histogram eyithuluzi elibaluleke kakhulu lokubukwa kwedatha, ukusetha ukulungiswa kwawo kubalulekile, ikakhulukazi uma i-histogram ngokwayo isetshenziselwa ukuhlaziya okwengeziwe. Ama-histograms ajwayele ukwenziwa ikhompuyutha ukuze ubone ngeso lengqondo ukuminyana kwedatha. Kulokhu okuthunyelwe, sihlola izibalo zokulingana kwabantu, ikakhulukazi sibheka ukuthi imigqomo kufanele inciphe kanjani njengoba idathasethi yethu ikhula. Sigqugquzelwe izinkambu eziseduze njenge-perturbation theory ku-physics kanye nokwandiswa kukaTaylor kwizibalo, sizothola indlela eqinile yokwakha ukuminyana.

Zonke izithombe ngezombhali

Ingemuva

Izilinganiso

I-intuition ilula: uma unedatha eyengeziwe, imininingwane eyengeziwe okufanele ukwazi ukuyibona. Uma ubheka isampula lokuphawuliwe okuyishumi, imigqomo emibili noma emithathu ebanzi cishe iyonke ongakwazi ukuyikhokhela ngaphambi kokuba ukubona ngeso lakho kube iqoqo elincane lezikhala ezingenalutho. Kodwa uma unombono oyizigidi eziyishumi, leyo migqomo ebanzi iqala ukuzwakala njengesithombe esinokulungiswa okuphansi. Ufuna “ukusondeza” ngokwandisa inani lemigqomo. Nokho, umbuzo uwukuthi: Kufanele sisilinganise kanjani ngempela lesi sinqumo?

Ku-physics, lapho sibhekene nesistimu eyinkimbinkimbi kakhulu ukuxazulula ngokuqondile, sivame ukuphendukela kuyo I-Perturbation Theory. Ku-Quantum Electrodynamics (QED), ngokwesibonelo, silinganisela ukusebenzisana okuyinkimbinkimbi ngokukunweba ngokuya nge-coupling constant encane, njengokushaja kwe-electron. e. Lawa “mandla okusebenzelana” ahlinzeka ngokulandelana kwemvelo kwezilinganiso zethu. Kodwa nge-histogram, iyini “inkokhelo” efaniswayo? Ingabe ikhona ipharamitha eyisisekelo elawula ukusebenzisana phakathi kwamaphoyinti ethu edatha ahlukene kanye nokusabalalisa okuyisisekelo esizama ukukulinganisela?

Izibalo zinikeza enye indlela: i Ukwandiswa kukaTaylor. Uma sicabanga ukuthi umsebenzi wokuminyana oyisisekelo ubushelelezi ngokwanele (ukuhlaziya), singawuchaza endaweni sisebenzisa okuphuma kuwo. Lokhu kuzwakala njengokuhola okuthembisayo njengoba ama-oda aphezulu angaboniswa ukuthi anyamalale. Nakuba singase sifune ukwamukela umkhawulo ekusabalaliseni kokuhlaziya, akucaci ukuthi lokhu kuholela kanjani kusayizi othile womgqomo.

Kungenjalo, singase siphathe inkinga njengokwandiswa kweMisebenzi Yesisekelo. Njengoba nje singamelela umsebenzi oqhubekayo ohlakaniphe ngocezu sisebenzisa i-Fourier transform noma i-Legender polynomials, singabuka imigqomo ye-histogram njengesethi yemisebenzi eyisisekelo. Ngokusebenzisa indlela enjalo singalinganisa umsebenzi ngokuya nge-L2. Kodwa le ndlela yethula isethi yayo yezithiyo. Siwabala kanjani ama-coefficient ale misebenzi ngempumelelo? Futhi okubaluleke kakhulu, sizanelisa kanjani izithiyo ezingokomzimba zamathuba omsebenzi wokuminyana? Ngokungafani nochungechunge olujwayelekile lwe-Fourier, umsebenzi wokuminyana kufanele ube uphozithivu ngokuqinile futhi ube ojwayelekile ube kowodwa. Sizobona kulokhu okulandelayo ukuthi indlela etholwe kuthiyori yolwazi inezici ezifanayo ekwandiseni imisebenzi yesisekelo.

Ulwazi Theory

Ezangaphambili & Ezingemuva

Ukuze uthole isingeniso sezibalo zeBayesian noma ithiyori yolwazi, umfundi ubhekiselwa kuye (Murphy, 2022). Ngendlela yaseBayesi, imodeli $P(X|theta)$lapho X yizinto ezibonakalayo esifuna ukumodela futhi $theta$ amapharamitha ethu, futhi iqukethe ukusatshalaliswa kwangaphambili 𝑃(𝜃|ℳ) okubonisa inkolelo yethu ekusabalaliseni ngaphambi kokuba idatha ibonwe. Ngemva kokuba idatha isibonwe, singalinganisela ukusatshalaliswa kwangemuva $P(theta | X)$

𝑃(𝜃|𝑋) = 𝑃(𝑋|𝜃)𝑃(𝜃|ℳ)/𝑃(𝑋)

Le nqubo inhle ngokwezibalo ngoba iphephile ngo-100% ekufakweni ngokweqile. Nokho, idinga isiyalo esiqinile: asivunyelwe ukukhetha imodeli yethu noma ngaphambi kwalokho ngemva kokubona idatha. Uma sisebenzisa idatha ukuze sinqume ukuthi yisiphi isakhiwo semodeli okufanele sisetshenziswe, siphula umqondo oyisisekelo we-inference.

Imodeli okungenzeka kakhulu inikezwe idatha iqhathaniswa nemodeli yesisindo

Ikhwalithi yemodeli ingabalwa ngekhompyutha ngokucabangela ukumangala kwayo (bona isb. (Vries, 2026))

log 𝑃(𝑋|ℳ) = −surprisal = ukunemba – inkimbinkimbi

Amamodeli anenani elidlulele lamapharamitha (ngoba umuntu angase alingeke ukuba afake lonke uhlobo lokusebenzelana okucatshangelwayo) angase azuze ukunemba okumangalisayo, kodwa “abulawa” isijeziso sobunzima bawo. Imodeli efanelekile ayiyona enemininingwane eminingi; iyona ethwebula ulwazi oluningi ngenani elincane lomthwalo ongadingekile.

Lapho ucubungula isethi yamamodeli, umuntu angakwazi ukubala amathuba emodeli ngayinye ngokuqhathanisa namamodeli acatshangelwayo.

𝑃(ℳ𝑖 ∣ 𝑋) ~ 𝑃(𝑋 | ℳ𝑖) 𝑃(ℳ𝑖)

Kuyalinga ukuvele ukhethe imodeli ngamathuba aphezulu bese uqhubekela phambili. Kodwa le ndlela “yokuwina ithatha konke” inezingozi:

• Ukushintshashintsha Kwezibalo: Idatha 𝑋 ingase iqukathe i-fluke engahleliwe eyenza imodeli ephansi kakhulu ibukeke iphakeme okwesikhashana.
• Isisindo Sesixuku: Kwesinye isikhathi, isamba samamodeli “amathuba amancane” amaningi empeleni siyawadlula amathuba emodeli eyodwa “engcono kakhulu”.

Ngenxa yalokhu, indlela eqine kakhudlwana iwukuthwala wonke amamodeli aye phambili, uwalinganise ngokwamathuba awo. Kubalulekile ukuqaphela ukuthi lokhu akuyona “ingxube” yamaqiniso ahlukene; sisacabanga ukuthi imodeli eyodwa kuphela eyiqiniso, kodwa sisebenzisa ukusabalalisa okugcwele kwamathuba ukuze siphendule ukungaqiniseki kwethu.

Ukuminyana

Ukuminyana kusetshenziswa indlela ye-Bayesian

Ukuphatha ukuminyana njengemodeli esemthethweni, sibuka imigqomo yayo ngayinye engu-𝐾 njengepharamitha. Ngokuqondile, sinikeza isisindo $w_k$ kumgqomo ngamunye, okubonisa ithuba lephoyinti ledatha eliwela kuleso sikhawu. Ngoba inani eliphelele lamathuba kufanele lifinyele kokukodwa ($sum_k w_k=1$), ukuminyana okunemigqomo engu-𝐾 kuchazwa ngamapharamitha angu-𝐾 −1 azimele, amamodeli anjalo abizwa nangokuthi izingxube. Ohlakeni lwethu lwaseBayesia, sidinga ukwabela ngaphambi kwalezi zisindo. Uma kubhekwa ukuthi sibhekene nezilinganiso zesigaba okufanele zifinyezwe kokukodwa, ukusatshalaliswa kwe-Dirichlet ukukhetha okungokwemvelo ngokwezibalo.

Ukukhetha ama-hyperparameters

Ukusatshalaliswa kwe-Dirichlet kulawulwa ama-hyperparameter, avame ukuchazwa ngokuthi 𝛼. Lawa manani amele “izibalo-mbumbulu” zethu – empeleni lokho esikholelwa ukuthi ukuminyana kubukeka kanjani ngaphambi kwethu
baye babona iphuzu lokuqala ledatha. Uma sithatha isikhathi esiqondile (lapho ubufakazi 𝑃(𝑋) bungashintshi), amasu amabili ayinhloko avelayo okukhetha 𝛼:

• 𝛼 =1/𝐾 (Inketho Encane): Lokhu kuvame ukusetshenziswa lapho silindele ukuthi idatha igxiliswe kakhulu. Kuthatha njengokubalulekile ukuthi iningi lemigqomo izobe ingenalutho, okwenza kube “ukukhangisa okumbalwa” ngaphambili.
• 𝛼 =1 (Ukukhetha Okufanayo): Eyaziwa nangokuthi ifulethi noma i-Laplace ngaphambili, lokhu kucabangela ukuthi konke ukusabalalisa okungaba khona kwezisindo kungenzeka ngokulinganayo. Ingeza okubonwayo “okubonakalayo” kuwo wonke umgqomo ngaphambi kokuba idatha yangempela ifike.

Ngenhloso yokwakha ukuminyana okujwayelekile, ukukhetha kwesibili 𝛼 = 1 kuvame ukuba yimvelo kakhulu. Ibonisa indawo yokuqala engathathi hlangothi lapho sicabanga ukuthi idatha isatshalaliswa ngokulinganayo kuso sonke isikhathi kuze kube yilapho ubufakazi bufakazela okunye.

Ngokuchaza imigqomo yethu ngale ndlela, siguqule “i-pixelation” ye-density yaba imodeli eqinile. Manje sinesethi emisiwe yamapharamitha (𝐾 − 1 izisindo) kanye nokwangaphambili okucacile (𝛼 = 1). Isinyathelo esilandelayo siwukusebenzisa idatha ukuze unqume inani eliphelele lemigqomo 𝐾 ngokulinganisa ukunemba kokulingana ngokumelene nobunkimbinkimbi bemingcele.

Isibonelo

Sicela ubheke idatha esithombeni esingezansi:

Uma sihambisana nemigqomo engu-8 sithola:

Umuntu angakubona kulokhu kuminyana ukuthi umgqomo ongakwesokudla ungaphezulu kukaziro nakuba kungekho amaphuzu edatha abekhona kulo mgqomo. Lokhu kuwumphumela wendlela ye-Bayesian elinganisela ukuminyana okukholelwayo ngokusekelwe enkolelweni yethu yangaphambili kanye nedatha esiyibonile.

Ngokufingqa, sithole ukuminyana sisebenzisa indlela yaseBayesian. Sichaze i-𝑃(𝜃) yangaphambili ebonisa ukulindela kwethu ukuminyana okufanayo. Sabe sesithatha idatha futhi sahlanganisa ingemuva 𝑃(𝜃|𝑋) elingaphansi kokuminyana okuwumphumela.

Ukuminyana okunesisindo

Ngokusebenzisa indlela yesigaba esidlule singenza ukuminyana sisebenzisa imigqomo engu-1, 2, 4, 8, 16, 32, 64, 128, 256, 512, kanye ne-1024. Imigqomo eminingi inikeza idatha enembe kakhudlwana kodwa futhi yethula izinto eziyinkimbinkimbi ezengeziwe. Njengoba kuxoxwe ngakho esigabeni esandulele, umuntu angasebenzisa ukunemba nobunkimbinkimbi ukubala ubufakazi bayo. Uma sibuka ukuminyana ngakunye njengemodeli, singabala ukuthi kungenzeka kube yiqiniso uma kuqhathaniswa nesethi yamamodeli esiwacabangelayo. Lokhu kuveza isibalo esingezansi:

Esigabeni esandulele kuxoxiwe ngaso ukuthi umuntu angakhetha imodeli “engcono kakhulu” kulokhu okungaba ukusetshenziswa kwemigqomo eyi-8. Nokho, kuphephe kakhudlwana ukuthatha isamba esinesisindo phezu kwawo wonke amamodeli. Lokhu
isivuno:

Kubalulekile ukuqaphela ukuthi ngokombono waseBayesia lokhu kungcono kakhulu esingakwenza. Qaphela futhi ukuthi kule grafu kukhona ukuminyana okukhona kwemigqomo eyi-1024. Okokugcina, umuntu angafakazela ukuthi ukuminyana kwama-oda aphezulu N kuzoncipha.

Ukuminyana ngemigqomo engalingani

Ukuminyana okutholwe ngenhla ngenhla kubukeka kuyi-blocky kancane okuvela ekukhetheni ukusebenzisa imigqomo elinganayo. Kukhona ezinye izinketho ezitholakalayo njengokuthatha ukuhlukaniswa okungahleliwe (nokunxephezela okwangaphambili kwakho). Lokhu kuveza igrafu engezansi:

Ukuminyana okunamabha wamaphutha

Manje ukuvala ukwakhiwa kokuminyana, kungase kube nentshisekelo ukubona ngeso lengqondo ukungaqiniseki kwethu kulokhu kuminyana. Nakuba kubiza ngokwezinombolo ukubala, isisho sokwenza ikhompuyutha ukuchezuka okujwayelekile kokuminyana siqonde ngokuphawulekayo (F. Pijlman, 2023)

$sigma_{P(x|X)}^2 = P(x|X) kwesokunxele( P(x|{x,X}) – P(x|X) kwesokudla)$

Lokhu kuveza ukuminyana okulandelayo:

Iziphetho

Siqale ngombuzo olula: Ingabe sikhona isisekelo sezibalo sokukhetha imigqomo ku-histogram? Njengoba umqondo wemigqomo uxhumanisa amaphuzu edatha nokuminyana, sifunde ukuthi kanjani
ukukhetha imigqomo yokuminyana.

Ukusebenzisa indlela ye-Bayesian (ithiyori yolwazi) umuntu angakwazi ukuhlanganisa ukuminyana ngaphandle kokukhathazeka ngokufakwa ngokweqile (imigqomo eminingi ebonisa imininingwane eminingi). Nakuba umuntu engakwazi ukubala “okungcono kakhulu” bin-width, sibonile ukuthi:

• Ukukala amamodeli kusivumela ukuthi sihlanganise izinqumo eziningi, sinikeze ukumelwa okushelelayo nokwethembeka kwedatha.
• I-Dirichlet Priors isinika indlela eqinile yokuveza imibono yethu yokuqala mayelana nokusabalalisa idatha.

Njengoba nje ithiyori yokuphazamisa ihlinzeka ngohlelo lokusebenzelana ngokomzimba, lolu hlaka lwase-Bayesian luhlinzeka ngesigaba sokulandelana sokulungiswa kwedatha. Ukulungiswa kukala ngokwemvelo njengoba idatha eyengeziwe itholakala. Qaphela ukuthi imibono enjalo ingasetshenziswa futhi lapho kufundwa amamodeli lapho umuntu enokwanda kokuxhumana.

Indlela yokuhlanganisa ukuminyana kwezinqumo ezihlukahlukene nayo yahlolisiswa uma kwenzeka kukhethwa imigqomo engahleliwe. Lokhu kuholele kuma-histograms abushelelezi angase abonakale engokwemvelo kudatha eminingi
amasethi.

Siphinde sethula ukusetshenziswa kokuchezuka okujwayelekile kuma-histogram. Nakuba ukubalwa kokuchezuka okujwayelekile kwatholakala kumamodeli e-Bayesian, inqubo yakho yokubala iphakamisa ukusebenza okubanzi. Kanjalo, kungaba okokubona ngeso lengqondo ukungaqiniseki okusele kokuminyana.

Ukubonga

Iphrojekthi ye-EdgeAI “Edge AI Technologies for Optimized Performance Embedded Processing” ithole uxhaso ku-Key Digital Technologies Joint Undertaking (KDT JU) ngaphansi kwesivumelwano soxhaso No. 101097300. I-KDT JU ithola ukwesekwa okuvela ohlelweni lwe-European Union's Horizon Europe, Austria, Latvia, France, Belgium, Belgium, Belgium kanye nohlelo lokusungula izinto ezintsha. eNetherlands, naseNorway.

Izithenjwa

• F. Pijlman, JL (2023). Ukuhlukahluka Kokwenzeka Kwedatha. 34/37.
• UMurphy, K. (2022). I-Probabilistic Machine Learning: Isingeniso. I-MIT Cindezela.
• Vries, B. d. (2026). I-Active Inference yama-Physical AI Agents. arXiv.

I-Bio

U-Fetze Pijlman unguSosayensi Oyinhloko kwa-Signify Research e-Eindhoven, e-Netherlands. Ukugxila kwakhe ocwaningweni kuhlanganisa ukufunda komshini okungenzeka kube khona, ukucabanga kwe-Bayesian, nokucubungula amasignali, enentshisekelo ethile ekusebenziseni lezi zinhlaka zezibalo ku-IoT, izinzwa, nezinhlelo ezihlakaniphile.