Mahi Tohatoha Noa
Ko te mahi tohatoha noa he ariā taketake i roto i ngā tatauranga me te tūponotanga e whai wāhi nui ana i roto i ngā mara mai i te ōhanga ki ngā pūtaiao pāpori, te ahupūngao, me te hangarau. He mea nui tēnei mahi nā te mea e whakaahua ana i te tohatoha taiao o ngā āhuatanga tūturu maha. Ka matapakihia e tēnei tuhinga te mahi tohatoha noa, ōna āhuatanga, ōna tono, me pēhea te whakamahi i roto i te tātari raraunga.
He aha te Mahi Tohatoha Noa?
Ko te tohatoha noa, e mōhiotia ana ko te tohatoha Gaussian, te tohatoha tūponotanga tino noa i roto i ngā tatauranga. He āhua pere tōna kōpiko, ā, he ōrite ki te toharite. Ka kiia tēnei tohatoha he 'noa' nā te mea he maha ngā taurangi matapōkere o te ao tūturu e tata ana ki tēnei tohatoha ina kohia he nui ngā raraunga.
E rua ngā tawhā e tautuhia ana te mahi tohatoha noa: te toharite (μ) me te paerewa rerekētanga (σ). Ko te toharite te mea e whakatau ana i te pokapū o te tohatoha, ko te paerewa rerekētanga te mea e whakatau ana i te whānui me te āhua o te pihi. Ka hoatu tēnei mahi e te tātai pāngarau e whai ake nei:
\[ f(x|\mu, \sigma) = \frac{1}{\sigma \sqrt{2\pi}} e^{ -\frac{(x-\mu)^2}{2\sigma^2}} \]
Kei hea:
– He taurangi matapōkere a \( x \)
– Ko te toharite te \( \mu \)
– Ko te paerewa rerekētanga ko \( \sigma \)
– Ko te \( \pi \) te pūmau pi (tata ki te 3.14159)
– Ko \( e \) te pūtake o te logarithm maori (tata ki te 2.71828)
Ngā Āhuatanga o te Tohatoha Noa
He maha ngā āhuatanga motuhake o te tohatoha noa e wehewehe ana i a ia mai i ētahi atu tohatoha tūponotanga:
1. Āhua ōrite: He āhua ōrite te tohatoha noa i te toharite. Ko te tikanga kei te taha mauī o te toharite te haurua o ngā raraunga, ā, kei te taha matau te haurua.
2. Te Tihi i te Toharite: He tihi tō te pihi i te toharite, ā, ka heke haere i te tere o te pikinga i a ia e neke atu ana i te toharite.
3. Te Tapeke Horahanga i raro i te Ānau: Ko te tapeke horahanga i raro i te ānau tohatoha noa ko te 1, e tohu ana i te tapeke tūponotanga.
4. Huinga Whakamātautau: Tata ki te 68% o ngā raraunga kei roto i te kotahi paerewa rerekētanga o te toharite, 95% kei roto i ngā paerewa rerekētanga e rua, me te 99.7% kei roto i ngā paerewa rerekētanga e toru. E mōhiotia ana tēnei ko te ture 68-95-99.7.
5. Korehere: Ko te pihi tohatoha noa koretake e whakatata atu ai ki te tuaka whakapae (tuaka-x), engari kāore e pā.
Ngā Taupānga Tohatoha Noa
He whānuitia ngā whakamahinga o te tohatoha noa i roto i ngā mara maha, anei ētahi tauira:
1. Ngā Pūtaiao Pāpori me te Hinengaro
I roto i ngā rangahau pāpori me te hinengaro, ka whakamahia te tohatoha noa hei whakaahua i te tohatoha o ngā uara pēnei i te IQ, ngā hua whakamātautau, me ngā āhuatanga o te tangata. Ko te whakaaro he maha ngā āhuatanga o te tangata he tohatoha noa, e āhei ai ngā kairangahau ki te mahi tātaritanga tatauranga.
2. Ōhanga me te Pūtea
I roto i ngā mākete ōhanga me ngā mākete pūtea, ka whakamahia te whakaaro tohatoha noa i roto i ngā tauira maha hei whakatau tata i te mōrearea, ngā hokinga mai o ngā haumitanga, me te piki me te heke o te mākete. Ko ngā tauira pēnei i te tauira Black-Scholes e whakamahi ana i te tohatoha noa ki ngā kōwhiringa utu me ētahi atu pānga pūtea.
3. Ngā Pūtaiao Taiao me te Hangarau
I roto i ngā pūtaiao taiao me te hangarau, ka whakamahia te tohatoha noa hei whakatauira i ngā hapa ine me ngā āhuatanga taiao. Hei tauira, i roto i te ahupūngao, ka whakamahia pinepine hei whakaahua i te nekehanga Brownian me te tohatoha o ngā hapa i roto i ngā whakamātautau.
4. Mana Kounga
I roto i te umanga whakanao, ka whakamahia te tohatoha noa mō ngā tikanga whakahaere kounga pēnei i ngā tūtohi whakahaere, hei āwhina i te whakatau mēnā kei roto i ngā rohe e whakaaetia ana te tukanga whakaputa, me whakarerekē rānei.
Te Whakatinanatanga o te Tohatoha Noa i roto i te Tātari Raraunga
Hei whakatinana i te tohatoha noa i roto i te tātari raraunga, me mārama koe ki te tatau i ngā tūponotanga me te whakarite paerewa i ngā raraunga. Mā te whakarite paerewa ka āwhina i te whakataurite i ngā raraunga mai i ngā tohatoha noa rerekē.
Tohu-Z
Ko te kaute-Z, te paerewa rerekētanga rānei, he tikanga e whakamahia ana hei ine i te tawhiti o te uara (x) mai i te toharite i roto i ngā waeine paerewa rerekētanga. Ka tatauhia te kaute-Z mā te whakamahi i te tātai:
\[ Z = \frac{(X – \mu)}{\sigma} \]
Ka taea te whakamahi i tēnei uara kaute-Z hei kimi i te tūponotanga, i te ōrau rānei o tētahi tohatoha noa paerewa (toharite=0 me te paerewa rerekētanga=1) mā te whakamahi i tētahi ripanga Z, i tētahi pūmanawa tatauranga rānei.
Kaupapa QQ
He taputapu whakairoiro te kauwhata QQ (Quantile-Quantile) hei aromatawai mēnā kei te whai tetahi huinga raraunga i tētahi tohatoha motuhake, pērā i te tohatoha noa. Mēnā kei te whai ngā raraunga i tētahi tohatoha noa, ka hanga he rārangi tika ngā pūwāhi i runga i te kauwhata QQ.
Whakamātautau Auautanga
He maha ngā whakamātautau tatauranga ka taea te whakamahi hei tirotiro i te ōritetanga o ngā raraunga, tae atu ki te whakamātautau Kolmogorov-Smirnov, te whakamātautau Shapiro-Wilk, me te whakamātautau Anderson-Darling. Mā ēnei whakamātautau ka taea te whakatau mēnā kei te whai ngā raraunga taupori taketake i tētahi tohatoha ōrite.
Tauira Tuturu
Hei whakamārama i te whakatinanatanga o te tohatoha noa, me whakaaro tātou ki tētahi tauira o te ao tūturu. Me kī he raraunga kaute whakamātautau tā tātou mai i te 1000 ākonga me te toharite o te 70 me te paerewa rerekētanga o te 10.
Te Tātai i te Tūponotanga
E hiahia ana mātou ki te mōhio ki te tūponotanga ka whiwhi tētahi ākonga i tētahi kaute i waenga i te 60 me te 80. Tuatahi, ka tatauhia e mātou te kaute-Z mō te 60 me te 80.
\[ Z_{60} = \frac{60 – 70}{10} = -1 \]
\[ Z_{80} = \frac{80 – 70}{10} = 1 \]
Mā te whakamahi i te ripanga-Z, ka kitea ko te tūponotanga o Z <= -1 he 0.1587, ā, ko te tūponotanga o Z <= 1 he 0.8413. Hei kimi i te tūponotanga o te kaute i waenga i te 60 me te 80, ka tangohia e mātou ēnei tūponotanga e rua: \[ P(60 <= X <= 80) = P(Z <= 1) - P(Z <= -1) = 0.8413 - 0.1587 = 0.6826 \] Nō reira, ko te tūponotanga o te whiwhi kaute a te ākonga i waenga i te 60 me te 80 he tata ki te 68.26%. Te Waihanga i tētahi Kauwhata QQ Mā te whakamahi i ngā pūmanawa tatauranga pēnei i a R, Python rānei, ka taea e tātou te waihanga i tētahi kauwhata QQ mō ā tātou raraunga kaute whakamātautau. Mena ka whai ā tātou raraunga kaute whakamātautau tūturu i tētahi tohatoha noa, ka hanga he rārangi tika ngā pūwāhi i runga i te kauwhata. Te Whakamātautau mō te Noa Hei whakamutunga, ka taea e tātou te whakahaere i te Whakamātautau Shapiro-Wilk hei tirotiro mena ka whai ā tātou raraunga kaute whakamātautau i tētahi tohatoha noa. Mā te whakamahi i ngā pūmanawa tatauranga, ka taea e tātou te whiwhi ngāwari i te uara-p mō tēnei whakamātautau. Mena he nui ake te uara-p i te taumata hiranga (ko te tikanga he 0.05), ka kore e taea e tātou te whakakore i te whakapae kore e kī ana he noa ā tātou raraunga. Whakatau Ko te mahi tohatoha noa te pou tuarā o te tātari tatauranga, ā, he whānui ngā tono puta noa i ngā momo marautanga. Ko ngā āhuatanga matua o te tohatoha—te ōritetanga, te tihi ki te toharite, me te ture 68-95-99.7—he mea pai mō ngā momo tātari raraunga maha. Mā te mārama ki te mahi a te tohatoha noa me te pehea e whakatinana ai i roto i te tātari raraunga, ka taea e ngā kairangahau me ngā tohunga te whakatau i ngā whakatau pakari me te tika mai i ā rātou raraunga. Ahakoa he matapae i ngā mōrearea pūtea, he aromatawai i ngā āhuatanga hinengaro, he whakahaere rānei i te kounga hanga, ka tukuna e te tohatoha noa he anga kaha mō te mārama me te tātari i te rerekētanga kei roto i ngā pūnaha uaua.
