Uara e tumanakohia ana mō te Tohatoha Noa
Ko te tohatoha noa, e mōhiotia ana ko te tohatoha Gaussian, tētahi o ngā tohatoha tūponotanga tino taketake i roto i ngā tatauranga, ā, e whakamahia pinepinetia ana i roto i ngā momo mara pūtaiao, tae atu ki te ōhanga, te hinengaro, te ahupūngao, me te koiora. Ko tētahi o ngā ariā matua o te tohatoha noa ko te uara e tumanakohia ana (te toharite), he tawhā matua e whakaahua ana i te wāhi o te pokapū o te tohatoha. Ka matapakihia whānuitia e tēnei tuhinga te uara e tumanakohia ana o te tohatoha noa, tae atu ki tōna whakamāramatanga, ngā āhuatanga, me ngā tono i roto i ngā momo mara.
1. Te Mārama ki te Tohatoha Noa
Ko te tohatoha noa he tohatoha tūponotanga tonu, he āhua pere, he ōrite hoki ki te toharite. Mā te pāngarau, ka taea te whakaatu i te tohatoha noa mā te mahi kiato tūponotanga e whai ake nei (pdf):
\[ f(x | \mu, \sigma^2) = \frac{1}{\sqrt{2\pi\sigma^2}} \exp \left( -\frac{(x – \mu)^2}{2\sigma^2} \right) \]
Kei hea:
– He taurangi matapōkere a \( x \).
– Ko te uara e tumanakohia ana, ko te toharite rānei o te tohatoha ko \( \mu \).
– Ko te \( \sigma \) te paerewa rerekētanga o te tohatoha.
– Ko \( \sigma^2 \) te rerekētanga o te tohatoha.
E rua ngā tawhā matua o te tohatoha noa: te toharite (\(\mu\)) me te paerewa rerekētanga (\(\sigma\)). 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, i te horapa rānei o te tohatoha.
2. Uara e Tumanakohia ana (Toharite)
Ko te uara e tumanakohia ana, e mōhiotia ana ko te tumanako, o te tohatoha tūponotanga te whakaaro pai rawa atu mō te pokapū o te tohatoha, inā koa i roto i te horopaki o te tohatoha noa. Ko te uara e tumanakohia ana o te taurangi matapōkere \( X \) e tohatoha noa ana me te toharite \( \mu \) me te rerekētanga \( \sigma^2 \) ko \(\mu\).
I roto i te tikanga, ko te uara e tumanakohia ana o tētahi taurangi matapōkere tonu \( X \) me te mahi kiato tūponotanga \( f \) ka tautuhia penei:
\[ E[X] = \int_{-\infty}^{\infty} xf(x) dx \]
Mō te tohatoha noa, ko te tikanga o tēnei ko te uara toharite, ko te uara e tumanakohia ana rānei (\(\mu\)) ko te pūwāhi kei reira te pihi tohatoha i tōna pūwāhi teitei rawa, ā, he ōrite te tohatoha.
3. Ngā Āhuatanga o te Uara e Tumanakohia ana
He maha ngā āhuatanga nui o te uara e tumanakohia ana i roto i te tohatoha noa e whai hua ana mō te māramatanga hohonu me te tono mahi:
1. Te ōritetanga:
He ōrite tino tika te tohatoha noa ki te toharite \(\mu\). Ko te tikanga o tēnei ko te haurua o ngā raraunga kei te taha maui o te toharite, ā, ko te haurua kei te taha matau o te toharite.
2. Te Uara Toharite e Tumanakohia ana:
I roto i te tohatoha noa, ko te toharite (\(\mu\)) te uara e tumanakohia ana, e whakaata ana i te toharite o ngā uara katoa ka taea e te taurangi matapōkere.
3. Ngā Tauoti Tauwehe Raina:
Mena he taurangi matapōkere a \( X \) me te tohatoha noa \( N(\mu, \sigma^2) \), ā, he tau pūmau a \( a \) me \( b \), ko te uara e tumanakohia ana o te taurangi matapōkere raina \( Y = aX + b \) ko \( E[Y] = aE[X] + b \). Mō te tohatoha noa, ka puta ko \( E[Y] = a\mu + b \).
4. Te Tāpiri i ngā Taurangi Matapōkere:
Mena he taurangi matapōkere motuhake a \( X_1 \) me \( X_2 \) e rua e tohatoha noa ana, ko te tapeke \( X = X_1 + X_2 \) ka tohatoha noa hoki me te toharite \( \mu_X = \mu_1 + \mu_2 \) me te rerekētanga \( \sigma_X^2 = \sigma_1^2 + \sigma_2^2 \).
4. Te Whakamahinga o te Uara e Tumanakohia ana i roto i te Tohatoha Noa
He maha ngā whakamahinga o te uara e tumanakohia ana i roto i te tohatoha noa i te ao tūturu, tae atu ki ēnei e whai ake nei:
1. Pūtea:
I roto i te tātari pūtea, ka whakamahia te uara e tumanakohia ana hei whakatau tata i te hokinga mai o tētahi putea haumitanga. Hei tauira, mēnā he tohatoha noa te hokinga mai o tētahi rawa, ka taea te whakamahi i te toharite o taua tohatoha hei whakaahua i te hokinga mai toharite e tumanakohia ana.
2. Inihuarangatia:
Ka whakamahia e ngā kamupene inihua te uara e tumanakohia ana hei whakatau tata i ngā kereme ā muri ake nei i runga i ngā raraunga o mua. E whakaarohia ana te tohatoha o ēnei kereme he tohatoha noa.
3. Te Kounga me te Tukanga Hanga:
I roto i te umanga whakanao, ka whakamahia e te mana whakahaere kounga te tohatoha noa hei whakatauira i te rerekētanga o te tukanga whakaputa, hei whakatau mena kei te pai te haere o te tukanga, mena rānei he hapa whakaputa.
4. Te Hinengaro me te Mātauranga:
Ka whakamahia te tohatoha noa hei whakaahua i te tohatoha o ngā kaute whakamātautau i roto i te ine mātauranga me te ine hinengaro. Ka āwhina i te whakataurite i ngā aromatawai me te mārama ki te tohatoha o ngā pūkenga puta noa i ngā taupori.
5. Whakamutunga
He ariā nui te uara e tumanakohia ana i roto i te tohatoha noa. Hei ine i te pokapū o te tohatoha, ka whakaratohia e te uara e tumanakohia ana he tirohanga ki te toharite o ngā raraunga i whakaputaina e tētahi tukanga matapōkere. I te ao tūturu, ka whakamahia ngā uara e tumanakohia ana i roto i ngā momo mara mō te whakatau kaupapa me te tātari raraunga. Ko te tohatoha noa, me ōna āhuatanga ōrite e tautuhia ana e te uara e tumanakohia ana me te paerewa rerekētanga, ka whakaratohia he tauira tūponotanga tino māmā, ngāwari hoki ki te whakatinana.
Mā te mārama ki te uara e tumanakohia ana i roto i te tohatoha noa, ka taea e tātou te tātari raraunga pai ake, te matapae, me te whakatau whai whakaaro nui ake i roto i ngā horopaki pakihi, pūtaiao, me te pāpori.
