unimore/triennale-appunti-steffo
unimore/triennale-appunti-steffo
1
Fork 0
mirror of https://github.com/Steffo99/unisteffo.git synced 2025-03-13 12:17:36 +00:00
Code Activity

0.8.6

Browse source
This commit is contained in:
Steffo 2020-08-26 18:16:43 +02:00
parent 3fd64d4320
commit fc513e8376
27 changed files with 412 additions and 121 deletions
Show stats Download patch file Download diff file Expand all files Collapse all files

1
docs/bundle.3fda8.esm.js.LICENSE.txt generated
View file

@ -1 +0,0 @@
/*! For license information please see bundle.3fda8.esm.js.LICENSE.txt */

6
docs/bundle.cdee3.js → docs/bundle.44838.js generated
View file

File diff suppressed because one or more lines are too long

0
docs/bundle.cdee3.js.LICENSE.txt → docs/bundle.44838.js.LICENSE.txt generated
View file

2
docs/bundle.cdee3.js.map → docs/bundle.44838.js.map generated
View file

File diff suppressed because one or more lines are too long

6
docs/bundle.3fda8.esm.js → docs/bundle.b7974.esm.js generated
View file

File diff suppressed because one or more lines are too long

1
docs/bundle.b7974.esm.js.LICENSE.txt generated Normal file
View file

@ -0,0 +1 @@
/*! For license information please see bundle.b7974.esm.js.LICENSE.txt */

2
docs/bundle.3fda8.esm.js.map → docs/bundle.b7974.esm.js.map generated
View file

@ -1 +1 @@
{"version":3,"sources":[],"names":[],"mappings":"","file":"bundle.3fda8.esm.js","sourceRoot":""}
{"version":3,"sources":[],"names":[],"mappings":"","file":"bundle.b7974.esm.js","sourceRoot":""}

2
docs/index.html generated
View file

@ -1 +1 @@
<!DOCTYPE html><html lang="it"><head><meta charset="utf-8"><title>appuntiweb</title><meta content="width=device-width,initial-scale=1" name="viewport"><link href="/favicon.ico" rel="icon" type="image/x-icon"><link href="/favicon.ico" rel="shortcut icon" type="image/x-icon"><style>body{background-color:#0d193b}</style><link rel="manifest" href="/manifest.json"><link href="/bundle.724d1.css" rel="preload" as="style"><noscript><link rel="stylesheet" href="/bundle.724d1.css"></noscript><script>function $loadcss(u,m,l){(l=document.createElement('link')).rel='stylesheet';l.href=u;document.head.appendChild(l)}$loadcss("/bundle.724d1.css")</script></head><body><script type="__PREACT_CLI_DATA__">{"preRenderData":{"url":"/"}}</script><script nomodule="">!function(){var e=document,t=e.createElement("script");if(!("noModule"in t)&&"onbeforeload"in t){var n=!1;e.addEventListener("beforeload",function(e){if(e.target===t)n=!0;else if(!e.target.hasAttribute("nomodule")||!n)return;e.preventDefault()},!0),t.type="module",t.src=".",e.head.appendChild(t),t.remove()}}();</script><script crossorigin="anonymous" src="/bundle.3fda8.esm.js" type="module"></script><script nomodule="" src="/polyfills.8f3b5.js"></script><script nomodule="" defer="defer" src="/bundle.cdee3.js"></script></body></html>
<!DOCTYPE html><html lang="it"><head><meta charset="utf-8"><title>appuntiweb</title><meta content="width=device-width,initial-scale=1" name="viewport"><link href="/favicon.ico" rel="icon" type="image/x-icon"><link href="/favicon.ico" rel="shortcut icon" type="image/x-icon"><style>body{background-color:#0d193b}</style><link rel="manifest" href="/manifest.json"><link href="/bundle.724d1.css" rel="preload" as="style"><noscript><link rel="stylesheet" href="/bundle.724d1.css"></noscript><script>function $loadcss(u,m,l){(l=document.createElement('link')).rel='stylesheet';l.href=u;document.head.appendChild(l)}$loadcss("/bundle.724d1.css")</script></head><body><script type="__PREACT_CLI_DATA__">{"preRenderData":{"url":"/"}}</script><script nomodule="">!function(){var e=document,t=e.createElement("script");if(!("noModule"in t)&&"onbeforeload"in t){var n=!1;e.addEventListener("beforeload",function(e){if(e.target===t)n=!0;else if(!e.target.hasAttribute("nomodule")||!n)return;e.preventDefault()},!0),t.type="module",t.src=".",e.head.appendChild(t),t.remove()}}();</script><script crossorigin="anonymous" src="/bundle.b7974.esm.js" type="module"></script><script nomodule="" src="/polyfills.29a91.js"></script><script nomodule="" defer="defer" src="/bundle.44838.js"></script></body></html>

4
docs/polyfills.8f3b5.js → docs/polyfills.29a91.js generated
View file

File diff suppressed because one or more lines are too long

2
docs/polyfills.8f3b5.js.map → docs/polyfills.29a91.js.map generated
View file

File diff suppressed because one or more lines are too long

4
docs/polyfills.b885f.esm.js → docs/polyfills.37ed5.esm.js generated
View file

File diff suppressed because one or more lines are too long

2
docs/polyfills.b885f.esm.js.map → docs/polyfills.37ed5.esm.js.map generated
View file

@ -1 +1 @@
{"version":3,"sources":[],"names":[],"mappings":"","file":"polyfills.b885f.esm.js","sourceRoot":""}
{"version":3,"sources":[],"names":[],"mappings":"","file":"polyfills.37ed5.esm.js","sourceRoot":""}

2
docs/push-manifest.json generated
View file

@ -1 +1 @@
{"/":{"bundle.724d1.css":{"type":"style","weight":1},"bundle.3fda8.esm.js":{"type":"script","weight":1}},"/AlgoritmiEStruttureDati":{"bundle.724d1.css":{"type":"style","weight":1},"bundle.3fda8.esm.js":{"type":"script","weight":1},"route-AlgoritmiEStruttureDati.chunk.e6f90.esm.js":{"type":"script","weight":0.9}},"/ApprendimentoSistemiArtificiali":{"bundle.724d1.css":{"type":"style","weight":1},"bundle.3fda8.esm.js":{"type":"script","weight":1},"route-ApprendimentoSistemiArtificiali.chunk.857ce.esm.js":{"type":"script","weight":0.9},"route-ApprendimentoSistemiArtificiali.chunk.55de3.css":{"type":"style","weight":0.9}},"/BasiDiDati":{"bundle.724d1.css":{"type":"style","weight":1},"bundle.3fda8.esm.js":{"type":"script","weight":1},"route-BasiDiDati.chunk.f1dc7.esm.js":{"type":"script","weight":0.9},"route-BasiDiDati.chunk.1d0a7.css":{"type":"style","weight":0.9}},"/CalcoloNumerico":{"bundle.724d1.css":{"type":"style","weight":1},"bundle.3fda8.esm.js":{"type":"script","weight":1},"route-CalcoloNumerico.chunk.3b3c9.esm.js":{"type":"script","weight":0.9},"route-CalcoloNumerico.chunk.782c4.css":{"type":"style","weight":0.9}},"/Fisica":{"bundle.724d1.css":{"type":"style","weight":1},"bundle.3fda8.esm.js":{"type":"script","weight":1},"route-Fisica.chunk.e2766.esm.js":{"type":"script","weight":0.9},"route-Fisica.chunk.ed9a8.css":{"type":"style","weight":0.9}},"/Home":{"bundle.724d1.css":{"type":"style","weight":1},"bundle.3fda8.esm.js":{"type":"script","weight":1},"route-Home.chunk.d3981.esm.js":{"type":"script","weight":0.9},"route-Home.chunk.b342d.css":{"type":"style","weight":0.9}},"/MingwInstall":{"bundle.724d1.css":{"type":"style","weight":1},"bundle.3fda8.esm.js":{"type":"script","weight":1},"route-MingwInstall.chunk.1eacd.esm.js":{"type":"script","weight":0.9}},"/NetLogo":{"bundle.724d1.css":{"type":"style","weight":1},"bundle.3fda8.esm.js":{"type":"script","weight":1},"route-NetLogo.chunk.18d97.esm.js":{"type":"script","weight":0.9},"route-NetLogo.chunk.1d0a7.css":{"type":"style","weight":0.9}},"/OttimizzazioneLineare":{"bundle.724d1.css":{"type":"style","weight":1},"bundle.3fda8.esm.js":{"type":"script","weight":1},"route-OttimizzazioneLineare.chunk.b7c6f.esm.js":{"type":"script","weight":0.9},"route-OttimizzazioneLineare.chunk.99830.css":{"type":"style","weight":0.9}},"/RipassoDiAlgebraLineare":{"bundle.724d1.css":{"type":"style","weight":1},"bundle.3fda8.esm.js":{"type":"script","weight":1},"route-RipassoDiAlgebraLineare.chunk.0ebef.esm.js":{"type":"script","weight":0.9},"route-RipassoDiAlgebraLineare.chunk.1d0a7.css":{"type":"style","weight":0.9}},"/Statistica":{"bundle.724d1.css":{"type":"style","weight":1},"bundle.3fda8.esm.js":{"type":"script","weight":1},"route-Statistica.chunk.68267.esm.js":{"type":"script","weight":0.9},"route-Statistica.chunk.9d494.css":{"type":"style","weight":0.9}},"/VlDiGeometria":{"bundle.724d1.css":{"type":"style","weight":1},"bundle.3fda8.esm.js":{"type":"script","weight":1},"route-VlDiGeometria.chunk.56154.esm.js":{"type":"script","weight":0.9}}}
{"/":{"bundle.724d1.css":{"type":"style","weight":1},"bundle.b7974.esm.js":{"type":"script","weight":1}},"/AlgoritmiEStruttureDati":{"bundle.724d1.css":{"type":"style","weight":1},"bundle.b7974.esm.js":{"type":"script","weight":1},"route-AlgoritmiEStruttureDati.chunk.e6f90.esm.js":{"type":"script","weight":0.9}},"/ApprendimentoSistemiArtificiali":{"bundle.724d1.css":{"type":"style","weight":1},"bundle.b7974.esm.js":{"type":"script","weight":1},"route-ApprendimentoSistemiArtificiali.chunk.857ce.esm.js":{"type":"script","weight":0.9},"route-ApprendimentoSistemiArtificiali.chunk.55de3.css":{"type":"style","weight":0.9}},"/BasiDiDati":{"bundle.724d1.css":{"type":"style","weight":1},"bundle.b7974.esm.js":{"type":"script","weight":1},"route-BasiDiDati.chunk.f1dc7.esm.js":{"type":"script","weight":0.9},"route-BasiDiDati.chunk.1d0a7.css":{"type":"style","weight":0.9}},"/CalcoloNumerico":{"bundle.724d1.css":{"type":"style","weight":1},"bundle.b7974.esm.js":{"type":"script","weight":1},"route-CalcoloNumerico.chunk.09ebc.esm.js":{"type":"script","weight":0.9},"route-CalcoloNumerico.chunk.8f997.css":{"type":"style","weight":0.9}},"/Fisica":{"bundle.724d1.css":{"type":"style","weight":1},"bundle.b7974.esm.js":{"type":"script","weight":1},"route-Fisica.chunk.e2766.esm.js":{"type":"script","weight":0.9},"route-Fisica.chunk.ed9a8.css":{"type":"style","weight":0.9}},"/Home":{"bundle.724d1.css":{"type":"style","weight":1},"bundle.b7974.esm.js":{"type":"script","weight":1},"route-Home.chunk.d3981.esm.js":{"type":"script","weight":0.9},"route-Home.chunk.b342d.css":{"type":"style","weight":0.9}},"/MingwInstall":{"bundle.724d1.css":{"type":"style","weight":1},"bundle.b7974.esm.js":{"type":"script","weight":1},"route-MingwInstall.chunk.1eacd.esm.js":{"type":"script","weight":0.9}},"/NetLogo":{"bundle.724d1.css":{"type":"style","weight":1},"bundle.b7974.esm.js":{"type":"script","weight":1},"route-NetLogo.chunk.18d97.esm.js":{"type":"script","weight":0.9},"route-NetLogo.chunk.1d0a7.css":{"type":"style","weight":0.9}},"/OttimizzazioneLineare":{"bundle.724d1.css":{"type":"style","weight":1},"bundle.b7974.esm.js":{"type":"script","weight":1},"route-OttimizzazioneLineare.chunk.b7c6f.esm.js":{"type":"script","weight":0.9},"route-OttimizzazioneLineare.chunk.99830.css":{"type":"style","weight":0.9}},"/RipassoDiAlgebraLineare":{"bundle.724d1.css":{"type":"style","weight":1},"bundle.b7974.esm.js":{"type":"script","weight":1},"route-RipassoDiAlgebraLineare.chunk.0ebef.esm.js":{"type":"script","weight":0.9},"route-RipassoDiAlgebraLineare.chunk.1d0a7.css":{"type":"style","weight":0.9}},"/Statistica":{"bundle.724d1.css":{"type":"style","weight":1},"bundle.b7974.esm.js":{"type":"script","weight":1},"route-Statistica.chunk.68267.esm.js":{"type":"script","weight":0.9},"route-Statistica.chunk.9d494.css":{"type":"style","weight":0.9}},"/VlDiGeometria":{"bundle.724d1.css":{"type":"style","weight":1},"bundle.b7974.esm.js":{"type":"script","weight":1},"route-VlDiGeometria.chunk.56154.esm.js":{"type":"script","weight":0.9}}}

186
docs/route-CalcoloNumerico.chunk.09ebc.esm.js generated Normal file
View file

File diff suppressed because one or more lines are too long

2
docs/route-CalcoloNumerico.chunk.3b3c9.esm.js.map → docs/route-CalcoloNumerico.chunk.09ebc.esm.js.map generated
View file

@ -1 +1 @@
{"version":3,"sources":[],"names":[],"mappings":"","file":"route-CalcoloNumerico.chunk.3b3c9.esm.js","sourceRoot":""}
{"version":3,"sources":[],"names":[],"mappings":"","file":"route-CalcoloNumerico.chunk.09ebc.esm.js","sourceRoot":""}

2
docs/route-CalcoloNumerico.chunk.3812b.js generated
View file

File diff suppressed because one or more lines are too long

1
docs/route-CalcoloNumerico.chunk.3812b.js.map generated
View file

File diff suppressed because one or more lines are too long

93
docs/route-CalcoloNumerico.chunk.3b3c9.esm.js generated
View file

File diff suppressed because one or more lines are too long

1
docs/route-CalcoloNumerico.chunk.782c4.css generated
View file

@ -1 +0,0 @@
.red__2y1B_{color:#ff7d7d}.orange__dD2kx{color:#ffbb7d}.yellow__OEpwl{color:#ffff7d}.lime__CVe41{color:#7dff7d}.cyan__26ZAg{color:#7dffff}.blue__LO7Xm{color:#7d7dff}.magenta__1Akee{color:#ff7dff}.example__2PzAa{color:#d3a1ff;padding:4px;border-radius:4px;margin:4px 0}.example__2PzAa,.example__2PzAa table{background-color:rgba(211,161,255,.05)}.example__2PzAa table{border-spacing:0;border:2px solid rgba(211,161,255,.1);border-collapse:collapse}.example__2PzAa table tbody td,.example__2PzAa table tbody th,.example__2PzAa table thead td,.example__2PzAa table thead th{padding:4px;border:1px solid rgba(211,161,255,.1)}.example__2PzAa table thead{background-color:rgba(211,161,255,.1);color:#fff}.menulist__2Cmnq{font-size:large}.menulist__2Cmnq small{font-size:small}

1
docs/route-CalcoloNumerico.chunk.8f997.css generated Normal file
View file

@ -0,0 +1 @@
.menulist__2Cmnq{font-size:large}.menulist__2Cmnq small{font-size:small}.red__2y1B_{color:#ff7d7d}.orange__dD2kx{color:#ffbb7d}.yellow__OEpwl{color:#ffff7d}.lime__CVe41{color:#7dff7d}.cyan__26ZAg{color:#7dffff}.blue__LO7Xm{color:#7d7dff}.magenta__1Akee{color:#ff7dff}.example__2PzAa{color:#d3a1ff;padding:4px;border-radius:4px;margin:4px 0}.example__2PzAa,.example__2PzAa table{background-color:rgba(211,161,255,.05)}.example__2PzAa table{border-spacing:0;border:2px solid rgba(211,161,255,.1);border-collapse:collapse}.example__2PzAa table tbody td,.example__2PzAa table tbody th,.example__2PzAa table thead td,.example__2PzAa table thead th{padding:4px;border:1px solid rgba(211,161,255,.1)}.example__2PzAa table thead{background-color:rgba(211,161,255,.1);color:#fff}

2
docs/route-CalcoloNumerico.chunk.b3074.js generated Normal file
View file

File diff suppressed because one or more lines are too long

1
docs/route-CalcoloNumerico.chunk.b3074.js.map generated Normal file
View file

File diff suppressed because one or more lines are too long

2
docs/sw-esm.js generated
View file

File diff suppressed because one or more lines are too long

2
docs/sw.js generated
View file

File diff suppressed because one or more lines are too long

2
package.json
View file

@ -1,7 +1,7 @@
{
"private": true,
"name": "appuntiweb",
"version": "0.8.4",
"version": "0.8.6",
"license": "AGPL-3.0-or-later",
"scripts": {
"start": "preact watch --template src/template.html",

2
size-plugin.json
View file

File diff suppressed because one or more lines are too long

202
src/routes/CalcoloNumerico/05_ApprossimazioneDatiSperimentali.js
View file

@ -1,6 +1,7 @@
import style from "./05_ApprossimazioneDatiSperimentali.less";
import {Fragment} from "preact";
import {Section, Panel, ILatex, BLatex, PLatex} from "bluelib";
import Example from "../../components/Example";
const r = String.raw;
@ -13,11 +14,208 @@ export default function (props) {
<p>
Interpolare dati sperimentali non fornisce quasi mai un modello del fenomeno.
</p>
<p>
Vogliamo costruire una <b>funzione di regressione</b> che, dati molti più dati del grado della funzione, minimizzi il quadrato della distanza tra i punti sperimentali e i punti della funzione di regressione.
</p>
<p>
Denominiamo:
</p>
<ul>
<li><ILatex>{r`{\color{Orange} f}`}</ILatex>: la <b>funzione "effettiva"</b> del fenomeno</li>
<li><ILatex>{r`{\color{Yellow} q}`}</ILatex>: la <b>funzione di regressione</b> che costruiamo per approssimarlo</li>
<li><ILatex>{r`{\color{Red} Q }`}</ILatex>: la <b>funzione "errore di regressione"</b> da minimizzare</li>
<li><ILatex>{r`(\ x_i, f(x_i)\ )`}</ILatex>: i <b>punti sperimentali</b></li>
</ul>
<p>
L'obiettivo è minimizzare l'<b>errore di approssimazione</b> <ILatex>{r`Q`}</ILatex>, ovvero:
</p>
<PLatex>{r`\min {\color{Red} Q } = \sum_{i = 1}^m (\ {\color{Yellow} q(x_i)} - {\color{Orange} f(x_i)}\ )^2 `}</PLatex>
</Panel>
</Section>
<Section>
<Panel title={""}>
<Panel title={"Regressione lineare"}>
<p>
Trova la <b>retta</b> <ILatex>{r`{\color{Yellow} q}`}</ILatex> che meglio approssima tutti gli <ILatex>{r`m`}</ILatex> dati sperimentali.
</p>
<p>
Essendo una retta, avrà <b>due parametri</b>: il termine noto <ILatex>{r`a_0`}</ILatex>, e la pendenza <ILatex>{`a_1`}</ILatex>.
</p>
<PLatex>{r`{\color{Yellow} q(x) } = a_0 + a_1 \cdot {\color{Green} x}`}</PLatex>
<p>
L'errore da minimizzare per ricavare i parametri sarà:
</p>
<PLatex>{r`
\min {\color{Red} Q } = \sum_{i = 1}^m ( {\color{Yellow} a_0 + a_1 \cdot x_i} - {\color{Orange} f(x_i)} )^2
`}</PLatex>
</Panel>
<Panel title={"Regressione lineare matriciale"}>
<p>
Possiamo costruire una <b>matrice di regressione</b> <ILatex>{r`A`}</ILatex> contenente tutti i <b>punti sperimentali</b>:
</p>
<PLatex>{r`
A =
\begin{pmatrix}
1 & x_1\\\\
1 & x_2\\\\
\vdots & \vdots\\\\
1 & x_m
\end{pmatrix}
`}</PLatex>
<p>
Inoltre, se costruiamo il <b>vettore dei parametri</b> <ILatex>{r`\alpha`}</ILatex>:
</p>
<PLatex>{r`
\alpha =
\begin{pmatrix}
a_0\\\\
a_1
\end{pmatrix}
`}</PLatex>
<p>
Avremo che:
</p>
<PLatex>{r`{\color{Yellow} q(x) } = A \cdot \alpha`}</PLatex>
<p>
Inoltre, potremo calcolare l'errore attraverso la norma:
</p>
<PLatex>{r`{\color{Red} Q } = \| A \cdot \alpha - y \|^2`}</PLatex>
</Panel>
</Section>
<Section>
<Panel title={"Regressione polinomiale"}>
<p>
Trova il <b>polinomio</b> <ILatex>{r`{\color{Yellow} q}`}</ILatex> di grado <ILatex>{r`n-1`}</ILatex> che meglio approssima tutti gli <ILatex>{r`m`}</ILatex> dati sperimentali.
</p>
<p>
Essendo un polinomio di grado <ILatex>{r`n-1`}</ILatex>, avrà <ILatex>{r`n`}</ILatex> parametri.
</p>
<PLatex>{r`{\color{Yellow} q(x) } = a_0 + a_1 \cdot {\color{Green} x} + a_2 \cdot {\color{Green} x^2} +\ \dots \ + a_{n-1} \cdot {\color{Green} x^{n-1}`}</PLatex>
<Example>
<p>
La regressione lineare è un caso particolare di regressione generale in cui i parametri sono 2!
</p>
</Example>
<p>
L'errore da minimizzare per ricavare i parametri sarà:
</p>
<PLatex>{r`
\min {\color{Red} Q} = \sum_{i = 1}^m ( {\color{Yellow} a_0 + a_1 \cdot x_i + a_2 \cdot x_i^2 +\ \dots \ + a_{n-1} \cdot x_i^{n-1}} - {\color{Orange} y_i} )^2
`}</PLatex>
</Panel>
<Panel title={"Regressione polinomiale matriciale"}>
<p>
Possiamo costruire una <b>matrice di regressione</b> <ILatex>{r`A`}</ILatex> contenente tutti i <b>punti sperimentali</b> a tutti i gradi del polinomio:
</p>
<PLatex>{r`
A =
\begin{pmatrix}
1 & x_1 & x_1^2 & \dots & x_1^{n-1} \\\\
1 & x_2 & x_2^2 & \dots & x_2^{n-1} \\\\
\vdots & \vdots & \vdots & \ddots & \vdots \\\\
1 & x_m & x_m^2 & \dots & x_m^{n-1}
\end{pmatrix}
`}</PLatex>
<p>
Inoltre, se costruiamo il <b>vettore dei parametri</b> <ILatex>{r`\alpha`}</ILatex>:
</p>
<PLatex>{r`
\alpha =
\begin{pmatrix}
a_0\\\\
a_1\\\\
\vdots\\\\
a_{n-1}
\end{pmatrix}
`}</PLatex>
<p>
Avremo che:
</p>
<PLatex>{r`{\color{Yellow} q(x) } = A \cdot \alpha`}</PLatex>
<p>
Inoltre, potremo calcolare l'errore attraverso la norma:
</p>
<PLatex>{r`{\color{Red} Q } = \| A \cdot \alpha - y \|^2`}</PLatex>
<Example>
Normalmente, i dati sono molti di più, ma se il numero di parametri <ILatex>{r`n`}</ILatex> fosse uguale al numero di dati <ILatex>{r`m`}</ILatex>, allora si otterrebbe il <b>polinomio di interpolazione</b>!
</Example>
</Panel>
</Section>
<Section>
<Panel title={"Regressione generale"}>
<p>
Trova i <b>coefficienti della combinazione lineare</b> <ILatex>{r`{\color{Yellow} q}`}</ILatex> che meglio approssima tutti gli <ILatex>{r`m`}</ILatex> dati sperimentali.
</p>
<PLatex>{r`{\color{Yellow} q(x) } = a_0 \cdot {\color{Green} \phi_0 (x)} + a_1 \cdot {\color{Green} \phi_1 (x)} + \dots + a_2 \cdot {\color{Green} \phi_2 (x)} +\ \dots\ + a_{n-1} \cdot {\color{Green} \phi_{n-1} (x)}`}</PLatex>
<Example>
<p>
La regressione polinomiale è un caso particolare di regressione generale in cui:
</p>
<PLatex>{r`{\color{Green} \phi_{n} (x)} = x^n`}</PLatex>
</Example>
<p>
L'errore da minimizzare per ricavare i parametri sarà:
</p>
<PLatex>{r`
\min {\color{Red} Q } = \sum_{i = 1}^m ( {\color{Yellow} a_0 \cdot \phi_0 (x) + a_1 \cdot \phi_1 (x) + \dots + a_2 \cdot \phi_2 (x) +\ \dots\ + a_{n-1} \cdot \phi_{n-1} (x)} - {\color{Orange} f(x_i)} )^2
`}</PLatex>
</Panel>
<Panel title={"Regressione polinomiale generale"}>
<p>
Possiamo costruire una <b>matrice di regressione</b> <ILatex>{r`A`}</ILatex> contenente tutti i <b>punti sperimentali</b> a tutti i gradi del polinomio:
</p>
<PLatex>{r`
A =
\begin{pmatrix}
\phi_0(x_1) & \phi_1(x_1) & \phi_2(x_1) & \dots & \phi_{n_1}(x_1) \\\\
\phi_0(x_2) & \phi_1(x_2) & \phi_2(x_2) & \dots & \phi_{n-1}(x_2) \\\\
\vdots & \vdots & \vdots & \ddots & \vdots \\\\
\phi_0(x_m) & \phi_1(x_m) & \phi_2(x_m) & \dots & \phi_{n-1}(x_m)
\end{pmatrix}
`}</PLatex>
<p>
Inoltre, se costruiamo il <b>vettore dei parametri</b> <ILatex>{r`\alpha`}</ILatex>:
</p>
<PLatex>{r`
\alpha =
\begin{pmatrix}
a_0\\\\
a_1\\\\
\vdots\\\\
a_{n-1}
\end{pmatrix}
`}</PLatex>
<p>
Avremo che:
</p>
<PLatex>{r`{\color{Yellow} q(x) } = A \cdot \alpha`}</PLatex>
<p>
Inoltre, potremo calcolare l'errore attraverso la norma:
</p>
<PLatex>{r`{\color{Red} Q } = \| A \cdot \alpha - y \|^2`}</PLatex>
</Panel>
</Section>
<Section title={"Trovare i parametri"}>
<Panel title={"Caso non degenere"}>
<p>
Caso che prevede che le colonne di <ILatex>{r`A`}</ILatex> siano <b>linearmente indipendenti</b>.
</p>
<p>
La soluzione <b>esiste</b> sempre, ed è <b>unica</b>.
</p>
<p>
Per trovarla:
</p>
<ul>
<li>Fattorizziamo <ILatex>{r`A = Q \cdot \begin{pmatrix} R\\ 0 \end{pmatrix}`}</ILatex>.</li>
<li>Calcoliamo <ILatex>{r`w = Q^T \cdot y`}</ILatex>.</li>
<li>Teniamo solo i primi <ILatex>n</ILatex> valori di <ILatex>{r`w`}</ILatex> e mettiamoli in <ILatex>{r`w_1`}</ILatex>.</li>
<li>Calcoliamo <ILatex>{r`R \cdot \alpha = w_1`}</ILatex>.</li>
</ul>
</Panel>
<Panel title={"Caso generale"}>
<p>
Caso che non preclude alcuna composizione di <ILatex>{r`A`}</ILatex>.
</p>
</Panel>
</Section>
</Fragment>