From defd85161873ccb27eea94d8589b20df2206c215 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Gi=C3=B9=20Marcer?= Date: Sun, 31 May 2020 16:16:56 +0200 Subject: [PATCH] lectures: remove everything about the previous pdf --- lectures/images/binomiale.png | Bin 13243 -> 0 bytes lectures/images/chi2.png | Bin 17521 -> 0 bytes lectures/images/correlazione.png | Bin 121976 -> 0 bytes lectures/images/exponential.png | Bin 17566 -> 0 bytes lectures/images/fischer.png | Bin 15174 -> 0 bytes lectures/images/gaussian.png | Bin 20091 -> 0 bytes lectures/images/kurtosis.png | Bin 45322 -> 0 bytes lectures/images/landau.png | Bin 18086 -> 0 bytes lectures/images/logistic.png | Bin 49517 -> 0 bytes lectures/images/poisson.png | Bin 14100 -> 0 bytes lectures/images/skewness.png | Bin 68248 -> 0 bytes lectures/images/student.png | Bin 18192 -> 0 bytes lectures/images/uniform.png | Bin 17163 -> 0 bytes lectures/images/wigner.png | Bin 21086 -> 0 bytes lectures/lectures.pdf | Bin 573801 -> 0 bytes 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z;Ft|W-{k_Mn1|a%Y}7KLOtzgjAJsQCXDs5$Xk_j^1a<*A@)K#Pp~NXrf+%b5VrxY$1q9$LKrXx8J{q{yf z^@2%W4GZ8!zKYHu;5F&z(uS%M)T>QnSnM)*(uHh} z^bp`=e$&PUSxwzJGpT>bZZqAIySQOy_IB>J2OiJf{B^|UuxY&TrE2K&f$31H(lpg` zwSCy8TiHqPth1u>x=zE+O)fDy&7vvq)VT(1(lmA2Y(5+>B{s*8xUAn!UYUa%{Bhl5 z&MLllb)4_ISvi!v_3cxwpwHNW3>@L8ktx-6g}t`UuJp0>(@wH z84Xuiu81t#KQ`VwU&9qg6CTQoNARx;Rma%Mu^R_0KoOC4({ zyp~Ev&D9nZNx;FAD;!N*{Bw3O^-82=x@HWT72*Iawb6`$025qXR^s;H|AJ$I(F~fmnMbX zx41bwU1xSXW_0s;+~+QrQRttnT7}*2|C3U@ysbUGQJzS95+OKD7!D-?gXJ|9Nd60@ Cf#|jX diff --git a/lectures/makefile b/lectures/makefile deleted file mode 100644 index 1773cbd..0000000 --- a/lectures/makefile +++ /dev/null @@ -1,17 +0,0 @@ -sections = $(shell find sections/ -name '*.md' | sort -V) - -pandoc = \ - @ pandoc $(1) \ - --toc \ - --standalone \ - --pdf-engine=xelatex \ - -F pandoc-crossref \ - -o $(2).pdf - -all: lectures.pdf - -clean: - rm lectures.pdf - -lectures.pdf: $(sections) images - $(call pandoc, $(sections), lessons) diff --git a/lectures/sections/0.md b/lectures/sections/0.md deleted file mode 100644 index b5aa1d4..0000000 --- a/lectures/sections/0.md +++ /dev/null @@ -1,23 +0,0 @@ ---- -title: Analisi statistica -subtitle: Riassunto delle slides -numbersections: true -documentclass: article -fontsize: 12pt - -geometry: - - width=150mm - - top=20mm - - bottom=30mm - -header-includes: | - ```{=latex} - % start new page with each section - \usepackage{etoolbox} - \usepackage{tikz} - \usepackage{amsmath} - \usepackage{amssymb} - \usepackage{siunitx} - \pretocmd{\section}{\clearpage}{}{} - ``` ---- diff --git a/lectures/sections/modulo-1.md b/lectures/sections/modulo-1.md deleted file mode 100644 index 92e409b..0000000 --- a/lectures/sections/modulo-1.md +++ /dev/null @@ -1,140 +0,0 @@ -# Calcolo numerico - -## Compilare un programma - -Per eseguire un programma, un computer fa questa cosa: legge le informazioni -dall'input e le mette nella memoria; poi da qui legge le istruzioni -sequenzialmente e se c'è qualche calcolo da fare, lo fa fare alla ALU -(aritmetic logic unit) e memorizza il risultato nella memoria; alla fine passa -tutto in output. -Per essere eseguibile, un programma deve essere scritto in codice macchina -e per questo si usano i linguaggi di programmazione che, attraverso il -compilatore, vengono tradotti in codice macchina: - -- codice sorgente, -- compilatore, -- codice oggetto, -- linker (aggiunge le librerie), -- codice eseguibile. - -## Errori di implementazione - -Quando si passa dal modello astratto a quello implementato, si distinguono -tre tipi di errori: - -- errori analitici: causati dalla necessità di una aritmetica discreta, - cioè il fatto che non si possa rappresentare una cosa continua e quindi - la si fa a punti; -- errori inerenti o algoritmici (o di round-off): causati dal numero finito di - cifre significative. Dipendono dall'algoritmo utilizzato, perché ci sono - metodi migliori di altri per evitare questo tipo di problemi. - -## Rappresentazione di un numero - -L'obiettivo è quello di tenere sotto controllo gli errori di round-off. -Per rappresentare un numero, esistono diverse rappresentazioni, tra cui quella -di Von Neumann: si dedicano $n$ cifre significative alla mantissa e poi un tot -anche alla caratteristica, cioè all'ordine di grandezza (che generalmente è in -base 2). -Con $n$ cifre significative in base 2, il numero più alto rappresentabile è -$2^n -1$ perché con due cifre (0 e 1), ci sono $2^n$ possibili numeri -rappresentabili e, se posti in ordine crescente, ognuno dista dal precedente -un'unità, con il minimo che vale 0 (tutti 0), per cui il numero più alto che si -può rappresentare è $2^n -1$: - -- 0 è l'1, -- 1 e il 2, -- ...$2^n -1$ è il $2^n$-esimo. - -questo numero corrisponde a un numero in base dieci di $m$ cifre significative, -per cui: - -$$ - m = \log(2^n -1) -$$ - -quindi per un numero con 24 bit per la mantissa, si hanno circa 7 cifre -significative in base 10. -In precisione semplice (floating point): - -- un bit per il segno, -- 23 bit per la mantissa (quindi 7 cifre significative), -- 8 bits per la caratteristica. - -In precisione doppia (duble) si raddoppiano i bite alla mantissa e le cifre -significative diventano 17, però il tempo per le moltiplicazioni è triplicato -e si occupa più memoria, quindi non è che convenga moltissimo. - -Esiste anche la rappresentazione fixed-point, in cui c'è un bit per il segno -e i restanti per il numero in sé, in cui da qualche parte c'è la virgola. -Nel caso del fixed pointd, l'errore relativo può variare moltissimo a seconda -del numero rappresentato, mentre per i float è lo stesso per ogni numero: la -densità relativa non è costante. -Quando un insieme di rappresentazione non è chiuso rispetto ad un'operazione, -il risultato presenta sicuramente errore di round-off e deve essere -approssimato. - - -## Errori algoritmici - -Per numeri troppo grandi o troppo piccoli che non si riesce a -rappresentare, si parla rispettivamente di overflow o underflow. Per questo -motivo, somme e soprattutto differenze di numeri grandi portano grandi errori -dovuti all'arrotondamento: bisogna cercare di evitarlo se possibile. Per -esempio, per calcolare un polinomio, conviene usare il metodi di -Ruffini-Horner perché riduce il numero di operazioni da effettuare: - -- metodo semplice: - - $$ - p(x) = a_0 + a_1x + a_2x^2 + ... +a_nx^n - $$ - - che sono n addizioni e n(n+1)/2 moltiplicazioni, quindi $\sim n^2$. -- metodo di R-H: - - $$ - p (x) = (((((a_n)x +a_{n-1})x +a_{n-2}x +... - $$ - - che sono n addizioni e n moltiplicazioni, quindi $\sim n$. - -Il problema si verifica anche se si sottraggono due numeri molto simili tra -loro, perché il risultato è un numero molto piccolo che potrebbe finire in -underflow (si parla di "cancellazione catastrofica"). - -\textcolor{orange}{oss:} Il costo computazionale della risoluzione di un -sistema lineare in $n$ equazioni è asintoticamente uguale al costo del -prodotto di due matrici $nxn$. Esistono algoritmi che non richiedono più di -$k \cdot n^{\alpha}$ operazioni, col il più piccolo valore noto di $\alpha$ -pari a 0.2375. - -Alcuni algoritmi sono definiti "instabili": accade quando gli errori di -round-off si accumulano fino a portare a risultati completamente errati. -Esistono problemi detti "mal condizionati" che con qualsiasi algoritmo danno -errori talmente elevati da rendere il risultato privo di significato. In -questi casi, piccole variazioni dei dati iniziali portano a grandi variazioni -nei risultati. Si chiama "numero di condizionamento del problema" il seguente -rapporto: - -$$ - \frac{\Delta r}{\Delta \alpha} = - \frac{\text{\% errore risultato}}{\text{\% errore dato iniziale}} -$$ - -che può quindi essere espresso in questo modo: - -$$ - \Delta \alpha = \frac{x + h - x}{x} = \frac{h}{x} - \hspace{100pt} - \Delta r = \frac{f(x + h)-f(x)}{f(x)} -$$ -$$ - \frac{\Delta r}{\Delta \alpha} = - \frac{f(x + h)-f(x)}{h} \cdot \frac{x}{f(x)} = - f'(x) \cdot \frac{x}{f(x)} -$$ - -Se questo numero è $<< 1$, allora il problema è poco sensibile ai dati -iniziali. - diff --git a/lectures/sections/modulo-2-1.md b/lectures/sections/modulo-2-1.md deleted file mode 100644 index 1b02c79..0000000 --- a/lectures/sections/modulo-2-1.md +++ /dev/null @@ -1,257 +0,0 @@ -# Probabilità - -## Concetto di probabilità - -Consideriamo un insieme S di eventi detto "spazio campionario" e due eventi -casuali A e B. Essi sono soggetti agli assiomi di Kolmogorov: - -$\forall A \subseteq S, 0 \leqslant P(A) \leqslant 1$ -$P(S) = 1$ -$A \cap B = \varnothing \Longrightarrow P(A \cup B) = P(A) + P(B)$ - -Si definisce probabilità condizionata di A dato B: - -$$ - P(A|B) = \frac{P(A \cap B)}{P(B)} -$$ - -Se due eventi sono indipendenti, allora vale che: - -$$ - P(A \cap B) = P(A) \cdot P(B) \Longrightarrow - P(A|B) = \frac{P(A) \cdot P(B)}{P(B)} = P(A) -$$ - -Esistono diversi approcci per definire la probabilità: - -- approccio frequentista: - - $$ - P(A) = \lim_{n \rightarrow +\infty} \frac{\#A}{n} - $$ - - ma c'è il problema che è impossibile fare un numero infinito di tentativi. -- approccio bayesiano: - utilizzare il teorema di Bayes, per esempio: - - $$ - P(\text{teoria|dati}) \propto P(\text{dati|teoria}) \cdot P(\text{teoria}) - $$ - si tratta quindi di un approccio "soggettivo", almeno per l'ultimo termine - dell'equazione qua sopra, perché la probabilità della validità della teoria - può essere data sulla base di ragionamenti e osservazioni riguardanti il - fenomeno di cui si sta parlando. - -\begin{tikzpicture} - \draw [thick, pink] (0,0) -- (1,0); - \draw [red] (1.5,0) circle [radius=0.1]; - \draw [thick, pink] (2,0) -- (3,0); - \draw [red] (3.5,0) circle [radius=0.1]; - \draw [thick, pink] (4,0) -- (5,0); - \draw [red] (5.5,0) circle [radius=0.1]; - \draw [thick, pink] (6,0) -- (7,0); - \draw [red] (7.5,0) circle [radius=0.1]; - \draw [thick, pink] (8,0) -- (9,0); - \draw [red] (9.5,0) circle [radius=0.1]; - \draw [thick, pink] (10,0) -- (11,0); - \draw [red] (11.5,0) circle [radius=0.1]; - \draw [thick, pink] (12,0) -- (13,0); - \draw [red] (13.5,0) circle [radius=0.1]; - \draw [thick, pink] (14,0) -- (15,0); -\end{tikzpicture} - -### Esempio di probabilità condizionata - -Si osservano dei decadimenti e se ne misurano indipendentemente $N_a$ e $N_b$; -in comune ne sono stati misurati $N_{ab}$. Possiamo stimare il numero totale -di eventi $N$ e l'efficienza totale $\epsilon$. - -$$ - P(a) = \frac{N_a}{N} \hspace{100pt} P(b) = \frac{N_b}{N} -$$ -$$ - P(ab) = \frac{N_{ab}}{N} = P(a) \cdot P(b) = \frac{N_a \cdot N_b}{N^2} - \Longrightarrow N = \frac{N_a \cdot N_b}{N_{ab}} -$$ -$$ - \epsilon = P(a \cup b) = P(a) + P(b) -P(a \cap b) = - \frac{N_a + N_b - N_{ab}}{N} -$$ - -\begin{tikzpicture} - \draw [thick, pink] (0,0) -- (1,0); - \draw [red] (1.5,0) circle [radius=0.1]; - \draw [thick, pink] (2,0) -- (3,0); - \draw [red] (3.5,0) circle [radius=0.1]; - \draw [thick, pink] (4,0) -- (5,0); - \draw [red] (5.5,0) circle [radius=0.1]; - \draw [thick, pink] (6,0) -- (7,0); - \draw [red] (7.5,0) circle [radius=0.1]; - \draw [thick, pink] (8,0) -- (9,0); - \draw [red] (9.5,0) circle [radius=0.1]; - \draw [thick, pink] (10,0) -- (11,0); - \draw [red] (11.5,0) circle [radius=0.1]; - \draw [thick, pink] (12,0) -- (13,0); - \draw [red] (13.5,0) circle [radius=0.1]; - \draw [thick, pink] (14,0) -- (15,0); -\end{tikzpicture} - -## Teorema di Bayes - -$$ - P(A|B) = \frac{P(A \cap B)}{P(B)} - \hspace{100pt} - P(B|A) = \frac{P(B \cap A)}{P(A)} -$$ -$$ - P(A|B) = \frac{P(B|A) \cdot P(A)}{P(B)} -$$ - -Dal teorema di Bayes è possibile dedurre il teorema della probabilità totale: -consideriamo un insieme $S$ diviso in sottoinsiemi disgiunti $A_i$ la cui -unione dà l'insieme di partenza: - -$$ - \cup_i A_i = S -$$ - -e consideriamo un insieme $B$ anch'esso interno ad $S$: - -$$ - B = B \cap S = B \cap (\cup_i A_i) = \cup_i (B \cap A_i) -$$ -$$ - \Longrightarrow P(B) = \sum_i P(B \cap A_i) -$$ -$$ - \Longrightarrow P(B) = \sum_i P(B|A_i)P(A_i) -$$ - -da cui, per uno specifico insieme $A_j$, attraverso il teorema di Bayes: - -$$ - P(A_j|B) = \frac{P(B|A_j) \cdot P(A_j)}{\sum_i P(B|A_i)P(A_i)} -$$ - -\begin{tikzpicture} - \draw [thick, pink] (0,0) -- (1,0); - \draw [red] (1.5,0) circle [radius=0.1]; - \draw [thick, pink] (2,0) -- (3,0); - \draw [red] (3.5,0) circle [radius=0.1]; - \draw [thick, pink] (4,0) -- (5,0); - \draw [red] (5.5,0) circle [radius=0.1]; - \draw [thick, pink] (6,0) -- (7,0); - \draw [red] (7.5,0) circle [radius=0.1]; - \draw [thick, pink] (8,0) -- (9,0); - \draw [red] (9.5,0) circle [radius=0.1]; - \draw [thick, pink] (10,0) -- (11,0); - \draw [red] (11.5,0) circle [radius=0.1]; - \draw [thick, pink] (12,0) -- (13,0); - \draw [red] (13.5,0) circle [radius=0.1]; - \draw [thick, pink] (14,0) -- (15,0); -\end{tikzpicture} - -### Esempio di probabilità totale: segnale e fondo - -Un rilevatore misura segnale e fondo con relative efficienze $P(R|S)$ e -$P(R|F)$. Se sono note a priori la probabilità di segnale e di fondo -$P(S)$ e $P(F)$, allora si può risalire alla probabilità, data una -misurazione, di aver misurato il segnale: - -$$ - P(S|R) = \frac{P(R|S) \cdot P(S)}{P(R|S) \cdot P(S) + P(R|F) \cdot P(F)} -$$ - -\begin{tikzpicture} - \draw [thick, pink] (0,0) -- (1,0); - \draw [red] (1.5,0) circle [radius=0.1]; - \draw [thick, pink] (2,0) -- (3,0); - \draw [red] (3.5,0) circle [radius=0.1]; - \draw [thick, pink] (4,0) -- (5,0); - \draw [red] (5.5,0) circle [radius=0.1]; - \draw [thick, pink] (6,0) -- (7,0); - \draw [red] (7.5,0) circle [radius=0.1]; - \draw [thick, pink] (8,0) -- (9,0); - \draw [red] (9.5,0) circle [radius=0.1]; - \draw [thick, pink] (10,0) -- (11,0); - \draw [red] (11.5,0) circle [radius=0.1]; - \draw [thick, pink] (12,0) -- (13,0); - \draw [red] (13.5,0) circle [radius=0.1]; - \draw [thick, pink] (14,0) -- (15,0); -\end{tikzpicture} - -### Esempio di probabilità totale: AIDS - -$$ - P(\text{AIDS}) = 0.001 \hspace{50pt} P(\text{no AIDS}) = 0.999 -$$ -$$ - P(\text{+|AIDS}) = 0.98 \hspace{50pt} P(\text{-|AIDS}) = 0.02 -$$ -$$ - P(\text{+|no AIDS}) = 0.03 \hspace{50pt} P(\text{-|no AIDS}) = 0.97 -$$ - -Quindi se il test è positivo, la probabilità di avere davvero preso l'AIDS è: - -$$ - P(\text{AIDS|+}) = \frac{P(\text{+|AIDS}) \cdot P(\text{AIDS})} - {P(\text{+|AIDS}) \cdot P(\text{AIDS}) + P(\text{+|no AIDS}) \cdot - P(\text{no AIDS})} = 0.032 -$$ - -\begin{tikzpicture} - \draw [thick, pink] (0,0) -- (1,0); - \draw [red] (1.5,0) circle [radius=0.1]; - \draw [thick, pink] (2,0) -- (3,0); - \draw [red] (3.5,0) circle [radius=0.1]; - \draw [thick, pink] (4,0) -- (5,0); - \draw [red] (5.5,0) circle [radius=0.1]; - \draw [thick, pink] (6,0) -- (7,0); - \draw [red] (7.5,0) circle [radius=0.1]; - \draw [thick, pink] (8,0) -- (9,0); - \draw [red] (9.5,0) circle [radius=0.1]; - \draw [thick, pink] (10,0) -- (11,0); - \draw [red] (11.5,0) circle [radius=0.1]; - \draw [thick, pink] (12,0) -- (13,0); - \draw [red] (13.5,0) circle [radius=0.1]; - \draw [thick, pink] (14,0) -- (15,0); -\end{tikzpicture} - -### Approccio bayesiano con una teoria - -Cominciamo com un esempio: con un esperimento è stata misurata la massa di -un elettrone e sono stati trovati i valori {$m_i$}, di cui il valore medio -è $m_e = \SI{520 \pm 10}{KeV}$. Si assume quindi che il valore vero sia -compreso tra \SI{510}{KeV} e \SI{530}{KeV} con una probabilità del 68% data -dal confidence level. Per cui, la probabilità che la massa vera sia proprio -\SI{520}{KeV} è data da: - -$$ - P(m_e|{m_i}) = \frac{}{} -$$ - -\textcolor{red}{**rivedere negli appunti questa formula**} - -Quando si fa una misurazione, si misurano $N$ valori e si calcolano il valore -medio $\bar{x}$ e la deviazione standard $\sigma$ e si dice che il risultato è: - -$$ - \mu = \bar{x} \pm \frac{\sigma}{\sqrt{N}} -$$ - -e di solito lo si interpreta dicendo che: - -$$ - P \left( \bar{x} - \frac{\sigma}{\sqrt{N}} \leqslant \mu \leqslant - \bar{x} + \frac{\sigma}{\sqrt{N}} \right) = 68\% -$$ - -ma in realtà quello che sappiamo è solo che: - -$$ - P \left( \mu - \frac{\sigma_{true}}{\sqrt{N}} \leqslant \bar{x} \leqslant - \mu + \frac{\sigma_{true}}{\sqrt{N}} \right) = 68\% -$$ - -**\textcolor{red}{WHAT?!?}** diff --git a/lectures/sections/modulo-2-2.md b/lectures/sections/modulo-2-2.md deleted file mode 100644 index 06bacc9..0000000 --- a/lectures/sections/modulo-2-2.md +++ /dev/null @@ -1,480 +0,0 @@ -# Statistica - -## Distribuzioni di probabilità - -Una funzione di densità di probabilità $f$ è definita in modo che la probabilità -che una variabile $x$ sia compresa tra $x$ e $x + dx$ sia data da: - -$$ - P(x \subset [x, x + dx]) = f(x)dx -$$ - -dunque vale che: - -$$ - \int\limits_{- \infty}^{+ \infty} dx f(x) = 1 -$$ - -Si definisce funzione cumulante: - -$$ - F(x) = \int\limits_{- \infty}^x dx' f(x') -$$ - -e quantile di ordine $\alpha$ il valore di $x$ per cui $F(x) = \alpha$. -Nel caso multidimensionale in cui si abbiano due o più variabili, si parla di -joint pdf: - -$$ - f(x, y) \hspace{30pt} \Longrightarrow \hspace{30pt} - \int\limits_{- \infty}^{+ \infty} \int\limits_{- \infty}^{+ \infty} - dx dy f(x, y) = 1 -$$ - -e si definiscono due distribuzioni marginali: - -$$ - f_x (x) = \int\limits_{- \infty}^{+ \infty} dy f(x, y) - \hspace{50pt} - f_y (y) = \int\limits_{- \infty}^{+ \infty} dx f(y, x) -$$ - -dunque due variabili $x$ e $y$ sono indipendenti se $f(x) = f_x(x) \cdot -f_y(y)$. Ora, se $A$ è l'evento di probabilità $f_x(x)dx$, mentre $B$ ha -probabilità $f_y(y)dy$, allora si possono definire le pdf condizionali come -segue: - -$$ - P(B|A) = \frac{P (A \cap B)}{P(A)} = \frac{f(x, y)dxdy}{f_x(x)dx} - \hspace{20pt} \Longrightarrow \hspace{20pt} h(y|x) = \frac{f(x, y)}{f_x(x)} -$$ - -per cui il teorema di Bayes diventa: - -$$ - g(x|y) = \frac{h(y|x)f_x(x)}{f_y(y)} -$$ - -\begin{tikzpicture} - \draw [thick, pink] (0,0) -- (1,0); - \draw [red] (1.5,0) circle [radius=0.1]; - \draw [thick, pink] (2,0) -- (3,0); - \draw [red] (3.5,0) circle [radius=0.1]; - \draw [thick, pink] (4,0) -- (5,0); - \draw [red] (5.5,0) circle [radius=0.1]; - \draw [thick, pink] (6,0) -- (7,0); - \draw [red] (7.5,0) circle [radius=0.1]; - \draw [thick, pink] (8,0) -- (9,0); - \draw [red] (9.5,0) circle [radius=0.1]; - \draw [thick, pink] (10,0) -- (11,0); - \draw [red] (11.5,0) circle [radius=0.1]; - \draw [thick, pink] (12,0) -- (13,0); - \draw [red] (13.5,0) circle [radius=0.1]; - \draw [thick, pink] (14,0) -- (15,0); -\end{tikzpicture} - -### Paradosso di Borel-Kolmogorov - -Si considerino dei punti distribuiti uniformemente sulla superficie del pianeta -Terra: ci si aspetterebbe che i punti siano uniformemente distribuiti anche -lungo un parallelo o un meridiano... ma consideriamo un meridiano: esso giace -per il 25% a nord del 45'esimo parallelo e quindi, secondo la logica di prima, -anche il 25% dei punti che si trovano su di esso. Però non è vero che il 45% -della superficie terrestre è al di sopra del 45'esimo parallelo! -Il paradosso è risolto perché non ci si può basare su un insieme di misura -nulla quale il meridiano (perché è unidimensionale). Lo si vede chiaramente -adottando la terminologia poc'anzi introdotta: -Se la distribuzione è uniforme, la probabilità di trovare un punto in una -certa superficie è dato dal rapporto tra l'angolo solido descritto da tale -superficie e l'angolo solido totale: - -$$ - f(\theta, \phi) d\theta d\phi= \frac{d\phi d\theta \cos(\theta)}{4 \pi} -$$ - -da cui è possibile determinare la due probabilità marginali: - -$$ - f_{\phi}(\phi) = \int\limits_{0}^{\pi} f(\theta, \phi) d\theta = - \int\limits_{0}^{\pi} \frac{\cos(\theta)}{4 \pi} = \frac{\cos(\theta)}{2} -$$ -$$ - f_{\theta}(\theta) = \int\limits_{0}^{2 \pi} f(\theta, \phi) d\phi = - \frac{1}{2 \pi} -$$ - -per cui si tratta di due costanti rispetto alle rispettive variabili. Da ciò -si può dunque dedurre che, mentre la densità lungo un parallelo è -effettivamente costante, lo stesso non si può dire riguardo a un meridiano. - -\begin{tikzpicture} - \draw [thick, pink] (0,0) -- (1,0); - \draw [red] (1.5,0) circle [radius=0.1]; - \draw [thick, pink] (2,0) -- (3,0); - \draw [red] (3.5,0) circle [radius=0.1]; - \draw [thick, pink] (4,0) -- (5,0); - \draw [red] (5.5,0) circle [radius=0.1]; - \draw [thick, pink] (6,0) -- (7,0); - \draw [red] (7.5,0) circle [radius=0.1]; - \draw [thick, pink] (8,0) -- (9,0); - \draw [red] (9.5,0) circle [radius=0.1]; - \draw [thick, pink] (10,0) -- (11,0); - \draw [red] (11.5,0) circle [radius=0.1]; - \draw [thick, pink] (12,0) -- (13,0); - \draw [red] (13.5,0) circle [radius=0.1]; - \draw [thick, pink] (14,0) -- (15,0); -\end{tikzpicture} - -Una funzione di una variabile casuale è essa stessa una variabile casuale. -Consideriamo la pdf $f(x)$ e una funzione $a(x)$ di cui si vuole trovare la pdf -$g(a)$. Nel caso in cui l'inversa di $a(x)$ sia univoca, definita $dS$ la -regione delle $x$ per cui $a \subset [a, a +da]$: - -$$ - g(a)da = \int\limits_{dS} dxf(x) - = \left| \int\limits_{x(a)}^{x(a +da)} f(x')dx' \right| - = \int\limits_{x(a) + \left| \frac{dx}{da} \right| da}^{x(a +da)} f(x')dx' -$$ - -Ovvero: - -$$ - g(a) = f(x(a)) \left| \frac{dx}{da} \right| -$$ - -e se $x(a)$ non è univoca, allora bisogna considerare tutti gli intervalli $dS$ -di $dx$ che corrispondono a $da$. -Nel caso di funzioni di $N$ variabili, siccome vale che: - -$$ - g(a')da' = \int \dots \int\limits_{dS} f(x_1 \dots x_N) dx_1 \dots dx_N -$$ - -con $dS$ regione dello spazio delle $x$ compreso tra le isosuperfici: - -$$ - a(\vec{x}) = a' \hspace{10pt} \wedge \hspace{10pt} a(\vec{x}) = a' + da' -$$ - -Nel caso in cui $z = x \cdot y$, si trova la convoluzione di Mellin: - -$$ - g(z)dz = \int\limits_{dS} dxdy f(x, y) - = \int\limits_{-\infty}^{+\infty} dx - \int\limits_{\frac{z}{x}}^{\frac{z + dz}{x}} dy f(x, y) -$$ - -**\textcolor{red}{Non ho capito questa parte...}** - -## Propagazione degli errori - -Consideriamo una variabile $x$ con pdf $f(x)$. Si definisce valore di -aspettazione o media (e lo si indica spesso con $\mu$): - -$$ - E[x] = \int dx f(x) x -$$ - -Nel caso di una variabile $y(x)$ con pdf $g(x)$, invece: - -$$ - E[y] = \int dy \cdot y \cdot g(y) = \int dx f(x) g(x) -$$ - -Mentre si definisce varianza (e la si indica spesso con $\sigma^2$, mentre -con deviazione standard si intende $\sigma$): - -$$ - V[x] = E[x - E[x]^2] = E[x^2] - \mu^2 -$$ - -Più in generale si definiscono 'momenti algebrici' $E[x^n] =\mu'_n$ con -$\mu'_1 = \mu$ e 'momenti centrali' $E[(x -\mu)^n] = \mu_n$ con $\mu_2 = -\sigma^2$. -Si definiscono inoltre due grandezze di correlazione. La covarianza: - -$$ - \text{cov} [x, y] = E[xy] - E[x]E[y] = E[xy] - \mu_x \mu_y -$$ - -che equivale a: - -\begin{align*} - \text{cov} [x, y] - &= E[(x -\mu_x)(y -\mu_y)] \\ - &= E[xy -x\mu_y -y\mu_x + \mu_x\mu_Y] \\ - &= E[xy] -\mu_y E[x] -\mu_x E[y] + \mu_x \mu_y \\ - &= E[xy] -\mu_y mu_x - \mu_x \mu_y + \mu_x \mu_y \\ - &= E[xy] - \mu_x \mu_y -\end{align*} - -Notare che se $x$ e $y$ sono indipendenti, allora $f(x, y) = f_x(x)f_y(y)$, -perciò: - -$$ - E[xy] = \int dx \int dy xy f(x, y) = \mu_x \mu_y - \hspace{20pt} \Longrightarrow \hspace{20pt} \text{cov} [x, y] = 0 -$$ - -e il coefficiente di correlazione: - -$$ - \rho_{xy} = \frac{\text{cov} [xy]}{\sigma_x \sigma_y} -$$ - -![Esempio di correlazione tra due -grandezze.](images/correlazione.png){width=70%} - -Anche se la $f(\vec{x})$ non è completamente nota, è comunque possibile stimare -il valore medio e la varianza di una grandezza $y(\vec{x})$ conoscendo solo le -stime di media e varianza della pdf. Espandiamo attraverso la serie di -Taylor: - -$$ - y(\vec{x}) = y(\vec{\mu}) + \sum_{i= 1}^N \left[ - \frac{\partial y}{\partial x_i} \right]_{\vec{x} - = \vec{\mu}} (x_i - \mu_i) -$$ -$$ - \Longrightarrow \hspace{20pt} E[y] = y(\vec{\mu}) - \Longleftarrow \hspace{20pt} E[x_i] = \mu_i -$$ - -Mentre per la varianza servono $E[y^2]$ ed $E[y]$. Sempre passando -attraverso uno sviluppo di Taylor attorno al valore medio: - -\begin{align*} - E[y^2] &= y^2(\vec{\mu}) + 2y(\vec{\mu}) \sum_{i = 1}^{N} - \left[ \frac{\partial y} {\partial x_i} \right]_{\vec{x} = \vec{\mu}} - E[x_i - \mu_i] \\ - &+ E \left[ \left( \sum_{i_1}^N - \left[ \frac{\partial y}{\partial x_i} \right]_{\vec{x} = \vec{\mu}} - (x_i - \mu_i) \right) \left( \sum_{j = 1}^N \left[ \frac{\partial y} - {\partial x_i} \right]_{\vec{x} = \vec{\mu}} (x_j - \mu_j) \right) \right] -\end{align*} - -Siccome il secondo termine si annulla sempre perché $E[x_i] = \mu_i$, allora -rimane che: - -$$ - V[y] = E[y^2] - E[y]^2 = \sigma_y^2 = \sum_{i,j = 1}^N \left[ - \frac{\partial y}{\partial x_i} - \frac{\partial y}{\partial x_j}\right]_{\vec{x} = \vec{\mu}} V_{ij} -$$ - -Con $V_{ij}$ che è la matrice di covarianza, che ha come entrate: - -$$ - V_{ij} = E[(x_i - \mu_i)(x_j - \mu_j)] = \rho_{ij} \sigma_i \sigma_j -$$ - -e quindi, nel caso in cui le variabili siano scorrelate, si ottiene che: - -$$ - V_{ij} = \sigma_i^2 \delta_{ij} - \hspace{20pt} \Longrightarrow \hspace{20pt} - \sigma_y^2 = \sum_{i = 1}^N \left[ \frac{\partial y}{\partial x_i} - \right]_{\vec{x} = \vec{\mu}}^2 \sigma_i^2 -$$ - -Cioè dice quanto cambia la $y$ al variare del 'dato iniziale' $\vec{x}$. -Ma quindi, per quanto visto prima: - -$$ - \text{cov} [x_i, x_j] = E[(x_i - \mu_i)(x_j - \mu_j)] = V_{ij} -$$ - -Più in generale, date $\vec{y}$ variabili dipendenti da $\vec{x}$, vale che: - -$$ - U = AVA^T \hspace{30pt} \text{con} \hspace{30pt} A_{ij} = \left[ - \frac{\partial y_i}{\partial x_j} \right]_{\vec{x} = \vec{\mu}} - \hspace{30pt} \text{e con} \hspace{30pt} U_{kl} = \text{cov}[y_k, y_l] -$$ - -dove $U$ è detta matrice di covarianza delle $y$. -Attenzione: quanto detto fin'ora, che descrive in che modo gli errori di -$\vec{x}$ influenzano $y$, vale solo nel caso in cui $y$ sia lineare nelle $x$. -Quindi, in casi come $y(x) = 1/x$, non si può fare questo discorso. - -\begin{tikzpicture} - \draw [thick, pink] (0,0) -- (1,0); - \draw [red] (1.5,0) circle [radius=0.1]; - \draw [thick, pink] (2,0) -- (3,0); - \draw [red] (3.5,0) circle [radius=0.1]; - \draw [thick, pink] (4,0) -- (5,0); - \draw [red] (5.5,0) circle [radius=0.1]; - \draw [thick, pink] (6,0) -- (7,0); - \draw [red] (7.5,0) circle [radius=0.1]; - \draw [thick, pink] (8,0) -- (9,0); - \draw [red] (9.5,0) circle [radius=0.1]; - \draw [thick, pink] (10,0) -- (11,0); - \draw [red] (11.5,0) circle [radius=0.1]; - \draw [thick, pink] (12,0) -- (13,0); - \draw [red] (13.5,0) circle [radius=0.1]; - \draw [thick, pink] (14,0) -- (15,0); -\end{tikzpicture} - -### Esempio - -Consideriamo: - -$$ - y = x_1 - x_2 -$$ -$$ - \text{con} \hspace{30pt} \mu_1 = \mu_2 = 10 \hspace{30pt} \wedge - \hspace{30pt} \sigma_1 = \sigma_2 = 1 -$$ - -allora abbiamo che $y = y(x_1, x_2)$, quindi: - -$$ - E[y] = y(\mu_1, \mu_2) = 10 - 10 = 0 -$$ -$$ - V[y] = \sum_{i, j = 1}^2 \left[ \frac{\partial y}{\partial x_i} - \frac{\partial y}{\partial x_j}\right]_{\vec{x} = \vec{\mu}} V_{ij} = - 1 \cdot V_{11} + 1 \cdot V_{22} -1 \cdot 2 \cdot V_{12} -$$ - -Se le correlazioni sono nulle, allora $V_{12} = 0 \Longrightarrow V[y] = 2 -\Longrightarrow \sigma_y = 1.4$, se invece $x_1$ e $x_2$ sono correlate, nel -caso in cui il coefficiente di correlazione sia unitario si ha che $V[y] = -0$. Quindi la correlazione può cambiare di molto le cose. - -\begin{tikzpicture} - \draw [thick, pink] (0,0) -- (1,0); - \draw [red] (1.5,0) circle [radius=0.1]; - \draw [thick, pink] (2,0) -- (3,0); - \draw [red] (3.5,0) circle [radius=0.1]; - \draw [thick, pink] (4,0) -- (5,0); - \draw [red] (5.5,0) circle [radius=0.1]; - \draw [thick, pink] (6,0) -- (7,0); - \draw [red] (7.5,0) circle [radius=0.1]; - \draw [thick, pink] (8,0) -- (9,0); - \draw [red] (9.5,0) circle [radius=0.1]; - \draw [thick, pink] (10,0) -- (11,0); - \draw [red] (11.5,0) circle [radius=0.1]; - \draw [thick, pink] (12,0) -- (13,0); - \draw [red] (13.5,0) circle [radius=0.1]; - \draw [thick, pink] (14,0) -- (15,0); -\end{tikzpicture} - -### Errori sistematici - -Consideriamo due grandezze $x_1$ e $x_2$ con un errore sistematico in comune -$S$: - -\begin{align*} - &x_1 = x_{1_0} + x_{1_s} \\ - &x_2 = x_{2_0} + x_{2_s} -\end{align*} - -si avrà che i termini con pedice $0$ sono indipendenti tra loro, mentre gli -altri due saranno correlati. Dato che gli errori si sommano in quadratura, la -matrice di covarianza sarà quindi: - -$$ -\text{cov}[x_1, x_2] = S^2 \hspace{30pt} \Longrightarrow \hspace{30pt} -V = \begin{pmatrix} -\sigma_1^2 + S^2 & S^2\\ -S^2 & \sigma_2^2 + S^2 -\end{pmatrix} -$$ - -perché: - -\begin{align*} - \text{cov}[x_1, x_2] &= E[x_1 x_2] - E[x_1]E[x_2] = \\ - &= E[(x_{1_0} + x_{1_s})(x_{2_0} + x_{2_s})] - E[x_{1_0} + x_{1_s}] - E[x_{2_0} + x_{2_s}] = \\ - &= E[x_{1_0}x_{2_0}] + E[x_{1_0}x_{2_s}] + E[x_{1_s}x_{2_0}] - + E[x_{1_s}x_{2_s}] + \\ - &\hspace{13pt} - E[x_{1_0}]E[x_{2_0}] - E[x_{1_0}]E[x_{2_2}] - - E[x_{1_s}]E[x_{2_0}] - E[x_{1_s}]E[x_{2_s}] = \\ - &= \mu_1 \mu_2 + \mu_1E[x_{2_s}] + E[x_{1_s}]\mu_2 +E[x_{1_s}x_{2_s}] + \\ - &\hspace{13pt} - \mu_1 \mu_2 - \mu_1 E[x_{2_s}] - E[x_{1_s}] \mu_2 - - E[x_{1_s}] E[x_{2_s}] = \\ - &= E[x_{1_s} x_{2_s}] - E[x_{1_s}] E[x_{2_s}] = \text{cov}[x_{1_s}, x_{2_s}] -\end{align*} - -### Trasformazione ortogonale - -Può tornare utile fare un cambio di variabile che permetta di ottenere una -matrice di covarianza delle $y$ diagonale. -Consideriamo le solite variabili $Y_i$ legate linearmente alle $x_j$: - -$$ - y_i = \sum_j = A^i_j x_j - \hspace{30pt} \Longrightarrow \hspace{30pt} - U_{ij} = \sum_{k,l} A_{ik} V_{kl} A^T_{lj} -$$ - -Si tratta quindi di diagonalizzare la matrice $U$: la soluzione è semplice, -la matrice $A$ è quella formata dagli autovalori di $V$. Questo concetto è -utile nel caso della scelta delle coordinate da utilizzare. -Se immaginiamo di star utilizzando le coordinate polari $\vec{x} = (x, y)$, la -matrice di covarianza sarà: - -$$ -V = \begin{pmatrix} -\sigma_1^2 & \rho \sigma_1 \sigma_2 \\ -\rho \sigma_1 \sigma_2 & \sigma_2^2 -\end{pmatrix} -$$ - -Diagonaliziamola: prima di tutto troviamo gli autovalori della matrice $V - -\lambda I$: - -$$ - \begin{vmatrix} - \sigma_1^2 - \lambda & \rho \sigma_1 \sigma_2 \\ - \rho \sigma_1 \sigma_2 & \sigma_2^2 - \lambda - \end{vmatrix} - = (\sigma_1^2 - \lambda) (\sigma_2^2 - \lambda) - \rho^2 \sigma_1^2 - \sigma_2^2 = 0 -$$ -\begin{align*} - &\Longrightarrow \hspace{30pt} - \lambda^2 -(\sigma_1^2 + \sigma_2^2) \lambda + \sigma_1^2 \sigma_2^2 - (1 - \rho^2) = 0 \\ - &\Longrightarrow \hspace{30pt} - \lambda_{1,2} = \frac{\sigma_1^2 + \sigma_2^2 \pm - \sqrt{\sigma_1^4 + \sigma_2^4 +2 \sigma_1^2 \sigma_2^2 - - 4 \sigma_1^2 \sigma_2^2 + 4\rho^2 \sigma_1^2 \sigma_2^2}}{2} = \\ - &\Longrightarrow \hspace{30pt} - \lambda_{1,2} = \frac{\sigma_1^2 + \sigma_2^2 \pm - \sqrt{(\sigma_1^2 - \sigma_2^2)^2 +4 \rho^2 \sigma_1^2 \sigma_2^2}}{2} -\end{align*} - -e ora calcoliamo gli autovettori: - -$$ - (V - \lambda I)\vec{r} = 0 - \hspace{30pt} \Longrightarrow \hspace{30pt} - \begin{pmatrix} - \sigma_1^2 - \lambda & \rho \sigma_1 \sigma_2 \\ - \rho \sigma_1 \sigma_2 & \sigma_2^2 - \lambda - \end{pmatrix} - \begin{pmatrix} - r_1 \\ - r_2 - \end{pmatrix} - = - \begin{pmatrix} - r_1 \\ - r_2 - \end{pmatrix} -$$ -$$ - \Longrightarrow \hspace{30pt} - \begin{cases} - (\sigma_1^2 - \lambda)r_1 + \rho \sigma_1 \sigma_2 r_2 = r_1 \\ - \rho \sigma_1 \sigma_2 r_1 + (\sigma_2^2 - \lambda)r_2 = r_2 - \end{cases} - \hspace{30pt} \Longrightarrow \hspace{30pt} - r_1 = \frac{}{} r_2 -$$ - -eccetera eccetera... diff --git a/lectures/sections/modulo-2-3.md b/lectures/sections/modulo-2-3.md deleted file mode 100644 index 43272d2..0000000 --- a/lectures/sections/modulo-2-3.md +++ /dev/null @@ -1,466 +0,0 @@ - Distribuzioni di probabilità - -## Distribuzione binomiale - -![Distribuzione binomiale: p = 0.3, n = 10, -N = 1000.](images/binomiale.png) - -Si considerino $N$ tentativi di un esperimento che può avere come esiti -soltanto successo o fallimento e che la probabilità di ogni successo sia $p$. -Definiamo $n$ il numero dei successi. Dunque la probabilità di ottenere $n$ -successi su $N$ tentativi totali è data da: - -$$ - P(N, n, p) = \binom{N}{n} p^n (1 - p)^{N -n} -$$ - -perché: - -- la probabilità che un successo si verifichi è $p$; -- la probabilità che $k$ successi si verifichino è data dal prodotto di tutte - le probabilità: $p^n$; -- lo stesso discorso vale per gli insuccessi: ognuno ha probabilità $(1_p)$ e - se ne verificano $N -n$; -- il termine binomiale rappresenta tutte le possibili permutazioni: il - concetto è semplice se si immagina di posizionare successi e fallimenti - all'interno di una griglia con $N$ possibili posizioni: un successo è un - pallino bianco e un fallimento è un pallino nero. In quanti posti posso - mettere il primo successo? $N$. E il secondo? $N -1$. E il terzo? E così - via, finché ho messo tutti i successi, che occupano $n$ posizioni, l'ultima - delle quali è stata scelta tra $N - (n -1)$ posizioni, per cui: - - $$ - N \cdot (N -1) \dots (N -n + 1) = - \frac{N \cdot (N -1) \dots (N -n + 1) \cdot 2 \cdot 1} - {(N -n) \cdot (N - n -1) \dots 2 \cdot 1} = \frac{N!}{(N -n)!} - $$ - - ma non bisogna considerare che poi tutte le posizioni delle palline nere - sono uguali, quindi va ulteriormente diviso per $n!$ per le stesse ragioni. - Da cui: - - $$ - \binom{N}{n} = \frac{N!}{n! (N -n)!} - $$ - -Per la normalizzazione, vale che: - -$$ - \sum_{n = 0}^N P(N, n, p) = 1 -$$ - -Possiamo definire un valore di aspettazione e una varianza: - -\begin{align*} - &E[n] = \sum_{n = 0}^N n P(N, n, p) = Np \\ - &V[n] = E[n^2] - E[n]^2 = Np(1 -p) -\end{align*} - -## Distribuzione multinomiale - -È la generalizzazione della pdf precedente nel caso in cui ci siano $m$ -possibili risultati, ciascuno con una probabilità $P_m$ di verificarsi. Per -esempio, è il caso di un istogramma riguardo al quale ci si domanda quale sia la -probabilità di trovarlo esattamente con quelle specifiche entrate. -La probabilità è conseguentemente data da: - -$$ - P(N, \vec{n}, \vec{p}) = \frac{N!}{n_1! n_2! \dots n_m!} - p_1^{n_1} p_2^{n_2} \dots p_m^{n_m} -$$ - -E come valore di aspettazione e deviazione standard si ottiene che: - -\begin{align*} - &E[n_i] = Np_i \\ - &V[n_i] = Np_i (1 -p_i) -\end{align*} - -### Legge dei grandi numeri - -La legge dei grandi numeri afferma che la media sperimentale di una variabile -$x$, per un numero di tentativi $N$ che tende all'infinito, si avvicina molto -alla media vera. Questa legge può essere utilizzata per stimare le probabilità -$P_i$ di una distribuzione muiltinomiale tramite le frequenze con cui i diversi -eventi si verificano. -Si consideri la frequenza $f_j$ con cui l'evento j-esimo si verifica, dato un -set di $N$ tentativi: - -$$ - f_j = \frac{1}{N} \sum_{i = 1}^N x_i = \frac{x_j}{N} -$$ - -dove $x_i$ è una variabile che vale 1 se l'evento j-esimo si è verificato e -vale 0 quando se ne è verificato un altro e $x_j$ è quindi il numero di volte -che l'evento j-esimo si è verificato. -A differenza di $P_j$, $f_j$ è una variabile casuale perché dipende da $x_j$ -che è la somma di variabili casuali. Definiamo valore medio: - -$$ - E(f_j) = \frac{E(x_j)}{N} = P_j -$$ - -e calcoliamo la varianza: - -$$ - V \left[ f_j \right] = V \left[ \frac{x_j}{N} \right] = - E \left[ \frac{x_j^2}{N^2} \right] - \left( E \left[ \frac{x_j}{N} - \right] \right)^2 = \frac{1}{N^2} V \left[ x_j \right] -$$ - -ora, $x_j$ è esattamente $n_j$ della multinomiale, perciò: - -$$ - V[x_j] = NP_j (1 - P_j) - \hspace{30pt} \Longrightarrow \hspace{30pt} - V[f_j] = \frac{1}{N} P_j (1 - P_j) \leqslant \frac{1}{N} -$$ - -## Distribuzione di Poisson - -![Distribuzione Poissoniana: $\nu$ = 1, -N = 1000.](images/poisson.png) - -Se si considera la distribuzione binomiale e ci si pone nel limite in cui il -numero di tentativi ripetuti tenda all'infinito e che la probabilità di -successo tenda a zero (con il vincolo che $N \cdot p = cost = \nu$), si ottiene -la distribuzione di Poisson: - -$$ - P(N, n, \nu) = \frac{\nu^n}{n!} e^{-\nu} -$$ - -con: - -\begin{align*} - &E[n] = \nu \\ - &V[n] = \nu -\end{align*} - -dove $\nu = NP = \frac{}{}$ numero medio di successi. -Quando $N$ è talmente grande da non essere definito (come nel caso in cui si -osservino i decadimenti di un atomo e dunque i tentativi sono le osservazioni, -che sono dunque continue), non è più possibile definire una probabilità di -successo per ogni evento (perché sarebbe nulla, da cui il motivo per cui -la Poissoniana è definita con questi due limiti), e quindi $\nu$ va definita in -un altro modo. Infatti la distribuzione di Poisson, trattandosi di un limite in -$N$ e $p$, non dipende più esplicitamente da queste due grandezze. -Nel caso in cui si osservi il decadimento di un atomo, si è soliti procedere in -questo modo: si suddivide il tempo di osservazione in intervalli (il che -significa aver suddiviso gli infiniti tentativi in sottoinsiemi di infiniti -tentativi) e si misura quante volte in ognuno di questi intervalli si verifica -un successo. L'esperimento è ora praticamente suddiviso in più esperimenti -minori da cui è possibile dedurre un numero medio frequentistico di successi. -Per esempio: - ----------------------------------------------------------- -# successi 0 1 2 3 4 5 6 7 -------------- ------ ----- ----- ---- ---- --- ----- ----- -# intervalli 1042 860 307 78 15 3 0 0 - -Poisson 1064 823 318 82 16 2 0.3 0.3 ----------------------------------------------------------- - -Table: Decadimento di un atomo. Il tempo di osservazione è stato suddiviso in -intervalli e per ogni intervallo è stato contato il numero di successi -osservati. - -\newpage - -Il numero medio di eventi è: - -$$ - \frac{1042 \cdot 0 + 860 \cdot 1 + 307 \cdot 2 + 78 \cdot 3 + 15 \cdot 4 - + 3 \cdot 5 + 0 \cdot 6 + 0 \cdot 7}{1064 + 860 + 307 + 78 + 15 + 3 - + 0 + 0} = 0.77 -$$ - -Da cui è possibile calcolare i valori sempre riportati nella tabella precedente. - -## Distribuzione uniforme - -![Distribuzione uniforme: $a = 0$, $b = 100$.](images/uniform.png) - -Una pdf di numeri che hanno tutti uguale probabilità di verificarsi è detta -uniforme: - -$$ - P (n, a, b) = \begin{cases} - \frac{1}{b - a} \hspace{30pt} a \leqslant x \leqslant b \\ - 0 \hspace{42pt} \text{altrove} - \end{cases} -$$ - -con: - -\begin{align*} - E[n] = \frac{1}{2} (a + b) \\ - V[n] = \frac{1}{12} (a + b)^2 -\end{align*} - -Se una variabile è distribuita secondo una pdf $f(x)$, la sua cumulante è -uniformemente distribuita. Intuitivamente è semplice perché basta vederla in -questo modo: si immagini il grafico della pdf; ogni volta che si estrae un -numero, questo cadrà in un punto casuale nell'area al di sotto della pdf, -lasciando uno spazio casuale alla sua sinistra (che è il valore della -cumulante) - -## Distribuzione Gaussiana e CLT - -![Distribuzione Gaussiana: $\mu = 30$, -$\sigma = 5$.](images/gaussian.png) - -La distribuzione Gaussiana (o normale) è definita come: - -$$ - P (x, \mu, \sigma) = \frac{1}{\sqrt{2 \pi} \sigma} - e^{\frac{(x - \mu)^2}{2 \sigma^2}} -$$ - -con: - -\begin{align*} - E[x] = \mu \\ - V[x] = \sigma^2 -\end{align*} - -La error function, che è la cumulativa di questa pdf, è molto utile in -laboratorio e i suoi valori sono tabulati. -Il teorema centrale del limite afferma che date $n$ variabili casuali -indipendenti distribuite con una pdf comune e varianze $\sigma_i^2$, nel -limite in qui $n \rightarrow + \infty$, la somma di queste variabili segue un -andamento gaussiano con valore medio la somma dei valori medi e varianza la -somma delle varianze. -Ciò può essere sfruttato per generare numeri casuali distribuiti secondo una -distribuzione normale. - - -Per grandi valori di $\mu$ (vale a dire qualche unità), la distribuzione di -Poisson tende a quella Gaussiana con $\mu = \nu$ e $\sigma = \sqrt{\nu}$. -Analogamente per $N \rightarrow + \infty$ la binomiale tende alla Gaussiana -con $\mu = Np$ e $\sigma = \sqrt{Np (1 - p)}$. - -## Distribuzione Gaussiana multivariata - -Nel caso multidimensionale, la pdf per il vettore $\vec{x} = {x_1 ... n_n}$ è -data da: - -$$ - f(\vec{x}, \vec{\mu}, V) = \frac{1}{(2 \pi)^{N/2} \mid V \mid^{1/2}} - \exp \left[ - \frac{1}{2} (\vec{x} - \vec{\mu})^t V^{-1} (\vec{x} - - \vec{\mu}) \right] -$$ - -con $E[x_i] = \mu_i$ e $\text{cov}[x_i, x_j] = V_{ij}$ - -## Media pesata - -Quando si hanno misure con diversi errori, vanno combinate attraverso il -concetto di media pesata: - -$$ - E[x] = \frac{\sum_{i = 1}^N \frac{x_i}{\sigma_i^2}}{\sum_{i = 1}^N - \frac{1}{\sigma_i^2}} -$$ -$$ - V[x] = \frac{1}{\sum_{i = 1}^N \frac{1}{\sigma_i^2}} -$$ - -Ma non ha senso mediare valori che non sono compatibili! - -## Distribuzione di Breit-Wigner - -![Distribuzione di Breit-Wigner: $x_0 = 20$, -$\Gamma = 10$.](images/wigner.png) - -Esistono alcune distribuzioni che hanno momenti non ben definiti e che per -questo si dicono "patologiche". Un esempio è la distribuzione di Breit-Wigner: - -$$ - f (x, \Gamma, x_o) = \frac{1}{\pi} \cdot - \frac{\Gamma/2}{\Gamma^2/4 + (x - x_0)^2} -$$ - -Un caso particolare è quello in cui $x_0 = 0$ e $\Gamma = 2$, caso in cui è -detta distribuzione di Cauchy: - -$$ - f(x, 2, 0) = f(x) = \frac{1}{\pi} \cdot \frac{1}{1 + x^2} -$$ - -Il valore medio e la varianza non sono definiti perché l'integrale è -divergente. Conviene usare la moda e l'ampiezza a mezza altezza, che sono -rispettivamente $x_0$ e $\Gamma$. -Nella libreria *GSL*, la pdf è scritta in questo modo: - -$$ - p(x) = \frac{1}{a \pi (1 + (x/a))^2} - \hspace{50pt} \Longrightarrow \hspace{50pt} - a = \Gamma/2 -$$ - -## Distribuzione di Landau - -Per una particella carica con $\beta = v/c$ che attraversa un materiale sottile -di spessore $d$, la perdita di energia $\Delta$ segue la distribuzione di -Landau: - -![Distribuzione di Landau.](images/landau.png) - -Ha una forma complicatissima che racchiude integrali, logaritmi... Anche in -questo caso non si possono definire i momenti algebrici perché l'integrale -diverge. - -## Distribuzione del chi-quadro - -![Distribuzione del $\chi^2$: $n = 5$.](images/chi2.png) - -Date $N$ grandezze distribuite ciascuna con una propria distribuzione -Gaussiana, la somma dei loro quadrati segue la distribuzione $\chi^2$. -Formalmente è definita così: - -$$ - f(z, n) \frac{1}{2^{n/2} \Gamma (n/2)} z^{n/2 - 1}e^{-z/2} -$$ - -\begin{align*} - &E[z] = n \\ - &V[z] = 2n -\end{align*} - -dove $z$ è la variabile e $n$ è il numero di fradi di libertà. -Quando si fa un esperimento e si campiona $y(x)$ e poi si fittano i dati -trovati con una funzione teorica $f(x)$, ciascun valore $y(x)$ si assume -distribuito come una gaussiana attorno al suo valore vero, che assumiamo -essere $f(x)$: dunque i residui, che sono la differenza $R(x)= y(x) - f(x)$, -sono ancora una gaussiana, ma centrata in zero. Il chi quadro è definito come: - -$$ - \sum_i \frac{[y(x_i) - f(x_i)]^2}{f(x_i)} - \hspace{50pt} \text{oppure} \hspace{50pt} - \sum_i \frac{[y(x_i) - f(x_i)]^2}{\sigma_i^2} -$$ - -Ne consegue che il chi quadro segua appunto la distribuzione del chi quadro. - -Nella libreria *GSL* la distribuzione $\chi^2$ corrisponde alla distribuzione -gamma con $a = n/2$ e $b = 2$. - -## Distribuzione esponenziale - -![Distribuzione del esponenziale: $\lambda = 3$.](images/exponential.png) - -$$ - f(x, \lambda) = \lambda e^{-\lambda x} -$$ - -\begin{align*} - &E[z] = \frac{1}{\lambda} \\ - &V[z] = \frac{1}{\lambda^2} -\end{align*} - -## Distribuzione t di Student - -![Distribuzione t di Student: $\nu = 3$.](images/student.png) - -È la distribuzione seguita dalla media di una popolazione gaussiana quando -la si stima con un piccolo campione e senza conoscere la deviazione standard. -Se $y_1$ è distribuita come una Gaussiana e $y_2$ come un $\chi^2$, se $\nu$ -sono i gradi di libertà, allora $x$ segue la t di Student: - -$$ - x = \frac{y_1}{\sqrt{\frac{y_2}{\nu}}} -$$ - -che è così definita: - -$$ - f(x, \nu) = \frac{\Gamma \left( \frac{\nu + 1}{2} \right)}{\sqrt{\nu \pi} - \Gamma \left( \frac{\nu}{2} \right)} \left( 1 + \frac{x^2}{\nu} - \right)^{- \frac{\nu + 1}{2}} -$$ - -\begin{align*} - &E[z] = 0 \\ - &V[z] = - \begin{cases} - \frac{\nu}{\nu - 2} \hspace{15pt} \nu \greater 2 \\ - \infty \hspace{30pt} \nu \leqslant 2 - \end{cases} -\end{align*} - - -## Distribuzione di Fischer-Snedecor - -![Distribuzione di Fischer: $n = 3$, $m = 4$.](images/fischer.png) - -Se si hanno due campioni $\vec{x}$ e $\vec{y}$ di variabili che seguono le -rispettive Gaussiane, si può usare la distribuzione di Fisher-Snedecor per -comparare le due varianze. Se nel primo caso le variabili sono $n$ e nel -secondo sono $m$, allora la distribuzione di Fisher con gradi di libertà -$n-1$ e $m-1$ dà la distribuzione del rapporto: - -$$ - \frac{S^2_x / S^2_y}{\sigma^2_x / \sigma^2_y} = - \frac{S^2_x / \sigma^2_x}{S^2_y / \sigma^2_y} -$$ - -con: - -$$ - S^2_x = \frac{1}{n -1} \sum_{i = 1}^n (x_i - \mu_i)^2 - \hspace{50pt} \text{,} \hspace{50pt} - S^2_y = \frac{1}{m -1} \sum_{i = 1}^m (y_i - \mu_i)^2 -$$ - -che quindi è il rapporto di due grandezze distribuite secondo il $chi^2$. -La definizione della pdf è complicata... - -## Funzione caratteristica - -Si definisce funzione caratteristica di una variabile $x$ distribuita secondo -una $f(x)$, la trasformata di Fourier di quest'ultima: - -$$ - \hat{f}(k) = E[e^{ikx}] = \int\limits_{-\infty}^{+\infty} dx f(x) e^{ikx} -$$ - -come per ogni trasformata, tutte le informazioni contenute nella funzione -originaria sono contenute anche nella funzione caratteristica, perché per -tornare alla prima è sufficiente calcolare la trasformata inversa: - -$$ - f(x) = \frac{1}{2 \pi} \int\limits_{-\infty}^{+\infty} dx \hat{f}(k) e^{-ikx} -$$ - -la funzione caratteristica è utile per semplificare alcuni conti. Se -$x_1 \dots x_N$ sono variabili casuali indipendenti: - -\begin{align*} - z = \sum_{i = 1}^N x_i - \hspace{20pt} \Longrightarrow \hspace{20pt} - \hat{f}_z(k) &= \int dx_1 \dots dx_N f_1(x_1) - \dots f_N(x_N) e^{ik \sum_{i=1}^N x_i} = \\ - &= \int dx_1 f_1(x_1) e^{ikx_1} \dots \int dx_N f_N(x_N) e^{ikx_N} = \\ - &= \hat{f}_1(k) \dots \hat{f}_N(k) -\end{align*} - -Inoltre vale anche che: - -$$ - \frac{d^m}{dk^m} \hat{f}(k) \big|_{k = 0} = - \frac{d^m}{dk^m} - \int\limits_{-\infty}^{+\infty} dx f(x) e^{ikx} \big|_{k = 0} = - i^m \int\limits_{-\infty}^{+\infty} dx f(x) e^{ikx} x^m \big|_{k = 0} = - i^m \mu_m = i^m E[x^m] -$$ - -che è il momento algebrico di ordine $m$. -Per esempio, nel caso di due variabili indipendenti $x$ e $y$ gaussiane, si -può notare subito che la loro somma è una gaussiana con $\mu = \mu_x + \mu_y$ e -$\sigma^2 = \sigma_x^2 + \sigma_y^2$. Analogamente per la Poissoniana. -Inoltre è facile osservare quale sia il comportamento delle pdf nei vari limiti -che abbiamo visto in precedenza: se si manda $N \rightarrow \infty$ mantenendo -il valore medio costante nella funzione caratteristica di una binomiale, si -ottiene la funzione caratteristica di una Poissoniana. Anche il teorema -centrale del limite si può dimostrare in questo modo. diff --git a/lectures/sections/modulo-3.md b/lectures/sections/modulo-3.md deleted file mode 100644 index 4b8d693..0000000 --- a/lectures/sections/modulo-3.md +++ /dev/null @@ -1,281 +0,0 @@ -# BPH - -## Statistica descrittiva - -Si possono distinguere due tipi di analisi dei dati: "model independent" -(statistica descrittiva) e "model dependent", che si basano su un modello -teorico. In questo capitolo studiamo quelli del primo tipo. -Alcuni argomenti tipici di statistica descrittiva sono: - -- test per stabilire se due datasets provengono dalla stessa distribuzione - $f(x)$; -- test per stabilire la correlazione tra due datasets (test di ipotesi); -- metodi per determinare i momenti di una distribuzione; -- metodi per lo smoothing dei dati sperimentali. - -### Momenti di una distribuzione - -Definire i momenti di una distribuzione ha senso quando gli eventi che la -costituiscono hanno la tendenza ad agglomerarsi attorno ad un valore centrale. -Se i dati sono discreti, si usano le seguenti definizioni: - -Media campionaria: se $n_i$ è la frequenza con cui si presenta ciascun valore -$x_j$: - -$$ - \bar{x} = \frac{1}{N} \sum_{i = 1}^N x_i - = \frac{1}{N} \sum_{j = 1}^{N'} n_j x_j -$$ - -Momento centrale di ordine $r$: - -$$ - V_r = \frac{1}{N} \sum_{j=1}^{N'} n_j^r (x_j - \bar{x})^r -$$ - -Esistono anche altri due valori "centrali", che nel caso continuo diventano: -mediana: - -$$ - \int\limits_{-\infty}^{x_{\text{med}}} dx \, f(x) = \frac{1}{2} -$$ - -moda: valore per cui $f(x)$ è massima, ovvero valore che si ripete con -maggiore frequenza. -Se la pdf ha code molto estese, è possibile che gli integrali non convergano -e questi valori non siano definiti. Per questo motivo la mediana è uno -stimatore del valore centrale più robusto della media. - -I momenti centrali definiscono il modo in cui i dati si distribuiscono attorno -al valore centrale: quanto sono "diffusi". Il primo è la varianza (si noti la -correzione di Bessel per cui $N \rightarrow N -1$ al denominatore): - -$$ - V = \frac{1}{N - 1} \sum_{i = 1}^N (x_i - \bar{x})^2 -$$ - -La skewness (letteralmente "asimmetria") descrive quanto i valori siano -distribuiti in modo disuniforme attorno al valore medio: - -$$ - \gamma = \frac{1}{\sigma^3} E[(x - \bar{x})^3] -$$ - -dove $\sigma$ è la deviazione standard. - -![Skewness.](images/skewness.png){width=12cm} - -Quanto una pdf è più o meno piccata rispetto ad una gaussuana è dato dalla -kurtosis ("curved", "arching"): - -$$ - K = \frac{1}{\sigma^4} E[(x -\bar{x})^4] -3 -$$ - -![Kurtosis.](images/kurtosis.png){width=8cm} - -Esiste una stima per le deviazioni standard di questi parametri nel caso di -distribizioni circa gaussiane: - -\begin{align*} - &V(\sigma^2) = \frac{2 \sigma^4}{N} \\ - &V(\gamma) \approx \frac{15}{N} \\ - &V(K) \approx \frac{96}{N} -\end{align*} - -### Smoothing dei dati - -Lo smoothing dei dati si rende necessario quando i dati sono corrotti da un -rumore casuale. Solitamente si attua una media su finestre che inglobano dati -contigui. Fare una media, però, significa abbassare inevitabilmente il valore -nei picchi, perché la maggior parte delle volte conservano l'area al di sotto -del picco e la posizione, ma non l'altezza. -Uno dei più efficienti metodi di smoothing è il filtro di Savitsky-Golay. - -Il segnale viene analizzato a gruppi di punti incentrati ciascuno in $Y_i$, con -$i$ che scorre su tutto l'array. Chiamiamo $y_0$ il punto centrale e $Y_N$ e -$Y_{-N}$ gli estremi. $Y_0$ viene sostituito con un valore calcolato in un modo -spiegato di seguito. Durante questo processo, i valori di $Y_i$ non vengono -sostituiti con $f_i$, bensì si crea un array parallelo che sarà poi quello -definitivo smoothato. -I valori di $Y_i$ si ottengono tramite un fit sui punti della finestra con -un polinomio di grado arbitrario $g$: $P_g(j)$. Il polinomio viene poi -valutato in zero e sostituito al valore di $y_0$. - -### Test di ipotesi - -Supponiamo di voler dimostrare che una certa variabile casuale $x$ segua una -pdf $f(x)$: questa è detta ipotesi nulla $H_0$. Se $f(x)$ non dipende da alcun -parametro, si parla di ipotesi semplice, altrimenti di dice composta. Oltre -alla ipotesi nulla si possono avere una o più ipotesi alternative $H_1$, -$H_2$... -Consideriamo il semplice caso in cui abbiamo una sola ipotesi alternativa -$H_1$ che proponga a sua volta una pdf. Per valutare l'accordo tra i dati e -un'ipotesi nulla si costruisce una statistica di test $t(x)$, che è una -variabile che dipende da $\vec{x}$ che definisco per determinare se l'ipotesi -nulla sia vera oppure no (vedi $t_{\text{cut}}$ oppure la discrepanza...) e che -segue a sua volta due pdf, una prevista da $H_0$ e una da $H_1$. - -\begin{center} -\begin{tikzpicture} - - \draw [thick, ->] (0,0) -- (12,0); - \draw [thick, ->] (0,0) -- (0,6); - \node [left] at (0,6) {g(t)}; - \node [below] at (12,0) {t}; - \draw [thick, dashed] (6,0) -- (6,6); - \node [below] at (6,0) {$t_{\text{cut}}$}; - \draw [thick, blue] (0,0) to [out = 20, in = 180] (3,5) - to [out = 0, in = 180] (8,0); - \draw [thick, red] (4,0) to [out = 20, in = 180] (7,3) - to [out = 0, in = 180] (11,0); - \node [blue] at (2.5, 2) {$g(t \, | \, H_0)$}; - \node [red] at (8, 1) {$g(t \, | \, H_1)$}; - -\end{tikzpicture} -\end{center} - -Si definisce 'significanza del criterio di test' $\alpha$ (mentre $(1 - -\alpha)$ è il 'livello di confidenza del criterio di test', o 'efficienza'): - -$$ - \alpha = \int\limits_{t_{\text{cut}}}^{+ \infty} dt \, g(t \, | \, H_0) -$$ - -mentre $\beta$ è chiamato 'potenza del test' (mentre $(1 - \beta)$ è -detto 'purezza'): - -$$ - \beta = \int\limits_{-\infty}^{t_{\text{cut}}} dt \, g(t \, | \, H_1) -$$ - -Si chiamano: - -- errore di prima specie: rigezione di $H_0$ qualora questa sia vera (con - relativa probabilità $P_1$); -- errore di seconda specie: accettazion di $H_0$ qualora questa sia falsa - (con relativa probabilità $P_2$); - -Per $t < t_{\text{cut}}$ deciso arbitrariamente, imponiamo che l'ipotesi -nulla sia verificata. Ne consgue che $\alpha = P_1$ e $\beta = P_2$. -La scelta migliore di $y_{\text{cut}}$ è quella che dà la massima purezza data -una certa efficienza. Nel caso 1D lo si ottiene automaticamente (vedi esempio), -altrimenti può essere complicato. - -Facciamo un esempio in cui applichiamo il lemma di Neyman-Pearson. -Immaginiamo di avere i valori $\vec{x} = (x_1 ... x_N)$ che appartengono ad -una distribuzione normale la cui varianza $\sigma$ è nota e si deve distinguere -tra due valori medi $\mu_0$ e $\mu_1$, cioé: - -$$ - H_0 = [\mu = \mu_0] - \hspace{100pt} - H_1 = [\mu = \mu_1] -$$ - -A questo punto le pdf previste da $H_0$ e $H_1$ sono due gaussiane centrate -ciascuna nel proprio valore medio. Secondo il lemma di cui sopra, dobbiamo -calcolare la Likelihood, che è la produttoria su tutte le misure effettuate -$x_i$ della pdf prevista di un'ipotesi calcolata in $x_i$: - -$$ - L(\vec{x}, \mu, \sigma) = \frac{1}{(\sigma \sqrt{2 \pi})^N} \Pi_{i=1}^N - N(x_i, \nu, \sigma) -$$ - -dove con $N$ si indica la distribuzione normale. Si tratta, cioè, della -probabilità di avere ottenuto quelle misure secondo l'ipotesi considerata. -Vorremo, quindi, che $L(H_0) >> L(H_1)$. A questo scopo si guarda $r$, -parametro previsto dal lemma, che vale: - -$$ - r = \frac{(L(\vec{x}) \, | \, H_0)}{(L(\vec{x}) \, | \, H_1)} - \hspace{30pt} \Longrightarrow \hspace{30pt} - \ln{r} = \ln{L(\vec{x}, \mu_0, \sigma)} - \ln{L(\vec{x}, \mu_0, \sigma)} -$$ - -Che deve essere a sua volta molto grande. La regione in cui si deve accettare -l'ipotesi nulla è infatti quella con $r > c$, dove $c$ deve ancora essere -valutato. - -$$ - \ln{r} = R(\vec{x}) > \ln{c} - \hspace{30pt} \Longrightarrow \hspace{30pt} - \vec{x} > (\text{oppure} <) \, g(c) = t_{\text{cut}} -$$ - -Per scegliere $k$, si impone che: - -$$ - P_1 = \alpha = Pr(\vec{x} > (\text{oppure} <) \, t_{\text{cut}} \, - | \, H_0) -$$ - -Quindi ciò che può essere scelto arbitrariamente, alla fine dei conti, è -$\alpha$, che solitamente si impone $= 5 \%$. - -### Discriminante lineare di Fisher - -In che modo si possono definire $f(t \, | \, H_0)$ e $f(t \, | \, H_1)$? Si -possono fare degli *ansatz* riguardo alla forma di $t$. Il modello di Fischer -utilizza una funzione lineare: - -$$ - t = \sum_{i = 1}^N a_i x_i = \vec{a} \cdot \vec{x} -$$ - -dove il vettore $\vec{a}$ è da determinare. Definiamo l'insieme dei valori medi -e delle "varianze" delle variabili misurate come segue: $\mu_{k, i}$ è il valore -medio della variabile $i$-esima secondo l'ipotesi $k$-esima: - -$$ - \mu_{k,i} = \int\limits_{-\infty}^{+\infty} dx_1 \dots dx_N - \, x_i f(\vec{x} \, | \, H_k) -$$ - -dove $k$ può quindi essere 0 o 1; mentre: - -$$ - (V_k)_{i,j} = \int\limits_{-\infty}^{+\infty} dx_1 \dots dx_N - \, (x_i - \mu_{k,i})(x_j - \mu_{k,j}) f(\vec{x} \, | \, H_k) -$$ - -Si può dimostrare che, per funzioni -gaussiane, la migliore statistica di test (ovvero che massimizza $1 - \beta$ -per un dato $a$) è quella per cui: - -$$ - \vec{a} = \frac{1}{w} (\vec{\nu}_0 - \vec{\nu}_1) - \hspace{40pt} \text{con} \hspace{40pt} - W_{i,j} = (V_0 + V_i)_{i,j} -$$ - -In genere si introduce anche un offset: - -$$ - t = a_0 + \sum_{i = 1}^N a_i x_i -$$ - -### Reti neuronali - -Si può dimostrare che se si usa il discriminante lineare di Fisher, allora dati -i dati $\vec{x}$, la probabilità che sia giusta $H_0$ è: - -$$ - P(H_0 | \vec{x}) = frac{1}{1 + e^{-t}} -$$ - -![Logistic function.](images/logistic.png){width=6cm} - -che è la funzione logistica. Se le due pdf $f(\vec{x} | H_0)$ e $f(\vec{x} | -H_1)$ non sono gaussiane, allora il discriminante lineare di Fisher non è più -ottimale e si può generalizzare $t(\vec{x})$ con un caso speciale di Artificial -Neural Network (ANN). -Supponiamo di prendere - -$$ - t(\vec{x}) = s_0 \left( a_0 \sum_{i = 1}^N a_i x_i \right) -$$ - -con $s$ detta funzione di attivazione e $a_0$ detta soglia. Siccome la sigmoide -è monotona, questa ANN è equivalente ad un test lineare.