From 8a4ec4c63eab4582004e46bd11a879def3d561d9 Mon Sep 17 00:00:00 2001 From: alek3y Date: Sat, 28 Sep 2024 19:44:49 +0200 Subject: [PATCH] ct0509: Add notes --- src/SUMMARY.md | 5 ++ src/ct0509/01/01/README.md | 41 ++++++++++++++ src/ct0509/01/01/assets/01.png | Bin 0 -> 82560 bytes src/ct0509/01/02/README.md | 97 +++++++++++++++++++++++++++++++++ src/ct0509/01/02/assets/01.png | Bin 0 -> 78249 bytes src/ct0509/01/README.md | 20 +++++++ src/ct0509/README.md | 1 + 7 files changed, 164 insertions(+) create mode 100644 src/ct0509/01/01/README.md create mode 100644 src/ct0509/01/01/assets/01.png create mode 100644 src/ct0509/01/02/README.md create mode 100644 src/ct0509/01/02/assets/01.png create mode 100644 src/ct0509/01/README.md create mode 100644 src/ct0509/README.md diff --git a/src/SUMMARY.md b/src/SUMMARY.md index 6e82d1a8..ffcbe8ce 100644 --- a/src/SUMMARY.md +++ b/src/SUMMARY.md @@ -4,6 +4,11 @@ # Terzo anno +- [Data e web mining](./ct0509/README.md) + - [Supervised learning](./ct0509/01/README.md) + - [k-NN](./ct0509/01/01/README.md) + - [Decision tree](./ct0509/01/02/README.md) + - [Calcolabilità e linguaggi formali](./ct0374/README.md) - [Linguaggi regolari](./ct0374/01/README.md) diff --git a/src/ct0509/01/01/README.md b/src/ct0509/01/01/README.md new file mode 100644 index 00000000..5576094e --- /dev/null +++ b/src/ct0509/01/01/README.md @@ -0,0 +1,41 @@ +# k-NN + +L'algoritmo di **$k$-nearest neighbors** prevede la classe di appartenenza semplicemente scegliendo quella **più frequente** tra i $k$ _feature vector_ (i.e. input) più vicini, secondo la loro [distanza euclidea](https://en.wikipedia.org/wiki/Euclidean_distance). + +Si vuole quindi **bilanciare** $k$: **non troppo piccolo** per evitare sensibilità al rumore, e **non troppo grande** per evitare costi computazionali elevati. +È anche possibile dare un **peso alla distanza** con $w = \frac{1}{d^2}$. + +Uno **svantaggio** della _distanza euclidea_ è che assume che ogni _feature_ (i.e. elemento del vettore) abbia lo stesso peso. +Questo si può aggirare **scalando** ogni colonna $j$ in modo che il suo intervallo +$$ +\left[\min_i A_{ij}, \max_i A_{ij}\right] \to [0, 1] +$$ +con `StandardScaler` o `MinMaxScaling`, che sono però entrambi sensibili agli _outliers_. + +Per esempio, usando dati casuali +```python +from sklearn.datasets import make_blobs +from sklearn.model_selection import train_test_split +from sklearn.preprocessing import MinMaxScaler +from sklearn.neighbors import KNeighborsClassifier + +X, X_labels = make_blobs( + n_samples=1000, centers=4, + random_state=12345 +) +X_train, X_test, y_train, y_test = train_test_split( + X, X_labels, + test_size=0.33, + random_state=42 +) + +scaler = MinMaxScaler() +scaler.fit(X_train) + +kNN = KNeighborsClassifier(n_neighbors=10) +kNN.fit(scaler.transform(X_train), y_train) +y_pred = kNN.predict(scaler.transform(X_test)) +``` +si ottiene la seguente classificazione, con le `X` normalizzate in $[0, 1]$: + +![Scatter plot 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suddivisione. + +Per esempio, dallo spazio con _feature vector_ $(x, y)$ + +![Dati in cluster opposti](assets/01.png) + +si può ricavare un albero simile a: + +```dot process +digraph { + node [shape=box] + edge [arrowsize=0.8] + + 0 [label="x ≤ 1.4" style=rounded] + 1 [label="y ≤ 1.5" style=rounded] + 2 [label="y ≤ 1.45" style=rounded] + 3 [label="Purple"] + 4 [label="Yellow"] + 5 [label="Yellow"] + 6 [label="Purple"] + + { + rank=same + _0 [shape=point width=0 height=inf style=invis] + 1 -> _0 -> 2 [style=invis] + } + + { + rank=same + _1, _2 [shape=point width=0 height=inf style=invis] + 3 -> _1 -> 4 [style=invis] + 5 -> _2 -> 6 [style=invis] + } + + 0 -> _0 [style=invis weight=100] + 1 -> _1 [style=invis weight=100] + 2 -> _2 [style=invis weight=100] + + 0 -> 1 [label="True"] + 0 -> 2 [label="False"] + 1 -> 3 [label="True"] + 1 -> 4 [label="False"] + 2 -> 5 [label="True"] + 2 -> 6 [label="False"] +} +``` + +## Classificazione + +Un algoritmo per la costruzione dell'albero è l'**algoritmo di Hunt** ricorsivo, che prende un _dataset_ $\mathcal{D}$: +```c +build_tree(D) + best_split, best_gain = null + for each feature f + for each threshold t + gain = split_goodness(f, t) // Riduzione dell'errore partizionando per (f <= t) + if gain >= best_gain + best_gain = gain + best_split = (f, t) + + if best_gain = 0 or other_stopping_criterion + μ = best_prediction(D) // Media/moda delle y di D per regressione/classificazione + return Leaf(μ) + + f, t = best_split + L = build_tree(filter(D, x : x[f] <= t)) + R = build_tree(filter(D, x : x[f] > t)) + return Node(L, R) +``` + +### Foglie + +La `best_prediction(D)` è definita come la moda delle $y$ di $(x, y) \in \mathcal{D}$, ovvero il **label più frequente**: +$$ +\argmin_\mu \mathrm{Error}(\mathcal{D}, \mu) = +\argmin_\mu \frac{1}{|\mathcal{D}|} \sum_{(x, y) \in \mathcal{D}} +\begin{cases} +0 & \text{se } \mu = y \\ +1 & \text{altrimenti} +\end{cases} +$$ + +Per semplicità, d'ora in poi si definisce 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per poter **valutare** successivamente il modello, per esempio: +```python +from sklearn.model_selection import train_test_split +from sklearn.metrics import accuracy_score + +X_train, X_test, y_train, y_test = train_test_split( + X, y, test_size=0.25, random_state=42 +) +model.fit(X_train, y_train) +y_pred = model.predict(X_test) +accuracy = accuracy_score(y_true=y_test, y_pred=y_pred) +``` diff --git a/src/ct0509/README.md b/src/ct0509/README.md new file mode 100644 index 00000000..0f8c62fb --- /dev/null +++ b/src/ct0509/README.md @@ -0,0 +1 @@ +# Data e web mining