# Classification and Representation

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7 月第一天!!! oh~~ Summer~~ GOGO!

## Classification : article

Classification:

• EX:
1. Email: Spam / Not Spam?
2. Online Transactions: Fraudulent (Yes / No)?
3. Tumor: Malignant / Benign?
`````` # binary classification problem
y ⊂ {0, 1}
0: "Negative Class" (e.g., benign tumor)
1: "Positive Class" (e.g., malignant tumor)

# multiclass classification problem
y ⊂ {0, 1, 2, 3, ...}
``````
• 使用 linear regression 來做 classification problems 通常不優!
從大神的圖可以看到，如果再加一數值進到資料集裡面，曲線又會再做更便兒~

• 讓我們再想想

`````` Classdification: y = 0 or 1

hθ(x) can be > 1 or < 0
``````

Logistic Regression is actually a classification algorithm:

Logistic Regression: 0 ≤ hθ(x) ≤ 1

## Hypothesis Representation : article

Logistic Regression Model:

• Want 0 ≤ hθ(x) ≤ 1
`````` hθ(x) = g ( θ^T * x )
``````
• Sigmoid function (Logistic function):
`````` g(z) = 1 / 1 + e^-z
``````

We Got:

• ``````hθ(x) = 1 / 1 + e^-(θ^T * x)
``````

Interpretation of Hypothesis Output:

hθ(x) = estimated probability that y = 1 on input x

• EX:
``````        | x0 |   |     1     |
if x = | x1 | = | tumorSize |
hθ(x) = 0.7
``````
• Tell patient that 70 % chance of tumor being malignant
• hθ(x) = P(y=1|x;θ), y = 0 or 1 “Probability that y = 1, given x, parameterized by θ”
`````` P(y=0|x;θ) + P(y=1|x;θ) = 1
P(y=0|x;θ) = 1 - P(y=1|x;θ)
``````

## Decision Boundary : article

Logistic regression:

``````hθ(x) = g ( θ^T * x )
g(z) = 1 / 1 + e^-z
``````
• y = 1 `(θ^T * x) ≥ 0`
• y = 0 `(θ^T * x) < 0`

Decision Boundary:

Non-linear decision boundaries:

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