Linear Model

Definition

A linear model computes an output from a weighted combination of input features.

Basic form:

score = w1*x1 + w2*x2 + ... + wn*xn + b

where:

x values = input features
w values = weights
b = bias

Linear model = weighted sum plus bias.

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Input Features

The input is a feature vector.

Example:

x = [x1, x2, x3]

Each component is a feature value.

Example:

x = [hours studied, attendance, prior score]

A linear model operates on feature vectors.

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Weights

Each feature gets a weight.

Example:

weights = [w1, w2, w3]

The weight controls how strongly the feature contributes to the score.

Example:

score = 2*x1 + 5*x2

Feature x2 has larger weight than x1.

Weights control feature influence.

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Bias

The bias is an added constant.

Example:

score = 2*x1 + 5*x2 + 3

The 3 is the bias.

It shifts the score even when feature values are zero.

Bias shifts the model's decision rule.

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Vector Form

The weighted part can be written as a dot product.

Example:

w = [2, 5]
x = [3, 4]

Dot product:

dot(w, x) = 2*3 + 5*4 = 26

With bias:

score = dot(w, x) + b

Linear models are built from dot products.

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Linear Model For Classification

For classification, the linear model produces a score.

Then a rule turns the score into a class.

Example:

if score >= 0:
  class 1
else:
  class -1

The model itself computes the score.

The decision rule converts score to class.

Linear classification uses a linear score plus a decision rule.

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Why "Linear"?

It is linear because each feature is multiplied by a weight and added.

Allowed:

w1*x1 + w2*x2

Not linear:

x1*x2
x1^2

Those introduce nonlinear relationships.

A linear model combines features by weighted addition.

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Final Summary

• A linear model computes a weighted sum plus bias.
• Inputs are feature vectors.
• Weights control feature influence.
• Bias shifts the score.
• The weighted part is a dot product.
• For classification, the score is converted into a class.
• Linear models combine features by weighted addition.