# Locally weighted Linear Regression

Linear regression is a supervised learning algorithm used for computing linear relationships between input (X) and output (Y).

The steps involved in ordinary linear regression are:

Training phase: Compute to minimize the cost. Predict output: for given query point ,  As evident from the image below, this algorithm cannot be used for making predictions when there exists a non-linear relationship between X and Y. In such cases, locally weighted linear regression is used. ### Locally Weighted Linear Regression:

Locally weighted linear regression is a non-parametric algorithm, that is, the model does not learn a fixed set of parameters as is done in ordinary linear regression. Rather parameters are computed individually for each query point . While computing , a higher “preference” is given to the points in the training set lying in the vicinity of than the points lying far away from .

The modified cost function is: where, is a non-negative “weight” associated with training point .
For s lying closer to the query point , the value of is large, while for s lying far away from the value of is small.

A typical choice of is: where, is called the bandwidth parameter and controls the rate at which falls with distance from Clearly, if is small is close to 1 and if is large is close to 0.

Thus, the training-set-points lying closer to the query point contribute more to the cost than the points lying far away from .

For example –

Consider a query point = 5.0 and let and be two points in the training set such that = 4.9 and = 3.0.
Using the formula with = 0.5:   Thus, the weights fall exponentially as the distance between and increases and so does the contribution of error in prediction for to the cost.

Consequently, while computing , we focus more on reducing for the points lying closer to the query point (having larger value of ). Steps involved in locally weighted linear regression are:

Compute to minimize the cost. Predict Output: for given query point , Points to remember:

• Locally weighted linear regression is a supervised learning algorithm.
• It a non-parametric algorithm.
• There exists No training phase. All the work is done during the testing phase/while making predictions.

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