# 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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## Two Machine Learning Fields

There are two sides to machine learning:

• Practical Machine Learning:This is about querying databases, cleaning data, writing scripts to transform data and gluing algorithm and libraries together and writing custom code to squeeze reliable answers from data to satisfy difficult and ill defined questions. It’s the mess of reality.
• Theoretical Machine Learning: This is about math and abstraction and idealized scenarios and limits and beauty and informing what is possible. It is a whole lot neater and cleaner and removed from the mess of reality.

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