Data Science R Regression Regression

Machine Learning Mastery: Locally weighted Linear Regression

By on Tuesday, June 15, 2021

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 theta to minimize the cost.
J(theta) = $sum_{i=1}^{m} (theta^Tx^{(i)} - y^{(i)})^2

Predict output: for given query point x,
return: theta^Tx


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 theta are computed individually for each query point x. While computing theta, a higher “preference” is given to the points in the training set lying in the vicinity of x than the points lying far away from x.

The modified cost function is:  J(theta) = $sum_{i=1}^{m} w^{(i)}(theta^Tx^{(i)} - y^{(i)})^2

where, w^{(i)} is a non-negative “weight” associated with training point x^{(i)}.
For x^{(i)}s lying closer to the query point x, the value of w^{(i)} is large, while for x^{(i)}s lying far away from x the value of w^{(i)} is small.

A typical choice of w^{(i)} is:  w^{(i)} = exp(frac{-(x^{(i)} - x)^2}{2tau^2})
where, tau is called the bandwidth parameter and controls the rate at which w^{(i)} falls with distance from x

Clearly, if |x^{(i)} - x| is small w^{(i)} is close to 1 and if |x^{(i)} - x| is large w^{(i)} is close to 0.

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

For example –

Consider a query point x = 5.0 and let x^{(1)} and x^{(2) be two points in the training set such that x^{(1)} = 4.9 and x^{(2)} = 3.0.
Using the formula  w^{(i)} = exp(frac{-(x^{(i)} - x)^2}{2tau^2})  with tau = 0.5:
w^{(1)} = exp(frac{-(4.9 - 5.0)^2}{2(0.5)^2}) = 0.9802
w^{(2)} = exp(frac{-(3.0 - 5.0)^2}{2(0.5)^2}) = 0.000335
So, J(theta) = 0.9802*(theta^Tx^{(1)} - y^{(1)}) + 0.000335*(theta^Tx^{(2)} - y^{(2)})
Thus, the weights fall exponentially as the distance between x and  x^{(i)} increases and so does the contribution of error in prediction for x^{(i)} to the cost.

Consequently, while computing theta, we focus more on reducing (theta^Tx^{(i)} - y^{(i)})^2 for the points lying closer to the query point (having larger value of w^{(i)}).

Steps involved in locally weighted linear regression are:

Compute theta to minimize the cost.  J(theta) = $sum_{i=1}^{m} w^{(i)}(theta^Tx^{(i)} - y^{(i)})^2
Predict Output: for given query point x,
return: theta^Tx

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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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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