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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