Linear Regression

Linear Regression

Goal: Find line of best fit

$\hat{y} = w_{0} + w_{1}x$

y $\approx \hat{y} + \epsilon$

x: feature

y: label

Learn weights that minimize the residuals

• Blue point: True value
• Red line: Positive residual
• Green line: Negative residual

Regression Evaluator

Measure "closeness" between label and prediction

Evaluation metrics:

• Loss: $(y - \hat{y})$
• Absolute loss: $|y - \hat{y}|$
• Squared loss: $(y - \hat{y})^2$

Evaluation metric: RMSE

$Error = (y_{i} - \hat{y_{i}})$

$SE = (y_{i} - \hat{y_{i}})^2$

$SSE = \sum_{i=1}^n (y_{i} - \hat{y_{i}})^2$

$MSE = \frac{1}{n}\sum_{i=1}^n (y_{i} - \hat{y_{i}})^2$

$RMSE = \sqrt{\frac{1}{n}\sum_{i=1}^n (y_{i} - \hat{y_{i}})^2}$

Train vs. Test RMSE

Which is more important? Why?

R2

Another measurement of "goodness of fit"

$SS_{\text{tot}}=\sum_{i=1}^n(y_{i}-{\bar {y}})^{2}$

$SS_{\text{res}}=\sum_{i=1}^n(y_{i}-\hat{y_{i}})^{2}$

$R^{2} = 1-{SS_{\rm {res}} \over SS_{\rm {tot}}}$

What is the range of R2?

Machine Learning Libraries

Can train multiple scikit-learn models in parallel with Joblib or Pandas UDFs, but what if our data or model get big...

ML Libraries in Spark

MLlib

Original Spark ML API based on RDDs

SparkML

Spark 2.0: Entered maintenance mode

Supported API moving forward

How to Handle Non-Numeric Features?

Categorical

• No intrinsic ordering
• e.g. Gender, Country, Occupation

Ordinal

• Relative ordering, but inconsistent spacing between categories
• e.g. Excellent, good, poor

One idea

Create single numerical feature to represent non-numeric one

Categorical features:

• Animal = {'Dog', 'Cat', 'Fish'}
• 'Dog' = 1, 'Cat' = 2, 'Fish' = 3

Implies Cats are 2x dogs!

One Hot Encoding (OHE)

Create a ‘dummy’ feature for each category

'Dog' => [1, 0, 0], 'Cat' => [0, 1, 0], 'Fish' => [0, 0, 1]

No spurious relationships!

Storage Space

Ok, so that works if we only have a few animal types, but what if we had a zoo?

Sparse Vectors

Size of vector, indices of non-zero elements, values


DenseVector(0, 0, 0, 7, 0, 2, 0, 0, 0, 0)
SparseVector(10, [3, 5], [7, 2])


Assumptions?

Linear relationship between X and Y

Features not correlated

Errors only in Y

Which ones are suited to Linear Regression?
Slide credit G. Calmattes