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# Regression

The most popular data science technique among data scientists is remodeling Logistic and linear regression. Due to their popularity, a lot of analysts even end up thinking that they are the only form of regressions. The ones who are slightly more involved think that they are the most important among all forms of regression analysis.

1. Linear Regression

Regression is a technique used to model and analyze the relationships between variables and oftentimes how they contribute and are related to producing a particular outcome together. A linear regression refers to a regression model that is completely made up of linear variables. Beginning with the simple case, Single Variable Linear Regression is a technique used to model the relationship between a single input independent variable (feature variable) and an output dependent variable using a linear model i.e., a line.

The more general case is Multi-Variable Linear Regression where a model is created for the relationship between multiple independent input variables (feature variables) and an output dependent variable. The model remains linear in that the output is a linear combination of the input variables. We can model a multi-variable linear regression as the following:

Y = a_1*X_1 + a_2*X_2 + a_3*X_3 ……. a_n*X_n + b

Where a_n are the coefficients, X_n are the variables and b is the bias. As we can see, this function does not include any non-linearities and so is only suited for modeling linearly separable data. It is quite easy to understand as we are simply weighing the importance of each feature variable X_n using the coefficient weights a_n. We determine these weights a_n and the bias b using a Stochastic Gradient Descent (SGD)

1. Polynomial Regression

When we want to create a model that is suitable for handling non-linearly separable data, we will need to use a polynomial regression. In this regression technique, the best fit line is not a straight line. It is rather a curve that fits into the data points. For a polynomial regression, the power of some independent variables is more than 1. For example, we can have something like:

Y = a_1*X_1 + (a_2)²*X_2 + (a_3)⁴*X_3 ……. a_n*X_n + b

We can have some variables have exponents, others without, and also select the exact exponent we want for each variable. However, selecting the exact exponent of each variable naturally requires some knowledge of how the data relates to the output.

1. Ridge Regression

A standard linear or polynomial regression will fail in the case where there is high collinearity among the feature variables. Collinearity is the existence of near-linear relationships among the independent variables. The presence of high weighing collinearity can be determined in a few different ways:

A regression coefficient is not significant even though, theoretically, that variable should be highly correlated with Y.

When you add or delete an X feature variable, the regression coefficients change dramatically.

Your X feature variables have high pairwise correlations (check the correlation matrix).

We can first look at the optimization function of a standard linear regression to gain some insight as to how ridge regression can help:

min || Xw – y ||²

Where X represents the feature variables, w represents the weights, and y represents the ground truth. Ridge Regression is a remedial measure taken to alleviate collinearity amongst regression predictor variables in a model. Collinearity is a phenomenon in which one feature variable in a multiple regression model can be linearly predicted from the others with a substantial degree of accuracy. Since the feature variables are so correlated in this way, the final regression model is quite restricted and rigid in its approximation i.e. it has high variance.

To alleviate this issue, Ridge Regression adds a small squared bias factor to the variables:

min || Xw — y ||² + z|| w ||²

Such a squared bias factor pulls the feature variable coefficients away from this rigidness, introducing a small amount of bias into the model but greatly reducing the variance.

1. Lasso Regression

Lasso Regression is quite similar to Ridge Regression in that both techniques have the same premise. We are again adding a biasing term to the regression optimization function to reduce the effect of collinearity and thus the model variance. However, instead of using a squared bias like ridge regression, lasso instead using an absolute value bias:

min || Xw — y ||² + z|| w ||

There are a few differences between the Ridge and Lasso regressions that essentially drawback to the differences in properties of the L2 and L1 regularization:

Built-in feature selection: is frequently mentioned as a useful property of the L1-norm, which the L2-norm does not. This is a result of the L1-norm, which tends to produce sparse coefficients. For example, suppose the model has 100 coefficients but only 10 of them have non-zero coefficients, this is effectively saying that “the other 90 predictors are useless in predicting the target values”. L2-norm produces non-sparse coefficients, so does not have this property. Thus one can say that Lasso regression is a form of “parameter selections” since the feature variables that aren’t selected will have a total weight of 0.

Sparsity: refers to that only very few entries in a matrix (or vector) are non-zero. L1-norm has the property of producing many coefficients with zero values or very small values with few large coefficients. This is connected to the previous point where Lasso performs a type of feature selection.

Computational efficiency: L1-norm does not have an analytical solution, but L2-norm does. This allows the L2-norm solutions to be calculated computationally efficiently. However, L1-norm solutions do have sparsity properties that allow them to be used along with sparse algorithms, which makes the calculation more computationally efficient.

1. ElasticNet Regression

ElasticNet is a hybrid of Lasso and Ridge Regression techniques. It is used both the L1 and L2 regularization taking on the effects of both techniques:

min || Xw — y ||² + z_1|| w || + z_2|| w ||²

A practical advantage of trading-off between Lasso and Ridge is that it allows Elastic-Net to inherit some of Ridge’s stability under rotation.

A few key points about ElasticNet Regression:

It encourages group effects in the case of highly correlated variables, rather than zeroing some of them out like Lasso.

There are no limitations on the number of selected variables.

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