Physicists often spend their lives trying to understand the fundamental laws of the universe, but when it comes to data science technology, we’re doing nothing different, just with fewer particle but with more compute power! Regression Analysis, for example, is trying to find the correlation between different variables, not unlike connecting the stars in a cosmic constellation.
Regression Analysis: The Fundamentals
In the ever-expanding data science industry, regression analysis is a statistical method that seeks to establish relationships between variables. Imagine attempting to hit the bull’s eye on a dartboard – regression analysis will tell you how one variable (the wind) will affect the other (the dart’s trajectory). This data science technique will enable you to model relationships, make predictions and uncover trends.
Birds Of the Same Feather – The Types of Linear Regression
Dealing with a complex set of numbers, algorithms, cold, hard statistics that require both intuition and rigorous math understanding – not something all of us possess, but nothing that cannot be acquired. To delve into the data science domain, let us observe and understand what the fundamental regression techniques do based on their types:
Linear Regression – This simple form of statistics is a statistical regression method that models a dependent variable and an independent variable. In simpler terms, models the relationship between data variables using a linear equation.
Linear Regression
y = β₀ + β₁x + ε
- Where β₀ is the intercept, β₁ is the slope, and ε is the error term
- Estimation: β̂ = (X'X)⁻¹X'y (using Ordinary Least Squares)
**Basic Equation: **
y = β₀ + β₁x + ε
- Where β₀ is the intercept, β₁ is the slope, and ε is the error term
- Estimation: β̂ = (X'X)⁻¹X'y (using Ordinary Least Squares)
Polynomial Regression – Consider this, dear reader, an extension of Linear regression. It fits a curved line using polynomial terms (x², x³, etc.) to capture nonlinear relationships between data entities. Especially useful when the data shows a curved or cyclical pattern, polynomial regression in data science is also prone to overfitting, anyway.
Polynomial Regression
y = β₀ + β₁x + β₂x² + ... + βₙxⁿ + ε
- n is the degree of the polynomial
- Higher degrees capture more complex curves
Multiple Linear Regression – In Multiple Linear Regression that includes multiple independent variables, modeling more complex real-world relationships. Think of it as keeping on adding the same least acquire principle in multiple dimensions, and you get the idea.
Multiple Linear Regression
y = β₀ + β₁x₁ + β₂x₂ + ... + βₚxₚ + ε
- p is the number of predictors
- Matrix form: Y = Xβ + ε
Ridge Regression – Ridge Regression addresses the multicollinearity (ooph!) of a system by adding a penalty term which is supposed to be square of coefficient magnitudes, preventing our coefficients from getting too excited but prevents them from becoming excessively large. Ridge Regression is used when dealing with multi-correlated features between the data, keeping all variables but at the same time reducing their impact on the model (L2 Regression)
Ridge Regression (L2)
min (||y - Xβ||² + λ||β||²)
- λ is the regularization parameter
- ||β||² is the L2 norm of coefficients
- Solution: β̂ᵣᵢᵈᵍᵉ = (X'X + λI)⁻¹X'y
Lasso Regression (L1 Regularization) – Widely known as the minimalist’s approach, when selecting features, it eliminates all unnecessary variables and redundant terms in the algorithm.
Lasso Regression (L1)
min (||y - Xβ||² + λ||β||₁)
CURIOUS MUCH?
Regression doesn’t just end there. There are several other types, some are even included in ensemble ML models. Let’s take a look at those:
Elastic Net Regression - Elastic Net Regression techniques often synthesize L1 and L2 models elucidated above, creating an entirely new regression model that can combine the best of both worlds. Practical applications include when dealing with multiple correlated features while wanting to maintain some degree of feature selection capabilities of L1 and L2.
Elastic Net Regression
min (||y - Xβ||² + λ₁||β||₁ + λ₂||β||²)
- Combines L1 and L2 penalties
- λ₁ and λ₂ control the strength of each penalty
Bayesian Regression – Any article on statistics in data science is incomplete without the mention of Bayesian Regression. This technique incorporates prior knowledge about parameters and produces probability distributions instead of point estimates. It also allows for uncertainty quantification in its predictions and can be updated as new data becomes available for ingestion and pre-processing.
Bayesian Regression
P(β|y) ∝ P(y|β) P(β)
- P(β) is the prior distribution
- P(y|β) is the likelihood
- P(β|y) is the posterior distribution
Quantile Regression – Quantile Regression models the relationship between variables at various quantiles of the response of the distribution, not just the mean. This is especially useful when a data scientist encounters extreme values or outliers or even when the relationship between variables varies across the distribution.
. Quantile Regression
min Σ ρτ(yi - xiᵀβ)
- ρτ(u) = u(τ - I(u < 0))
- τ is the desired quantile
- I() is the indicator function
Feature Engineering – Using domain knowledge to create new variables from existing ones and math transformations form the foundation of feature engineering. It can also create interaction terms, polynomial features and overall, enhance model performance.
Cross Validation – Cross validation is used for assessing the performance of a model by repeatedly splitting training available datasets into training and testing sets and then deploying them in testing. This helps overfitting and provides a more accurate estimate of how well the model will generalize to new data. This does not require statistical analysis.
Model Selection - This is where the data scientists come in. The process of choosing between the different types of regression models based on performance metrics and problem requirements helps them choose the right model for the right task. This involves balancing the complexity of the model with predictive accuracy when comparing interpretability and computing power requirements. This does not require statistical analysis.
Gaussian Process Regression – The Gaussian process is not parametric. It approaches the model so that the target variables as just a sample from a multivariate distribution of this nature. This regression technique is specialized in providing uncertainty estimates naturally and can capture complex multi-linear relationships without specifying a functional form.
Gaussian Process Regression
f(x) ~ GP(m(x), k(x,x'))
- m(x) is the mean function
- k(x,x') is the covariance function
- Prediction: y* = K*(K + σ²I)⁻¹y
Neural Network Regression – NNR uses artificial neural networks to create models of complex relationships between input and output data without assuming a fixed form. Just as flexible as the brain is, neural networks do not assume any specific fixed form, but reconfigure themselves to solve the problem at hand, without human intervention. Neural Network Regression, a relatively new concept, has the power to automatically learn feature representations and capture highly irregular patterns, if the dataset is large enough and has been carefully tuned.
Neural Network Regression
y = f(Wₙ⋅f(... f(W₁x + b₁)...) + bₙ)
where,
Wᵢ are weight matrices
bᵢ are bias vectors
f is the activation function
Multicollinearity – Multicollinearity occurs when independent variables in a regression model are highly correlated to one another. This often leads to unstable and unreliable coefficient estimates, increasing the difficulty levels and reducing the model’s performance. This does not require statistical analysis.
Heteroscedasticity – Heteroscedasticity is a condition where the residuals of an error is not constant across all the values of the variables. This can lead to inefficient parameter estimates and error prone standards, invalidating the tests altogether. This does not require statistical analysis.
Overfitting- We’ve saved the most common for the last. Overfitting, a common error among new and inexperienced data scientists or even expert ones, occurs when a model learns the training data too well, including the noise it contains and the outlying fluctuations. The consequences are many: poor generalization of pre-processed data, even if the test performed is perfect but the output is indictive of a poor performance. This does not require statistical analysis.
CONCLUSION
Becoming the master of regression analysis in particular, and statistics in general are highly sought after by most aspiring data scientists willing to enter the field. With the demand for skilled and certified shows no signs of stopping or even slowing, these data science certifications often spell rapid career growth for the incumbent. The data science industry urgently needs talented engineers NOW!
Take your first step, embark on this fascinating journey and remember, by harnessing the power of regression analysis, you will have the power to unlock secrets within big data, driving innovation and progress.
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