Regression analysis is the methodology that attempts to establish a relationship between a dependent variable and a single or multiple independent variable.
Regression natively is a statistical concept, but it is finding its applications in many business-related fields such as finance, investment, stock markets, as well as in areas such as science and engineering.
There are some up-and-coming applications of regression analyses in the form of data science, machine learning, and Artificial Intelligence that mark the future of humanity.
Terminologies related to Regression
To understand types of regression analysis, understanding the related terminologies is useful.
Outliers are visible with data plots on a graph. In regression analysis(1), the outliers are points on the graph that fall significantly outside the cloud made up of other points. The outlier points are essential because they can heavily influence the outcome of a regression analysis.To understand this concept, let’s suppose that a building is filled with professionals with average financial backgrounds in terms of their earnings.
They all have a mean salary of around one hundred thousand dollars a year. Suddenly, Bill Gates and Jeff Bezos step into the building, and once you include the salaries of these two billionaires, the mean salary becomes drastically inaccurate.The salaries of these two well-known gentlemen are the outliers in this example.
In regression analysis, looking at the correlation between two or more input variables, it is observable that when the addition of one or more input variables takes place, the model fails to make things more transparent about the real world.
It is crucial to find out how input variables relate to each other.Measuring the multicollinearity of the regression model is a way to find the relationship between input variables.As an instance, you may come across a model in which you are willing to find out what determines the salary of a person at a particular of age.Independent variables (factors) such as educational background, age, and many other factors that influences average salary of an individual are brought under consideration.
But, before you go any further and throw every factor under the Sun in your model, you need to know how they correlate (inter-associate). If the multicollinearity goes too high, it causes disturbance of data and the model falls apart.
Heteroscedasticity (sometimes spelled as heteroskedasticity) occurs when the reading of a variable’s standard error (SE) measured over a given time is not constant.
Any regression analysis running on such data that exhibits heteroscedasticity gives, at the very least, biased coefficients and ruin the results.
The overfitting in a regression analysis is the occurrence when the variables start to show random errors rather than efficiently describing the relationship among the variables. The overfitting produces a lot of noise rather than the true representation of the population. The outcome of the model is not realistic any more.You need to make your model as close to reality as possible.As an example of outfitting from the real world. The best possible word that describes outfitting from the real-world example is “over-generalization.” When error or the bias is increasing, the realistic values are unable to be determined as an outcome.
Underfitting happens when the number of variables scarcely fits a given model, and the output does not remain accurate. To have successful results from a regression analysis, you need the optimum values of the variables, so the model obtained is close to reality.In short, when the variables are not optimized, or the model does not fit the data efficiently, it is called an underfit.
Types of Regression Analysis
There are two types of variables in any form of Regression. One is the independent variables, or they are also called explanatory variables, they are used for inputs. The other type of variable is a dependent variable, also known as the predictor. It is the value that you are trying to find out or the outcome of the model.
The following describes the different types of regression analysis.
Linear regression deals with two types of variables. One variable is called an independent variable, and the other kind of variable is the dependent variable.
The independent variable varies along the x-axis of the cartesian plane, and the dependent variable varies along the y-axis. These variables are “x” and “y,” respectively. The value of y depends on x. When x changes, the “y” either increase or decrease.
There are two types of Linear Regression.
- Simple Linear Regression
- Multiple Linear Regression
- Simple Linear Regression: In simple Linear Regression, there is only one dependent variable and one dependent variable.
The equation for simple linear regression is y=β_0+β_1 xHere, x represents the independent variable, is the slope of the regression line, and is the y-intercept. “y” is the dependent variable or the outcome.
- Multiple Linear Regression: In Multiple Linear Regression, the dependent variable is one, but you have multiple independent variables.
The following equation represents the Multiple Linear Regression,y= β_0+β_1 x_1+⋯β_n x_n+ εHere, y is the dependent variable, is the y-intercept. denote the multiple independent variables in the model. is the “bias” or “error.” The minimization of bias or error is our primary objective in order to create a model close to the real-world situation.
The Multivariate Regression is different from Multiple Linear Regression in the sense that it has multiple dependent variables with the input of multiple independent variables. The dependent variables ( y_1,y_2 ,y_3 …. y_n) are in different formulae. And it has more than one independent variables ( x_1, x_2, ….x_m ) to predict the Ys. In Multivariate Regression the data that is used, is mostly of the same type as in other types of Regression Analysis.
Logistics Regression is the second most popular form of Regression after linear Regression, and its uses span biostatistics, medicine, and social sciences.
The logistics regression deals with Boolean values such as,
- true or false
- yes or no
- big or small
- one or zero
The Logistics Regression is used in the classification of objects such as an email is “spam” or “not spam.”
In short, there is one outputs in logistic Regression that can either be “True” or “False.” Moreover, there can be a single input or multiple inputs in a Logistics Regression models.
There are cases when we have to deal with variables whose relationship is non-linear. In such a case, our model is a curve, not a line unlike in Linear Regression. Thus, we have another form of Regression known as polynomial Regression.
The equation of polynomial Regression is the ascending powers of the input variable x, a generalization of which is below.
y= β_0+ β_1 x+〖β 〗_2 x^2+〖 β〗_3 〖 x〗^3+⋯β_n x^n+ ε
The Quantile Regression definition is very different from what it is in practice. The quantile is another name of the median in statistics.
Quantile is the point or line that splits the output data into two equal parts. Imagine some data set in the form of a line on the y-axis. The data set is divided into exactly two equal pieces. The value of the quantile is 0.5 or 50% at the point of the split.
On the same note, the two equally divided pieces of data are equally split again along the y-axis. This time we got the data split into four equal parts, and the new split points at the lower y-axis of the graph are 0.25 or 25%.
Similarly, the upper y-axis split quantile is 0.75 or 75%. In general, the quantiles are just lines or points that split data into equal chunks or groups.
Quantiles spit data in a hundred equally sized groups. But, in the real world, the definition of quantile is much more flexible.
Quantile regression is useful when there is a presence of high heteroscedasticity in the model, and linear Regression is not accurate enough to predict the outcome because the linear model relies on mean values and quantiles can be more precise with median values.
The ridge regression employs a technique that is called “Regularization.” The regularization is appropriate for models that fail on testing data but pass on the training data.
Ridge regression works best when most of the variables in the model are useful.
When sample data is showing multi collinearity, two unwanted things happen,
- The Least Square estimates of coefficients of the predictor variables give high errors.
- There is inflation in standard errors.
Ridge Regression is a technique for the stabilization of the regression coefficients in the presence of multicollinearity.
Lasso stands for “Least Absolute Shrinkage and Selection Operator.” Lasso Regression performs best when you have a lot of useless variables. Lasso Regression resembles Ridge regression, but some differences make it unique.
The Ridge Regression and Lasso Regression have applications to the same scenarios in which multicollinearity is present. However, Ridge Regression is suitable for long term predictions.
The Lasso Regression applies shrinkage to the data. The data values become shrink towards a central point like the median or the mean.
The simplification and sparseness of data models are the functions where Lasso Regression does the best. In other words, the data models should have the optimum parameters for accurate outcomes.
Principal Component Regression (PCR)
The Principal Component Analysis has an application to the x variable, reducing the dimensionality of the data. It involves the extraction of data set with most variations in an iterative process.
Since the process is iterative so it can analyze a multi-dimensional data set, the Principal Component Regression overcomes the dimensionality and collinearity problems present in ordinary Least Squares Regression.
Elastic Net Regression
Elastic Net Regression simplifies down a model for the ease of interpretation. A model can have tons of variables (aka parameters); they can range up to millions in specific models. In such a model, it is not possible to determine which variables are useful and which are useless.
In such a case, you do not know which regression type to choose from Ridge Regression and Lasso regression. Here, the Elastic Net Regression comes into play to simplify the model.
The Elastic-Net Regression combines a Ridge Regression penalty with the Lasso Regression penalty and gives the best of both worlds. It also works better with correlated variables.
Partial Least Squares (PLS)
The partial Least Squares considers both the explanatory and the dependent variables. The underlying principle of this type of Regression is that x and y variables go through decomposition into latent structures in an iterative process.
PLS can deal with multicollinearity. It takes into account the data structures related to x and y, providing you with elaborately visual results for the interpretation of data. Several variables can come into consideration.
Support Vector Regression
The Support Vector Regression (SVR) is an algorithm that works with a continuous function. It is in contrast with Support Vector Machine in this sense that Support Vector Machine (SVM) deals with classification problems. SVR predicts continuous ordered variables.
In simple Regression, the emphasis has to be on minimizing the error while Support Vector Regression finds out the threshold of the error.
The Logistics Regression deals with two categories, but in Ordinal Regression (aka Ordinal Logistics Regression), three or more categories come into play with the assumption of unambiguous ordering.
Ordinal Regression helps to predict an ordinal dependent variable when one or more independent variables are present.
In Poisson Regression, the count or rate at which the event occurs is the main point of focus.
We measure the rate at which event occurs in Poisson Regression. In other words, we model the number of times the event occurs (count) over time. In Poisson Regression, the time is constant, and we measure the count of the event.
Negative Binomial Regression
It is useful to model the discrete (count) data set. On the same note, Negative Binomial Regression helps when the data has a higher variance compared to the mean that is the data’s dispersion is too much when you plot it.
The Negative Binomial Model does not assume that the variable is equal to mean as the model based on Poisson Regression makes.
Quasi Poisson Regression
The Quasi Poisson Regression is the generalization of Poisson Regression. As mentioned before, the Poisson Regression Model hinges on a usually unfair assumption that variance is equal to mean.
The Quasi Poisson Model comes in play when the variance is the linear function of mean, and it is also higher than the mean. It is the scenario when Quasi Poisson is more appropriate to be applicable.
Cox Regression (aka Proportional Hazards Regression) investigates the effects of several variables for the duration of time a specified event takes to occur.
Consider the following events where Cox Regression can be found useful,
- The time it took for a second heart attack after the first heart attack.
- The time it took for the second accident after the first accident.
- The time it took after the cancer detection till death.
The time-to-event data is vital for the application of cox regression.
Tobit Regression comes in handy in the estimation of a linear relationship when censoring is found in the dependent variable. Censoring is the observation of all independent variables. The actual account of the value of the dependent variable is in only in a restricted range of observations.
The Bayesian Regression is based on probability distribution rather than on point estimation. As a result, the output or the “y” is not a single value. It is a probability distribution. As we know that probability distribution is a mathematical function and not a value. The probability distribution gives possible outcomes in an experiment.
When we compose the formulation of the linear regression model based upon the probability distribution, we get the following expression.
y ˜ N(β^T X,σ^2 I)
- The output (y) is calculated from a normal Gaussian Distribution depending upon mean and variance.
- The transpose (T) of the weight matrix (β) is obtained by multiplying it with the predictor matrix (X).
- The variance is the standard deviation squared (σ^2 ) multiplied by the Identity matrix (I).
(The multi-dimensional formulation of the model is under consideration)
Least Absolute Deviation (LAD) Regression
The Least Absolute Deviation is the most widely known alternative to the Least Square method to analyze the linear models. We know that in the Least Square method, we minimize the sum of the squared errors, but in LAD, we minimize the sum of absolute values of errors. It tries to find a function that closely fits a set of data.
In a case where our data is simple, the Least Absolute Deviation is a straight line in two-dimensional Cartesian Plane.
The formulation of the Least Absolute is very straightforward to understand. Let’s suppose our data set consists of two variable points ( (x_i ,y_i) and the i=1,2,3,4,5……n.
Our objective is to find a function f such that is approximately equal to (~) as shown below.
f(x_i ) ~ y_i
The claim is that the function f is of a specific form containing some parameters that we need to calculate. The point to note here is that the function f can have I number of x parameters (or independent variables or explanatory variables).
We will attempt to find out the values of parameters that will minimize the following sum of the absolute values of the errors (or residuals).
S = ∑_(i=1)^n▒〖|y_i 〗-f(x_(i) )
Ecological Regression is instrumental mostly in subjects like political sciences and history. The technique allows us to take counts on a macro level and come up with predictions on a micro- level.
The Ecological Regression can determine the voting behavior of individuals between different factions and groups of societies. The estimation is based on data that is collected from previous accounts.
The ecological data is based on counts in a particular region, groups, objects or, over time. In short, the aggregate data helps us to learn about the behavior narrowed down to individuals.
What is the Regression Analysis Used For?
The regression analysis is useful in obtaining several business objectives.
One of the most prominent applications is the predictive analysis that allows forecasting of specific business events more accurately. One type of predictive analysis is the “demand analysis,” which measures the increase in the sales of a product. The success of a newly launched product, as well as running products, can be positioned correctly in the market.
As another example, Regression Analysis has applications in the advertisement of products and services. It is predictable with Regression Analysis that how many shoppers are likely to come across an advertisement. It helps the sales and marketing professionals set the bid value of promotional materials.
Regression Analysis is also a helpful tool for insurance companies. Insurance companies use it to find out the credit of policyholders and estimate the number of claims likely to be put forward from their clients.
Organizations make serious decisions using Regression Analysis to optimize their operations.
Data-driven decisions can rule out questionable decisions, inaccurate guesswork with gut feelings, and corporate politics.
The Regressive Analysis is converting the art of management into a science. As an example, it is possible to relate the wait time of a caller with the number of complaints in a call center or a customer care department.
Decision Making Support
The organizations today have loads of data relating to finance, marketing, operations, and many other departments. The top decision-makers are leaning more towards data analytics and data science to make more informed decisions with the elimination of guesswork.
With the help of Regression Analysis, big data can undergo compression for action-oriented lean information opening the path to more accurate decision making. The regression analysis does not remove or replace managers; instead, it puts a potent tool in their hands to make more impactful and efficient decisions than ever before.
Regression Analysis also helps identify intuitive errors in judgment and decision making for business managers.
As an example, a store manager may decide to keep the store open at night time for which he decides to hire new staff.
The Regression Analysis can accurately indicate that considering the expenses of the staff and the total sales that it generates at night time cannot have mutual justification. Thus, quantitative application of Regression Analysis enables to rule out bad decision making.
Companies understand and acknowledge the value of data and what can be achieved by the techniques of regression analysis, but many fail to convert this data into actionable insights. Driving insights from raw data is not an easy task. A report by Forrester claims that 74% of companies want to decide with data inputs, but only 29% succeed in obtaining analytics that can allow them to make fruitful decisions.
One critical case study from the business world is Konica Minolta. Konica was one of the most successful manufacturers of cameras. In 2000, most photographers and camera enthusiasts shifted to digital cameras.
The top decision-making body at Konica did not take decisions fast enough as a result by 2004 when Konica launched its first camera, most of the competitors like Nikon and Canon had well established themselves in the new digital camera market. As a result, in 2006, the company suffered such heavy losses that it sold out much of its technology and assets to Sony.
If Konica had the insights from the raw commercial and market data processed through regression analysis and similar techniques, Konica would have been able to make the right decision at the right time.
Data regression analysis providing actionable insights puts sheer power in the hands of decision makers that can be game changers in the real world.
How to Pick the Right Regression Model?
There are hundreds of types of Regressions, and we have covered the most popular types.
The real world is very complex, and the model creators measure many variables but include only a few in the model. The analysts exclude the independent variables that have very little to no impact on the dependent variable or the outcome.
When selecting a regression model, the following simple fact should be kept in mind to maintain balance by putting the correct number of independent variables in the regression equation.
- Too few independent variables, the unspecified model becomes bias.
- Too many independent variables, the unspecified model loses its precision.
- Just the Right model comes into creation when the math terms are not biased and are the most precise.
The Regression Analysis has its origins in statistics that is a science hundred years old, but it is recently has gained the spotlight of attention as the big data is exploding. Regression Analysis is finding its way through statistics in data analytics, data science, and their applications in almost all organizations.
The Regression Models created with Regression Analysis are an indispensable tool for the enhanced provision of predictability, operation efficiency, well-informed decision making, prevention of error, averting wrong decisions, and better insights.