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Why Ordinary Least Square (OLS) is a Bad Option to Work With?
Introduction
Ordinary least squares is a well−liked and often used method for linear regression analysis (OLS). For data analysis and prediction, however, it is not always the best option. OLS has several limitations and presumptions that, if not properly addressed, might provide biased and false results. The drawbacks and restrictions of OLS will be covered in this article, along with some reasons why it might not be the ideal choice for all datasets and applications. We will also look at additional regression analysis approaches and methodologies that can get around OLS's drawbacks and deliver more accurate and trustworthy findings.
Data scientists and analysts may choose the best approaches for their unique datasets and applications by being aware of the drawbacks of OLS and investigating alternatives.
This will enhance the precision and quality of their predictions and analysis.
What is Ordinary Least Square (OLS)?
OLS seeks the line that best fits a given set of data points by lowering the sum of the squared residuals between the expected and actual values. The residuals are the disparities between the predicted and actual values. The best−fit line has the lowest sum of squared residuals. For OLS to operate, the residuals must be regularly distributed and have a fixed variance.
OLS is a straightforward and uncomplicated approach that doesn't need any complicated calculations or underlying data presumptions. Yet it's important to realize that OLS has several drawbacks.
Disadvantages of using Ordinary Least Square (OLS)
Some of the disadvantages of Ordinary Least Squares (OLS) are:
Responsive to outliers: The performance of the model can be severely impacted by outliers in the data since OLS is so sensitive to them. The regression coefficient estimates using OLS may be skewed if the dataset contains outliers.
OLS takes the linearity of the connection between the independent and dependent variables as given. The OLS model could yield erroneous results if this presumption is not true.
Assumption of normality and constant variance of errors: OLS assumes that the model's errors are both normally distributed and have a constant variance. It can provide skewed estimates of the regression coefficients and incorrect predictions if the errors are not normally distributed or have a non−constant variance.
Cannot handle categorical variables: OLS is not suitable for handling categorical variables or interactions between variables, which can be important factors in many real−world applications. In such cases, alternative regression techniques such as logistic regression or decision trees may be more appropriate.
Overfitting: OLS can suffer from overfitting if the model is too complex or if there are too many predictor variables. This can lead to poor generalization performance on new data.
OLS presupposes that the predictor variables are not substantially associated with one another. If there is multicollinearity or a significant correlation between the predictor variables, the regression coefficient estimations may be unstable.
Lack of robustness: OLS is not resilient to assumption violations. Slight changes in the assumptions might cause huge changes in the regression coefficient estimations.
The Alternative of using Ordinary Least Square (OLS)
Despite these shortcomings, OLS is still a popular approach in regression analysis due to its simplicity and convenience of use. Nevertheless, various other strategies can be utilized to circumvent OLS's restrictions. These are some examples:
One alternative to using Ordinary Least Squares (OLS) is to use a robust regression method, such as the Huber regression or the M−estimator. These methods are designed to handle outliers and heavy−tailed distributions that violate the assumptions of OLS.
Huber regression is a hybrid between OLS and M−estimation. It uses a loss function that is quadratic for small errors and linear for large errors, making it less sensitive to outliers than OLS. The parameter that determines the switch between the quadratic and linear loss is called the tuning constant. If the tuning constant is set to zero, Huber regression is equivalent to OLS. As the tuning constant increases, the estimator becomes more robust to outliers.
M−estimators are a type of robust regression approach that focuses on minimizing a certain objective function. The objective function is a hybrid of a loss and a weight function. The loss function calculates the difference between observed and predicted data, but the weight function gives more weight to observations that are less likely to be outliers.
The iteratively reweighted least squares (IRLS) estimator is a common Mestimator. The IRLS estimator updates the weight function repeatedly based on the current estimate of the model parameters. The weight function distributes bigger weights to observations with larger residuals in each iteration, reducing the impact of outliers on parameter estimations.
Another alternative to OLS is to use a nonparametric regression method, such as kernel regression or spline regression. Nonparametric regression methods do not assume a specific functional form for the relationship between the predictor variables and the response variable. Instead, they estimate the relationship using flexible functions that can adapt to the data.
Kernel regression works by averaging the response variable over neighboring observations, weighted by a kernel function that assigns larger weights to closer observations. The bandwidth parameter of the kernel function determines the degree of smoothing in the estimated function.
Spline regression works by dividing the predictor variable range into intervals and fitting a separate polynomial function to each interval. The degree of the polynomial and the number of intervals are determined by cross−validation or other model selection criteria. Spline regression can capture nonlinear relationships between the predictor variables and the response variable, but it may be less interpretable than parametric regression models.
Conclusion
To summarize, while OLS is a popular linear regression analysis approach, it has various limitations and assumptions that might lead to biased and erroneous conclusions. Understanding the limits of OLS and exploring other approaches, such as robust regression, ridge regression, and LASSO regression, that can overcome these constraints and produce more accurate and consistent results, are critical. Data scientists and analysts may increase the quality and accuracy of their predictions and analysis by using these alternative approaches, as well as making educated judgments about which methods to utilize for their datasets and applications.
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