Least Squared Error vs. Mean Squared Error

Attribute	Least Squared Error	Mean Squared Error
Definition	Minimizes the sum of the squares of the differences between the observed and predicted values	Calculates the average of the squares of the differences between the observed and predicted values
Optimization	Uses calculus to find the optimal parameters that minimize the error	Also uses calculus to find the optimal parameters that minimize the error
Application	Commonly used in regression analysis	Commonly used in evaluating the performance of predictive models
Formula	$$\sum_{i=1}^{n} (y_i - \hat{y}_i)^2$$	$$\frac{1}{n} \sum_{i=1}^{n} (y_i - \hat{y}_i)^2$$

Introduction

When it comes to evaluating the accuracy of a model, two common metrics used in regression analysis are Least Squared Error (LSE) and Mean Squared Error (MSE). While both metrics are used to measure the difference between the actual values and the predicted values of a model, they have some key differences in terms of their calculation and interpretation.

Calculation

Least Squared Error is calculated by taking the sum of the squared differences between the actual values and the predicted values. This means that each individual error is squared before being summed up. On the other hand, Mean Squared Error is calculated by taking the average of the squared differences between the actual values and the predicted values. This means that the total squared error is divided by the number of observations in the dataset.

Interpretation

One key difference between LSE and MSE lies in their interpretation. LSE gives equal weight to all errors, regardless of their magnitude. This means that outliers or extreme values can have a significant impact on the overall error. In contrast, MSE gives more weight to larger errors due to the squaring of the differences. This can make MSE more sensitive to outliers and extreme values in the dataset.

Robustness

Another important aspect to consider when comparing LSE and MSE is their robustness to outliers. LSE can be heavily influenced by outliers since it does not give more weight to larger errors. This means that a single outlier can have a disproportionate impact on the overall error. On the other hand, MSE is more robust to outliers due to the squaring of the errors. This means that larger errors are penalized more in MSE, making it less sensitive to outliers.

Optimization

When it comes to optimizing a model, both LSE and MSE can be used as the objective function. However, the choice between the two metrics can depend on the specific goals of the analysis. If the goal is to minimize the total error without considering the distribution of errors, LSE may be more appropriate. On the other hand, if the goal is to minimize the impact of larger errors and outliers, MSE may be a better choice.

Application

In practice, both LSE and MSE are commonly used in regression analysis to evaluate the performance of a model. LSE is often used when the focus is on minimizing the total error, while MSE is preferred when the goal is to penalize larger errors more heavily. Ultimately, the choice between LSE and MSE depends on the specific requirements of the analysis and the desired outcome.