How do residuals relate to the assumptions of linear regression?

Residuals are the differences between the observed and predicted values, serving as a measure of how well your model fits the data and its predictive power. They help identify outliers that may impact the model's efficiency. Analyzing residuals is an effective method for validating a linear model's assumptions of normality, linearity, and homoscedasticity.

The following relationships show how to calculate the residual for a data point: y = y' + e y' = (mx + b) y: observed value y': predicted value (result of linear regression) e: remainder (can be 0, positive or negative) Finally: e = y - y'

Residuals in linear regression help verify key model assumptions: linearity, independence, homoscedasticity, and normality. Linearity is checked by plotting residuals against predicted values; no patterns should appear. Independence is assessed through the random scatter in time-sequential plots. Homoscedasticity is indicated by constant residual variance across predicted values, checked via scatter plots. Normality is verified with a Q-Q plot, where residuals should align with a normal distribution line. Multicollinearity, though primarily about predictors, can influence residual behavior and variance. Proper residual analysis ensures the reliability of regression outcomes.

In simple ML terms, a residual is the difference between the actual value (y) and the predicted value (y'). It can be calculated as (y - y'). In the context of linear regression, for example, we aim to find a linear equation (a line) that represents the relationship between the data points. Suppose we draw a line randomly; we calculate the residuals as the differences between the actual data points and the corresponding points on the line.

Residuals in linear regression analysis reveal how well the model fits the data. It validates the model's assumptions are met: 1. There is a linear relationship between dependent and independent variables.? 2. The residuals are independent and uncorrelated with one another. 3. The variance of the residuals is constant (homoscedasticity). 4. Residuals are normally distributed; this is especially important for smaller sample sizes.

If when plotting the residuals vs the predicted values we observe that the points are shown randomly around a horizontal line at the value 0, it indicates that there is a linear dependence between the variables. But if any systematic pattern like a curve is observed, it means that the model is not correctly capturing the relationship between the independent variable and the dependent variable or that there is no linear relationship between them. A Q-Q plot of the residuals would also help in this analysis.

The goal of using residuals in linear regression is to fit a line (by adjusting its slope and intercept) that minimizes these residuals, typically by minimizing the sum of the squared residuals. This process helps us find the line that best fits the data, avoiding overfitting and ensuring the model generalizes well to new data. If the data follows a linear relationship, the residuals will be randomly distributed around 0, without any discernible pattern. However, if the residuals display a systematic pattern, such as a curve, it suggests that the relationship between the variables is non-linear, indicating that a linear model may not be appropriate.

Plotting the residuals against the predicted values will help assess the relationship between them. Spotting any clear pattern (a curve) suggests non-linearity, as the model might not properly capture the relationship.? Using a different model or transforming the variables are ways to capture that underlying relationship. Performing a linearity check helps to assess if there are any non-linearity issues that need to be addressed.

The reliability of a regression model requires that there is independence of the residuals, which means that the value of one residual should not influence another residual (with the exception of time series). If a plot of residuals over time is drawn and some patterns such as curves are observed, it is an indicator that the residuals could be related, which is not convenient for the model. To validate it we can use the Durtbin Watson test that measures the autocorrelation of the residuals. Values close to 2 indicate that there is no autocorrelation (independence of the residuals), otherwise it indicates autocorrelation (there is no independence of the residuals) so the model requires adjustments.

A residual plot is an excellent tool for determining whether residuals are independent, which is crucial for reliable regression analysis. Ideally, the error from one prediction should not influence another. The exception to this is time series data, where one observation affects another. You can check for independence by plotting residuals against time variables or any spatial variables present. A systematic pattern in the plot indicates a lack of independence.

Independence tests help check that the assumption of independent residuals is not violated, as this can lead to poor prediction.? Making a residuals vs. time plot can reveal there is no systematic pattern, and residuals are uncorrelated to one another. Analysis in time series has specific techniques for assessing autocorrelation among the residuals. There are several methods and tools to validate the regression model in order to draw reliable inferences from it.

The concept of homoscedasticity refers to the existence of the same variance of the residuals in all the independent variables. This can be verified by making a graph of residuals vs predicted values. The data points must be randomly around a horizontal line that cuts the Y axis (residuals) at 0. Otherwise, if well-defined patterns are observed, there is heterosedasticity and requires using variable transformation techniques such as the application of the logarithm function.

Homoscedasticity is another assumption that you need to check for.?The spread of the residuals (errors) should be consistent across the values of dependent variables.?You can view plots of residuals vs. predictors for constant variance. There are a few methods to use like the White Test and the Breusch-Pagan test. If this assumption is violated, there are a few ways to address it. You can perform different transformations on the variables and help to achieve homoscedasticity. Outliers can influence your data so can either be removed or transformed accordingly. Robust regression methods are useful for handling outliers. You can also fit a generalized linear model or run a weighted least squares regression.

While normality of residuals is not strictly necessary for linear regression coefficients to be unbiased and consistent, it is crucial for conducting hypothesis tests and constructing reliable confidence intervals. Departures from normality in residuals may indicate model misspecification or influential outliers, compromising the validity of statistical inferences drawn from the regression analysis. Therefore, assessing normality of residuals helps ensure the reliability and validity of the linear regression model's results and interpretations.

Addressing residual issues in linear regression involves adjusting the model or employing alternative techniques. For non-linearity, consider variable transformations or polynomial terms. Autocorrelation among residuals can be managed through time-series modeling or adjustments for serial correlation. Heteroscedasticity may require variable transformations, weighted least squares, or robust standard errors. Non-normal residuals can be addressed with transformations, robust regression methods, or outlier identification.