How can you address underfitting in linear regression?

All the models including linear regression are based on certain assumptions. If the assumptions are aligned with data, then the model can represent the data well. Else we have a concern. Linear regression has assumptions of linearity, homoscedasticity, normality etc. There are various tests to see if data follows these assumptions. If we find that data follows these assumptions then we are good. Else we have to do transformations of the data, however if model still doesn't fit, then it's better to explore other models.

Underfitting in linear regression models occurs when the model is too simple to capture the complexity of the data. This usually occurs when the relationship between the predictor variables and the response variable is more complex than the model can represent. Here are some strategies to address underfitting in linear regression: ? Increase Model Complexity ? Include Additional Variables ? Regularization ? Cross Validation ? Adjust Hyperparameters ? Consider Other Models Identify Outliers and Anomalies: Remember that the choice of strategy will depend on the specific nature of your data and the problem you are addressing. It is advisable to test different approaches and evaluate model performance to determine the most effective strategy.

Residual plots are a great way to determine if you have some extra work to do before continuing with a more simple model like linear regression. Patterns that appear such as a set of negative points, then positive points, the negative points can indicate a non-linear relationship (polynomial or exponential) or may indicate a pattern break in the data where one linear model describes the first part of data and a second linear model describes the second part. Digging into residuals early can save a lot of time later!

As an early career Researcher, this has happened to me a lot of times. I believe it is very important to check the hypothesis first and decide on your outcome variable before jumping on to the regression. Cleaning dataset and removing outlier values is extremely crucial. Regarding the regression, check if your outcome variable follows the basic assumptions of OLS model. Also, before directly jumping to interpret the results, check if you have covered enough variables to predict an outcome. Also if you have non-zer values, other models like poison or negative binomial may fit better (depending upon your predictor variable).

By reducing noise in the data and increasing the feature numbers in the data set. It was the question asked by my examination panel during the first year of my PhD. This question also encouraged me to increase the duration of training data which is an important factor to tackle underfitting.

1. Check if the training dataset is having noise. Apply proper data processing including missing data imputation, data transformation etc. to remove noise. 2. Check if all the basic assumptions of linear regression hold good (linearity, independence, normality, homoscedasticity). Also check if other assumptions are valid e.g. no multicollinearity in data, predictors are normally distributed. 3. Add more independent variables in the model 4. Increase model complexity (e.g. use hyperparameter tuning. GridSearchCV and RandomSearchCV can tune the parameters better. Also try decreasing regularization if you have used any). You can also try polynomial fitting and check if the underfitting gets resolved but please be cautious of overfitting.

Think before adding complexity, because it must add value to make sense. Add features that cover characteristics of the data not covered by other features (=> avoid multicollinearity). Go beyond linear relationships if you can explain why that relationship is non-linear. A model you can't understand risks being brittle or, worse, useless for the customer.

Increase Training Data to address underfitting. Gather more data if possible as a larger dataset may help the model better understand the underlying patterns in the data.

Underfitting occurs when the model is too simple to capture the underlying patterns in the data. By introducing additional, relevant predictors, you increase the model's complexity, allowing it to better fit the data. This could mean including interaction terms to capture the relationship between variables, or incorporating polynomial features to model non-linear relationships. However, caution is necessary to avoid overfitting, where the model becomes too complex. It’s a delicate balance; the goal is to enhance the model’s predictive power without losing its ability to generalize to new data. Regular evaluation using cross-validation can help in fine-tuning the number and nature of features to optimize the model's performance.

By using domain knowledge, you can extract more information from the data, create features that capture the underlying patterns and relationships, and select the most important features for your model.

Underfitting occurs when the model is too simplistic, failing to capture the underlying trends in the data. Enhancing complexity can involve adding more variables, introducing polynomial terms, or considering interaction effects between predictors. This allows the model to better adapt to the intricacies of the data. However, it's crucial to balance complexity with the risk of overfitting, where the model becomes overly tailored to the training data, losing generalizability. Techniques like cross-validation can help in assessing the right level of complexity. Regularly evaluating model performance with new data is essential to ensure it accurately reflects the underlying patterns without being overly complex.

1.?Boosting the?model's?complexity:?Increasing?the?model's?complexity?is?one?strategy?to?combat?underfitting. To?achieve?this,?expand?your?linear?regression?equation?by?include?more?features?or?polynomial?terms. As?a?result,?the?model?can?represent?more?complex?correlations?between?the?variables. 2.?Feature?engineering:?In?certain?cases,?it?may?not?be?possible?to?fully?capture?the?underlying?relationships?with?the?features?that?are?currently?available. In?these?situations,?you?can?modify,?combine,?or?scale?the?current?features?to?produce?new?ones. This?could?benefit?the?model.

Sometimes, parsimonious models work well because they are simple models with great explanatory predictive power. So, we can reduce the number of parameters in a model.

One thing I have found helpful, in a medical study, researchers aim to predict the likelihood of a rare disease based on various patient characteristics. A simple linear model fails to provide accurate predictions due to the intricate relationships between genetic factors, lifestyle, and environmental exposures. So by Increasing model complexity by employing advanced machine learning techniques, such as ensemble methods or deep learning, allows the model to capture intricate nonlinear relationships. This results in a more accurate disease prediction model, aiding early diagnosis and intervention.

One thing I've found very useful is an exponential model because the majority of phenomena are exponential. I also recommend cross validation as a powerful tool, but you must note that this will require a powerful machine.

Don't forget to do analysis because people tends to skip the EDA and jump straightforward to boosting tree methods. Linear Regression is one of the most well studied method in statistics and can be super powerful if the assumption are met (which to be honest rarely happens , but not impossible)

One of the most important issues regarding model selection is about the tree based models. As boosted and bagged forest models are battle tested with many benchmark datasets, they give the impression as the primary choice. But be careful whether your data and task are applicable with a tree based model, such as necessity for extrapolation etc.

There are several alternative methods that can be considered instead of linear regression in economic models. For example, one method to use is logistic regression, which is employed when the dependent variable is binary or categorical. Logistic regression is estimated using the maximum likelihood method, and the parameters are chosen to maximize the likelihood of observing the data.

From an econometric research point of view, addressing linear regression underfitting involves, exploring models aligning with econometric principles: Generalized Linear Models (GLMs) for non-normal responses. Panel Data Models like fixed/random effects for multi-dimensional data. Instrumental Variable Regression to tackle endogeneity. Time Series Models (e.g., ARIMA, VAR) for temporal data. Quantile Regression for diverse impacts across outcome distributions. ...and many more... Each model's choice should be guided by data characteristics, research questions, and underlying economic theories, with performance evaluated using econometric criteria.

At the basis, you should first look whether the thing you are predicting is numerical or categorical. Don't apply a regression model when predicting a categorical and don't apply a classifier (including logistic regression) when predicting something on a scale. If this basic rule is not followed, everything you do is bogus.

There is another important metric. The Akaike information criterion (AIC) compares the fit of different regression models. AIC estimates the relative amount of information lost by a given model: lesser the loss, higher the quality of that model. Once you’ve fit several regression models, you can compare the AIC value of each model. The lower the AIC, the better the model fit. However, AIC tells nothing about the absolute quality of a model, only the quality relative to other models. Similarly, there is BIC which stands for Bayesian Information Criterion. BIC is a model based on Bayesian principles on complexity measure but places a greater penalty on models with more parameters. BIC may be preferred for larger sample sizes.

Try various techniques to address underfitting and evaluate their performance. This involves experimenting with different feature sets, model complexities, or entirely different algorithms. Evaluate each approach using metrics like cross-validation, R-squared, or mean squared error to determine which method best mitigates underfitting while maintaining generalizability.

In a recent project on patient treatment responses, linear regression initially underfitted. To tackle this, I experimented with data transformations, like logarithmic scaling, and added features such as demographics and treatment history. I then shifted to a random forest model, utilizing Python's sklearn for its nonlinear handling. Seaborn was invaluable for visualizing data and model fits. K-fold cross-validation, employed for validation, improved robustness against overfitting. This process not only enhanced accuracy but also illuminated patient response patterns, emphasizing the need for flexibility in modeling and rigorous validation. PS. could also use R (randomForest and caret packages)

Experimentation and evaluation are fundamental processes that enable us to test ideas, generate knowledge, make informed decisions, and improve our understanding of the world. They are essential tools for scientific progress, innovation, and societal advancement.

Experimentation and evaluation are critical in refining models and making informed decisions. Let's consider this exemple, a hospital adopts a machine learning model to predict the likelihood of patient readmission. The model is trained on historical patient data, and its performance is initially evaluated using standard metrics. Upon deploying the model, the hospital monitors its performance in a real clinical environment. The evaluation exposes unexpected challenges, such as changes in patient demographics or shifts in treatment protocols. Through continuous evaluation and iteration, the model is adapted to maintain accuracy and reliability in the ever-evolving healthcare landscape.

In my prespective any model is assamption which missing specific data , so if you have too much underfiting you shuld add more explanition. Also, most of the models not supposed to be liner, they can be ljner on part of them so you can split the data into 2 or more segment to check how your model react in each.

Reducing regularization in linear regression models can help deal with underfitting. Regularization techniques, like L1 or L2, add a term to the loss function that penalizes the complexity of the model. This encourages the model to favor simpler or smoother solutions. However, if a model is already underfitting, increasing regularization can worsen the issue. It prevents the model from becoming complex enough to capture all the relationships in the data. Therefore, reducing regularization relaxes the constraints on model complexity, allowing the model to learn the patterns in the data more freely, which may decrease underfitting.

Les 5 principaux proposés suffisent largement pour résoudre le problème. La recherche effrénée de la perfection me semble contreproductive puisque elle est plus guidée par nos appréhensions et non par l’observation et l’analyse des données dont on dispose. On veut tellement arriver absolument à nos fins qu’on cède au vice qui gangrène la réputation des statisticiens à l’effet qu’ils peuvent faire dire aux chiffres ce qu’ils veulent.

Transforming features, such as through logarithmic or exponential adjustments, can linearize relationships in skewed data. Adding more relevant features can provide a fuller representation of the underlying phenomena. Addressing data noise and randomness with smoothing techniques, and ensuring data quality and relevance are also key. Additionally, removing outliers, possibly using percentile-based methods or other techniques, can improve model accuracy by focusing on more representative data points. If these approaches don't yield improvements, considering alternative, non-linear models like decision trees or neural networks might be necessary, always validating the model to ensure it generalizes effectively beyond the training data.

Linear regression model performance can also be improved by transforming the variables prior to training. Transformations include normal cumulative distribution, polynomial and logistic functions. One strategy to maximize performance is for each variable calculate the optimal parameters of each transformation function that minimizes AIC value using an optimization function such as optim() in R. Select the best transformations and use the transformed variables for your linear regression model.