Linear Regression Models

I've seen 150+ data science candidates in the last 3 years. When I ask "How does linear regression actually work?", 90% say "It finds the best fit line." Then I ask: "How does it find that line?" Silence. You can run the code. But if you can't explain Ordinary Least Squares, you don't really understand regression. Here's what's actually happening under the hood: 𝐓𝐡𝐞 𝐏𝐫𝐨𝐛𝐥𝐞𝐦 𝐎𝐋𝐒 𝐒𝐨𝐥𝐯𝐞𝐬 You have scattered data points. You need a line that "fits" them best. But what does "best fit" actually mean? → Closest to all points? → Touches the most points? → Minimizes total error? OLS chooses option 3: minimize total error. But not just any error, the sum of SQUARED errors. 𝐖𝐡𝐲 "𝐋𝐞𝐚𝐬𝐭 𝐒𝐪𝐮𝐚𝐫𝐞𝐬"? Think about prediction errors for house prices: → Predicted $300K, Actual $310K → Error = -$10K → Predicted $350K, Actual $340K → Error = +$10K If you just add these errors: -10K + 10K = 0. Looks perfect, but both predictions were wrong. Squaring fixes this: → (-10K)² = 100M → (+10K)² = 100M → Total squared error = 200M Now you see the real cost of being wrong. 𝐇𝐨𝐰 𝐎𝐋𝐒 𝐀𝐜𝐭𝐮𝐚𝐥𝐥𝐲 𝐖𝐨𝐫𝐤𝐬 𝟏. 𝐌𝐞𝐚𝐬𝐮𝐫𝐞 𝐯𝐞𝐫𝐭𝐢𝐜𝐚𝐥 𝐝𝐢𝐬𝐭𝐚𝐧𝐜𝐞 For each data point, OLS measures the vertical distance from the point to your proposed line. This is your "residual" or prediction error. 𝟐. 𝐒𝐪𝐮𝐚𝐫𝐞 𝐞𝐚𝐜𝐡 𝐝𝐢𝐬𝐭𝐚𝐧𝐜𝐞 Why square? Three reasons: → Eliminates negative values (errors don't cancel out) → Penalizes large errors exponentially more than small ones → Makes the math solvable (you can take derivatives) 𝟑. 𝐒𝐮𝐦 𝐚𝐥𝐥 𝐬𝐪𝐮𝐚𝐫𝐞𝐝 𝐝𝐢𝐬𝐭𝐚𝐧𝐜𝐞𝐬 Add up all those squared errors. This is your "cost" or "loss" function. 𝟒. 𝐅𝐢𝐧𝐝 𝐭𝐡𝐞 𝐥𝐢𝐧𝐞 𝐭𝐡𝐚𝐭 𝐦𝐢𝐧𝐢𝐦𝐢𝐳𝐞𝐬 𝐭𝐡𝐚𝐭 𝐬𝐮𝐦 OLS uses calculus to find the slope (m) and intercept (b) where the sum of squared errors is smallest possible. 𝐖𝐡𝐲 "𝐎𝐫𝐝𝐢𝐧𝐚𝐫𝐲"? Because it's the simplest, most straightforward method. No fancy tricks. Just pure math. Other methods exist (Weighted Least Squares, Generalized Least Squares), but OLS is the foundation. Master this, and everything else makes sense. 𝐓𝐡𝐞 𝐇𝐢𝐝𝐝𝐞𝐧 𝐂𝐨𝐬𝐭 𝐨𝐟 "𝐒𝐪𝐮𝐚𝐫𝐞𝐝" 𝐄𝐫𝐫𝐨𝐫𝐬 This is where most models break in production. One outlier at $10M when everything else is under $100K? → Error = $10M - $100K = $9.9M → Squared error = ($9.9M)² = 98 trillion That one point now dominates your entire model. This is why OLS is sensitive to outliers. One bad data point can destroy everything. 𝐖𝐡𝐚𝐭 𝐭𝐨 𝐝𝐨 𝐚𝐛𝐨𝐮𝐭 𝐢𝐭: → Remove outliers (if they're genuine errors) → Use robust regression (minimizes absolute error instead) → Apply transformations (log scale to compress large values) → Use regularization (Ridge/Lasso to limit coefficient sizes) ♻️ Repost if you think understanding fundamentals beats memorizing code

*** Assumptions of Linear Regression *** ~ Linear Regression is a powerful statistical technique, but it relies on a few key assumptions to be valid. Here they are: 1. Linearity What It Means: The relationship between the dependent variable Y and the independent variable(s) X must be linear. Why It Matters: If the relationship is not linear, the predictions and insights from the model will be misleading. Detection & Remedies: * Use scatterplots to inspect the relationship between variables visually. * If the relationship appears non-linear, apply polynomial or other non-linear transformations to the independent variables. 2. Independence What It Means: The observations in the dataset should be independent of each other. Why It Matters: Violating this assumption (e.g., in time-series data where observations are dependent over time) can lead to underestimated standard errors and unreliable significance tests. Detection & Remedies: * Use the Durbin-Watson test to detect autocorrelation in residuals. * Consider time-series models or mixed-effect models for dependent observations. 3. Homoscedasticity What It Means: The variance of the residuals should be constant across all levels of the independent variable(s). Why It Matters: If this assumption is violated (heteroscedasticity), it can lead to inefficient estimates and invalid statistical tests. Detection & Remedies: * Plot residuals versus fitted values to check for patterns. * Use the Breusch-Pagan or White test for formal testing. * Transform the dependent variable or use robust standard errors to correct heteroscedasticity. 4. Normality of Residuals What It Means: The residuals (errors) of the model should be normally distributed. Why It Matters: The normality of residuals is essential for hypothesis testing and constructing confidence intervals. Detection & Remedies: * Use Q-Q plots to assess normality visually. * Apply formal statistical tests like the Shapiro-Wilk test. * Transform the dependent variable or use bootstrapping if residuals are not normally distributed. 5. No Multicollinearity What It Means: The independent variables should not be highly correlated. Why It Matters: High multicollinearity inflates the variances of the coefficient estimates and makes the model unstable. Detection & Remedies: * Calculate the Variance Inflation Factor (VIF) for each predictor. * Remove or combine correlated predictors or use dimensionality reduction techniques like Principal Component Analysis (PCA). ~ Conclusion By ensuring these assumptions are met, you can trust the insights and predictions provided by your Linear Regression model. --- B. Noted

Linear Regression in Excel with Python and Copilot 🔗 https://lnkd.in/gAcCmx8h Regression has been around forever, but it’s still one of the most useful tools in modern analytics. And now, with Copilot and Python in Excel, you can build, interpret, and visualize sophisticated regression models without leaving your spreadsheet. This post walks you through building a linear regression from scratch in Excel, step by step, using a fuel economy dataset. Here’s what you’ll learn 👇 📈 How to run simple and multiple linear regressions in Python directly in Excel. 🧮 How to interpret coefficients, evaluate model fit with R-squared and RMSE, and visualize predicted vs. actual values. 🔍 How to check model assumptions using residual plots and identify potential issues like nonlinearity or heteroskedasticity. 💡 How to make real-world predictions and understand why regression still matters for business decision-making today. If you’ve ever wanted to go beyond Excel’s built-in tools and use regression to make smarter, data-driven predictions, this post shows how Copilot makes it intuitive and powerful.

🚀 Linear Regression Ultimate Cheat Sheet When I was learning Machine Learning, linear regression looked simple at first, but the assumptions, evaluation metrics, and diagnostics were confusing. So I created this visual cheat sheet to clearly explain linear regression from fundamentals to model evaluation using Scikit Learn. 👉 What this cheat sheet covers - Linear regression equation and intuition - Key assumptions like linearity and homoscedasticity - End to end Scikit Learn workflow - Train test split, fitting, and prediction - Evaluation metrics like MAE, MSE, RMSE, and R2 - Residual analysis to diagnose model issues - Improving models using feature engineering - Regularization with Ridge and Lasso This is a practical quick reference for interviews, projects, and anyone learning Machine Learning step by step. Feel free to save and share with someone revising ML basics. I share simple AI, ML, DL, LLM, RAG, Agentic AI, and AI agent cheat sheets regularly. Follow me if you want to learn AI concepts clearly without confusion. #MachineLearning #LinearRegression #AI #ML #DataScience #ScikitLearn #Python #MLModels #AIForBeginners #TechLearning #AppliedAI

Leaving EDA Behind: I Just Mastered Linear Regression, the Y=βX+ϵ Story. 📉🚗 We've finally entered the world of Inferential Data Analysis with Linear Regression—and what a foundation it is! In economics (where we’re always solving for that OLS equation!), in finance, and in every corner of statistics, this simple model is the starting point for predictive modeling. The exercise was fitting car Speed (the dependent variable, Y) to engine Power (the independent variable, X), using data from johanneslederer.com. The Story of Three Models The best part of this chapter was seeing how choosing the right model specification dramatically improved the fit: Model 1: The Straight Line (Figure 1) - We started with simple Ordinary Least Squares (OLS), fitting a classic straight line: Y=β0+β1X. - Visually, the line was okay , but it clearly under-predicted speed at high power levels. The model was structurally sound, but the linear assumption was too rigid for the physics of car performance. - Result: RMSE was high, and the R2 reflected a mediocre fit. Model 2: The Power Law (Figure 2) - To account for diminishing returns (you get more speed from the first 50kW than the last 50kW), we used a data transformation: Y=β0+β1(X1/3). This is common in real-world modeling to capture non-linear relationships using linear math. - This transformation immediately hugged the data much better , reflecting a more realistic physics-based model. - Result: The Root Mean Squared Error (RMSE) dropped significantly, a big win for predictive accuracy! Model 3: Multiple Regression (Figure 3) - Finally, we combined the best of both worlds: Y=β0+β1X+β2(X1/3). This is Multiple Linear Regression, where the model learns how much weight to put on the raw power versus the transformed power. - This model achieved the most accurate fit by far! The resulting R2 was highest, proving that adding complexity only pays off if it's the right kind of complexity. The True Test: Holdout Validation - The ultimate lesson was moving beyond fitting the data we already had. We performed Holdout Cross-Validation—estimating the model on a training set and testing its performance on unseen data (the test set). This is the only way to ensure the model isn't just overfitting noise. - By validating on the test set, we confirmed our model's low RMSE and high R2 are genuinely predictive. This small exercise sets the stage for every complex model in finance and risk analysis, proving that the simple concepts of β estimation and residual analysis are the foundation of all advanced modeling. We're now ready to tackle logistic regression next! 🤓 #LinearRegression #InferentialStatistics #MachineLearning #DataScience #Econometrics #OLSLove #Python

No…no! Linear Regression doesn’t assume a linear relationship between the Target and the Predictors! I’ve said this countless times in my Stat lectures, and I’ll say it again here. Take this equation: Y = b0 + b1X + b2X² At first glance, it looks quadratic, surely not “linear,” right? Plot it in 2D, and yes, it curves. But here’s the catch - you’re looking at it in the wrong space. Because this equation has three variables: Y, X1 = X, X2 = X² Plot it in 3D, and what you’ll see is not a curve at all. The curve straightens itself into a plane. Beautiful, isn’t it? Nothing nonlinear about it! "When you torture a plane to live its life in a lower dimension, it bends. But in its true space, it’s perfectly flat." I have made an illustration for you hoping it'd help. Then what's the linearity assumption in Linear Regression? - Linear Regression is linear in the parameters (b0, b1, b2…), not necessarily in the raw predictors (X1, X2…). So next time someone says “linear regression only works with straight lines,” you know what to tell them. And like I always say - DON'T FALL FOR RANDOM BLOGS. Linear regression deserves better. #Statistics #DataScience

𝐃𝐚𝐭𝐚 𝐒𝐜𝐢𝐞𝐧𝐜𝐞 𝐈𝐧𝐭𝐞𝐫𝐯𝐢𝐞𝐰 𝐐𝐮𝐞𝐬𝐭𝐢𝐨𝐧: After fitting a linear regression model, you notice that the residuals are not normally distributed. How would you diagnose the cause of this issue? A go-to plot for Linear Regression is the Residual vs Fitted plot. It plots the residuals (the differences between the observed values and the predicted values) against the fitted values (the values predicted by the model). Its a good tool to check for non-linearity, heteroscedasticity, and outliers. A few simple patterns to check for - 𝘙𝘢𝘯𝘥𝘰𝘮 𝘚𝘤𝘢𝘵𝘵𝘦𝘳 (𝘐𝘥𝘦𝘢𝘭 𝘊𝘢𝘴𝘦): If the residuals are randomly scattered around 0, this suggests that the model fits the data well and that the assumptions of linearity and homoscedasticity (constant variance of residuals) are reasonably satisfied 𝘊𝘶𝘳𝘷𝘦𝘥 𝘗𝘢𝘵𝘵𝘦𝘳𝘯 (𝘕𝘰𝘯-𝘓𝘪𝘯𝘦𝘢𝘳𝘪𝘵𝘺): A curved or systematic pattern in the residuals suggests that the relationship between the predictors and the target variable is not linear. This indicates that the model may be missing key non-linear relationships. Solution: Consider adding polynomial terms (e.g., square or cubic terms) or trying transformations (e.g., log or square root) on either the dependent or independent variables. 𝘍𝘶𝘯𝘯𝘦𝘭 𝘚𝘩𝘢𝘱𝘦 (𝘏𝘦𝘵𝘦𝘳𝘰𝘴𝘤𝘦𝘥𝘢𝘴𝘵𝘪𝘤𝘪𝘵𝘺): A funnel shape in the residuals (where the spread of residuals increases or decreases as fitted values increase) indicates heteroscedasticity, meaning the residual variance changes across the range of fitted values. Solution: You can address heteroscedasticity by applying transformations to the dependent variable (e.g., log-transforming the target variable) or using Weighted Least Squares (WLS) regression. 𝘖𝘶𝘵𝘭𝘪𝘦𝘳𝘴 𝘰𝘳 𝘏𝘪𝘨𝘩 𝘓𝘦𝘷𝘦𝘳𝘢𝘨𝘦 𝘗𝘰𝘪𝘯𝘵𝘴: Residuals far from the horizontal axis or points that significantly deviate from the bulk of other points may be outliers or influential points that disproportionately affect the model. Solution: Investigate these points further using measures like Cook's distance or leverage values. Outliers might be removed or treated depending on the context. 𝘏𝘰𝘳𝘪𝘻𝘰𝘯𝘵𝘢𝘭 𝘉𝘢𝘯𝘥𝘴 𝘸𝘪𝘵𝘩 𝘕𝘰 𝘚𝘵𝘳𝘶𝘤𝘵𝘶𝘳𝘦: Points are uniformly spread with no visible clustering or patterns, which is the desired case. What it Means: If residuals form random, horizontal bands around 0 with no discernible pattern, the model is likely correctly specified, and linearity and homoscedasticity assumptions are likely satisfied. For more questions, grab a copy of Decoding ML Interviews - a book with 100+ ML questions here - https://lnkd.in/gc76-4eP 𝐋𝐢𝐤𝐞/𝐂𝐨𝐦𝐦𝐞𝐧𝐭 to see more such content. 𝗙𝗼𝗹𝗹𝗼𝘄 Karun Thankachan for all things Data Science.

𝗟𝗶𝗻𝗲𝗮𝗿 𝗥𝗲𝗴𝗿𝗲𝘀𝘀𝗶𝗼𝗻 𝗘𝘅𝗽𝗹𝗮𝗶𝗻𝗲𝗱 (𝗹𝗶𝗸𝗲 𝗮 𝗿𝗲𝗮𝗹 𝗲𝗻𝗴𝗶𝗻𝗲𝗲𝗿) Most beginners think Linear Regression is just a formula. It’s not. It’s your first real predictive system. 𝗪𝗵𝗮𝘁 𝗶𝘁 𝗮𝗰𝘁𝘂𝗮𝗹𝗹𝘆 𝗱𝗼𝗲𝘀 Linear Regression learns a straight-line relationship between input and output. Example: House size → Price Study hours → Marks Experience → Salary You give historical data. It learns a rule. Then it predicts future values. 𝗧𝗵𝗲 𝗰𝗼𝗿𝗲 𝗶𝗱𝗲𝗮 𝒚 = 𝒎𝒙 + 𝒃 Where: y ⤷ prediction x ⤷ input feature m ⤷ slope (how strongly x affects y) b ⤷ intercept (base value) In ML terms: Prediction = (weight × input) + bias That’s it. Everything else is optimization. 𝗛𝗼𝘄 𝘁𝗵𝗲 𝗺𝗼𝗱𝗲𝗹 𝗹𝗲𝗮𝗿𝗻𝘀 It starts with random weights. Then repeats this loop: Predict → Measure error → Adjust weights → Repeat The error is calculated using Mean Squared Error. Weights are updated using Gradient Descent. In simple words: It keeps nudging the line until predictions fit data. 𝗙𝗹𝗼𝘄 Data ↳ Model guesses ↳ Error calculated ↳ Weights updated ↳ Better guesses ↳ Repeat Eventually: Best-fit line achieved. 𝗪𝗵𝗲𝗿𝗲 𝗶𝘁 𝗶𝘀 𝘂𝘀𝗲𝗱 𝗶𝗻 𝗿𝗲𝗮𝗹 𝗹𝗶𝗳𝗲 Salary prediction Sales forecasting Demand estimation Risk scoring Baseline ML models Almost every ML pipeline starts here. Even deep learning engineers use Linear Regression as a sanity check. 𝗦𝗶𝗺𝗽𝗹𝗲 𝗣𝘆𝘁𝗵𝗼𝗻 𝗘𝘅𝗮𝗺𝗽𝗹𝗲 from sklearn.linear_model import LinearRegression model = LinearRegression() model.fit(X_train, y_train) prediction = model.predict([[5]]) print(prediction) That’s production-grade regression in 4 lines. 𝗪𝗵𝗲𝗻 𝗶𝘁 𝘄𝗼𝗿𝗸𝘀 𝘄𝗲𝗹𝗹 ⤷ Relationship is roughly linear ⤷ Data is clean ⤷ Outliers are controlled 𝗪𝗵𝗲𝗻 𝗶𝘁 𝗯𝗿𝗲𝗮𝗸𝘀 ⤷ Complex nonlinear patterns ⤷ Heavy outliers ⤷ Feature interactions That’s when trees or neural nets step in. 𝗧𝗵𝗲 𝗯𝗶𝗴 𝗹𝗲𝘀𝘀𝗼𝗻 Linear Regression teaches you: How models learn How loss works How optimization behaves How features influence predictions If you truly understand this… everything else in ML becomes easier. 𝗧𝗟;𝗗𝗥 Linear Regression isn’t basic. It’s foundational. It shows how machines turn data into decisions. Master this properly, and half of ML stops feeling mysterious. --- 📕 400+ 𝗗𝗮𝘁𝗮 𝗦𝗰𝗶𝗲𝗻𝗰𝗲 𝗥𝗲𝘀𝗼𝘂𝗿𝗰𝗲𝘀: https://lnkd.in/gv9yvfdd

Linear Regression is a staple in clinical trial analysis. Here is how the linear regression test statistics (the F-Test) really works: Recall that most statistical tests follow the form (Observed Data - Expected Data) - well the F-Test does this but instead of using the word "data" we use the word "Statistical Model". F-Test = Observed Model - Model assuming the null hypothesis is true. Put another way - the F-Test allows us to assume the null hypothesis that one continuous variable (like the amount of drug received in mg) does NOT impact a second variable (like blood pressure in mmHg) and to then determine if you null hypothesis should be rejected. Our null hypothesis is that there is no slope to the line through the plotted data with BP on the Y axis and the amount of drug received on the X axis (the horizontal white line in the plot attached to this post). The F-Test looks like this: SSR/SSE The numerator is known as the Sum or Squares for Regression (SSR) and this is just the sum of the squared distances between our regression model (the purple line in the image attached to this post) and the null line (the white horizontal line) at every point of observed data. SSR is just a measure of how far away our predicted model (the purple line) is from the average of all the data points we collected. The closer our model is to that average, the smaller are numerator and the smaller our F-Statistic. Recall that a small test statistic is typically related to a large p-value (we don't reject the null hypothesis). The denominator is known as the Sum of Square for Errors (SSE) and this is just the sum of the squared distances of our data points from our estimated regression line. SSE is really just a measure of how far off (the variance) our prediction line is from the actual data we recorded. Understanding that linear regression test statistics are really just subtracting the observed model by what was expected help us to better interpret what the test statistic is telling us. In this case, a significant F-Test tells us that our estimated line (purple line) is closer (on average) to the actual data we observed than the null hypothesis (white line) is. LinkedIn is not the place for full lessons on these test statistics as they do require several modules that build on top of each other, but it is a great place for quick introductions like this one on the F-Statistic. For a much more detailed description (+ examples!) of the F-Statistic and other regression measures - give me a follow and check out my website (link in the bio). I believe that better statistics leads to better research which in turn leads to a better world. My goal is to provide material to help you start and grow your understanding of statistics so that you can contribute to better research 💪. Happy Monday