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2477Why Centering Can Simplify OLS Algebra 7Why does centering features and targets often make OLS derivations cleaner when an intercept is present?机器学习中等essay未尝试免费2478Residual Orthogonality to Features 8Why must the OLS residual vector be orthogonal to every column of the design matrix at the optimum?机器学习中等derivation未尝试面试订阅2479Why Multicollinearity Hurts Coefficient Stability More Than Fit 10Why can severe multicollinearity make coefficients unstable even when training predictions barely change?机器学习中等essay未尝试面试订阅2480Orthogonal Features Give Coordinatewise Coefficients 9Suppose two features x1 and x2 are centered and orthogonal. Derive the OLS coefficients in terms of x1 T y, x2 T y, ||x1|| 2, and ||x2|| 2.机器学习困难derivation未尝试面试订阅2481Adding an Orthogonal Zero-Signal Feature 11If a new centered feature z is orthogonal to both the existing centered design and the response y, what coefficient does OLS assign to z?机器学习简单derivation未尝试免费2482Projection Interpretation of the Fitted Vector 12In one sentence, what geometric object is X beta hat in OLS?机器学习简单derivation未尝试免费2483Why Centering Leaves Slopes Unchanged 13Why does centering x and y leave the fitted slope unchanged in simple OLS with an intercept?机器学习中等derivation未尝试面试订阅2484Response Scaling 14If every target is multiplied by c, what happens to the OLS coefficient vector and intercept?机器学习困难derivation未尝试面试订阅2485Why Gradient Descent and Closed Form Agree 15Why do exact gradient descent convergence and the normal-equation solution agree for OLS?机器学习困难derivation未尝试面试订阅2486Intercept From Means 17In a simple regression with intercept, xbar = 3, ybar = 11, and beta hat = 2. What is alpha hat?机器学习简单数值题未尝试免费2487Prediction Invariance Under Equivalent Parameterizations 16Why can two different coefficient vectors produce exactly the same OLS predictions when the design is rank-deficient?机器学习中等derivation未尝试面试订阅2488Why Residual Mean Is Zero With an Intercept 18Why must OLS residuals sum to zero whenever an intercept is included?机器学习困难derivation未尝试面试订阅2489A Constant Shift in One Feature 19If a feature x is replaced by x+k in a regression that already includes an intercept, what happens to the slope on x and the intercept?机器学习中等derivation未尝试面试订阅2490Why OLS Can Still Predict Well Under Misspecification 20Why can OLS remain a useful predictor even when the true data-generating process is not exactly linear?机器学习困难essay未尝试面试订阅2491Solve a Two-Feature No-Intercept OLS System 21For a no-intercept regression with X T X = [[4, 1], [1, 9]] and X T y = [10, 19], what is beta hat?机器学习简单数值题未尝试免费2492Why Feature Scaling Helps Gradient Descent More Than Closed Form 22Why is feature scaling often crucial for gradient-descent training of OLS even though the closed-form solution itself is scale-equivariant?机器学习简单essay未尝试免费2493Projection Error Is Orthogonal to the Fitted Subspace 23Why is y - X beta hat orthogonal to every fitted vector Xv?机器学习中等derivation未尝试面试订阅2494Centered Simple Regression Through the Origin 24After centering x and y in simple regression with an intercept, what optimization problem remains for the slope?机器学习中等derivation未尝试面试订阅2495When OLS Predictions Are Unique 25Even if the coefficient vector is not unique, why is the OLS fitted prediction X beta hat still unique?机器学习困难derivation未尝试面试订阅2496Orthogonal-Design Ridge Coefficient 1In a one-feature orthogonal design with x T x = d and x T y = s, derive the ridge coefficient as a function of lambda.机器学习简单derivation未尝试免费