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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2498Lasso Zero Threshold in an Orthogonal Coordinate 3In an orthogonal one-feature problem with x T x = d and score z = x T y, for what lambdas does the lasso coefficient become exactly zero?机器学习中等derivation未尝试免费2504Compute a Ridge Coefficient Numerically 9In an orthogonal coordinate with d = 9, z = 18, and lambda = 3, what is the ridge coefficient?机器学习中等数值题未尝试免费2514Equivalent L2 Radius in One Dimension 19In one dimension, if the ridge solution equals beta hat lambda, what radius t makes the constrained problem min RSS(beta) subject to |beta| <= t share the same optimizer?机器学习困难derivation未尝试面试订阅2516Coordinate-Descent Update for a Positive Orthogonal Lasso Coordinate 21In an orthogonal coordinate with d = 5, z = 11, and lambda = 3, what coefficient does one exact lasso coordinate-descent update return?机器学习简单数值题未尝试免费2518Ridge Shrinkage Ratio Numerically 23In an orthogonal coordinate with d = 6 and lambda = 2, what fraction of the OLS coefficient remains under ridge?机器学习中等derivation未尝试面试订阅2520Why L1 and L2 Pull Differently Near Zero 25Why does L1 regularization create a stronger qualitative push toward exact zero than L2 regularization near the origin?机器学习困难derivation未尝试面试订阅2525One Newton Step for an Intercept-Only Logistic ModelAn intercept-only logistic model is fit to 7 positives and 3 negatives. Starting from b 0 = 0, what is one Newton step b 1 for minimizing the negative log-likelihood?机器学习困难数值题未尝试面试订阅2540Intercept Shift for a Deployment Prior ChangeA logistic model was trained under class prior 0.5 and has intercept -0.4. At deployment the base rate falls to 0.2 while feature likelihood ratios are assumed unchanged. What adjusted intercept should be used?机器学习困难数值题未尝试面试订阅2576Why Feature Subsampling Helps When One Predictor Dominates 12Why can random feature subsampling improve a forest when one very strong predictor would otherwise appear at the top of almost every tree?机器学习简单essay未尝试免费