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1787Ridge Effective Degrees of Freedom 2A standardized ridge model has singular-value squares d j 2 = [16, 4] and penalty lambda = 4. What is the effective degrees of freedom tr(S lambda) = sum d j 2/(d j 2+lambda)?统计中等derivation未尝试免费1813Validity or Stationarity Check 3Consider the proposal An AR(1) with phi = 1.03. Is it valid from a stationarity / autocorrelation perspective?统计中等数值题未尝试免费1819Ljung-Box InterpretationWhat null hypothesis does the Ljung-Box test target in a return series, and what practical concern does rejection raise?统计中等derivation未尝试免费1826MA(1) Lag-1 Correlation 1A microstructure noise model uses Y t = e t + 0.5 e (t-1). What is its lag-1 autocorrelation rho(1)?统计中等derivation未尝试面试订阅1949Derive the Optimizer for Log Utility Minus Quadratic Cost 4For parameters a>0 and b>0, derive the unique maximizer of F(x)=a ln(1+x)-b x 2 on x>-1.数学中等derivation未尝试免费1959Large Cushion Interior Minimum 14The reciprocal term is strong enough that the optimum sits well away from the boundary. Minimize H(x) = 1 x 2 + 36/(1+x) over x > -1.数学困难derivation未尝试免费1997Strict Convexity of a Barrier-Regularized Cost 2The desk wants a direct curvature argument, not a vague appeal to 'it looks bowl-shaped'. Show that f(q) = 3 q 2 + 1/(1-q) is strictly convex on q<1.数学简单derivation未尝试免费2006Minimum Ridge for a Sharper Quartic Approximation 11A local approximation is even more non-convex, so the repair parameter must rise accordingly. A local PnL model is h(q)=q 4-10q 2+lambda q 2. What is the smallest lambda that makes h globally convex?数学简单数值题未尝试免费2007Three-Sleeve Convex Balance-Sheet Penalty 12The third sleeve is much more expensive on its own, but the aggregate term still preserves convexity. Prove that F(w 1,w 2,w 3) = 1w 1 2 + 4w 2 2 + 9w 3 2 + 2(w 1+w 2+w 3) 2 is convex.数学简单derivation未尝试免费2019Convexity of Logistic Loss 24Show that ell(z)=ln(1+e -z ) is convex on R.数学中等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2499Soft-Thresholded Lasso Coefficient 4In an orthogonal one-feature problem with x T x = d and x T y = z > 0, derive the lasso coefficient when 0 < lambda < z.机器学习中等derivation未尝试面试订阅2508Why Elastic Net Keeps the Lasso Threshold but Adds Ridge Shrinkage 14Why does elastic net still need |z| to clear an L1 threshold before a coordinate activates, but then shrink the active coefficient more than lasso does?机器学习中等derivation未尝试面试订阅3266Best Rank-One Reconstruction Error From Singular ValuesA centered data matrix has singular values \sigma 1\ge\sigma 2\ge\cdots\ge\sigma r. What is the Frobenius-norm reconstruction error of the best rank-one approximation?数学中等derivation未尝试面试订阅3268Recovering Correlation Strength from a Known First PCA covariance matrix has the form \Sigma=egin pmatrix 5&c\c&5\end pmatrix . You are told that the first principal component points along (1,1) and has variance 8. What is c?数学中等derivation未尝试面试订阅3269One-PC Reconstruction of a Two-Asset MoveLet \Sigma=egin pmatrix 9&-3\-3&9\end pmatrix , whose first principal direction is along (1,-1). For the observed move x=(2,-1), what are the rank-1 reconstruction using only the first principal component and the residual?数学中等derivation未尝试面试订阅3270How Standardization Changes the PCA SpectrumA raw covariance matrix is \Sigma=egin pmatrix 9&6\6&9\end pmatrix . After standardizing each coordinate to unit variance, what are the eigenvalues of the correlation matrix, and what fraction of standardized variance does the first component explain?数学中等derivation未尝试面试订阅3272How Large Can the Third Singular Value Be?A data matrix has singular values 7, 4, and s. You plan to keep rank 2 and want to retain at least 95\% of total squared Frobenius energy. What is the largest allowable value of s?数学困难derivation未尝试面试订阅3273Which Portfolio Is Better Aligned with the Low-Variance PC?A covariance matrix has eigenvalues 25 and 1, with first eigenvector proportional to (2,1) and second eigenvector proportional to (1,-2). Compare the variances of portfolios p 1=(1,-2) and p 2=(2,1).数学中等derivation未尝试面试订阅3277Covariance of PCA ScoresIf data are centered and projected onto orthonormal eigenvectors of the covariance matrix, what is the covariance matrix of the resulting PCA score vector?数学中等derivation未尝试面试订阅