Predicting Multivariate Response in Linear Regression Model
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Bibliographic record
Abstract
Abstract Predicting a multivariate response vector in a linear multivariate regression model requires the estimate of the matrix of regression parameters. Stein (Stein, C. (1973 Stein, C. 1973. Estimation of the mean of a multivariate normal distribution. Proc. Prague Symp. Asymp. Statist., : 345–381. [Google Scholar]). Estimation of the mean of a multivariate normal distribution. Proc. Prague Symp. Asymp. Statist. 345–381), van der Merwe and Zidek (van der Merwe, A., Zidek, J.V. (1980 van der Merwe, A. and Zidek, J. V. 1980. Multivariate regression analysis and canonical variates. Canadian Journal of Statistics, 8: 27–39. [Crossref] , [Google Scholar]). Multivariate regression analysis and canonical variates. Canadian Journal of Statistics 8:27–39), Bilodeau and Kariya (Bilodeau, M., Kariya, T. (1989 Bilodeau, M. and Kariya, T. 1989. Minimax estimators in the normal MANOVA model. Journal of Multivariate Analysis, 28: 260–270. [Crossref], [Web of Science ®] , [Google Scholar]). Minimax estimators in the normal MANOVA model. Journal of Multivariate Analysis 28:260–270) and Konno (Konno, Y. (1990 Konno, Y. 1990. On estimation of a matrix of mean. Unpublished manuscript [Google Scholar]). On estimation of a matrix of mean. Unpublished manuscript; Konno, Y. (1991 Konno, Y. 1991. On estimation of a matrix of normal means with unknown covariance matrix. J. Multi. Analysis, 36: 44–55. [Crossref], [Web of Science ®] , [Google Scholar]). On estimation of a matrix of normal means with unknown covariance matrix. J. Multi. Analysis 36:44–55) have shown that their shrinkage estimators perform better than the least squares estimator. Recently, Breiman and Friedman (Breiman, L., Friedman, J. H. (1997 Breiman, L. and Friedman, J. H. 1997. Predicting multivariate responses in multiple regression. J. Roy. Statist. Soc. Ser. B, 59: 3–54. [Crossref] , [Google Scholar]). Predicting multivariate responses in multiple regression. J. Roy. Statist. Soc. Ser. B 59:3–54) proposed another class of shrinkage estimators, called C&W-GCV estimators. Through extensive simulations, they have showed that their C&W-GCV estimator performs better than the FICYREG estimator of van der Merwe and Zidek (van der Merwe, A., Zidek, J. V. (1980 van der Merwe, A. and Zidek, J. V. 1980. Multivariate regression analysis and canonical variates. Canadian Journal of Statistics, 8: 27–39. [Crossref] , [Google Scholar]). Multivariate regression analysis and canonical variates. Canadian Journal of Statistics 8:27–39), the reduced rank regression method of Anderson (Anderson, T. W. (1951 Anderson, T. W. 1951. Estimating linear restrictions on regression coefficients for multivariate normal distribution. Ann. Math. Statist., 22: 327–351. (Correction in Ann. Statist. (1980), 8, 1400)[Crossref] , [Google Scholar]). Estimating linear restrictions on regression coefficients for multivariate normal distribution. Ann. Math. Statist., 22:327–351 (Correction in Ann. Statist. (1980), 8, 1400). Estimating linear restrictions on regression coefficients for multivariate normal distribution. Ann. Math. Statist. 22:327–351. (Correction in Ann. Statist. (1980), 8, 1400)), the component-wise ridge regression and the partial least squares. They, however, did not include in their comparisons, the minimax estimators of Bilodeau and Kariya (Bilodeau, M., Kariya, T. (1989 Bilodeau, M. and Kariya, T. 1989. Minimax estimators in the normal MANOVA model. Journal of Multivariate Analysis, 28: 260–270. [Crossref], [Web of Science ®] , [Google Scholar]). Minimax estimators in the normal MANOVA model. Journal of Multivariate Analysis 28:260–270) and Konno (Konno, Y. (1990 Konno, Y. 1990. On estimation of a matrix of mean. Unpublished manuscript [Google Scholar]). On estimation of a matrix of mean. Unpublished manuscript; Konno, Y. (1991 Konno, Y. 1991. On estimation of a matrix of normal means with unknown covariance matrix. J. Multi. Analysis, 36: 44–55. [Crossref], [Web of Science ®] , [Google Scholar]). On estimation of a matrix of normal means with unknown covariance matrix. J. Multi. Analysis 36:44–55). In this article, we compare C&W-GCV estimator with two invariant minimax estimators and show that C&W-GCV does not perform as well as the two minimax estimators unless the number of response variables is fairly small compared to the number of independent variables and the sample size is small.
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Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.006 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from it