Revision #1768 → #2285 · back to history
modifiedLinear regression model449cd1ea13c9
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| anchors | [{"section":"Formulation","snippet":"a linear regression model assumes that the relationship between the dependent variable y and the vector of regressors x is linear"},{"type":"math_alttext","value":"{\\displaystyle y_{i}=\\beta _{0}+\\beta _{1}x_{i1}+\\cdots +\\beta _{p}x_{ip}+\\varepsilon _{i}=\\mathbf {x} _{i}^{\\mathsf {T}}{\\boldsymbol {\\beta }}+\\varepsilon _{i},\\qquad i=1,\\ldots ,n,}"}] | — |
modifiedBall tossed in the air532ef9987c50
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| anchors | [{"section":"Example","snippet":"Consider a situation where a small ball is being tossed up in the air"},{"type":"math_alttext","value":"{\\displaystyle h_{i}=\\beta _{1}t_{i}+\\beta _{2}t_{i}^{2}+\\varepsilon _{i},}"},{"type":"math_alttext","value":"{\\displaystyle h_{i}=\\mathbf {x} _{i}^{\\mathsf {T}}{\\boldsymbol {\\beta }}+\\varepsilon _{i}.}"}] | — |
modifiedGroup effecta16289970e6b
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| anchors | [{"section":"Group effects","snippet":"is defined as a linear combination of their parameters"},{"type":"math_alttext","value":"{\\displaystyle \\xi (\\mathbf {w} )=w_{1}\\beta _{1}+w_{2}\\beta _{2}+\\dots +w_{q}\\beta _{q},}"}] | — |
modifiedMeaningful group effectd27882af3117
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| anchors | [{"section":"Group effects","snippet":"is said to be meaningful if the underlying simultaneous changes of the"},{"type":"math_alttext","value":"{\\displaystyle \\xi (\\mathbf {w} )=w_{1}\\beta _{1}+w_{2}\\beta _{2}+\\dots +w_{q}\\beta _{q},}"}] | — |
modifiedMinimum-variance unbiased estimator of group effect69cc66fb5413
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| anchors | [{"section":"Group effects","snippet":"and its minimum-variance unbiased linear estimator is"},{"type":"math_alttext","value":"{\\displaystyle y'=\\beta _{1}'x_{1}'+\\cdots +\\beta _{p}'x_{p}'+\\varepsilon .}"},{"type":"math_alttext","value":"{\\displaystyle \\xi '(\\mathbf {w} )=w_{1}\\beta _{1}'+w_{2}\\beta _{2}'+\\dots +w_{q}\\beta _{q}',}"},{"type":"math_alttext","value":"{\\displaystyle {\\hat {\\xi }}'(\\mathbf {w} )=w_{1}{\\hat {\\beta }}_{1}'+w_{2}{\\hat {\\beta }}_{2}'+\\dots +w_{q}{\\hat {\\beta }}_{q}',}"}] | — |
| note | Mathlib has general decision-theoretic estimators/risk (Mathlib.Probability.Decision.Risk.Defs) but no unbiasedness or MVUE-of-group-effect result. | Mathlib has general decision-theoretic estimators/risk infrastructure but no unbiasedness or MVUE-of-group-effect result. |
modifiedLeast-squares optimum parameter314078402892
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| anchors | [{"section":"Least-squares estimation and related techniques","snippet":"the optimum parameter vector is defined as such that minimizes the sum of mean squared loss"},{"type":"math_alttext","value":"{\\displaystyle {\\vec {\\hat {\\beta }}}={\\underset {\\vec {\\beta }}{\\mbox{arg min}}}\\,L\\left(D,{\\vec {\\beta }}\\right)={\\underset {\\vec {\\beta }}{\\mbox{arg min}}}\\sum _{i=1}^{n}\\left({\\vec {\\beta }}\\cdot {\\vec {x_{i}}}-y_{i}\\right)^{2}}"}] | — |
| mathlib.match_kind | — | generalization |
modifiedConvex loss minimized at gradient zero0592a3d5699b
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| mathlib.match_kind | — | generalization |
| note | Mathlib proves a local minimum of a convex function is a global minimum (and Fermat's theorem gives gradient zero at minima), giving the general principle but not this regression-specific statement. | Mathlib proves a local minimum of a convex function is a global minimum (with Fermat's theorem giving gradient zero), supplying the general principle but not this regression-specific statement. |
modifiedNormal equations (gradient zero solution)8e33e81285c8
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| mathlib.match_kind | — | generalization |