\( \newcommand{\A} {\mathbb{A}} \newcommand{\rma} {\mathrm{a}} \newcommand{\rmA} {\mathrm{A}} \newcommand{\ba} {\mathbf{a}} \newcommand{\bA} {\mathbf{A}} \newcommand{\cA} {\mathcal{A}} \newcommand{\bcA} {\boldsymbol{\cA}} \newcommand{\tnsrA} {\underline{\bA}} \newcommand{\wta} {\widetilde{a}} \newcommand{\wtA} {\widetilde{A}} \newcommand{\wtrma} {\widetilde{\rma}} \newcommand{\wtrmA} {\widetilde{\rmA}} \newcommand{\wtba} {\widetilde{\ba}} \newcommand{\wtbA} {\widetilde{\bA}} \newcommand{\wha} {\widehat{a}} \newcommand{\whA} {\widehat{A}} \newcommand{\whrma} {\widehat{\rma}} \newcommand{\whrmA} {\widehat{\rmA}} \newcommand{\whba} {\widehat{\ba}} \newcommand{\whbA} {\widehat{\bA}} \newcommand{\whcA} {\widehat{\cA}} \newcommand{\whtnsrA} {\widehat{\tnsrA}} \newcommand{\B} {\mathbb{B}} \newcommand{\rmb} {\mathrm{b}} \newcommand{\rmB} {\mathrm{B}} \newcommand{\bb} {\mathbf{b}} \newcommand{\bB} {\mathbf{B}} \newcommand{\cB} {\mathcal{B}} \newcommand{\bcB} {\boldsymbol{\cB}} \newcommand{\tnsrB} {\underline{\bB}} \newcommand{\wtb} {\widetilde{b}} \newcommand{\wtB} {\widetilde{B}} \newcommand{\wtrmb} {\widetilde{\rmb}} \newcommand{\wtrmB} {\widetilde{\rmB}} \newcommand{\wtbb} {\widetilde{\bb}} \newcommand{\wtbB} {\widetilde{\bB}} \newcommand{\whb} {\widehat{b}} \newcommand{\whB} {\widehat{B}} \newcommand{\whrmb} {\widehat{\rmb}} \newcommand{\whrmB} {\widehat{\rmB}} \newcommand{\whbb} {\widehat{\bb}} \newcommand{\whbB} {\widehat{\bB}} \newcommand{\whcB} {\widehat{\cB}} \newcommand{\whtnsrB} {\widehat{\tnsrB}} \newcommand{\C} {\mathbb{C}} \newcommand{\rmc} {\mathrm{c}} \newcommand{\rmC} {\mathrm{C}} \newcommand{\bc} {\mathbf{c}} \newcommand{\bC} {\mathbf{C}} \newcommand{\cC} {\mathcal{C}} \newcommand{\bcC} {\boldsymbol{\cC}} \newcommand{\tnsrC} {\underline{\bC}} \newcommand{\wtc} {\widetilde{c}} \newcommand{\wtC} {\widetilde{C}} \newcommand{\wtrmc} {\widetilde{\rmc}} \newcommand{\wtrmC} {\widetilde{\rmC}} \newcommand{\wtbc} {\widetilde{\bc}} \newcommand{\wtbC} {\widetilde{\bC}} \newcommand{\whc} {\widehat{c}} \newcommand{\whC} {\widehat{C}} \newcommand{\whrmc} {\widehat{\rmc}} \newcommand{\whrmC} {\widehat{\rmC}} \newcommand{\whbc} {\widehat{\bc}} \newcommand{\whbC} {\widehat{\bC}} \newcommand{\whcC} {\widehat{\cC}} \newcommand{\whtnsrC} {\widehat{\tnsrC}} \newcommand{\D} {\mathbb{D}} \newcommand{\rmd} {\mathrm{d}} \newcommand{\rmD} {\mathrm{D}} \newcommand{\bd} {\mathbf{d}} \newcommand{\bD} {\mathbf{D}} \newcommand{\cD} {\mathcal{D}} \newcommand{\bcD} {\boldsymbol{\cD}} \newcommand{\tnsrD} {\underline{\bD}} \newcommand{\wtd} {\widetilde{d}} \newcommand{\wtD} {\widetilde{D}} \newcommand{\wtrmd} {\widetilde{\rmd}} \newcommand{\wtrmD} {\widetilde{\rmD}} \newcommand{\wtbd} {\widetilde{\bd}} \newcommand{\wtbD} {\widetilde{\bD}} \newcommand{\whd} {\widehat{d}} \newcommand{\whD} {\widehat{D}} \newcommand{\whrmd} {\widehat{\rmd}} \newcommand{\whrmD} {\widehat{\rmD}} \newcommand{\whbd} {\widehat{\bd}} \newcommand{\whbD} {\widehat{\bD}} \newcommand{\whcD} {\widehat{\cD}} \newcommand{\whtnsrD} {\widehat{\tnsrD}} \newcommand{\E} {\mathbb{E}} \newcommand{\rme} {\mathrm{e}} \newcommand{\rmE} {\mathrm{E}} \newcommand{\be} {\mathbf{e}} \newcommand{\bE} {\mathbf{E}} \newcommand{\cE} {\mathcal{E}} \newcommand{\bcE} {\boldsymbol{\cE}} \newcommand{\tnsrE} {\underline{\bE}} \newcommand{\wte} {\widetilde{e}} \newcommand{\wtE} {\widetilde{E}} \newcommand{\wtrme} {\widetilde{\rme}} \newcommand{\wtrmE} {\widetilde{\rmE}} \newcommand{\wtbe} {\widetilde{\be}} \newcommand{\wtbE} {\widetilde{\bE}} \newcommand{\whe} {\widehat{e}} \newcommand{\whE} {\widehat{E}} \newcommand{\whrme} {\widehat{\rme}} \newcommand{\whrmE} {\widehat{\rmE}} \newcommand{\whbe} {\widehat{\be}} \newcommand{\whbE} {\widehat{\bE}} \newcommand{\whcE} {\widehat{\cE}} \newcommand{\whtnsrE} {\widehat{\tnsrE}} \newcommand{\F} {\mathbb{F}} \newcommand{\rmf} {\mathrm{f}} \newcommand{\rmF} {\mathrm{F}} \newcommand{\bff} {\mathbf{f}} \newcommand{\bF} {\mathbf{F}} \newcommand{\cF} {\mathcal{F}} \newcommand{\bcF} {\boldsymbol{\cF}} \newcommand{\tnsrF} {\underline{\bF}} \newcommand{\wtf} {\widetilde{f}} \newcommand{\wtF} {\widetilde{F}} \newcommand{\wtrmf} {\widetilde{\rmf}} \newcommand{\wtrmF} {\widetilde{\rmF}} \newcommand{\wtbf} {\widetilde{\bf}} \newcommand{\wtbF} {\widetilde{\bF}} \newcommand{\whf} {\widehat{f}} \newcommand{\whF} {\widehat{F}} \newcommand{\whrmf} {\widehat{\rmf}} \newcommand{\whrmF} {\widehat{\rmF}} \newcommand{\whbf} {\widehat{\bf}} \newcommand{\whbF} {\widehat{\bF}} \newcommand{\whcF} {\widehat{\cF}} \newcommand{\whtnsrF} {\widehat{\tnsrF}} \newcommand{\G} {\mathbb{G}} \newcommand{\rmg} {\mathrm{g}} \newcommand{\rmG} {\mathrm{G}} \newcommand{\bg} {\mathbf{g}} \newcommand{\bG} {\mathbf{G}} \newcommand{\cG} {\mathcal{G}} \newcommand{\bcG} {\boldsymbol{\cG}} \newcommand{\tnsrG} {\underline{\bG}} \newcommand{\wtg} {\widetilde{g}} \newcommand{\wtG} {\widetilde{G}} \newcommand{\wtrmg} {\widetilde{\rmg}} \newcommand{\wtrmG} {\widetilde{\rmG}} \newcommand{\wtbg} {\widetilde{\bg}} \newcommand{\wtbG} {\widetilde{\bG}} \newcommand{\whg} {\widehat{g}} \newcommand{\whG} {\widehat{G}} \newcommand{\whrmg} {\widehat{\rmg}} \newcommand{\whrmG} {\widehat{\rmG}} \newcommand{\whbg} {\widehat{\bg}} \newcommand{\whbG} {\widehat{\bG}} \newcommand{\whcG} {\widehat{\cG}} \newcommand{\whtnsrG} {\widehat{\tnsrG}} \newcommand{\bbH} {\mathbb{H}} \newcommand{\rmh} {\mathrm{h}} \newcommand{\rmH} {\mathrm{H}} \newcommand{\bh} {\mathbf{h}} \newcommand{\bH} {\mathbf{H}} \newcommand{\cH} {\mathcal{H}} \newcommand{\bcH} {\boldsymbol{\cH}} \newcommand{\tnsrH} {\underline{\bH}} \newcommand{\wth} {\widetilde{h}} \newcommand{\wtH} {\widetilde{H}} \newcommand{\wtrmh} {\widetilde{\rmh}} \newcommand{\wtrmH} {\widetilde{\rmH}} \newcommand{\wtbh} {\widetilde{\bh}} \newcommand{\wtbH} {\widetilde{\bH}} \newcommand{\whh} {\widehat{h}} \newcommand{\whH} {\widehat{H}} \newcommand{\whrmh} {\widehat{\rmh}} \newcommand{\whrmH} {\widehat{\rmH}} \newcommand{\whbh} {\widehat{\bh}} \newcommand{\whbH} {\widehat{\bH}} \newcommand{\whcH} {\widehat{\cH}} \newcommand{\whtnsrH} {\widehat{\tnsrH}} \newcommand{\I} {\mathbb{I}} \newcommand{\rmi} {\mathrm{i}} \newcommand{\rmI} {\mathrm{I}} \newcommand{\bi} {\mathbf{i}} \newcommand{\bI} {\mathbf{I}} \newcommand{\cI} {\mathcal{I}} \newcommand{\bcI} {\boldsymbol{\cI}} \newcommand{\tnsrI} {\underline{\bI}} \newcommand{\wti} {\widetilde{i}} \newcommand{\wtI} {\widetilde{I}} \newcommand{\wtrmi} {\widetilde{\rmi}} \newcommand{\wtrmI} {\widetilde{\rmI}} \newcommand{\wtbi} {\widetilde{\bi}} \newcommand{\wtbI} {\widetilde{\bI}} \newcommand{\whi} {\widehat{i}} \newcommand{\whI} {\widehat{I}} \newcommand{\whrmi} {\widehat{\rmi}} \newcommand{\whrmI} {\widehat{\rmI}} \newcommand{\whbi} {\widehat{\bi}} \newcommand{\whbI} {\widehat{\bI}} \newcommand{\whcI} {\widehat{\cI}} \newcommand{\whtnsrI} {\widehat{\tnsrI}} \newcommand{\J} {\mathbb{J}} \newcommand{\rmj} {\mathrm{j}} \newcommand{\rmJ} {\mathrm{J}} \newcommand{\bj} {\mathbf{j}} \newcommand{\bJ} {\mathbf{J}} \newcommand{\cJ} {\mathcal{J}} \newcommand{\bcJ} {\boldsymbol{\cJ}} \newcommand{\tnsrJ} {\underline{\bJ}} \newcommand{\wtj} {\widetilde{j}} \newcommand{\wtJ} {\widetilde{J}} \newcommand{\wtrmj} {\widetilde{\rmj}} \newcommand{\wtrmJ} {\widetilde{\rmJ}} \newcommand{\wtbj} {\widetilde{\bj}} \newcommand{\wtbJ} {\widetilde{\bJ}} \newcommand{\whj} {\widehat{j}} \newcommand{\whJ} {\widehat{J}} \newcommand{\whrmj} {\widehat{\rmj}} \newcommand{\whrmJ} {\widehat{\rmJ}} \newcommand{\whbj} {\widehat{\bj}} \newcommand{\whbJ} {\widehat{\bJ}} \newcommand{\whcJ} {\widehat{\cJ}} \newcommand{\whtnsrJ} {\widehat{\tnsrJ}} \newcommand{\K} {\mathbb{K}} \newcommand{\rmk} {\mathrm{k}} \newcommand{\rmK} {\mathrm{K}} \newcommand{\bk} {\mathbf{k}} \newcommand{\bK} {\mathbf{K}} \newcommand{\cK} {\mathcal{K}} \newcommand{\bcK} {\boldsymbol{\cK}} \newcommand{\tnsrK} {\underline{\bK}} \newcommand{\wtk} {\widetilde{k}} \newcommand{\wtK} {\widetilde{K}} \newcommand{\wtrmk} {\widetilde{\rmk}} \newcommand{\wtrmK} {\widetilde{\rmK}} \newcommand{\wtbk} {\widetilde{\bk}} \newcommand{\wtbK} {\widetilde{\bK}} \newcommand{\whk} {\widehat{k}} \newcommand{\whK} {\widehat{K}} \newcommand{\whrmk} {\widehat{\rmk}} \newcommand{\whrmK} {\widehat{\rmK}} \newcommand{\whbk} {\widehat{\bk}} \newcommand{\whbK} {\widehat{\bK}} \newcommand{\whcK} {\widehat{\cK}} \newcommand{\whtnsrK} {\widehat{\tnsrK}} \newcommand{\bbL} {\mathbb{L}} \newcommand{\rml} {\mathrm{l}} \newcommand{\rmL} {\mathrm{L}} \newcommand{\bl} {\mathbf{l}} \newcommand{\bL} {\mathbf{L}} \newcommand{\cL} {\mathcal{L}} \newcommand{\bcL} {\boldsymbol{\cL}} \newcommand{\tnsrL} {\underline{\bL}} \newcommand{\wtl} {\widetilde{l}} \newcommand{\wtL} {\widetilde{L}} \newcommand{\wtrml} {\widetilde{\rml}} \newcommand{\wtrmL} {\widetilde{\rmL}} \newcommand{\wtbl} {\widetilde{\bl}} \newcommand{\wtbL} {\widetilde{\bL}} \newcommand{\whl} {\widehat{l}} \newcommand{\whL} {\widehat{L}} \newcommand{\whrml} {\widehat{\rml}} \newcommand{\whrmL} {\widehat{\rmL}} \newcommand{\whbl} {\widehat{\bl}} \newcommand{\whbL} {\widehat{\bL}} \newcommand{\whcL} {\widehat{\cL}} \newcommand{\whtnsrL} {\widehat{\tnsrL}} \newcommand{\M} {\mathbb{M}} \newcommand{\rmm} {\mathrm{m}} \newcommand{\rmM} {\mathrm{M}} \newcommand{\bm} {\mathbf{m}} \newcommand{\bM} {\mathbf{M}} \newcommand{\cM} {\mathcal{M}} \newcommand{\bcM} {\boldsymbol{\cM}} \newcommand{\tnsrM} {\underline{\bM}} \newcommand{\wtm} {\widetilde{m}} \newcommand{\wtM} {\widetilde{M}} \newcommand{\wtrmm} {\widetilde{\rmm}} \newcommand{\wtrmM} {\widetilde{\rmM}} \newcommand{\wtbm} {\widetilde{\bm}} \newcommand{\wtbM} {\widetilde{\bM}} \newcommand{\whm} {\widehat{m}} \newcommand{\whM} {\widehat{M}} \newcommand{\whrmm} {\widehat{\rmm}} \newcommand{\whrmM} {\widehat{\rmM}} \newcommand{\whbm} {\widehat{\bm}} \newcommand{\whbM} {\widehat{\bM}} \newcommand{\whcM} {\widehat{\cM}} \newcommand{\whtnsrM} {\widehat{\tnsrM}} \newcommand{\N} {\mathbb{N}} \newcommand{\rmn} {\mathrm{n}} \newcommand{\rmN} {\mathrm{N}} \newcommand{\bn} {\mathbf{n}} \newcommand{\bN} {\mathbf{N}} \newcommand{\cN} {\mathcal{N}} \newcommand{\bcN} {\boldsymbol{\cN}} \newcommand{\tnsrN} {\underline{\bN}} \newcommand{\wtn} {\widetilde{n}} \newcommand{\wtN} {\widetilde{N}} \newcommand{\wtrmn} {\widetilde{\rmn}} \newcommand{\wtrmN} {\widetilde{\rmN}} \newcommand{\wtbn} {\widetilde{\bn}} \newcommand{\wtbN} {\widetilde{\bN}} \newcommand{\whn} {\widehat{n}} \newcommand{\whN} {\widehat{N}} \newcommand{\whrmn} {\widehat{\rmn}} \newcommand{\whrmN} {\widehat{\rmN}} \newcommand{\whbn} {\widehat{\bn}} \newcommand{\whbN} {\widehat{\bN}} \newcommand{\whcN} {\widehat{\cN}} \newcommand{\whtnsrN} {\widehat{\tnsrN}} \newcommand{\bbO} {\mathbb{O}} \newcommand{\rmo} {\mathrm{o}} \newcommand{\rmO} {\mathrm{O}} \newcommand{\bo} {\mathbf{o}} \newcommand{\bO} {\mathbf{O}} \newcommand{\cO} {\mathcal{O}} \newcommand{\bcO} {\boldsymbol{\cO}} \newcommand{\tnsrO} {\underline{\bO}} \newcommand{\wto} {\widetilde{o}} \newcommand{\wtO} {\widetilde{O}} \newcommand{\wtrmo} {\widetilde{\rmo}} \newcommand{\wtrmO} {\widetilde{\rmO}} \newcommand{\wtbo} {\widetilde{\bo}} \newcommand{\wtbO} {\widetilde{\bO}} \newcommand{\who} {\widehat{o}} \newcommand{\whO} {\widehat{O}} \newcommand{\whrmo} {\widehat{\rmo}} \newcommand{\whrmO} {\widehat{\rmO}} \newcommand{\whbo} {\widehat{\bo}} \newcommand{\whbO} {\widehat{\bO}} \newcommand{\whcO} {\widehat{\cO}} \newcommand{\whtnsrO} {\widehat{\tnsrO}} \newcommand{\bbP} {\mathbb{P}} \newcommand{\rmp} {\mathrm{p}} \newcommand{\rmP} {\mathrm{P}} \newcommand{\bp} {\mathbf{p}} \newcommand{\bP} {\mathbf{P}} \newcommand{\cP} {\mathcal{P}} \newcommand{\bcP} {\boldsymbol{\cP}} \newcommand{\tnsrP} {\underline{\bP}} \newcommand{\wtp} {\widetilde{p}} \newcommand{\wtP} {\widetilde{P}} \newcommand{\wtrmp} {\widetilde{\rmp}} \newcommand{\wtrmP} {\widetilde{\rmP}} \newcommand{\wtbp} {\widetilde{\bp}} \newcommand{\wtbP} {\widetilde{\bP}} \newcommand{\whp} {\widehat{p}} \newcommand{\whP} {\widehat{P}} \newcommand{\whrmp} {\widehat{\rmp}} \newcommand{\whrmP} {\widehat{\rmP}} \newcommand{\whbp} {\widehat{\bp}} \newcommand{\whbP} {\widehat{\bP}} \newcommand{\whcP} {\widehat{\cP}} \newcommand{\whtnsrP} {\widehat{\tnsrP}} \newcommand{\Q} {\mathbb{Q}} \newcommand{\rmq} {\mathrm{q}} \newcommand{\rmQ} {\mathrm{Q}} \newcommand{\bq} {\mathbf{q}} \newcommand{\bQ} {\mathbf{Q}} \newcommand{\cQ} {\mathcal{Q}} \newcommand{\bcQ} {\boldsymbol{\cQ}} \newcommand{\tnsrQ} {\underline{\bQ}} \newcommand{\wtq} {\widetilde{q}} \newcommand{\wtQ} {\widetilde{Q}} \newcommand{\wtrmq} {\widetilde{\rmq}} \newcommand{\wtrmQ} {\widetilde{\rmQ}} \newcommand{\wtbq} {\widetilde{\bq}} \newcommand{\wtbQ} {\widetilde{\bQ}} \newcommand{\whq} {\widehat{q}} \newcommand{\whQ} {\widehat{Q}} \newcommand{\whrmq} {\widehat{\rmq}} \newcommand{\whrmQ} {\widehat{\rmQ}} \newcommand{\whbq} {\widehat{\bq}} \newcommand{\whbQ} {\widehat{\bQ}} \newcommand{\whcQ} {\widehat{\cQ}} \newcommand{\whtnsrQ} {\widehat{\tnsrQ}} \newcommand{\R} {\mathbb{R}} \newcommand{\rmr} {\mathrm{r}} \newcommand{\rmR} {\mathrm{R}} \newcommand{\br} {\mathbf{r}} \newcommand{\bR} {\mathbf{R}} \newcommand{\cR} {\mathcal{R}} \newcommand{\bcR} {\boldsymbol{\cR}} \newcommand{\tnsrR} {\underline{\bR}} \newcommand{\wtr} {\widetilde{r}} \newcommand{\wtR} {\widetilde{R}} \newcommand{\wtrmr} {\widetilde{\rmr}} \newcommand{\wtrmR} {\widetilde{\rmR}} \newcommand{\wtbr} {\widetilde{\br}} \newcommand{\wtbR} {\widetilde{\bR}} \newcommand{\whr} {\widehat{r}} \newcommand{\whR} {\widehat{R}} \newcommand{\whrmr} {\widehat{\rmr}} \newcommand{\whrmR} {\widehat{\rmR}} \newcommand{\whbr} {\widehat{\br}} \newcommand{\whbR} {\widehat{\bR}} \newcommand{\whcR} {\widehat{\cR}} \newcommand{\whtnsrR} {\widehat{\tnsrR}} \newcommand{\bbS} {\mathbb{S}} \newcommand{\rms} {\mathrm{s}} \newcommand{\rmS} {\mathrm{S}} \newcommand{\bs} {\mathbf{s}} \newcommand{\bS} {\mathbf{S}} \newcommand{\cS} {\mathcal{S}} \newcommand{\bcS} {\boldsymbol{\cS}} \newcommand{\tnsrS} {\underline{\bS}} \newcommand{\wts} {\widetilde{s}} \newcommand{\wtS} {\widetilde{S}} \newcommand{\wtrms} {\widetilde{\rms}} \newcommand{\wtrmS} {\widetilde{\rmS}} \newcommand{\wtbs} {\widetilde{\bs}} \newcommand{\wtbS} {\widetilde{\bS}} \newcommand{\whs} {\widehat{s}} \newcommand{\whS} {\widehat{S}} \newcommand{\whrms} {\widehat{\rms}} \newcommand{\whrmS} {\widehat{\rmS}} \newcommand{\whbs} {\widehat{\bs}} \newcommand{\whbS} {\widehat{\bS}} \newcommand{\whcS} {\widehat{\cS}} \newcommand{\whtnsrS} {\widehat{\tnsrS}} \newcommand{\T} {\mathbb{T}} \newcommand{\rmt} {\mathrm{t}} \newcommand{\rmT} {\mathrm{T}} \newcommand{\bt} {\mathbf{t}} \newcommand{\bT} {\mathbf{T}} \newcommand{\cT} {\mathcal{T}} \newcommand{\bcT} {\boldsymbol{\cT}} \newcommand{\tnsrT} {\underline{\bT}} \newcommand{\wtt} {\widetilde{t}} \newcommand{\wtT} {\widetilde{T}} \newcommand{\wtrmt} {\widetilde{\rmt}} \newcommand{\wtrmT} {\widetilde{\rmT}} \newcommand{\wtbt} {\widetilde{\bt}} \newcommand{\wtbT} {\widetilde{\bT}} \newcommand{\wht} {\widehat{t}} \newcommand{\whT} {\widehat{T}} \newcommand{\whrmt} {\widehat{\rmt}} \newcommand{\whrmT} {\widehat{\rmT}} \newcommand{\whbt} {\widehat{\bt}} \newcommand{\whbT} {\widehat{\bT}} \newcommand{\whcT} {\widehat{\cT}} \newcommand{\whtnsrT} {\widehat{\tnsrT}} \newcommand{\U} {\mathbb{U}} \newcommand{\rmu} {\mathrm{u}} \newcommand{\rmU} {\mathrm{U}} \newcommand{\bu} {\mathbf{u}} \newcommand{\bU} {\mathbf{U}} \newcommand{\cU} {\mathcal{U}} \newcommand{\bcU} {\boldsymbol{\cU}} \newcommand{\tnsrU} {\underline{\bU}} \newcommand{\wtu} {\widetilde{u}} \newcommand{\wtU} {\widetilde{U}} \newcommand{\wtrmu} {\widetilde{\rmu}} \newcommand{\wtrmU} {\widetilde{\rmU}} \newcommand{\wtbu} {\widetilde{\bu}} \newcommand{\wtbU} {\widetilde{\bU}} \newcommand{\whu} {\widehat{u}} \newcommand{\whU} {\widehat{U}} \newcommand{\whrmu} {\widehat{\rmu}} \newcommand{\whrmU} {\widehat{\rmU}} \newcommand{\whbu} {\widehat{\bu}} \newcommand{\whbU} {\widehat{\bU}} \newcommand{\whcU} {\widehat{\cU}} \newcommand{\whtnsrU} {\widehat{\tnsrU}} \newcommand{\V} {\mathbb{V}} \newcommand{\rmv} {\mathrm{v}} \newcommand{\rmV} {\mathrm{V}} \newcommand{\bv} {\mathbf{v}} \newcommand{\bV} {\mathbf{V}} \newcommand{\cV} {\mathcal{V}} \newcommand{\bcV} {\boldsymbol{\cV}} \newcommand{\tnsrV} {\underline{\bV}} \newcommand{\wtv} {\widetilde{v}} \newcommand{\wtV} {\widetilde{V}} \newcommand{\wtrmv} {\widetilde{\rmv}} \newcommand{\wtrmV} {\widetilde{\rmV}} \newcommand{\wtbv} {\widetilde{\bv}} \newcommand{\wtbV} {\widetilde{\bV}} \newcommand{\whv} {\widehat{v}} \newcommand{\whV} {\widehat{V}} \newcommand{\whrmv} {\widehat{\rmv}} \newcommand{\whrmV} {\widehat{\rmV}} \newcommand{\whbv} {\widehat{\bv}} \newcommand{\whbV} {\widehat{\bV}} \newcommand{\whcV} {\widehat{\cV}} \newcommand{\whtnsrV} {\widehat{\tnsrV}} \newcommand{\W} {\mathbb{W}} \newcommand{\rmw} {\mathrm{w}} \newcommand{\rmW} {\mathrm{W}} \newcommand{\bw} {\mathbf{w}} \newcommand{\bW} {\mathbf{W}} \newcommand{\cW} {\mathcal{W}} \newcommand{\bcW} {\boldsymbol{\cW}} \newcommand{\tnsrW} {\underline{\bW}} \newcommand{\wtw} {\widetilde{w}} \newcommand{\wtW} {\widetilde{W}} \newcommand{\wtrmw} {\widetilde{\rmw}} \newcommand{\wtrmW} {\widetilde{\rmW}} \newcommand{\wtbw} {\widetilde{\bw}} \newcommand{\wtbW} {\widetilde{\bW}} \newcommand{\whw} {\widehat{w}} \newcommand{\whW} {\widehat{W}} \newcommand{\whrmw} {\widehat{\rmw}} \newcommand{\whrmW} {\widehat{\rmW}} \newcommand{\whbw} {\widehat{\bw}} \newcommand{\whbW} {\widehat{\bW}} \newcommand{\whcW} {\widehat{\cW}} \newcommand{\whtnsrW} {\widehat{\tnsrW}} \newcommand{\X} {\mathbb{X}} \newcommand{\rmx} {\mathrm{x}} \newcommand{\rmX} {\mathrm{X}} \newcommand{\bx} {\mathbf{x}} \newcommand{\bX} {\mathbf{X}} \newcommand{\cX} {\mathcal{X}} \newcommand{\bcX} {\boldsymbol{\cX}} \newcommand{\tnsrX} {\underline{\bX}} \newcommand{\wtx} {\widetilde{x}} \newcommand{\wtX} {\widetilde{X}} \newcommand{\wtrmx} {\widetilde{\rmx}} \newcommand{\wtrmX} {\widetilde{\rmX}} \newcommand{\wtbx} {\widetilde{\bx}} \newcommand{\wtbX} {\widetilde{\bX}} \newcommand{\whx} {\widehat{x}} \newcommand{\whX} {\widehat{X}} \newcommand{\whrmx} {\widehat{\rmx}} \newcommand{\whrmX} {\widehat{\rmX}} \newcommand{\whbx} {\widehat{\bx}} \newcommand{\whbX} {\widehat{\bX}} \newcommand{\whcX} {\widehat{\cX}} \newcommand{\whtnsrX} {\widehat{\tnsrX}} \newcommand{\Y} {\mathbb{Y}} \newcommand{\rmy} {\mathrm{y}} \newcommand{\rmY} {\mathrm{Y}} \newcommand{\by} {\mathbf{y}} \newcommand{\bY} {\mathbf{Y}} \newcommand{\cY} {\mathcal{Y}} \newcommand{\bcY} {\boldsymbol{\cY}} \newcommand{\tnsrY} {\underline{\bY}} \newcommand{\wty} {\widetilde{y}} \newcommand{\wtY} {\widetilde{Y}} \newcommand{\wtrmy} {\widetilde{\rmy}} \newcommand{\wtrmY} {\widetilde{\rmY}} \newcommand{\wtby} {\widetilde{\by}} \newcommand{\wtbY} {\widetilde{\bY}} \newcommand{\why} {\widehat{y}} \newcommand{\whY} {\widehat{Y}} \newcommand{\whrmy} {\widehat{\rmy}} \newcommand{\whrmY} {\widehat{\rmY}} \newcommand{\whby} {\widehat{\by}} \newcommand{\whbY} {\widehat{\bY}} \newcommand{\whcY} {\widehat{\cY}} \newcommand{\whtnsrY} {\widehat{\tnsrY}} \newcommand{\Z} {\mathbb{Z}} \newcommand{\rmz} {\mathrm{z}} \newcommand{\rmZ} {\mathrm{Z}} \newcommand{\bz} {\mathbf{z}} \newcommand{\bZ} {\mathbf{Z}} \newcommand{\cZ} {\mathcal{Z}} \newcommand{\bcZ} {\boldsymbol{\cZ}} \newcommand{\tnsrZ} {\underline{\bZ}} \newcommand{\wtz} {\widetilde{z}} \newcommand{\wtZ} {\widetilde{Z}} \newcommand{\wtrmz} {\widetilde{\rmz}} \newcommand{\wtrmZ} {\widetilde{\rmZ}} \newcommand{\wtbz} {\widetilde{\bz}} \newcommand{\wtbZ} {\widetilde{\bZ}} \newcommand{\whz} {\widehat{z}} \newcommand{\whZ} {\widehat{Z}} \newcommand{\whrmz} {\widehat{\rmz}} \newcommand{\whrmZ} {\widehat{\rmZ}} \newcommand{\whbz} {\widehat{\bz}} \newcommand{\whbZ} {\widehat{\bZ}} \newcommand{\whcZ} {\widehat{\cZ}} \newcommand{\whtnsrZ} {\widehat{\tnsrZ}} \newcommand{\balpha} {\boldsymbol{\alpha}} \newcommand{\wtalpha} {\widetilde{\alpha}} \newcommand{\whalpha} {\widehat{\alpha}} \newcommand{\wtbalpha} {\widetilde{\balpha}} \newcommand{\whbalpha} {\widehat{\balpha}} \newcommand{\bbeta} {\boldsymbol{\beta}} \newcommand{\wtbeta} {\widetilde{\beta}} \newcommand{\whbeta} {\widehat{\beta}} \newcommand{\wtbbeta} {\widetilde{\bbeta}} \newcommand{\whbbeta} {\widehat{\bbeta}} \newcommand{\bgamma} {\boldsymbol{\gamma}} \newcommand{\wtgamma} {\widetilde{\gamma}} \newcommand{\whgamma} {\widehat{\gamma}} \newcommand{\wtbgamma} {\widetilde{\bgamma}} \newcommand{\whbgamma} {\widehat{\bgamma}} \newcommand{\bdelta} {\boldsymbol{\delta}} \newcommand{\wtdelta} {\widetilde{\delta}} \newcommand{\whdelta} {\widehat{\delta}} \newcommand{\wtbdelta} {\widetilde{\bdelta}} \newcommand{\whbdelta} {\widehat{\bdelta}} \newcommand{\bepsilon} {\boldsymbol{\epsilon}} \newcommand{\wtepsilon} {\widetilde{\epsilon}} \newcommand{\whepsilon} {\widehat{\epsilon}} \newcommand{\wtbepsilon}{\widetilde{\bepsilon}} \newcommand{\whbepsilon}{\widehat{\bepsilon}} \newcommand{\veps} {\varepsilon} \newcommand{\bveps} {\boldsymbol{\veps}} \newcommand{\wtveps} {\widetilde{\veps}} \newcommand{\whveps} {\widehat{\veps}} \newcommand{\wtbveps} {\widetilde{\bveps}} \newcommand{\whbveps} {\widehat{\bveps}} \newcommand{\bEta} {\boldsymbol{\eta}} \newcommand{\wteta} {\widetilde{\eta}} \newcommand{\wheta} {\widehat{\eta}} \newcommand{\wtbEta} {\widetilde{\bEta}} \newcommand{\whbEta} {\widehat{\bEta}} \newcommand{\btheta} {\boldsymbol{\theta}} \newcommand{\wttheta} {\widetilde{\theta}} \newcommand{\whtheta} {\widehat{\theta}} \newcommand{\wtbtheta} {\widetilde{\btheta}} \newcommand{\whbtheta} {\widehat{\btheta}} \newcommand{\bvtheta} {\boldsymbol{\vartheta}} \newcommand{\wtvtheta} {\widetilde{\vartheta}} \newcommand{\whvtheta} {\widehat{\vartheta}} \newcommand{\wtbvtheta} {\widetilde{\bvtheta}} \newcommand{\whbvtheta} {\widehat{\bvtheta}} \newcommand{\biota} {\boldsymbol{\iota}} \newcommand{\wtiota} {\widetilde{\iota}} \newcommand{\whiota} {\widehat{\iota}} \newcommand{\wtbiota} {\widetilde{\biota}} \newcommand{\whbiota} {\widehat{\biota}} \newcommand{\bkappa} {\boldsymbol{\kappa}} \newcommand{\wtkappa} {\widetilde{\kappa}} \newcommand{\whkappa} {\widehat{\kappa}} \newcommand{\wtbkappa} {\widetilde{\bkappa}} \newcommand{\whbkappa} {\widehat{\bkappa}} \newcommand{\blambda} {\boldsymbol{\lambda}} \newcommand{\wtlambda} {\widetilde{\lambda}} \newcommand{\whlambda} {\widehat{\lambda}} \newcommand{\wtblambda} {\widetilde{\blambda}} \newcommand{\whblambda} {\widehat{\blambda}} \newcommand{\bmu} {\boldsymbol{\mu}} \newcommand{\wtmu} {\widetilde{\mu}} \newcommand{\whmu} {\widehat{\mu}} \newcommand{\wtbmu} {\widetilde{\bmu}} \newcommand{\whbmu} {\widehat{\bmu}} \newcommand{\bnu} {\boldsymbol{\nu}} \newcommand{\wtnu} {\widetilde{\nu}} \newcommand{\whnu} {\widehat{\nu}} \newcommand{\wtbnu} {\widetilde{\bnu}} \newcommand{\whbnu} {\widehat{\bnu}} \newcommand{\bxi} {\boldsymbol{\xi}} \newcommand{\wtxi} {\widetilde{\xi}} \newcommand{\whxi} {\widehat{\xi}} \newcommand{\wtbxi} {\widetilde{\bxi}} \newcommand{\whbxi} {\widehat{\bxi}} \newcommand{\bpi} {\boldsymbol{\pi}} \newcommand{\wtpi} {\widetilde{\pi}} \newcommand{\whpi} {\widehat{\pi}} \newcommand{\wtbpi} {\widetilde{\bpi}} \newcommand{\whbpi} {\widehat{\bpi}} \newcommand{\bvpi} {\boldsymbol{\varpi}} \newcommand{\wtvpi} {\widetilde{\varpi}} \newcommand{\whvpi} {\widehat{\varpi}} \newcommand{\wtbvpi} {\widetilde{\bvpi}} \newcommand{\whbvpi} {\widehat{\bvpi}} \newcommand{\brho} {\boldsymbol{\rho}} \newcommand{\wtrho} {\widetilde{\rho}} \newcommand{\whrho} {\widehat{\rho}} \newcommand{\wtbrho} {\widetilde{\brho}} \newcommand{\whbrho} {\widehat{\brho}} \newcommand{\bvrho} {\boldsymbol{\varrho}} \newcommand{\wtvrho} {\widetilde{\varrho}} \newcommand{\whvrho} {\widehat{\varrho}} \newcommand{\wtbvrho} {\widetilde{\bvrho}} \newcommand{\whbvrho} {\widehat{\bvrho}} \newcommand{\bsigma} {\boldsymbol{\sigma}} 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Data captured from the real world can be of different types, such as text, numerical, and categorical. Before feeding this raw data into a machine learning system, we need to ensure that it is compatible with our algorithm. This is where a “Data Pre-Processor” comes in, converting the data into a suitable format for the machine learning system.

Data Sample: A data sample is an observation containing all the information related to the problem at hand or an instance of the task. Here are some examples:

  • Task: Identifying Dogs and Cats in an Image
    • Data Sample: A collection of images of dogs and cats, each labeled to indicate whether the image contains a dog or a cat.
  • Task: Predicting Stock Prices
    • Data Sample: All previous price history of the particular stock.

Depending on the task, a data sample can consist of several types of data grouped together. For instance, to predict the price of a house, a data sample may include features like square footage, number of bedrooms, number of bathrooms, township name, year built, and house price.

Types of Data

A typical data sample is made up of several different data types grouped together. The common data types are:

  • Numerical Data: Quantitative data represented in numbers, such as house prices or the number of bedrooms.
  • Text Data: Data represented by strings or documents, such as product reviews, news articles, or social media posts. AI algorithms like ChatGPT, Meta AI, and Gwork use this kind of data.
  • Categorical Data: Qualitative data usually divided into categories, such as images of dogs and cats, or gender (male or female).

A data sample may contain only numerical data or a combination of various types.

Independent and Dependent Variables in Data

The variables comprising data samples can be classified into two categories:

Independent Variables: Also known as features, these variables play a key role in making predictions or solving the machine learning problem. For example, to predict the price of a house, features like square footage, number of bedrooms, and number of bathrooms are considered.

Dependent Variables: These variables contain the information or answer to the problem we are solving. For a house price prediction problem, this would be the house price. For a classification problem, it would indicate whether a picture contains a dog or a cat.

Structure of Numerical Data

In most applications, numerical data can be discrete or continuous. Discrete data is usually countable, while continuous data is quantifiable. For instance, pixels in a digital image are discrete data, whereas the temperature of a room is continuous.

Since computers are digital machines and can only handle discrete data, various methods are used to represent continuous data digitally. For example, digital temperature can be represented by a real-valued scalar $x \in \R$.

When dealing with a machine learning application that predicts temperature, the history of temperature readings needs to be stored. This kind of data is represented by a vector $\bx$. Another example of vector data is GPS coordinates, representing the location of an object in 3D space. Mathematically, this is expressed as $\bx \in \R^d$, where $\bx$ contains real numbers in a $d$-dimensional space.

Other representations of numerical data include:

  • Matrix Data: This is typically 2D data and is often used to represent grayscale images, expressed as $\bX \in \R^{d \times k}$, where $d$ are the rows and $k$ are the columns. Each pixel value ranges from 0 to 255, with 0 representing true black and 255 representing true white.
  • 3D Array: Used to represent color images (e.g., RGB), where each color is a 2D matrix. The full-colored image is represented as $\bX \in \R^{d \times k \times 3}$.
  • 4D Array: If a time dimension is added to a 3D array, it represents colored videos, expressed as $\bX \in \R^{d \times k \times t \times 3}$, where $t$ represents time.

Structure of Categorical Data

Categorical data represents categories or qualitative information. Examples include:

  • Gender: {male, female, non-binary, prefer not to say, others}
  • Fruit Types: {apples, bananas, strawberries}
  • Spice Levels: {mild, medium, hot, extra hot}
  • Eye Colors: {brown, black, green}
  • Transport Methods: {bus, car, train}
  • Netflix Ratings: {poor, fair, good, excellent}

From these examples, we observe that some categorical data have meaningful order, such as Netflix Ratings and Spice Levels, known as ordinal data. Other categorical data, like eye color and fruit types, are nominal and have no inherent order.

Machine learning algorithms do not understand alphabets; they understand numbers. Therefore, we need to convert categorical data into numbers using encoding methods:

One-Hot Encoding: Used for nominal data with no order. For instance, encoding eye colors (brown, black, green) results in [1, 0, 0], [0, 1, 0], [0, 0, 1].

Label Encoding: Used for ordinal data, such as volume levels (low, medium, high), encoded as 1, 2, 3.

Data Pre-Processing

We have seen one kind of pre-processing for categorical data: encoding, which transforms alphabetical categorical data into numbers using either label encoding (for ordered data) or one-hot encoding (for nominal data).

There are other kinds of data pre-processing methods used in machine learning problems:

  • Imputation: Filling in missing values using statistical methods.
  • Standardization: Ensuring features in data samples are on the same scale, such as converting weights measured in different units to a common scale.
  • Flattening: Converting a 2D matrix (e.g., an image) into a 1D array.
  • Stemming: Primarily for text data, breaking down sentences into individual words or tokens.

By understanding and properly structuring raw data and variables, we can ensure that machine learning algorithms can effectively process and learn from the data.