\usepackage{amsmath} % Added to get math % \usepackage{amsthm} % Added to get theorems \usepackage{xspace} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % favorite macros %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Define different operators \DeclareMathOperator{\rank}{rank} \DeclareMathOperator{\spanv}{span} \DeclareMathOperator{\prank}{prank} \DeclareMathOperator{\erank}{erank} \DeclareMathOperator{\pspanv}{pspan} \DeclareMathOperator{\range}{range} \DeclareMathOperator{\sign}{sign} \DeclareMathOperator*{\argmin}{arg\,min} \DeclareMathOperator*{\argmax}{arg\,max} \DeclareMathOperator{\vectorize}{vec} \DeclareMathOperator{\tr}{tr} \DeclareMathOperator{\diag}{diag} \DeclareMathOperator{\SNR}{SNR} \DeclareMathOperator{\MSE}{MSE} \DeclareMathOperator{\RSS}{RSS} % \DeclareMathOperator{\if}{if} % This might be used elsewhere % \DeclareMathOperator{\real}{Re} % Use \Re % \DeclareMathOperator{\imag}{Im} % Use \I \DeclareMathOperator{\sothat}{s.t.} % Define Probability, Expectation, Variance, Covariance, Median, Mode \DeclareMathOperator{\Prob}{\mathsf{P}} % Probability \DeclareMathOperator{\E}{\mathbf{E}} % Expectation \DeclareMathOperator{\var}{var} % Variance \DeclareMathOperator{\Var}{Var} % Variance \DeclareMathOperator{\cov}{cov} % Covariance \DeclareMathOperator{\Cov}{Cov} % Covariance \DeclareMathOperator{\cor}{cor} % Correlation \DeclareMathOperator{\Cor}{Cor} % Correlation \DeclareMathOperator{\Cost}{Cost} % Cost function \DeclareMathOperator{\mean}{mean} % Mean \DeclareMathOperator{\median}{median} % Median \DeclareMathOperator{\pdf}{\mathbf{p}} % probability density function \DeclareMathOperator{\df}{\mathbf{P}} % cumulative distribution function \DeclareMathOperator{\chisquare}{\chi^{2}} % chi-square distribution function \DeclareMathOperator{\invchisquare}{\chi^{-2}} % inverse chi-square distribution function \DeclareMathOperator{\Gammafunc}{Gamma} % Gamma function \DeclareMathOperator{\Binomial}{Binomial(n,p)} % Binomial distribution \DeclareMathOperator{\binomial}{\mathcal{B}(n,p)} % binomial distribution \DeclareMathOperator{\normal}{\mathcal{N}(\mu,\sigma^2)} % normal distribution \DeclareMathOperator{\normall}{\mathcal{N}(\mu,\lambda)} % normal distribution with lambda variance % Other \newcommand{\degree}{\ensuremath{^\circ}} \providecommand{\commentt}[1]{\emph{#1} } \renewcommand{\emph}{\textit} % Used to keep as italic in theorems \newcommand{\ie}{\emph{i.e.}\xspace} \newcommand{\eg}{\emph{e.g.}\xspace} \newcommand{\Ie}{\emph{I.e.}\xspace} \newcommand{\Eg}{\emph{E.g.}\xspace} \newcommand{\iec}{\emph{i.e.}, } \newcommand{\egc}{\emph{e.g.}, } \newcommand{\ied}{\emph{i.e\@.} } \newcommand{\egd}{\emph{e.g\@.} } \newcommand{\etc}{\emph{etc.}\xspace} \newcommand{\etcc}{\emph{etc.}, } \newcommand{\etcd}{\emph{etc\@.} } \newcommand{\etal}{\emph{et al.}\xspace} \newcommand{\etalc}{\emph{et al.}, } \newcommand{\etald}{\emph{et al.} } \newcommand{\invivo}{\emph{in vivo}\xspace} \newcommand{\Invivo}{\emph{In vivo}\xspace} \newcommand{\invitro}{\emph{in vitro}\xspace} \newcommand{\insilico}{\emph{in silico}\xspace} \newcommand{\Insilico}{\emph{In silico}\xspace} \newcommand{\apriori}{\emph{a priori}\xspace} \newcommand{\aposteriori}{\emph{a posteriori}\xspace} \newcommand{\Apriori}{\emph{A priori}\xspace} \newcommand{\Aposteriori}{\emph{A posteriori}\xspace} \newcommand{\aprioric}{\emph{a priori}, } \newcommand{\aposterioric}{\emph{a posteriori}, } \newcommand{\defacto}{\emph{de facto}\xspace} \newcommand{\perse}{\emph{per se}\xspace} \newcommand{\persec}{\emph{per se}, } \newcommand{\persed}{\emph{per se\@.} } \newcommand{\viceversa}{\emph{vice versa}\xspace} \newcommand{\yeast}{\textit{S\@. cerevisiae}\xspace} \newcommand{\Yeast}{\textit{Saccharomyces cerevisiae}\xspace} \newcommand{\coli}{\textit{E\@. coli}\xspace} \newcommand{\Coli}{\textit{Escherichia coli}\xspace} \newcommand{\lasso}{LASSO\xspace} \newcommand{\matlab}{MATLAB\xspace} \newcommand{\iid}{{i.i.d.}\ } % This command already exist in BMC class \providecommand{\pp}[1]{p.~#1} % Used for indicating page numbers in references % \providecommand{\pp}[1]{} % Used for removing page numbers in references \providecommand{\s}[1]{ch.~#1} % Used for indicating section/chapter numbers in references % \providecommand{\s}[1]{} % Used for removing section/chapter numbers in references \def\inv{{^{-1}}}% inverse \providecommand{\abs}[1]{\left\lvert#1\right\rvert} % abs \providecommand{\norm}[1]{\left\lVert#1\right\rVert} % norm \providecommand{\bignorm}[1]{\big\lVert#1\big\rVert} % norm \providecommand{\Bignorm}[1]{\Big\lVert#1\Big\rVert} % norm \providecommand{\mnorm}[1]{\left\lVert#1\right\rVert} % matrix norm \providecommand{\bigmnorm}[1]{\big\lVert#1\big\rVert} % matrix norm \providecommand{\Bigmnorm}[1]{\Big\lVert#1\Big\rVert} % matrix norm \providecommand{\rv}[1]{\mathrm{#1}} % Mark of random variables \providecommand{\set}[1]{\mathcal{#1}} % Mark of variables denoting sets \providecommand{\mean}[1]{\ensuremath{\widehat{#1}}} % Mark of mean value \providecommand{\subs}[1]{\ensuremath{\textrm{#1}}} % Mark of substrates \providecommand{\conc}[1]{\ensuremath{\left[{#1}\right]}} % Mark of concentration \providecommand{\figvar}[1]{\ensuremath{\mathsf{#1}}} % Mark figure variables \renewcommand{\breve}{} % Used to remove the breve marks \def\pb{\\*[0.42cm]} % Vector marker \def\b0{{\boldsymbol{0}}} \def\bA{{\boldsymbol{A}}} \def\ba{{\boldsymbol{a}}} \def\bB{{\boldsymbol{B}}} \def\bb{{\boldsymbol{b}}} \def\bC{{\boldsymbol{C}}} \def\bc{{\boldsymbol{c}}} \def\bD{{\boldsymbol{D}}} \def\bd{{\boldsymbol{d}}} \def\bE{{\boldsymbol{E}}} \def\be{{\boldsymbol{e}}} \def\bF{{\boldsymbol{F}}} \def\bbf{{\boldsymbol{f}}} \def\bG{{\boldsymbol{G}}} \def\bg{{\boldsymbol{g}}} \def\bH{{\boldsymbol{H}}} \def\bh{{\boldsymbol{h}}} \def\bI{{\boldsymbol{I}}} \def\bi{{\boldsymbol{i}}} \def\bJ{{\boldsymbol{J}}} \def\bj{{\boldsymbol{j}}} \def\bK{{\boldsymbol{K}}} \def\bk{{\boldsymbol{k}}} \def\bL{{\boldsymbol{L}}} \def\bl{{\boldsymbol{l}}} \def\bM{{\boldsymbol{M}}} \def\bm{{\boldsymbol{m}}} \def\bN{{\boldsymbol{N}}} \def\bn{{\boldsymbol{n}}} \def\bO{{\boldsymbol{O}}} \def\bo{{\boldsymbol{o}}} \def\bP{{\boldsymbol{P}}} \def\bp{{\boldsymbol{p}}} \def\bQ{{\boldsymbol{Q}}} \def\bq{{\boldsymbol{q}}} \def\bR{{\boldsymbol{R}}} \def\br{{\boldsymbol{r}}} \def\bs{{\boldsymbol{s}}} \def\bS{{\boldsymbol{S}}} \def\bT{{\boldsymbol{T}}} \def\bt{{\boldsymbol{t}}} \def\bU{{\boldsymbol{U}}} \def\bu{{\boldsymbol{u}}} \def\bV{{\boldsymbol{V}}} \def\bv{{\boldsymbol{v}}} \def\bW{{\boldsymbol{W}}} \def\bw{{\boldsymbol{w}}} \def\bx{{\boldsymbol{x}}} \def\bX{{\boldsymbol{X}}} \def\bY{{\boldsymbol{Y}}} \def\by{{\boldsymbol{y}}} \def\bZ{{\boldsymbol{Z}}} \def\bz{{\boldsymbol{z}}} \def\balpha{{\boldsymbol{\alpha}}} \def\bbeta{{\boldsymbol{\beta}}} \def\bgamma{{\boldsymbol{\gamma}}} \def\bdelta{{\boldsymbol{\delta}}} \def\bepsilon{{\boldsymbol{\epsilon}}} \def\bvarepsilon{{\boldsymbol{\varepsilon}}} \def\bzeta{{\boldsymbol{\zeta}}} \def\boldeta{{\boldsymbol{\eta}}} \def\btheta{{\boldsymbol{\theta}}} \def\bvartheta{{\boldsymbol{\vartheta}}} \def\biota{{\boldsymbol{\iota}}} \def\bkappa{{\boldsymbol{\kappa}}} \def\blambda{{\boldsymbol{\lambda}}} \def\bmu{{\boldsymbol{\mu}}} \def\bnu{{\boldsymbol{\nu}}} \def\bxi{{\boldsymbol{\xi}}} \def\bpi{{\boldsymbol{\pi}}} \def\bvarpi{{\boldsymbol{\varpi}}} \def\brho{{\boldsymbol{\rho}}} \def\bvarrho{{\boldsymbol{\varrho}}} \def\bsigma{{\boldsymbol{\sigma}}} \def\bvarsigma{{\boldsymbol{\varsigma}}} \def\btau{{\boldsymbol{\tau}}} \def\bupsilon{{\boldsymbol{\upsilon}}} \def\bphi{{\boldsymbol{\phi}}} \def\bvarphi{{\boldsymbol{\varphi}}} \def\bchi{{\boldsymbol{\chi}}} \def\bpsi{{\boldsymbol{\psi}}} \def\bomega{{\boldsymbol{\omega}}} \def\bGamma{{\boldsymbol{\Gamma}}} \def\bDelta{{\boldsymbol{\Delta}}} \def\bTheta{{\boldsymbol{\Theta}}} \def\bLambda{{\boldsymbol{\Lambda}}} \def\bXi{{\boldsymbol{\Xi}}} \def\bPi{{\boldsymbol{\Pi}}} \def\bSigma{{\boldsymbol{\Sigma}}} \def\bUpsilon{{\boldsymbol{\Upsilon}}} \def\bPhi{{\boldsymbol{\Phi}}} \def\bPsi{{\boldsymbol{\Psi}}} \def\bOmega{{\boldsymbol{\Omega}}} % Matrix marker \def\m0{{\boldsymbol{0}}} \def\mA{{\boldsymbol{A}}} \def\ma{{\boldsymbol{a}}} \def\mB{{\boldsymbol{B}}} \def\mb{{\boldsymbol{b}}} \def\mC{{\boldsymbol{C}}} \def\mc{{\boldsymbol{c}}} \def\mD{{\boldsymbol{D}}} \def\md{{\boldsymbol{d}}} \def\mE{{\boldsymbol{E}}} \def\me{{\boldsymbol{e}}} \def\mF{{\boldsymbol{F}}} \def\mf{{\boldsymbol{f}}} \def\mG{{\boldsymbol{G}}} \def\mg{{\boldsymbol{g}}} \def\mH{{\boldsymbol{H}}} \def\mh{{\boldsymbol{h}}} \def\mI{{\boldsymbol{I}}} \def\mi{{\boldsymbol{i}}} \def\mJ{{\boldsymbol{J}}} \def\mj{{\boldsymbol{j}}} \def\mK{{\boldsymbol{K}}} \def\mk{{\boldsymbol{k}}} \def\mL{{\boldsymbol{L}}} \def\ml{{\boldsymbol{l}}} \def\mM{{\boldsymbol{M}}} \def\mm{{\boldsymbol{m}}} \def\mN{{\boldsymbol{N}}} \def\mn{{\boldsymbol{n}}} \def\mO{{\boldsymbol{O}}} \def\mo{{\boldsymbol{o}}} \def\mP{{\boldsymbol{P}}} \def\mp{{\boldsymbol{p}}} \def\mQ{{\boldsymbol{Q}}} \def\mq{{\boldsymbol{q}}} \def\mR{{\boldsymbol{R}}} \def\mr{{\boldsymbol{r}}} \def\ms{{\boldsymbol{s}}} \def\mS{{\boldsymbol{S}}} \def\mT{{\boldsymbol{T}}} \def\mt{{\boldsymbol{t}}} \def\mU{{\boldsymbol{U}}} \def\muu{{\boldsymbol{u}}} \def\mV{{\boldsymbol{V}}} \def\mv{{\boldsymbol{v}}} \def\mW{{\boldsymbol{W}}} \def\mw{{\boldsymbol{w}}} \def\mx{{\boldsymbol{x}}} \def\mX{{\boldsymbol{X}}} \def\mY{{\boldsymbol{Y}}} \def\my{{\boldsymbol{y}}} \def\mZ{{\boldsymbol{Z}}} \def\mz{{\boldsymbol{z}}} \def\malpha{{\boldsymbol{\alpha}}} \def\mbeta{{\boldsymbol{\beta}}} \def\mgamma{{\boldsymbol{\gamma}}} \def\mdelta{{\boldsymbol{\delta}}} \def\mepsilon{{\boldsymbol{\epsilon}}} \def\mvarepsilon{{\boldsymbol{\varepsilon}}} \def\mzeta{{\boldsymbol{\zeta}}} \def\mateta{{\boldsymbol{\eta}}} \def\mtheta{{\boldsymbol{\theta}}} \def\mvartheta{{\boldsymbol{\vartheta}}} \def\miota{{\boldsymbol{\iota}}} \def\mkappa{{\boldsymbol{\kappa}}} \def\mlambda{{\boldsymbol{\lambda}}} \def\mmu{{\boldsymbol{\mu}}} \def\mnu{{\boldsymbol{\nu}}} \def\mxi{{\boldsymbol{\xi}}} \def\mpi{{\boldsymbol{\pi}}} \def\mvarpi{{\boldsymbol{\varpi}}} \def\mrho{{\boldsymbol{\rho}}} \def\mvarrho{{\boldsymbol{\varrho}}} \def\msigma{{\boldsymbol{\sigma}}} \def\mvarsigma{{\boldsymbol{\varsigma}}} \def\mtau{{\boldsymbol{\tau}}} \def\mupsilon{{\boldsymbol{\upsilon}}} \def\mphi{{\boldsymbol{\phi}}} \def\mvarphi{{\boldsymbol{\varphi}}} \def\mchi{{\boldsymbol{\chi}}} \def\mpsi{{\boldsymbol{\psi}}} \def\momega{{\boldsymbol{\omega}}} \def\mGamma{{\boldsymbol{\Gamma}}} \def\mDelta{{\boldsymbol{\Delta}}} \def\mTheta{{\boldsymbol{\Theta}}} \def\mLambda{{\boldsymbol{\Lambda}}} \def\mXi{{\boldsymbol{\Xi}}} \def\mPi{{\boldsymbol{\Pi}}} \def\mSigma{{\boldsymbol{\Sigma}}} \def\mUpsilon{{\boldsymbol{\Upsilon}}} \def\mPhi{{\boldsymbol{\Phi}}} \def\mPsi{{\boldsymbol{\Psi}}} \def\mOmega{{\boldsymbol{\Omega}}} % Symbols for number sets \def\bmF{{\mathbb{F}}} \def\bmR{{\mathbb{R}}} \def\bmC{{\mathbb{C}}} \def\bmZ{{\mathbb{Z}}} \def\bmN{{\mathbb{N}}}