prompt_front: Now, answer the following question based on the text below, only give me the answer and do not output any other words.

Question: How many brother does Njoroge have?

Read the following text and answer briefly.

Context 1: Weep Not, Child is a 1964 novel by Kenyan author Ngũgĩ wa Thiong'o  It was his first novel, published in 1964 under the name James Ngugi  It was among the African Writers Series  It was the first English language|English novel to be published by an East African  Thiong'o's works deal with the relationship between Africans and white settlers in colonial Kenya, and are heavily critical of colonial rule  Specifically, Weep Not, Child deals with the Mau Mau Uprising, and "the bewildering dispossession of an entire people from their ancestral land " Ngũgĩ wrote the novel while he was a student at Makerere University 

The book is divided into two parts and eighteen chapters  Part one deals mostly with the education of Njoroge, while part two deals with the rising Mau Mau movement 

Plot summary

Njoroge, a little boy, is urged to attend school by his mother  He is the first one of his family able to go to school  His family lives on the land of Jacobo, an African made rich by his dealings with white settlers, namely Mr  Howlands, the most powerful land owner in the area  Njoroge's brother Kamau works as an apprentice to a carpenter, while Boro, the eldest living son, is troubled by his experiences while in forced service during World War II, including witnessing the death of his elder brother  Ngotho, Njoroge's father and a respected man in the surrounding area, tends Mr  Howlands' crops, but is motivated by his passion to preserve his ancestral land, rather than for any compensation or loyalty 

One day, black workers call for a strike to obtain higher wages  Ngotho is ambivalent about participating in the strike because he fears he will lose his job  However, he decides to go to the gathering, even though his two wives do not agree  At the demonstration, there are calls for higher wages  Suddenly, the white police inspector brings Jacobo to the gathering to pacify the native people  Jacobo tries to put an end to the strike  Ngotho attacks Jacobo, and the result is a riot where two people are killed  Jacobo survives and swears revenge  Ngotho loses his job and Njoroge’s family is forced to move  Njoroge’s brothers fund his education and seem to lose respect for their father 

Mwihaki, Jacobo's daughter and Njoroge's best friend, enters a girls' only boarding school, leaving Njoroge relatively alone  He reflects upon her leaving, and realizes that he was embarrassed by his father's actions towards Jacobo  For this reason, Njoroge is not upset by her exit and their separation  Njoroge switches to another school 

For a time, everyone's attention is focused on the upcoming trial of Jomo Kenyatta – a revered leader of the movement  Many blacks think that he is going to bring forth Kenya’s independence  But Jomo loses the trial and is imprisoned  This results in further protests and greater suppression of the black population 

Jacobo and a white landowner, Mr  Howlands, fight against the rising activities of the Mau Mau, an organization striving for Kenyan economic, political, and cultural independence  Jacobo accuses Ngotho of being the leader of the Mau Mau and tries to imprison the whole family  Meanwhile, the situation in the country is deteriorating  Six black men are taken out of their houses and executed in the woods 

One day Njoroge meets Mwihaki again, who has returned from boarding school  Although Njoroge had planned to avoid her due to the conflict between their fathers, their friendship is unaffected  Njoroge passes an important exam that allows him to advance to High School  His village is proud of him, and collects money to pay Njoroge's High School tuition 

Several months later, Jacobo is murdered in his office by a member of the Mau Mau  Mr  Howlands has Njoroge removed from school for questioning  Both father and son are brutally beaten before release and Ngotho is left barely alive  Although there doesn't seem to be a connection between Njoroge's family and the murder, it is eventually revealed that Njoroge's brothers are behind the assassination, and that Boro is the real leader of the Mau Mau  Ngotho soon dies from his injuries and Njoroge finds out that his father was protecting his brothers  Kamau has been imprisoned for life  Only Njoroge and his two mothers remain free, and Njoroge is left as the sole provider of his two mothers  Njoroge fears that he cannot make ends meet; he gives up hope of continuing in school and loses faith in God 

Njoroge asks Mwihaki's for support, but she is angry because of her father’s death  When he finally pledges his love to her, she refuses to leave with him, realizing her obligation to Kenya and her mother  Njoroge decides to leave town and makes an attempt at suicide; however, he fails when his mothers find him before he is able to hang himself  The novel closes with Njoroge feeling hopeless, and ashamed of cowardice 

Characters in Weep Not, Child
 Njoroge: the main character of the book whose main goal throughout the book is to become as educated as possible 
 Ngotho: Njoroge's father  He works for Mr Howlands and is respected by him until he attacks Jacobo at a workers strike  He is fired and the family is forced to move to another section of the country  Over the course of the book his position as the central power of the family weakened, to the point where his self-realization that he has spent his whole life waiting for the prophecy (that proclaims the blacks will be returned their land) to come true rather than fighting for Kenyan independence, leads to his depression 
 Nyokabi and Njeri: the two wives of Ngotho  Njeri is Ngotho's first wife, and mother of Boro, Kamau, and Kori  Nyokabi is his second wife, and the mother of Njoroge and Mwangi 
 Njoroge has four brothers: Boro, Kamau, Kori and Mwangi (who is Njoroge's only full brother, who died in World War II) 
 Boro: Son of Njeri who fights for the Allies in World War II  Upon returning his anger against the colonial government is compounded by their confiscation of the his land  Boro's anger and position as eldest son leads him to question and ridicule Ngotho, which eventually defeats their father's will (upon realizing his life was wasted waiting and not acting)  It is eventually revealed that Boro is the leader of the Mau Mau (earlier alluded to as "entering politics") and murders Mr Howlands  He is caught by police immediately after and is scheduled to be executed by the book's end  It is highly likely that it is also Boro who kills Jacobo 
 Mwihaki: Njoroge's best friend (and later develops into his love interest)  Daughter of Jacobo  When it is revealed that his family killed Jacobo (most likely Boro), Mwihaki distances herself from Njoroge, asking for time to mourn her father and care for her mother 
 Jacobo: Mwihaki's father and an important landowner  Chief of the village 
 Mr  Howlands: A white settler who emigrated to colonial Kenya and now owns a farm made up of land that originally belonged to Ngotho's ancestors  Has three children: Peter who died in World War II before the book's beginning, a daughter who becomes a missionary, and Stephen who met Njoroge while the two were in high school 

Themes and motifs
Weep Not, Child integrates Gikuyu mythology and the ideology of nationalism that serves as catalyst for much of the novel's action  The novel explores the negative aspects of colonial rule over Kenya  Njoroge's aspiration to attend university is frustrated by both the violence of the Mau Mau rebels and the violent response of the colonial government  This disappointment leads to his alienation from his family and ultimately his suicide attempt 

The novel also ponders the role of saviours and salvation  The author notes in his The River Between: "Salvation shall come from the hills  From the blood that flows in me, I say from the same tree, a son shall rise  And his duty shall be to lead and save the people " Jomo Kenyatta, the first prime minister of Kenya, is immortalised in Weep Not, Child  The author says, "Jomo had been his (Ngotho's) hope  Ngotho had come to think that it was Jomo who would drive away the white man  To him, Jomo stood for custom and traditions purified by grace of learning and much travel " Njoroge comes to view Jomo as a messiah who will win the struggle against the colonial government 
Context 2: \section{Introduction}
Given a data set and a model with some unknown parameters, the inverse problem aims to find the values of the model parameters that best fit the data   
In this work, in which we focus on systems of interacting elements,
 the inverse problem concerns  the statistical inference
 of the underling interaction network and of its coupling coefficients from observed data on the dynamics  of the system  
 Versions of this problem are encountered in physics, biology (e g , \cite{Balakrishnan11,Ekeberg13,Christoph14}), social sciences and finance (e g ,\cite{Mastromatteo12,yamanaka_15}), neuroscience (e g , \cite{Schneidman06,Roudi09a,tyrcha_13}), just to cite a few, and are becoming more and more important due to the increase in the amount of data available from these fields \\
 \indent 
 A standard approach used in statistical inference is to predict the interaction couplings by maximizing the likelihood function 
 This technique, however, requires the evaluation of the  
 
   partition function that, in the most general case, concerns a number of computations scaling exponentially with the system size 
  
  
    Boltzmann machine learning uses Monte Carlo sampling to compute the gradients of the Log-likelihood looking for stationary points \cite{Murphy12} but this method is computationally manageable only for small systems  A series of faster approximations, such as naive mean-field, independent-pair approximation \cite{Roudi09a, Roudi09b}, inversion of TAP equations \cite{Kappen98,Tanaka98}, small correlations expansion \cite{Sessak09}, adaptive TAP \cite{Opper01}, adaptive  cluster expansion \cite{Cocco12} or Bethe approximations \cite{Ricci-Tersenghi12, Nguyen12}  have, then, been developed  These techniques take as input means and correlations of observed variables and most of them assume a fully connected graph as underlying connectivity network, or expand  around it by perturbative dilution   In most cases, network reconstruction turns out to  be not accurate for small data sizes and/or when couplings are  strong or, else, if the original interaction network is sparse \\
\indent
 A further method, substantially improving performances for small data, is the so-called Pseudo-Likelyhood Method (PLM) \cite{Ravikumar10}  In Ref  \cite{Aurell12} Aurell and Ekeberg performed a comparison between PLM and some of the just mentioned mean-field-based algorithms on the pairwise interacting Ising-spin  ($\sigma = \pm 1$) model, showing how PLM performs sensitively better, especially on sparse graphs and in the high-coupling limit, i e , for low temperature 
 
 In this work, we aim at performing statistical inference  on a model whose interacting variables are continuous $XY$ spins, i e , $\sigma \equiv \left(\cos \phi,\sin \phi\right)$ with $\phi \in [0, 2\pi )$  The developed tools can, actually, be also straightforward applied  to the $p$-clock model  \cite{Potts52} where the phase $\phi$ takes discretely equispaced $p$ values  in the $2 \pi$ interval, $\phi_a =  a 2 \pi/p$, with $a= 0,1,\dots,p-1$  The $p$-clock model, else called vector Potts model, gives a hierarchy of discretization of the $XY$ model as $p$ increases  For $p=2$, one recovers the Ising model, for $p=4$ the Ashkin-Teller model \cite{Ashkin43}, for $p=6$ the ice-type model \cite{Pauling35,Baxter82} and the eight-vertex model \cite{Sutherland70,Fan70,Baxter71} for $p=8$   
It turns out to be very useful also for numerical implementations of the continuous $XY$ model  
Recent analysis on the multi-body $XY$ model has shown that for a limited number of discrete phase values ($p\sim 16, 32$) the thermodynamic critical properties of the $p\to\infty$ $XY$ limit are promptly recovered \cite{Marruzzo15, Marruzzo16}   
Our main motivation to study statistical inference is that these kind of models have recently turned out to be rather useful in describing the behavior of optical systems, 
including standard mode-locking lasers \cite{Gordon02,Gat04,Angelani07,Marruzzo15} and random lasers \cite{Angelani06a,Leuzzi09a,Antenucci15a,Antenucci15b,Marruzzo16}  
In particular, the inverse problem on the pairwise XY model analyzed here might be of help in recovering images from light propagated through random media  


      This paper is organized as follows: in Sec  \ref{sec:model} we introduce the general model  and we discuss its  derivation also  as a model for light transmission through random scattering media  
    In Sec  \ref{sec:plm} we introduce the PLM with $l_2$ regularization  and with decimation, two variants of the PLM respectively introduced in Ref  \cite{Wainwright06} and \cite{Aurell12} for the inverse Ising problem  
    Here, we analyze these techniques for continuous $XY$ spins and we test them on thermalized data generated by Exchange Monte Carlo numerical simulations of the original model dynamics  In Sec  \ref{sec:res_reg} we  present the results related to the PLM-$l_2$  In Sec  \ref{sec:res_dec} the results related to the PLM with decimation are reported and its performances are compared to the PLM-$l_2$ and to a variational mean-field method analyzed in Ref  \cite{Tyagi15}  In Sec  \ref{sec:conc}, we outline conclusive remarks and perspectives 

        \section{The leading $XY$ model}
      \label{sec:model}
 The leading model we are considering is defined, for a system of $N$ angular $XY$ variables, by the Hamiltonian 
 \begin{equation}
  \mathcal{H} = - \sum_{ik}^{1,N} J_{ik} \cos{\left(\phi_i-\phi_k\right)} 
  \label{eq:HXY}
 
  \end{equation} 
  
  The $XY$ model is well known in statistical mechanics, displaying important physical
  insights, starting from the Berezinskii-Kosterlitz-Thouless
  transition in two dimensions\cite{Berezinskii70,Berezinskii71,Kosterlitz72} and moving to, e g , the
  transition of liquid helium to its superfluid state \cite{Brezin82}, the roughening transition of the interface of a crystal in equilibrium with its vapor \cite{Cardy96}  In presence of disorder and frustration \cite{Villain77,Fradkin78} the model has been adopted to describe synchronization problems as the Kuramoto model \cite{Kuramoto75} and in the theoretical modeling of Josephson junction arrays \cite{Teitel83a,Teitel83b} and arrays of coupled lasers \cite{Nixon13} 
  Besides several derivations and implementations of the model in quantum and classical physics, equilibrium or out of equilibrium, ordered or fully frustrated systems, Eq  (\ref{eq:HXY}), in its generic form,
  has found applications also in other fields  A rather fascinating example being the behavior of starlings flocks  \cite{Reynolds87,Deneubourg89,Huth90,Vicsek95, Cavagna13} 
  Our interest on the $XY$ model resides, though, in optics  Phasor and phase models with pairwise and  multi-body interaction terms can, indeed, describe the behavior of electromagnetic modes in both linear and nonlinear optical systems in the analysis of problems such as light propagation and lasing \cite{Gordon02, Antenucci15c, Antenucci15d}  As couplings are strongly frustrated, these models turn out to be especially useful to the study of optical properties in random media \cite{Antenucci15a,Antenucci15b}, as in the noticeable case of random lasers \cite{Wiersma08,Andreasen11,Antenucci15e}  and they might as well be applied to linear scattering problems, e g , propagation of waves in opaque systems or disordered fibers  
  
  
  \subsection{A propagating wave model}
  We briefly mention a derivation of the model as a proxy for the propagation of light through random linear media  
 Scattering of light is held responsible to obstruct our view and make objects opaque  Light rays, once that they enter the material, only exit  after getting scattered multiple times within the material  In such a disordered medium, both the direction and the phase of the propagating waves are random  Transmitted light 
 yields a disordered interference pattern typically  having low intensity, random phase and almost no resolution, called a speckle  Nevertheless, in recent years it has been realized that disorder is rather a blessing in disguise \cite{Vellekoop07,Vellekoop08a,Vellekoop08b}  Several experiments have made it possible to control the behavior of light and other optical processes in a given random disordered medium,  
 by exploiting, e g ,  the tools developed for wavefront shaping to control the propagation of light and to engineer the confinement of light \cite{Yilmaz13,Riboli14} 
 \\
 \indent
 In a linear dielectric medium, light propagation can be described through a part of the scattering matrix, the transmission matrix $\mathbb{T}$, linking the outgoing to the incoming fields  
 Consider the case in which there are $N_I$ incoming channels and $N_O$ outgoing ones; we can indicate with $E^{\rm in,out}_k$ the input/output electromagnetic field phasors of channel $k$  In the most general case, i e , without making any particular assumptions on the field polarizations, each light mode and its polarization polarization state can be represented by means of the $4$-dimensional Stokes vector  Each $ t_{ki}$ element of $\mathbb{T}$, thus, is a $4 \times 4$ M{\"u}ller matrix  If, on the other hand, we know that the source is polarized and the observation is made on the same polarization, one can use a scalar model and adopt Jones calculus \cite{Goodman85,Popoff10a,Akbulut11}:
   \begin{eqnarray}
 E^{\rm out}_k = \sum_{i=1}^{N_I}  t_{ki} E^{\rm in}_i \qquad \forall~ k=1,\ldots,N_O
 \label{eq:transm}
 \end{eqnarray}
  We recall that the elements of the transmission matrix are random complex coefficients\cite{Popoff10a}  For the case of completely unpolarized modes, we can also use a scalar model similar to Eq  \eqref{eq:transm}, but whose variables are  the intensities of the outgoing/incoming fields, rather than the fields themselves \\ 
In the following, for simplicity, we will consider Eq  (\ref{eq:transm}) as our starting point,
where $E^{\rm out}_k$, $E^{\rm in}_i$ and $t_{ki}$ are all complex scalars  
If Eq  \eqref{eq:transm} holds for any $k$, we can write:
  \begin{eqnarray}
  \int \prod_{k=1}^{N_O} dE^{\rm out}_k \prod_{k=1}^{N_O}\delta\left(E^{\rm out}_k - \sum_{j=1}^{N_I}  t_{kj} E^{\rm in}_j \right) = 1
  \nonumber
  \\
  \label{eq:deltas}
  \end{eqnarray}

 Observed data are a noisy representation of the true values of the fields  Therefore, in inference problems it is statistically more meaningful to take that noise into account in a probabilistic way, 
 rather than looking  at the precise solutions of the exact equations (whose parameters are unknown)  
 To this aim we can introduce Gaussian distributions whose limit for zero variance are the Dirac deltas in Eq  (\ref{eq:deltas}) 
 Moreover, we move to consider the ensemble of all possible solutions of Eq  (\ref{eq:transm}) at given $\mathbb{T}$, looking at  all configurations of input fields  We, thus, define the function:
 
   \begin{eqnarray}
  Z &\equiv &\int_{{\cal S}_{\rm in}} \prod_{j=1}^{N_I}  dE^{\rm in}_j \int_{{\cal S}_{\rm out}}\prod_{k=1}^{N_O} dE^{\rm out}_k 
  \label{def:Z}
\\
    \times
  &&\prod_{k=1}^{N_O}
   \frac{1}{\sqrt{2\pi \Delta^2}}  \exp\left\{-\frac{1}{2 \Delta^2}\left|
  E^{\rm out}_k -\sum_{j=1}^{N_I}  t_{kj} E^{\rm in}_j\right|^2
\right\} 
\nonumber
 \end{eqnarray}
   We stress that the integral of Eq  \eqref{def:Z} is not exactly a Gaussian integral  Indeed, starting from Eq  \eqref{eq:deltas}, two constraints on the electromagnetic field intensities must be taken into account  

  The space of solutions is  delimited by the total power ${\cal P}$ received by system, i e , 
  ${\cal S}_{\rm in}: \{E^{\rm in} |\sum_k I^{\rm in}_k = \mathcal{P}\}$, also implying  a constraint on the total amount of energy that is transmitted through the medium, i  e , 
  ${\cal S}_{\rm out}:\{E^{\rm out} |\sum_k I^{\rm out}_k=c\mathcal{P}\}$, where the attenuation factor  $c<1$ accounts for total losses 
  As we will see more in details in the following, being interested in inferring the transmission matrix through the PLM, we can omit to explicitly include these terms in Eq  \eqref{eq:H_J} since they do not depend on $\mathbb{T}$ not adding any information on the gradients with respect to the elements of $\mathbb{T}$ 
  
 Taking the same number of incoming and outcoming channels, $N_I=N_O=N/2$, and  ordering the input fields in the first $N/2$ mode indices and the output fields in the last $N/2$ indices, we can drop the ``in'' and ``out'' superscripts and formally write $Z$  as a partition function
    \begin{eqnarray}
        \label{eq:z}
 && Z =\int_{\mathcal S} \prod_{j=1}^{N} dE_j \left(   \frac{1}{\sqrt{2\pi \Delta^2}} \right)^{N/2} 
 \hspace*{- 4cm} \exp\left\{
  -\frac{ {\cal H} [\{E\};\mathbb{T}] }{2\Delta^2}
  \right\}
  \\
&&{\cal H} [\{E\};\mathbb{T}] =
-  \sum_{k=1}^{N/2}\sum_{j=N/2+1}^{N} \left[E^*_j t_{jk} E_k + E_j t^*_{kj} E_k^* 
\right]
 \nonumber
\\
&&\qquad\qquad \qquad + \sum_{j=N/2+1}^{N} |E_j|^2+ \sum_{k,l}^{1,N/2}E_k
U_{kl} E_l^*
 \nonumber
 \\
 \label{eq:H_J}
 &&\hspace*{1 88cm } = - \sum_{nm}^{1,N} E_n J_{nm} E_m^*
 \end{eqnarray}
 where ${\cal H}$ is a real-valued function by construction, we have introduced the effective input-input coupling matrix
\begin{equation}
U_{kl} \equiv \sum_{j=N/2+1}^{N}t^*_{lj} t_{jk} 
 \label{def:U}
 \end{equation}
 and the whole interaction matrix reads (here $\mathbb{T} \equiv \{ t_{jk} \}$)
 \begin{equation}
 \label{def:J}
 \mathbb J\equiv \left(\begin{array}{ccc|ccc}
 \phantom{()}&\phantom{()}&\phantom{()}&\phantom{()}&\phantom{()}&\phantom{()}\\
 \phantom{()}&-\mathbb{U} \phantom{()}&\phantom{()}&\phantom{()}&{\mathbb{T}}&\phantom{()}\\
\phantom{()}&\phantom{()}&\phantom{()}&\phantom{()}&\phantom{()}&\phantom{()}\\
 \hline
\phantom{()}&\phantom{()}&\phantom{()}&\phantom{()}&\phantom{()}&\phantom{()}\\
 \phantom{()}& \mathbb   T^\dagger&\phantom{()}&\phantom{()}& - \mathbb{I} &\phantom{()}\\
\phantom{a}&\phantom{a}&\phantom{a}&\phantom{a}&\phantom{a}&\phantom{a}\\
 \end{array}\right)
 \end{equation}
 
 Determining the electromagnetic complex amplitude configurations that minimize the {\em cost function} ${\cal H}$, Eq   (\ref{eq:H_J}),  means to maximize the overall distribution peaked around the solutions of the transmission Eqs  (\ref{eq:transm})  As the variance $\Delta^2\to 0$, eventually, the initial set of Eqs  (\ref{eq:transm}) are recovered  The ${\cal H}$ function, thus, plays the role of an Hamiltonian and  $\Delta^2$ the role of a noise-inducing temperature  The exact numerical problem corresponds to the zero temperature limit of the statistical mechanical problem  Working with real data, though, which are noisy, a finite ``temperature''
  allows for a better representation of the ensemble of solutions to the sets of equations of continuous variables  
  

  
  
  Now, we can express every phasor in Eq  \eqref{eq:z}  as $E_k = A_k e^{\imath \phi_k}$  As a working hypothesis we will consider the intensities $A_k^2$ as either homogeneous or as \textit{quenched} with respect to phases 
The first condition occurs, for instance, to the input intensities $|E^{\rm in}_k|$ produced by a phase-only spatial light modulator (SLM) with homogeneous illumination \cite{Popoff11} 
With \textit{quenched} here we mean, instead, that the intensity of each mode is the same for every solution of Eq  \eqref{eq:transm} at fixed $\mathbb T$ 
We stress that, including intensities in the model does not preclude the inference analysis but it is out of the focus of the present work and will be considered elsewhere  

If all intensities are uniform in input and in output, this amount to a constant rescaling for each one of the four sectors of matrix $\mathbb J$ in Eq  (\ref{def:J}) that will not change the properties of the matrices 
For instance, if the original transmission matrix is unitary, so it will be the rescaled one and the matrix $\mathbb U$ will be  diagonal 
Otherwise, if intensities are \textit{quenched}, i e , they can be considered as constants in Eq  (\ref{eq:transm}),
they are inhomogeneous with respect to phases  The generic Hamiltonian element will, therefore, rescale as 
  \begin{eqnarray}
  E^*_n J_{nm} E_m = J_{nm} A_n A_m e^{\imath (\phi_n-\phi_m)} \to J_{nm} e^{\imath (\phi_n-\phi_m)}
  \nonumber
  \end{eqnarray}
  and the properties of the original  $J_{nm}$ components are not conserved  in the rescaled one  In particular, we have no argument, anymore, to possibly set the rescaled $U_{nm}\propto \delta_{nm}$ 
  Eventually, we end up with the complex couplings $XY$ model, whose real-valued Hamiltonian is written as
 \begin{eqnarray}
  \mathcal{H}& = &  - \frac{1}{2} \sum_{nm} J_{nm} e^{-\imath (\phi_n - \phi_m)}  + \mbox{c c } 
    \label{eq:h_im}
\\    &=&  - \frac{1}{2} \sum_{nm} \left[J^R_{nm} \cos(\phi_n - \phi_m)+
  J^I_{nm}\sin (\phi_n - \phi_m)\right] 
  \nonumber
 \end{eqnarray}
where $J_{nm}^R$ and $J_{nm}^I$ are the real and imaginary parts of $J_{nm}$  Being $\mathbb J$  Hermitian, $J^R_{nm}=J^R_{mn}$ is symmetric and $J_{nm}^I=-J_{mn}^I$ is skew-symmetric 

\begin{comment}
\textcolor{red}{
F: comment about quenched:
I think that to obtain the XY model, it is not necessary that the intensities are strictly quenched (that is also a quite unfeasible situation, I guess) 
Indeed eq (2) does not deal with the dynamics of the modes, but just connect the in and out ones 
For this, what it is necessary to have the XY model, it is that the intensities are always the same on the different samples
(so that the matrix $t_{ij}$ is the same for different phase data)  If the intensities are fixed, then they can be incorporated in $t_{ij}$ and eq (2) can be written just for phases as described  \\
}
\end{comment}


  \section{Pseudolikelihood Maximization}
  \label{sec:plm}
The inverse problem consists in the reconstruction of the parameters $J_{nm}$ of the Hamiltonian, Eq  (\ref{eq:h_im})   
Given a set of $M$ data configurations of $N$ spins
 $\bm\sigma = \{ \cos \phi_i^{(\mu)},\sin \phi_i^{(\mu)} \}$, $i = 1,\dots,N$ and $\mu=1,\dots,M$, we want to \emph{infer} the couplings:
 \begin{eqnarray}
\bm \sigma  \rightarrow  \mathbb{J} 
\nonumber
 \end{eqnarray}
 With this purpose in mind,
 in the rest of this section we implement the working equations for the techniques used  
 In order to test our methods, we generate the input data, i e , the configurations, by Monte-Carlo simulations of the model 
 The joint probability distribution of the $N$ variables $\bm{\phi}\equiv\{\phi_1,\dots,\phi_N\}$, follows the Gibbs-Boltzmann distribution:
 \begin{equation}\label{eq:p_xy}
 P(\bm{\phi}) = \frac{1}{Z} e^{-\beta \mathcal{H\left(\bm{\phi}\right)}} \quad \mbox{ where } \quad Z = \int \prod_{k=1}^N d\phi_k  e^{-\beta \mathcal{H\left(\bm{\phi}\right)}}  
 \end{equation}
 and where we denote $\beta=\left( 2\Delta^2 \right)^{-1}$ with respect to Eq  (\ref{def:Z}) formalism 
 In order to stick to usual statistical inference notation, in the following we will rescale the couplings by a factor $\beta / 2$: $\beta J_{ij}/2 \rightarrow J_{ij}$  
 The main idea of the PLM is to work with the conditional probability distribution of one variable $\phi_i$ given all other variables, 
 $\bm{\phi}_{\backslash i}$:
 
  \begin{eqnarray}
	\nonumber
   P(\phi_i | \bm{\phi}_{\backslash i}) &=& \frac{1}{Z_i} \exp \left \{ {H_i^x (\bm{\phi}_{\backslash i})
  	\cos \phi_i + H_i^y (\bm{\phi}_{\backslash i}) \sin \phi_i } \right \}
	\\
 \label{eq:marginal_xy}
	&=&\frac{e^{H_i(\bm{\phi}_{\backslash i}) \cos{\left(\phi_i-\alpha_i(\bm{\phi}_{\backslash i})\right)}}}{2 \pi I_0(H_i)}
  \end{eqnarray}
  where $H_i^x$ and $H_i^y$ are defined as
   \begin{eqnarray}
   H_i^x (\bm{\phi}_{\backslash i})  &=& \sum_{j (\neq i)} J^R_{ij} \cos \phi_j  - \sum_{j  (\neq i) } J_{ij}^{I} \sin \phi_j \phantom{+  h^R_i} \label{eq:26} \\
   H_i^y  (\bm{\phi}_{\backslash i})  &=&  \sum_{j  (\neq i)} J^R_{ij}  \sin \phi_j  + \sum_{j  (\neq i) } J_{ij}^{I}   \cos \phi_j \phantom{ + h_i^{I} }\label{eq:27}
   \end{eqnarray}
and $H_i= \sqrt{(H_i^x)^2 + (H_i^y)^2}$, $\alpha_i = \arctan H_i^y/H_i^x$   and we introduced the modified Bessel function of the first kind:
  \begin{equation}
  \nonumber
   I_k(x) = \frac{1}{2 \pi}\int_{0}^{2 \pi} d \phi e^{x \cos{ \phi}}\cos{k \phi}
  \end{equation}
  
   Given $M$ observation samples $\bm{\phi}^{(\mu)}=\{\phi^\mu_1,\ldots,\phi^\mu_N\}$, $\mu = 1,\dots, M$, the
   pseudo-loglikelihood for the variable $i$ is given by the logarithm of Eq  (\ref{eq:marginal_xy}),
   \begin{eqnarray}
   \label{eq:L_i}
   L_i &=& \frac{1}{M}  \sum_{\mu = 1}^M  \ln P(\phi_i^{(\mu)}|\bm{\phi}^{(\mu)}_{\backslash i})
   \\
   \nonumber
   & =&  \frac{1}{M}  \sum_{\mu = 1}^M  \left[ H_i^{(\mu)} \cos( \phi_i^{(\mu)} - \alpha_i^{(\mu)}) - \ln  2 \pi I_0\left(H_i^{(\mu)}\right)\right] \,  
    \end{eqnarray}
The underlying idea of PLM is that an approximation of the true parameters of the model is obtained for values that maximize the functions $L_i$ 
The specific maximization scheme differentiates the different techniques 


   
   
   \subsection{PLM with $l_2$ regularization}
   Especially for the case of sparse graphs, it is useful to add a regularizer, which prevents the maximization routine to move towards high values of 
   $J_{ij}$ and $h_i$ without converging  We will adopt an $l_2$ regularization so that the Pseudolikelihood function (PLF) at site $i$  reads:
   \begin{equation}\label{eq:plf_i}
  {\cal L}_i = L_i
  - \lambda \sum_{i \neq j} \left(J_{ij}^R\right)^2 - \lambda \sum_{i \neq j} \left(J_{ij}^I\right)^2  
   \end{equation}
   with $\lambda>0$ 
   Note that the values of $\lambda$ have to be chosen arbitrarily, but not too large, in order not to overcome $L_i$ 
   The standard implementation of the PLM consists in maximizing each ${\cal L}_i$, for $i=1\dots N$, separately  The expected values of the couplings are then:
   \begin{equation}
   \{ J_{i j}^*\}_{j\in \partial i} :=  \mbox{arg max}_{ \{ J_{ij} \}}
    \left[{\cal L}_i\right]
   \end{equation}
   In this way, we obtain two estimates for the coupling $J_{ij}$, one from maximization of ${\cal L}_i$, $J_{ij}^{(i)}$, and another one from ${\cal L}_j$, say $J_{ij}^{(j)}$ 
    Since the original Hamiltonian of the $XY$ model is Hermitian, we know that the real part of the couplings is symmetric while the imaginary part is skew-symmetric  
   
     The final estimate for $J_{ij}$ can then be obtained  averaging the two results:
  
  
  
   \begin{equation}\label{eq:symm}
   J_{ij}^{\rm inferred} = \frac{J_{ij}^{(i)} + \bar{J}_{ij}^{(j)}}{2} 
   \end{equation}
   where with $\bar{J}$ we indicate the complex conjugate 
   It is worth noting that the pseudolikelihood $L_i$, Eq  \eqref{eq:L_i}, is characterized by the
   following properties: (i) the normalization term of Eq \eqref{eq:marginal_xy} can be
   computed analytically at odd with the {\em full} likelihood case that
   in general require a computational time which scales exponentially
   with the size of the systems; (ii) the $\ell_2$-regularized pseudolikelihood
   defined in Eq \eqref{eq:plf_i} is strictly concave (i e  it has a single
   maximizer)\cite{Ravikumar10}; (iii) it is consistent, i e  if $M$ samples are
   generated by a model $P(\phi | J*)$ the maximizer tends to $J*$
   for $M\rightarrow\infty$\cite{besag1975}  Note also that (iii) guarantees that  
   $|J^{(i)}_{ij}-J^{(j)}_{ij}| \rightarrow 0$ for $M\rightarrow \infty$ 
   In Secs  \ref{sec:res_reg}, \ref{sec:res_dec} 
   we report the results obtained and we analyze the performances of the PLM having taken the configurations from Monte-Carlo simulations of models whose details are known 
   

   
   \subsection{PLM with decimation}
   Even though the PLM with $l_2$-regularization allows to dwell the inference towards the low temperature region and in the low sampling case with better performances that mean-field methods, in some situations some couplings are overestimated and not at all symmetric  Moreover, in the technique there is the bias of the $l_2$ regularizer 
   Trying to overcome these problems, Decelle and Ricci-Tersenghi introduced a new method \cite{Decelle14}, known as PLM + decimation: the algorithm maximizes the sum of the $L_i$,
   \begin{eqnarray}
    {\cal L}\equiv \frac{1}{N}\sum_{i=1}^N \mbox{L}_i
    \end{eqnarray}  
    and, then, it recursively set to zero couplings which are estimated very small  We expect that as long as we are setting to zero couplings that are unnecessary to fit the data, there should be not much changing on ${\cal L}$  Keeping on with decimation, a point is reached where ${\cal L}$ decreases abruptly indicating  that relevant couplings are being decimated and under-fitting is taking place 
   Let us define  by $x$  the fraction of non-decimated couplings  To have a quantitative measure for the halt criterion of the decimation process, a tilted ${\cal L}$ is defined as,
   \begin{eqnarray}
  \mathcal{L}_t &\equiv& \mathcal{L}  - x \mathcal{L}_{\textup{max}} - (1-x) \mathcal{L}_{\textup{min}} \label{$t$PLF} 
   \end{eqnarray}
   where 
   \begin{itemize}
   \item $\mathcal{L}_{\textup{min}}$ is the pseudolikelyhood of a model with independent variables  In the XY case: $\mathcal{L}_{\textup{min}}=-\ln{2 \pi}$ 
   \item
   $\mathcal{L}_{\textup{max}}$ is the pseudolikelyhood in the fully-connected model and it is maximized over all the $N(N-1)/2$ possible couplings  
   \end{itemize}
   At the first step, when $x=1$, $\mathcal{L}$ takes value $\mathcal{L}_{\rm max}$ and  $\mathcal{L}_t=0$  On the last step, for an empty graph, i e , $x=0$, $\mathcal{L}$ takes the value $\mathcal{L}_{\rm min}$ and, hence, again $\mathcal{L}_t =0$  
   In the intermediate steps, during the decimation procedure, as $x$ is decreasing from $1$ to $0$, one observes firstly that $\mathcal{L}_t$ increases linearly and, then, it displays an abrupt decrease indicating that from this point on relevant couplings are being decimated\cite{Decelle14}  In Fig  \ref{Jor1-$t$PLF} we give an instance of this behavior for the 2D short-range XY model with ordered couplings  We notice that the maximum point of $\mathcal{L}_t$ coincides with the minimum point of the reconstruction error, the latter defined as 
   \begin{eqnarray}\label{eq:errj}
   \mbox{err}_J \equiv \sqrt{\frac{\sum_{i<j} (J^{\rm inferred}_{ij} -J^{\rm true}_{ij})^2}{N(N-1)/2}} \label{err}
   \end{eqnarray}
  We stress that the ${\cal L}_t$ maximum is obtained ignoring the underlying graph, while the err$_J$ minimum can be evaluated once the true graph has been reconstructed  
   
     \begin{figure}[t!]
   	\centering
   	\includegraphics[width=1\linewidth]{Jor1_dec_tPLF_new eps}
   	\caption{The tilted likelyhood ${\cal L}_t$ curve and the reconstruction error vs the number of decimated couplings  for an ordered, real-valued J on 2D XY model with $N=64$ spins  The peak of ${\cal L}_t$ coincides with the dip of the error }  
   	\label{Jor1-$t$PLF}
   \end{figure} 
       
   
   In the next sections we will show the results obtained on the  $XY$ model analyzing the performances of the two methods and comparing them also with a mean-field method \cite{Tyagi15} 
   
  
    
   \section{Inferred couplings with PLM-$l_2$}
    \label{sec:res_reg}
    \subsection{$XY$ model with real-valued couplings}
    
    In order to obtain the vector of couplings, $J_{ij}^{\rm inferred}$ the function $-\mathcal{L}_i$ is minimized through the vector of derivatives ${\partial \mathcal{L}_i}/\partial J_{ij}$  The process is repeated for all the couplings obtaining then a fully connected adjacency matrix  The results here presented are obtained with $\lambda = 0 01$ 
   For the minimization we have used the MATLAB routine \emph{minFunc\_2012}\cite{min_func}  
      
      \begin{figure}[t!]
    	\centering
    	\includegraphics[width=1\linewidth]{Jor11_2D_l2_JR_soJR_TPJR}
    	\caption{Top panels: instances of single site coupling reconstruction for the case of $N=64$ XY spins on a 2D lattice with ordered $J$ (left column) and bimodal distributed $J$ (right column)  
    	
    	Bottom panels: sorted couplings }
    	\label{PL-Jor1}
    \end{figure}

   
To produce the data by means of numerical Monte Carlo simulations a system with $N=64$  spin variables  is considered on a deterministic 2D lattice with periodic boundary conditions  
Each spin has then connectivity $4$, i e , we expect to infer an adjacency matrix with $N c = 256$ couplings different from zero  
The dynamics of the simulated model is based on the Metropolis algorithm and parallel tempering\cite{earl05} is used to speed up the thermalization of the system 
The thermalization is tested looking at the average energy over logarithmic time windows and
the acquisition of independent configurations
starts only after the system is well thermalized 

   For the values of the couplings we considered two cases: an ordered case, indicated in the figure as  $J$ ordered (e g , left column of Fig  \ref{PL-Jor1}) where the couplings can take values $J_{ij}=0,J$, with $J=1$, 
   and a quenched disordered case, indicated in the figures as  $J$ disordered (e g , right column of Fig  \ref{PL-Jor1})
   where the couplings can take also  negative values, i e , 
    $J_{ij}=0,J,-J$, with a certain probability   The results here presented were obtained with bimodal distributed $J$s: 
   
    $P(J_{ij}=J)=P(J_{ij}=-J)=1/2$   The performances of the PLM have shown not to depend on $P(J)$  
   
    We recall that in Sec  \ref{sec:plm} we used the temperature-rescaled notation, i e , $J_{ij}$ stands for $J_{ij}/T$  
   
    To analyze the performances of the PLM, in Fig  \ref{PL-Jor1} the inferred couplings, $\mathbb{J}^R_{\rm inf}$, are shown on top of the original couplings,  $\mathbb{J}^R_{\rm true}$ 
     The first figure (from top) in the left column shows  the $\mathbb{J}^R_{\rm inf}$ (black) and the $\mathbb{J}^R_{\rm tru}$ (green) for a given spin
     at temperature $T/J=0 7$ and number of samples  $M=1024$  PLM appears to reconstruct the correct couplings, though zero couplings are always given a small inferred non-zero value  
     In the left column of Fig   \ref{PL-Jor1},  both the $\mathbb{J}^R_{\rm{inf}}$ and the $\mathbb{J}^R_{\rm{tru}}$ are sorted in decreasing order and plotted on top of each other  
     We can clearly see that $\mathbb{J}^R_{\rm inf}$ reproduces the expected step function  Even though the jump is smeared, the difference between inferred couplings corresponding to the set of non-zero couplings 
     and to the set of zero couplings can be clearly appreciated 
     Similarly, the plots in the right column of Fig  \ref{PL-Jor1} show the results obtained for the case with  bimodal disordered couplings, for the same working temperature and number of samples  
     In particular, note that the algorithm infers half positive and half negative couplings, as expected 
     
     
\begin{figure}
\centering
\includegraphics[width=1\linewidth]{Jor11_2D_l2_errJ_varT_varM}
\caption{Reconstruction error $\mbox{err}_J$, cf  Eq  (\ref{eq:errj}), plotted as a function of temperature (left) for three values of the number of samples $M$ and  as a function $M$ (right) for three values of temperature in the ordered system, i e , $J_{ij}=0,1$  
The system size is $N=64$ }
\label{PL-err-Jor1}
\end{figure}

In order to analyze the effects of the number of samples and of the temperature regimes, we plot in Fig  \ref{PL-err-Jor1} the reconstruction error, Eq  (\ref{err}), as a function of temperature for three different sample sizes $M=64,128$ and $512$  
The error is seen to sharply rise al low temperature, incidentally, in the ordered case, for  $T<T_c \sim 0 893$, which is the Kosterlitz-Thouless transition temperature of the 2XY model\cite{Olsson92}  
 However, we can see that if only $M=64$ samples are considered, $\mbox{err}_J$ remains high independently on the working temperature  
 In the right plot of Fig  \ref{PL-err-Jor1},  $\mbox{err}_J$ is plotted as a function of $M$ for three different working temperatures $T/J=0 4,0 7$ and $1 3$  As we expect, 
 $\mbox{err}_J$  decreases as $M$ increases  This effect was observed also with mean-field inference techniques on the same model\cite{Tyagi15} 

To better understand the performances of the algorithms, in Fig  \ref{PL-varTP-Jor1} we show several True Positive (TP) curves obtained for various values of $M$ at three different temperatures $T$  As $M$ is large and/or temperature is not too small,  we are able to reconstruct correctly all the couplings present in the system (see bottom plots) 
The True Positive curve displays how many times the inference method finds a true link of the original network  as a function of the index of the vector of sorted absolute value of reconstructed couplings $J_{ij}^{\rm inf}$  
The index $n_{(ij)}$ represents the related spin couples $(ij)$  The TP curve is obtained as follows: 
first  the values $|J^{\rm inf}_{ij}|$ are sorted in descending order and  the spin pairs $(ij)$ are ordered according to the sorting position of $|J^{\rm inf}_{ij}|$  Then,
     	a cycle over the ordered set of pairs $(ij)$, indexed by $n_{(ij)}$, is performed, comparing with the original network coupling $J^{\rm true}_{ij}$ and verifying whether it is zero or not  The true positive curve is computed as
\begin{equation}
\mbox{TP}[n_{(ij)}]= \frac{\mbox{TP}\left[n_{(ij)}-1\right] (n_{ij}-1)+ 1 -\delta_{J^{\rm true}_{ij},0}}{n_{(ij)}}
\end{equation}
As far as $J^{\rm true}_{ij} \neq 0$, TP$=1$  As soon as the true coupling of a given $(ij)$ couple in the sorted list is zero, the TP curve departs from one  
In our case, where the connectivity per spin of the original system is  $c=4$ and there are $N=64$ spins, we know that we will have  $256$ non-zero couplings   
     	If the inverse problem is successful, hence, we expect a steep decrease of the TP curve when  $n_{ij}=256$ is overcome 

In Fig  \ref{PL-varTP-Jor1}
it is  shown that,  almost independently of $T/J$, the TP score improves as $M$ increases  Results are plotted for three different temperatures, $T=0 4,1$ and $2 2$, with increasing number of samples $M = 64, 128,512$ and $1024$ (clockwise)  
We can clearly appreciate the improvement in temperature if the size of the data-set is not very large: for small $M$, $T=0 4$ performs better  
When $M$ is high enough (e g , $M=1024$), instead, the TP curves do not appear to be strongly influenced by the temperature 

\begin{figure}[t!]
	\centering
	\includegraphics[width=1\linewidth]{Jor11_2D_l2_TPJR_varT_varM}
	\caption{TP  curves  for 2D short-range ordered $XY$ model with $N=64$ spins at three different values of $T/J$ with increasing - clockwise from top - $M$ }
	\label{PL-varTP-Jor1} 
\end{figure} 

\subsection{$XY$ model with complex-valued couplings}
For the complex $XY$ we have to contemporary  infer $2$ apart coupling matrices,  $J^R_{i j}$ and $J^I_{i j}$   As before, a system of $N=64$ spins is considered on a 2D lattice 
For the couplings we have considered both ordered and bimodal disordered cases 
In Fig  \ref{PL-Jor3}, a single row of the matrix $J$ (top) and the whole sorted couplings (bottom) are displayed for the ordered model (same legend as in Fig  \ref{PL-Jor1}) for the real, $J^R$ (left column), and the imaginary part, $J^I$   

\begin{figure}[t!]
	\centering
\includegraphics[width=1\linewidth]{Jor3_l2_JRJI_soJRJI_TPJRJI}
	\caption{Results related to the ordered complex XY model with $N=64$ spins on a  2D lattice  Top: instances of  single site reconstruction for the real, JR (left column), and
		the imaginary, JI (right column), part of $J_{ij}$  Bottom: sorted values of JR (left) and JI (right) }
		
	\label{PL-Jor3}
\end{figure}
 
 
  \section{PLM with Decimation}
 \label{sec:res_dec}
 

\begin{figure}[t!]
   	\centering
   	\includegraphics[width=1\linewidth]{Jor1_dec_tPLF_varT_varM}
    	\caption{Tilted Pseudolikelyhood, ${\cal L}_t$, plotted as a function of decimated couplings  Top: Different ${\cal L}_t$ curves obtained for different values of $M$ plotted on top of each other  Here $T=1 3$  The black line indicates the expected number of decimated couplings, $x^*=(N (N-1) - N c)/2=1888$  As we can see, as $M$ increases, the maximum point of ${\cal L}_t$ approaches $x^*$  Bottom: Different ${\cal L}_t$ curves obtained for different values of T with $M=2048$  We can see that, with this value of $M$, no differences can be appreciated on the maximum points of the different  ${\cal L}_t$ curves }
   	\label{var-$t$PLF}
   \end{figure}

\begin{figure}[t!]
    \centering
    \includegraphics[width=1\linewidth]{Jor1_dec_tPLF_peak_statistics_varM_prob eps}
    	\caption{Number of most likely decimated couplings, estimated by the maximum point of $\mathcal{L}_t$, as a function of the number of samples $M$  We can clearly see that the maximum point of $\mathcal{L}_t$ tends toward $x^*$, which is the right expected number of zero couplings in the system }  
    	\label{PLF_peak_statistics}
    \end{figure}
      
      For the ordered real-valued XY model we show in Fig  \ref{var-$t$PLF}, top panel, the outcome on the tilted pseudolikelyhood, $\mathcal{L}_t$ Eq  \eqref{$t$PLF}, of the progressive decimation: from a fully connected lattice  down to an empty lattice  The figure shows the behaviour of $\mathcal{L}_t$ for three different data sizes $M$  A clear data size dependence of the maximum point of  $\mathcal{L}_t$, signalling the most likely value for decimation, is shown  For small $M$ the most likely number of  couplings is overestimated and for increasing $M$ it tends to the true value, as displayed in Fig  \ref{PLF_peak_statistics}   In the bottom panel of Fig  \ref{var-$t$PLF} we display instead different 
 $\mathcal{L}_t$ curves obtained for three different values of $T$ 
  Even though the values of $\mathcal{L}_t$ decrease with increasing temperature, the value of the most likely number of decimated couplings appears to be quite independent on $T$ with $M=2048$ number of samples 
In Fig  \ref{fig:Lt_complex} we eventually display the tilted pseudolikelyhood for a 2D network with complex valued ordered couplings, where the decimation of the real and imaginary coupling matrices proceeds in parallel, that is, 
when a real coupling is small enough to be decimated its imaginary part is also decimated, and vice versa 
One can see that though the apart errors for the real and imaginary parts are different in absolute values, they display the same dip, to be compared with the maximum point of $\mathcal{L}_t$ 
     
       \begin{figure}[t!]
    	\centering
      	\includegraphics[width=1\linewidth]{Jor3_dec_tPLF_new}
      	\caption{Tilted Pseudolikelyhood, ${\cal L}_t$, plotted with the reconstruction errors for the XY model with $N=64$ spins on a 2D lattice  These results refer to the case of  ordered and complex valued couplings  The full (red) line indicates ${\cal L}_t$  The dashed (green) 
      		and the dotted (blue) lines show the reconstruction errors (Eq  \eqref{eq:errj}) obtained for the real and the imaginary couplings respectively  We can see that both ${\rm err_{JR}}$ and ${\rm err_{JI}}$ have a minimum at $x^*$ }
          	\label{fig:Lt_complex}
    \end{figure}

\begin{figure}[t!]
   	\centering
   	\includegraphics[width=1\linewidth]{Jor1_dec_JR_soJR_TPJR}
   	\caption{XY model on a 2D lattice with $N=64$ sites and real valued couplings  The graphs show the inferred (dashed black lines) and true couplings (full green lines) plotted on top of each other  The left and right columns refer to the
   		 cases of ordered and bimodal disordered couplings, respectively  Top figures: single site reconstruction, i e , one row of the matrix $J$  Bottom figures: couplings are plotted sorted in descending order }  
   	\label{Jor1_dec}
   \end{figure}
   
\begin{figure}[t!]
    	\centering
    	\includegraphics[width=1\linewidth]{Jor3_dec_JRJI_soJRJI_TPJRJI}
    	\caption{XY model on a 2D lattice with $N=64$ sites and ordered complex-valued couplings 
    		The inferred and true couplings are plotted on top of each other  The left and right columns show the real and imaginary parts, respectively, of the couplings  Top figures refer to a single site reconstruction, i e , one row of the matrix $J$  Bottom figures report the couplings sorted in descending order }
    	\label{Jor3_dec}
    \end{figure}
    
       







       \begin{figure}[t!]
     	\centering
     	\includegraphics[width=1\linewidth]{MF_PL_Jor1_2D_TPJR_varT}
     	\caption{True Positive curves obtained with the three techniques: PLM with decimation, (blue) dotted line,  PLM with $l_2$ regularization, (greed) dashed line, and mean-field, (red) full line   These results refer to real valued ordered couplings with $N=64$ spins on a 2D lattice  The temperature is here $T=0 7$ while the four graphs refer to different sample sizes: $M$ increases clockwise }
     	\label{MF_PL_TP}
     \end{figure}
    
    \begin{figure}[t!]
    	\centering
    	\includegraphics[width=1\linewidth]{MF_PL_Jor1_2D_errJ_varT_varM}
    	\caption{Variation of reconstruction error, ${\rm err_J}$, with respect to temperature as obtained with the three different techniques, see Fig  \ref{MF_PL_TP}, for four different sample size:  clockwise from top   $M=512,1024, 2048$ and $4096$ } 
    	\label{MF_PL_err}
    \end{figure}
    
     Once the most likely network has been identified through the decimation procedure, we perform the same analysis displayed in Fig  \ref{Jor1_dec}  for ordered and then quenched disordered real-valued couplings
and in Fig  \ref{Jor3_dec} for  complex-valued ordered couplings   In comparison to the results shown in Sec  \ref{sec:res_reg},
  the PLM with decimation leads to rather cleaner results  In Figs  \ref{MF_PL_err} and \ref{MF_PL_TP} we compare the performances of the PLM with decimation in respect to ones of the PLM with $l_2$-regularization  These two techniques are also analysed in respect to a mean-field technique previously implemented on the same XY systems\cite{Tyagi15} 
  
    For what concerns the network of connecting links, in Fig  \ref{MF_PL_TP} we compare the TP curves obtained with the three techniques  The results refer to the case of ordered and real valued couplings, but similar behaviours were obtained for the other cases analysed  
  The four graphs are related to different sample sizes, with $M$ increasing clockwise  When $M$ is high enough, all techniques reproduce the true network  
  However, for lower values of $M$ the performances of the PLM with $l_2$ regularization and with decimation drastically overcome those ones of the previous mean field technique  
  In particular, for $M=256$ the PLM techniques still reproduce the original network while the mean-field method fails to find more than half of the couplings  
  When $M=128$, the network is clearly reconstructed only through the PLM with decimation while the PLM with $l_2$ regularization underestimates the couplings  
  Furthermore, we notice that the PLM method with decimation is able to clearly infer the network of interaction even when $M=N$ signalling that it could be considered also in the under-sampling regime $M<N$   
 
 
In Fig  \ref{MF_PL_err} we compare the temperature behaviour of the reconstruction error 
In can be observed that for all temperatures and for all sample sizes  the reconstruction error, ${\rm err_J}$, (plotted here in log-scale) obtained with the PLM+decimation is always smaller than 
that one obtained with the other techniques  The temperature behaviour of ${\rm err_J}$ agrees with the one already observed for Ising spins in \cite{Nguyen12b} and for XY spins  in \cite{Tyagi15} with a mean-field approach:  ${\rm err_J}$ displays a minimum around $T\simeq 1$ and then it increases for very lower $T$; however,
 the error obtained with the PLM with decimation is several times smaller  than the error estimated by the other methods 



 
 

     
     \section{Conclusions}
     \label{sec:conc}


Different statistical inference methods have been applied to the inverse problem of the XY model 
After a short review of techniques based on pseudo-likelihood and their formal generalization to the model we have tested their performances against data generated by means of Monte Carlo numerical simulations of known instances
with diluted, sparse, interactions 

The main outcome is that the best performances are obtained by means of the  pseudo-likelihood method combined with decimation  Putting to zero (i e , decimating) very weak bonds, this technique turns out to be very precise for  problems whose real underlying interaction network is sparse, i e , the number of couplings per variable does not scale with number of variables 
The PLM + decimation method is compared to the PLM + regularization method, with $\ell_2$ regularization and to a mean-field-based method  The behavior of the quality of the network reconstruction is analyzed by looking at the overall sorted couplings and at the single site couplings, comparing them with the real network, and at the true positive curves in all three approaches  In the PLM +decimation method, moreover, the identification of the number of decimated bonds at which the tilted pseudo-likelihood is maximum allows for a precise estimate of the total number of bonds  Concerning this technique, it is also shown that the network with the most likely number of bonds is also the one of least reconstruction error, where not only the prediction of the presence of a bond is estimated but also its value 

The behavior of the inference quality in temperature and in the size of data samples is also investigated, basically confirming the low $T$ behavior hinted by Nguyen and Berg \cite{Nguyen12b} for the Ising model  In temperature, in particular, the reconstruction error curve displays a minimum at a low temperature, close to the critical point in those cases in which a critical behavior occurs, and a sharp increase as temperature goes to zero  The decimation method, once again, appears to enhance this minimum of the reconstruction error of almost an order of magnitude with respect to other methods 
 
The techniques displayed and the results obtained in this work can be of use in any of the many systems whose theoretical representation is given by Eq  \eqref{eq:HXY} or Eq  \eqref{eq:h_im}, some of which are recalled in Sec  \ref{sec:model}  In particular, a possible application can be the field of light waves propagation through random media and the corresponding problem of the  reconstruction of an object seen through an opaque medium or a disordered optical fiber \cite{Vellekoop07,Vellekoop08a,Vellekoop08b, Popoff10a,Akbulut11,Popoff11,Yilmaz13,Riboli14} 

 	



ground_truth: Four.

prompt_front: Now, answer the following question based on the text below, only give me the answer and do not output any other words.

Question: What are some fields in which the inverse problem is encountered?

Read the following text and answer briefly.

Context 4: \section{Introduction}

Spectral line surveys have revealed that high-mass star-forming
regions are rich reservoirs of molecules from simple diatomic species
to complex and larger molecules (e g ,
\citealt{schilke1997b,hatchell1998b,comito2005,bisschop2007}) 
However, there have been rarely studies undertaken to investigate the
chemical evolution during massive star formation from the earliest
evolutionary stages, i e , from High-Mass Starless Cores (HMSCs) and
High-Mass Cores with embedded low- to intermediate-mass protostars
destined to become massive stars, via High-Mass Protostellar Objects
(HMPOs) to the final stars that are able to produce Ultracompact H{\sc
  ii} regions (UCH{\sc ii}s, see \citealt{beuther2006b} for a recent
description of the evolutionary sequence)  The first two evolutionary
stages are found within so-called Infrared Dark Clouds (IRDCs)   While
for low-mass stars the chemical evolution from early molecular
freeze-out to more evolved protostellar cores is well studied (e g ,
\citealt{bergin1997,dutrey1997,pavlyuchenkov2006,joergensen2007}),
it is far from clear whether similar evolutionary patterns are present
during massive star formation 

To better understand the chemical evolution of high-mass star-forming
regions we initiated a program to investigate the chemical properties
from IRDCs to UCH{\sc ii}s from an observational and theoretical
perspective  We start with single-dish line surveys toward a large
sample obtaining their basic characteristics, and then perform
detailed studies of selected sources using interferometers on smaller
scales  These observations are accompanied by theoretical modeling of
the chemical processes   Long-term goals are the chemical
characterization of the evolutionary sequence in massive star
formation, the development of chemical clocks, and the identification
of molecules as astrophysical tools to study the physical processes
during different evolutionary stages  Here, we present an initial
study of the reactive radical ethynyl (C$_2$H) combining single-dish
and interferometer observations with chemical modeling   Although
C$_2$H was previously observed in low-mass cores and Photon Dominated
Regions (e g , \citealt{millar1984,jansen1995}), so far it was not
systematically investigated in the framework of high-mass star
formation 

\section{Observations}
\label{obs}

The 21 massive star-forming regions were observed with the Atacama
Pathfinder Experiment (APEX) in the 875\,$\mu$m window in fall 2006 
We observed 1\,GHz from 338 to 339\,GHz and 1\,GHz in the image
sideband from 349 to 350\,GHz   The spectral resolution was
0 1\,km\,s$^{-1}$, but we smoothed the data to
$\sim$0 9\,km\,s$^{-1}$  The average system temperatures were around
200\,K, each source had on-source integration times between 5 and 16
min  The data were converted to main-beam temperatures with forward
and beam efficiencies of 0 97 and 0 73, respectively
\citep{belloche2006}  The average $1\sigma$ rms was 0 4\,K   The main
spectral features of interest are the C$_2$H lines around 349 4\,GHz
with upper level excitation energies $E_u/k$ of 42\,K (line blends of
C$_2$H$(4_{5,5}-3_{4,4})$ \& C$_2$H$(4_{5,4}-3_{4,3})$ at
349 338\,GHz, and C$_2$H$(4_{4,4}-3_{3,3})$ \&
C$_2$H$(4_{4,3}-3_{3,2})$ at 349 399\,GHz)  The beam size was $\sim
18''$ 

The original Submillimeter Array (SMA) C$_2$H data toward the
HMPO\,18089-1732 were first presented in \citet{beuther2005c}  There
we used the compact and extended configurations resulting in good
images for all spectral lines except of C$_2$H  For this project, we
re-worked on these data only using the compact configuration  Because
the C$_2$H emission is distributed on larger scales (see
\S\ref{results}), we were now able to derive a C$_2$H image  The
integration range was from 32 to 35\,km\,s$^{-1}$, and the achieved
$1\sigma$ rms of the C$_2$H image was 450\,mJy\,beam$^{-1}$   For more
details on these observations see \citet{beuther2005c} 

\section{Results}
\label{results}

The sources were selected to cover all evolutionary stages from IRDCs
via HMPOs to UCH{\sc ii}s  We derived our target list from the samples
of \citet{klein2005,fontani2005,hill2005,beltran2006}   Table
\ref{sample} lists the observed sources, their coordinates, distances,
luminosities and a first order classification into the evolutionary
sub-groups IRDCs, HMPOs and UCH{\sc ii}s based on the previously
available data  Although this classification is only based on a
limited set of data, here we are just interested in general
evolutionary trends  Hence, the division into the three main classes
is sufficient 

Figure \ref{spectra} presents sample spectra toward one source of each
evolutionary group  While we see several CH$_3$OH lines as well as
SO$_2$ and H$_2$CS toward some of the HMPOs and UCH{\sc ii}s but not
toward the IRDCs, the surprising result of this comparison is the
presence of the C$_2$H lines around 349 4\,GHz toward all source types
from young IRDCs via the HMPOs to evolved UCH{\sc ii}s   Table
\ref{sample} lists the peak brightness temperatures, the integrated
intensities and the FWHM line-widths of the C$_2$H line blend at
349 399\,GHz  The separation of the two lines of 1 375\,MHz already
corresponds to a line-width of 1 2\,km\,s$^{-1}$  We have three C$_2$H
non-detections (2 IRDCs and 1 HMPO), however, with no clear trend with
respect to the distances or the luminosities (the latter comparison is
only possible for the HMPOs)  While IRDCs are on average colder than
more evolved sources, and have lower brightness temperatures, the
non-detections are more probable due to the relatively low sensitivity
of the short observations (\S\ref{obs})  Hence, the data indicate
that the C$_2$H lines are detected independent of the evolutionary
stage of the sources in contrast to the situation with other
molecules   When comparing the line-widths between the different
sub-groups, one finds only a marginal difference between the IRDCs and
the HMPOs (the average $\Delta v$ of the two groups are 2 8 and
3 1\,km\,s$^{-1}$)   However, the UCH{\sc ii}s exhibit significantly
broader line-widths with an average value of 5 5\,km\,s$^{-1}$ 

Intrigued by this finding, we wanted to understand the C$_2$H spatial
structure during the different evolutionary stages   Therefore, we
went back to a dataset obtained with the Submillimeter Array toward
the hypercompact H{\sc ii} region IRAS\,18089-1732 with a much higher
spatial resolution of $\sim 1''$ \citep{beuther2005c}   Albeit this
hypercompact H{\sc ii} region belongs to the class of HMPOs, it is
already in a relatively evolved stage and has formed a hot core with a
rich molecular spectrum   \citet{beuther2005c} showed the spectral
detection of the C$_2$H lines toward this source, but they did not
present any spatially resolved images  To recover large-scale
structure, we restricted the data to those from the compact SMA
configuration (\S\ref{obs})  With this refinement, we were able to
produce a spatially resolved C$_2$H map of the line blend at
349



ground_truth: Physics, biology, social sciences, finance, and neuroscience are some fields in which the inverse problem is encountered.

prompt_front: Now, answer the following question based on the text below, only give me the answer and do not output any other words.

Question: What is the advantage of decorrelating the data before running the PLS algorithm?

Read the following text and answer briefly.

Context 3: Paper Info

Title: Efficient nonparametric estimation of Toeplitz covariance matrices
Publish Date: March 20, 2023
Author List: Karolina Klockmann (from Department of Statistics and Operations Research, Universität Wien), Tatyana Krivobokova (from Department of Statistics and Operations Research, Universität Wien)

Figure

Figure 1: Spectral density functions (first row) and autocovariance functions (second row) for examples 1, 2, 3 
Figure 2: Distance between the first atom and the first center of mass of aquaporin (left) and the opening diameter y t over time t (right) 
black line in the left plot) confirms that the covariance matrix estimated with our VST-DCT method almost completely decorrelates the channel diameter Y on the training data set Next, we estimated the regression coefficients β with the usual PLS algorithm, ignoring the dependence in the data Finally, we estimated β with PLS that takes into account dependence using our covariance estimator Σ Based on these regression coefficient estimators, the prediction on the test set was calculated The plot on the right side of Figure 2 shows the Pearson correlation between the true channel diameter on the test set and the prediction on the same test set based on raw (grey) and decorrelated data (black) 
Figure 3: On the left, the auto-correlation function of Y (grey) and of Σ−1/2 Y (black), where Σ is estimated with the VST-DCT method; On the right, correlation between the true values on the test data set and prediction based on partial least squares (in grey) and corrected partial least squares (black) 
Uniform distributionThe observations follow a uniform distribution with covariance matrices Σ 1 , Σ 2 , Σ 3 of examples 1, 2, 3, i e , Y i = Σ 1/2 j X i , j = 1, 3, with X 1 ,    the parameter innov of the R function arima sim is used to pass the innovations X 1 ,    , X n i i d Table4, 5 and 6 show respectively the results for (A) p = 5000, n = 1, (B) p = 1000, n = 50 and (C) p = 5000, n = 10 
(A) p = 5000, n = 1: Errors of the Toeplitz covariance matrix and the spectral density estimators with respect to the spectral and L 2 norm, respectively, as well as the average computation time of the covariance estimators in seconds for one Monte Carlo sample (last column) 
(C) p = 5000, n = 10: Errors of the Toeplitz covariance matrix and the spectral density estimators with respect to the spectral and L 2 norm, respectively, as well as the average computation time of the covariance estimators in seconds for one Monte Carlo sample (last column) 
(A) p = 5000, n = 1: Errors of the Toeplitz covariance matrix and the spectral density estimators with respect to the spectral norm and the L 2 norm, respectively Average computation time of the covariance estimators in seconds for one Monte Carlo sample (last column) 
(B) p = 1000, n = 50: Errors of the Toeplitz covariance matrix and the spectral density estimators with respect to the spectral norm and the L 2 norm, respectively Average computation time of the covariance estimators in seconds for one Monte Carlo sample (last column) 
(C) p = 5000, n = 10: Errors of the Toeplitz covariance matrix and the spectral density estimators with respect to the spectral norm and the L 2 norm, respectively Average computation time of the covariance estimators in seconds for one Monte Carlo sample (last column) 

abstract

A new nonparametric estimator for Toeplitz covariance matrices is proposed  This estimator is based on a data transformation that translates the problem of Toeplitz covariance matrix estimation to the problem of mean estimation in an approximate Gaussian regression  The resulting Toeplitz covariance matrix estimator is positive definite by construction, fully data-driven and computationally very fast 
Moreover, this estimator is shown to be minimax optimal under the spectral norm for a large class of Toeplitz matrices  These results are readily extended to estimation of inverses of Toeplitz covariance matrices  Also, an alternative version of the Whittle likelihood for the spectral density based on the Discrete Cosine Transform (DCT) is proposed 
The method is implemented in the R package vstdct that accompanies the paper 

Introduction

Estimation of covariance and precision matrices is a fundamental problem in statistical data analysis with countless applications in the natural and social sciences  Covariance matrices with a Toeplitz structure arise in the study of stationary stochastic n = 1, to the best of our knowledge, there is no fully data-driven approach for selecting the banding/tapering/thresholding parameter 
suggested first to split the time series into non-overlapping subseries and then apply the cross-validation criterion of   However, it turns out that the right choice of the subseries length is crucial for this approach, but there is no data-based method available for this  In this work, an alternative way to estimate a Toeplitz covariance matrix and its inverse is chosen 
Our approach exploits the one-to-one correspondence between Toeplitz covariance matrices and their spectral densities  First, the given data are transformed into approximate Gaussian random variables whose mean equals to the logarithm of the spectral density  Then, the log-spectral density is estimated by a periodic smoothing spline with a data-driven smoothing parameter 
Finally, the resulting spectral density estimator is transformed into an estimator for Σ or its inverse  It is shown that this procedure leads to an estimator that is fully data-driven, automatically positive definite and achieves the minimax optimal convergence rate under the spectral norm over a large class of Toeplitz covariance matrices 
In particular, this class includes Toeplitz covariance matrices that correspond to long-memory processes with bounded spectral densities  Moreover, the computation is very efficient, does not require iterative or resampling schemes and allows to apply any inference and adaptive estimation procedures developed in the context of nonparametric Gaussian regression 
Estimation of the spectral density from a stationary time series is a research topic with a long history  Earlier nonparametric methods are based on smoothing of the (log-)periodogram, which itself is not a consistent estimator   Another line of nonparametric methods for estimating the spectral density is based on the Whittle likelihood, which is an ap-proximation to the exact likelihood of the time series in the frequency domain 
For example, estimated the spectral density from a penalized Whittle likelihood, while used polynomial splines to estimate the log-spectral density function maximizing the Whittle likelihood  Recently, Bayesian methods for spectral density estimation have been proposed (see , but these may become very computationally intensive in large samples due to posterior sampling 
The minimax optimal convergence rate for nonparametric estimators of Hölder continuous spectral densities from Gaussian stationary time series was obtained by under the L p , 1 ≤ p ≤ ∞, norm  Only few works on spectral density estimation show the optimality of the corresponding estimators  In particular, and derived convergence rates of their estimators for the log-spectral density under the L 2 norm, while neglecting the Whittle likelihood approximation error 
In general, most works on spectral density estimation do not exploit further the close connection to the corresponding Toeplitz covariance matrix estimation  In particular, an upper bound for the L ∞ risk of a spectral density estimator automatically provides an upper bound for the risk of the corresponding Toeplitz covariance matrix estimator under the spectral norm 
This fact is used to establish the minimax optimality of our nonparametric estimator for the Toeplitz covariance matrices  The main contribution of this work is to show that our proposed spectral density estimator is not only numerically very efficient, but also achieves the minimax optimal rate in the L ∞ norm, which in turn ensures the minimax optimality of the corresponding Toeplitz covariance matrix estimator 
The paper is structured as follows  In Section 2, the model is introduced and ap-proximate diagonalization of Toeplitz covariance matrices with the discrete cosine transform is discussed  Moreover, an alternative version of the Whittle's likelihood is proposed  In Section 3, new estimators for the Toeplitz covariance matrix and the precision matrix are derived, while in Section 4 their theoretical properties are presented 
Section 5 contains simulation results, Section 6 presents a real data example, and Section 7 closes the paper with a discussion  The proofs are given in the appendix to the paper 

Set up and diagonalization of Toeplitz matrices

Let Y 1 ,       , Y n i i d  ∼ N p (0 p , Σ), where Σ is a (p × p)-dimensional positive definite covariance matrix with a Toeplitz structure, that is, Σ = {σ |i−j| } p i,j=1 0  The sample size n may tend to infinity or to be a constant  The case n = 1 corresponds to a single observation of a stationary time series, and in this case the data are simply denoted by Y ∼N p (0 p , Σ) 
The dimension p is assumed to grow  The spectral density function f , corresponding to a Toeplitz covariance matrix Σ, is given by so that for f ∈ L 2 (−π, π) the inverse Fourier transform implies Hence, Σ is completely characterized by f , and the non-negativity of the spectral density function implies the positive definiteness of the covariance matrix 
Moreover, the decay of the autocovariance σ k is directly connected to the smoothness of f   Finally, the convergence rate of a Toeplitz covariance estimator and that of the corresponding spectral density estimator are directly related via Σ ≤ f ∞ := sup x∈ |f (x)|, where • denotes the spectral norm (see  
As in , we introduce a class of positive definite Toeplitz covariance matrices with Hölder continuous spectral densities  For β = γ + α > 0, where The optimal convergence rate for estimating Toeplitz covariance matrices over P β (M 0 , M 1 ) depends crucially on β  It is well known that the k-th Fourier coefficient of a function whose γ-th derivative is α-Hölder continuous decays at least with order O(k −β ) (see  
Hence, β determines the decay rate of the autocovariances σ k , which are the Fourier coefficients of the spectral density f , as k → ∞  In particular, this implies that for β ∈ (0, 1], the class P β (M 0 , M 1 ) includes Toeplitz covariance matrices corresponding to long-memory processes with bounded spectral densities, since the sequence of corresponding autocovariances is not summable 
A connection between Toeplitz covariance matrices and their spectral densities is further exploited in the following lemma  Lemma 1  Let Σ ∈ P β (M 0 , M 1 ) and let x j = (j − 1)/(p − 1), j = 1,    , p, then where δ i,j is the Kroneker delta, O(•) terms are uniform over i, j = 1,       , p and divided by √ 2 when i, j ∈ {1, p} is the Discrete Cosine Transform I (DCT-I) matrix 
The proof can be found in Appendix A 1  This result shows that the DCT-I matrix approximately diagonalizes Toeplitz covariance matrices and that the diagonalization error depends to some extent on the smoothness of the corresponding spectral density  In the spectral density literature the discrete Fourier transform (DFT) matrix
, where i is the imaginary unit, is typically employed to approximately diagonalize Toeplitz covariance matrices  Using the fact that introduced an approximation for the likelihood of a single Gaussian stationary time series (case n = 1), the so-called Whittle likelihood (1) The quantity , where F j denotes the j-th column of F , is known as the periodogram at the j-th Fourier frequency 
Note that due to periodogram symmetry, only p/2 data points I 1 ,    , I p/2 are available for estimating the mean f (2πj/p), j = 1,       , p/2 , where x denotes the largest integer strictly smaller than x  The Whittle likelihood has become a popular tool for parameter estimation of stationary time series, e g , for nonparametric and parametric spectral density estimation or for estimation of the Hurst exponent, see e g , ;  
Lemma 1 yields the following alternative version of the Whittle likelihood where W j = (D t j Y ) 2   Note that this likelihood approximation is based on twice as many data points W j as the standard Whittle likelihood  Thus, it allows for a more efficient use of the data Y to estimate the parameter of interest, such as the spectral density or the Hurst parameter 
Equations ( ) or (2) invite for the estimation of f by maximizing the (penalized) likelihood over certain linear spaces (e g , spline spaces), as suggested e g , in or   However, such an approach requires well-designed numerical methods to solve the corresponding optimization problem, since the spectral density in the second term of (1) or ( ) is in the denominator, which does not allow to obtain a closed-form expression for the estimator and often leads to numerical instabilities 
Also, the choice of the smoothing parameter becomes challenging  Therefore, we suggest an alternative approach that allows the spectral density to be estimated as a mean in an approximate Gaussian regression  Such estimators have a closed-form expression, do not require an iterative optimization algorithm and a smoothing parameter can be easily obtained with any conventional criterion 
First Hence, for W j = (D t j Y ) 2 , j = 1,       , p it follows with Lemma 1 that where Γ(a, b) denotes a gamma distribution with a shape parameter a and a scale parameter b  Note that the random variables W 1 ,       , W p are only asymptotically independent  Obviously, E(W j ) = f (πx j ) + O(1), j = 1,      
, p  To estimate f from W 1 ,       , W p , one could use a generalized nonparametric regression framework with a gamma distributed response, see e g , the classical monograph by   However, this approach requires an iterative procedure for estimation, e g , a Newton-Raphson algorithm, with a suitable choice for the smoothing parameter at each iteration step 
Deriving the L ∞ rate for the resulting estimator is also not a trivial task  Instead, we suggest to employ a variance stabilizing transform of that converts the gamma regression into an approximate Gaussian regression  In the next section we present the methodology in more detail for a general setting with n ≥ 1 

Methodology

For Y i ∼ N p (0 p , Σ), i = 1,       , n, it was shown in the previous section that with Lemma 1 the data can be transformed into gamma distributed random variables     , n, j = 1,       , p, where for each fixed i the random variable W i,j has the same distribution as W j given in (3)  Now the approach of Cai et al  ( ) is adapted to the setting n ≥ 1 
First, the transformed data points W i,j are binned, that is, fewer new variables     , T   Note that the number of observations in a bin is m = np/T   In Theorem 1 in Section 4, we show that setting T = p υ for any υ ∈ ((4 − 2 min{β, 1})/3, 1) leads to the minimax optimal rate for the spectral density estimator 
To simplify the notation, m is handled as an integer (otherwise, one can discard several observations in the last bin)  Next, applying the variance stabilizing transform (VST) ∼ where H(y) = {φ(m/2) + log (2y/m)} / √ 2 and φ is the digamma function (see   Now, the scaled and shifted log-spectral density H(f ) can be estimated with a periodic smoothing spline
where h > 0 denotes a smoothing parameter, q ∈ N is the penalty order and S per (2q − 1) a space of periodic splines of degree 2q − 1  The smoothing parameter h can be chosen either with generalized cross-validation (GCV) as derived in or with the restricted maximum likelihood, see   Once an estimator H(f ) is obtained, application of the inverse transform function H −1 (y) = m exp √ 2y − φ (m/2) /2 yields the spectral density estimator f = H −1 H(f )  
Finally, using the inverse Fourier transform leads to the fol- The precision matrix Ω is estimated by the inverse Fourier transform of the reciprocal of the spectral density estimator, i e , Ω = (ω |i−j| ) p i,j=1 with ωk = The estimation procedure for Σ and Ω can be summarised as follows  1  Data Transformation:
where D is the (p × p)-dimensional DCT-I matrix as given in Lemma 1 and D j is its j-th column  2  Binning: Set T = p υ for any υ ∈ ((4 − 2 min{β, 1})/3, 1) and calculate W i,j , k = 1,       , T 

VST:

where k are asymptotically i i d  Gaussian variables  Inverse VST: Estimate the spectral density f with f = H −1 H(f ) , where Note that Σ and Ω are positive definite matrices by construction, since their spectral density functions f and f −1 are non-negative, respectively  Unlike the banding and tapering estimators, the autocovariance estimators σk are controlled by a single smoothing parameter h, which can be estimated fully data-driven with several available automatic methods, which are numerically efficient and well-studied 
In addition, one can also use methods for adaptive mean estimation, see e g , , which in turn leads to adaptive Toeplitz covariance matrix estimation  All inferential procedures developed in the Gaussian regression context can also be adopted accordingly 

Theoretical Properties

In this section, we study the asymptotic properties of the estimators f , Σ and Ω  The results are established under the asymptotic scenario where p → ∞ and p/n → c ∈ (0, ∞], that is, the dimension p grows, while the sample size n either remains fixed or also grows but not faster than p  This corresponds to the asymptotic scenario when the sample covariance matrix is inconsistent 
Let f be the spectral density estimator defined in Section 3, i e , f = m exp{ √ 2 H(f ) − φ(m/2)}/2, where H(f ) is given in (4), m = np/T and φ is the digamma function  Furthermore, let Σ be the Toeplitz covariance matrix estimator and Ω the corresponding precision matrix defined in equations ( ) and (6), respectively 
The following theorem shows that both Σ and Ω attain the minimax optimal rate of convergence over the class and hT → ∞, then with T = p υ for any υ ∈ ((4 − 2 min{β, 1})/3, 1) and q = max{1, γ}, the spectral density estimator f , the corresponding covariance matrix estimator Σ and the precision matrix estimator Ω satisfy sup
For h {log(np)/(np)}   The proof of Theorem 1 can be found in the Appendix A 3 and is the main result of our work  The most important part of this proof is the derivation of the convergence rate for the spectral density estimator f under the L ∞ norm  In the original work, established an L 2 rate for a wavelet nonparametric mean estimator in a gamma regression where the data are assumed to be independent 
In our work, the spectral density estimator f is based on the gamma distributed data W i,1 ,       , W i,p , which are only asymptotically independent  Moreover, the mean of these data is not exactly f (πx 1 ),       , f (πx p ), but is corrupted by the diagonalization error given in Lemma 1  This error adds to the error that arises via binning and VST and that describes the deviation from a Gaussian distribution, as derived in  
Finally, we need to obtain an L ∞ rather than an L 2 rate for our spectral density estimator  Overall, the proof requires different tools than those used in   To get the L ∞ rate for f , we first derive that for the periodic smoothing spline estimator H(f ) of the log-spectral density  To do so, we use a closed-form expression of its effective kernel obtained in , thereby carefully treating various (dependent) errors that describe deviations from a Gaussian nonparametric regression with independent errors and mean f (πx i ) 
Note also that although the periodic smoothing spline estimator is obtained on T binned points, the rate is given in terms of the vector dimension p  Then, using the Cauchy-Schwarz inequality and a mean value argument, this rate is translated into the L ∞ rate for the spectral density estimator f   To obtain the rate for the Toeplitz covariance matrix estimator is enough to note that

Simulation Study

In this section, we compare the performance of the proposed Toeplitz covariance estimator, denoted as VST-DCT, with the tapering estimator of and with the sample covariance matrix  We consider Gaussian vectors Y 1 ,    , (3) such that the corresponding spectral density is Lipschitz continuous but not differentiable: f (x) = 1 44{| sin(x + 0 5π)| 1 7 + 0 45} 
In particular, var(Y i ) = 1 44 in all three examples  Figure shows the spectral densities and the corresponding autocorrelation functions for the three examples  A Monte Carlo simulation with 100 iterations is performed using R (version 4 1 2, seed 42)  For our VST-DCT estimator, we use a cubic periodic spline, i e , q = 2 is set in (4) 
The binning parameters are set to T = 500 bins with m = 10 points for (A) and T = 500 bins with m = 100 points for both (B) and (C)  To select the regularisation parameter for our estimator, we implemented the restricted maximum likelihood (ML) method, generalized cross validation (GCV) and the corresponding oracle versions, i e , as if Σ were known The tapering parameter
where T ap k (Σ ν 2 ) is the tapering estimator of with parameter k  If n = 1, that is, under scenario (A), suggest to split the time series Y into l non-overlapping subseries of length p/l and then proceed as before to select the tuning parameter k  To the best of our knowledge, there is no data-driven method for selecting this parameter l 
Using the true covariance matrix Σ, we selected l = 30 subseries for the example 1 and l = 15 subseries for the exam-ples 2 and 3  The parameter k can then be chosen by cross-validation as above  We employ this approach under scenario (A) instead of an unavailable fully data-driven criterion and name it semi-oracle 
Finally, for all three scenarios (A), (B) and (C), the oracle tapering parameter is computed using grid search for each Monte Carlo sample as kor = arg min k=2,3,   ,p/2 T ap k ( Σ) − Σ , where Σ is the sample covariance matrix  To speed up the computation, one can replace the spectral norm by the 1 norm, as suggested by  
In Tables , the errors of the Toeplitz covariance estimators with respect to the spectral norm and the computation time for one Monte Carlo iteration are given for scenarios (A), (B) and (C), respectively  To illustrate the goodness-of-fit of the spectral density, the L 2 norm f − f 2 is also computed 
The results show that the tapering and VST-DCT estimator perform overall similar in terms of the spectral norm risk  This is not surprising as both estimators are proved to be rate-optimal  Moreover, both the tapering and VST-DCT estimators are clearly superior to the inconsistent sample Toeplitz covariance matrix 
A closer look at the numbers shows that the VST-DCT method has better constants, i e , VST-DCT estimators have somewhat smaller errors in the spectral norm than the tapering estimators across all examples, but especially under scenario (C)  The oracle estimators show similar behaviour, but are slightly less variable compared to the data-driven estimators 
In general, both the tapering and VST-DCT estimators perform best for example 1, second best for example 3 and worst for example 2, which traces back to functions complexity  In terms of computational time, both methods are similarly fast for scenarios (A) and (B)  For scenario (C), the tapering method is much slower due to the multiple high-dimensional matrix multiplications in the cross-validation method 
It is expected that for larger p the tapering estimator is much more computationally intensive than the corresponding VST-DCT estimator  (1) polynomial σ ( (  To test how robust our approach is to deviations from the Gaussian assumption, we simulated the data from gamma and uniform distributions and conducted a simulation study for the same scenarios and examples 
The results are very similar to those of the Gaussian distribution, see supplementary materials for the details 

Application to Protein Dynamics

We revisit the data analysis of protein dynamics performed in Krivobokova et al  (2012) and   We consider data generated by the molecular dynamics (MD) simulations for the yeast aquaporin (Aqy1) -the gated water channel of the yeast Pichi pastoris  MD simulations are an established tool for studying biological systems at the atomic level on timescales of nano-to microseconds 
The data are given as Euclidean coordinates of all 783 atoms of Aqy1 observed in a 100 nanosecond time frame, split into 20 000 equidistant observations  Additionally, the diameter of the channel y t at time t is given, measured by the distance between two centers of mass of certain residues of the protein 
The aim of the analysis is to identify the collective motions of the atoms responsible for the channel opening  In order to model the response variable y t , which is a distance, based on the motions of the protein atoms, we chose to represent the protein structure by distances between atoms and certain fixed base points instead of Euclidean coordinates 
That is, we calculated where A t,i ∈ R 3 , i = 1,       , 783 denotes the i-th atom of the protein at time t, B j ∈ R 3 , j = 1, 2, 3, 4, is the j-th base point and d(•, •) is the Euclidean distance  Figure shows the diameter y t and the distance between the first atom and the first center of mass  It can therefore be concluded that a linear model Y = Xβ + holds, where
  This linear model has two specific features which are intrinsic to the problem: first, the observations are not independent over time and second, X t is high-dimensional at each t and only few columns of X are relevant for Y   have shown that the partial least squares (PLS) algorithm performs exceptionally well on this type of data, leading to a small-dimensional and robust representation of proteins, which is able to identify the atomic dynamics relevant for Y  
Singer et al  ( ) studied the convergence rates of the PLS algorithm for dependent observations and showed that decorrelating the data before running the PLS algorithm improves its performance  Since Y is a linear combination of columns of X, it can be assumed that Y and all columns of X have the same correlation structure 
Hence, it is sufficient to estimate Σ = cov(Y ) to decorrelate the data for the PLS algorithm, i e , Σ −1/2 Y = Σ −1/2 Xβ + Σ −1/2 results in a standard linear regression with independent errors  Our goal now is to estimate Σ and compare the performance of the PLS algorithm on original and decorrelated data 
For this purpose, we divided the data set into a training and a test set (each with p = 10 000 observations)  First, we tested whether the data are stationary  The augmented Dickey-Fuller test confirmed stationarity for Y with a p-value< 0 01  The Hurst exponent of Y is 0 85, indicating moderate long-range dependence supported by a rather slow decay of the sample autocovariances (see grey line in the left plot of Figure ) 
Therefore, we set q = 1 for the VST-DCT estimator to match the low smoothness of the corresponding spectral density  Moreover, the smoothing parameter is selected with the restricted maximum likelihood method and T = 550 bins are used  Obviously, the performance of the PLS algorithm on the decorrelated data is significantly better for the small number of components 
In particular, with just one PLS component, the correlation between the true opening diameter on the test set and its prediction that takes into account the dependence in the data is already 0 54, while it is close to zero for PLS that ignores the dependence in the data  showed that the estimator of β based on one PLS component is exactly the ensemble-weighted maximally correlated mode (ewMCM), which is defined as the collective mode of atoms that has the highest probability to achieve a specific alteration of the response Y  
Therefore, an accurate estimator of this quantity is crucial for the interpretation of the results and can only be achieved if the dependence in the data is taken into account  Estimating Σ with a tapered covariance estimator has two practical problems  First, since we only have a single realization of a time series Y (n = 1), there is no datadriven method for selecting the tapering parameter 
Second, the tapering estimator turned out not to be positive definite for the data at hand  To solve the second problem, we truncated the corresponding spectral density estimator ftap to a small positive value, i e , f + tap = max{ ftap , 1/ log(p)} (see   To select the tapering parameter with cross-validation, we experimented with different subseries lengths and found that the tapering estimator is very sensitive to this choice 
For example, estimating the tapered covariance matrix based on subseries of length 8/15/30 yields a correlation of 0 42/0 53/0 34 between the true diameter and the first PLS component, respectively  Altogether, our proposed estimator is fully data-driven, fast even for large sample sizes, automatically positive definite and can handle certain long-memory processes 
In contrast, the tapering estimator is not data-driven and must be manipulated to become positive definite  Our method is implemented in the R package vstdct 

Discussion

In this paper, we proposed a simple, fast, fully data-driven, automatically positive definite and minimax optimal estimator of Toeplitz covariance matrices from a large class that also includes covariance matrices of certain long-memory processes  Our estimator is derived under the assumption that the data are Gaussian 
However, simulations show that the suggested approach yields robust estimators even when the data are not normally distributed  In the context of spectral density estima- , for mixing processes (see Theorem 5 3 of Rosenblatt, 2012), as well as for non-linear processes (see   Since DFT and DCT matrices are closely related, we expect that equation (3) also holds asymptotically for these non-Gaussian time series, but consider a rigorous analysis to be beyond the scope of this paper 
In fact, our numerical experiments have even shown that if the spectral density is estimated from W j = f (πx j ) + j , that is, as if W j were Gaussian instead of gamma distributed, then the resulting spectral density estimator has almost the same L ∞ risk (and hence the corresponding covariance matrix has almost the same spectral norm) 
Of course, such an estimator would lead to a wrong inference about f (πx j ), since the growing variance of W j would be ignored  Since our approach translates Toeplitz covariance matrix estimation into a mean estimation in an approximate Gaussian nonparametric regression, all approaches developed in the context of Gaussian nonparametric regression, such as (locally)
adaptive estimation, as well as the corresponding (simultaneous) inference, can be directly applied  Bayesian tools for adaptive estimation and inference in Gaussian nonparametric regression as proposed in can also be employed 

Appendix

Throughout the appendix, we denote by c, c 1 , C, C 1 ,       etc  generic constants, that are independent of n and p  To simplify the notation, the constants are sometimes skipped and we write for less than or equal to up to constants  We embed the p-dimensional Toeplitz matrix Σ = toep(σ 0 ,       , σ p−1 ) in a (2p − 2)dimensional circulant matrix Σ = toep(σ 0 ,      
, σ p−1 , σ p−2 ,       , σ 1 )  Then, Σ = with the conjugate transpose U * , and Λ is a diagonal matrix with the k-th diagonal value for k = 1,    , p given by Furthermore, Σ = V * ΛV , where V ∈ C (2p−2)×p contains the first p columns of U   In particular, b(j, r) = b(j, 2p−r) and c(j, r) = −c(j, 2p−r) for r = p+1,      
, 2p−2  Together, we have (A 1) Some calculations show that for r = 1,       p Using the Taylor expansion of cot(x) for 0 < |x| < π one obtains for r = 1,       p where the O term does not depend on j and the hidden constant does not depend on r, p  If i = j, equations (A 1) -(A 3) imply where the O terms do not depend on j 
Since the complex exponential function is Lipschitz continuous with constant L = 1, it holds λ r = λ j + L r,j |r − j|p −1 where −1 ≤ L r,j ≤ 1 is a constant depending on r, j  Then, , it is sufficient to consider j = 1,    , p − 1  We begin with first sum  For a shorter notation, we use k := r − 1 and l := j − 1 in the following 
Then, summing the squares of the first term in (A 4) for l = 0,    , p−2 on sums of reciprocal powers  If p is even, then the residual terms are given by where φ and φ (1) denote the digamma function and its derivative  If p is odd, similar remainder terms can be derived  To see that R i (l, p) = O(p −1 ) for i = 1, 2, 3 and uniformly in l we use that asymptotically φ(x)∼ log(x)−1/(2x) and
The mixed term are both of the order p −1   Furthermore, since the harmonic sum diverges at a rate of log(p)  Finally, λ j = f (x j )+O{log(p)p −β } by the uniform approximation properties of the discrete Fourier series for Hölder continuous functions (see   All together, we have shown that (DΣD) j,j = where the O terms are uniform over j = 1,    , p 
Case i = j and |i − j| is even In this case, (DΣD) i,j = a i a j uniformly in i, j  To show that a i a j 2p−2 r=1 λ r c(i, r)c(j, r) = O(p −1 ), we proceed similarly as before  Setting k=r−1, l=j−1, m=i−1 and using that l =m and |l−m| is even, one obtains where for even p the residual terms are given by If p is odd, analogous residual terms can be derived 
Using similar techniques as before, one can show that the two residual terms and the remaining mixed and square terms vanish at a rate of the order O(p −1 ) and uniformly in i, j  Case i = j and |i − j| is odd |r − i| and |r − j| are either odd and even, or even and odd  Without loss of generality, assume that |r − i| is even 
Then, (DΣD) i,j = a i a j 2p−2 r=1 λ r b(i, r)c(j, r)  Since b(i, •) is an even function, c(j, •) is an odd function and λ r = λ 2p−r , it follows (DΣD) i,j = 0  The structure of the proof is as follows  First, we derive the L ∞ rate of the periodic smoothing spline estimator H(f )  Then, using the Cauchy-Schwarz inequality and a mean value argument, the convergence rate of the spectral density estimator f is
∞ the first claim of the theorem follows  Finally, we prove the second statement on the precision matrices  For the sake of clarity, some technical lemmas used in the proof are listed separately in A 4  hT → ∞, then with T = p υ for any υ ∈ ((4 − 2 min{1, β})/3, 1), the estimator H(f ) described in Section 3 with q = max{1, γ} satisfies
Proof : Application of the triangle inequality yields a bias-variance decomposition Set T = 2T − 2 and x k = (k − 1)/ T for k = 1,    , T   Using Lemma 4, we can write where Mirroring and renumerating ζ k , η k , k is similar as for Y * k , k = 1,    , T   Using the above representation, one can write First we reduce the supremum to a maximum over a finite number of points 
If q > 1, then W (•, x k ) is Lipschitz continuous with constant L > 0  In this case, it holds almost surely that sup ) is a piecewise linear function with knots at x j = j/ T   The factor (ζ k + ξ k ) can be considered as stochastic weights that do not affect the piecewise linear property  Thus, the supremum is attained at one of the knots x j = j/ T , j = 1,    , T , and (A 7) is also valid for q = 1 
Again with (a + b) 2 ≤ 2a 2 + 2b 2 we obtain We start with bounding   This requires a bound on 1 • ψ 2 denotes the sub-Gaussian norm  In case of a Gaussian random variable the norm equals to the variance  Thus with Lemma 2 and Lemma 4, we obtain Lemma 1 6 of ) then yields Recall that T = p υ for some fixed υ ∈ ((4 − 2 min{1, β})/3, 1) 
Using the inequality log(x) ≤ x a /a one can find constants x υ , C υ > 0 depending on υ but not on n, p such that log(2 T ) log(p) Next, we derive a bound for the second term The exponential decay property of the kernel K stated in Lemma 2 yields The first term in (A 9) can be bounded again with Lemma 1 6 of  
We use the fact that for not necessarily independent random variables X 1 ,    , X N R and R > 0 are constants  This is a consequence of Lemma 1 of which yields , it follows that N i=1 a i X i has a subGaussain distribution and the subGaussian norm is bounded by 2R( N i=1 a 2 i ) 1/2   See for further details on the subgaussian distribution 
T h   For the second inequality Lemma 2(ii) is used  Applying Lemma 1 6 of then yields To bound the second term in (A 9), we use the moment bounds for ξ k derived in Lemma 4  Then, for all integers > 1 Combining the error bounds (A 10) and (A 11) and choosing R=m −1/2 gives By assumption T = p υ and m = np (1−υ) for some fixed υ ∈ ((4 − 2 min{1, β})/3, 1) 
If is an integer such that ≥ 1/(1 − υ), then where we used log(x) ≤ x a /a with a = 1/(4 )  Consider 1/2 < β ≤ 1 and let 0 < χ < 1 be a constant  Applying log(x) ≤ x a /a twice with a = χ/(2 ) yields For any fixed υ∈((4 − 2 min{1, β})/3, 1) one can find an integer which is independent of n, p such that the right side of (A 12) holds 
Since p/n → c ∈ (0, ∞] and thus n/p = O(1) and p −1 = O(n −1 ), it follows for satisfying (A 12) that In total, choosing an integer Using the representation in Lemma 4 once more gives for each x ∈ [0, 1] The bounds on k in Lemma 4 imply Consider the case that β ≥ 1  In particular, q = γ and f (q) is α-Hölder continuous 
Since f is a periodic function with f (x) ∈ [δ, M 0 ] and H(y) ∝ φ(m/2)+ log (2y/m), it follows that {H(f )} (q) is also α-Hölder continuous  Extending g := H(f ) to the entire real line, we get Expanding g(t) in a Taylor series around x and using that h −1 K h is a kernel of order 2q, see Lemma 2(iii), it follows that for any x ∈ [0, 1]
where ξ x,t is a point between x and t  Using the fact that the kernel K h decays exponentially and that g (q) is α-Hölder continuous on [δ, M 0 ] with some constant L, the logarithm is Lipschitz continuous on a compact interval, it follows g = H(f ) is β-Hölder continuous  Expanding g to the entire line and using Lemma 2(iii) with
In a similar way as before, one obtains Note that T −β =o(h β ) as β > 1/2, T h → ∞ and h → 0 by assumption  Since the derived bounds are uniform for x ∈ [0, 1] it holds Putting the bounds A 13 and A 14 together gives If h > 0 such that h → 0 and hT → ∞, then with T = p υ for any υ ∈ ((4 − 2 min{1, β})/3, 1), the estimator f described in Section 3 with q = max{1, γ} satisfies
Proof : By the mean value theorem, it holds for some function g between H(f ) and To show that the second term on the right hand side of (A 15) is negligible we use the moment generating function of H(f ) ∞   In the next paragraph, we derive the asymptotic order of E[exp{λ H(f ) ∞ }] for n, p → ∞, where λ > 0 may depend on n, p or not 
By the exponential decay property of the kernel K stated in Lemma 2 holds First, H(f ) ∞ is bounded with the maximum over a finite number of points  Calculating the derivative of s : Since δ δx s(x) > 0 almost surely for x = x k , the extrema occur at x k , k = 1,    , T   Thus, for λ > 0 the moment generating function of H(f ) ∞ is bounded by
Let M j = ( T h) −1 T k=1 γ h (x j , x k ), which by Lemma 2 is bounded uniformly in j by some global constant M > 0  By the convexity of the exponential function we obtain √ 2 and by assumption 0 ≤ δ ≤ f ≤ M 0   Using Lemma 3, Q k can be written as a sum of m = np/T independent gamma random variables, i e 
The moment generating function of | log(X)| when X follows a Γ(a, b)-distribution is given by where Γ(a) is the gamma function and γ(a, b) is the lower incomplete gamma function  In particular, To derive the asymptotic order of E[exp{λ H(f ) ∞ }] for n, p → ∞ we first establish the asymptotic order of the ratio Γ(a + t)/Γ(a) for a → ∞ 
We distinguish the two cases where t is independent of a and where t linearly depends on a  Thus, for 0 < t < a and t independent of a, equation (A 17) implies for a → ∞ that Γ(a + t)/Γ(a) = O(a t )  Similarly, it can be seen that Γ(a − t)/Γ(a) = O(a −t )  If 0 < t < a and t linearly depends on a, i e  t = ca for some constant c ∈ (0, 1), then we get Γ(a ± t)/Γ(a) = O(a ±t exp{a}) for a → ∞ 
Hence, for a fixed λ not depending on n, p and such that 0 < λ < m/( √ 2M j ) we get for sufficiently large n, p If λ = cm such that 0 < λ < m/( √ 2M j ), then for sufficiently large n, p b∈{cδ/m,cM 0 /m} (bm/2) for some constant L > 1  Set K = min j=1,   , T 1/( √ 2M j ) which is a constant independent of n, p  Altogether, we showed that for 0 < λ < Km and n, p → ∞
Bounding the right hand side of (A 15) for some constants c 0 , c 1 > 0 and n, p → ∞ Since g lies between H(f ) and H(f ), and f almost surely pointwise  Thus, for C > f ∞ = M 0 it holds where c 1 := H(C − M 0 )  Applying Markov inequality for t = cm with c ∈ (0, K) and C = 2L 4/c + M 0 where c, K, L are the constants in gives
Together with Proposition 1 follows Using the fact that the spectral norm of a Toeplitz matrix is upper bounded by the sup norm of its spectral density we get sup According to the mean value theorem, for a function g between H(f ) and H(f ), it holds that some constant c 1 > 0 not depending on n, p  Chosing the same constant C as in section A 3 2 it follows
Noting that 1/f ∞ ≤ 1/δ and 2/m exp {φ(m/2)} ∈ [0 25, 1] for m ≥ 1, (A 18) implies for some constants c 2 , c 3 > 0 and n, p → ∞ Since the derived bounds hold for each Σ(f ) ∈ F β , we get all together sup This section states some technical lemmata needed for the proof of Theorem 1  The proofs can be found in the supplementary material 
The first lemma lists some properties of the kernel K h and its extension K h on the real line  The proof is based on   Lemma 2  Let h > 0 be the bandwith parameter depending on N   (i) There are constants 0 < C < ∞ and 0 < γ < 1 such that for all for x, t ∈ [0, 1] Lemma 3 states that the sum of the correlated gamma random variables in each bin can be rewritten as a sum of independent gamma random variables 
for i = 1,    , n and j = (k − 1)m + 1,    , km, and x j = (j − 1)/(2p − 2)  Finally, Lemma 4 gives explicit bounds for the stochastic and deterministic errors of the variance stabilizing transform  Thus, it quantifies the difference to an exact Gaussian regression setting  This result is a generalization of Theorem 1 of Cai et al 
(2010) adapted to our setting with n ≥ 1 observations and correlated observations  √ 2 can be written as where for the proof of the first statement  Furthermore, for x, t ∈ [0, 1] holds In particular, for some constants C 1 , C 2 > 0 depending on γ ∈ (0, 1) but not on h and x, it holds h (iii) See Lemma 15 of with p = 2q − 1 
It is sufficient to show the statement for n = 1 by independence of the Y i   Then, the number of points per bin is m = p/T   For simplicity, the index i is skipped in the following  First, we write Q k as a matrix-vector product and refactor it so that it corresponds to a sum of independent scaled χ 2 random variables 
In the second step, we calculate the scaling factors  Let E (km) be a diagonal matrix with ones on the (k − 1)m + 1,    , km-th entries and otherwise zero diagonal elements  Then, By Theorem 1 of for the gamma distribution it follows where Wi,j iid  ∼ Γ(1/2, 2 f (x * k )) and such that Cov( Wi,j , Wi,h ) = Cov(W i,j , W i,h ) for j = (k − 1)p/T + 1,    , kp/T and h ∈ {1,    , p} \ {(k − 1)p/T + 1,    , kp/T } 
Let θ be the maximum difference of the observations' means in each bin  Then, θ = max are defined via quantile coupling, it holds Z k = Φ −1 {F Q( Qk )} (see   Furthermore, define the uniform random variables Let ρ = Cov(Z k , Z l )  Then, the identity implies F Z,Z (x, y) − Φ(x)Φ(y) ≥ 0 for all x, y ∈ R ⇐⇒ ρ ≥ 0, (see  
Since Cov( Qk , Ql ) ≥ 0 and the ratio of two densities is non-negative, x = − 2/m, it follows that f Q(x) is monotone decreasing for x ≥ − 2/m  Furthermore, F Q(− m/2) ≤ 0 5 for all m ∈ N as f Q(x) is right-skewed  In particular, − m/2 ≤ F −1 Q (1/2) for all m ∈ N  Finally, since f Q(− 2/m) → φ(0) for m → ∞ there is a constant c > 0 not depending on m such that
The simulation study in Section 5 is performed in the same way, but with the uniform and the gamma distribution instead of the Gaussian distribution 



ground_truth: Decorrelating the data before running the PLS algorithm improves the performance of the algorithm.