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 1: 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 
Context 3: \section{introduction}
In the past few years, the synthesis of ferromagnetic semiconductors has become a major challenge for spintronics  Actually, growing a magnetic and semiconducting material could lead to promising advances like spin injection into non magnetic semiconductors, or electrical manipulation of carrier induced magnetism in magnetic semiconductors \cite{ohno00,Bouk02}  Up to now, major efforts have focused on diluted magnetic semiconductors (DMS) in which the host semiconducting matrix is randomly substituted by transition metal (TM) ions such as Mn, Cr, Ni, Fe or Co \cite{Diet02}  However Curie temperatures ($T_{C}$) in DMS remain rather low and TM concentrations must be drastically raised in order to increase $T_{C}$ up to room temperature  That usually leads to phase separation and the formation of secondary phases  It was recently shown that phase separation induced by spinodal decomposition could lead to a significant increase of $T_{C}$ \cite{Diet06,Fuku06}  For semiconductors showing $T_{C}$ higher than room temperature one can foresee the fabrication of nanodevices such as memory nanodots, or nanochannels for spin injection  Therefore, the precise control of inhomogeneities appears as a new challenge which may open a way to industrial applications of ferromagnetism in semiconductors 

The increasing interest in group-IV magnetic semiconductors can also be explained by their potential compatibility with the existing silicon technology  In 2002, carrier mediated ferromagnetism was reported in  MBE grown Ge$_{0 94}$Mn$_{0 06}$ films by Park \textit{et al } \cite{Park02}  The maximum critical temperature was 116 K  Recently many publications indicate a significant increase of $T_{C}$ in Ge$_{1-x}$Mn$_{x}$ material depending on growth conditions \cite{Pint05,Li05,tsui03}  Cho \textit{et al } reported a Curie temperature as high as 285 K \cite{Cho02}  
Taking into account the strong tendency of Mn ions to form intermetallic compounds in germanium, a detailed investigation of the nanoscale structure is required  Up to now, only a few studies have focused on the nanoscale composition in Ge$_{1-x}$Mn$_{x}$ films  Local chemical inhomogeneities have been recently reported by Kang \textit{et al } \cite{Kang05} who evidenced a micrometer scale segregation of manganese in large Mn rich stripes  Ge$_3$Mn$_5$ as well as Ge$_8$Mn$_{11}$ clusters embedded in a germanium matrix have been reported by many authors  However, Curie temperatures never exceed 300 K \cite{Bihl06,Morr06,Pass06,Ahle06}  Ge$_3$Mn$_5$ clusters exhibit a Curie temperature of 296 K \cite{Mass90}  This phase frequently observed in Ge$_{1-x}$Mn$_{x}$ films is the most stable (Ge,Mn) alloy  The other stable compound Ge$_8$Mn$_{11}$ has also been observed in nanocrystallites surrounded with pure germanium \cite{Park01}  Ge$_8$Mn$_{11}$ and Ge$_3$Mn$_5$ phases are ferromagnetic but their metallic character considerably complicates their potential use as spin injectors 
Recently, some new Mn-rich nanostructures have been evidenced in Ge$_{1-x}$Mn$_{x}$ layers  Sugahara \textit{et al } \cite{Sugh05} reported the formation of high Mn content (between 10 \% and 20 \% of Mn) amorphous Ge$_{1-x}$Mn$_x$ precipitates  in a Mn-free germanium matrix  Mn-rich coherent cubic clusters were observed by Ahlers \textit{et al } \cite{Ahle06} which exhibit a Curie temperatures below 200 K  Finally, high-$T_{C}$ ($>$ 400 K) Mn-rich nanocolumns have been evidenced \cite{Jame06} which could lead to silicon compatible room temperature operational devices \\
In the present paper, we investigate the structural and magnetic properties of Ge$_{1-x}$Mn$_x$ thin films for low growth temperatures ($<$ 200$^{\circ}$C) and low Mn concentrations (between 1 \% and 11 \%)  By combining TEM, x-Ray diffraction and SQUID magnetometry, we could identify different magnetic phases  We show that depending on growth conditions, we obtain either Mn-rich nanocolumns or Ge$_{3}$Mn$_{5}$ clusters embedded in a germanium matrix  We discuss the structural and magnetic properties of these nanostructures as a function of manganese concentration and growth temperature  We also discuss the magnetic anisotropy of nanocolumns and  
Ge$_3$Mn$_5$ clusters  

\section{Sample growth}

Growth was performed using solid sources molecular beam epitaxy (MBE) by co-depositing Ge and Mn evaporated from standard Knudsen effusion cells   Deposition rate was low ($\approx$ 0 2 \AA s$^{-1}$)  Germanium substrates were epi-ready Ge(001) wafers with a residual n-type doping and resistivity of 10$^{15}$ cm$^{-3}$ and 5 $\Omega cm$ respectively  After thermal desorption of the surface oxide, a 40 nm thick Ge buffer layer was grown at 250$^{\circ}$C, resulting in a 2 $\times$ 1 surface reconstruction as observed by reflection high energy electron diffraction (RHEED) (see Fig  1a)  Next, 80 nm thick Ge$_{1-x}$Mn$_{x}$ films were subsequently grown  at low substrate temperature (from 80$^{\circ}$C to 200$^{\circ}$C)  Mn content has been determined by x-ray fluorescence measurements performed on thick samples ($\approx$ 1 $\mu m$ thick) and complementary Rutherford Back Scattering (RBS) on thin Ge$_{1-x}$Mn$_{x}$ films grown on silicon  Mn concentrations range from 1 \% to 11\% Mn 

For Ge$_{1-x}$Mn$_{x}$ films grown at substrate temperatures below 180$^{\circ}$C, after the first monolayer (ML) deposition, the  2 $\times$ 1 surface reconstruction almost totally disappears  After depositing few MLs, a slightly diffuse 1 $\times$ 1 streaky RHEED pattern and a very weak 2 $\times$ 1 reconstruction (Fig  1b) indicate a predominantly two-dimensional growth  For growth temperatures above 180$^{\circ}$C additional spots appear in the RHEED pattern during the Ge$_{1-x}$Mn$_{x}$ growth (Fig  1c)  These spots may correspond to the formation of very small secondary phase crystallites  The nature of these crystallites will be discussed below 

Transmission electron microscopy (TEM) observations were performed using a JEOL 4000EX microscope with an acceleration voltage of 400 kV  Energy filtered transmission electron microscopy (EFTEM) was done using a JEOL 3010 microscope equipped with a Gatan Image Filter   Sample preparation was carried out by standard mechanical polishing and argon ion milling for cross-section investigations and plane views were prepared by wet etching with H$_3$PO$_4$-H$_2$O$_2$ solution \cite{Kaga82} 

\begin{figure}[htb]
    \center
    \includegraphics[width= 29\linewidth]{ /fig1a eps}
    \includegraphics[width= 29\linewidth]{ /fig1b eps}
    \includegraphics[width= 29\linewidth]{ /fig1c eps}
    \caption{RHEED patterns recorded during the growth of Ge$_{1-x}$Mn$_{x}$ films: (a) 2 $\times$ 1 surface reconstruction of the germanium buffer layer  (b) 1 $\times$ 1 streaky RHEED pattern obtained at low growth temperatures ($T_g<$180$^{\circ}$C)  (c) RHEED pattern of a sample grown at $T_g=$180$^{\circ}$C  The additional spots reveal the presence of Ge$_3$Mn$_5$ clusters at the surface of the film }
\label{fig1}
\end{figure}

\section{Structural properties \label{structural}}

\begin{figure}[htb]
    \center
	\includegraphics[width= 49\linewidth]{ /fig2a eps}
	\includegraphics[width= 49\linewidth]{ /fig2b eps}
	 \includegraphics[width= 49\linewidth]{ /fig2c eps}
	 \includegraphics[width= 49\linewidth]{ /fig2d eps}
    \caption{Transmission electron micrographs of a Ge$_{1-x}$Mn$_{x}$ film grown at 130$^{\circ}$C and containing 6 \% of manganese  (a) cross-section along the [110] axis : we clearly see the presence of nanocolumns elongated along the growth axis  (b) High resolution image of the interface between the Ge$_{1-x}$Mn$_{x}$ film and the Ge buffer layer  The Ge$_{1-x}$Mn$_{x}$ film exhibits the same diamond structure as pure germanium  No defect can be seen which could be caused by the presence of nanocolumns  (c) Plane view micrograph performed on the same sample confirms the columnar structure and gives the density and size distribution of nanocolumns  (d) Mn chemical map obtained by energy filtered transmission electron microcopy (EFTEM)  The background was carefully substracted from pre-edge images  Bright areas correspond to Mn-rich regions }
\label{fig2}
\end{figure}

In samples grown at 130$^{\circ}$C and containing 6 \% Mn, we can observe vertical elongated nanostructures \textit{i e } nanocolumns as shown in Fig  2a  Nanocolumns extend through the whole Ge$_{1-x}$Mn$_{x}$ film thickness  From the high resolution TEM image shown in Fig  2b, we deduce their average diameter around 3 nm  Moreover in Fig  2b, the interface between the Ge buffer layer and the Ge$_{1-x}$Mn$_{x}$ film is flat and no defect propagates from the interface into the film  The Ge$_{1-x}$Mn$_{x}$ film is a perfect single crystal in epitaxial relationship with the substrate  In Fig  2c is shown a plane view micrograph of the same sample confirming the presence of nanocolumns in the film  From this image, we can deduce the size and density of nanocolumns  The nanocolumns density is 13000 $\rm{\mu m}^{-2}$ with a mean diameter of 3 nm which is coherent with cross-section measurements  In order to estimate the chemical composition of these nanocolumns, we further performed chemical mapping using EFTEM  In Fig  2d we show a cross sectional Mn chemical map of the Ge$_{1-x}$Mn$_{x}$ film  This map shows that the formation of nanocolumns is a consequence of Mn segregation  Nanocolumns are Mn rich and the surrounding matrix is Mn poor  However, it is impossible to deduce the Mn concentration in Ge$_{1-x}$Mn$_{x}$ nanocolumns from this cross section  Indeed, in cross section observations, the columns diameter is much smaller than the probed film thickness and the signal comes from the superposititon of the Ge matrix and Mn-rich nanocolumns  In order to quantify Mn concentration inside the nanocolumns and inside the Ge matrix, EELS measurements (not shown here) have been performed in a plane view geometry \cite{Jame06}  These observations revealed that the matrix Mn content is below 1 \% (detection limit of our instrument)  Measuring the surface occupied by the matrix and the nanocolumns in plane view TEM images, and considering the average Mn concentration in the sample (6 \%), we can estimate the Mn concentration in the nanocolumns  The Mn concentration measured by EELS being between 0\% and 1\%, we can conclude that the Mn content in the nanocolumns is between 30 \% and 38 \% \\
For samples grown between 80$^\circ$C and 150$^\circ$C cross section and plane view TEM observations reveal the presence of Mn rich nanocolumns surrounded with a Mn poor Ge matrix  In order to investigate the influence of Mn concentration on the structural properties of Ge$_{1-x}$Mn$_{x}$ films, ten samples have been grown at 100$^\circ$C and at 150$^\circ$C with Mn concentrations of 1 3 \%, 2 3 \%, 4 \%, 7 \% and 11 3 \%  Their structural properties have been investigated by plane view TEM observations  

\begin{figure}[htb]
    \center
    \includegraphics[width= 98\linewidth]{ /fig3a eps}
	\includegraphics[width= 45\linewidth]{ /fig3b eps}
		\includegraphics[width= 45\linewidth]{ /fig3c eps}
    \caption{Nanocolumns size and density as a function of growth conditions  Samples considered have been grown at 100$^{\circ}$C and 150$^{\circ}$C respectively  (a) Mn concentration dependence of the size distribution  (b) columns density as a function of Mn concentration  (c) Volume fraction of the nanocolumns as a function of Mn concentration }
 \label{fig3}
\end{figure}

For samples grown at 100$^\circ$C with Mn concentrations below 5 \% the nanocolumns mean diameter is 1 8$\pm$0 2 nm  The evolution of columns density as a fonction of Mn concentration is reported in figure 3b  By increasing the Mn concentration from 1 3 \% to 4 \% we observe a significant increase of the columns density from 13000 to 30000 $\rm{\mu m}^{-2}$  For Mn concentrations higher than 5 \% the density seems to reach a plateau corresponding to 35000 $\rm{\mu m}^{-2}$ and their diameter slightly increases from 1 8 nm at 4 \% to 2 8 nm at 11 3 \%  By plotting the volume fraction occupied by the columns in the film as a function of Mn concentration, we observe a linear dependence for Mn contents below 5 \%  The non-linear behavior above 5 \% may indicate that the mechanism of Mn incorporation is different in this concentration range, leading to an increase of Mn concentration in the columns or in the matrix  For samples grown at 100$^\circ$C, nanocolumns are always fully coherent with the surrounding matrix (Fig  4a)  

Increasing the Mn content in the samples grown at 150$^\circ$C from 1 3 \% to 11 3 \% leads to a decrease of the columns density (fig 3b)  Moreover, their average diameter increases significantly and size distributions become very broad (see Fig  3a)  For the highest Mn concentration (11 3 \%) we observe the coexistence of very small columns with a diameter of 2 5 nm and very large columns with a diameter of 9 nm  In samples grown at 150$^\circ$C containing 11 3 \% of Mn, the crystalline structure of nanocolumns is also highly modified  In plane view TEM micrographs, one can see columns exhibiting several different crystalline structures  We still observe some columns which are fully coherent with the Ge matrix like in the samples grown at lower temperature  Nevertheless, observations performed on these samples grown at 150$^\circ$C and with 11 3\% Mn reveal some uniaxially \cite{Jame06} or fully relaxed columns exhibiting a misfit of 4 \% between the matrix and the columns and leading to misfit dislocations at the interface between the column and the matrix (see fig  4b)  Thus we can conclude that coherent columns are probably in strong compression and the surrounding matrix in tension  On the same samples (T$_g$=150$^{\circ}$C, 11 3\% Mn), we also observe a large number of highly disordered nanocolumns leading to an amorphous like TEM contrast(fig  4c) 

\begin{figure}[htb]
    \center
   \includegraphics[width= 31\linewidth]{ /fig4a eps}
	\includegraphics[width= 31\linewidth]{ /fig4b eps}
	\includegraphics[width= 31\linewidth]{ /fig4c eps}
    \caption{Plane view high resolution transmission electron micrographs of different types of nanocolumns : (a) typical structure of a column grown at 100$^{\circ}$C  The crystal structure is exactly the same as germanium   (b) Partially relaxed nanocolumn  One can see dislocations at the interface between the columns and the matrix leading to stress relaxation  (c) Amorphous nanocolumn  These columns are typical in samples grown at 150$^{\circ}$C with high Mn contents }
 \label{fig4}
\end{figure}

In conclusion, we have evidenced a complex mechanism of Mn incorporation in Mn doped Ge films grown at low temperature  In particular Mn incorporation is highly inhomogeneous  For very low growth temperatures (below 120$^\circ$C) the diffusion of Mn atoms leads to the formation of Mn rich, vertical nanocolumns  Their density mostly depends on Mn concentration and their mean diameter is about 2 nm  These results can be compared with the theoretical predictions of Fukushima \textit{et al } \cite{Fuku06}: they proposed a model of spinodal decomposition in (Ga,Mn)N and (Zn,Cr)Te based on layer by layer growth conditions and a strong pair attraction between Mn atoms which leads to the formation of nanocolumns  This model may also properly describe the formation of Mn rich nanocolumns in our samples  Layer by layer growth conditions can be deduced from RHEED pattern evolution during growth  For all the samples grown at low temperature, RHEED observations clearly indicate two-dimensional growth  Moreover, Ge/Ge$_{1-x}$Mn$_{x}$/Ge heterostructures have been grown and observed by TEM (see Fig  5)  Ge$_{1-x}$Mn$_{x}$/Ge (as well as Ge/Ge$_{1-x}$Mn$_{x}$) interfaces are very flat and sharp thus confirming a two-dimensional, layer by layer growth mode  Therefore we can assume that the formation of Mn rich nanocolumns is a consequence of 2D-spinodal decomposition 

\begin{figure}[htb]
    \center
	\includegraphics[width= 7\linewidth]{ /fig5 eps}
    \caption{Cross section high resolution micrograph of a Ge/Ge$_{1-x}$Mn$_{x}$/Ge/Ge$_{1-x}$Mn$_{x}$/Ge heterostructure  This sample has been grown at 130 $^{\circ}$C with 6\% Mn  Ge$_{1-x}$Mn$_{x}$ layers are 15 nm thick and Ge spacers 5 nm thick  We clearly see the sharpness of both Ge$_{1-x}$Mn$_{x}$/Ge and Ge/Ge$_{1-x}$Mn$_{x}$ interfaces  Mn segregation leading to the columns formation already takes place in very thin Ge$_{1-x}$Mn$_{x}$ films }
\label{fig5}
\end{figure}

For growth temperatures higher than 160$^\circ$C, cross section TEM and EFTEM observations (not shown here) reveal the coexistence of two Mn-rich phases: nanocolumns and Ge$_{3}$Mn$_{5}$ nanoclusters embedded in the germanium matrix  A typical high resolution TEM image is shown in figure 6  
Ge$_{3}$Mn$_{5}$ clusters are not visible in RHEED patterns for temperatures below 180$^\circ$C  To investigate the nature of these clusters, we performed x-ray diffraction in $\theta-2\theta$ mode  Diffraction scans were acquired on a high resolution diffractometer using the copper K$_\alpha$ radiation and on the GMT station of the BM32 beamline at the European Synchrotron Radiation Facility (ESRF)  Three samples grown at different temperatures and/or annealed at high temperature were investigated  The two first samples are Ge$_{1-x}$Mn$_{x}$ films grown at 130$^\circ$C and 170$^\circ$C respectively  The third one has been grown at 130$^\circ$C and post-growth annealed at 650$^\circ$C  By analysing x-ray diffraction spectra, we can evidence two different crystalline structures  For the sample grown at 130$^\circ$C, the $\theta-2\theta$ scan only reveals the (004) Bragg peak of the germanium crystal, confirming the good epitaxial relationship between the layer and the substrate, and the absence of secondary phases in the film in spite of a high dynamics of the order of 10$^7$  For both samples grown at 170$^\circ$C and annealed at 650$^\circ$C, $\theta-2\theta$ spectra are identical  In addition to the (004) peak of germanium, we observe three additional weak peaks  The first one corresponds to the (002) germanium forbidden peak which probably comes from a small distortion of the germanium crystal, and the two other peaks are respectively attributed to the (002) and (004) Bragg peaks of a secondary phase  The $c$ lattice parameter of Ge$_3$Mn$_5$ hexagonal crystal is 5 053 \AA \ \cite{Fort90} which is in very good agreement with the values obtained from diffraction data for both (002) and (004) lines assuming that the $c$ axis of Ge$_3$Mn$_5$ is along the [001] direction of the Ge substrate 

\begin{figure}[htb]
    \center
	\includegraphics[width= 7\linewidth]{ /fig6 eps}
	\caption{Cross section high resolution transmission electron micrograph of a sample grown at 170$^{\circ}$C  We observe the coexistence of two different Mn-rich phases: Ge$_{1-x}$Mn$_{x}$ nanocolumns and Ge$_3$Mn$_5$ clusters }
\label{fig6}
\end{figure}

In summary, in a wide range of growth temperatures and Mn concentrations, we have evidenced a two-dimensional spinodal decomposition leading to the formation of Mn-rich nanocolumns in Ge$_{1-x}$Mn$_{x}$ films  This decomposition is probably the consequence of: $(i)$ a strong pair attraction between Mn atoms, $(ii)$ a strong surface diffusion of Mn atoms in germanium even at low growth temperatures and $(iii)$ layer by layer growth conditions  We have also investigated the influence of growth parameters on the spinodal decomposition: at low growth temperatures (100$^{\circ}$C), increasing the Mn content leads to higher columns densities while at higher growth temperatures (150$^{\circ}$C), the columns density remains nearly constant whereas their size increases drastically  By plotting the nanocolumns density as a function of Mn content, we have shown that the mechanism of Mn incorporation in Ge changes above 5 \% of Mn  Finally, using TEM observations and x-ray diffraction, we have shown that Ge$_3$Mn$_5$ nanoclusters start to form at growth temperatures higher than 160$^\circ$C 

\section{Magnetic properties \label{magnetic}}

We have thoroughly investigated the magnetic properties of thin Ge$_{1-x}$Mn$_{x}$ films for different growth temperatures and Mn concentrations  In this section, we focus on Mn concentrations between 2 \% and 11 \%  We could clearly identify four different magnetic phases in Ge$_{1-x}$Mn$_{x}$ films : diluted Mn atoms in the germanium matrix, low $T_{C}$ nanocolumns ($T_{C}$ $\leq$ 170 K), high $T_{C}$ nanocolumns ($T_{C}$ $\geq$ 400 K) and Ge$_{3}$Mn$_{5}$ clusters ($T_{C}$ $\thickapprox$ 300 K)  The relative weight of each phase clearly depends on the growth temperature and to a lesser extend on Mn concentration  For low growth temperature ($<$ 120$^{\circ}$C), we show that nanocolumns are actually made of four uncorrelated superparamagnetic nanostructures  Increasing T$_{g}$ above 120$^{\circ}$C, we first obtain continuous columns exhibiting low $T_{C}$ ($<$ 170 K) and high $T_{C}$ ($>$ 400 K) for $T_{g}\approx$130$^{\circ}$C  The larger columns become ferromagnetic \textit{i e } $T_{B}>T_{C}$  Meanwhile Ge$_{3}$Mn$_{5}$ clusters start to form  Finally for higher $T_{g}$, the magnetic contribution from Ge$_{3}$Mn$_{5}$ clusters keeps increasing while the nanocolumns signal progressively disappears 

\begin{figure}[htb]
\center
   \includegraphics[width= 6\linewidth]{ /fig7a eps}
   \includegraphics[width= 3\linewidth]{ /fig7b eps}
\caption{(a) Temperature dependence of the saturation magnetization (in $\mu_{B}$/Mn) of Ge$_{0 93}$Mn$_{0 07}$ samples for different growth temperatures



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

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 does a media application determine the context of an event?

Read the following text and answer briefly.

Context 2: 2015-05-14 Assigned to ROVI GUIDES, INC  reassignment ROVI GUIDES, INC  MERGER (SEE DOCUMENT FOR DETAILS)  Assignors: TV GUIDE, INC 
2015-05-14 Assigned to UV CORP  reassignment UV CORP  MERGER (SEE DOCUMENT FOR DETAILS)  Assignors: UNITED VIDEO PROPERTIES, INC 
2015-05-14 Assigned to TV GUIDE, INC  reassignment TV GUIDE, INC  MERGER (SEE DOCUMENT FOR DETAILS)  Assignors: UV CORP 
Methods and systems are described herein for quickly and easily displaying supplemental information about an event occurring in a media asset  In some embodiments, a media application may use a content-recognition module to determine the context of an event and distribute itemized tasks to multiple entities in order to generate the supplemental information about the event 
While viewing media assets (e g , a television program), users may wish to learn more information about an event (e g , a statement made by a person appearing in the media asset, the validity of a claim in an advertisement, etc ) occurring in the media asset  While some media assets allow a user to select additional options or added features (e g , pop-up biographies about the cast and crew), when the added features appear and what topic the added features concern are determined by the content producer and not the user  Furthermore, as the added feature is derived from the content producer, the added feature may be biased or may present limited viewpoints about an event  Therefore, added features provided by a content producer may not provide the added information about an event that a user desires 
In order to gain the added information that a user desires, the user may use additional devices (e g , a laptop computer) to search (e g , using an Internet search engine) for more information about the event  However, without knowing the proper context (e g , who said the statement, what was the tone of the statement, when was the statement said, etc ) of the event or what search terms to use to describe the context of the event (e g , how to describe the tone of the statement), a user may not be able to determine (even using a search engine) more information about the event  Moreover, the use of general search terms may not provide the accuracy or precision needed by the user  Furthermore, even if a user may eventually determine the information, the effort and time required may distract the user from the media asset 
Accordingly, methods and systems are described herein for quickly and easily displaying supplemental information about an event occurring in a media asset  In some embodiments, a media application may use a content-recognition module to determine the context of an event in a media asset and distribute itemized tasks to multiple users in order to generate the supplemental information about the event  The context-recognition module prevents the user from being distracted from the media asset (e g , while the user attempts to describe the context of the event or search for information about the event)  In addition, by distributing tasks to multiple entities (e g , crowd-sourcing), the media application may collect large amounts of information in relatively short periods of time (or in real-time) and aggregate and/or filter the information to generate the supplemental information about the event based on multiple viewpoints and/or sources  By using multiple viewpoints and/or sources, the media application enhances the completeness (e g , by providing unbiased information) and accuracy of the supplemental information 
For example, when a statement or action is made by a character or person appearing on a media asset (e g , a television program), a user may request supplemental information about the statement or action  In response, the media application may determine the context of the statement (e g , who said the statement and to what the statement was referring) or action (e g , what was the reason for the action)  After determining the context of the statement or action, the media application may itemize into tasks the additional information it requires in order to generate the supplemental information  The media application may then transmit requests including the tasks to a plurality of other users  Based on the responses from the plurality of other users, the media application may generate the supplemental information for display to the user 
In some embodiments, a media application may use multiple types of content-recognition modules and/or algorithms to determine the context of an event  For example, the media application may process data associated with the event in order to determine the context of an event  In some embodiments, processing the various types of data may include cross-referencing the data in a database indicating the different contexts the event may have 
In some embodiments, a media application may generate supplemental information about an event in a media asset in response to a user request  In order to generate the supplemental information, the media application may transmit, to multiple users, a request for additional information regarding a context of an event shown in a media asset  Upon receiving messages from the plurality of users that include the requested additional information, the media application may generate the supplemental information associated with the context of the event based on the messages 
It should be noted, the systems and/or methods described above may be applied to, or used in accordance with, other systems, methods and/or apparatuses 
FIG  9 is a flowchart of illustrative steps for generating supplemental information based on additional information provided by a plurality of users in accordance with some embodiments of the disclosure 
Accordingly, methods and systems are described herein for quickly and easily displaying supplemental information about an event occurring in a media asset  The methods and systems described herein alleviate the need for a user to determine the proper context (e g , who said a statement, what was the tone of the statement, when was the statement said, etc ) of an event in a media asset, or the search terms to use to describe the event (e g , the proper search terms to describe the tone of the statement), in order to determine more information about the event  In addition, the methods and systems increase the completeness and accuracy of the information compared to information gathered using traditional searching methods (e g , an Internet search engine), without distracting the user from the media asset 
In some embodiments, a media application may receive a user input from a user device for supplemental information about the context of an event shown in a media asset  The media application may determine additional information required to generate the supplemental information about the context of the event shown in a media asset, and transmit requests for the additional information to one or more users  The media application may receive one or more messages, which include the requested additional information, from the one or more users and generate the supplemental information based on the one or more message  The media application may then instruct the user device to display the supplemental information 
As used herein, “supplemental information” refers to any information related to or associated with an event in a media asset  For example, supplemental information may include, but is not limited to, the verification of a statement or claim in a media asset, further descriptions and/or information about objects or entities shown and/or described in a media asset, and/or any other information, including, but not limited to, a video or audio segment, that may interest a user about an event in a media asset  In some embodiments, the media application may generate supplemental information based on one or more pieces of additional information 
As used herein, “additional information” refers to any information used to generate supplemental information  For example, in an embodiment in which supplement information is the verification of a statement made by a person displayed in a media asset, and a request for the additional information from the media application includes a request for a fact needed to verify the factual basis of the statement, the additional information may be the fact used to verify the statement  For example, if an advertisement claims to have the best product on the market, the media application may use additional information such as the name of the product in question, a list of all other products in the market, and the results of a comparison study of the product in question to all other products to determine whether or not the product is actually the “best” product on the market  Additionally or alternatively, the media application may request industry and/or user reviews related to the event (e g , reviews indicating the quality of the product)  The media application may then use the information in the reviews to generate the supplemental information 
As used herein, an “event” is any action (e g , a verbal statement, opinion and/or physical movement), segment (e g , a portion of a news broadcast featuring a particular topic), or other occurrence during a media asset that may be of particular interest to a user  For example, in some embodiments an event may be a statement or gesture made by a character or person in a media asset affirming or denying a claim 
As referred to herein, the terms “media asset” and “content” should be understood to mean an electronically consumable user asset, such as television programming, as well as pay-per-view programs, on-demand programs (as in video-on-demand (VOD) systems), Internet content (e g , streaming content, downloadable content, Webcasts, etc ), video clips, audio, content information, pictures, rotating images, documents, playlists, websites, articles, books, electronic books, blogs, advertisements, chat sessions, social media, applications, games, and/or any other media or multimedia and/or combination of the same  Media applications also allow users to navigate among and locate content  As referred to herein, the term “multimedia” should be understood to mean content that utilizes at least two different content forms described above, for example, text, audio, images, video, or interactivity content forms  Content may be recorded, played, displayed or accessed by user equipment devices, but can also be part of a live performance 
As referred to herein, the phrase “user equipment device,” “user equipment,” “user device,” “electronic device,” “electronic equipment,” “media equipment device,” or “media device” should be understood to mean any device for accessing the content described above, such as a television, a Smart TV, a set-top box, an integrated receiver decoder (IRD) for handling satellite television, a digital storage device, a digital media receiver (DMR), a digital media adapter (DMA), a streaming media device, a DVD player, a DVD recorder, a connected DVD, a local media server, a BLU-RAY player, a BLU-RAY recorder, a personal computer (PC), a laptop computer, a tablet computer, a WebTV box, a personal computer television (PC/TV), a PC media server, a PC media center, a hand-held computer, a stationary telephone, a personal digital assistant (PDA), a mobile telephone, a portable video player, a portable music player, a portable gaming machine, a smart phone, or any other television equipment, computing equipment, or wireless device, and/or combination of the same 
In some embodiments, the user equipment device may have a front facing screen and a rear facing screen, multiple front screens, or multiple angled screens  In some embodiments, the user equipment device may have a front facing camera and/or a rear facing camera  On these user equipment devices, users may be able to navigate among and locate the same content available through a television  Consequently, media may be available on these devices, as well  The media provided may be for content available only through a television, for content available only through one or more of other types of user equipment devices, or for content available both through a television and one or more of the other types of user equipment devices  The media applications may be provided as on-line applications (i e , provided on a website), or as stand-alone applications or clients on user equipment devices  Various devices and platforms that may implement media applications are described in more detail below 
In some embodiments, a media application may transmit, to a plurality of users, a request for additional information regarding a context of an event shown in a media asset  As used herein, a “plurality of users” may include, but is not limited to any device, entity, or source of information that may process a request for additional information  For example, the plurality of users may include a person operating a user equipment device  In some embodiments, the person may receive (e g , via e-mail, Internet posting, advertisement, or any other applicable information delivery method) the request from the media application for additional information, and in response generate a message (e g , via a return e-mail, an answer to the Internet posting, a user input in the advertisement, or any other applicable method of transmitting information) that includes the additional information  It should be noted that in some embodiments, transmitting a request to a plurality of users may also include querying one or more databases (e g , an Internet search engine or any other storage device, including, but not limited to, databases containing previously generated supplemental information and/or additional information) or consulting one or more data gathering services (e g , a intelligent personal assistant application) for the additional information 
In some embodiments, a media application may use a content-recognition module or algorithm to determine the context of an event and distribute itemized tasks to multiple users in order to generate the supplemental information about the event  The content-recognition module may use object recognition techniques such as edge detection, pattern recognition, including, but not limited to, self-learning systems (e g , neural networks), optical character recognition, on-line character recognition (including but not limited to, dynamic character recognition, real-time character recognition, intelligent character recognition), and/or any other suitable technique or method to determine the objects and/or characteristics in media assets  For example, the media application may receive media assets in the form of a video  The video may include a series of frames  For each frame of the video, the media application may use a content-recognition module or algorithm to determine the context (e g , the person that is speaking or a facial gesture affirming or denying a statement) of an event occurring during the frame or series of frames 
In some embodiments, the content-recognition module or algorithm may also include speech recognition techniques, including but not limited to Hidden Markov Models, dynamic time warping, and/or neural networks (as described above) to translate spoken words into text  The content-recognition module may also use other techniques for processing audio and/or visual data  For example, the media application may monitor the volume of a statement in a media asset to determine the tone of the statement (e g , a high volume may indicate an angry tone) 
In addition, the media application may use multiple types of optical character recognition and/or fuzzy logic, for example, when determining the context of a keyword(s) retrieved from data (e g , media data, translated audio data, subtitle data, user-generated data, etc ) associated with the media asset (or when cross-referencing various types of data with databases indicating the different contexts of events as described below)  For example, the particular data field may be a textual data field  Using fuzzy logic, the system may determine two fields and/or values to be identical even though the substance of the data field or value (e g , two different spellings) is not identical  In some embodiments, the system may analyze particular data fields of a data structure or media asset frame for particular values or text  The data fields could be associated with characteristics, additional information, and/or any other data required for the function of the embodiments described herein  Furthermore, the data fields could contain values (e g , the data fields could be expressed in binary or any other suitable code or programming language) 
As used herein, the “context” of an event refers to the set of circumstances or facts that surround a particular event that influence or affect the meaning of the event  For example, when determining the context of a written and/or spoken statement, the media application may determine who or what authored/stated the statement, the written and/or spoken words and/or other statements that preceded and/or followed the statement, the tone of the statement, and/or any other conditions that may alter the connotation of the statement 
FIG  1 shows an illustrative example of a media application that may be used to display supplemental information in accordance with some embodiments of the disclosure  Display 100 illustrates a display on a user device displaying a media asset  Display 108 illustrates a display featuring supplemental information as described and/or generated in FIGS  6-9  It should be noted that display 100 and display 108 may be presented on any of the devices shown in FIGS  3-4  For example, in some embodiments, display 100 and display 108 may be displayed on user equipment 402, 404, and/or 406 (FIG  4) 
In FIG  1, display 100 represents a display of a media asset (e g , a streaming television program) on a user device (e g , user equipment 402, 404, and/or 406 (FIG  4))  Display 100 includes entity 102 and entity 104  In display 100, entity 104 is currently speaking as indicated by event 106  As shown in FIG  1, event 106 is a statement (e g , “We export a lot of coal”) by a person in the media asset 
In some embodiments, display 108 represents the continued display of the media asset on a user device, after a user has requested supplemental information about event 106  For example, a media application may have received a user input (e g , via user input interface 310 (FIG  3)) while entity 104 was speaking  Using the systems and methods described herein (e g , FIGS  6-9), the media application generated supplemental information 110  Supplemental information 110 represents more information about event 106 
For example, the media application (e g , media application 206 (FIG  2)) may have determined the context of event 106  Specifically, the media application may determine via a content-recognition module or algorithm the words spoken and/or actions by the person during the event  Additionally or alternatively, the media application may analyze the words and/or action during a predetermined amount of time (e g , ten seconds) before and/or after the event (e g , in order to better understand the context of the event)  Furthermore, by cross-referencing the words and/or other information obtained by the content-recognition module (e g , as discussed below in relation to FIG  5) with a database, the content-recognition module determines that the term “we,” the person in the media asset refers to an organization or body  The content-recognition module or algorithm may also determine that the term “export” refers to shipping goods out of a country  The content-recognition module or algorithm may also determine that the term “a lot” refers to a particular numerical amount  Finally, the content-recognition module or algorithm may also determine that the term “coal” refers to a mineral of fossilized carbon 
The content-recognition module or algorithm may also determine the relationships between words and/or other information obtained by the content-recognition module  For example, by processing the relationship between the words, the media application determines that event 106 is a statement regarding a particular amount of a particular substance shipped out of a particular country  Therefore, the media application determines that the request for supplemental information is likely a request to determine the validity of the statement  The media application then generates the supplemental information 
The media application may also have stored supplemental information generated by previous requests (e g , supplemental information generated in response to the same or different user viewing the media asset at an earlier date), and display the supplemental information again during the event (either in response to a user input requesting supplemental information or automatically without a user requesting supplemental information) 
FIG  2 shows an illustrative example of a system that may be used to generate supplemental information (e g , supplemental information 110 (FIG  1)) based on additional information provided by a plurality of users in accordance with some embodiments of the disclosure  For example, in some embodiments, system 200 may be used to generate supplemental information (e g , supplemental information 110 (FIG  1)) on a display (e g , display 108 (FIG  1)) of a user device (e g , user equipment 402, 404, and/or 406 (FIG  4))  It should be noted that in some embodiments, the devices shown in FIG  2 may correspond to one or more devices in FIGS  3-4 
FIG  2 shows system 200  In system 200, a user is currently accessing a media asset on display 202  In some embodiments, display 202 may correspond to display 100 (FIG  1))  During an event (e g , event 106 (FIG  1)) a user may have requested supplemental information about an event (e g , event 106 (FIG  1)) in display 202 using user device 204  Media application 206, which in some embodiments, may be implemented on user device 204 or at a remote location (e g , supplemental information source 418 (FIG  4)), receives the request for supplemental information 
Media application 206 determines the context of the event (e g , who said the statement making up the event and to what the statement was referring)  After determining the context of the statement, the media application may itemize into one or more tasks, additional information (e g , facts) it requires in order to generate the supplemental information (e g , a verification or correction of the factual basis of the statement)  For example, if the event is a statement about the amount of coal that is exported from the United States (e g , as described in relation to FIG  1 above), media application 206 may determine the fact required to generate the supplemental information is the exact numerical amount of coal that is exported from the United States  The media application may then transmit requests for the additional information (e g , a request for the exact numerical amount of coal that is exported from the United States) to a plurality of other users 
In FIG  2, users operating user device 208, user device 210, and user device 212 represent a plurality of users  Having determined the additional information it requires in order to generate the supplemental information, media application 206 requests the additional information from the plurality of users  In system 200, media application 206 has transmitted the same task (e g , the same question) to each of the plurality of users  In some embodiments, one or more of the users may receive different tasks  For example, by breaking the additional information into small, independent tasks, media application 206 may increase the speed (e g , multiple users may work concurrently to solve different parts of a problem) and accuracy (e g , reducing the tasks to smaller, less complex problems reduces the chance of human error) of the additional information returned by the plurality of users 
In addition, by breaking the additional information into small, independent tasks, the plurality of users may not know to what they are contributing (enhancing the privacy of the user that requested the supplemental information), however, the plurality of users can still be effective in their individual tasks  In addition, by breaking the additional information into small, independent tasks, the media application may more easily outsource the requests for additional information  For example, one or more of the tasks used to generate the additional information may be the same as one or more of the tasks used to generate other additional information (e g , additional information used to generate different supplemental information in response to a request for supplemental information about the same or a different event issued by the same or a different user)  The response to each of the request and/or the additional information may be stored (e g , on any of the devices accessible by communications network 414 (FIG  4)) for subsequent retrieval 
Based on the responses, transmitted as messages including the additional information, from the plurality of other users, media application 206 may generate the supplemental information (e g , supplemental information 110 (FIG  1)) for display to the user on the user device 204  For example, media application may aggregate, append, and/or compare the additional information in each of the messages received from the plurality of users  The supplemental information may then be generated based on the aggregated, appended, and/or compared additional information (e g , as described in FIG  9 below) 
In some embodiments, the plurality of users may receive summary information about the event with the request for additional information  (e g , a video clip of a portion or segment of the media asset, a textual description, etc ), which may help the plurality of users provide additional information  For example, in some embodiments, the media application may instead of (or in addition to) determining the context of an event, determine a particular portion of the event that would be needed for the plurality of users to provide additional information about the event 
For example, the media application may use progress information associated with the progress of the media asset (e g , line 506 (FIG  5)) to determine at what point during the progression of the media asset the event occurred, and in response, transmit a portion of the media asset beginning ten second before that point and ending ten seconds after that point  For example, if the event is a statement made by a character or person in a media asset, the media application may determine when the statement began (e g , the point of progress of the media asset in which the statement began) and ended  The media application may then include the portion containing the entire statement (and the event) in the request for additional information sent to the plurality of users 
The selected portion may include any amount of summary information that the media application determines is necessary for the user or any one of the plurality of users to understand the main action sequence  This summary information (e g , a portion of the media asset) may be included with the request for additional information (e g , in a file transmitted with the request), or may be included with the generated supplemental information as a reference for the user  For example, the media application may select a segment of the play length of the media asset or a particular scene of the media asset, which includes the event, for to display to the plurality of users along with the request for additional information 
For example, if an event (e g , a statement) was in response to a question, the media application may also determine when the question began and ended, and send the entire question (or the play length of the media asset corresponding to the question) to the plurality of users as well  After determining the portion to provide to the plurality of users (e g , a segment including the ten seconds before and the ten seconds after the event), the media application may provide the summary information of the event and any other material needed by the plurality of users to understand the event and/or request for supplemental information from the user 
In some embodiments, a portion of the media asset containing the event, as selected by the media application, may also include any amount of the play length of the media asset, or any amount of scenes or segments from the media asset  In some embodiments, the portion may include segments of the play length of the media asset or scenes from the media asset that are not adjacent during the normal playback of the media asset  For example, in some embodiments, a portion of the media asset may include one or more sequences or scenes of interest to the plurality of users, even though the particular sequences or scenes are featured at different points in the play length of the media asset  The media application may determine the segments or scenes to include based on a content recognition file (e g , data structure 500 (FIG  5)) describing the media asset  For example, if a plot point or other information, which may be relevant to an event is displayed earlier in the media asset, the summary information may include a portion of the media asset displaying the plot point 
In some embodiments, the length of a portion may be determined based on the genre of the media asset  In some embodiments, the length of the portion may depend on a user profile for the user or for anyone of the plurality of users  For example, a user profile and/or a content recognition file (e g , data structure 500 (FIG  5)) may indicate that a particular user may require more or less additional content  For example, the user may be aware of particular characters or plot points in the media asset and, therefore, may not require the additional content to introduce those aspects 
In some embodiments, the plurality of users may receive a particular user interface, which organizes the data about the event (e g , a clip of the actual event, summary information about the event, information about the request for supplemental information issued by the user, etc )  The interface may also include an automatic submission form, which may be used to generate a message, which is sent to the media application 
In some embodiments, the media application may also receive user input from the user requesting the supplemental information that further affects the generation of supplemental information by the media application  For example, the user may request the supplemental information includes particular information (e g , the factual basis of a statement), may request a multimedia format of the supplemental information (e g , textual description, a video clip, etc ), may request a form of the supplemental information (e g , a short description about the event, an Internet link to other sources of information on the event, or a true or false designation about the event) by entering user inputs (e g , via user input interface 310 (FIG  3)) 
It should be noted that any information or process referred to in this disclosure that is referred to as being in response to a user input may alternatively and/or additionally be performed automatically by the media application (e g , via control circuitry 304 (FIG  3))  For example, in some embodiments, a user may request a true or false designation (e g , an on-screen pop-up box indicating whether an event was true or false)  Additionally and/or alternatively, in some embodiments, the true or false designation may appear automatically based on predetermined settings indicating to the media application to display a true or false designation in response to detecting an event 
In some embodiments, an indicator that supplemental information has previously been generated or is currently ready to generate (e g , a plurality of users are available) may be displayed to a user (e g , on display 100 (FIG  1) during the event)  The indicator may also indicate the particular information, the multimedia format, and/or the form of supplemental information that is available  An indicator may also appear with the supplemental information (e g , supplemental information 110 (FIG  1)), which allows the user to request additional supplemental information or provide feedback/responses (e g , rating the quality of the supplemental information) to the media application and/or plurality of users 
In some embodiments, a user may also access (e g , via selection of an indicator and/or automatically upon the supplemental information being generated) summary information about the event  For example, in some embodiments (e g , when the supplemental information is not generated in real-time), the media asset may have progressed to a different point by the time the supplemental information is ready for display  Therefore, the media application may need to provide a video clip of the event or other summary information, so that the user remembers about what or why the supplemental information was requested 
FIG  3 is a block diagram of an illustrative user equipment device in accordance with some embodiments of the disclosure  It should be noted that the components shown in FIG  3 may be used to store, receive, transmit, and/or display the media assets, additional information, and/or supplemental information as described herein  For example, media application 206 (FIG  2) may be implemented on user equipment device 300, and may issue instructions (e g , displaying supplemental information 110 (FIG  1)) via control circuitry 304 
Users may access media assets and the media application (and its display screens described above and below) from one or more of their user equipment devices  FIG  3 shows a generalized embodiment of illustrative user equipment device 300  More specific implementations of user equipment devices are discussed below in connection with FIG  4  User equipment device 300 may receive content and data via input/output (hereinafter “I/O”) path 302  I/O path 302 may provide content (e g , broadcast programming, on-demand programming, Internet content, content available over a local area network (LAN) or wide area network (WAN), and/or other content) and data to control circuitry 304, which includes processing circuitry 306 and storage 308  Control circuitry 304 may be used to send and receive commands, requests, and other suitable data using I/O path 302  I/O path 302 may connect control circuitry 304 (and specifically processing circuitry 306) to one or more communications paths (described below)  I/O functions may be provided by one or more of these communications paths, but are shown as a single path in FIG  3 to avoid overcomplicating the drawing 
Control circuitry 304 may be based on any suitable processing circuitry such as processing circuitry 306  As referred to herein, processing circuitry should be understood to mean circuitry based on one or more microprocessors, microcontrollers, digital signal processors, programmable logic devices, field-programmable gate arrays (FPGAs), application-specific integrated circuits (ASICs), etc , and may include a multi-core processor (e g , dual-core, quad-core, hexa-core, or any suitable number of cores) or supercomputer  
Context 3:  The magnetic field is applied in the film plane  The inset shows the temperature dependence of a sample grown at 130$^{\circ}$C and annealed at 650$^{\circ}$C for 15 minutes  After annealing, the magnetic signal mostly arises from Ge$_{3}$Mn$_{5}$ clusters  (b) ZFC-FC measurements performed on Ge$_{0 93}$Mn$_{0 07}$ samples for different growth temperatures  The in-plane applied field is 0 015 T  The ZFC peak at low temperature ($\leq$150 K) can be attributed to the superparamagnetic nanocolumns  This peak widens and shifts towards high blocking temperatures when increasing growth temperature  The second peak above 150 K in the ZFC curve which increases with increasing growth temperature is attributed to superparamagnetic Ge$_{3}$Mn$_{5}$ clusters  The increasing ZFC-FC irreversibility at $\approx$ 300 K is due to the increasing contribution from large ferromagnetic Ge$_{3}$Mn$_{5}$ clusters  The nanocolumns signal completely vanishes after annealing at 650$^{\circ}$C for 15 minutes }
\label{fig7}
\end{figure}

In Fig  7a, the saturation magnetization at 2 Tesla in $\mu_{B}$/Mn of Ge$_{1-x}$Mn$_{x}$ films with 7 \% of Mn is plotted as a function of temperature for different growth temperatures ranging from $T_{g}$=90$^{\circ}$C up to 160$^{\circ}$C  The inset shows the temperature dependence of the magnetization at 2 Tesla after annealing at 650$^{\circ}$C during 15 minutes  Figure 7b displays the corresponding Zero Field Cooled - Field Cooled (ZFC-FC) curves recorded at 0 015 Tesla  In the ZFC-FC procedure, the sample is first cooled down to 5 K in zero magnetic field and the susceptibility is subsequently recorded at 0 015 Tesla while increasing the temperature up to 400 K (ZFC curve)  Then, the susceptibility is recorded under the same magnetic field while decreasing the temperature down to 5 K (FC curve)  Three different regimes can be clearly distinguished  \\
For $T_{g}\leq$120$^{\circ}$C, the temperature dependence of the saturation magnetization remains nearly the same while increasing growth temperature  The overall magnetic signal vanishing above 200 K is attributed to the nanocolumns whereas the increasing signal below 50 K originates from diluted Mn atoms in the surrounding matrix  The Mn concentration dependence of the saturation magnetization is displayed in figure 8  For the lowest Mn concentration (4 \%), the contribution from diluted Mn atoms is very high and drops sharply for higher Mn concentrations (7 \%, 9 \% and 11 3 \%)  Therefore the fraction of Mn atoms in the diluted matrix decreases with Mn concentration probably because Mn atoms are more and more incorporated in the nanocolumns  In parallel, the Curie temperature of nanocolumns increases with the Mn concentration reaching 170 K for 11 3 \% of Mn  This behavior may be related to different Mn compositions and to the increasing diameter of nanocolumns (from 1 8 nm to 2 8 nm) as discussed in section \ref{structural} 

\begin{figure}[htb]
\center
   \includegraphics[width= 7\linewidth]{ /fig8 eps}
    \caption{Temperature dependence of the saturation magnetization (in $\mu_{B}$/Mn) of Ge$_{1-x}$Mn$_{x}$ films grown at 100$^{\circ}$C plotted for different Mn concentrations: 4 1 \%; 7 \%; 8 9 \% and 11 3 \% }
\label{fig8}
\end{figure}

ZFC-FC measurements show that the nanocolumns are superparamagnetic  The magnetic signal from the diluted Mn atoms in the matrix is too weak to be detected in susceptibility measurements at low temperature  In samples containing 4 \% of Mn, ZFC and FC curves superimpose down to low temperatures  As we do not observe hysteresis loops at low temperature, we believe that at this Mn concentration nanocolumns are  superparamagnetic in the whole temperature range and the blocking temperature cannot be measured  For higher Mn contents, the ZFC curve exhibits a very narrow peak with a maximum at the blocking temperature of 15 K whatever the Mn concentration and growth temperature (see Fig  7b)  Therefore the anisotropy barrier distribution is narrow and assuming that nanocolumns have the same magnetic anisotropy, this is a consequence of the very narrow size distribution of the nanocolumns as observed by TEM  To probe the anisotropy barrier distribution, we have performed ZFC-FC measurements but instead of warming the sample up to 400 K, we stopped at a lower temperature $T_{0}$  

\begin{figure}[htb]
\center
   \includegraphics[width= 6\linewidth]{ /fig9 eps}
\caption{Schematic drawing of the anisotropy barrier distribution n($E_{B}$) of superparamagnetic nanostructures  If magnetic anisotropy does not depend on the particle size, this distribution exactly reflects their magnetic size distribution  In this drawing the blocking temperature ($T_{B}$) corresponds to the distribution maximum  At a given temperature $T_{0}$ such that 25$k_{B}T_{0}$ falls into the anisotropy barrier distribution, the largest nanostructures with an anisotropy energy larger than 25$k_{B}T_{0}$ are blocked whereas the others are superparamagnetic }
\label{fig9}
\end{figure}

If this temperature falls into the anisotropy barrier distribution as depicted in Fig  9, the FC curve deviates from the ZFC curve  Indeed the smallest nanostructures have become superparamagnetic at $T_{0}$ and when decreasing again the temperature, their magnetization freezes along a direction close to the magnetic field and the FC susceptibility is higher than the ZFC susceptibility  Therefore any irreversibility in this procedure points at the presence of superparamagnetic nanostructures  The results are given in Fig  10a  ZFC and FC curves clearly superimpose up to $T_{0}$=250 K thus the nanocolumns are superparamagnetic up to their Curie temperature and no Ge$_{3}$Mn$_{5}$ clusters could be detected  Moreover for low $T_{0}$ values, a peak appears at low temperature in FC curves which evidences strong antiferromagnetic interactions between the nanocolumns \cite{Chan00} 

\begin{figure}[htb]
\center
    \includegraphics[width= 35\linewidth]{ /fig10a eps}
    \includegraphics[width= 63\linewidth]{ /fig10b eps}
\caption{(a) ZFC-FC measurements performed on a Ge$_{0 887}$Mn$_{0 113}$ sample grown at 115$^{\circ}$C  The in-plane applied field is 0 015 T  Magnetization was recorded up to different T$_{0}$ temperatures: 30 K, 50 K, 100 K, 150 K, 200 K and 250 K  Curves are shifted up for more clarity  (b) ZFC-FC curves for in-plane and out-of-plane applied fields (0 015 T) }
\label{fig10}
\end{figure}

In order to derive the magnetic size and anisotropy of the Mn-rich nanocolumns embedded in the Ge matrix, we have fitted the inverse normalized in-plane (resp  out-of-plane) susceptibility: $\chi_{\parallel}^{-1}$ (resp  $\chi_{\perp}^{-1}$)  The corresponding experimental ZFC-FC curves are reported in Fig  10b  Since susceptibility measurements are performed at low field (0 015 T), the matrix magnetic signal remains negligible  In order to normalize susceptibility data, we need to divide the magnetic moment by the saturated magnetic moment recorded at 5 T  However the matrix magnetic signal becomes very strong at 5 T and low temperature so that we need to subtract it from the saturated magnetic moment using a simple Curie function  From Fig  10b, we can conclude that nanocolumns are isotropic  Therefore to fit experimental data we use the following expression well suited for isotropic systems or cubic anisotropy: $\chi_{\parallel}^{-1}= \chi_{\perp}^{-1}\approx 3k_{B}T/M(T)+\mu_{0}H_{eff}(T)$  $k_{B}$ is the Boltzmann constant, $M=M_{s}v$ is the magnetic moment of a single-domain nanostructure (macrospin approximation) where $M_{s}$ is its magnetization and $v$ its volume  The in-plane magnetic field is applied along $[110]$ or $[-110]$ crystal axes  Since the nanostructures Curie temperature does not exceed 170 K, the temperature dependence of the saturation magnetization is also accounted for by writting $M(T)$  Antiferromagnetic interactions between nanostructures are also considered by adding an effective field estimated in the mean field approximation \cite{Fruc02}: $\mu_{0}H_{eff}(T)$ 
The only fitting parameters are the maximum magnetic moment (\textit{i e } at low temperature) per nanostructure: $M$ (in Bohr magnetons $\mu_{B}$) and the maximum interaction field (\textit{i e } at low temperature): $\mu_{0}H_{eff}$ 

\begin{figure}[htb]
\center
   \includegraphics[width= 7\linewidth]{ /fig11 eps}
\caption{Temperature dependence of the inverse in-plane (open circles) and out-of-plane (open squares) normalized susceptibilities of a Ge$_{0 887}$Mn$_{0 113}$ sample grown at 115$^{\circ}$C  Fits were performed assuming isotropic nanostructures or cubic anisotropy  Dashed line is for in-plane susceptibility and solid line for out-of-plane susceptibility }
\label{fig11}
\end{figure}

In Fig  11, the best fits lead to $M\approx$1250 $\mu_{B}$ and $\mu_{0}H_{eff}\approx$102 mT for in-plane susceptibility and $M\approx$1600 $\mu_{B}$ and $\mu_{0}H_{eff}\approx$98 mT for out-of-plane susceptibility  It gives an average magnetic moment of 1425 $\mu_{B}$ per column and an effective interaction field of 100 mT  Using this magnetic moment and its temperature dependence, magnetization curves could be fitted using a Langevin function and $M(H/T)$ curves superimpose for $T<$100 K  However, from the saturated magnetic moment of the columns and their density (35000 $\rm{\mu m}^{-2}$), we find almost 6000 $\mu_{B}$ per column  Therefore, for low growth temperatures, we need to assume that nanocolumns are actually made of almost four independent elongated magnetic nanostructures  The effective field for antiferromagnetic interactions between nanostructures estimated from the susceptibility fits is at least one order of magnitude larger than what is expected from pure magnetostatic coupling  This difference may be due to either an additional antiferromagnetic coupling through the matrix which origin remains unexplained or to the mean field approximation which is no more valid in this strong coupling regime  As for magnetic anisotropy, the nanostructures behave as isotropic magnetic systems or exhibit a cubic magnetic anisotropy  First we can confirm that nanostructures are not amorphous otherwise shape anisotropy would dominate leading to out-of-plane anisotropy  We can also rule out a random distribution of magnetic easy axes since the nanostructures are clearly crystallized in the diamond structure and would exhibit at least a cubic anisotropy (except if the random distribution of Mn atoms within the nanostructures can yield random easy axes)  Since the nanostructures are in strong in-plane compression (their lattice parameter is larger than the matrix one), the cubic symmetry of the diamond structure is broken and magnetic cubic anisotropy is thus unlikely  We rather believe that out-of-plane shape anisotropy is nearly compensated by in-plane magnetoelastic anisotropy due to compression leading to a \textit{pseudo} cubic anisotropy  From the blocking temperature (15 K) and the magnetic volume of the nanostructures , we can derive their magnetic anisotropy constant using $Kv=25k_{B}T_{B}$: K$\approx$10 kJ m$^{-3}$ which is of the same order of magnitude as shape anisotropy 

\begin{figure}[htb]
\center
    \includegraphics[width= 35\linewidth]{ /fig12a eps}
    \includegraphics[width= 63\linewidth]{ /fig12b eps} 
\caption{(a) ZFC-FC measurements performed on a Ge$_{0 93}$Mn$_{0 07}$ sample grown at 122$^{\circ}$C  The in-plane applied field is 0 015 T  Magnetization was recorded up to different T$_{0}$ temperatures: 50 K, 100 K, 150 K, 200 K and 250 K  Curves are shifted up for more clarity  (b) ZFC-FC curves for in-plane and out-of-plane applied fields  (0 015 T) }
\label{fig12}
\end{figure}

For growth temperatures $T_{g}\geq$120$^{\circ}$C and Mn concentrations $\geq$ 7 \%, samples exhibit a magnetic signal above 200 K corresponding to Ge$_{3}$Mn$_{5}$ clusters (see Fig  7a)  As we can see, SQUID measurements are much more sensitive to the presence of Ge$_{3}$Mn$_{5}$ clusters, even at low concentration, than TEM and x-ray diffraction used in section \ref{structural}  We also observe a sharp transition in the ZFC curve (see Fig  7b, Fig  12a and 12b): the peak becomes very large and is shifted towards high blocking temperatures (the signal is maximum at $T=$23 K)  This can be easily understood as a magnetic percolation of the four independent nanostructures obtained at low growth temperatures into a single magnetic nanocolumn  Therefore the magnetic volume increases sharply as well as blocking temperatures  At the same time, the size distribution widens as observed in TEM  In Fig  12a, we have performed ZFC-FC measurements at different $T_{0}$ temperatures  The ZFC-FC irreversibility is observed up to the Curie temperature of $\approx$120 K meaning that a fraction of nanocolumns is ferromagnetic (\textit{i e } $T_{B}\geq T_{C}$) 
In Fig  12b, in-plane and out-of-plane ZFC curves nearly superimpose for $T\leq$150 K due to the isotropic magnetic behavior of the nanocolumns: in-plane magnetoelastic anisotropy is still compensating out-of-plane shape anisotropy  Moreover the magnetic signal above 150 K corresponding to Ge$_{3}$Mn$_{5}$ clusters that start to form in this growth temperature range is strongly anisotropic  This perpendicular anisotropy confirms the epitaxial relation: (0002) Ge$_{3}$Mn$_{5}$ $\parallel$ (002) Ge discussed in Ref \cite{Bihl06}  The magnetic easy axis of the clusters lies along the hexagonal $c$-axis which is perpendicular to the film plane 

\begin{figure}[ht]
\center
   \includegraphics[width= 35\linewidth]{ /fig13a eps}
   \includegraphics[width= 63\linewidth]{ /fig13b eps} 
\caption{(a) ZFC-FC measurements performed on a Ge$_{0 887}$Mn$_{0 113}$ sample grown at 145$^{\circ}$C  The in-plane applied field is 0 015 T  Magnetization was recorded up to different T$_{0}$ temperatures: 50 K, 100 K, 150 K, 200 K, 250 K and 300 K  Curves are shifted up for more clarity  (b) ZFC-FC curves for in-plane and out-of-plane applied fields (0 015 T) }
\label{fig13}
\end{figure}

For growth temperatures $T_{g}\geq$145$^{\circ}$C the cluster magnetic signal dominates (Fig  13b)  Superparamagnetic nanostructures are investigated performing ZFC-FC measurements at different $T_{0}$ temperatures (Fig  13a)  The first ZFC peak at low temperature \textit{i e } $\leq$ 150 K is attributed to  low-$T_{C}$ nanocolumns ($T_{C}\approx$130 K)  This peak is wider than for lower growth temperatures and its maximum is further shifted up to 30 K  These results are in agreement with TEM observations: increasing $T_{g}$ leads to larger nanocolumns (\textit{i e } higher blocking temperatures) and wider size distributions  ZFC-FC irreversibility is observed up to the Curie temperature due to the presence of ferromagnetic columns  The second peak above 180 K in the ZFC curve is attributed to Ge$_{3}$Mn$_{5}$ clusters and the corresponding ZFC-FC irreversibility persisting up to 300 K means that some clusters are ferromagnetic  We clearly evidence the out-of-plane anisotropy of Ge$_{3}$Mn$_{5}$ clusters and the isotropic magnetic behavior of nanocolumns (Fig  13b)  In this growth temperature range, we have also investigated the Mn concentration dependence of magnetic properties  

\begin{figure}[ht]
\center
    \includegraphics[width= 49\linewidth]{ /fig14a eps}
    \includegraphics[width= 49\linewidth]{ /fig14b eps} 
\caption{Temperature dependence of the saturation magnetization (in $\mu_{B}$/Mn) of Ge$_{1-x}$Mn$_{x}$ films grown at 150$^{\circ}$C plotted for different Mn concentrations: 2 3 \%; 4 \%; 7 \%; 9 \%; 11 3 \%  (b) ZFC-FC measurements performed on Ge$_{1-x}$Mn$_{x}$ films grown at 150$^{\circ}$C  The in-plane applied field is 0 025 T for 2 3 \% and 4 \% and 0 015 T for 8 \% and 11 3 \%  }
\label{fig14}
\end{figure}

In Fig  14a, for low Mn concentrations (2 3 \% and 4 \%) the contribution from diluted Mn atoms in the germanium matrix to the saturation magnetization is very high and nearly vanishes for higher Mn concentrations (7 \%, 9 \% and 13 \%) as observed for low growth temperatures  Above 7 \%, the magnetic signal mainly comes from nanocolumns and Ge$_{3}$Mn$_{5}$ clusters  We can derive more information from ZFC-FC measurements (Fig  14b)  Indeed, for 2 3 \% of Mn, ZFC and FC curves nearly superimpose down to low temperature meaning that nanocolumns are superparamagnetic in the whole temperature range  Moreover the weak irreversibility arising at 300 K  means that some Ge$_{3}$Mn$_{5}$ clusters have already formed in the samples even at very low Mn concentrations  For 4 \% of Mn, we can observe a peak with a maximum at the blocking temperature (12 K) in the ZFC curve  We can also derive the Curie temperature of nanocolumns: $\approx$45 K  The irresversibility arising at 300 K still comes from Ge$_{3}$Mn$_{5}$ clusters  Increasing the Mn concentration above 7 \% leads to: higher blocking temperatures (20 K and 30 K) due to larger nanocolumns and wider ZFC peaks due to wider size distributions in agreement with TEM observations (see Fig  3a)  Curie temperatures also increase (110 K and 130 K) as well as the contribution from Ge$_{3}$Mn$_{5}$ clusters \\
Finally when increasing $T_{g}$ above 160$^{\circ}$C, the nanocolumns magnetic signal vanishes and only Ge$_{3}$Mn$_{5}$ clusters and diluted Mn atoms coexist  The overall magnetic signal becomes comparable to the one measured on annealed samples in which only Ge$_{3}$Mn$_{5}$ clusters are observed by TEM (see Fig  7a) \\
The magnetic properties of high-$T_{C}$ nanocolumns obtained for $T_{g}$ close to 130$^{\circ}$C are discussed in detail in Ref \cite{Jame06} \\
In conclusion, at low growth temperatures ($T_{g}\leq$120$^{\circ}$C), nanocolumns are made of almost 4 independent elongated magnetic nanostructures  For $T_{g}\geq$120$^{\circ}$C, these independent nanostructures percolate into a single nanocolumn sharply leading to higher blocking temperatures  Increasing $T_{g}$ leads to larger columns with a wider size distribution as evidenced by ZFC-FC measurements and given by TEM observations  In parallel, some Ge$_{3}$Mn$_{5}$ clusters start to form and their contribution increases when increasing $T_{g}$  Results on magnetic anisotropy seems counter-intuitive  Indeed Ge$_{3}$Mn$_{5}$ clusters exhibit strong out-of-plane anisotropy whereas nanocolumns which are highly elongated magnetic structures are almost isotropic  This effect is probably due to compensating in-plane magnetoelastic coupling (due to the columns compression) and out-of-plane shape anisotropy  

\section{Conclusion}

In this paper, we have investigated the structural and magnetic properties of thin Ge$_{1-x}$Mn$_{x}$ films grown by low temperature molecular beam epitaxy  A wide range of growth temperatures and Mn concentrations have been explored  All the samples contain Mn-rich nanocolumns as a consequence of 2D-spinodal decomposition  However their size, crystalline structure and magnetic properties depend on growth temperature and Mn concentration  For low growth temperatures, nanocolumns are very small (their diameter ranges between 1 8 nm for 1 3 \% of Mn and 2 8 nm for 11 3 \% of Mn), their Curie temperature is rather low ($<$ 170 K) and they behave as almost four uncorrelated superparamagnetic nanostructures  Increasing Mn concentration leads to higher columns densities while diameters remain nearly unchanged  For higher growth temperatures, the nanocolumns mean diameter increases and their size distribution widens  Moreover the 4 independent magnetic nanostructures percolate into a single magnetic nanocolumn  Some columns are ferromagnetic even if Curie temperatures remain quite low  In this regime, increasing Mn concentration leads to larger columns while their density remains nearly the same  In parallel, Ge$_{3}$Mn$_{5}$ nanoclusters start to form in the film with their $c$-axis perpendicular to the film plane  In both temperature regimes, the Mn incorporation mechanism in the nanocolumns and/or in the matrix changes above 5 \% of Mn and nanocolumns exhibit an isotropic magnetic behaviour due to the competing effects of out-of-plane shape anisotropy and in-plane magnetoelastic coupling  Finally for a narrow range of growth temperatures around 130$^{\circ}$C, nanocolumns exhibit Curie temperatures higher than 400 K  Our goal is now to investigate the crystalline structure inside the nanocolumns, in particular the position of Mn atoms in the distorted diamond structure, which is essential to understand magnetic and future transport properties in Ge$_{1-x}$Mn$_{x}$ films 

\section{Aknowledgements}
The authors would like to thank Dr  F  Rieutord for grazing incidence x-ray diffraction measurements performed on the GMT station of BM32 beamline at the European Synchrotron Radiation Facility 





ground_truth: It uses a content-recognition module or algorithm.

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 potential applications of ferromagnetic semiconductors?

Read the following text and answer briefly.

Context 4: 32% Asian, 0 06% Pacific Islander, 0 79% from other races, and 1 16% from two or more races   1 94% of the population were Hispanic or Latino of any race  37 1% were of German, 12 9% Swedish, 12 1% American, 6 7% English and 6 3% Irish ancestry according to Census 2000 

There were 11,205 households, out of which 33 00% had children under the age of 18 living with them, 62 50% were married couples living together, 6 00% had a female householder with no husband present, and 28 90% were non-families  25 50% of all households were made up of individuals, and 11 80% had someone living alone who was 65 years of age or older   The average household size was 2 49 and the average family size was 2 99 

In the county, the population was spread out, with 25 40% under the age of 18, 10 30% from 18 to 24, 25 20% from 25 to 44, 21 80% from 45 to 64, and 17 30% who were 65 years of age or older   The median age was 38 years  For every 100 females there were 95 90 males   For every 100 females age 18 and over, there were 92 90 males 

The median income for a household in the county was $41,138, and the median income for a family was $48,243  Males had a median income of $33,530 versus $21,175 for females  The per capita income for the county was $18,921   About 4 20% of families and 6 60% of the population were below the poverty line, including 5 20% of those under age 18 and 8 10% of those age 65 or over 

Government

Presidential elections
McPherson county is often carried by Republican candidates  The last time a Democratic candidate has carried this county was in 1964 by Lyndon B  Johnson 

Laws
Following amendment to the Kansas Constitution in 1986, the county remained a prohibition, or "dry", county until 1996, when voters approved the sale of alcoholic liquor by the individual drink with a 30 percent food sales requirement 

Education

Colleges
 McPherson College in McPherson
 Bethany College in Lindsborg
 Central Christian College in McPherson

Unified school districts
 Smoky Valley USD 400
 McPherson USD 418
 Canton-Galva USD 419
 Moundridge USD 423
 Inman USD 448

School district office in neighboring county
 Goessel USD 411
 Little River-Windom USD 444

Museums
 Birger Sandzén Memorial Gallery in Lindsborg
 McCormick-Deering Days Museum in Inman
 McPherson Museum in McPherson
 Lindsborg Old Mill & Swedish Heritage Museum in Lindsborg
 Kansas Motorcycle Museum in Marquette

Communities

Cities

 Canton
 Galva
 Inman
 Lindsborg
 Marquette
 McPherson (county seat) 
 Moundridge
 Windom

Unincorporated communities
† means a Census-Designated Place (CDP) by the United States Census Bureau 
 Conway
 Elyria†
 Groveland
 Johnstown
 New Gottland
 Roxbury†

Ghost towns
 Alta Mills
 Battle Hill
 Christian
 Doles Park
 Elivon
 King City
 Sweadal

Townships
McPherson County is divided into twenty-five townships   The cities of Lindsborg and McPherson are considered governmentally independent and are excluded from the census figures for the townships   In the following table, the population center is the largest city (or cities) included in that township's population total, if it is of a significant size 

See also
 List of people from McPherson County, Kansas
 National Register of Historic Places listings in McPherson County, Kansas
 McPherson Valley Wetlands
 Maxwell Wildlife Refuge

References

Notes

Further reading

 Wheeler, Wayne Leland  "An Analysis of Social Change in a Swedish-Immigrant Community: The Case of Lindsborg, Kansas " (PhD dissertation, University of Missouri-Columbia; ProQuest Dissertations Publishing, 1959  5905657) 

County
 Through the Years: A Pictorial History of McPherson County; McPherson Sentinel' Heritage House Publishing Co; 1992 
 McPherson County First Courthouse Built About 1869 or 1870; Lindsborg News-Record; March 30, 1959 
 Pioneer Life and Lore of McPherson County, Kansas; Edna Nyquist; Democratic-Opinion Press; 1932 
 A History of the Church of the Brethren in Kansas (includes McPherson College history); Elmer LeRoy Craik; McPherson Daily; Republican Press; 397 pages; 1922 
 Portrait and Biographical Record of Dickinson, Saline, McPherson, and Marion Counties, Kansas; Chapman Bros; 614 pages; 1893 
 Standard Atlas of McPherson County, Kansas; Geo  A  Ogle & Co; 82 pages; 1921 
 Plat Book of McPherson County, Kansas; North West Publishing Co; 50 pages; 1903 
 Edwards' Atlas of McPherson County, Kansas; John P  Edwards; 51 pages; 1884 

Trails
 The Story of the Marking of the Santa Fe Trail by the Daughters of the American Revolution in Kansas and the State of Kansas; Almira Cordry; Crane Co; 164 pages; 1915  (Download 4MB PDF eBook)
 The National Old Trails Road To Southern California, Part 1 (LA to KC); Automobile Club Of Southern California; 64 pages; 1916  (Download 6 8MB PDF eBook)

Mennonite Settlements
 Impact of Mennonite settlement on the cultural landscape of Kansas; Brenda Martin; Kansas State University; 1985/1988  
 Mennonite settlement : the relationship between the physical and cultural environment; Susan Movle; University of Utah; 1975/1886 
 Status of Mennonite women in Kansas in their church and home relationships; Eva Harshbarger; Bluffton College; 1925/1945 

External links

County
 
 McPherson County - Directory of Public Officials
Historical
 , from Hatteberg's People'' on KAKE TV news
Maps
 McPherson County Maps: Current, Historic, KDOT
 Kansas Highway Maps: Current, Historic, KDOT
 Kansas Railroad Maps: Current, 1996, 1915, KDOT and Kansas Historical Society

 
Kansas counties
1867 establishments in Kansas




ground_truth: Spin injection into non magnetic semiconductors, or electrical manipulation of carrier induced magnetism in magnetic semiconductors.