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: Does DUO contain more instances per image than COCO?

Read the following text and answer briefly.

Context 1: \section{Introduction}
Underwater robot picking is to use the robot to automatically capture sea creatures like holothurian, echinus, scallop, or starfish in an open-sea farm where underwater object detection is the key technology for locating creatures  Until now, the datasets used in this community are released by the Underwater Robot Professional Contest (URPC$\protect\footnote{Underwater Robot Professional Contest: {\bf http://en cnurpc org} }$) beginning from 2017, in which URPC2017 and URPC2018 are most often used for research  Unfortunately, as the information listed in Table \ref{Info}, URPC series datasets do not provide the annotation file of the test set and cannot be downloaded after the contest  
Therefore, researchers \cite{2020arXiv200511552C,2019arXiv191103029L} first have to divide the training data into two subsets, including a new subset of training data and a new subset of testing data, and then train their proposed method and other \emph{SOTA} methods  On the one hand, training other methods results in a significant increase in workload  On the other hand, different researchers divide different datasets in different ways, 
\begin{table}[t]
\renewcommand\tabcolsep{3 5pt}
\caption{Information about all the collected datasets  * denotes the test set's annotations are not available  \emph{3} in Class means three types of creatures are labeled, \emph{i e ,} holothurian, echinus, and scallop  \emph{4} means four types of creatures are labeled (starfish added)  Retention represents the proportion of images that retain after similar images have been removed }
\centering 
\begin{tabular}{|l|c|c|c|c|c|}
\hline
Dataset&Train&Test&Class&Retention&Year \\ 
\hline 
URPC2017&17,655&985*&3&15\%&2017 \\
\hline
URPC2018&2,901&800*&4&99\%&2018 \\
\hline
URPC2019&4,757&1,029*&4&86\%&2019 \\
\hline
URPC2020$_{ZJ}$&5,543&2,000*&4&82\%&2020 \\
\hline
URPC2020$_{DL}$&6,575&2,400*&4&80\%&2020 \\
\hline
UDD&1,827&400&3&84\%&2020 \\
\hline 

\end{tabular}
\label{Info}
\end{table}
\begin{figure*}[htbp]
\begin{center}
\includegraphics[width=1\linewidth]{example pdf}
\end{center}
   \caption{Examples in DUO, which show a variety of scenarios in underwater environments }
\label{exam}
\end{figure*}
causing there is no unified benchmark to compare the performance of different algorithms 
In terms of the content of the dataset images, there are a large number of similar or duplicate images in the URPC datasets  URPC2017 only retains 15\% images after removing similar images compared to other datasets  Thus the detector trained on URPC2017 is easy to overfit and cannot reflect the real performance 
For other URPC datasets, the latter also includes images from the former, \emph{e g }, URPC2019 adds 2,000 new images compared to URPC2018; compared with URPC2019, URPC2020$_{ZJ}$ adds 800 new images  The URPC2020$_{DL}$ adds 1,000 new images compared to the URPC2020$_{ZJ}$  It is worth mentioning that the annotation of all datasets is incomplete; some datasets lack the starfish labels and it is easy to find error or missing labels  \cite{DBLP:conf/iclr/ZhangBHRV17} pointed out that although the CNN model has a strong fitting ability for any dataset, the existence of dirty data will significantly weaken its robustness 
Therefore, a reasonable dataset (containing a small number of similar images as well as an accurate annotation) and a corresponding recognized benchmark are urgently needed to promote community development 


To address these issues, we introduce a dataset called Detecting Underwater Objects (DUO) by collecting and re-annotating all the available underwater datasets  It contains 7,782 underwater images after deleting overly similar images and has a more accurate annotation with four types of classes (\emph{i e ,} holothurian, echinus, scallop, and starfish)  
Besides, based on the MMDetection$\protect\footnote{MMDetection is an open source object detection toolbox based on PyTorch  {\bf https://github com/open-mmlab/mmdetection}}$ \cite{chen2019mmdetection} framework, we also provide a \emph{SOTA} detector benchmark containing efficiency and accuracy indicators, providing a reference for both academic research and industrial applications  It is worth noting that JETSON AGX XAVIER$\protect\footnote{JETSON AGX XAVIER is an embedded development board produced by NVIDIA which could be deployed in an underwater robot  Please refer {\bf https://developer nvidia com/embedded/jetson-agx-xavier-developer-kit} for more information }$ was used to assess all the detectors in the efficiency test in order to simulate robot-embedded environment  DUO will be released in https://github com/chongweiliu soon 

In summary, the contributions of this paper can be listed as follows 

  $\bullet$ By collecting and re-annotating all relevant datasets, we introduce a dataset called DUO with more reasonable annotations as well as a variety of underwater scenes 

  $\bullet$ We provide a corresponding benchmark of \emph{SOTA} detectors on DUO including efficiency and accuracy indicators which could be a reference for both academic research and industrial applications  


\pagestyle{empty}
\section{Background}
In the year of 2017, underwater object detection for open-sea farming is first proposed in the target recognition track of Underwater Robot Picking Contest 2017$\protect\footnote{From 2020, the name has been changed into Underwater Robot Professional Contest which is also short for URPC }$ (URPC2017) which aims to promote the development of theory, technology, and industry of the underwater agile robot and fill the blank of the grabbing task of the underwater agile robot  The competition sets up a target recognition track, a fixed-point grasping track, and an autonomous grasping track  The target recognition track concentrates on finding the {\bf high accuracy and efficiency} algorithm which could be used in an underwater robot for automatically grasping 

The datasets we used to generate the DUO are listed below  The detailed information has been shown in Table \ref{Info} 

  {\bf URPC2017}: It contains 17,655 images for training and 985 images for testing and the resolution of all the images is 720$\times$405  All the images are taken from 6 videos at an interval of 10 frames  However, all the videos were filmed in an artificial simulated environment and pictures from the same video look almost identical  
  
   {\bf URPC2018}: It contains 2,901 images for training and 800 images for testing and the resolutions of the images are 586$\times$480, 704$\times$576, 720$\times$405, and 1,920$\times$1,080  The test set's annotations are not available  Besides, some images were also collected from an artificial underwater environment 
  
  {\bf URPC2019}: It contains 4,757 images for training and 1029 images for testing and the highest resolution of the images is 3,840$\times$2,160 captured by a GOPro camera  The test set's annotations are also not available and it contains images from the former contests 
  
  {\bf URPC2020$_{ZJ}$}: From 2020, the URPC will be held twice a year  It was held first in Zhanjiang, China, in April and then in Dalian, China, in August  URPC2020$_{ZJ}$ means the dataset released in the first URPC2020 and URPC2020$_{DL}$ means the dataset released in the second URPC2020  This dataset contains 5,543 images for training and 2,000 images for testing and the highest resolution of the images is 3,840$\times$2,160  The test set's annotations are also not available 
  
  {\bf URPC2020$_{DL}$}: This dataset contains 6,575 images for training and 2,400 images for testing and the highest resolution of the images is 3,840$\times$2,160  The test set's annotations are also not available 
  
  {\bf UDD \cite{2020arXiv200301446W}}: This dataset contains 1,827 images for training and 400 images for testing and the highest resolution of the images is 3,840$\times$2,160  All the images are captured by a diver and a robot in a real open-sea farm 

\begin{figure}[t]
\begin{center}
\includegraphics[width=1\linewidth]{pie pdf}
\end{center}
   \caption{The proportion distribution of the objects in DUO }
\label{pie}
\end{figure}



\begin{figure*}
  \centering
  \subfigure[]{\includegraphics[width=3 45in]{imagesize pdf}}
  \subfigure[]{\includegraphics[width=3 45in]{numInstance pdf}}
  \caption{(a) The distribution of instance sizes for DUO; (b) The number of categories per image }
  \label{sum}
\end{figure*}
\section{Proposed Dataset}

\subsection{Image Deduplicating}
As we explained in Section 1, there are a large number of similar or repeated images in the series of URPC datasets  Therefore, it is important to delete duplicate or overly similar images and keep a variety of underwater scenarios when we merge these datasets together  Here we employ the Perceptual Hash algorithm (PHash) to remove those images  PHash has the special property that the hash value is dependent on the image content, and it remains approximately the same if the content is not significantly modified  Thus we can easily distinguish different scenarios and delete duplicate images within one scenario  

After deduplicating, we obtain 7,782 images (6,671 images for training; 1,111 for testing)  The retention rate of the new dataset is 95\%, which means that there are only a few similar images in the new dataset  Figure \ref{exam} shows that our dataset also retains various underwater scenes 

\subsection{Image Re-annotation}
Due to the small size of objects and the blur underwater environment, there are always missing or wrong labels in the existing annotation files  In addition, some test sets' annotation files are not available and some datasets do not have the starfish annotation  In order to address these issues, we follow the next process which combines a CNN model and manual annotation to re-annotate these images  Specifically, we first train a detector (\emph{i e ,} GFL \cite{li2020generalized}) with the originally labeled images  After that, the trained detector predicts all the 7,782 images  We treat the prediction as the groundtruth and use it to train the GFL again  We get the final GFL prediction called {\bf the coarse annotation}  Next, we use manual correction to get the final annotation called {\bf the fine annotation}  Notably, we adopt the COCO \cite{Belongie2014} annotation form as the final format 
\subsection{Dataset Statistics}
{\bf The proportion of classes}: The total number of objects is 74,515  Holothurian, echinus, scallop, and starfish are 7,887, 50,156, 1,924, and 14,548, respectively  Figure \ref{pie} shows the proportion of each creatures where echinus accounts for 67 3\% of the total  The whole data distribution shows an obvious long-tail distribution because the different economic benefits of different seafoods determine the different breed quantities 

{\bf The distribution of instance sizes}: Figure \ref{sum}(a) shows an instance size distribution of DUO  \emph{Percent of image size} represents the ratio of object area to image area, and \emph{Percent of instance} represents the ratio of the corresponding number of objects to the total number of objects  Because of these small creatures and high-resolution images, the vast majority of objects occupy 0 3\% to 1 5\% of the image area 

{\bf The instance number per image}: Figure \ref{sum}(b) illustrates the number of categories per image for DUO  \emph{Number of instances} represents the number of objects one image has, and \emph{ Percentage of images} represents the ratio of the corresponding number of images to the total number of images  Most images contain between 5 and 15 instances, with an average of 9 57 instances per image 

{\bf Summary}:
In general, smaller objects are harder to detect  For PASCAL VOC \cite{Everingham2007The} or COCO \cite{Belongie2014}, roughly 50\% of all objects occupy no more than 10\% of the image itself, and others evenly occupy from 10\% to 100\%  
In the aspect of instances number per image, COCO contains 7 7 instances per image and VOC contains 3  In comparison, DUO has 9 57 instances per image and most instances less than 1 5\% of the image size 
Therefore, DUO contains almost exclusively massive small instances and has the long-tail distribution at the same time, which means it is promising to design a detector to deal with massive small objects and stay high efficiency at the same time for underwater robot picking 

\section{Benchmark}
Because the aim of underwater object detection for robot picking is to find {\bf the high accuracy and efficiency} algorithm, we consider both the accuracy and efficiency evaluations in the benchmark as shown in Table \ref{ben} 

\subsection{Evaluation Metrics}
Here we adopt the standard COCO metrics (mean average precision, \emph{i e ,} mAP) for the accuracy evaluation and also provide the mAP of each class due to the long-tail distribution 

{\bf AP} -- mAP at IoU=0 50:0 05:0 95 

{\bf AP$_{50}$} --  mAP at IoU=0 50 

{\bf AP$_{75}$} --  mAP at IoU=0 75  

{\bf AP$_{S}$} --   {\bf AP} for small objects of area smaller than 32$^{2}$ 

{\bf AP$_{M}$} --   {\bf AP} for objects of area between 32$^{2}$ and 96$^{2}$ 

{\bf AP$_{L}$} --   {\bf AP} for large objects of area bigger than 96$^{2}$ 

{\bf AP$_{Ho}$} --  {\bf AP} in holothurian 

{\bf AP$_{Ec}$} --  {\bf AP} in echinus 

{\bf AP$_{Sc}$} --  {\bf AP} in scallop 

{\bf AP$_{St}$} --  {\bf AP} in starfish 


For the efficiency evaluation, we provide three metrics:

{\bf Param } --  The parameters of a detector 

{\bf FLOPs} --  Floating-point operations per second 

{\bf FPS} --  Frames per second 

Notably, {\bf FLOPs} is calculated under the 512$\times$512 input image size and {\bf FPS} is tested on a JETSON AGX XAVIER under MODE$\_$30W$\_$ALL  

\subsection{Standard Training Configuration}
We follow a widely used open-source toolbox, \emph{i e ,} MMDetection (V2 5 0) to produce up our benchmark  During the training, the standard configurations are as follows:

  $\bullet$ We initialize the backbone models (\emph{e g ,} ResNet50) with pre-trained parameters on ImageNet \cite{Deng2009ImageNet} 

  $\bullet$ We resize each image into 512 $\times$ 512 pixels both in training and testing  Each image is flipped horizontally with 0 5 probability during training 

  $\bullet$ We normalize RGB channels by subtracting 123 675, 116 28, 103 53 and dividing by 58 395, 57 12, 57 375, respectively 

  $\bullet$ SGD method is adopted to optimize the model  The initial learning rate is set to be 0 005 in a single GTX 1080Ti with batchsize 4 and is decreased by 0 1 at the 8th and 11th epoch, respectively  WarmUp \cite{2019arXiv190307071L} is also employed in the first 500 iterations  Totally there are 12 training epochs 

  $\bullet$ Testing time augmentation (\emph{i e ,} flipping test or multi-scale testing) is not employed 



\subsection{Benchmark Analysis}
Table \ref{ben} shows the benchmark for the \emph{SOTA} methods  Multi- and one- stage detectors with three kinds of backbones (\emph{i e ,} ResNet18, 50, 101) give a comprehensive assessment on DUO  We also deploy all the methods to AGX to assess efficiency 

In general, the multi-stage (Cascade R-CNN) detectors have high accuracy and low efficiency, while the one-stage (RetinaNet) detectors have low accuracy and high efficiency  However, due to recent studies \cite{zhang2019bridging} on the allocation of more reasonable positive and negative samples in training, one-stage detectors (ATSS or GFL) can achieve both high accuracy and high efficiency 

\begin{table*}[htbp]
\renewcommand\tabcolsep{3 0pt}

\begin{center}
\caption{Benchmark of \emph{SOTA} detectors (single-model and single-scale results) on DUO  FPS is measured on the same machine with a JETSON AGX XAVIER under the same MMDetection framework, using a batch size of 1 whenever possible  R: ResNet } 
\label{ben}
\begin{tabular}{|l|l|c|c|c|ccc|ccc|cccc|}
\hline
Method&Backbone&Param &FLOPs&FPS&AP&AP$_{50}$&AP$_{75}$&AP$_{S}$&AP$_{M}$&AP$_{L}$&AP$_{Ho}$&AP$_{Ec}$&AP$_{Sc}$&AP$_{St}$ \\ 
\hline 
\emph{multi-stage:} &&&&&&&&&&&&&& \\

\multirow{3}{*}{Faster R-CNN \cite{Ren2015Faster}}
&R-18&28 14M&49 75G&5 7&50 1&72 6&57 8&42 9&51 9&48 7&49 1&60 1&31 6&59 7\\
&R-50&41 14M&63 26G&4 7&54 8&75 9&63 1&53 0&56 2&53 8&55 5&62 4&38 7&62 5\\
&R-101&60 13M&82 74G&3 7&53 8&75 4&61 6&39 0&55 2&52 8&54 3&62 0&38 5&60 4\\
\hline

\multirow{3}{*}{Cascade R-CNN \cite{Cai_2019}}
&R-18&55 93M&77 54G&3 4&52 7&73 4&60 3&\bf 49 0&54 7&50 9&51 4&62 3&34 9&62 3\\
&R-50&68 94M&91 06G&3 0&55 6&75 5&63 8&44 9&57 4&54 4&56 8&63 6&38 7&63 5\\
&R-101&87 93M&110 53G&2 6&56 0&76 1&63 6&51 2&57 5&54 7&56 2&63 9&41 3&62 6\\
\hline

\multirow{3}{*}{Grid R-CNN \cite{lu2019grid}}
&R-18&51 24M&163 15G&3 9&51 9&72 1&59 2&40 4&54 2&50 1&50 7&61 8&33 3&61 9\\
&R-50&64 24M&176 67G&3 4&55 9&75 8&64 3&40 9&57 5&54 8&56 7&62 9&39 5&64 4\\
&R-101&83 24M&196 14G&2 8&55 6&75 6&62 9&45 6&57 1&54 5&55 5&62 9&41 0&62 9\\
\hline

\multirow{3}{*}{RepPoints \cite{yang2019reppoints}}
&R-18&20 11M&\bf 35 60G&5 6&51 7&76 9&57 8&43 8&54 0&49 7&50 8&63 3&33 6&59 2\\
&R-50&36 60M&48 54G&4 8&56 0&80 2&63 1&40 8&58 5&53 7&56 7&65 7&39 3&62 3\\
&R-101&55 60M&68 02G&3 8&55 4&79 0&62 6&42 2&57 3&53 9&56 0&65 8&39 0&60 9\\
\hline 
\hline 
\emph{one-stage:} &&&&&&&&&&&&&& \\
\multirow{3}{*}{RetinaNet \cite{Lin2017Focal}}
&R-18&19 68M&39 68G&7 1&44 7&66 3&50 7&29 3&47 6&42 5&46 9&54 2&23 9&53 8\\
&R-50&36 17M&52 62G&5 9&49 3&70 3&55 4&36 5&51 9&47 6&54 4&56 6&27 8&58 3\\
&R-101&55 16M&72 10G&4 5&50 4&71 7&57 3&34 6&52 8&49 0&54 6&57 0&33 7&56 3\\
\hline 

\multirow{3}{*}{FreeAnchor \cite{2019arXiv190902466Z}}
&R-18&19 68M&39 68G&6 8&49 0&71 9&55 3&38 6&51 7&46 7&47 2&62 8&28 6&57 6\\
&R-50&36 17M&52 62G&5 8&54 4&76 6&62 5&38 1&55 7&53 4&55 3&65 2&35 3&61 8\\
&R-101&55 16M&72 10G&4 4&54 6&76 9&62 9&36 5&56 5&52 9&54 0&65 1&38 4&60 7\\
\hline 

\multirow{3}{*}{FoveaBox \cite{DBLP:journals/corr/abs-1904-03797}}
&R-18&21 20M&44 75G&6 7&51 6&74 9&57 4&40 0&53 6&49 8&51 0&61 9&34 6&59 1\\
&R-50&37 69M&57 69G&5 5&55 3&77 8&62 3&44 7&57 4&53 4&57 9&64 2&36 4&62 8\\
&R-101&56 68M&77 16G&4 2&54 7&77 3&62 3&37 7&57 1&52 4&55 3&63 6&38 9&60 8\\
\hline 

\multirow{3}{*}{PAA \cite{2020arXiv200708103K}}
&R-18&\bf 18 94M&38 84G&3 0&52 6&75 3&58 8&41 3&55 1&50 2&49 9&64 6&35 6&60 5\\
&R-50&31 89M&51 55G&2 9&56 8&79 0&63 8&38 9&58 9&54 9&56 5&66 9&39 9&64 0\\
&R-101&50 89M&71 03G&2 4&56 5&78 5&63 7&40 9&58 7&54 5&55 8&66 5&42 0&61 6\\
\hline 

\multirow{3}{*}{FSAF \cite{zhu2019feature}}
&R-18&19 53M&38 88G&\bf 7 4&49 6&74 3&55 1&43 4&51 8&47 5&45 5&63 5&30 3&58 9\\
&R-50&36 02M&51 82G&6 0&54 9&79 3&62 1&46 2&56 7&53 3&53 7&66 4&36 8&62 5\\
&R-101&55 01M&55 01G&4 5&54 6&78 7&61 9&46 0&57 1&52 2&53 0&66 3&38 2&61 1\\
\hline 

\multirow{3}{*}{FCOS \cite{DBLP:journals/corr/abs-1904-01355}}
&R-18&\bf 18 94M&38 84G&6 5&48 4&72 8&53 7&30 7&50 9&46 3&46 5&61 5&29 1&56 6\\
&R-50&31 84M&50 34G&5 4&53 0&77 1&59 9&39 7&55 6&50 5&52 3&64 5&35 2&60 0\\
&R-101&50 78M&69 81G&4 2&53 2&77 3&60 1&43 4&55 4&51 2&51 7&64 1&38 5&58 5\\
\hline 

\multirow{3}{*}{ATSS \cite{zhang2019bridging}}
&R-18&\bf 18 94M&38 84G&6 0&54 0&76 5&60 9&44 1&56 6&51 4&52 6&65 5&35 8&61 9\\
&R-50&31 89M&51 55G&5 2&58 2&\bf 80 1&66 5&43 9&60 6&55 9&\bf 58 6&67 6&41 8&64 6\\
&R-101&50 89M&71 03G&3 8&57 6&79 4&65 3&46 5&60 3&55 0&57 7&67 2&42 6&62 9\\
\hline 

\multirow{3}{*}{GFL \cite{li2020generalized}}
&R-18&19 09M&39 63G&6 3&54 4&75 5&61 9&35 0&57 1&51 8&51 8&66 9&36 5&62 5\\
&R-50&32 04M&52 35G&5 5&\bf 58 6&79 3&\bf 66 7&46 5&\bf 61 6&55 6&\bf 58 6&\bf 69 1&41 3&\bf 65 3\\
&R-101&51 03M&71 82G&4 1&58 3&79 3&65 5&45 1&60 5&\bf 56 3&57 0&\bf 69 1&\bf 43 0&64 0\\


\hline 
\end{tabular}
\end{center}
\end{table*}
Therefore, in terms of accuracy, the accuracy difference between the multi- and the one- stage methods in AP is not obvious, and the AP$_{S}$ of different methods is always the lowest among the three size AP  For class AP, AP$_{Sc}$ lags significantly behind the other three classes because it has the smallest number of instances  In terms of efficiency, large parameters and FLOPs result in low FPS on AGX, with a maximum FPS of 7 4, which is hardly deployable on underwater robot  Finally, we also found that ResNet101 was not significantly improved over ResNet50, which means that a very deep network may not be useful for detecting small creatures in underwater scenarios  

Consequently, the design of high accuracy and high efficiency detector is still the main direction in this field and there is still large space to improve the performance 
In order to achieve this goal, a shallow backbone with strong multi-scale feature fusion ability can be proposed to extract the discriminant features of small scale aquatic organisms; a specially designed training strategy may overcome the DUO's long-tail distribution, such as a more reasonable positive/negative label sampling mechanism or a class-balanced image allocation strategy within a training batch 

\section{Conclusion}
In this paper, we introduce a dataset (DUO) and a corresponding benchmark to fill in the gaps in the community  DUO contains a variety of underwater scenes and more reasonable annotations  Benchmark includes efficiency and accuracy indicators to conduct a comprehensive evaluation of the \emph{SOTA} decoders  The two contributions could serve as a reference for academic research and industrial applications, as well as promote community development 
Context 2: 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 



ground_truth: Yes, DUO has 9.57 instances per image while COCO contains 7.7.

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

Discussion

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

Appendix

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



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

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 did the London Directory proclaim to contain?

Read the following text and answer briefly.

Context 3: A Brief History of Benjamin Franklin's Residences on Craven Street, London: 1757 - 1775 - Journal of the American Revolution
Benjamin Franklin House, 36 Craven St, London  (Photo by Elliott Brown | Wikimedia Commons)
If one looked into Benjamin Franklin’s time on Craven Street, they might initially believe he lived at 36 Craven Street the entirety of his two stays in London based on the plethora of articles on the internet that say so  If they dug a little deeper they might read that he lived at No  27 Craven Street, previously numbered 7, but now numbered 36; or that he lived exclusively at No  7 Craven Street; or that he lived in multiple residences on Craven Street; or that he moved out of No  36 to another house on Craven Street and then moved back into No  36 the last year of his residence  What is one to believe with all of the conflicting accounts? What does the historical record have to say about Franklin’s time on Craven Street?
Figure 1  Spur Alley 1685  “A map of the parish of St Martins in the Fields, taken from ye last survey, with additions (1685)”  (© The British Library Board, Shelfmark: Maps Crace Port  13 2, Item number: 2)
Before Craven Street existed there was Spur Alley, a narrow passageway sandwiched between the Hungerford Market to the north (now Charing Cross Station) and Scotland Yard and the Northumberland House and Garden to the south  It was flanked on both ends by major thoroughfares, the Strand on the west, connecting Westminster to London by road, and the River Thames on the east, not only connecting the two cities to each other and to Southwark on the south side of the Thames, but connecting the entire metropolis to the rest of the world  Being located in the City of Westminster, Spur Alley had escaped the devastation of the Great Fire of London in 1666 leaving its wooden structures, built in the early part of seventeenth century, intact, but also in dire need of restoration or demolition  “The ratebooks show that during the last thirty years or so of their existence the houses in Spur Alley were in a very bad condition  Few of them were rated at more than a few shillings and many of them were unoccupied ”[1] The landowner, William, 5th Baron Craven, desiring to increase the profitability of his assets, tore down the derelict structures on Spur Alley around 1730 and leased the newly established lots to builders  By 1735, twenty brick houses in the Georgian style had been built on the west side and sixteen on the east side of the way now called Craven Street [2]
Figure 2  Craven Street 1746  (John Rocque London, Westminster and Southwark, First Edition 1746, Motco Enterprises Limited, motco com)
Letters to Franklin during his residence with Mrs  Margaret Stevenson, his landlady on Craven Street, were addressed rather vaguely; “Craven Street/Strand”, “Mrs  Stevensons in Craven Street”, or “Benjamin Franklin Esqr ” are but a few examples  Letters from Franklin referenced “London,” or sometimes “Cravenstreet,” but never included a number  Despite the absence of numbered addresses in Franklin’s correspondence, there was a sense of one’s place in the neighborhood based on entries in the Westminster Rate Books (tax assessments)  The Rate Books did not list house numbers during Franklin’s time there, but they did list the residents of Craven Street in a particular order that became the default numbering system for the street  Number one was associated with the first resident listed under “Craven Street” in the Rate Books and was the northernmost house on the west side of the street  The numbers increased counter-clockwise down the west side and up the east side in accordance with the list of residents  In 1748, the first year of Margaret Stevenson’s (Stevens in the Rate Books for that year) residence on Craven Street, she is listed as the twenty-seventh resident, the second house north of Court Street (later Craven Court, now Craven Passage) on the east side of the street [3]
In 1766, Parliament passed the London Paving and Lighting Act (6 Geo  3 c  26), “An act for the better paving, cleansing, and enlightening, the city of London, and the liberties thereof; and for preventing obstructions and annoyances within the same; and for other purposes therein mentioned ”[4] One of the other purposes therein mentioned was the numbering of houses  With an aim to bring order to the chaotic numbering systems or lack thereof on London streets the Act provided that “… the said commissioners … may also cause every house, shop, or warehouse, in each of the said streets, lanes, squares, yards, courts, alleys, passages, and places, to be marked or numbered, in such manner as they shall judge most proper for distinguishing the same ”[5] This was quite an undertaking that took years to accomplish  It was a decade later before numbered addresses on Craven Street in the City of Westminster appeared in The London Directory (1776)  The London Directory and its competitors were published primarily by booksellers or printers to supplement their income and were highly profitable  To say they were competitive is an understatement  “Some of the most hotly disputed struggles over copyright in the century concerned guidebooks  Many were optimistically emblazoned with a royal license and a notice that the work had been entered at Stationers’ Hall  Various struggles between rival guides intensified as the potential for profits became clear ”[6] The London Directory boldly proclaimed to contain “An ALPHABETICAL LIST OF THE NAMES and PLACES of ABODE of the MERCHANTS and PRINCIPAL TRADERS of the Cities of LONDON and WESTMINSTER, the Borough of SOUTHWARK, and their Environs, with the Number affixed to each House ”[7] Kent’s Directory made a similar proclamation: “An Alphabetical LIST OF THE Names and Places of Abode OF THE DIRECTORS of COMPANIES, Persons in Public Business, MERCHANTS, and other eminent TRADERS in the Cities of London and Westminster, and Borough of Southwark WITH THE NUMBERS as they are affixed to their Houses agreeable to the late Acts of Parliament ”[8] Mrs  Stevenson wasn’t included in the directories because she didn’t meet the criteria of being a merchant or trader, not because she was a woman  Although it is rare to see women listed in the directories, some examples do exist [9] If Mrs  Stevenson had appeared in the directories in 1776 it would not have been on Craven Street as she had moved to Northumberland Court, a stone’s throw away, the previous year [10] A comparison of Craven Street residents whose names and addresses do appear in the directories with the same residents as they appear in the Westminster Rate Books determines if the numbering systems were congruent  For the most part they were  For example, Joseph Bond at No  30, William Rowles at No  31, Samuel Sneyd at No  32, and Jonathan Michie at No  35 in The London Directory coincide with their places of residence in the Westminster Rate Books; however, errors did occur  The 1776 edition of The London Directory lists Brown & Whiteford, wine merchants, at No  9 Craven Street while the Westminster Rate Books list them as the twenty-ninth residents  Obviously, it makes no sense to have Brown & Whiteford at No  9 in The London Directory and their next-door neighbor, Joseph Bond, at No  30  The same error appears in Baldwin’s The New Complete Guide for 1783  The New Complete Guide may have “borrowed” the error from The London Directory  It was not uncommon for the owner of one directory to copy entries from another to save both time and money  Beginning in 1778 and contrary to The London Directory, Kent’s Directory faithfully followed the numbering system of the Westminster Rate Books in all of its editions and listed Brown & Whiteford at No  29 as did Bailey’s Northern Directory in 1781  Perhaps realizing their error, The London Directory changed their listing of Brown & Whiteford from No  9 to No  29 in their 1783 edition and maintained that listing thereafter 
Sometime prior to 1792, the embankment on the Thames at the south end of Craven Street had been sufficiently extended allowing for the construction of ten new houses below the original houses: “ … four houses, Nos  21–24, were built on the west side, and six houses, Nos  25–30, on the east side of the way ”[11] In a note in the same report, the new numbering system is explained  “The houses in the street, which had previously been numbered consecutively down the west side and up the east side, were then renumbered on the same system to include the additional houses ”[12] Because the new houses (21-24) on the west side were built below the existing houses (1-20), houses 1-20 retained their original numbering 
Figure 4  Craven Street 1799  (Richard Horwood’s Map of London, Westminster and the Borough of Southwark 1799, Motco Enterprises Limited, motco com)
One would think that the numbers of the sixteen original houses on the east side, Nos  21 – 36, would simply increase by ten with the addition of the ten new houses, but such was not the case; they increased by nine  How could that be? The only possible explanation is that No  21 of the original houses was demolished to make way for the construction of the northernmost of the six new houses on the east side (No  30)  Evidence of No  21’s demolition appears in the lease granted to Charles Owen by William, 7th Baron Craven, in 1792, which describes No  22 as: “All that messuage in Craven Street late in the occupation of Francis Deschamps undertaker … being the Southernmost house in the Old Buildings on the East Side of the said Street numbered with the No  22 ”[13] The lease describes No  22 as being the southernmost house in the old buildings on the east side of Craven Street  Clearly the house previously at No  21 did not exist when the lease granted to Charles Owen was written in 1792 as it used to be the southernmost house  It is also worth noting that in 1790, The London Directory listed Jacob Life at No  21 (original numbering)  In 1791-2, it listed him at No  6  With No  21 vacated, it would allow for its demolition and the construction of the tenth new house  By utilizing lot No  21 for the new construction, only nine additional lots were needed to build the ten houses, hence, Margaret Stevenson’s former residence at 27 became 36 (27 + 9) in the renumbering and not 37 
For nearly a century and a half after Franklin departed London for America in March of 1775 the scales were tipped heavily in favor of his residence having been No  7 Craven Street  As early as 1807 in London; Being An Accurate History And Description Of The British Metropolis And Its Neighborhood, Volume 4, one would have read: “In Craven Street is a house, No  7, remarkable for having been the residence of Dr  Benjamin Franklin [14] In 1815, the identical phrase appeared in The Beauties of England and Wales [15] After 23 editions of not mentioning Franklin, his name finally appeared in the 24th edition of The Picture of London in 1826: “The house, No  7, Craven Street, in the Strand, was once the residence of Dr  Benjamin Franklin ”[16] In 1840, Jared Sparks referred to Franklin’s Craven Street residence appearing in London guide books in his voluminous The Works of Benjamin Franklin: “In the London Guide Books, ‘No  7, Craven Street,’ is still indicated as the house in which Dr  Franklin resided ”[17] In 1846, George Gulliver F R S , in his book, The Works of William Hewson, wrote: “She [Polly] had been upon terms of the warmest friendship with Dr  Franklin
Figure 5  No  7 Craven Street with Memorial Tablet  (Photo courtesy of British History Online, and the Survey of London)
since she was eighteen years of age  That eminent philosopher resided with her mother, Mrs  Margaret Stevenson, at No  7, Craven Street, Strand, during the fifteen years of his abode in London ”[18] Guide books mentioning Franklin at No  7 continued to proliferate throughout the century: Handbook for London; Past and Present, Volume I (1849);”[19] Handbook for Modern London (1851);”[20] The Town; Its Memorable Characters and Events (1859);”[21] London and Its Environs (1879) [22] There was an anomaly when London In 1880 Illustrated With Bird’s-Eye Views of the Principal Streets, Sixth Edition (1880) placed Franklin at 27 Craven street [23] The anomaly lasted for six years until his place of residence was changed to No  7 in the revised edition, London  Illustrated by Eighteen Bird’s-Eye Views of the Principal Streets (1886) [24] London Past and Present; Its History, Associations, and Traditions, Volume 1 (1891), copied the 1849 Handbook for London almost word-for-word and included, “The house is on the right from the Strand ”[25] In October of 1867, The Society of Arts in London declared that: “In order to show how rich the metropolis is in the memory of important personages and events, which it would be desirable to mark by means of tablets on houses, the Council have caused an alphabetical list to be prepared, … ”[26] Franklin had been elected a corresponding member to the Society in 1756 and was a popular choice among Council members deciding who they were to memorialize [27] By January of 1870, a tablet honoring him was affixed to the house they believed to have been his residence while in London, No  7 Craven Street in the Strand on the west side of the street [28] A majority of historians writing about Franklin in the nineteenth and early twentieth century placed him at No  7: O  L  Holley, The Life of Benjamin Franklin (1848); E  M  Tomkinson, Benjamin Franklin (1885); John Torrey Morse, Benjamin Franklin (1891); Paul Elmer More, Benjamin Franklin (1900); John S  C  Abbot, Benjamin Franklin (1903); Sydney George Fisher, The True Benjamin Franklin (1903)  A notable exception is D  H  Montgomery’s His Life Written by Himself published in 1896  He has Franklin at No  27 Craven Street  It seems then that depending upon the source, Franklin was thought to have lived at either No  7 or No  27, but not both, the overwhelming majority favoring No  7  As late as 2011, Franklin is still mentioned as living at No  7 [29]
In 1913, No  7 was scheduled to be torn down  An article in the March 1914 edition of The Book News Monthly, describes the situation:
As is well known to informed American pilgrims, it has been possible for all admirers of the famous philosopher and statesman to pay their respects to his memory before that house, No  7 Craven Street, just off the Strand, which was his chief home during his two sojourns in the British capital, but even as these lines are being written the London newspapers are recording that that interesting shrine is soon to be pulled down to make room for a restaurant  It is some mitigation of this misfortune to remember that at the most the Craven Street house was nothing more than a reproduction of the one in which Franklin had his suite of four rooms, for the structure has been rebuilt since Franklin’s time  When, then, some one makes a piteous plea that at least the philosopher’s bedroom shall be preserved, the soothing answer is that the apartment in question is only a replica of that in which the illustrious American enjoyed his well-earned slumbers in 1757-62 and 1764-75  The restaurant-builder, however, with an eye doubtless to possible American patronage, has assured the world that every effort will be made to preserve as much as possible of the entire structure [30]
Concerned with the possible demolition of Franklin’s residence, the Royal Society of Arts (formerly the Society of Arts[31]) initiated an inquiry into the matter [32] The London County Council, having taken over the responsibility of placing memorial tablets on notable houses from the Royal Society, was charged with the investigation  It ultimately fell to Sir George Laurence Gomme, a clerk to the Council, to come up with a response  A few years earlier Sir George had discovered Margaret Stevenson residing at No  27 Craven Street in the Westminster Rate Books  He must have wondered why No  7 on the west side of Craven Street was being celebrated as Franklin’s residence when the evidence clearly showed otherwise 
Sir George and his staff examined the various London directories discussed earlier and came up with a novel explanation for the discrepancy  They concluded that there had been two numbering systems on Craven Street  An anonymous author echoes Sir George’s conclusion about the two numbering systems in an article in The Journal of the Royal Society of Arts:
…an inspection of the directories of that time proves that there were at least two systems of numbering in Craven Street before the erection of the additional houses  According to one of these the numbers started from the top (Strand end) on the west side of the street, and ran down to the bottom to No  20, then crossed over and went back to the Strand along the east side – 21 to 36  According to the other system, the east side of the street was numbered from the bottom upwards, starting at No 1  This was not apparently in general use, but there is evidence that this numbering was at all events occasionally used 
The evidence of these two systems of numbering, and for believing that Mrs  Stevenson’s house was first No  7 under the oldest system, next No  27 under the second system, and finally No  36 under the latest and existing system, is to be found in the various directories and the Westminster rate-books [33]
The “evidence” mentioned above consisted of The London Directory’s listing of Brown & Whiteford at No  9: “The rate-books for 1781 and 1786 show the house next but one to the north of Mrs  Stevenson’s house as in the occupation of Brown and ‘Whiteford,’ while the old directories mention the business of the firm as wine merchants, and give their address as 9, Craven Street – then a little later, down to 1791, as 29, Craven Street  Curiously enough, in the years 1778 to 1780, or 1781, Lowndes gives it as No  9, and Kent as 29 ”[34] Ignoring Kent’s Directory having Brown and Whiteford as 29 and The London Directory (Lowndes) having Brown and Whiteford “a little later” as 29, and knowing that Mrs  Stevenson lived two doors south of them, Sir George concluded that her house must have been numbered 7, even though there is no listing in any of the directories of her residence ever being No  7  He surmised that the No  7 on the west side of Craven Street with the memorial tablet thought to have been Franklin’s residence had simply been confused with number 7 (27) on the east side  Again from The Journal of the Royal Society of Arts:
Taking all the evidence together, there cannot be any doubt whatever that Mrs  Stevenson’s house, in which Franklin lodged, was the house two doors north from Craven Court, first numbered 7, afterwards 27, and finally 36, and consequently that the house in which Franklin lived was that now numbered 36, not the one now numbered 7, on which the tablet is placed [35]
A response to The Royal Society of Arts was issued: “… the London County Council … informed the Society that it had made a mistake and that No  36 Craven street was the building that deserved commemoration ”[36] The Society accepted the Council’s conclusion, and despite assurances of preservation by the restaurant builder, No  7 was torn down the following year 
Sir George’s assertion “that Mrs  Stevenson’s house, in which Franklin lodged, was the house two doors north from Craven Court” was correct, however, his assertion that it was “first numbered 7, afterwards 27”, was not  It was only by association with the errant entry of Brown & Whiteford at No  9 from 1776-1782 in The London Directory that Mrs  Stevenson’s address was conjured to be No  7  The problem with associating her address exclusively with that of Brown & Whiteford at No  9 during those years is that, as previously demonstrated, The London Directory also listed four other Craven Street residents, Bond, Rowles, Sneyd, and Michie, who’s addresses did conform to the numbering system in The Westminster Rate Books  If Brown & Whiteford at No  9 was indicative of a numbering system different from The Westminster Rate Books, Bond, Rowles, Sneyd, and Michie would have been listed as Nos  10, 11, 12, and 15, respectively  So on one hand Sir George was relying on the Westminster Rate Books to establish Mrs  Stevenson at No  27 and on the other hand he was dismissing the Westminster Rate Books to establish her at No  7  Instead of using the anomalous listing of Brown & Whiteford at No  9, he could have just as easily, and more logically, used the Bond et al  listings, or the post-1782 Brown & Whiteford listing in the London Directory at No  29 to establish Mrs  Stevenson at No  27  Even if there had been two numbering systems, his assertion that No  27 was first numbered 7 would still be false  The earliest numbering system was the Westminster Rate Books dating from the early 1730s when the houses were constructed  Brown & Whiteford at No  9 didn’t appear until 46 years later and then only for a brief period 
There is ample evidence in Franklin’s correspondence and in a memoir by Polly Hewson (Mrs  Stevenson’s daughter) that Benjamin and Mrs  Stevenson lived in not one, but two houses on Craven Street  On July 6, 1772, Polly wrote to Benjamin from her house at Broad Street North in London: “My Mother I must tell you went off last friday week, took our little Boy with her and left Mr  Hewson [Polly’s husband, William] the care of her House [27 Craven Street]  The first thing he did was pulling down a part of it in order to turn it to his own purpose, and advantage we hope  This Demolition cannot affect you, who at present are not even a Lodger [Benjamin was traveling at the time], your litterary apartment remains untouch’d, the Door is lock’d …”[37] In a memoir about her husband written after his death Polly writes: “He [William Hewson] began his Lectures Sept  30, 1772, in Craven-street, where he had built a Theatre adjoining a house which he intended for the future residence of his family ”[38] On October 7, 1772, Benjamin wrote to his son William: “I am very well  But we [Mrs  Stevenson and I] are moving to another House in the same street; and I go down tomorrow to Lord LeDespencer’s to [stay a] Week till things are settled ”[39] To his son-in-law, Richard Bache, on the same day he wrote: “We are moving to another House in the [street] leaving this to Mr  Hewson ”[40] Writing to a friend on October 30, 1772 he explained: “I should sooner have answered your Questions but that in the Confusion of my Papers, occasioned by removing to another House, I could not readily find the Memorandums …”[41] On November 4, 1772 Benjamin informed his wife Deborah of the move  “We are removed to a more convenient House in the same street, Mrs  Stevenson having accommodated her Son-in-Law with that we lived in  The Removing has been a troublesome Affair, but is now over ”[42]
An agreement had been struck between the parties  Margaret and Benjamin would move to another house on Craven Street and allow Polly and William to move into No  27, the large yard behind the house being spacious enough to accommodate the anatomy school William wished to build [43] Perhaps the idea was inspired by Margaret’s next-door neighbor at No  26, Dr  John Leake, a man-midwife and founder of the Westminster Lying-in Hospital, who had built a theater adjoining his residence in which he practiced anatomy and taught midwifery [44]
After Margaret and Benjamin vacated No  27, Polly, William, their son William Jr , and William’s younger sister, Dorothy Hewson, took up residence there [45] In the 1773 Westminster Rate Books for Craven Street, Mrs  Stevenson’s (Stephenson in the Rate Books) name has been crossed out and replaced with “William Hewson ”[46] Further proof that the Hewsons had indeed moved into 27 Craven Street has been confirmed by the discovery of human and animal remains buried in the basement of No  36 (formerly No  27 and now the Benjamin Franklin House), a by-product of the dissections that took place at William’s anatomy school [47]
So what house on Craven Street did Mrs  Stevenson and Benjamin move into after vacating No  27? An examination of the Westminster Rate Books for the years 1774 and 1775 reveal them living not at No  7 on the west side of Craven Street as one might expect from the overwhelming consensus of nineteenth century guidebooks and biographies, but surprisingly at No  1 [48] The controversy of No  7 being torn down was all for naught as it had never been Franklin’s residence  Sir George was correct on that point  Unfortunately, No  1 was torn down as well in the early part of the twentieth century  The first time No  1 is mentioned as Franklin’s second residence is in the Survey of London: Volume 18, St Martin-in-The-Fields II: the Strand published by the London County Council in 1937, ironically the same County Council that had declared No  36 as Franklin’s only residence twenty-four years earlier 
From 1748 until 1772 Margaret ‘Stephenson’ occupied this house [No  27 (36)], and it was there that Benjamin Franklin settled after his arrival in London in 1757 as Agent to the General Assembly of Pennsylvania … In October, 1772, Mrs  Stevenson and Franklin removed to No  1, Craven Street (now demolished), and No  36 was for the next two years occupied by William Hewson, surgeon, who had married Mary Stevenson [49]
In the spring of 1774, William Hewson died unexpectedly of septicemia two weeks after cutting himself while dissecting a cadaver  Polly was left to care for their two young sons and was pregnant with a daughter she would give birth to in August of the same year  Is it possible that Margaret and Benjamin moved back into No  27 to assist Polly after the death of her husband as suggested in The Americanization of Benjamin Franklin?[50]
If the Westminster Rate Books are to be believed, the answer is no  For the year 1774, the Rate Books list Margaret Stevenson at No  1 and William Hewson at No  27  For the year 1775, they list Margaret Stevenson at No  1 and Magnus Falkner (Falconer/Falconar) at No  27  Magnus was William’s assistant at the anatomy school and fiancé to William’s sister, Dorothy  On his death bed, William instructed Polly, “let Mr  Falconar be my successor ”[51] Magnus would immediately take over the running of the anatomy school and continue William’s unfinished research  Four months later, he and Dorothy would marry [52] Essentially only two things changed at 27 Craven Street after William’s death: Polly gave birth to her daughter, and Magnus replaced William as the lease holder, so even if Margaret and Benjamin had wished to move back into No  27, there would have been no room for them  It is also interesting to note that considering the multiple times Benjamin wrote of his move out of No  27 (and complained of it), he never once mentioned moving back into No  27 in any of his correspondence after Mr  Hewson’s death 
Figure 6  No  36 Craven Street  (Photo courtesy of David Ross, britainexpress com)
In sum, based on the Westminster Rate Books[53] and Franklin’s correspondence, Mrs  Stevenson is known to have resided at No  27 (36) Craven Street from 1748 to 1772  It follows that, aside from the two years Franklin spent in Philadelphia from 1762 to 1764, he resided there from 1757 to 1772  Franklin’s correspondence also reveals that in the autumn of 1772, he and Mrs  Stevenson moved to another house on Craven Street  The 1773 Westminster Rate Books show her name crossed off at No  27 and William Hewson’s inserted  The following year the Rate Books list her at No  1 Craven Street  Evidence for Mrs  Stevenson and Benjamin remaining at No  1 after William’s death appears in the Westminster Rate Books for 1775 which have Mrs  Stevenson still residing at No  1 and Magnus Falkner residing at No  27  Further evidence can be construed from the lack of any mention of a move back into No  27 in Franklin’s correspondence  Despite the many theories one could devise as to why Franklin was thought to have lived at No  7 Craven Street by so many guide books and Franklin biographers of the nineteenth century, one thing is certain; at some point after Franklin’s departure to America in March of 1775, and no later than 1807, someone mistakenly associated him with No  7 on the west side of Craven Street, and it soon became his de facto residence  Credit must go to D  H  Montgomery in 1896 and Sir George in 1913 for setting the record partially straight by placing Franklin at No  27(36)  In 1937, the London County Council gave us the first accurate account of Franklin’s residences on Craven Street in the Survey of London at No  27(36) and No  1  It has been shown conclusively that No  27 was never previously numbered 7  It was, however, renumbered 36 in 1792 after ten additional houses were built at the southern end of the street and remains No  36 to this day 
[1] “Craven Street and Hungerford Lane”, in Survey of London: Volume 18, St Martin-in-the-Fields II: the Strand, ed  G H Gater and E P Wheeler (London, 1937), 27-39, Early History of the Site 
http://www british-history ac uk/survey-london/vol18/pt2/pp27-39
[2] “England, Westminster Rate Books, 1634-1900,” from database with images, Craven Street – 1735, FamilySearch from database by FindMyPast and images digitized by FamilySearch; citing Westminster City Archives, London 
[3] Ibid , Craven Street – 1748 
[4] The Statutes at Large, From Magna Charta to the End of the Eleventh Parliament of Great Britain  Anno 1761 Continued, Vol  XXVII, ed  Danby Pickering, (Cambridge, John Archdeacon, 1767), 96 
[6] James Raven, Publishing Business in Eighteenth-Century England, (Woodbridge: The Boydell Press, 2014), 201 
[7] The London Directory For the Year 1776, Ninth Edition, (London: T  Lowndes, 1776), title page 
[8] Kent’s Directory For the Year 1778, Forty-Sixth Edition, (London: Richard and Henry Causton, 1778), title page 
[9] A listing in Kent’s Directory for the Year 1882 on p  28 reveals, “Brown Sarah, Leather-seller, 1, Westmoreland-buildings, Aldersgate-street”, and in Kent’s Directory for the Year 1883 on p  175, “Whiteland Mary, Wine & Brandy Mercht  Jermyn-str  St  James ”
[10] “The Papers of Benjamin Franklin,” Sponsored by The American Philosophical Society and Yale University, Digital Edition by The Packard Humanities Institute, 22:263a 
http://franklinpapers org/franklin
Mrs  Stevenson wrote to Benjamin Franklin a letter from her new home at 75 Northumberland Court on November 16, 1775: “In this Court I have a kind friend, Mr  Lechmoen he comes and seats with me and talks of you with a hiy regard and friendship ”
[11] Survey of London, Early History of the Site 
[12] Survey of London, Footnotes/n 10 
[13] Survey of London, Historical Notes/No  31 
[14] David Hughson, LL D , London; Being An Accurate History And Description Of The British Metropolis And Its Neighbourhood, To Thirty Miles Extent, From An Actual Perambulation, Vol  IV, (London: W  Stratford, 1807), 227 
[15] The Reverend Joseph Nightingale, The Beauties of England and Wales: Or, Original Delineations, Topographical, Historical, and Descriptive, of Each County, Vol  X, Part III, Vol  II (London: J  Harris; Longman and Co ; J  Walker; R  Baldwin; Sherwood and Co ; J  and J  Cundee; B  and R  Crosby and Co ; J Cuthell; J  and J  Richardson; Cadell and Davies; C  and J  Rivington; and G  Cowie and Co , 1815), 245 
[16] John Britton, F S A  & Co , ed , The Original Picture of London, Enlarged and Improved: Being A Correct Guide For The Stranger, As Well As For the Inhabitant, To The Metropolis Of The British Empire Together With A Description Of The Environs, The Twenty-Fourth Edition (London: Longman, Rees, Orme, Brown, and Green, 1826), 479 
[17] Jared Sparks, The Works of Benjamin Franklin, Vol  VII, (Philadelphia: Childs & Peterson, 1840), 151 
[18] George Gulliver, F R S , The Works of William Hewson, F  R  S , (London: Printed for the Sydenham Society, MDCCCXLVI), xx 
[19] Peter Cunningham, Handbook for London; Past and Present, Vol  I, (London: John Murray, 1849), 245 
[20] F  Saunders, Memories of the Great Metropolis: or, London, from the Tower to the Crystal Palace, (New York: G P  Putnam, MDCCCLII), 138 
[21] Leigh Hunt, The Town; Its Memorable Characters and Events, (London: Smith, Elder and Co , 1859), 185 
[22] K  Baedeker, London and Its Environs, Including Excursions To Brighton, The Isle of Wight, Etc : Handbook For Travelers, Second Edition, (London: Dulau and Co , 1879), 133 
[23] Herbert Fry, London In 1880 Illustrated With Bird’s-Eye Views of the Principal Streets, Sixth Edition, (New York: Scribner, Welford, & Co , 1880), 50 
[24] Herbert Fry, London  Illustrated By Eighteen Bird’s-Eye Views of the Principal Streets, (London: W  H  Allen and Co , 1886), 40 
[25] Henry B  Wheatley, F S A , London Past and Present; Its History, Associations, and Traditions, Vol  1, (London: John Murray, New York: Scribner & Welford, 1891), 473 
[26] The Journal of the Society of Arts, Vol  XV, No  778, (October 18, 1867): 717 
[27] D  G  C  Allen, “Dear and Serviceable to Each Other: Benjamin Franklin and the Royal Society of Arts,” American Philosophical Society, Vol  144, No  3, (September 2000): 248-249 
Franklin was a corresponding member in 1756 because he was still residing in Philadelphia  He became an active member the following year when he moved to London 
[28] The Journal of the Society of Arts, Vol  XVIII, No  894, (Jan  7, 1870): 137 
“Since the last announcement, the following tablets have been affixed on houses formerly occupied by – Benjamin Franklin, 7 Craven-street, Strand, W C ”
[29] Franklin in His Own Time, eds  Kevin J  Haytes and Isabelle Bour, (Iowa City, University of Iowa Press, 2011), xxxvii 
“Takes lodgings with Margaret Stevenson at No  7 Craven Street ” It is unknown if the editors are referring to No  7 on the west side of Craven Street or No  36 on the east side using Sir George’s explanation of No  36 being previously numbered 7 
[30] Henry C  Shelly, “American Shrines on English Soil, III  In the Footprints of Benjamin Franklin,” in The Book News Monthly, September, 1913 to August, 1914, (Philadelphia: John Wanamaker, 1914), 325 
[31] The Journal of the Royal Society of Arts, Vol  LVI, No  2,880, (Jan  31, 1908): 245 
http://babel hathitrust org/cgi/pt?id=mdp 39015058423073;view=1up;seq=251
“His Majesty the King, who is Patron of the Society, has granted permission to the Society to prefix to its title the term ‘Royal,’ and the Society will consequently be known in future as the ‘Royal Society of Arts ’”
[32] Nineteenth Annual Report, 1914, of the American Scenic and Historic Preservation Society, (Albany: J  B  Lyon Company, 1914), 293 
http://babel hathitrust org/cgi/pt?id=wu 89072985302;view=1up;seq=4;size=150
[33] The Journal of the Society of Arts, Vol  LXII, No  3,183, (Nov  21, 1913): 18 
http://babel hathitrust org/cgi/pt?id=mdp 39015058422968;view=1up;seq=26
[36] Allen, “Dear and Serviceable,” 263-264 
[37] Papers of Benjamin Franklin, 19:20 
[38] Thomas Joseph Pettigrew, F  L  S , Memoirs of the Life and Writings of the Late John Coakley Lettsom With a Selection From His Correspondence, Vol  I, (London: Nichols, Son, and Bentley, 1817), 144 of Correspondence 
[39] Papers of Benjamin Franklin, 19:321b 
[40] Ibid , 19:314 
[41] Ibid , 19:353a 
[43] Simon David John Chaplin, John Hunter and the ‘museum oeconomy’, 1750-1800, Department of History, King’s College London  Thesis submitted for the degree of Doctor of Philosophy of the University of London , 202 
“Following Falconar’s death [1778] the lease [27 Craven Street] was advertised, and the buildings were described as:
A genteel and commodious house, in good Repair, with Coach-house and Stabling for two Horses…consisting of two rooms and light closets on each floor, with outbuildings in the Yard, a Museum, a Compleat Theatre, and other conveniences  (Daily Advertiser, 27 August 1778)”
[44] Simon Chaplin, “Dissection and Display in Eighteenth-Century London,” in Anatomical Dissection in Enlightenment England and Beyond: Autopsy, Pathology and Display, ed  Dr  Piers Mitchell, (Burlington: Ashgate Publishing Company, 2012), 108 
“Given that a nearby building at 35 [ No  26 in Franklin’s time] was occupied by the man-midwife John Leake, who advertised lectures – including lessons in the art of making preparations – at his ‘theatre’ between 1764 and 1788, it is possible that some facilities were shared  In both cases, however, the buildings [Leake’s residence at No  26 and Hewson’s residence next door at 27] served a dual function as domestic accommodation and as sites for lecturing and dissection ”
[45] George Gulliver, F R S , The Works of William Hewson, F  R  S , (London: Printed for the Sydenham Society, MDCCCXLVI), xviii 
[46] Westminster Rate Books, Craven Street – 1773, courtesy of the City of Westminster Archives 
[47] S W  Hillson et al , “Benjamin Franklin, William Hewson, and the Craven Street Bones,” Archaeology International, Vol  2, (Nov  22, 1998): 14-16 
http://dx doi org/10 5334/ai 0206
[48] Westminster Rate Books, Craven Street – 1774, 1775, courtesy of the City of Westminster Archives 
[49] Survey of London, Historical Notes/No  36, Craven Street (not sourced) 
[50] Gordon S  Wood, The Americanization of Benjamin Franklin, (New York: The Penguin Press, 2004), 261 
[51] Pettigrew, Memoirs, 146 of Correspondence 
[52] http://founders archives gov/documents/Franklin/01-22-02-0178, note 7  “Falconar married Hewson’s sister five months after the Doctor’s death; most of the Craven Street circle attended the wedding, and BF gave away the bride: Polly to Barbara Hewson, Oct  4, 1774, APS” (American Philosophical Society); “England Marriages, 1538–1973 ,” database, FamilySearch (https://familysearch org/ark:/61903/1:1:V52W-TGS : accessed September 15, 2015), Magnus Falconar and Dorothy Hewson, September 12, 1774; citing Saint Martin In The Fields, Westminster, London, England, reference ; FHL microfilm 561156, 561157, 561158, 942 B4HA V  25, 942 B4HA V  66 
[53] I chose to rely on the Westminster Rate Books for the numbering system on Craven Street  The books were consistent throughout the eighteenth century in the ordering of residents on the street and were used as the basis for the 1792 re-numbering  For the most part, commercial directories aligned with them as well  If by chance a directory didn’t initially align, it would inevitably produce future editions that did 
Benjamin Franklin, Benjamin Franklin House, London
More from David Turnquist
If one looked into Benjamin Franklin’s time on Craven Street, they might   
I think it’s very ironic that on the street maps included in your excellent article, Craven Street is so close to Scotland Yard  Because following the back and forth juxtapositions of numbers 7, 27 and 36 Craven Street (throw in 75 Northumberland Court and 1 Craven Street, too) was a case that could confound Sherlock Holmes 
Excellent job of deciphering street renumbering material spanning sixty years, including that of a wrong house number (# 7) being erroneously identified and then perpetuated in subsequent street map printings  It’s gratifying at least to know that the present day #36 Craven Street is the correct house for Ben Franklin tourists to visit  Except for #1 Craven Street for the last three years Franklin was in London, but we won’t get into that 
Again, excellent article, David!



ground_truth: An alphabetical list of names and places of abode of the merchants and principal traders of the cities of London and Westminster, the Borough of Southwark, and their environs, with the number affixed to each house.