├── .gitattributes ├── .gitignore ├── README.md ├── Representational_similarity_analysis.Rmd ├── Representational_similarity_analysis.nb.html ├── dimensions.csv ├── holdists.csv └── neural_pattern_similarity.csv /.gitattributes: -------------------------------------------------------------------------------- 1 | # Auto detect text files and perform LF normalization 2 | * text=auto 3 | 4 | # Custom for Visual Studio 5 | *.cs diff=csharp 6 | 7 | # Standard to msysgit 8 | *.doc diff=astextplain 9 | *.DOC diff=astextplain 10 | *.docx diff=astextplain 11 | *.DOCX diff=astextplain 12 | *.dot diff=astextplain 13 | *.DOT diff=astextplain 14 | *.pdf diff=astextplain 15 | *.PDF diff=astextplain 16 | *.rtf diff=astextplain 17 | *.RTF diff=astextplain 18 | -------------------------------------------------------------------------------- /.gitignore: -------------------------------------------------------------------------------- 1 | 2 | # R-specific files 3 | *.Rhistory 4 | .Rdata 5 | 6 | # Windows image file caches 7 | Thumbs.db 8 | ehthumbs.db 9 | 10 | # Folder config file 11 | Desktop.ini 12 | 13 | # Recycle Bin used on file shares 14 | $RECYCLE.BIN/ 15 | 16 | # Windows Installer files 17 | *.cab 18 | *.msi 19 | *.msm 20 | *.msp 21 | 22 | # Windows shortcuts 23 | *.lnk 24 | 25 | # Matlab temporary files 26 | *.asv 27 | 28 | # ========================= 29 | # Operating System Files 30 | # ========================= 31 | 32 | # OSX 33 | # ========================= 34 | 35 | .DS_Store 36 | .AppleDouble 37 | .LSOverride 38 | 39 | # Thumbnails 40 | ._* 41 | 42 | # Files that might appear in the root of a volume 43 | .DocumentRevisions-V100 44 | .fseventsd 45 | .Spotlight-V100 46 | .TemporaryItems 47 | .Trashes 48 | .VolumeIcon.icns 49 | 50 | # Directories potentially created on remote AFP share 51 | .AppleDB 52 | .AppleDesktop 53 | Network Trash Folder 54 | Temporary Items 55 | .apdisk -------------------------------------------------------------------------------- /README.md: -------------------------------------------------------------------------------- 1 | # Representational similarity analysis 2 | ### Tutorial for Methods in Neuroscience at Dartmouth (MIND) 2018 3 | 4 | Representational similarity analysis (RSA) is statistical technique based on analyzing second-order isomorphisms. That rather than directly analyzing the relationship between one measure and another, RSA instead computes some measure of similarity within each measure and then compares these similarities to each other. RSA was pioneered by [Kriegeskorte, Mur, and Bandettini (2008, Frontiers in System Neuroscience)](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC2605405/) and has since become a popular method for analyzing neuroimaging data. Much of this popularity is driven by the fact that - because RSA focuses on second-order isomorphisms (i.e., similarities) - it is an incredibly flexible analytic technique, capable linking disparate measures of brain and behavior. 5 | 6 | ![Kriegeskorte, Mur, and Bandettini (2008)](http://www.mrc-cbu.cam.ac.uk//personal/nikolaus.kriegeskorte/fig5_kriegeskorte_RSA_FNS.gif) 7 | 8 | In the context of fMRI, RSA usually takes the form of a correlation or regression between neural pattern similarity and a task, rating, or model. In this tutorial we will learn how to conduct these confirmatory RSAs as well as how 9 | to perform complementary exploratory analyses. 10 | 11 | ### Installation 12 | 13 | First, if you have not already done so, please install [R](https://cran.r-project.org/) and [RStudio](https://www.rstudio.com/products/rstudio/download/#download). 14 | 15 | Clone this repository to your local machine, and open "Representational_similarity_analysis.Rmd" with RStudio. From the "Run" dropdown menu in the upper left pane, select "Run All". When the script completes, save the .Rmd file, and then select "Preview" to view the R Notebook. To make changes, edit the code in the .Rmd file, re-run the corresponding code-chunk(s), and then save changes - the preview will update automatically. 16 | 17 | Alternatively, a static version of the R Notebook can be viewed directly through your web browser by downloading the file ending in ".nb.html" and then opening it with the browser of your choice. Viewing this version does not require R/RStudio, but changes cannot be made to the code. 18 | 19 | 20 | 21 | -------------------------------------------------------------------------------- /Representational_similarity_analysis.Rmd: -------------------------------------------------------------------------------- 1 | --- 2 | title: "Representational similarity analysis (MIND 2018)" 3 | author: "Mark A. Thornton, Ph. D." 4 | date: "August 1, 2018" 5 | output: html_notebook 6 | --- 7 | 8 | ```{r setup, include=FALSE} 9 | knitr::opts_chunk$set(echo = TRUE) 10 | ``` 11 | 12 | ## Introduction to RSA 13 | 14 | Representational similarity analysis (RSA) is statistical technique based on analyzing second-order isomorphisms. That rather than directly analyzing the relationship between one measure and another, RSA instead computes some measure of similarity within each measure and then compares these similarities to each other. RSA was pioneered by [Kriegeskorte, Mur, and Bandettini (2008, Frontiers in System Neuroscience)](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC2605405/) and has since become a popular method for analyzing neuroimaging data. Much of this popularity is driven by the fact that - because RSA focuses on second-order isomorphisms (i.e., similarities) - it is an incredibly flexible analytic technique, capable linking disparate measures of brain and behavior. 15 | 16 | ![Kriegeskorte, Mur, and Bandettini (2008)](http://www.mrc-cbu.cam.ac.uk//personal/nikolaus.kriegeskorte/fig5_kriegeskorte_RSA_FNS.gif) 17 | 18 | In the context of fMRI, RSA usually takes the form of a correlation or regression between neural pattern similarity and a task, rating, or model. In this tutorial we will learn how to conduct these confirmatory RSAs as well as how 19 | to perform complementary exploratory analyses. 20 | 21 | ## How to measure similarity? 22 | 23 | There are many ways to measure the similarity (or distance) between data objects. While the nature of the data in question partially constrains which metrics are appropriate, often we are still left with a choice of several different distance metrics for measuring a space. This section will explore distance metrics with the aim of building intuition for how to make such choices. 24 | 25 | The most common distance metrics used to measure similarity in fMRI data are mean distance, Euclidean distance, and correlation distance. In this section, we will simulate some data to illustrate the relationships between these difference metrics. 26 | 27 | 28 | 29 | ```{r, echo=F, results="hide", include=F} 30 | # load packages 31 | if(!require(MASS)) install.packages("MASS"); require(MASS) 32 | if(!require(lattice)) install.packages("lattice"); require(lattice) 33 | if(!require(rasterVis)) install.packages("rasterVis"); require(rasterVis) 34 | if(!require(psych)) install.packages("psych"); require(psych) 35 | if(!require(pracma)) install.packages("pracma"); require(pracma) 36 | if(!require(nnls)) install.packages("nnls"); require(nnls) 37 | if(!require(smacof)) install.packages("smacof"); require(smacof) 38 | if(!require(Rtsne)) install.packages("Rtsne"); require(Rtsne) 39 | ``` 40 | 41 | ```{r} 42 | # data generation 43 | set.seed(1) 44 | sigmat <- matrix(c(1,0,.8,0,0,1,0,.8,.8,0,1,0,0,.8,0,1),nrow=4) 45 | dat <- mvrnorm(200,c(0,0,1,1),Sigma = sigmat) 46 | 47 | # plot data 48 | layout(matrix(1:4,2,2)) 49 | for (i in 1:4){ 50 | plot(dat[,i],type="o",pch=20,ylim=c(-4,4),ylab="Activity",xlab=paste("Condition",as.character(i))) 51 | } 52 | 53 | ``` 54 | 55 | The data we've simulated here isn't particularly realistic, but it is ideally suited to display the differences between the three distance metrics in question. You can think of each of the four variables as the mean activity 56 | in a given condition in an fMRI experiment, across a 200 voxel region. 57 | 58 | ### Mean distance 59 | We'll begin by calculating mean distance - which is simply the differences in means between the four conditions. This metric discards any information in the pattern across voxels, and is most similar to a standard univariate fMRI analysis. The bar graph and heatmap below illustrate how these means differ. 60 | 61 | ```{r} 62 | # mean distance 63 | cmeans<-apply(dat,2,mean) # calculate means of each variable (object) 64 | barplot(cmeans) # plot means 65 | dmat1<-as.matrix(dist(cmeans)) # calculate distance between means 66 | levelplot(dmat1) # heatmap of distances 67 | ``` 68 | 69 | ### Euclidean distance 70 | Next we'll examine Euclidean distance. This distance corresponds to the "real" distance we use most frequently in our everyday lives. The twist is that, instead of the familiar 3-D space we all inhabit, here we're calculating Euclidean distance in an N-dimensional space, where N = # of voxels (in this case, 200). The heatmap and scatter plot matrix below illustrates the Euclidean distance between the four conditions. 71 | 72 | ```{r} 73 | # Euclidean distance 74 | dmat2 <- as.matrix(dist(t(dat))) 75 | levelplot(dmat2) 76 | pairs(dat) 77 | ``` 78 | 79 | ### Correlation distance 80 | Correlation distance is perhaps the most common metric used in fMRI analysis. This metric discards the mean completely (data are implicitly z-scores prior to correlation, or else it's just covariance) so all information comes from the pattern instead. Since correlation is naturally a similarity rather than dissimilarity measure, the "distance" part is basically just a sign flip: 1-R. Again, the heatmap illustrates the results. 81 | 82 | ```{r} 83 | # correlation distance 84 | dmat3 <- 1-cor(dat) 85 | levelplot(dmat3) 86 | ``` 87 | 88 | ### Comparing distance metrics 89 | Now let's put these three distance metrics together. In the heatmaps below, you can see the lawful relationship between the three: mean and correlation distance are completely orthogonal to each other, but both "contribute" to the Euclidean distance. Correlation distance tends to be preferred because RSA (and MVPA in general) is often done in direct contrast to univariate analyses. Since the univariate analyses already reflect mean distance, it makes sense to over orthogonal information using correlation distance when doing RSA. However, if you're simply seeking an agnostic measure of neural similarity, and don't particularly care whether it is drive by the mean or the pattern, Euclidean distance might be a good bet. 90 | 91 | ```{r} 92 | # combined plot 93 | dmat2<-dmat2/max(dmat2) 94 | rlist <- list(raster(dmat1),raster(dmat2),raster(dmat3)) 95 | names(rlist)<-c("Mean","Euclidean","Correlation") 96 | levelplot(stack(rlist),layout=c(3,1),at=seq(0,1,.01)) 97 | 98 | ``` 99 | 100 | Does this result mean that measuring neural similarity with correlations allows you to completely remove univariate signal from your results? Unfortunately not. In this toy example, we knew the exact borders of our region of interest. In reality, it will never be that clean: activity/patterns will be aliased with respect to your feature selection. For example, imagine that there is a single active blob in your data, but your region of interest is a bit too big and therefor the blob only fills 80% of the ROI. The remaining 20% of voxels will not change their activity across conditions. In the process, the difference between these two populations of voxels will induce correlations - as well as just mean activity differences - across conditions. Again, this might not matter depending on the conclusion you want to draw, but if your conclusion relies heavily on the results not being driven by univariate signal, this is bad news. 101 | 102 | ## RSA: NHST, effect sizes, cross-validation, and model selection 103 | 104 | In this section, we will introduce RSA proper, and deal with some core issues surrounding it. In particular, we will examine how to test the significance of RSA results at the item-analysis and group levels. We will also hear a couple caveats about RSA effect sizes, and learn how to perform cross-validation and model selection. 105 | 106 | Instead of the toy data used above, for this section we will turn to real fMRI, rating, and text data from a previous study: [Thornton & Mitchell, 2017, Cerebral Cortex](http://markallenthornton.com/cv/Thornton&Mitchell_CC_2017.pdf). In this study, participants performed a social judgment task in which they mentalized about 60 famous people. On each trial, participants would judgment how well a particular statement, such as "would like to learn karate," would apply to a particular target person, such as Bill Nye. This procedure repeated across the course of the study, fully crossing the 60 targets with 12 items. After preprocessing, the GLM was used to average across all trials featuring the same target person. The resulting regression coefficient patterns were z-scored by voxel to remove the influence of the global background pattern, and then correlated with each other to estimate their similarity. That's where we start. 107 | 108 | 109 | ![Regions of reliability target person-related activity, within which we analyze patterns](http://markallenthornton.com/images/pmap.png) 110 | 111 | First, we'll read in the estimates of neural pattern similarity we want to predict. These data consist of vectorized (lower-triangular) 112 | correlation matrices between z-scored patterns within the set of voxels in the figure above. 113 | 114 | ```{r} 115 | # load neural data 116 | ndat <- read.csv("neural_pattern_similarity.csv") 117 | dim(ndat) 118 | 119 | # average: 120 | nsim <- rowMeans(scale(ndat)) 121 | ``` 122 | 123 | Next we'll load in a set of trait dimensions on which separate participants rated the famous target people: 124 | 125 | ```{r fig.height=10, fig.width=8} 126 | 127 | # read in dimensions 128 | pdims <- read.csv("dimensions.csv") 129 | pnames <- as.character(pdims$name) 130 | pdims <- scale(pdims[,2:14]) 131 | rownames(pdims)<-pnames 132 | levelplot(t(pdims),xlab="",ylab="",scales=list(x=list(rot=45))) 133 | ``` 134 | 135 | 136 | Finally, we'll read in some holistic similarity measures: explicit ratings of the pairwise similarity between target people, and 137 | a measure of textual similarity derived from taking a bag-of-words approach to the target people's Wikipedia pages. These have been 138 | reverse-coded (i.e., as distances) so we'll flip their signs to make them similarities again. 139 | ```{r} 140 | holdists <- read.csv("holdists.csv") 141 | explicit <- 100-holdists$holistic 142 | text <- 2-holdists$text 143 | 144 | ``` 145 | 146 | ### Our first RSA! 147 | 148 | Now that we have the requisite data load into R, let's run our first RSA analysis. This will consist of a simple correlation between the average neural pattern similarity and explicit ratings of interpersonal similarity. 149 | 150 | ```{r fig.height=7, fig.width=7} 151 | cor(nsim,explicit) 152 | plot(explicit,nsim,xlab="Rated similarity",ylab="Neural similarity",pch=20) 153 | abline(lm(nsim~explicit),col="red",lwd=2) 154 | 155 | ``` 156 | 157 | As you can see, there's a fairly substantial correlation between how similar MTurkers think famous people are to each other, and similarity between patterns of brain activity elicited by mentalizing about those people. The Pearson correlation between the two values is r = 0.4. (I'll note in passing that Kriegeskorte and colleagues recommend using Spearman correlations instead of Pearson. Their reasoning is sound, but I've never seen this make a substantial difference in practice.) 158 | 159 | ```{r} 160 | cor(nsim,explicit,method = "spearman") 161 | 162 | ``` 163 | 164 | ### Null hypothesis significance testing 165 | 166 | How can we tell if this correlation is statistically significant? A parametric correlation test assumes that every observation is independent, and will thus give us a fixed-effects p-value. 167 | 168 | ```{r} 169 | cor.test(nsim,explicit) 170 | ``` 171 | 172 | We can see that the results of this test are wildly significant (i.e., p = 0 to within machine precision). However, if are stimuli are just s small sample from a large population, we may want a p-value that reflects the dependencies in the similarity matrices. To achieve this, we can instead use a permutation test. As with any permutation test, it is important that we permute at the level of independent observations. In this case, that means permuting the rows and columns of one of our similarity matrices with respect to the other. 173 | 174 | ```{r} 175 | # turn vectorized neural similiarity matrix back into a matrix 176 | sqnsim <- squareform(nsim) 177 | 178 | # set random seed for reproducible results 179 | set.seed(1) 180 | nperm <- 5000 # set permutation count 181 | nppl <- dim(sqnsim)[1] # number of target people 182 | permcor <- rep(NA,nperm) # preallocate results 183 | for (i in 1:nperm){ 184 | sel <- sample(nppl) # permutation vector 185 | rnsim <- squareform(sqnsim[sel,sel]) # permute matrix and re-vectorize neural similarity 186 | permcor[i] <- cor(rnsim,explicit) # calculate permuted correlation 187 | } 188 | 189 | # calculate p-value 190 | mean(abs(permcor) > cor(nsim,explicit)) 191 | 192 | # visualize 193 | hist(abs(permcor),xlim=c(0,.4),main="Permuted null versus actual correlation") 194 | abline(v=cor(nsim,explicit),col="red",lwd=2) 195 | 196 | ``` 197 | 198 | As you can see, in this instance our permutation results remain the same (p effectively = 0) as in the parametric case, but bear in mind that this will not always be the case! 199 | 200 | Permutation testing resolved the issue of dependency within the similarity matrix, but the correlation we're examining is still effectively an item-analysis (since we averaged neural similarity across participants). Thus, the p-value we obtain above doesn't strictly-speaking license inference to other samples of participants, only to other samples of famous targets. To say something more general about social perceivers, we must conduct a random effect analysis. The easiest way to do this is by taking the summary statistic approach. 201 | 202 | ```{r} 203 | # correlate explicit ratings with each imaging participant's neural similarity 204 | ncors <- cor(explicit,ndat) 205 | 206 | # parametric one-sample t-test (note use of Fisherized correlations) 207 | t.test(atanh(ncors)) 208 | ``` 209 | 210 | In the code above, we correlated each imaging participant's neural similarity with the explicit ratings. We then computed a one-sample t-test on the resulting correlation coefficients to determine if the average was greater than zero. Note that correlation coefficients have a nonlinear, non-normal distribution, and thus should be normalized via Fisher's z transformation (hyperbolic arctangent) prior to any parametric tests. We can also compute a non-parametric alternative via bootstrapping (below), in which case this transformation is not critical to obtaining valid NHST. 211 | 212 | ```{r} 213 | # bootstrap 95% CI 214 | bootres <- replicate(5000,mean(sample(ncors,length(ncors),T))) 215 | quantile(bootres,c(.025,.975)) 216 | 217 | # visualize result 218 | plot(1:29,sort(ncors),xlab="Participant (sorted)",ylab="Correlation (r)", 219 | pch=20,xlim=c(1,30)) 220 | points(30,mean(ncors),col="red",cex=2,pch=20) 221 | segments(30,quantile(bootres,.975),30,quantile(bootres,.025),col="red",lwd=2) 222 | abline(h=0) 223 | ``` 224 | 225 | Note that the correlation is greater than zero for all but one participant in this data set, so formal NHST is scarcely necessary to reject the null hypothesis. 226 | 227 | Although analyzing data at the group level permits inference to the population, the best possible NHST setup would involve accounting for both participant and item random effects (both intercepts and slopes for all predictors). Unfortunately, due to the complexity of the random effects structure resulting from similarity matrices, the ideal maximal mixed effects model will very rarely converge. As a result, it's effectively impossible to get the correct NHST value for most RSAs. However, there are a few alternatives. One possibility is [prevalence-based inference](https://arxiv.org/abs/1512.00810). Another option is to switch over to Bayesian statistics, since Bayesian mixed effects models can often be made to converge when (Re)ML models won't. These options are beyond the scope of this tutorial, but later we will look at one other option to mitigate this issue: cross-validation. 228 | 229 | ### Effect sizes 230 | 231 | Effect sizes in RSA are relatively straight-forward. For item-level analyses like the first one we examined, the effect size is simply the correlation between the two similarity measures. However, it's worth bearing in mind the reliability of your measures when you do so. Neural data can often be quite noisy, and this can make it appear as if the association between neural similarity and another measure is smaller than it really is. One way to deal with this is through correlation disattenuation: 232 | 233 | ```{r} 234 | # compute raw correlation 235 | cor(nsim,explicit) 236 | # compute inter-participant Cronbach's alpha for neural data 237 | rel <- alpha(ndat) 238 | # standardized alpha 239 | rel$total$std.alpha 240 | # compute disattenuated correlation 241 | cor(nsim,explicit)/sqrt(rel$total$std.alpha) 242 | ``` 243 | 244 | In this example, the correlation between neural and explicit similarity increased from 0.4 to 0.56 when we adjusted for the mediocre reliability of the neural data. The same could be done for the ratings, if we had the individual participant data from which to calculate its reliability. Note only the raw correlation should be used for NHST purposes. Also, remember that the reliability we calculated is only an estimate of the true variance in the data. When that estimate itself is noisy (e.g., with small samples) it clean lead to nonsensical results, like correlations greater than one. Thus, although this approach can be helpful, it should be applied with caution. 245 | 246 | There is another reason besides reliability why RSA correlations might be smaller than they should be: converting raw data into a similarity matrix inherently discards information. As a result, on average an RSA correlation will be the square-root of the correlation in the underlying dimensions used to create the similarity matrices. 247 | 248 | ![](http://markallenthornton.com/images/rsacor.png) 249 | 250 | If you care about variance-explained in similarities themselves, then this isn't really a problem. However, if you care more about variance in the variables underlying your similarity estimates, you could arguably improve you effect size estimates by taking the square root of the RSA correlations you obtain. 251 | 252 | #### Group level effect sizes 253 | 254 | At the group level, we can calculate familiar test-statistics like Cohen's d. 255 | 256 | ```{r} 257 | # Compute Cohen's d 258 | mean(atanh(ncors))/sd(atanh(ncors)) 259 | ``` 260 | 261 | Here, for instance, we can see that the Cohen's d for the association between explicit similarity ratings and neural similarity is 2.16. The standard rule of thumb for Cohen's d states that a "large effect" is 0.8, so at 250%+ of that, we've got a whopper here! 262 | 263 | It's also possible to estimate a wide-range of other standardized effect size measures in the context of RSAs based multiple regressions or mixed effects models. However, those measures aren't RSA-specific, so won't consider them here. 264 | 265 | ### Cross-validation 266 | 267 | Significance testing offers some nice hypothetical guarantees about inference to a population. However, cross-validation offers a much more concrete way of assessing how well your model can actually predict new data. In cross-validation, a model is trained on some subset of data, and then tested on left-out data, iteratively treating each subset as test and training set. Let's conduct a cross-validated multiple regression RSA, predicting neural similarity from explicit ratings and text! 268 | 269 | ```{r fig.height=7, fig.width=7} 270 | # plot item-level scatterplot matrix 271 | pairs(data.frame(neural=nsim,explicit,text),pch=20) 272 | 273 | # rescale neural similarities to positive values 274 | ndatp <- ndat-min(ndat) 275 | 276 | # conduct leave-one-participant-out cross-validation 277 | xvars <- cbind(1,explicit,text) 278 | nsub <- dim(ndat)[2] 279 | cvperf <- rep(NA,nsub) 280 | for (i in 1:nsub){ 281 | fit <- nnls(xvars,rowMeans(ndatp[,-i])) # fit using non-negative least squares 282 | cvperf[i] <- cor(fit$fitted,ndatp[,i]) 283 | } 284 | mean(cvperf) 285 | 286 | ``` 287 | Here we can see that together, these two predictors can achieve a cross-validated performance of r = .11. Note that there are many other measures for assessing such performance, such as RMSE (root-mean-square-error), but correlation is a relatively interpretable choice. However, how can we contextual this performance? An r = .11 is obviously not great in absolute terms, but what's the best we could have done? We can answer this by calculating a noise ceiling. In this context, that simply means the average correlation between each participant and the average of the other participants. 288 | 289 | ```{r fig.height=7, fig.width=7} 290 | noise <- rep(NA,nsub) 291 | for (i in 1:nsub){ 292 | noise[i] <- cor(ndat[,i],rowMeans(ndat[,-i])) 293 | } 294 | mean(noise) # noise ceiling 295 | mean(cvperf)/mean(noise) # performance as fraction of noise ceiling 296 | ``` 297 | 298 | The noise ceiling for this data is only r = 0.129. This means that, given the amount of heterogeneity between participants, even a hypothetical perfect model couldn't do better than r = 0.129. If we divide our actual performance by the noise ceiling, we can see that together, explicit ratings and textual similarity achieve 86% of the performance of a perfect theory. Although the calculations are somewhat different due to the cross-validation, you might realize that what we've just done is conceptually the same as the correlation disattenuation we performed earlier. Thus, although noise ceilings can be helpful, they should be interpreted with a grain of salt because they are only estimates and they carry their own assumptions with them. For instance, in the example above, we're implicitly assuming that all meaningful variance is at the group level (i.e., there are no meaningful idiosyncrasies), which is almost certainly not true. 299 | 300 | More generally, using different cross-validation schemes can be very useful for estimating generalization across different "boundaries" in your data. In the example above, we only cross-validated with respect to participants, but in the case below, we also cross-validate with respect to mental states. 301 | 302 | ```{r} 303 | 304 | # define function to translate vector selector to matrix selector and vectorize 305 | rsasel <- function(selvec){ 306 | nobj <- length(selvec) 307 | selmat <- matrix(F,nobj,nobj) 308 | selmat[selvec,selvec] <- T 309 | diag(selmat)<-0 310 | return(squareform(selmat)) 311 | } 312 | 313 | 314 | # split data by target people 315 | set.seed(1) 316 | targsel <- sample(c(rep(T,30),rep(F,30))) 317 | targsel1 <- rsasel(targsel)==1 318 | targsel2 <- rsasel(!targsel)==1 319 | ndatp1 <- ndatp[targsel1,] 320 | ndatp2 <- ndatp[targsel2,] 321 | xvars <- cbind(1,explicit,text) 322 | xvars1 <- xvars[targsel1,] 323 | xvars2 <- xvars[targsel2,] 324 | 325 | 326 | # conduct leave-one-participant-out and split-half-by-target cross-validation 327 | nsub <- dim(ndat)[2] 328 | cvperf <- matrix(NA,nsub,2) 329 | for (i in 1:nsub){ 330 | fit <- nnls(xvars1,rowMeans(ndatp1[,-i])) 331 | cvperf[i,1] <- cor(xvars2 %*% fit$x,ndatp2[,i]) 332 | fit <- nnls(xvars2,rowMeans(ndatp2[,-i])) 333 | cvperf[i,2] <- cor(xvars1 %*% fit$x,ndatp1[,i]) 334 | } 335 | mean(cvperf) 336 | 337 | ``` 338 | 339 | The performance of the model dropped to r = .108 (vs r = .111) when we ask it to generalize to a new set of target people. However, this is a very slightly reduction in performance, suggesting that this model could potentially generalize quite well to new target people. In practice, we should probably repeat the process above multiple times with different split halves (or another cross-validation scheme) to ensure that the results we see aren't a fluke. Note that cross-validation can thus give us an answer which is difficult to obtain through significance testing (since, as discussed previously, it can be difficult to get a maximal mixed effects model to converge for such data). 340 | 341 | ### Model selection 342 | 343 | In the examples so far, we've worked with models that have only one or two predictors. In many studies, researchers try to explain neural similarity using many more dimensions. As models get more complex, how can we tell when we have the best RSA model to explain brain activity? Model selection is not a unique problem for RSA, but it is an important one. There are many good ways to pick the best set of dimensions for a regression model. Traditional methods include stepwise and best subset regression. However, here we'll give an example of a simple exhaustive search through a relatively small set of dimensions. 344 | 345 | Among the various predictors we asked participants to rate famous people on are the cardinal dimensions of the Big 5: openness, conscientiousness, extraversion, agreeableness, and neuroticism. What combination of these five predictors is best for explaining 346 | the neural pattern similarity between representations of famous people? 347 | 348 | ```{r} 349 | # select appropriate dimensions from rating matrix 350 | big5 <- pdims[,c(2,5,8,10,11)] 351 | big5d <- -apply(big5,2,dist) # convert to distances and sign flip 352 | big5d <- big5d - min(big5d) # ensure positivity 353 | 354 | # enumerate all possible combinations 355 | b5combs <- list() 356 | ind <- 1 357 | for (i in 1:5){ 358 | ccombs <- combn(5,i) 359 | for (j in 1:dim(ccombs)[2]){ 360 | b5combs[[ind]] <- ccombs[,j] 361 | ind <- ind + 1 362 | } 363 | } 364 | 365 | # cycle through combinations, with LOO-CV 366 | nc <- length(b5combs) 367 | perf <- matrix(NA,nc,nsub) 368 | for (i in 1:nc){ 369 | xvars <- cbind(1,big5d[,b5combs[[i]]]) 370 | for (j in 1:nsub){ 371 | fit <- nnls(xvars,rowMeans(ndatp[,-j])) # fit using non-negative least squares 372 | perf[i,j] <- cor(fit$fitted,ndatp[,j]) 373 | } 374 | } 375 | # calculate model with best performance 376 | mperf <- rowMeans(perf) 377 | colnames(big5)[b5combs[[which(mperf==max(mperf))[1]]]] 378 | 379 | ``` 380 | 381 | The best model consists of just 3 of the Big 5: openness, conscientousness, and extraversion. In practice, if we wanted to then assess the performance of this model (e.g., relativel to the noise ceiling), we would be well-advised to separate the model selection and evaluation steps. In this case, we were only examining 31 possible models (combinations of 5 variables) but the space of possible models grows very rapidly as the number of possible predictors increases. This means that with just a few more dimensions, we would be at serious risk of overfitting by model selection (rather than the typical case of model selection via parameter estimation). Using independent data for model selection and evaluation mitigates this risk. 382 | 383 | ## Exploration and visualization 384 | 385 | So far we have mainly examined representational similarity from a confirmatory perspsective: trying to fit particular models to the observed neural data. However, exploratory analysis - including data visualization - is essential to complement these confirmatory approaches. In this section we will explore several important exploratory and visualization techniques. 386 | 387 | ### Multidimensional scaling 388 | 389 | Multidimensional scaling (MDS) is one of the most powerful techniques for visualizing and exploring similarity data. The basic idea is to take a distance matrix and use it to reconstruct a configuration of points that embody those distances as accurately as possible within an N-dimensional space. In practice, N usually is 2 (sometimes 3) since this makes the results easy to visualize. There are many implementations in R, but we'll be using the smacof package. 390 | 391 | #### Trival example: airline distances between French cities 392 | 393 | ```{r fig.height=8, fig.width=8} 394 | layout(mat=matrix(1)) 395 | mdfit <- smacofSym(Guerry,2) 396 | plot(mdfit) # a reconstructed map of France! 397 | ``` 398 | Note that the orientation of an MDS plot is arbitrary: this is one of the major limitations of the method. However, different configurations can be rotated into the same orientation via Procrutes algorithm. 399 | 400 | #### MDS on neural pattern similarity 401 | Now that we've seen how MDS works, let's apply it to our fMRI data. 402 | ```{r fig.height=8, fig.width=8} 403 | # name similarity matrix for plotting 404 | rownames(sqnsim) <- pnames 405 | colnames(sqnsim) <- pnames 406 | 407 | # flip sign and make positive 408 | psqnsim <- -sqnsim 409 | psqnsim <- psqnsim - min(psqnsim) 410 | 411 | # fit MDS 412 | mdfit1 <- smacofSym(psqnsim,2) 413 | plot(mdfit1) 414 | 415 | ``` 416 | 417 | The closer to people are in the figure above, the more similar the neural pattern they elicit. How can we tell if this configuration is a good fit for the data? The key statistic for assessing MDS fit is called "stress". It's a measure of how far each point is in the configuration relative to where it "should" be, given the input distance matrix. A typical rule of thumb is "stress > .15 = bad" but this is not really a good approach because stress depends heavily on the number of 'objects' in the distance matrix. Stress per point (SPP) is more helpful, as it can illustrate which particular points are ill-fit. 418 | 419 | ```{r fig.height=8, fig.width=8} 420 | mdfit1$stress 421 | plot(mdfit1,plot.type="stressplot") 422 | 423 | ``` 424 | 425 | The overall stress of this configuration is pretty bad. Nancy Grace, Michael Jordan, Jimmy Fallon, and Bob Marley are particularly 'misplaced' in the configuration plot, relative to their actual neural similarity to other targets. Let's try to get a better fit! One way to do this is to relax the assumption that your similarity measure is a metric variable, and instead treat it as ordinal. 426 | 427 | ```{r fig.height=4, fig.width=8} 428 | 429 | # fit MDS 430 | mdfit2 <- smacofSym(psqnsim,2,"ordinal") 431 | mdfit2$stress 432 | 433 | # Shepard plots are another helpful visual diagnostic to examine fit 434 | layout(mat = t(matrix(c(1,2)))) 435 | plot(mdfit1,plot.type="Shepard","Ratio fit") 436 | plot(mdfit2,plot.type="Shepard","Ordinal fit") 437 | ``` 438 | 439 | Unfortunately, switching to an ordinal MDS hasn't helped much - our data are just too high dimensional to reproduce their similarities on a 2-dimensional manifold! In a moment, we'll take a look at a different manifold-learning technique that might fair a bit better. However, let's first examine one nice addition to MDS: biplots. As I mentioned earlier, the axes of an MDS are arbitrary. Biplots are arrows added to an MDS which indicate the correlation of each dimension with the axes of the plot. They help to interpret the meaning behind the MDS plot. 440 | 441 | ![A heavily embellished MDS, including biplots](http://markallenthornton.com/images/person-space.png) 442 | 443 | #### t-SNE 444 | t-SNE, or t-distributed stochastic neighbor embedding, is another popular manifold learning/dimensionality reduction algorithm. Where MDS tries to preserve the global structure of the data, t-SNE ties to preserve the local structure. Thus long-range distances in an MDS configuration are quite meaningful, but you're not guaranteed to end up with true nearest-neighbors being represented as such in the configuration. In contrast, in t-SNE the long range distances don't mean much, but nearest neighbor structure will be preserved as much as possible. 445 | 446 | ```{r fig.height=8, fig.width=8} 447 | 448 | # fit t-SNE and plot 449 | set.seed(1) 450 | tres <- Rtsne(as.dist(psqnsim),perplexity=4) 451 | plot(tres$Y,pch=20,xlab="",ylab="") 452 | text(tres$Y[,1],tres$Y[,2]+2,pnames,cex=.75) 453 | ``` 454 | 455 | You can see here that the results of t-SNE tend to look a bit more cluster-like. How much this is true depends on the "perplexity" parameter of the algorithm. This parameter can be difficult to set because different settings reveal different structure in the data. There is an excellent introduction to setting perplexity and interpreting t-SNE can be found [here](https://distill.pub/2016/misread-tsne/). 456 | 457 | #### Hierarchical clustering 458 | Clustering is another powerful technique for exploring similarity data. Clustering is a complex topic of its own, so we won't go into too much detail here, but let's take a quick look at one method: hierarchical clustering. It can take two main forms: agglomerative (bottom-up) merging of points into larger and larger cluster, and divisive (top-down) breaking of clusters eventually yielding individual points. Both forms operate directly on distance matrices rather than data matrices, and both yield the familiar tree-like structure of dendrograms. Agglomerative clustering tends to be faster and more common, so we'll focus on it. Hierarchical clustering is particularly valuable for data exploration because it pairs well with a powerful visualization technique - the dendrogram. 459 | 460 | ```{r fig.height=10, fig.width=8} 461 | 462 | hc <- hclust(as.dist(psqnsim),method = "ward.D2") 463 | plot(as.dendrogram(hc),horiz=T,xlim=c(4,-1)) 464 | ``` 465 | 466 | ## Conclusion 467 | 468 | Today we've learned how to conduct representational similarity analyses of fMRI data. First, we learned several methods for calculating the similarity between data objects. Next, we learned about RSA proper, including methods for NHST, effect sizes, cross-validation, and model selection. Finally, we learned about several techniques for exploring and visualizing similarity data, including multidimensional scaling, t-SNE, and hierarchical clustering. Together these methods should give you an excellent position from which to start analyzing neural similarity data! 469 | 470 | 471 | -------------------------------------------------------------------------------- /dimensions.csv: -------------------------------------------------------------------------------- 1 | name,agency,agreeableness,attractiveness,competence,conscientiousness,dominance,experience,extraversion,intelligence,neuroticism,openness,trustworthiness,warmth 2 | Cesar Chavez,71.06521739,47.51162791,42.03278689,71.31372549,60.73584906,75.01851852,76.125,66,62.96296296,53.91666667,53.76785714,46.01886792,43.7755102 3 | Bill Nye,74.94642857,80.46153846,47.15151515,81.22222222,75.546875,49.64788732,79.53061224,71.36231884,88.15254237,38.73684211,76.73333333,79.37704918,72.58333333 4 | Pablo Picasso,66.05454545,56.37037037,47.42647059,76.95454545,44.3968254,56.07142857,75.14,26.09230769,79.15625,66.56140351,78.58730159,58.96721311,53 5 | William Shakespeare,70.68965517,63.44642857,50.47826087,84.20588235,69.10769231,63.20833333,82.90384615,42.51470588,82.89230769,54.80327869,70.6875,65.19047619,58.34920635 6 | Abraham Lincoln,76.74576271,78.87931034,43.15492958,89.61764706,83.95384615,83.09722222,87.51923077,50,82.828125,26.46666667,65.33846154,84.09375,69.64615385 7 | Mahatma Gandhi,74.43103448,79.69811321,43.66666667,86.35820896,88.44615385,56.07352941,88.01923077,44.29230769,81.74603175,14.25,68.56923077,83.93548387,81.44262295 8 | Nancy Grace,58.36956522,28.60869565,37.15517241,48.78181818,40.70588235,64.83606557,47.37209302,76.09090909,47.22,69.24,32.58490566,37.16363636,32.22807018 9 | Thomas Edison,80.5862069,61.55357143,41.39130435,88.50746269,76.64615385,72.07042254,83.84313725,38.73529412,89.27692308,43.01639344,76.01538462,69.25396825,51.75 10 | Kim Kardashian,48.17857143,23.66071429,61.6,23.54411765,21.46031746,41.77464789,29.90384615,87.51388889,27.96875,73.53225806,52.01538462,22.74193548,28.28125 11 | Sidney Poitier,70.29545455,67.97222222,61.32142857,70.78431373,68.75,62.30434783,71.10526316,54.83333333,73.58139535,30.38095238,63.02222222,66.88636364,67.45652174 12 | Warren Buffett,81.59615385,61.59574468,36.20634921,79.88333333,82.03225806,77.58333333,72.79591837,49.2,81.26229508,29.86792453,53.28070175,62.90740741,59.63333333 13 | Pharrell Williams,61.8372093,58.24444444,64.26315789,60.59649123,54.42105263,44.4137931,57.83333333,74.94915254,57.98076923,40.09259259,63.37037037,55.74,63.14 14 | Cameron Diaz,65.27586207,60.35714286,79.08450704,55.98529412,43.22222222,40.62857143,60.2,75.63768116,52.55555556,48.14516129,56.33870968,54.29508197,62.375 15 | Edward Snowden,67.35185185,54.02,47.21666667,68.55932203,58.5862069,54.5,64.84444444,34.65517241,71.57142857,53.52941176,62.59649123,50.3220339,44.81818182 16 | Bill Gates,86.16666667,73.51724138,41.76056338,87.91304348,89.38461538,79.66666667,80.59615385,40.76388889,91.75384615,26.24590164,65.87692308,67.46875,69.328125 17 | Bob Marley,61.98275862,68.24561404,57.68115942,66.6119403,48.328125,54.35714286,67.29411765,58.76811594,58.22222222,41.62295082,73.55555556,64.7,71.578125 18 | Muhammad Ali,68.43103448,62.5,59.69565217,71.60606061,55.625,76.32857143,75.09803922,69.52941176,57.12698413,44.68852459,59.12121212,65.11290323,61.19047619 19 | Ashton Kutcher,62.57894737,55.92727273,66.84057971,55.30769231,36.71428571,43.33802817,51.07692308,79.92647059,53.3015873,50.53225806,63.734375,43.25806452,57.69230769 20 | Vladimir Putin,76.54385965,21.90740741,31.17910448,56.203125,51.8852459,87.390625,67.33333333,60.07692308,64.9516129,67.48275862,32.46774194,19.3015873,16.73015873 21 | Melanie Griffith,65.70833333,52.925,57.78947368,55.2745098,43.23076923,36.87931034,51.84210526,66.16666667,50.55555556,57.75,53.14893617,52.73913043,58.09615385 22 | Jimmy Fallon,64.64285714,71.01818182,59.53846154,68.47692308,41.72580645,44.26153846,64.08,83.61538462,60.50793651,39.31666667,72.6984127,62.73333333,70 23 | Albert Einstein,80.47457627,76.48275862,43.32857143,94.14925373,83.49230769,69.14084507,85.80769231,30.8115942,97.83076923,43.50819672,78.29230769,83.78125,63.16923077 24 | Bill Clinton,76.41666667,66.18965517,53.84285714,75.10294118,60.72307692,75.63888889,79.53846154,77.54166667,78.46153846,41.68852459,66.57575758,52.80952381,65.72307692 25 | Dwayne Johnson,71.98039216,71.21276596,68.69491525,68.3559322,52.375,69.95238095,61.60465116,78.38095238,59.71666667,34.56603774,63.85964912,63,71.05172414 26 | Will Smith,71.96610169,65.5862069,70.14285714,69.54411765,58.49206349,57.76388889,69.05769231,83.63888889,63.64615385,35,70.26153846,65.328125,71.72307692 27 | Elizabeth II,69.7,52.95348837,46.5483871,72,70.70588235,79.36065574,73.04255319,44.54347826,63.41071429,41.16,41.54545455,58.86792453,56.6 28 | Stephen Hawking,72.60344828,68.69642857,32.86567164,85.07692308,86.13114754,65.65151515,81.79166667,37.10294118,93.12903226,29.79661017,69.18032787,76.91666667,57.76666667 29 | Cate Blanchett,69.11764706,55.04761905,74.69230769,66.74074074,57.0754717,42.52727273,60.48837209,57.18,59.32,38.81818182,66.75,62.28888889,63.67924528 30 | Amelia Earhart,69.71428571,66.92727273,53.30434783,80.109375,59.67213115,61.91304348,74.98039216,47.45901639,69.34920635,34.56666667,75.42424242,73.06557377,64.50819672 31 | George W. Bush,61.4,45.96551724,33.46478873,50.36764706,50.53846154,70.05633803,65.51923077,66.08333333,54.27692308,47.29032258,41.36363636,41.25,50.01538462 32 | John Travolta,64.74137931,58.79310345,57.64285714,55.29850746,46.0483871,52.83098592,64.47058824,70.13043478,53.52307692,48.01612903,55.59090909,51.37096774,58.64615385 33 | Michael Jordan,72.25862069,65.52631579,64.11428571,75.20588235,62.8,73.72222222,67.1372549,68.25,60.41538462,30.62903226,60.12121212,61.390625,57.0625 34 | John Legend,64.20833333,63.29268293,62.42307692,65.03846154,60.7254902,45.98148148,61.94871795,60.83333333,61.16,29.68,64.34693878,57.67346939,63.65217391 35 | "Martin Luther King, Jr.",74.24137931,84.70689655,59.49295775,90.42028986,87.06153846,78.13888889,84.75,77.88732394,82.09230769,27.53225806,73.89393939,81.84375,80.56923077 36 | Matthew McConaughey,69.07017544,60.30909091,72.1641791,62.3125,45.9,52.84126984,62.53061224,72.86363636,53.65,38.76271186,65.96825397,57.17241379,68.06349206 37 | Franklin D. Roosevelt,77.51724138,71.55172414,44.22058824,83.85294118,79.390625,83.61428571,82.71153846,61.1,84.32307692,29.86666667,62.81538462,70.6875,67.95384615 38 | Snoop Dogg,62.20689655,57.48275862,46.01449275,54.72463768,28.15384615,56.65277778,59.34615385,77.13888889,49.44444444,46.80327869,65.13846154,47.3968254,56.07692308 39 | Bruce Lee,68.94915254,68.01851852,64.1,76.24615385,70.06349206,71.48571429,72.70588235,52.6,68.6031746,38.08333333,60.28571429,72.03174603,63.078125 40 | Robin Thicke,51.14893617,31.44,52.55,40.74137931,26.38983051,38.10526316,43.93023256,74.42857143,40.26315789,62.22807018,54.17647059,30.81034483,40.96551724 41 | Mark Zuckerberg,76.23214286,49.94230769,41.67692308,74.81538462,66.84126984,66.37313433,63.43137255,42.65217391,81.64516129,52.75409836,62.74193548,40.69491525,40.68852459 42 | Robert De Niro,75.3559322,62.51785714,57.44927536,73.50746269,60.66129032,68.7761194,68.75510204,64.84507042,67.328125,44.32258065,61.38709677,64.23333333,63.37704918 43 | Howard Hughes,63.75,44.83783784,43.47368421,75.43137255,51.27083333,65.84782609,62.57777778,30.19047619,72.36170213,71.8,51.42,45.28888889,40.22 44 | Bill Murray,70.25862069,75.24074074,52.05882353,68.96923077,42.53225806,53.44444444,71.39215686,71.13432836,68.046875,42.41666667,69.828125,67.32786885,73.77419355 45 | Megan Fox,55.81818182,48.33962264,79.05797101,40.53968254,30.88709677,40.22727273,44.22,71.57352941,43.83870968,54.25,52.69491525,42.66071429,46.86666667 46 | Ernest Hemingway,66.73214286,60.05555556,49.92307692,78.203125,54.80952381,60.63492063,84.75,29.62903226,79.40322581,61.45454545,64.12068966,64.30508475,52.11864407 47 | James Dean,58.61538462,55.78,77.32835821,60.01724138,34.22413793,57.38333333,57.7826087,57.22033898,56.08474576,53.63636364,62.18333333,54.44642857,55.42105263 48 | Justin Bieber,44.74576271,17.72413793,40.08571429,20.25,10.75384615,26.70833333,27.23076923,81.54166667,20.61538462,80.69354839,52.44615385,16.15873016,19.56923077 49 | Martin Scorsese,72.77358491,56.04255319,41.27419355,75.51851852,62.11320755,67.89285714,71.91304348,50.83928571,72.80769231,47.4375,67.70175439,56.41509434,52.45283019 50 | Barack Obama,70.41666667,60.20689655,57.54929577,67.11764706,75.8,71.47222222,77.44230769,73.64788732,72.83076923,31.70967742,60.60606061,51.609375,54.72307692 51 | Bruno Mars,64.61403509,52.60784314,61.14516129,60.24590164,46.11666667,39.5,51.1875,73.90322581,56.64912281,39.79310345,61.16071429,55.66666667,60.09090909 52 | Theodore Roosevelt,79.37931034,71.0862069,46.26470588,84.10294118,80.75,83.63380282,84.38461538,63.38571429,82.55384615,32.16666667,64.09230769,73.47619048,64.328125 53 | Ryan Gosling,68.33333333,61.18867925,75.06060606,60.68333333,51.35483871,43.44444444,57.82222222,58.63333333,55.45901639,38.14545455,59.95,59.50877193,64.44827586 54 | Ellen DeGeneres,74.18965517,73.36842105,57.77142857,72.60606061,59.27692308,52.43055556,70.44,85.18309859,66.875,38.03225806,78.29230769,70.58064516,74.625 55 | Mark Wahlberg,67.90740741,61.22222222,69.27941176,65.26984127,51.3220339,61.06060606,63.71428571,68.875,57.15873016,42.95081967,63.75806452,59.06779661,64.44262295 56 | Winston Churchill,75.98305085,65.05769231,39.62686567,84.66666667,76,83.44285714,83.52941176,56.4,82.80952381,35.64912281,53.86153846,70.29032258,58.19672131 57 | Justin Timberlake,70.43103448,64.82142857,70.67142857,65.04411765,52.234375,48.81944444,61.96153846,80.875,59.24615385,41.01612903,65.34848485,53.33870968,67.546875 58 | Keanu Reeves,67.64912281,62.09090909,65.91428571,57.88709677,49.83333333,42.89855072,55.32653061,46.40298507,56.78688525,45.83870968,59.765625,62.16393443,61.796875 59 | Harrison Ford,76.74137931,67.44827586,68.08571429,76.04615385,62.14516129,62.5,67.84313725,59.94117647,64.71875,32.88333333,63.5625,69.80952381,64.87301587 60 | Liza Minnelli,65.44230769,49.95744681,45.26153846,52.46666667,36.70175439,49.09836066,58.84444444,74.11320755,52.38888889,69.76,61.23214286,52.10909091,55.33333333 61 | Kevin Bacon,68.89090909,62.58490566,58.47826087,66.2,55.09259259,50.12698413,63.62745098,66.26984127,57.44444444,37.9,58.51666667,63.48333333,66.04761905 62 | -------------------------------------------------------------------------------- /holdists.csv: -------------------------------------------------------------------------------- 1 | holistic,text 2 | 70.22361111,1.664120642 3 | 59.3625,1.604135322 4 | 67.3,1.652000146 5 | 29.3,1.529474217 6 | 39.5281746,1.506284056 7 | 62.75138889,1.65431402 8 | 57.76111111,1.559012794 9 | 73.53611111,1.68616266 10 | 63.18888889,1.703616684 11 | 65.82777778,1.589394325 12 | 73.16111111,1.729820514 13 | 80.41111111,1.743551004 14 | 58.50138889,1.57935809 15 | 54.00138889,1.574924792 16 | 56.24444444,1.600691411 17 | 55.77222222,1.616944946 18 | 73.77222222,1.689810341 19 | 46.12638889,1.55812801 20 | 73.68888889,1.756531532 21 | 69.48055556,1.687397491 22 | 57.07777778,1.582164582 23 | 54.98055556,1.499771685 24 | 68.05,1.673150795 25 | 68.175,1.712490402 26 | 61.30396825,1.581424239 27 | 68.9031746,1.605268355 28 | 75.58472222,1.741583689 29 | 51.68888889,1.546493845 30 | 65.93888889,1.508309641 31 | 67.98650794,1.713202386 32 | 67.09166667,1.648022306 33 | 71.41210317,1.64427883 34 | 30.12936508,1.424089782 35 | 66.93888889,1.717009244 36 | 44.97777778,1.496550584 37 | 77.98333333,1.664470976 38 | 53.60555556,1.624042817 39 | 74.93888889,1.704632353 40 | 73.62638889,1.589032993 41 | 74.36944444,1.673957194 42 | 63.13888889,1.603593999 43 | 76.43888889,1.679326427 44 | 72.80496032,1.718994444 45 | 65.27222222,1.600344762 46 | 69.59761905,1.624424318 47 | 70.70277778,1.671400067 48 | 68.6531746,1.679910842 49 | 46.16805556,1.486107774 50 | 75.07380952,1.686682327 51 | 56.96666667,1.497855097 52 | 76.06388889,1.696707034 53 | 69.16111111,1.636519016 54 | 64.85555556,1.688015747 55 | 34.31388889,1.536174064 56 | 70.3,1.712101558 57 | 71.06388889,1.660823779 58 | 67.42996032,1.618691422 59 | 70.97460317,1.671099671 60 | 70.80694444,1.678423881 61 | 61.86388889,1.65788086 62 | 59.67777778,1.707195016 63 | 54.30138889,1.698428513 64 | 52.53888889,1.68868888 65 | 73.78333333,1.588007195 66 | 33.38888889,1.629664639 67 | 84.05555556,1.60707774 68 | 62.09365079,1.678609096 69 | 65.09444444,1.666124108 70 | 73.63888889,1.6974337 71 | 77.33888889,1.747929607 72 | 54.52638889,1.664396129 73 | 33.23888889,1.606901709 74 | 74.17638889,1.695805187 75 | 71.73888889,1.672154658 76 | 69.01111111,1.607305961 77 | 85.98888889,1.659361468 78 | 77.58888889,1.74052795 79 | 61.73888889,1.572398641 80 | 19.83888889,1.585862339 81 | 70.83888889,1.622958574 82 | 70.61666667,1.621432127 83 | 67.34444444,1.666227361 84 | 83.13333333,1.686495258 85 | 24.99444444,1.579622457 86 | 77.21666667,1.694285533 87 | 55.33888889,1.612926172 88 | 75.67638889,1.634551702 89 | 76.83888889,1.673490421 90 | 77.61666667,1.675254224 91 | 74.85555556,1.641919447 92 | 69.63888889,1.640593456 93 | 69.28888889,1.690172289 94 | 68.07777778,1.654098631 95 | 80.03888889,1.648735688 96 | 77.05138889,1.642502188 97 | 78.02638889,1.664431541 98 | 53.03333333,1.640839253 99 | 75.82222222,1.678750942 100 | 65.42301587,1.64428247 101 | 58.13888889,1.562508968 102 | 76.33888889,1.641539385 103 | 62.18888889,1.663873631 104 | 80.92777778,1.657669307 105 | 81.97777778,1.629008782 106 | 69.68888889,1.684003869 107 | 63.63888889,1.63874075 108 | 75.82083333,1.673079197 109 | 59.93888889,1.654286459 110 | 75.8,1.670573599 111 | 59.38888889,1.524493108 112 | 72.13888889,1.639422172 113 | 54.43888889,1.64954993 114 | 73.28888889,1.62889667 115 | 68.41111111,1.634986499 116 | 62.06746032,1.589651217 117 | 73.78174603,1.640738824 118 | 72.35,1.646240255 119 | 32.55,1.463850512 120 | 56.06388889,1.523845692 121 | 66.10555556,1.478260731 122 | 78.78611111,1.621092743 123 | 44.21666667,1.510485382 124 | 81.38333333,1.675319299 125 | 57.58888889,1.719034407 126 | 70.41111111,1.544259352 127 | 57.41388889,1.686705758 128 | 71.13333333,1.710805898 129 | 67.55,1.578387724 130 | 63.46269841,1.546595586 131 | 59.48055556,1.571556672 132 | 68.38888889,1.554319902 133 | 69.38333333,1.679760894 134 | 82.54305556,1.515800609 135 | 71.25138889,1.72131679 136 | 61.82777778,1.669772446 137 | 44.48888889,1.465912283 138 | 76.87638889,1.570102109 139 | 68.35,1.665376483 140 | 65.43888889,1.688213313 141 | 77.82777778,1.484679233 142 | 55.10555556,1.528634791 143 | 64.23055556,1.685287162 144 | 61.66111111,1.506854496 145 | 70.07777778,1.565442904 146 | 69.53888889,1.734222982 147 | 71.71666667,1.618103931 148 | 60.17698413,1.661211195 149 | 67.13333333,1.504205736 150 | 59.61388889,1.733858661 151 | 69.32777778,1.488593769 152 | 68.72777778,1.610006823 153 | 65.58472222,1.509420105 154 | 65.09861111,1.632126731 155 | 71.71666667,1.563863532 156 | 63.88333333,1.637231223 157 | 55.61666667,1.498968938 158 | 60.95972222,1.659872325 159 | 68.93194444,1.644480991 160 | 29.21666667,1.395049814 161 | 65.55,1.54712405 162 | 74.27916667,1.639946315 163 | 51.34166667,1.589132836 164 | 72.66666667,1.550175643 165 | 61.71666667,1.592018981 166 | 72.77222222,1.466762524 167 | 66.40138889,1.608600285 168 | 59.05,1.634068111 169 | 66.1125,1.635495086 170 | 62.71666667,1.439142812 171 | 57.2,1.667914685 172 | 60.27222222,1.60214451 173 | 62.07380952,1.632826063 174 | 57.75634921,1.639901135 175 | 63.95555556,1.664221554 176 | 47.79305556,1.521655666 177 | 52.49444444,1.500683135 178 | 80.93095238,1.590150594 179 | 44.15,1.531151406 180 | 84.82222222,1.717962873 181 | 51.10555556,1.673709389 182 | 68.71031746,1.567856402 183 | 72.75555556,1.670540232 184 | 77.57777778,1.70831404 185 | 71.77222222,1.585369771 186 | 54.07083333,1.58850809 187 | 62.01527778,1.535650621 188 | 71.7,1.565895503 189 | 72.16111111,1.695434458 190 | 76.43888889,1.570004169 191 | 70.81388889,1.691535072 192 | 68.60555556,1.667475964 193 | 37.96111111,1.491763386 194 | 71.43888889,1.586299273 195 | 78.23888889,1.647494391 196 | 64.32777778,1.70059592 197 | 49.66111111,1.496544874 198 | 54.6,1.511187392 199 | 67.85555556,1.64786001 200 | 58.75555556,1.533952599 201 | 70.175,1.597638229 202 | 70.82777778,1.676286938 203 | 74.05,1.612896573 204 | 68.22460317,1.647055943 205 | 54.21666667,1.504808603 206 | 63.56111111,1.693071012 207 | 61.23888889,1.551116583 208 | 71.96666667,1.615155771 209 | 67.51527778,1.512984217 210 | 72.28710317,1.635687824 211 | 68.60555556,1.60175274 212 | 61.55,1.636788097 213 | 62.25555556,1.549987393 214 | 61.22361111,1.632398624 215 | 75.55694444,1.620527338 216 | 16.32777778,1.459988511 217 | 63.26388889,1.539195098 218 | 80.05,1.646390307 219 | 41.2781746,1.536649147 220 | 71.64444444,1.615037913 221 | 72.55,1.615252102 222 | 54.59444444,1.496453203 223 | 71.93333333,1.569319675 224 | 70.66666667,1.651633418 225 | 66.82777778,1.621171207 226 | 40.21666667,1.480936201 227 | 67.09444444,1.63491832 228 | 70.05,1.540382239 229 | 58.72460317,1.617509079 230 | 60.1452381,1.605185367 231 | 66.93888889,1.604589625 232 | 18.43194444,1.288701786 233 | 68.82083333,1.571988299 234 | 32.63888889,1.462014449 235 | 85.20138889,1.702293447 236 | 50.73650794,1.725858235 237 | 58.76031746,1.443249207 238 | 74.28888889,1.719833974 239 | 81.95972222,1.761360121 240 | 47.90714286,1.343050711 241 | 57.68888889,1.521398337 242 | 71.93888889,1.561642051 243 | 62.85138889,1.430934565 244 | 78.16805556,1.705096361 245 | 51.18888889,1.293401883 246 | 76.50138889,1.791871808 247 | 72.18888889,1.681279658 248 | 39.56388889,1.404558405 249 | 21.00138889,1.220056169 250 | 72.12638889,1.603485492 251 | 74.89027778,1.714056842 252 | 47.66111111,1.403230548 253 | 64.65138889,1.488695544 254 | 76.86746032,1.721262368 255 | 51.10138889,1.421913909 256 | 34.12638889,1.197169127 257 | 80.33888889,1.728282875 258 | 67.94583333,1.543677997 259 | 72.76031746,1.657664187 260 | 27.09861111,1.255604193 261 | 72.88888889,1.739746676 262 | 15.83888889,1.100760609 263 | 80.96388889,1.623824027 264 | 74.42103175,1.524643332 265 | 84.09960317,1.642456473 266 | 72.82083333,1.537375601 267 | 72.93194444,1.652685574 268 | 62.66031746,1.461716232 269 | 70.93888889,1.664995326 270 | 79.56388889,1.660032822 271 | 42.79305556,1.447496583 272 | 63.85138889,1.574153971 273 | 86.31388889,1.64248292 274 | 69.81388889,1.608701317 275 | 23.47638889,1.257517379 276 | 80.73055556,1.639440249 277 | 13.96388889,1.123359158 278 | 75.18888889,1.626322379 279 | 63.83888889,1.631055021 280 | 73.75138889,1.654738856 281 | 13.18194444,1.171786591 282 | 74.02638889,1.659102423 283 | 72.76388889,1.650001684 284 | 67.01924603,1.590263264 285 | 72.80496032,1.66472841 286 | 71.81388889,1.677282501 287 | 75.3406746,1.563626198 288 | 54.71666667,1.474426838 289 | 86.72222222,1.726038143 290 | 47.35555556,1.729501886 291 | 58.12638889,1.47559491 292 | 77.06388889,1.72085434 293 | 71.18888889,1.764118178 294 | 52.14722222,1.381914078 295 | 48.77222222,1.517780069 296 | 53.00138889,1.539481505 297 | 51.05555556,1.452083248 298 | 79.05,1.70687315 299 | 75.24444444,1.35158487 300 | 78.89424603,1.782780094 301 | 75.49444444,1.711152928 302 | 33.56111111,1.367529436 303 | 54.8625,1.375188083 304 | 69.02460317,1.616774812 305 | 69.77222222,1.720544886 306 | 46.28611111,1.411504764 307 | 38.2,1.455092904 308 | 73.27222222,1.728552078 309 | 52.53611111,1.431452185 310 | 65.08472222,1.361173657 311 | 73.43888889,1.741429803 312 | 67.46666667,1.592549134 313 | 67.23253968,1.653722333 314 | 15.77222222,1.185949143 315 | 66.43888889,1.756085473 316 | 33.45555556,1.288070784 317 | 80.87777778,1.616012068 318 | 61.93888889,1.522352237 319 | 84.08174603,1.649713066 320 | 73.82777778,1.505841627 321 | 73.32777778,1.666667556 322 | 71.28888889,1.476151421 323 | 65.06388889,1.708355431 324 | 82.19583333,1.608804556 325 | 54.7375,1.426262826 326 | 73.32777778,1.564720956 327 | 86.4875,1.631899613 328 | 63.0281746,1.583388216 329 | 50.4,1.373018518 330 | 78.08472222,1.610581348 331 | 40.90555556,1.272838686 332 | 81.54603175,1.626722463 333 | 65.92222222,1.633557136 334 | 72.25138889,1.636790902 335 | 29.99444444,1.213305755 336 | 77.41111111,1.675363567 337 | 63.88333333,1.627905121 338 | 61.2781746,1.597654945 339 | 75.86746032,1.665000726 340 | 74.41111111,1.672155986 341 | 73.17698413,1.593599049 342 | 55.48333333,1.608010166 343 | 58.33472222,1.74951631 344 | 68.93888889,1.553694866 345 | 66.14027778,1.705608848 346 | 52.74444444,1.729149901 347 | 63.31388889,1.439995928 348 | 62.3,1.55006951 349 | 72.9875,1.614115617 350 | 73.21666667,1.518471068 351 | 59.88333333,1.588781594 352 | 59.78015873,1.560335591 353 | 51.39424603,1.736604177 354 | 58.28809524,1.585589871 355 | 71.82777778,1.55422886 356 | 64.97063492,1.513948839 357 | 66.46666667,1.624246574 358 | 66.10555556,1.690666964 359 | 71.14027778,1.556311012 360 | 76.32083333,1.553920522 361 | 53.99444444,1.702684174 362 | 60.81388889,1.548545106 363 | 49.09166667,1.530558135 364 | 66.37638889,1.65237165 365 | 73.93194444,1.607406582 366 | 66.18888889,1.634994463 367 | 70.48333333,1.488085227 368 | 60.73888889,1.72204356 369 | 68.27916667,1.575028144 370 | 75.20555556,1.539115532 371 | 74.38333333,1.571887311 372 | 63.23253968,1.600209272 373 | 65.95277778,1.512646994 374 | 62.55,1.652897306 375 | 69.10555556,1.546194632 376 | 60.48055556,1.602761344 377 | 54.10555556,1.510399862 378 | 69.18888889,1.516150661 379 | 70.00138889,1.548587378 380 | 62.39722222,1.576126877 381 | 62.42301587,1.652593775 382 | 58.89444444,1.540175629 383 | 69.84166667,1.59746438 384 | 57.10555556,1.537130274 385 | 75.36746032,1.568290565 386 | 44.16111111,1.529213399 387 | 57.75634921,1.549507752 388 | 65.98055556,1.540626987 389 | 66.27222222,1.624473031 390 | 69.10555556,1.621024252 391 | 69.47857143,1.584830584 392 | 56.08472222,1.616393327 393 | 64.36111111,1.611281836 394 | 82.43888889,1.689912303 395 | 58.11031746,1.687824573 396 | 55.25,1.444019236 397 | 75.88888889,1.662052436 398 | 84.40555556,1.715948612 399 | 67.38333333,1.496110577 400 | 25.07638889,1.478951363 401 | 77.02638889,1.572254816 402 | 75.18888889,1.519127588 403 | 79.46666667,1.639746602 404 | 63.03611111,1.463595925 405 | 81.89424603,1.747530093 406 | 79.59444444,1.667031599 407 | 11.93888889,1.40903845 408 | 55.88888889,1.479697579 409 | 81.51746032,1.624121854 410 | 76.63888889,1.680049317 411 | 70.70555556,1.508974013 412 | 23.53888889,1.464688145 413 | 81.07222222,1.691937847 414 | 48.38888889,1.419295597 415 | 57.72638889,1.49566925 416 | 80.93888889,1.707570214 417 | 78.13333333,1.581967114 418 | 73.10555556,1.630578869 419 | 51.13888889,1.465714565 420 | 78.46746032,1.695549589 421 | 46.73888889,1.429291074 422 | 81.38888889,1.613684932 423 | 81.40138889,1.514396352 424 | 81.4547619,1.628369156 425 | 36.74444444,1.510697211 426 | 79.51111111,1.660543389 427 | 40.20138889,1.412435043 428 | 76.55138889,1.6294805 429 | 82.22638889,1.630848319 430 | 48.44444444,1.466852997 431 | 78.78888889,1.546390388 432 | 84.21388889,1.621183543 433 | 69.70277778,1.557977661 434 | 60.48888889,1.490428548 435 | 84.62777778,1.632488134 436 | 42.03888889,1.402826766 437 | 81.13888889,1.619101454 438 | 72.73888889,1.622586414 439 | 75.37638889,1.629996077 440 | 35.34444444,1.432526585 441 | 77.78888889,1.633129853 442 | 76.43888889,1.594146058 443 | 61.76031746,1.570846808 444 | 74.82777778,1.61654063 445 | 76.89444444,1.667109542 446 | 68.96666667,1.703255873 447 | 72.76111111,1.65578818 448 | 47.03888889,1.617876105 449 | 41.45555556,1.643745408 450 | 74.74444444,1.670407566 451 | 79.62638889,1.657169797 452 | 71.63888889,1.662174981 453 | 69.83888889,1.678064389 454 | 39.06666667,1.508950636 455 | 80.8625,1.691311649 456 | 46.11944444,1.600806452 457 | 62.80138889,1.594087318 458 | 88.18888889,1.679663109 459 | 63.38888889,1.668261765 460 | 59.71111111,1.644569717 461 | 56.17222222,1.609789722 462 | 74.95555556,1.660908815 463 | 87.63888889,1.668576744 464 | 50.74444444,1.667868596 465 | 78.23888889,1.676503339 466 | 73.43888889,1.673144463 467 | 65.83888889,1.641636856 468 | 68.45138889,1.634599738 469 | 55.51825397,1.601286036 470 | 86.53333333,1.671708737 471 | 59.6,1.666298258 472 | 86.78888889,1.683805742 473 | 43.48888889,1.584430072 474 | 75.98888889,1.670993213 475 | 46.82777778,1.584575323 476 | 65.22222222,1.575957442 477 | 71.81388889,1.645488932 478 | 67.55138889,1.637378774 479 | 76.12638889,1.604799927 480 | 23.77638889,1.616798354 481 | 84.12222222,1.66610683 482 | 76.78888889,1.640066469 483 | 24.93333333,1.534828194 484 | 72.88888889,1.700862578 485 | 58.93888889,1.635966048 486 | 53.82777778,1.616174344 487 | 85.03888889,1.674836144 488 | 61.08333333,1.638678421 489 | 58.23888889,1.556918929 490 | 53.43888889,1.608437862 491 | 82.27222222,1.692098553 492 | 50.63888889,1.547268557 493 | 66.08888889,1.59545128 494 | 76.48888889,1.648167704 495 | 56.49444444,1.585981681 496 | 67.58888889,1.630187827 497 | 52.68333333,1.723813426 498 | 56.60555556,1.693399001 499 | 41.68888889,1.563661349 500 | 66.30555556,1.739263612 501 | 61.64126984,1.699003015 502 | 56.08174603,1.652915063 503 | 50.98888889,1.693360197 504 | 52.38333333,1.685939228 505 | 72.60555556,1.720388403 506 | 33.35555556,1.563927227 507 | 59.17698413,1.648365892 508 | 69.73055556,1.674278902 509 | 63.16507937,1.714486064 510 | 49.78412698,1.650154364 511 | 35.81984127,1.52922499 512 | 68.12777778,1.700383389 513 | 64.78412698,1.67096919 514 | 59.08888889,1.441630006 515 | 59.04603175,1.685485323 516 | 58.35555556,1.735955293 517 | 54.85555556,1.529702962 518 | 60.4031746,1.702813608 519 | 57.83888889,1.622119771 520 | 45.10555556,1.697364279 521 | 40.18888889,1.467335149 522 | 55.72460317,1.704103367 523 | 65.06388889,1.682446474 524 | 58.15555556,1.635285339 525 | 66.83888889,1.674813856 526 | 62.23055556,1.714867693 527 | 48.68888889,1.509546275 528 | 63.80555556,1.673184089 529 | 46.1531746,1.553401576 530 | 64.02222222,1.623373078 531 | 51.14126984,1.659292475 532 | 44.48888889,1.614866157 533 | 71.53412698,1.673495166 534 | 54.18888889,1.454601626 535 | 58.95972222,1.708964508 536 | 56.32638889,1.658595215 537 | 62.48055556,1.697341054 538 | 58.52777778,1.529892608 539 | 45.02222222,1.571778502 540 | 46.23888889,1.618280163 541 | 59.71666667,1.692228665 542 | 58.10555556,1.649908172 543 | 51.41805556,1.541833205 544 | 31.29603175,1.52643598 545 | 43.02222222,1.507561424 546 | 46.67222222,1.455487249 547 | 69.09960317,1.660790137 548 | 66.48888889,1.694205351 549 | 55.52222222,1.413196939 550 | 14.12638889,1.339069908 551 | 80.8,1.581517338 552 | 72.58888889,1.513713038 553 | 65.05,1.635938464 554 | 63.78888889,1.398227522 555 | 68.87638889,1.7576686 556 | 65.75555556,1.633986441 557 | 56.43888889,1.451711571 558 | 39.43888889,1.378799571 559 | 76.60853175,1.604139488 560 | 62.75,1.657326213 561 | 72.62638889,1.492307934 562 | 58.60555556,1.492892297 563 | 78.13333333,1.681547354 564 | 63.17638889,1.468533293 565 | 46.92222222,1.372939457 566 | 64.07638889,1.683682554 567 | 61.46388889,1.527228631 568 | 71.28412698,1.612720524 569 | 71.47777778,1.438671595 570 | 72.03888889,1.70233774 571 | 46.73888889,1.389700196 572 | 74.18333333,1.581676522 573 | 74.38174603,1.547681272 574 | 64.88888889,1.626685237 575 | 32.98888889,1.377936148 576 | 67.36388889,1.652902888 577 | 23.01666667,1.430453185 578 | 73.32638889,1.643047807 579 | 81.8515873,1.607934723 580 | 65.96666667,1.494156941 581 | 80.13888889,1.546551431 582 | 77.08888889,1.573558637 583 | 52.6125,1.601730839 584 | 51.97777778,1.35738767 585 | 76.38333333,1.561742293 586 | 51.83888889,1.387778785 587 | 63.93194444,1.620313176 588 | 65.03888889,1.593877564 589 | 60.79603175,1.61270298 590 | 60.6,1.401037829 591 | 60.66111111,1.60257839 592 | 67.88333333,1.601161487 593 | 49.96746032,1.564375529 594 | 65.54603175,1.652919901 595 | 61.81666667,1.65702509 596 | 50.58888889,1.676666409 597 | 73.48650794,1.675332116 598 | 69.80138889,1.651985196 599 | 25.40555556,1.473383841 600 | 53.71666667,1.637845726 601 | 41.51666667,1.659273513 602 | 82.08472222,1.673597719 603 | 70.0281746,1.689992096 604 | 54.11944444,1.531488692 605 | 79.94444444,1.643015608 606 | 73.77222222,1.69865162 607 | 41.82777778,1.626487388 608 | 19.93888889,1.63612831 609 | 82.07638889,1.678882676 610 | 75.05,1.633923192 611 | 61.25138889,1.674708902 612 | 76.38333333,1.662018665 613 | 75.03888889,1.700716327 614 | 68.01388889,1.66212858 615 | 44.36388889,1.644943318 616 | 15.43888889,1.367699186 617 | 62.21666667,1.692156105 618 | 52.48888889,1.677731674 619 | 81.49444444,1.703886448 620 | 25.73888889,1.36134149 621 | 70.83888889,1.628688041 622 | 22.27222222,1.247946818 623 | 66.52222222,1.613391489 624 | 62.66388889,1.65669297 625 | 61.98353175,1.656738536 626 | 62.11388889,1.637730578 627 | 54.24047619,1.611888257 628 | 57.2781746,1.63474815 629 | 65.20873016,1.664918478 630 | 36.28888889,1.360062274 631 | 62.25138889,1.620494671 632 | 57.61111111,1.669545741 633 | 13.27916667,1.329882942 634 | 74.91388889,1.689979558 635 | 56.07380952,1.600421048 636 | 47.23888889,1.568786387 637 | 51.41111111,1.570747046 638 | 77.10555556,1.68532175 639 | 17.82638889,1.240086631 640 | 47.48888889,1.609592477 641 | 72.93888889,1.618033567 642 | 56.53888889,1.544625934 643 | 62.33888889,1.636347644 644 | 77.91805556,1.754742899 645 | 71.35555556,1.71495873 646 | 68.43888889,1.688080881 647 | 71.78888889,1.717358524 648 | 19.35,1.606113476 649 | 82.93888889,1.728003683 650 | 22.47638889,1.486438923 651 | 39.55,1.614407082 652 | 83.28888889,1.720033858 653 | 77.10555556,1.731483032 654 | 53.22222222,1.667670254 655 | 40.85555556,1.493153783 656 | 82.57222222,1.704840444 657 | 78.83888889,1.72513449 658 | 16.43888889,1.485598053 659 | 75.22777778,1.722132291 660 | 77.83472222,1.723707996 661 | 40.18888889,1.523111861 662 | 66.49444444,1.713416378 663 | 48.74246032,1.641629351 664 | 82.21666667,1.744745178 665 | 34.77222222,1.470889926 666 | 84.8,1.737505148 667 | 54.00555556,1.646076097 668 | 78.55793651,1.645534668 669 | 47.59365079,1.63404724 670 | 70.27222222,1.678332609 671 | 44.98333333,1.536097249 672 | 71.23888889,1.66285807 673 | 38.95972222,1.530830412 674 | 19.33888889,1.535855825 675 | 73.25138889,1.651897934 676 | 59.67222222,1.642628553 677 | 50.40555556,1.637722845 678 | 55.80138889,1.500714333 679 | 66.42222222,1.704261508 680 | 58.74444444,1.608050265 681 | 80.85555556,1.726643705 682 | 37.21388889,1.494840646 683 | 24.53888889,1.601042546 684 | 34.06388889,1.549938552 685 | 76.88333333,1.742264696 686 | 41.13888889,1.600994238 687 | 42.48888889,1.468034858 688 | 40.4031746,1.562052849 689 | 38.4531746,1.55586228 690 | 50.05555556,1.484073433 691 | 53.32083333,1.436771111 692 | 68.77222222,1.572980226 693 | 60.01388889,1.457243641 694 | 66.64027778,1.664469797 695 | 54.91507937,1.277820961 696 | 77.43888889,1.77947019 697 | 70.75833333,1.667178866 698 | 55.5,1.402016043 699 | 59.04206349,1.288327753 700 | 77.91210317,1.617250269 701 | 72.38333333,1.709382755 702 | 74.55,1.465481758 703 | 57.36666667,1.458697482 704 | 75.23055556,1.743393191 705 | 65.175,1.413221671 706 | 63.28611111,1.278793234 707 | 70.12638889,1.71922004 708 | 72.99444444,1.582057431 709 | 70.41805556,1.617829551 710 | 57.65416667,1.315618031 711 | 70.36666667,1.739909476 712 | 62.33333333,1.332408488 713 | 76.80138889,1.583549702 714 | 72.89722222,1.524872476 715 | 74.75138889,1.601209442 716 | 52.18194444,1.374578555 717 | 76.99444444,1.643899152 718 | 57.13888889,1.453803421 719 | 72.81388889,1.670212428 720 | 76.09166667,1.54851549 721 | 68.93888889,1.466409516 722 | 64.40138889,1.548192018 723 | 70.53611111,1.568461298 724 | 67.06388889,1.595541197 725 | 61.9875,1.330313178 726 | 72.37638889,1.584172217 727 | 64.175,1.3462756 728 | 69.00138889,1.611033395 729 | 68.48333333,1.604341523 730 | 67.46666667,1.608596822 731 | 56.71666667,1.30698701 732 | 59.29444444,1.633895637 733 | 68.11944444,1.631338077 734 | 63.51031746,1.555894382 735 | 78.61746032,1.655709529 736 | 62.01666667,1.6568564 737 | 71.25138889,1.600894422 738 | 73.57638889,1.54816143 739 | 71.89722222,1.608632478 740 | 73.31388889,1.483722465 741 | 76.45674603,1.750680121 742 | 65.45277778,1.624044808 743 | 33.81388889,1.477721504 744 | 52.87638889,1.443239384 745 | 73.35138889,1.597435863 746 | 67.07777778,1.623856274 747 | 69.93888889,1.532896533 748 | 34.12638889,1.484411074 749 | 75.59365079,1.676824183 750 | 67.99603175,1.525723566 751 | 54.18888889,1.473022687 752 | 75.12638889,1.697666747 753 | 58.8625,1.559116788 754 | 69.4656746,1.609796692 755 | 58.65416667,1.487418528 756 | 70.1531746,1.695904649 757 | 50.61388889,1.487658511 758 | 72.53888889,1.605715336 759 | 77.50138889,1.549396376 760 | 80.08174603,1.638544793 761 | 16.52916667,1.334187004 762 | 69.3515873,1.64538417 763 | 26.8031746,1.483797009 764 | 63.18888889,1.614070397 765 | 78.43888889,1.601231549 766 | 57.07777778,1.534609279 767 | 79.26388889,1.579903919 768 | 83.31388889,1.546841867 769 | 58.25138889,1.611359042 770 | 57.87638889,1.462271507 771 | 76.89722222,1.565047096 772 | 52.9531746,1.459418262 773 | 71.13888889,1.610387619 774 | 65.78888889,1.576334599 775 | 73.12638889,1.599226856 776 | 46.37638889,1.474308593 777 | 71.22638889,1.605078561 778 | 65.12638889,1.575934152 779 | 56.0906746,1.569257169 780 | 65.85853175,1.64403891 781 | 64.87638889,1.618239001 782 | 40.61746032,1.531525927 783 | 61.41111111,1.669960729 784 | 85.25138889,1.550527686 785 | 68.85853175,1.712851929 786 | 67.05694444,1.576672937 787 | 79.32083333,1.512935384 788 | 74.17996032,1.575934761 789 | 53.78888889,1.61166079 790 | 39.38333333,1.587639435 791 | 83.31388889,1.491529747 792 | 81.2531746,1.535845748 793 | 72.50138889,1.684277604 794 | 68.83888889,1.507779144 795 | 80.93888889,1.574833726 796 | 65.59166667,1.660167836 797 | 55.81388889,1.607721531 798 | 28.6531746,1.474880973 799 | 46.45277778,1.506026809 800 | 61.08888889,1.714399653 801 | 82.46388889,1.558151844 802 | 20.86388889,1.45666265 803 | 60.32281746,1.54510253 804 | 50.97460317,1.437669643 805 | 76.84166667,1.565709343 806 | 73.44583333,1.611560995 807 | 69.07638889,1.541500008 808 | 65.62638889,1.617381968 809 | 69.18888889,1.586860343 810 | 67.31388889,1.503052899 811 | 54.62460317,1.558183672 812 | 37.25138889,1.46050486 813 | 69.7156746,1.586041267 814 | 62.95277778,1.564810189 815 | 37.34166667,1.422437588 816 | 74.73888889,1.549435599 817 | 67.65416667,1.572984558 818 | 58.76388889,1.577519167 819 | 62.91210317,1.56833377 820 | 71.96666667,1.49996559 821 | 35.91388889,1.443588202 822 | 60.21388889,1.568664285 823 | 60.4031746,1.591151802 824 | 64.51031746,1.501094247 825 | 63.95138889,1.641741993 826 | 66.87222222,1.692403249 827 | 73.53888889,1.46276294 828 | 76.34861111,1.756692126 829 | 67.06666667,1.62212906 830 | 71.18888889,1.478690008 831 | 67.56388889,1.423693763 832 | 28.68888889,1.486153685 833 | 31.72777778,1.649643795 834 | 79.80555556,1.479503199 835 | 74.38888889,1.509261312 836 | 74.77222222,1.692443975 837 | 68.18888889,1.483119932 838 | 73.35138889,1.449114916 839 | 73.13888889,1.685293766 840 | 22.52777778,1.44656355 841 | 56.34365079,1.622256849 842 | 51.22777778,1.385392186 843 | 60.08888889,1.712917278 844 | 73.48888889,1.434431085 845 | 48.93888889,1.547913001 846 | 31.72460317,1.500256081 847 | 73.24603175,1.568115911 848 | 75.9,1.550023358 849 | 72.02222222,1.633292171 850 | 71.12638889,1.46819417 851 | 72.51388889,1.65427339 852 | 72.85138889,1.576386465 853 | 63.78888889,1.459874429 854 | 65.18888889,1.508718402 855 | 74.51388889,1.592079366 856 | 68.87638889,1.537040806 857 | 53.53888889,1.485795151 858 | 58.70555556,1.553648147 859 | 67.53888889,1.377200778 860 | 70.48888889,1.588553387 861 | 68.38888889,1.580061261 862 | 60.1531746,1.530664433 863 | 56.75,1.420301409 864 | 71.58888889,1.619849788 865 | 68.53888889,1.610720025 866 | 57.18174603,1.581883291 867 | 70.33888889,1.587035297 868 | 68.03333333,1.62853171 869 | 78.7375,1.690790631 870 | 42.3,1.59102218 871 | 24.82777778,1.589047173 872 | 84.4,1.660636908 873 | 73.9468254,1.678928894 874 | 38.98888889,1.632318771 875 | 22.05,1.578971119 876 | 84.77222222,1.674599222 877 | 79.23333333,1.653467058 878 | 36.87638889,1.637744344 879 | 68.72222222,1.666989168 880 | 68.38333333,1.680957932 881 | 28.20555556,1.589070341 882 | 65.32777778,1.648753538 883 | 47.70873016,1.617599037 884 | 79.66111111,1.677595129 885 | 19.81666667,1.584703391 886 | 85.51111111,1.683011393 887 | 39.80555556,1.611099453 888 | 71.75634921,1.596305772 889 | 34.37539683,1.656404251 890 | 45.71666667,1.560829484 891 | 48.05,1.59623628 892 | 65.90138889,1.618805143 893 | 34.04305556,1.538125723 894 | 29.82083333,1.541064959 895 | 71.60555556,1.629228794 896 | 40.01666667,1.595099316 897 | 26.77222222,1.615311768 898 | 53.54305556,1.666112033 899 | 64.18333333,1.648049523 900 | 50.49444444,1.637422366 901 | 80.27222222,1.679225688 902 | 22.11666667,1.536304337 903 | 35.82777778,1.491315171 904 | 18.85555556,1.544724938 905 | 75.425,1.691137939 906 | 26.00555556,1.613343459 907 | 18.4,1.541330682 908 | 28.67698413,1.556457824 909 | 48.7156746,1.619460163 910 | 36.85555556,1.550619009 911 | 80.32281746,1.78506419 912 | 83.05694444,1.668168268 913 | 73.66388889,1.417125849 914 | 49.68888889,1.237758258 915 | 75.81388889,1.590252342 916 | 78.9875,1.689334704 917 | 44.49444444,1.356726919 918 | 81.13730159,1.472203917 919 | 80.55793651,1.705742162 920 | 76.11944444,1.41703735 921 | 37.62638889,1.167996608 922 | 75.50138889,1.726892766 923 | 79.18194444,1.553870889 924 | 83.17698413,1.639139532 925 | 75.8,1.327786627 926 | 75.48888889,1.733501268 927 | 42.73888889,1.192442948 928 | 82.12638889,1.576430583 929 | 73.09960317,1.532410523 930 | 72.73353175,1.638456743 931 | 68.93888889,1.45653253 932 | 73.14722222,1.645278372 933 | 67.53888889,1.460030937 934 | 78.12638889,1.665964711 935 | 85.50138889,1.634726393 936 | 71.7781746,1.493919444 937 | 74.6031746,1.579276769 938 | 70.50833333,1.574852985 939 | 70.75138889,1.579206668 940 | 42.05,1.22205917 941 | 83.06388889,1.606797997 942 | 50.56388889,1.263444584 943 | 81.7031746,1.660536977 944 | 80.30138889,1.616180115 945 | 79.00138889,1.643546044 946 | 37.00833333,1.201034063 947 | 81.12638889,1.6320414 948 | 85.00138889,1.632315775 949 | 72.43888889,1.573764803 950 | 77.3406746,1.667659272 951 | 79.58888889,1.679666574 952 | 47.50138889,1.661707368 953 | 79.37638889,1.742558676 954 | 75.06388889,1.777203131 955 | 59.79305556,1.686595783 956 | 50.12638889,1.552220888 957 | 73.60555556,1.721616541 958 | 79.40138889,1.739636359 959 | 32.70972222,1.462259816 960 | 68.56388889,1.738621453 961 | 70.70972222,1.771128728 962 | 36.07777778,1.48919314 963 | 77.8406746,1.727627741 964 | 70.2781746,1.682336656 965 | 82.79603175,1.784407795 966 | 48.91111111,1.47368603 967 | 81.8406746,1.761794316 968 | 67.33472222,1.686774393 969 | 69.75138889,1.681788273 970 | 64.23888889,1.668060369 971 | 69.62638889,1.719453706 972 | 53.43888889,1.498658134 973 | 69.49444444,1.713539446 974 | 48.9656746,1.546298916 975 | 45.29305556,1.597516871 976 | 69.25138889,1.697132151 977 | 54.56388889,1.64114445 978 | 60.62638889,1.703619367 979 | 52.1531746,1.557875458 980 | 70.43888889,1.739138253 981 | 66.81388889,1.66136754 982 | 78.61746032,1.773806762 983 | 58.37638889,1.532294256 984 | 31.68888889,1.567275748 985 | 43.68888889,1.567475172 986 | 74.79305556,1.770199512 987 | 49.81388889,1.637543167 988 | 46.48055556,1.513416621 989 | 47.36746032,1.554124634 990 | 44.42996032,1.511948854 991 | 53.84166667,1.429892482 992 | 78.23888889,1.667208737 993 | 65.3,1.666476286 994 | 45.27638889,1.557319115 995 | 33.70555556,1.582881135 996 | 85.66111111,1.653637548 997 | 79.82777778,1.635471219 998 | 64.46666667,1.654091207 999 | 73.93888889,1.630220982 1000 | 72.91388889,1.680080757 1001 | 44.92638889,1.560196941 1002 | 61.71666667,1.624922698 1003 | 50.86746032,1.518323124 1004 | 77.88333333,1.657620274 1005 | 42.00555556,1.609759612 1006 | 70.34603175,1.675892917 1007 | 54.1,1.526506414 1008 | 67.44603175,1.62597207 1009 | 43.08174603,1.506462463 1010 | 65.05694444,1.583101617 1011 | 65.38333333,1.572895771 1012 | 72.98174603,1.620993137 1013 | 24.50138889,1.484483848 1014 | 51.39722222,1.563516157 1015 | 74.32777778,1.626442018 1016 | 57.51031746,1.539984236 1017 | 59.74444444,1.496275699 1018 | 54.06388889,1.573105584 1019 | 62.44444444,1.649868965 1020 | 56.05,1.533072898 1021 | 71.37638889,1.624155805 1022 | 31.62936508,1.553289378 1023 | 19.67638889,1.46365701 1024 | 36.37638889,1.512519848 1025 | 70.27222222,1.666435438 1026 | 29.21388889,1.431068687 1027 | 35.38333333,1.570183196 1028 | 55.36746032,1.592535894 1029 | 58.7156746,1.460123366 1030 | 52.58888889,1.560972561 1031 | 63.96388889,1.455424389 1032 | 82.5,1.595773254 1033 | 77.31666667,1.65456667 1034 | 72.925,1.471978675 1035 | 16.52222222,1.266102539 1036 | 83.12638889,1.68009411 1037 | 48,1.398966705 1038 | 68.12638889,1.448261915 1039 | 83.58888889,1.696503461 1040 | 75.17777778,1.563096094 1041 | 75.58968254,1.623735065 1042 | 49.76111111,1.387170232 1043 | 79.98888889,1.709736654 1044 | 52.93888889,1.395281143 1045 | 82.78888889,1.608989118 1046 | 79.20972222,1.477515137 1047 | 84.55793651,1.589872404 1048 | 52.64444444,1.493649844 1049 | 82.35555556,1.619992974 1050 | 62.16746032,1.436058414 1051 | 77.87638889,1.653578178 1052 | 85.98888889,1.604349705 1053 | 54.60555556,1.379460845 1054 | 76.86666667,1.529825918 1055 | 87.98888889,1.600878283 1056 | 72.43888889,1.550849115 1057 | 55.48888889,1.435952522 1058 | 85.98333333,1.589794967 1059 | 50.18888889,1.37071961 1060 | 83.82777778,1.595144109 1061 | 74.33888889,1.591888769 1062 | 82.72460317,1.597762804 1063 | 35.75555556,1.319194086 1064 | 80.73888889,1.644064156 1065 | 79.48888889,1.592766277 1066 | 73.13888889,1.578999799 1067 | 75.1531746,1.611518112 1068 | 77.15,1.643280132 1069 | 73.76746032,1.595869644 1070 | 66.89027778,1.65260595 1071 | 48.38333333,1.392945405 1072 | 70.70138889,1.465202412 1073 | 80.12936508,1.693259985 1074 | 68.51388889,1.431435971 1075 | 37.12638889,1.035873052 1076 | 69.47638889,1.682384154 1077 | 64.25833333,1.551732469 1078 | 75.47460317,1.591227002 1079 | 44.57777778,1.262527123 1080 | 68.36388889,1.718335473 1081 | 24.96388889,1.159650421 1082 | 68.72638889,1.583511873 1083 | 77.06388889,1.564635038 1084 | 74.50138889,1.629387189 1085 | 55.55694444,1.453680257 1086 | 68.89722222,1.618518415 1087 | 63.46031746,1.45368326 1088 | 67.56388889,1.625824202 1089 | 79.75138889,1.650069405 1090 | 57.25833333,1.507849492 1091 | 73.17638889,1.531139147 1092 | 71.62638889,1.580192926 1093 | 60.87638889,1.579941497 1094 | 11.61388889,1.076711764 1095 | 74.33571429,1.597398966 1096 | 35.87638889,1.215533016 1097 | 72.9531746,1.634279591 1098 | 66.36388889,1.575993409 1099 | 70.12638889,1.628302752 1100 | 35.97361111,1.265164005 1101 | 66.05138889,1.651530526 1102 | 71.75138889,1.616507286 1103 | 64.68888889,1.561079431 1104 | 70.57281746,1.639384204 1105 | 70.11388889,1.632348424 1106 | 21.41666667,1.633563349 1107 | 80.77222222,1.588201925 1108 | 80.40555556,1.597843177 1109 | 62.29305556,1.633431543 1110 | 73.67638889,1.596197667 1111 | 72.06746032,1.588464673 1112 | 40.37222222,1.635249714 1113 | 34.31388889,1.510411452 1114 | 51.73253968,1.62314634 1115 | 64.58888889,1.589960727 1116 | 36.33333333,1.643831892 1117 | 79.68333333,1.585720674 1118 | 41.13888889,1.547301374 1119 | 28.88888889,1.547706668 1120 | 49.71666667,1.590586981 1121 | 72.51031746,1.589621564 1122 | 51.32777778,1.638665814 1123 | 66.43333333,1.570389601 1124 | 45.62638889,1.548528708 1125 | 57.25138889,1.615153916 1126 | 73.67460317,1.583653443 1127 | 51.27222222,1.562639405 1128 | 58.05,1.577457496 1129 | 57.12638889,1.576583423 1130 | 57.01111111,1.591930921 1131 | 54.22222222,1.570142806 1132 | 72.80138889,1.570866616 1133 | 53.78888889,1.592461449 1134 | 56.18888889,1.556263061 1135 | 34.26388889,1.560827176 1136 | 73.09444444,1.559756328 1137 | 53.67222222,1.559558555 1138 | 38.29444444,1.596491355 1139 | 30.33888889,1.580387418 1140 | 70.04603175,1.558800167 1141 | 48.58888889,1.602132268 1142 | 80.88333333,1.658250589 1143 | 77.31666667,1.65885757 1144 | 44.96666667,1.584872546 1145 | 73.10138889,1.663173651 1146 | 77.97361111,1.668470473 1147 | 33.63333333,1.522090965 1148 | 29.60555556,1.655591086 1149 | 27.91210317,1.575063514 1150 | 51.77222222,1.685759965 1151 | 26.37222222,1.541575966 1152 | 77.30555556,1.685628042 1153 | 26.25555556,1.586495117 1154 | 46.73055556,1.601902583 1155 | 38.96269841,1.611131263 1156 | 59.32777778,1.609376246 1157 | 50.66111111,1.556056527 1158 | 63.52638889,1.595863719 1159 | 39.79305556,1.58050547 1160 | 52.3625,1.600367093 1161 | 69.49444444,1.641793548 1162 | 40.63888889,1.582151671 1163 | 50.71666667,1.596106899 1164 | 46.0281746,1.591298058 1165 | 44.61666667,1.634247899 1166 | 33.71666667,1.591205825 1167 | 76.10555556,1.696040272 1168 | 39.42222222,1.534088334 1169 | 39.38888889,1.547965575 1170 | 27.10555556,1.549929063 1171 | 71.27222222,1.694367435 1172 | 26.49444444,1.57858585 1173 | 27.58333333,1.479166876 1174 | 24.08968254,1.500835414 1175 | 56.57281746,1.54229099 1176 | 42.78333333,1.442537342 1177 | 76.39444444,1.482209497 1178 | 67.62638889,1.636985546 1179 | 62.05,1.458746676 1180 | 48.46666667,1.385481889 1181 | 83.16111111,1.688239621 1182 | 82.96666667,1.595358553 1183 | 76.79603175,1.655127287 1184 | 67.82777778,1.415418338 1185 | 81.24444444,1.720388814 1186 | 36.13333333,1.341861209 1187 | 84.16111111,1.592506905 1188 | 83.34166667,1.537383657 1189 | 80.0281746,1.600896649 1190 | 78.88333333,1.522956302 1191 | 83.22361111,1.626072215 1192 | 71.78888889,1.494792317 1193 | 82.87638889,1.642856441 1194 | 73.6125,1.6260521 1195 | 77.77222222,1.428682662 1196 | 82.75138889,1.572403092 1197 | 84.16111111,1.570256526 1198 | 80.37638889,1.617383444 1199 | 42.85,1.393441479 1200 | 84.71666667,1.612823304 1201 | 49.93888889,1.379110035 1202 | 84.68888889,1.628542195 1203 | 70.27222222,1.624858548 1204 | 77.43888889,1.614075042 1205 | 32.38333333,1.211924503 1206 | 80.49444444,1.622425622 1207 | 85.05,1.593437572 1208 | 81.24047619,1.568702077 1209 | 71.77222222,1.623924416 1210 | 76.38333333,1.674392825 1211 | 80.83472222,1.667820426 1212 | 60.86111111,1.466400864 1213 | 70.62638889,1.481242676 1214 | 77.03888889,1.701132759 1215 | 75.91111111,1.581333901 1216 | 76.13333333,1.615778507 1217 | 58.60555556,1.440272777 1218 | 72.06111111,1.699447177 1219 | 61.68888889,1.486988868 1220 | 77.98888889,1.612317238 1221 | 78.76388889,1.499482959 1222 | 81.71666667,1.600111512 1223 | 51.69444444,1.503112402 1224 | 77.53888889,1.651067866 1225 | 63.02916667,1.489340117 1226 | 79.68888889,1.609544786 1227 | 83.57083333,1.578167386 1228 | 49.49444444,1.456481173 1229 | 80.28888889,1.556144038 1230 | 82.76388889,1.614521014 1231 | 72.43888889,1.583837393 1232 | 68.51388889,1.462728492 1233 | 80.95138889,1.616106239 1234 | 63.90138889,1.427020405 1235 | 81.28888889,1.567955309 1236 | 71.13888889,1.558276934 1237 | 77.74444444,1.610226489 1238 | 48.7375,1.408073374 1239 | 73.38888889,1.629505961 1240 | 81.03888889,1.590494427 1241 | 70.56031746,1.555154922 1242 | 77.24047619,1.61302372 1243 | 74.33888889,1.618433602 1244 | 65.79305556,1.676979475 1245 | 76.18888889,1.715537585 1246 | 46.43888889,1.545854176 1247 | 79.42698413,1.677275501 1248 | 70.27222222,1.623900837 1249 | 78.41111111,1.721405988 1250 | 43.75555556,1.463187706 1251 | 82.98055556,1.705917336 1252 | 66.85555556,1.666924275 1253 | 70.05555556,1.612379636 1254 | 63.18888889,1.697547058 1255 | 72.6531746,1.681334545 1256 | 51.29305556,1.431859997 1257 | 68.75555556,1.650394643 1258 | 52.20972222,1.524417479 1259 | 27.83174603,1.577916848 1260 | 79.76031746,1.67787912 1261 | 64.83888889,1.58134183 1262 | 70.31388889,1.685589813 1263 | 56.56984127,1.370635299 1264 | 71.31388889,1.696609096 1265 | 71.58472222,1.634939646 1266 | 78.04305556,1.676022229 1267 | 43.93888889,1.513878404 1268 | 33.88333333,1.544403645 1269 | 38.23888889,1.578849083 1270 | 71.32777778,1.686812297 1271 | 57.49444444,1.628477541 1272 | 52.27222222,1.519607638 1273 | 39.18888889,1.493427 1274 | 47.42222222,1.456743711 1275 | 52.60555556,1.428590925 1276 | 64.87638889,1.417699289 1277 | 80.68333333,1.635327925 1278 | 75.36666667,1.525323272 1279 | 68.17698413,1.628635 1280 | 50.12777778,1.403646565 1281 | 71.16388889,1.717350052 1282 | 55.78888889,1.377492896 1283 | 79.53888889,1.563952729 1284 | 72.77638889,1.49193917 1285 | 75.76428571,1.59401459 1286 | 62.45,1.525147704 1287 | 73.82777778,1.64882015 1288 | 53.24603175,1.311943032 1289 | 75.87638889,1.648814193 1290 | 71.25833333,1.617323333 1291 | 66.21666667,1.384610673 1292 | 69.50138889,1.471459118 1293 | 81.58888889,1.55188766 1294 | 70.43888889,1.51759303 1295 | 63.18888889,1.427813755 1296 | 73.55,1.571767282 1297 | 58.53888889,1.352616535 1298 | 79.6531746,1.612653958 1299 | 47.77777778,1.590429362 1300 | 78.50138889,1.582729533 1301 | 58.93888889,1.365626313 1302 | 76.33888889,1.602918956 1303 | 75.02916667,1.608591439 1304 | 64.03412698,1.480581585 1305 | 55.58174603,1.583013909 1306 | 73.16111111,1.656626659 1307 | 70.23888889,1.689510063 1308 | 72.18888889,1.521670493 1309 | 74.18888889,1.597665174 1310 | 54.23055556,1.29552806 1311 | 58.03888889,1.707980322 1312 | 29.38888889,1.118075413 1313 | 71.75138889,1.59135338 1314 | 77.31388889,1.570591403 1315 | 68.25138889,1.635371497 1316 | 63.43888889,1.435218741 1317 | 67.20277778,1.641934674 1318 | 52.65555556,1.449832788 1319 | 61.87638889,1.659814113 1320 | 80.00138889,1.655043229 1321 | 60.51825397,1.49856178 1322 | 74.26388889,1.591774155 1323 | 74.87638889,1.573093503 1324 | 66.7484127,1.579357119 1325 | 38.78888889,1.039635342 1326 | 70.52222222,1.611432886 1327 | 27.52638889,1.222510338 1328 | 72.07638889,1.649029574 1329 | 74.76388889,1.590961701 1330 | 66.31388889,1.661166665 1331 | 32.12638889,1.252454759 1332 | 75.22638889,1.633513634 1333 | 71.56388889,1.634851122 1334 | 52.67103175,1.553282279 1335 | 64.22460317,1.647845557 1336 | 67.43888889,1.664103184 1337 | 75.19444444,1.684640792 1338 | 55.5281746,1.635606037 1339 | 80.20555556,1.697081273 1340 | 20.33888889,1.468857025 1341 | 83.08888889,1.726489496 1342 | 67.48888889,1.60525865 1343 | 57.5031746,1.632236296 1344 | 41.16904762,1.637188416 1345 | 66.11388889,1.660549964 1346 | 21.88333333,1.536215884 1347 | 63.07460317,1.637607373 1348 | 23.81388889,1.480751827 1349 | 54.02916667,1.604429159 1350 | 75.43888889,1.662687585 1351 | 32.37222222,1.586553198 1352 | 48.52222222,1.589736815 1353 | 28.87638889,1.592144414 1354 | 65.88888889,1.679986921 1355 | 56.88333333,1.614747794 1356 | 77.51111111,1.6969903 1357 | 33.33888889,1.575463595 1358 | 48.93888889,1.54469902 1359 | 31.96388889,1.523181417 1360 | 76.55555556,1.730703247 1361 | 39.73888889,1.596939273 1362 | 28.53888889,1.494893116 1363 | 16.01825397,1.528083795 1364 | 39.81190476,1.519243956 1365 | 38.15555556,1.493846519 1366 | 58.86746032,1.641850037 1367 | 51.16111111,1.546982233 1368 | 59.30138889,1.688943199 1369 | 82.46111111,1.546889898 1370 | 46.05138889,1.588735757 1371 | 61.04305556,1.564695335 1372 | 75.70079365,1.601224728 1373 | 64.77222222,1.592797712 1374 | 68.77222222,1.660073813 1375 | 66.42460317,1.533998234 1376 | 62.77222222,1.579920627 1377 | 73.51527778,1.647116848 1378 | 74.21666667,1.554674452 1379 | 74.78888889,1.55896493 1380 | 69.18888889,1.599242885 1381 | 71.18888889,1.590245389 1382 | 52.91666667,1.528595629 1383 | 49.29305556,1.588089558 1384 | 73.06111111,1.520418878 1385 | 72.97222222,1.640662417 1386 | 68.64444444,1.580125198 1387 | 65.55,1.603944458 1388 | 64.43888889,1.530582646 1389 | 63.51111111,1.598634386 1390 | 61.98333333,1.629388356 1391 | 59.85952381,1.578803014 1392 | 72.6531746,1.604383286 1393 | 68.68888889,1.653328161 1394 | 57.51031746,1.616654052 1395 | 48.08968254,1.618398791 1396 | 76.41031746,1.63326349 1397 | 38.39920635,1.425351933 1398 | 66.28412698,1.630634834 1399 | 36.47460317,1.279642871 1400 | 70.29603175,1.581785033 1401 | 66.93888889,1.614934803 1402 | 72.03888889,1.649761161 1403 | 57.8406746,1.593169084 1404 | 68.17698413,1.555571028 1405 | 59.22460317,1.615713859 1406 | 56.8,1.642546324 1407 | 43.26924603,1.390826997 1408 | 64.58174603,1.607069697 1409 | 57.58888889,1.57730626 1410 | 15.35853175,1.287415499 1411 | 75.01031746,1.628561665 1412 | 44.08174603,1.622739018 1413 | 57.81388889,1.547123623 1414 | 55.78174603,1.535183376 1415 | 76.81388889,1.620609557 1416 | 27.26924603,1.308924863 1417 | 52.31746032,1.615716096 1418 | 60.55793651,1.609925659 1419 | 63.70079365,1.464673311 1420 | 58.69285714,1.583142927 1421 | 75.47638889,1.751630517 1422 | 38.93333333,1.233695726 1423 | 67.68333333,1.566056668 1424 | 69.425,1.49454641 1425 | 74.98353175,1.605457216 1426 | 74.175,1.495517513 1427 | 77.88333333,1.622502072 1428 | 71.27222222,1.438922811 1429 | 69.49246032,1.65378206 1430 | 83.93888889,1.61138983 1431 | 53.66111111,1.444750004 1432 | 70.88174603,1.511387273 1433 | 87.25138889,1.57529344 1434 | 68.44583333,1.555553148 1435 | 36.84444444,1.300977177 1436 | 72.84861111,1.574015177 1437 | 49.09444444,1.230902094 1438 | 84.88888889,1.605020209 1439 | 58.79444444,1.602328609 1440 | 74.62638889,1.578695042 1441 | 33.16111111,1.254628802 1442 | 79.52222222,1.640019491 1443 | 76.60555556,1.628704431 1444 | 68.7484127,1.567296038 1445 | 78.0906746,1.631997314 1446 | 74.24444444,1.648346636 1447 | 78.08333333,1.731175114 1448 | 56.53888889,1.672763467 1449 | 58.47222222,1.642168209 1450 | 52.08174603,1.673758008 1451 | 60.22638889,1.637691794 1452 | 39.65,1.474282254 1453 | 60.48888889,1.681752766 1454 | 36.88888889,1.539124659 1455 | 49.45138889,1.540399146 1456 | 59.6031746,1.653535116 1457 | 34.11666667,1.616432975 1458 | 55.18333333,1.685131331 1459 | 35.32083333,1.48797588 1460 | 68.68888889,1.677602136 1461 | 62.28888889,1.666542548 1462 | 73.00138889,1.712311104 1463 | 17.94444444,1.514524673 1464 | 44.83888889,1.544453868 1465 | 23.03888889,1.544595507 1466 | 70.46111111,1.716036191 1467 | 28.63888889,1.63734126 1468 | 20.42222222,1.481009103 1469 | 24.5531746,1.529275639 1470 | 59.90714286,1.550520685 1471 | 35.18888889,1.410756103 1472 | 84.23888889,1.601810049 1473 | 83.12638889,1.520421676 1474 | 80.22460317,1.62654007 1475 | 74.86666667,1.469409953 1476 | 74.03333333,1.615996676 1477 | 58.23888889,1.408628135 1478 | 78.82777778,1.657637129 1479 | 81.04305556,1.621936027 1480 | 44.83888889,1.417142623 1481 | 77.49444444,1.558700576 1482 | 86.47638889,1.590525341 1483 | 72.18888889,1.575954996 1484 | 39.28888889,1.160300695 1485 | 85.90416667,1.600013125 1486 | 13.48888889,1.00290482 1487 | 78.91388889,1.629149963 1488 | 75.78888889,1.621683298 1489 | 74.49444444,1.631971708 1490 | 13.48333333,1.075630308 1491 | 78.42222222,1.639741867 1492 | 78.21111111,1.625459796 1493 | 70.99603175,1.552663414 1494 | 76.09960317,1.643395998 1495 | 71.88333333,1.663641433 1496 | 71.62460317,1.528665552 1497 | 38.89920635,1.355225229 1498 | 68.39444444,1.560190561 1499 | 69.22777778,1.637059543 1500 | 71.13888889,1.520500114 1501 | 59.90138889,1.573279827 1502 | 52.07638889,1.582357996 1503 | 78.19444444,1.564789927 1504 | 58.23888889,1.54010049 1505 | 33.90138889,1.35999569 1506 | 64.98888889,1.616950504 1507 | 61.33888889,1.575167299 1508 | 35.03333333,1.357378223 1509 | 79.73888889,1.565719457 1510 | 63.88888889,1.558727109 1511 | 55.58888889,1.562710476 1512 | 50.61388889,1.496978356 1513 | 78.34444444,1.598208448 1514 | 36.58888889,1.361684367 1515 | 53.08888889,1.552874878 1516 | 66.08174603,1.555162515 1517 | 64.79603175,1.479346782 1518 | 62.73888889,1.620103831 1519 | 75.3531746,1.597231547 1520 | 79.51031746,1.539370513 1521 | 64.2484127,1.58186606 1522 | 73.93888889,1.47062156 1523 | 71.47460317,1.567114593 1524 | 69.08472222,1.56467958 1525 | 76.7484127,1.492995024 1526 | 59.97222222,1.479493531 1527 | 76.87638889,1.603718704 1528 | 67.85555556,1.513076398 1529 | 74.21388889,1.556813766 1530 | 71.63333333,1.559968879 1531 | 77.32638889,1.487459519 1532 | 74.71031746,1.525676001 1533 | 70.41388889,1.576794632 1534 | 50.9656746,1.533548155 1535 | 73.31984127,1.507292356 1536 | 71.21388889,1.600934636 1537 | 54.88888889,1.483643185 1538 | 51.08174603,1.509114726 1539 | 68.34365079,1.564356748 1540 | 64.48888889,1.560557147 1541 | 58.45972222,1.59538979 1542 | 67.74047619,1.648168877 1543 | 66.74444444,1.591844326 1544 | 69.54603175,1.618299435 1545 | 50.08174603,1.546465663 1546 | 62.92103175,1.57414217 1547 | 57.60555556,1.547380797 1548 | 16.78809524,1.279091268 1549 | 61.54603175,1.623656789 1550 | 71.43174603,1.622448805 1551 | 12.07777778,1.27307384 1552 | 80.79603175,1.628796229 1553 | 35.73650794,1.559590319 1554 | 56.78809524,1.561582041 1555 | 49.23650794,1.513898385 1556 | 74.61944444,1.602297731 1557 | 18.03412698,1.21345732 1558 | 47.25833333,1.609193906 1559 | 62.93888889,1.594933949 1560 | 65.17698413,1.476995291 1561 | 59.43888889,1.609338801 1562 | 70.93888889,1.623213748 1563 | 36.33472222,1.475023255 1564 | 65.78611111,1.598144823 1565 | 73.07083333,1.529578784 1566 | 71.99444444,1.524454192 1567 | 71.83888889,1.542447258 1568 | 68.01527778,1.508772685 1569 | 63.37638889,1.614956092 1570 | 59.01666667,1.442163366 1571 | 71.37638889,1.547191257 1572 | 70.91111111,1.466169861 1573 | 53.92301587,1.566136879 1574 | 60.81666667,1.546367694 1575 | 66.37638889,1.567751501 1576 | 68.32777778,1.490268496 1577 | 59.38333333,1.570501993 1578 | 63.51388889,1.549865179 1579 | 61.41805556,1.58402686 1580 | 64.33174603,1.621287826 1581 | 62.33888889,1.583494466 1582 | 56.72222222,1.599365887 1583 | 42.90416667,1.485964059 1584 | 65.18888889,1.541910399 1585 | 68.66111111,1.610956047 1586 | 39.68888889,1.506604422 1587 | 77.38333333,1.652370053 1588 | 25.93888889,1.317964847 1589 | 69.49444444,1.628492511 1590 | 72.18888889,1.619675392 1591 | 69.60555556,1.608225259 1592 | 43.63888889,1.435683002 1593 | 64.79444444,1.519352134 1594 | 30.3,1.498441205 1595 | 66.05,1.632970117 1596 | 64.27777778,1.60786535 1597 | 41.27222222,1.501092332 1598 | 23.78809524,1.49112539 1599 | 68.31984127,1.523044928 1600 | 50.28333333,1.460933759 1601 | 64.48888889,1.594333847 1602 | 73.28888889,1.564939063 1603 | 45.42638889,1.40379435 1604 | 54.38888889,1.464122431 1605 | 79.00138889,1.561132497 1606 | 50.76388889,1.507370227 1607 | 62.72222222,1.466617925 1608 | 75.06388889,1.569072307 1609 | 55.34444444,1.405874753 1610 | 68.92222222,1.560932191 1611 | 63.22638889,1.560023801 1612 | 63.93888889,1.562418047 1613 | 54.38333333,1.415156623 1614 | 60.31388889,1.568678386 1615 | 65.0031746,1.548648123 1616 | 58.82460317,1.480830877 1617 | 53.83888889,1.568809392 1618 | 66.93888889,1.619740728 1619 | 60.12638889,1.533770742 1620 | 63.69583333,1.590817379 1621 | 60.87638889,1.532018109 1622 | 72.29305556,1.613709501 1623 | 45.00138889,1.527097054 1624 | 68.73888889,1.625125383 1625 | 70.09861111,1.604521551 1626 | 69.93888889,1.628133973 1627 | 40.52638889,1.497585502 1628 | 34.85138889,1.519074857 1629 | 34.50138889,1.487297889 1630 | 70.37638889,1.660860458 1631 | 52.81388889,1.562319577 1632 | 43.88333333,1.46699168 1633 | 34.92103175,1.480077734 1634 | 56.53710317,1.484397646 1635 | 49.53888889,1.455851828 1636 | 83.31388889,1.569410786 1637 | 56.40714286,1.509422295 1638 | 46.85555556,1.544802374 1639 | 57.97460317,1.56760762 1640 | 74.38888889,1.630857922 1641 | 57.09761905,1.53320038 1642 | 81.87638889,1.613581499 1643 | 43.61349206,1.448246746 1644 | 41.10138889,1.511329849 1645 | 48.25138889,1.433268935 1646 | 81.06388889,1.631128237 1647 | 54.79305556,1.540739603 1648 | 48.38888889,1.49432643 1649 | 58.81388889,1.515503541 1650 | 51.58174603,1.530315031 1651 | 60.40138889,1.471626952 1652 | 69.35,1.452630888 1653 | 81.06388889,1.589718137 1654 | 55.57083333,1.553336176 1655 | 61.62777778,1.476427282 1656 | 72.57083333,1.576044943 1657 | 43.77222222,1.376317132 1658 | 77.73888889,1.575792344 1659 | 68.83888889,1.583919337 1660 | 65.19781746,1.567579825 1661 | 44.77222222,1.357248085 1662 | 64.71388889,1.608348854 1663 | 68.27222222,1.540759316 1664 | 54.18492063,1.534325872 1665 | 63.49444444,1.603892156 1666 | 67.60555556,1.609742416 1667 | 50.48888889,1.57293053 1668 | 62.03888889,1.49657073 1669 | 76.43888889,1.559923928 1670 | 60.90555556,1.560219058 1671 | 67.61666667,1.509512761 1672 | 37.91388889,1.479804057 1673 | 64.23888889,1.532260105 1674 | 37.38174603,1.489958849 1675 | 75.01666667,1.531106282 1676 | 44.32777778,1.568052848 1677 | 36.35138889,1.50644409 1678 | 35.21031746,1.483306094 1679 | 63.04305556,1.476351763 1680 | 50.78888889,1.541719407 1681 | 72.68888889,1.608016671 1682 | 71.92638889,1.547501156 1683 | 27.71666667,1.307978068 1684 | 85.10138889,1.611630588 1685 | 60.16031746,1.555715565 1686 | 51.12222222,1.524138894 1687 | 52.1125,1.512574279 1688 | 84.46666667,1.582779605 1689 | 18.25,1.259544093 1690 | 55.66388889,1.586063891 1691 | 79.32777778,1.577424723 1692 | 59.28710317,1.493312264 1693 | 61.05,1.621115824 1694 | 65.07083333,1.603783954 1695 | 68.6531746,1.546055312 1696 | 70.58174603,1.532428531 1697 | 50.04603175,1.434700087 1698 | 61.95277778,1.523247506 1699 | 39.08472222,1.514338603 1700 | 59.35555556,1.565481984 1701 | 66.81388889,1.585105041 1702 | 55.43888889,1.458779657 1703 | 32.08968254,1.470389821 1704 | 59.72460317,1.407499182 1705 | 49.26388889,1.457712283 1706 | 62.27222222,1.587325966 1707 | 37.43888889,1.256340556 1708 | 72.10555556,1.646229901 1709 | 54.88888889,1.540486465 1710 | 70.06388889,1.628035207 1711 | 33.85,1.292811317 1712 | 60.28888889,1.611713767 1713 | 69.28888889,1.618869731 1714 | 60.97460317,1.554102407 1715 | 71.52460317,1.6376061 1716 | 64.88888889,1.654831898 1717 | 80.41507937,1.587347932 1718 | 54.89722222,1.496568211 1719 | 52.93888889,1.536146076 1720 | 50.02222222,1.501653349 1721 | 81.32777778,1.596163816 1722 | 19.55,1.257871038 1723 | 63.25,1.547163798 1724 | 65.68492063,1.594812266 1725 | 56.55793651,1.427832717 1726 | 70.87777778,1.590555489 1727 | 73.20138889,1.591098604 1728 | 74.68888889,1.601021222 1729 | 71.37638889,1.59271815 1730 | 20.86666667,1.149646321 1731 | 74.63888889,1.636394673 1732 | 75.21666667,1.612381381 1733 | 63.66746032,1.541439185 1734 | 73.93095238,1.615655319 1735 | 69.02916667,1.620185525 1736 | 54.93888889,1.526171598 1737 | 20.3625,1.459663627 1738 | 75.82222222,1.621389901 1739 | 25.21666667,1.525109372 1740 | 23.47222222,1.413473412 1741 | 34.36746032,1.469070918 1742 | 72.04206349,1.48465821 1743 | 39.01111111,1.406484749 1744 | 46.97638889,1.523520903 1745 | 69.20555556,1.62373625 1746 | 41.43888889,1.473973451 1747 | 50.83888889,1.543577309 1748 | 57.52460317,1.531289818 1749 | 36.19285714,1.427515818 1750 | 61.33888889,1.475318102 1751 | 71.64722222,1.61462798 1752 | 35.72638889,1.509792741 1753 | 21.48888889,1.454781318 1754 | 18.36746032,1.504025052 1755 | 62.72460317,1.525945732 1756 | 39.18888889,1.453024702 1757 | 73.27222222,1.636508055 1758 | 74.10555556,1.61313658 1759 | 54.30396825,1.554299897 1760 | 71.18492063,1.644138583 1761 | 69.41111111,1.660876597 1762 | 32.58888889,1.558200791 1763 | 61.26031746,1.56283285 1764 | 57.08968254,1.450398457 1765 | 53.18888889,1.584770131 1766 | 27.78888889,1.473526941 1767 | 59.55,1.511404832 1768 | 37.13888889,1.458695334 1769 | 70.36746032,1.536755814 1770 | 36.01031746,1.479376593 1771 | 58.49603175,1.462965021 1772 | --------------------------------------------------------------------------------