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Inference for Totals and Weighted Sums from Finite Spatial Populations

Matt Higham, Jay M. Ver Hoef, Bryce M. Frank

2022-12-11


Introduction

The sptotal package was developed for predicting a weighted sum, most commonly a mean or total, from a finite number of sample units in a fixed geographic area. Estimating totals and means from a finite population is an important goal for both academic research and management of environmental data. One naturally turns to classical sampling methods, such as simple random sampling or stratified random sampling. Classical sampling methods depend on probability-based sample designs and are robust. Very few assumptions are required because the probability distribution for inference comes from the sample design, which is known and under our control. For design-based methods, sample plots are chosen at random, they are measured or counted, and inference is obtained from the probability of sampling those units randomly based on the design (e.g., Horwitz-Thompson estimation). As an alternative, we will use model-based methods, specifically geostatistics, to accomplish the same goals. Geostatistics does not rely on a specific sampling design. Instead, when using geostatistics, we assume the data were produced by a stochastic process with parameters that can be estimated. The relevant theory is given by Ver Hoef (2008). The sptotal package puts much of the code and plots in Ver Hoef (2008) in easily accessible, convenient functions.

In the sptotal package, our goal is to estimate some linear function of all of the sample units, call it \(\tau(\mathbf{z}) = \mathbf{b}^\prime \mathbf{z}\), where \(\mathbf{z}\) is a vector of the realized values for all the sample units and \(\mathbf{b}\) is a vector of weights. By “realized,” we mean that whatever processes produced the data have already happened, and that, if we had enough resources, we could measure them all, obtaining a complete census. If \(\tau(\mathbf{z})\) is a population total, then every element of \(\mathbf{b}\) contains a \(1\). Generally, \(\mathbf{b}\) can contain any set of weights that we would like to multiply times each value in a population, and then these are summed, yielding a weighted sum.

The vector \(\mathbf{b}\) contains the weights that we would apply if we could measure or count every observation, but, because of cost consideration, we usually only have a sample.

Data

Prior to using the sptotal package, the data needs to be in R in the proper format. For this package, we assume that your data set is a data.frame() object, described below.

Data Frame Structure

Data input for the sptotal package is a data.frame. The basic information required to fit a spatial linear model, and make predictions, are the response variable, covariates, the x- and y-coordinates, and a column of weights. You can envision your whole population of possible samples as a data.frame organized as follows,

where the red rectangle represents the column of the response variable, and the top part, colored in red, are observed locations, and the lower part, colored in white, are the unobserved values. To the right, colored in blue, are possibly several columns containing covariates thought to be predictive for the response value at each location. Covariates must be known for both observed and unobserved locations, and the covariates for unobserved locations are shown as pale blue below the darker blue covariates for observed locations above. It is also possible that there are no available covariates.

The data.frame must have x- and y-coordinates, and they are shown as two columns colored in green, with the coordinates for the unobserved locations shown as pale green below the darker green coordinates for the observed locations above. The data.frame can have a column of weights. If one is not provided, we assume a column of all ones so that the prediction is for the population total. The column of weights is purple, with weights for the observed locations a darker shade, above the lighter shade of purple representing weights for unsampled locations. Finally, the data.frame may contain columns that are not relevant to predicting the weighted sum. These columns are represented by the orange color, with the sampled locations a darker shade, above the unsampled locations with the lighter shade.

Of course, the data do not have to be in exactly this order, either in terms of rows or columns. Sampled and unsampled rows can be intermingled, and columns of response variable, covariates, coordinates, and weights can be also be intermingled. The figure above is an idealized graphic of the data. However, this figure helps envision how the data are used and illustrate the goal. We desire a weighted sum, where the weights (in the purple column) are multiplied with the response variable (red/white) column, and then summed. Because some of the response values are unknown (the white values in the response column), covariates and spatial information (obtained from the x- and y-coordinates) are used to predict the unobserved (white) values. The weights (purple) are then applied to both the observed response values (red), and the predicted response values (white), to obtain a weighted sum. Because we use predictions for unobserved response values, it is important to assess our uncertainty, and the software provides both an estimate of the weighted sum, mean, or total for the response variable as well as its estimated prediction variance.

Simulated Data Creation

To demonstrate the package, we created some simulated data so they are perfectly behaved, and we know exactly how they were produced. Here, we give a brief description before using the main features of the package. To get started, install the package

install.packages("sptotal")

and then type

library(sptotal)

Type

data(simdata)

and then simdata will be available in your workspace. To see the first six observations of simdata, type

head(simdata)
#>       x     y         X1          X2         X3          X4          X5
#> 1 0.025 0.975 -0.8460525  0.11866907 -0.2123901  0.38430607  0.08154129
#> 2 0.025 0.925 -0.6583116 -0.07686491 -0.9001410 -1.24774376  1.46631630
#> 3 0.025 0.875  0.2222961 -0.22803942  0.2820468  0.20560677  0.48713665
#> 4 0.025 0.825 -0.5433925  0.56894993 -0.9839629 -0.04950434 -0.78195604
#> 5 0.025 0.775 -0.7550155 -0.72592167 -0.4217208  0.26767033  0.40493269
#> 6 0.025 0.725 -0.1786784  0.33452155 -1.2134533  2.18704575 -0.54903128
#>           X6         X7 F1 F2        Z   wts1 wts2
#> 1  1.0747592 -0.0252824  3  3 15.94380 0.0025    0
#> 2  0.1299263  1.4651052  2  5 15.04616 0.0025    0
#> 3 -0.2537515  0.2682010  2  3 14.52765 0.0025    0
#> 4 -0.3259937  0.7858140  2  5 12.13401 0.0025    0
#> 5 -1.2284475  1.2944342  2  2 11.75260 0.0025    0
#> 6 -1.0366099  0.7938890  1  4 11.58142 0.0025    0

simdata is a data frame with 400 observations. The spatial coordinates are numeric variables in columns named x and y. We created 7 continuous covariates, X1 through X7. The variables X1 through X5 were all created using the rnorm() function, so they are all standard normal variates that are independent between and within variable. Variables X6 and X7 were independent from each other, but spatially autocorrelated within, each with a variance parameter of 1, an autocorrelation range parameter of 0.2 from an exponential model, and a small nugget effect of 0.01. The variables F1 and F2 are factor variables with 3 and 5 levels, respectively. The variable Z is the response. Data were simulated from the model

\[\begin{align*} Z_i = 10 & + 0 \cdot X1_i + 0.1 \cdot X2_i + 0.2 \cdot X3_i + 0.3 \cdot X4_i + \\ & 0.4 \cdot X5_i + 0.4 \cdot X6_i + 0.1 \cdot X7_i + F1_i + F2_i + \delta_i + \varepsilon_i \end{align*}\]

where factor levels for F1 have effects \(0, 0.4, 0.8\), and factor levels for F2 have effects \(0, 0.1, 0.2, 0.3, 0.4\). The random errors \(\{\delta_i\}\) are spatially autocorrelated from an exponential model,

\[ \textrm{cov}(\delta_i,\delta_j) = 2*\exp(-d_{i,j}) \]

where \(d_{i,j}\) is Euclidean distance between locations \(i\) and \(j\). In geostatistics terminology, this model has a partial sill of 2 and a range of 1. The random errors \(\{\varepsilon_i\}\) are independent with variance 0.02, and this variance is called the nugget effect. Two columns with weights are included, wts1 contains 1/400 for each row, so the weighted sum will yield a prediction of the overall mean. The column wts2 contains a 1 for 25 locations, and 0 elsewhere, so the weighted sum will be a prediction of a total in the subset of 25 locations.

The spatial locations of simdata are in a \(20 \times 20\) grid uniformly spaced in a box with sides of length 1,

require(ggplot2)
ggplot(data = simdata, aes(x = x, y = y)) + geom_point(size = 3) +
  geom_point(data = subset(simdata, wts2 == 1), colour = "red",
    size = 3)

The locations of the 25 sites where wts2 is equal to one are shown in red.

We have simulated the data for the whole population. This is convenient, because we know the true means and totals. In order to compare with the prediction from the sptotal package, let’s find the true population total

sum(simdata[ ,'Z'])
#> [1] 4834.326

as well as the total in the subset of 25 sites

sum(simdata[ ,'wts2'] * simdata[ ,'Z'])
#> [1] 273.3751

However, we will now sample from this population to provide a more realistic setting where we can measure only a part of the whole population. In order to make results reproducible, we use the set.seed command, along with sample. The code below will replace some of the response values with NA to represent the unsampled sites.

set.seed(1)
# take a random sample of 100
obsID <- sample(1:nrow(simdata), 100)
simobs <- simdata
simobs$Z <- NA
simobs[obsID, 'Z'] <- simdata[obsID, 'Z']

We now have a data set where the whole population is known, simdata, and another one, simobs, where 75% of the response variable of the population has been replaced by NA. Next we show the sampled sites as solid circles, while the missing values are shown as open circles, and we use red again to show the sites within the small area of 25 locations.

ggplot(data = simobs, aes(x = x, y = y)) +
  geom_point(shape = 1, size = 2.5, stroke = 1.5) +
  geom_point(data = subset(simobs, !is.na(Z)), shape = 16, size = 3.5) +
  geom_point(data = subset(simobs, !is.na(Z) & wts2 == 1), shape = 16,
    colour = "red", size = 3.5) +
  geom_point(data = subset(simobs, is.na(Z) & wts2 == 1), shape = 1,
    colour = "red", size = 2.5, stroke = 1.5)

We will use the simobs data to illustrate use of the sptotal package.

Using the sptotal Package

After your data is in a similar format to simobs, using the sptotal package occurs in two primary stages. In the first, we fit a spatial linear model. This stage estimates spatial regression coefficients and spatial autocorrelation parameters. In the second stage, we predict the unsampled locations for the response value, and create a prediction for the weighted sum (e.g. the total) of all response variable values, both observed and predicted. To show how the package works, we demonstrate on ideal, simulated data. Then, we give a realistic example on moose data and a second example on lakes data to provide further insight and documentation. The moose example also has a section on data preparation steps.

Fitting a Spatial Linear Model: slmfit

We continue with our use of the simulated data, simobs, to illustrate fitting the spatial linear model. The spatial model-fitting function is slmfit (spatial-linear-model-fit), which uses a formula like many other model-fitting functions in R (e.g., the lm() function). To fit a basic spatial linear model we use

slmfit_out1 <- slmfit(formula = Z ~ X1 + X2 + X3 + X4 + X5 +
                        X6 + X7 + F1 + F2, 
                      data = simobs, xcoordcol = 'x',
                      ycoordcol = 'y',
                      CorModel = "Exponential")

The documentation describes the arguments in more detail, but as mentioned earlier, the linear model includes a formula argument, and the data.frame that is being used as a data set. We also need to include which columns contain the \(x\)- and \(y\)-coordinates, which are arguments to xcoordcol and ycoordcol, respectively. In the above example, we specify 'x' and 'y' as the column coordinates arguments since the names of the coordinate columns in our simulated data set are 'x' and 'y'. We also need to specify a spatial autocorrelation model, which is given by the CorModel argument. As with many other linear model fits, we can obtain a summary of the model fit,

summary(slmfit_out1)
#> 
#> Call:
#> Z ~ X1 + X2 + X3 + X4 + X5 + X6 + X7 + F1 + F2
#> 
#> Residuals:
#>     Min      1Q  Median      3Q     Max 
#> -1.9390 -0.6271  0.3338  1.2520  2.8137 
#> 
#> Coefficients:
#>             Estimate Std. Error t value Pr(>|t|)    
#> (Intercept) 11.36965    1.03775  10.956  < 2e-16 ***
#> X1          -0.05596    0.06400  -0.874  0.38437    
#> X2           0.02661    0.06606   0.403  0.68814    
#> X3           0.18292    0.06469   2.828  0.00583 ** 
#> X4           0.26487    0.05741   4.613    1e-05 ***
#> X5           0.38434    0.06022   6.382  < 2e-16 ***
#> X6           0.47612    0.11198   4.252    5e-05 ***
#> X7           0.02893    0.11761   0.246  0.80625    
#> F12          0.29596    0.15154   1.953  0.05407 .  
#> F13          0.70853    0.13136   5.394  < 2e-16 ***
#> F22          0.15384    0.17073   0.901  0.37008    
#> F23          0.19804    0.17828   1.111  0.26973    
#> F24          0.25492    0.20024   1.273  0.20641    
#> F25          0.39748    0.23691   1.678  0.09703 .  
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> 
#> Covariance Parameters:
#>              Exponential Model
#> Nugget            1.009265e-06
#> Partial Sill      2.930385e+00
#> Range             5.891474e-01
#> 
#> Generalized R-squared: 0.5996812

The output looks similar to the summary of a standard lm object, but there is some extra output at the end that gives our fitted covariance parameters. Plotting slmfit_out1 gives a semi-variogram of the residuals along with the fitted model:

plot(slmfit_out1)

Note that the fitted curve may not appear to fit the empirical variogram perfectly for a couple of reasons. First, only pairs of points that have a distance between 0 and one-half the maximum distance are shown. Second, the fitted model is estimated using REML, which may give different results than using weighted least squares.

We can also examine a histogram of the residuals as well as a histogram of the cross-validation (leave-one-out) residuals:

residraw <- residuals(slmfit_out1)
qplot(residraw, bins = 20) + xlab("Residuals")
#> Warning: `qplot()` was deprecated in ggplot2 3.4.0.

residcv <- residuals(slmfit_out1, cross.validation = TRUE)
qplot(residcv, bins = 20) + xlab("CV Residuals")

There is still one somewhat large cross-validation residual for an observed count that is larger than what would be predicted from a model without that particular count. The cause of this somewhat large residual can be attributed to random chance because we know that the data was simulated to follow all assumptions.

Prediction: predict

After we have obtained a fitted spatial linear model, we can use the predict() function to construct a data frame of predictions for the unsampled sites. By default, the predict() function assumes that we are predicting the population total and outputs this predicted total, the prediction variance for the total, a 90% prediction interval for the total, and some basic summary information about the number of sites sampled, the total number of units counted, etc. We name this object pred_obj in the chunk below and also construct a 90% confidence interval for the total.

pred_obj <- predict(slmfit_out1, conf_level = 0.90)
pred_obj

We predict a total of 4817 units in this simulated region with 90% confidence bounds of (4779, 4856). The prediction interval is fairly small because we simulated data that were highly correlated, increasing precision in prediction for unobserved sites. We can see that the prediction of the total is close to the true value of 4834.326, and the true value is within the prediction interval.

To access the data.frame that was input into slmfit, but is now appended with site-by-site predictions and site-by-site prediction variances, we can use pred_obj$Pred_df. This data set might be particularly useful if you would like to generate your own map with site-by-site predictions using other tools. The site-by-site predictions for density are given by the variable name_of_response_pred_density while the site-by-site predictions for counts are given by name_of_response_pred_count. These two columns will only differ if you have provided a column for areas of each site.

prediction_df <- pred_obj$Pred_df
head(prediction_df[ ,c("x", "y", "Z", "Z_pred_density")])

Examining results: plot()

Finally, to get a basic plot of the predictions, we can use the plot() function.

plot(pred_obj)

The map shows the distribution of the response across sampled and unsampled sites. Its purpose is simply to give the user a very quick idea of the distribution of the response. For example, we see from the plot that the predicted response is low in the upper-right region of the graph, is high in the middle of the region and in the upper-left corner of the region, and is low again at the lower portion of the area of interest. However, using the prediction data frame generated from the predict() function, you can use ggplot2 or any other plotting package to construct your own map that may be more useful in your context.

Prediction for a Small Area of Interest

Spatial prediction can be used to estimate means and totals over finite populations from geographic regions, but can also be used for the special case of estimating a mean or total in a small area of interest. The term small area estimation refers to making an inference on a smaller geographic area within the overall study area. There may be few or no samples within that small area, so that estimation by classical sampling methods may not be possible or variances become exceedingly large.

If we want to predict a quantity other than the population total, then we need to specify the column in our data set that has the appropriate prediction weights in a wtscol argument. For example, we might want to predict the total for a small area of interest. if we want to predict the total for the 25 sites in coloured in red, then we can use

pred_obj2 <- predict(slmfit_out1, wtscol = "wts2")
print(pred_obj2)
#> Prediction Info:
#>   Prediction    SE 90% LB 90% UB
#> Z      282.2 7.342  270.1  294.3
#>   Numb. Sites Sampled Total Numb. Sites Total Observed Average Density
#> Z                 100               400           1220            12.2

Recall that the true total for this small area was 273.4. We see that this is close to our prediction of 282.2 and is also within the bounds of our prediction interval.

Real Data Examples

Moose Abundance from Aerial Surveys

The simulated data example assumes that the coordinates are a Transverse Mercator projection (TM), that the vector of the response is numeric and has NA values for sites that were not sampled, and that the areas of each site sampled are all the same. For this example, we consider a data set on moose abundance in Alaska obtained from Alaska Department of Fish and Game, Division of Wildlife Conservation. Each observation corresponds to a moose counted at a particular site, but operational constraints do not permit all sites to be counted. We begin by loading the data into R.

data(AKmoose_df)
AKmoose_df
#>     elev_mean strat surveyed total            x            y       lon      lat
#> 0    560.3333     L        0    NA  38.98384825 1.301806e+02 -147.8750 63.71667
#> 1    620.4167     L        0    NA  34.86652773 1.302284e+02 -147.9583 63.71667
#> 2    468.9167     L        1     0  30.74963291 1.302815e+02 -148.0417 63.71667
#> 3    492.7500     L        0    NA  26.63241710 1.303400e+02 -148.1250 63.71667
#> 4    379.5833     L        0    NA  22.51526115 1.304038e+02 -148.2083 63.71667
#> 5    463.7500     L        0    NA  38.94319469 1.264665e+02 -147.8750 63.68333
#> 6    456.4375     L        0    NA  34.82103091 1.265143e+02 -147.9583 63.68333
#> 7    358.9375     L        0    NA  30.69929374 1.265674e+02 -148.0417 63.68333
#> 8    333.1875     L        0    NA  26.57723416 1.266260e+02 -148.1250 63.68333
#> 9    257.8750     L        0    NA  22.45523537 1.266899e+02 -148.2083 63.68333
#> 10   417.6250     L        0    NA  38.90255070 1.227521e+02 -147.8750 63.65000
#> 11   362.3125     L        1     0  34.77554479 1.228000e+02 -147.9583 63.65000
#> 12   265.7500     L        0    NA  30.64896544 1.228532e+02 -148.0417 63.65000
#> 13   269.3125     L        0    NA  26.52206419 1.229118e+02 -148.1250 63.65000
#> 14   225.0000     L        0    NA  22.39522272 1.229758e+02 -148.2083 63.65000
#> 15   172.7500     M        1     0  18.26882464 1.230451e+02 -148.2917 63.65000
#> 16   398.6250     L        0    NA  38.86191861 1.190378e+02 -147.8750 63.61666
#> 17   279.6250     L        1     0  34.73007200 1.190857e+02 -147.9583 63.61666
#> 18   227.8750     L        0    NA  30.59865236 1.191390e+02 -148.0417 63.61666
#> 19   197.8750     L        0    NA  26.46691037 1.191976e+02 -148.1250 63.61666
#> 20   194.7500     L        0    NA  22.33522813 1.192616e+02 -148.2083 63.61666
#> 21   167.9375     M        0    NA  18.20398971 1.193311e+02 -148.2917 63.61666
#> 22   204.0000     L        0    NA  14.07244432 1.194059e+02 -148.3750 63.61666
#> 23   619.6000     L        1     0  47.09442203 1.152440e+02 -147.7083 63.58333
#> 24   479.7500     L        0    NA  42.95803072 1.152812e+02 -147.7917 63.58333
#> 25   350.5000     L        0    NA  38.82130074 1.153237e+02 -147.8750 63.58333
#> 26   286.4167     L        0    NA  34.68461462 1.153716e+02 -147.9583 63.58333
#> 27   207.5833     L        0    NA  30.54835739 1.154250e+02 -148.0417 63.58333
#> 28   181.5833     L        0    NA  26.41177586 1.154837e+02 -148.1250 63.58333
#> 29   174.3333     L        0    NA  22.27525505 1.155478e+02 -148.2083 63.58333
#> 30   164.5833     M        0    NA  18.13917747 1.156172e+02 -148.2917 63.58333
#> 31   188.4000     L        0    NA  14.00279297 1.156921e+02 -148.3750 63.58333
#> 32   515.0500     L        0    NA  47.06349078 1.115299e+02 -147.7083 63.55000
#> 33   358.8750     L        0    NA  42.92226236 1.115671e+02 -147.7917 63.55000
#> 34   364.1250     L        0    NA  38.78069481 1.116096e+02 -147.8750 63.55000
#> 35   217.5625     L        0    NA  34.63917159 1.116576e+02 -147.9583 63.55000
#> 36   191.5625     L        0    NA  30.49807620 1.117110e+02 -148.0417 63.55000
#> 37   172.6250     L        0    NA  26.35665755 1.117697e+02 -148.1250 63.55000
#> 38   162.0625     M        0    NA  22.21529857 1.118339e+02 -148.2083 63.55000
#> 39   164.5625     L        0    NA  18.07438426 1.119034e+02 -148.2917 63.55000
#> 40   167.5500     L        0    NA  13.93316208 1.119784e+02 -148.3750 63.55000
#> 41   543.8750     L        0    NA  67.76480465 1.077107e+02 -147.2917 63.51667
#> 42   577.3125     L        0    NA  63.61824587 1.077209e+02 -147.3750 63.51667
#> 43   593.1875     L        0    NA  59.47169973 1.077365e+02 -147.4583 63.51667
#> 44   612.0000     L        0    NA  55.32555017 1.077575e+02 -147.5417 63.51667
#> 45   630.5625     L        0    NA  51.17904516 1.077839e+02 -147.6250 63.51667
#> 46   455.8500     L        0    NA  47.03256864 1.078158e+02 -147.7083 63.51667
#> 47   319.5625     L        0    NA  42.88650452 1.078530e+02 -147.7917 63.51667
#> 48   289.2500     L        0    NA  38.74010081 1.078956e+02 -147.8750 63.51667
#> 49   211.6875     L        0    NA  34.59374144 1.079436e+02 -147.9583 63.51667
#> 50   181.8125     L        0    NA  30.44781080 1.079970e+02 -148.0417 63.51667
#> 51   164.1250     M        1     0  26.30155543 1.080558e+02 -148.1250 63.51667
#> 52   163.5000     M        1     0  22.15536074 1.081200e+02 -148.2083 63.51667
#> 53   174.9375     L        0    NA  18.00961012 1.081896e+02 -148.2917 63.51667
#> 54   186.3500     L        1     0  13.86355163 1.082646e+02 -148.3750 63.51667
#> 55   503.6923     L        0    NA  76.06048918 1.039922e+02 -147.1250 63.48333
#> 56   620.5000     L        0    NA  71.90908579 1.039916e+02 -147.2083 63.48333
#> 57   466.6667     L        0    NA  67.75806458 1.039965e+02 -147.2917 63.48333
#> 58   447.9375     L        0    NA  63.60667113 1.040067e+02 -147.3750 63.48333
#> 59   597.9375     L        0    NA  59.45529032 1.040223e+02 -147.4583 63.48333
#> 60   655.8750     L        0    NA  55.30430652 1.040434e+02 -147.5417 63.48333
#> 61   566.3750     L        0    NA  51.15296682 1.040698e+02 -147.6250 63.48333
#> 62   507.7000     L        0    NA  47.00165560 1.041017e+02 -147.7083 63.48333
#> 63   375.7500     L        0    NA  42.85075722 1.041389e+02 -147.7917 63.48333
#> 64   261.4375     L        0    NA  38.69951878 1.041815e+02 -147.8750 63.48333
#> 65   198.4375     L        0    NA  34.54832466 1.042296e+02 -147.9583 63.48333
#> 66   173.5000     M        0    NA  30.39755970 1.042831e+02 -148.0417 63.48333
#> 67   158.6250     M        0    NA  26.24646955 1.043419e+02 -148.1250 63.48333
#> 68   170.1667     L        0    NA  22.09544005 1.044062e+02 -148.2083 63.48333
#> 69   190.5714     L        0    NA  17.94485504 1.044758e+02 -148.2917 63.48333
#> 70   219.0769     L        0    NA  13.79396170 1.045509e+02 -148.3750 63.48333
#> 71   434.9375     L        0    NA  84.37549449 1.002954e+02 -146.9583 63.45000
#> 72   347.5000     L        0    NA  80.21965006 1.002840e+02 -147.0417 63.45000
#> 73   406.3750     L        0    NA  76.06341713 1.002780e+02 -147.1250 63.45000
#> 74   482.6500     L        0    NA  71.90718051 1.002774e+02 -147.2083 63.45000
#> 75   354.3750     L        0    NA  67.75132649 1.002823e+02 -147.2917 63.45000
#> 76   458.6667     L        0    NA  63.59509981 1.002925e+02 -147.3750 63.45000
#> 77   656.5833     L        0    NA  59.43888574 1.003082e+02 -147.4583 63.45000
#> 78   583.9167     L        0    NA  55.28306963 1.003292e+02 -147.5417 63.45000
#> 79   549.4167     L        0    NA  51.12689618 1.003557e+02 -147.6250 63.45000
#> 80   384.0667     L        0    NA  46.97075167 1.003876e+02 -147.7083 63.45000
#> 81   285.6667     L        0    NA  42.81502044 1.004248e+02 -147.7917 63.45000
#> 82   218.3333     L        0    NA  38.65894871 1.004675e+02 -147.8750 63.45000
#> 83   178.7500     L        0    NA  34.50292126 1.005156e+02 -147.9583 63.45000
#> 84   164.2500     M        0    NA  30.34732291 1.005691e+02 -148.0417 63.45000
#> 85   160.0000     M        0    NA  26.19139991 1.006280e+02 -148.1250 63.45000
#> 86   171.1875     L        0    NA  22.03553651 1.006924e+02 -148.2083 63.45000
#> 87   202.7000     L        0    NA  17.88011903 1.007621e+02 -148.2917 63.45000
#> 88   213.3750     L        0    NA  13.72439227 1.008372e+02 -148.3750 63.45000
#> 89   425.2500     L        0    NA  92.71015129 9.662025e+01 -146.7917 63.41667
#> 90   422.3125     L        0    NA  88.54912778 9.659803e+01 -146.8750 63.41667
#> 91   419.7500     L        0    NA  84.38808476 9.658122e+01 -146.9583 63.41667
#> 92   362.3125     L        1     0  80.22740946 9.656983e+01 -147.0417 63.41667
#> 93   298.5625     L        0    NA  76.06634422 9.656384e+01 -147.1250 63.41667
#> 94   370.1500     L        0    NA  71.90527628 9.656327e+01 -147.2083 63.41667
#> 95   361.0625     L        0    NA  67.74459040 9.656811e+01 -147.2917 63.41667
#> 96   499.3750     L        0    NA  63.58353190 9.657836e+01 -147.3750 63.41667
#> 97   567.0000     L        0    NA  59.42248602 9.659402e+01 -147.4583 63.41667
#> 98   430.5000     L        1     0  55.26183802 9.661510e+01 -147.5417 63.41667
#> 99   423.7500     L        0    NA  51.10083322 9.664159e+01 -147.6250 63.41667
#> 100  315.4500     L        0    NA  46.93985688 9.667349e+01 -147.7083 63.41667
#> 101  239.8750     L        0    NA  42.77929423 9.671080e+01 -147.7917 63.41667
#> 102  210.0625     L        0    NA  38.61839062 9.675352e+01 -147.8750 63.41667
#> 103  161.7500     M        1     0  34.45753079 9.680166e+01 -147.9583 63.41667
#> 104  167.3750     M        1     5  30.29710196 9.685521e+01 -148.0417 63.41667
#> 105  171.7500     L        0    NA  26.13634651 9.691417e+01 -148.1250 63.41667
#> 106  176.5625     L        0    NA  21.97565167 9.697854e+01 -148.2083 63.41667
#> 107  206.0500     L        0    NA  17.81540216 9.704832e+01 -148.2917 63.41667
#> 108  191.6875     L        0    NA  13.65484337 9.712352e+01 -148.3750 63.41667
#> 109  626.5555     L        0    NA 126.05677151 9.327908e+01 -146.1250 63.38334
#> 110  554.2222     L        1     7 121.89125903 9.321350e+01 -146.2083 63.38334
#> 111  318.1250     L        0    NA 117.72606674 9.315335e+01 -146.2917 63.38334
#> 112  238.8750     L        0    NA 113.56043764 9.309861e+01 -146.3750 63.38334
#> 113  242.3750     L        0    NA 109.39475788 9.304928e+01 -146.4583 63.38334
#> 114  388.3125     L        0    NA 105.22941315 9.300537e+01 -146.5417 63.38334
#> 115  332.3333     L        0    NA 101.06364792 9.296688e+01 -146.6250 63.38334
#> 116  369.0667     L        0    NA  96.89784735 9.293381e+01 -146.7083 63.38334
#> 117  347.8333     L        0    NA  92.73239864 9.290615e+01 -146.7917 63.38334
#> 118  296.5000     L        0    NA  88.56654473 9.288391e+01 -146.8750 63.38334
#> 119  291.3571     L        0    NA  84.40067131 9.286709e+01 -146.9583 63.38334
#> 120  287.1250     L        0    NA  80.23516557 9.285568e+01 -147.0417 63.38334
#> 121  287.1250     L        0    NA  76.06927044 9.284969e+01 -147.1250 63.38334
#> 122  388.1500     L        0    NA  71.90337162 9.284912e+01 -147.2083 63.38334
#> 123  469.2500     L        0    NA  67.73785630 9.285396e+01 -147.2917 63.38334
#> 124  471.3750     L        0    NA  63.57196741 9.286422e+01 -147.3750 63.38334
#> 125  362.5625     L        0    NA  59.40609114 9.287990e+01 -147.4583 63.38334
#> 126  362.6875     L        0    NA  55.24061368 9.290099e+01 -147.5417 63.38334
#> 127  401.3125     L        0    NA  51.07477798 9.292750e+01 -147.6250 63.38334
#> 128  401.9000     L        0    NA  46.90897122 9.295943e+01 -147.7083 63.38334
#> 129  229.0000     L        0    NA  42.74357858 9.299677e+01 -147.7917 63.38334
#> 130  162.1250     L        0    NA  38.57784452 9.303954e+01 -147.8750 63.38334
#> 131  197.0625     M        0    NA  34.41215472 9.308772e+01 -147.9583 63.38334
#> 132  190.3750     L        0    NA  30.24689485 9.314131e+01 -148.0417 63.38334
#> 133  183.4375     L        0    NA  26.08130940 9.320032e+01 -148.1250 63.38334
#> 134  187.6875     L        0    NA  21.91578353 9.326475e+01 -148.2083 63.38334
#> 135  191.0000     L        0    NA  17.75070439 9.333459e+01 -148.2917 63.38334
#> 136  189.4375     L        0    NA  13.58531502 9.340986e+01 -148.3750 63.38334
#> 137  604.0833     L        0    NA 126.11764670 8.956511e+01 -146.1250 63.35000
#> 138  558.4500     L        0    NA 121.94730484 8.949947e+01 -146.2083 63.35000
#> 139  304.8125     L        0    NA 117.77728363 8.943927e+01 -146.2917 63.35000
#> 140  257.7500     L        0    NA 113.60682519 8.938448e+01 -146.3750 63.35000
#> 141  205.5625     L        0    NA 109.43631612 8.933511e+01 -146.4583 63.35000
#> 142  247.3750     L        0    NA 105.26614253 8.929116e+01 -146.5417 63.35000
#> 143  199.6875     L        0    NA 101.09554802 8.925264e+01 -146.6250 63.35000
#> 144  250.0000     L        1     0  96.92491868 8.921953e+01 -146.7083 63.35000
#> 145  248.8750     L        0    NA  92.75464066 8.919186e+01 -146.7917 63.35000
#> 146  245.4375     L        0    NA  88.58395751 8.916960e+01 -146.8750 63.35000
#> 147  232.1429     L        0    NA  84.41325535 8.915276e+01 -146.9583 63.35000
#> 148  300.6667     L        0    NA  80.24292082 8.914134e+01 -147.0417 63.35000
#> 149  398.0000     L        0    NA  76.07219596 8.913535e+01 -147.1250 63.35000
#> 150  483.8667     L        0    NA  71.90146842 8.913477e+01 -147.2083 63.35000
#> 151  435.9167     L        0    NA  67.73112381 8.913962e+01 -147.2917 63.35000
#> 152  403.0000     L        0    NA  63.56040569 8.914989e+01 -147.3750 63.35000
#> 153  293.9167     L        0    NA  59.38970019 8.916558e+01 -147.4583 63.35000
#> 154  318.8333     L        0    NA  55.21939343 8.918669e+01 -147.5417 63.35000
#> 155  376.8333     L        0    NA  51.04872897 8.921323e+01 -147.6250 63.35000
#> 156  369.8000     L        0    NA  46.87809295 8.924518e+01 -147.7083 63.35000
#> 157  171.9167     M        1     0  42.70787147 8.928256e+01 -147.7917 63.35000
#> 158  171.6667     M        0    NA  38.53730812 8.932536e+01 -147.8750 63.35000
#> 159  219.0833     L        0    NA  34.36678901 8.937358e+01 -147.9583 63.35000
#> 160  203.5000     L        0    NA  30.19670025 8.942722e+01 -148.0417 63.35000
#> 161  192.3333     L        0    NA  26.02628544 8.948628e+01 -148.1250 63.35000
#> 162  183.0833     L        0    NA  21.85593020 8.955077e+01 -148.2083 63.35000
#> 163  181.8000     L        0    NA  17.68602210 8.962067e+01 -148.2917 63.35000
#> 164  195.7500     L        0    NA  13.51580329 8.969600e+01 -148.3750 63.35000
#> 165  478.5000     L        0    NA 134.52825666 8.599880e+01 -145.9583 63.31667
#> 166  346.7500     L        0    NA 130.35360672 8.592227e+01 -146.0417 63.31667
#> 167  386.2500     L        0    NA 126.17850383 8.585115e+01 -146.1250 63.31667
#> 168  301.4615     L        0    NA 122.00333452 8.578546e+01 -146.2083 63.31667
#> 169  191.5000     L        0    NA 117.82848533 8.572520e+01 -146.2917 63.31667
#> 170  171.0769     L        0    NA 113.65319898 8.567036e+01 -146.3750 63.31667
#> 171  144.5000     L        0    NA 109.47786203 8.562095e+01 -146.4583 63.31667
#> 172  153.8750     L        0    NA 105.30286102 8.557697e+01 -146.5417 63.31667
#> 173  152.1250     L        0    NA 101.12743866 8.553841e+01 -146.6250 63.31667
#> 174  170.3889     L        0    NA  96.95198099 8.550528e+01 -146.7083 63.31667
#> 175  177.0000     L        0    NA  92.77687608 8.547758e+01 -146.7917 63.31667
#> 176  179.6875     L        1     0  88.60136512 8.545530e+01 -146.8750 63.31667
#> 177  246.0000     L        0    NA  84.42583516 8.543844e+01 -146.9583 63.31667
#> 178  334.1250     L        0    NA  80.25067327 8.542702e+01 -147.0417 63.31667
#> 179  412.1250     L        0    NA  76.07512062 8.542102e+01 -147.1250 63.31667
#> 180  496.7222     L        0    NA  71.89956478 8.542044e+01 -147.2083 63.31667
#> 181  452.1111     L        0    NA  67.72439332 8.542530e+01 -147.2917 63.31667
#> 182  343.0625     L        0    NA  63.54884740 8.543557e+01 -147.3750 63.31667
#> 183  298.1250     L        0    NA  59.37331409 8.545128e+01 -147.4583 63.31667
#> 184  226.0625     L        0    NA  55.19817996 8.547241e+01 -147.5417 63.31667
#> 185  227.0000     L        0    NA  51.02268769 8.549896e+01 -147.6250 63.31667
#> 186  195.9000     L        0    NA  46.84722383 8.553095e+01 -147.7083 63.31667
#> 187  148.8125     M        0    NA  42.67217495 8.556836e+01 -147.7917 63.31667
#> 188  217.8125     M        1     0  38.49678374 8.561119e+01 -147.8750 63.31667
#> 189  251.2500     L        0    NA  34.32143675 8.565946e+01 -147.9583 63.31667
#> 190  189.3750     L        0    NA  30.14652104 8.571314e+01 -148.0417 63.31667
#> 191  174.6875     L        0    NA  25.97127781 8.577226e+01 -148.1250 63.31667
#> 192  174.2222     L        0    NA  21.79609511 8.583680e+01 -148.2083 63.31667
#> 193  182.7778     L        0    NA  17.62135899 8.590676e+01 -148.2917 63.31667
#> 194  194.5625     L        0    NA  13.44631218 8.598216e+01 -148.3750 63.31667
#> 195  452.7500     L        0    NA 134.59874437 8.228521e+01 -145.9583 63.28333
#> 196  259.8125     L        1     0 130.41926857 8.220860e+01 -146.0417 63.28333
#> 197  237.2500     L        0    NA 126.23933941 8.213742e+01 -146.1250 63.28333
#> 198  186.4375     L        0    NA 122.05934335 8.207168e+01 -146.2083 63.28333
#> 199  141.8500     L        0    NA 117.87966888 8.201137e+01 -146.2917 63.28333
#> 200  116.3333     L        0    NA 113.69955635 8.195648e+01 -146.3750 63.28333
#> 201  122.6667     L        0    NA 109.51939322 8.190703e+01 -146.4583 63.28333
#> 202  127.0000     L        0    NA 105.33956650 8.186301e+01 -146.5417 63.28333
#> 203  149.3333     L        0    NA 101.15931799 8.182442e+01 -146.6250 63.28333
#> 204  151.5000     L        0    NA  96.97903420 8.179126e+01 -146.7083 63.28333
#> 205  151.4500     L        0    NA  92.79910363 8.176353e+01 -146.7917 63.28333
#> 206  169.8125     L        0    NA  88.61876656 8.174123e+01 -146.8750 63.28333
#> 207  247.7500     L        0    NA  84.43841000 8.172436e+01 -146.9583 63.28333
#> 208  356.5625     L        0    NA  80.25842247 8.171293e+01 -147.0417 63.28333
#> 209  390.8125     L        0    NA  76.07804424 8.170692e+01 -147.1250 63.28333
#> 210  450.8750     L        0    NA  71.89766231 8.170635e+01 -147.2083 63.28333
#> 211  345.1000     L        0    NA  67.71766521 8.171120e+01 -147.2917 63.28333
#> 212  300.9375     L        0    NA  63.53729320 8.172149e+01 -147.3750 63.28333
#> 213  268.5000     L        0    NA  59.35693381 8.173721e+01 -147.4583 63.28333
#> 214  188.1250     L        0    NA  55.17697452 8.175835e+01 -147.5417 63.28333
#> 215  177.0000     L        0    NA  50.99665564 8.178493e+01 -147.6250 63.28333
#> 216  147.5000     M        1     6  46.81636566 8.181695e+01 -147.7083 63.28333
#> 217  165.5000     M        0    NA  42.63649108 8.185439e+01 -147.7917 63.28333
#> 218  281.9375     L        0    NA  38.45627372 8.189726e+01 -147.8750 63.28333
#> 219  231.5625     L        0    NA  34.27610056 8.194556e+01 -147.9583 63.28333
#> 220  175.1250     L        1     5  30.09635860 8.199929e+01 -148.0417 63.28333
#> 221  168.3125     L        1     2  25.91628966 8.205846e+01 -148.1250 63.28333
#> 222  174.5500     L        0    NA  21.73628023 8.212306e+01 -148.2083 63.28333
#> 223  187.1875     L        0    NA  17.55671878 8.219308e+01 -148.2917 63.28333
#> 224  189.7500     L        0    NA  13.37684567 8.226854e+01 -148.3750 63.28333
#> 225  405.5833     M        0    NA 134.66921114 7.857162e+01 -145.9583 63.25000
#> 226  344.1333     M        0    NA 130.48491091 7.849495e+01 -146.0417 63.25000
#> 227  165.3750     L        0    NA 126.30015691 7.842371e+01 -146.1250 63.25000
#> 228  129.9375     M        1     0 122.11533654 7.835791e+01 -146.2083 63.25000
#> 229  130.0000     M        0    NA 117.93083723 7.829755e+01 -146.2917 63.25000
#> 230  108.2778     M        0    NA 113.74589993 7.824261e+01 -146.3750 63.25000
#> 231  111.4375     M        1     0 109.56091207 7.819312e+01 -146.4583 63.25000
#> 232  119.3750     L        0    NA 105.37626106 7.814906e+01 -146.5417 63.25000
#> 233  132.1875     L        0    NA 101.19118786 7.811044e+01 -146.6250 63.25000
#> 234  133.7857     L        0    NA  97.00607937 7.807725e+01 -146.7083 63.25000
#> 235  137.0667     L        0    NA  92.82132456 7.804950e+01 -146.7917 63.25000
#> 236  208.2500     L        0    NA  88.63616283 7.802718e+01 -146.8750 63.25000
#> 237  306.2500     L        0    NA  84.45098212 7.801030e+01 -146.9583 63.25000
#> 238  393.6667     L        0    NA  80.26617037 7.799885e+01 -147.0417 63.25000
#> 239  328.0000     L        1     0  76.08096699 7.799284e+01 -147.1250 63.25000
#> 240  329.0625     L        0    NA  71.89576041 7.799226e+01 -147.2083 63.25000
#> 241  279.5500     L        0    NA  67.71093910 7.799712e+01 -147.2917 63.25000
#> 242  245.0000     L        0    NA  63.52574244 7.800742e+01 -147.3750 63.25000
#> 243  261.4375     L        0    NA  59.34055839 7.802315e+01 -147.4583 63.25000
#> 244  182.8750     L        0    NA  55.15577438 7.804432e+01 -147.5417 63.25000
#> 245  153.5000     M        0    NA  50.97063132 7.807092e+01 -147.6250 63.25000
#> 246  135.9000     M        0    NA  46.78551666 7.810296e+01 -147.7083 63.25000
#> 247  202.2500     M        1     0  42.60081782 7.814043e+01 -147.7917 63.25000
#> 248  297.0000     L        0    NA  38.41577574 7.818334e+01 -147.8750 63.25000
#> 249  229.4375     L        0    NA  34.23077785 7.823169e+01 -147.9583 63.25000
#> 250  166.9375     M        1     0  30.04621207 7.828547e+01 -148.0417 63.25000
#> 251  176.5000     L        0    NA  25.86131785 7.834468e+01 -148.1250 63.25000
#> 252  181.3000     L        0    NA  21.67648411 7.840934e+01 -148.2083 63.25000
#> 253  184.0000     L        0    NA  17.49209777 7.847942e+01 -148.2917 63.25000
#> 254  187.8333     L        0    NA  13.30739981 7.855495e+01 -148.3750 63.25000
#> 255  241.3333     L        0    NA   9.12277713 7.863591e+01 -148.4583 63.25000
#> 256  328.5000     L        0    NA   4.93861863 7.872230e+01 -148.5417 63.25000
#> 257  266.4167     L        1     3   0.75416383 7.881413e+01 -148.6250 63.25000
#> 258  262.5882     L        1     0 134.73965692 7.485806e+01 -145.9583 63.21667
#> 259  262.0000     L        0    NA 130.55053370 7.478132e+01 -146.0417 63.21667
#> 260  202.5833     L        0    NA 126.36095628 7.471002e+01 -146.1250 63.21667
#> 261  107.8333     M        0    NA 122.17131206 7.464416e+01 -146.2083 63.21667
#> 262  107.8750     M        1     0 117.98199034 7.458375e+01 -146.2917 63.21667
#> 263  120.4500     L        1     0 113.79222970 7.452877e+01 -146.3750 63.21667
#> 264  114.0625     L        0    NA 109.60241855 7.447923e+01 -146.4583 63.21667
#> 265  116.4375     L        0    NA 105.41294470 7.443513e+01 -146.5417 63.21667
#> 266  120.3750     L        0    NA 101.22304821 7.439648e+01 -146.6250 63.21667
#> 267  124.3750     L        0    NA  97.03311649 7.436326e+01 -146.7083 63.21667
#> 268  158.2632     L        0    NA  92.84353888 7.433548e+01 -146.7917 63.21667
#> 269  260.1765     L        0    NA  88.65355391 7.431315e+01 -146.8750 63.21667
#> 270  324.4375     L        0    NA  84.46354948 7.429625e+01 -146.9583 63.21667
#> 271  405.5000     L        0    NA  80.27391496 7.428479e+01 -147.0417 63.21667
#> 272  250.0000     L        0    NA  76.08388886 7.427878e+01 -147.1250 63.21667
#> 273  229.2500     L        0    NA  71.89385908 7.427820e+01 -147.2083 63.21667
#> 274  196.0000     L        0    NA  67.70421500 7.428307e+01 -147.2917 63.21667
#> 275  147.5833     L        0    NA  63.51419513 7.429337e+01 -147.3750 63.21667
#> 276  136.4167     L        0    NA  59.32418785 7.430912e+01 -147.4583 63.21667
#> 277  134.1667     M        0    NA  55.13458156 7.433030e+01 -147.5417 63.21667
#> 278  132.6667     M        0    NA  50.94461476 7.435693e+01 -147.6250 63.21667
#> 279  160.2000     M        0    NA  46.75467684 7.438899e+01 -147.7083 63.21667
#> 280  330.1667     L        0    NA  42.56515519 7.442650e+01 -147.7917 63.21667
#> 281  290.5000     L        0    NA  38.37528983 7.446944e+01 -147.8750 63.21667
#> 282  181.0000     M        1     0  34.18546863 7.451783e+01 -147.9583 63.21667
#> 283  167.9167     M        1     0  29.99607947 7.457165e+01 -148.0417 63.21667
#> 284  180.4167     L        0    NA  25.80636242 7.463092e+01 -148.1250 63.21667
#> 285  191.2667     L        0    NA  21.61670480 7.469563e+01 -148.2083 63.21667
#> 286  190.0833     L        0    NA  17.42749601 7.476578e+01 -148.2917 63.21667
#> 287  200.5625     L        1     1  13.23797464 7.484137e+01 -148.3750 63.21667
#> 288  285.3750     L        0    NA   9.04852898 7.492240e+01 -148.4583 63.21667
#> 289  271.7500     L        0    NA   4.85954693 7.500886e+01 -148.5417 63.21667
#> 290  280.2353     L        0    NA   0.67026862 7.510077e+01 -148.6250 63.21667
#> 291  300.6250     L        0    NA 139.00433264 7.122676e+01 -145.8750 63.18333
#> 292  164.2000     L        1     0 134.81008168 7.114451e+01 -145.9583 63.18333
#> 293  136.6875     M        0    NA 130.61613690 7.106771e+01 -146.0417 63.18333
#> 294  123.7500     M        1     3 126.42173754 7.099635e+01 -146.1250 63.18333
#> 295  105.9375     M        0    NA 122.22727186 7.093043e+01 -146.2083 63.18333
#> 296  110.3333     L        0    NA 118.03312817 7.086996e+01 -146.2917 63.18333
#> 297  119.0000     L        0    NA 113.83854566 7.081494e+01 -146.3750 63.18333
#> 298  120.0000     L        0    NA 109.64391263 7.076535e+01 -146.4583 63.18333
#> 299  129.0000     L        0    NA 105.44961738 7.072122e+01 -146.5417 63.18333
#> 300  130.0000     L        0    NA 101.25489908 7.068253e+01 -146.6250 63.18333
#> 301  144.6250     L        0    NA  97.06014553 7.064928e+01 -146.7083 63.18333
#> 302  244.2500     L        1     0  92.86574656 7.062149e+01 -146.7917 63.18333
#> 303  275.7500     L        0    NA  88.67093981 7.059913e+01 -146.8750 63.18333
#> 304  295.2500     L        0    NA  84.47611409 7.058222e+01 -146.9583 63.18333
#> 305  263.6250     L        0    NA  80.28165824 7.057075e+01 -147.0417 63.18333
#> 306  187.8750     L        0    NA  76.08680987 7.056473e+01 -147.1250 63.18333
#> 307  170.0625     L        0    NA  71.89195832 7.056416e+01 -147.2083 63.18333
#> 308  142.4000     L        0    NA  67.69749290 7.056902e+01 -147.2917 63.18333
#> 309  126.8750     L        0    NA  63.50265126 7.057934e+01 -147.3750 63.18333
#> 310  123.6875     L        0    NA  59.30782220 7.059510e+01 -147.4583 63.18333
#> 311  126.8125     M        1     3  55.11339406 7.061630e+01 -147.5417 63.18333
#> 312  131.0000     M        0    NA  50.91860596 7.064295e+01 -147.6250 63.18333
#> 313  232.6000     L        0    NA  46.72384624 7.067504e+01 -147.7083 63.18333
#> 314  380.7500     L        0    NA  42.52950320 7.071258e+01 -147.7917 63.18333
#> 315  297.4375     L        0    NA  38.33481600 7.075556e+01 -147.8750 63.18333
#> 316  171.6875     M        0    NA  34.14017295 7.080399e+01 -147.9583 63.18333
#> 317  175.0000     M        0    NA  29.94596285 7.085786e+01 -148.0417 63.18333
#> 318  179.5625     M        0    NA  25.75142338 7.091718e+01 -148.1250 63.18333
#> 319  196.1500     M        0    NA  21.55694434 7.098194e+01 -148.2083 63.18333
#> 320  200.4375     L        0    NA  17.36291353 7.105215e+01 -148.2917 63.18333
#> 321  243.1875     L        0    NA  13.16857015 7.112780e+01 -148.3750 63.18333
#> 322  227.5000     L        0    NA   8.97430199 7.120890e+01 -148.4583 63.18333
#> 323  229.8750     L        0    NA   4.78049882 7.129544e+01 -148.5417 63.18333
#> 324  251.5000     L        1     0   0.58639840 7.138743e+01 -148.6250 63.18333
#> 325  210.6923     L        0    NA 139.07956127 6.751309e+01 -145.8750 63.15000
#> 326  152.5556     L        0    NA 134.88048894 6.743077e+01 -145.9583 63.15000
#> 327  116.6667     M        0    NA 130.68172427 6.735390e+01 -146.0417 63.15000
#> 328  103.5625     M        1     2 126.48250411 6.728248e+01 -146.1250 63.15000
#> 329  104.8750     L        0    NA 122.28321715 6.721650e+01 -146.2083 63.15000
#> 330  113.6250     L        0    NA 118.08425366 6.715598e+01 -146.2917 63.15000
#> 331  125.1250     L        1     0 113.88485043 6.710091e+01 -146.3750 63.15000
#> 332  160.5500     L        0    NA 109.68539669 6.705129e+01 -146.4583 63.15000
#> 333  214.3333     L        0    NA 105.48628120 6.700712e+01 -146.5417 63.15000
#> 334  155.9167     L        0    NA 101.28674224 6.696839e+01 -146.6250 63.15000
#> 335  184.8333     L        1     9  97.08716855 6.693512e+01 -146.7083 63.15000
#> 336  210.3333     L        0    NA  92.88794888 6.690730e+01 -146.7917 63.15000
#> 337  310.2632     L        0    NA  88.68832150 6.688492e+01 -146.8750 63.15000
#> 338  234.4375     L        0    NA  84.48867517 6.686799e+01 -146.9583 63.15000
#> 339  152.8125     L        0    NA  80.28939915 6.685652e+01 -147.0417 63.15000
#> 340  146.3125     L        0    NA  76.08973017 6.685049e+01 -147.1250 63.15000
#> 341  143.1250     L        0    NA  71.89005851 6.684992e+01 -147.2083 63.15000
#> 342  125.1000     M        0    NA  67.69077243 6.685479e+01 -147.2917 63.15000
#> 343  120.0000     M        1     4  63.49111018 6.686511e+01 -147.3750 63.15000
#> 344  121.0625     M        0    NA  59.29146051 6.688088e+01 -147.4583 63.15000
#> 345  127.2500     L        0    NA  55.09221218 6.690210e+01 -147.5417 63.15000
#> 346  173.5000     L        0    NA  50.89260345 6.692878e+01 -147.6250 63.15000
#> 347  354.5000     L        0    NA  46.69302308 6.696090e+01 -147.7083 63.15000
#> 348  323.5625     L        0    NA  42.49385982 6.699847e+01 -147.7917 63.15000
#> 349  185.3125     M        1     2  38.29435194 6.704149e+01 -147.8750 63.15000
#> 350  182.2500     L        1     2  34.09488820 6.708996e+01 -147.9583 63.15000
#> 351  186.7500     L        0    NA  29.89585732 6.714387e+01 -148.0417 63.15000
#> 352  186.8125     L        0    NA  25.69649761 6.720324e+01 -148.1250 63.15000
#> 353  209.7000     M        1     0  21.49719730 6.726806e+01 -148.2083 63.15000
#> 354  220.9375     M        0    NA  17.29834664 6.733833e+01 -148.2917 63.15000
#> 355  232.9375     L        0    NA  13.09918243 6.741405e+01 -148.3750 63.15000
#> 356  227.0625     L        1     0   8.90009391 6.749522e+01 -148.4583 63.15000
#> 357  237.2500     L        0    NA   4.70146979 6.758183e+01 -148.5417 63.15000
#> 358  278.6316     L        0    NA   0.50254843 6.767390e+01 -148.6250 63.15000
#> 359  223.0000     L        0    NA 139.15476741 6.379943e+01 -145.8750 63.11666
#> 360  178.8333     L        0    NA 134.95087615 6.371704e+01 -145.9583 63.11666
#> 361  117.8667     M        1     0 130.74729203 6.364011e+01 -146.0417 63.11666
#> 362  103.2500     M        0    NA 126.54325250 6.356862e+01 -146.1250 63.11666
#> 363  105.3125     L        0    NA 122.33914621 6.350259e+01 -146.2083 63.11666
#> 364  112.2500     L        0    NA 118.13536385 6.344202e+01 -146.2917 63.11666
#> 365  124.4375     L        0    NA 113.93114134 6.338690e+01 -146.3750 63.11666
#> 366  264.4000     L        0    NA 109.72686835 6.333724e+01 -146.4583 63.11666
#> 367  339.3750     L        0    NA 105.52293406 6.329303e+01 -146.5417 63.11666
#> 368  162.0625     L        0    NA 101.31857587 6.325427e+01 -146.6250 63.11666
#> 369  192.5000     L        0    NA  97.11418247 6.322097e+01 -146.7083 63.11666
#> 370  215.9375     L        0    NA  92.91014455 6.319312e+01 -146.7917 63.11666
#> 371  207.0625     L        0    NA  88.70569800 6.317073e+01 -146.8750 63.11666
#> 372  187.6667     L        0    NA  84.50123198 6.315379e+01 -146.9583 63.11666
#> 373  126.2500     L        0    NA  80.29713723 6.314230e+01 -147.0417 63.11666
#> 374  119.5833     M        0    NA  76.09264960 6.313627e+01 -147.1250 63.11666
#> 375  120.1667     M        0    NA  71.88815827 6.313569e+01 -147.2083 63.11666
#> 376  120.0000     M        0    NA  67.68405397 6.314057e+01 -147.2917 63.11666
#> 377  120.0000     M        1     0  63.47957255 6.315090e+01 -147.3750 63.11666
#> 378  120.9333     M        1     2  59.27510371 6.316669e+01 -147.4583 63.11666
#> 379  128.5625     M        1     0  55.07103714 6.318793e+01 -147.5417 63.11666
#> 380  224.0000     L        0    NA  50.86660872 6.321462e+01 -147.6250 63.11666
#> 381  271.8500     L        0    NA  46.66220915 6.324677e+01 -147.7083 63.11666
#> 382  170.7500     L        0    NA  42.45822711 6.328437e+01 -147.7917 63.11666
#> 383  168.1875     M        1     3  38.25390000 6.332743e+01 -147.8750 63.11666
#> 384  215.0625     L        0    NA  34.04961700 6.337594e+01 -147.9583 63.11666
#> 385  184.9333     L        0    NA  29.84576728 6.342990e+01 -148.0417 63.11666
#> 386  207.8333     L        0    NA  25.64158827 6.348932e+01 -148.1250 63.11666
#> 387  214.0667     M        0    NA  21.43746864 6.355420e+01 -148.2083 63.11666
#> 388  220.5833     M        0    NA  17.23379906 6.362452e+01 -148.2917 63.11666
#> 389  231.2500     L        1     0  13.02981546 6.370030e+01 -148.3750 63.11666
#> 390  242.5833     L        0    NA   8.82590753 6.378154e+01 -148.4583 63.11666
#> 391  257.5000     L        0    NA   4.62246439 6.386823e+01 -148.5417 63.11666
#> 392  318.3750     L        0    NA   0.41872354 6.396038e+01 -148.6250 63.11666
#> 393  192.5625     L        0    NA 139.22994674 6.008600e+01 -145.8750 63.08333
#> 394  148.0625     L        1     0 135.02123777 6.000354e+01 -145.9583 63.08333
#> 395  104.7895     M        0    NA 130.81283641 5.992654e+01 -146.0417 63.08333
#> 396  103.5385     M        0    NA 126.60397923 5.985500e+01 -146.1250 63.08333
#> 397  107.9167     L        0    NA 122.39505533 5.978891e+01 -146.2083 63.08333
#> 398  118.6667     L        0    NA 118.18645582 5.972829e+01 -146.2917 63.08333
#> 399  124.1875     L        0    NA 113.97741575 5.967312e+01 -146.3750 63.08333
#> 400  177.1667     L        0    NA 109.76832521 5.962342e+01 -146.4583 63.08333
#> 401  179.7778     L        0    NA 105.55957384 5.957917e+01 -146.5417 63.08333
#> 402  181.1875     L        0    NA 101.35039815 5.954038e+01 -146.6250 63.08333
#> 403  187.7500     L        0    NA  97.14118777 5.950705e+01 -146.7083 63.08333
#> 404  156.7500     L        0    NA  92.93233231 5.947918e+01 -146.7917 63.08333
#> 405  214.7778     L        0    NA  88.72306829 5.945677e+01 -146.8750 63.08333
#> 406  153.7222     M        0    NA  84.51378482 5.943981e+01 -146.9583 63.08333
#> 407  106.6250     M        1     0  80.30487357 5.942832e+01 -147.0417 63.08333
#> 408  105.9375     M        1     3  76.09556798 5.942228e+01 -147.1250 63.08333
#> 409  116.2500     M        0    NA  71.88625972 5.942170e+01 -147.2083 63.08333
#> 410  120.0556     M        0    NA  67.67733791 5.942658e+01 -147.2917 63.08333
#> 411  125.0556     L        0    NA  63.46803904 5.943692e+01 -147.3750 63.08333
#> 412  126.3846     L        0    NA  59.25875274 5.945272e+01 -147.4583 63.08333
#> 413  130.2500     M        1     0  55.04986865 5.947398e+01 -147.5417 63.08333
#> 414  141.2500     M        0    NA  50.84062326 5.950070e+01 -147.6250 63.08333
#> 415  142.5333     M        0    NA  46.63140621 5.953287e+01 -147.7083 63.08333
#> 416  143.9167     M        1     0  42.42260711 5.957050e+01 -147.7917 63.08333
#> 417  194.5000     L        0    NA  38.21346248 5.961360e+01 -147.8750 63.08333
#> 418  180.2500     L        0    NA  34.00436143 5.966215e+01 -147.9583 63.08333
#> 419  185.6923     L        0    NA  29.79569561 5.971616e+01 -148.0417 63.08333
#> 420  200.2778     L        0    NA  25.58669851 5.977563e+01 -148.1250 63.08333
#> 421  217.0556     L        0    NA  21.37776177 5.984056e+01 -148.2083 63.08333
#> 422  228.7500     M        1     0  17.16927449 5.991094e+01 -148.2917 63.08333
#> 423  242.5000     L        0    NA  12.96047322 5.998679e+01 -148.3750 63.08333
#> 424  259.5000     L        0    NA   8.75174708 6.006810e+01 -148.4583 63.08333
#> 425  309.6111     L        0    NA   4.54348714 6.015486e+01 -148.5417 63.08333
#> 426  400.0000     L        0    NA   0.33492851 6.024709e+01 -148.6250 63.08333
#> 427  167.0625     L        0    NA 139.30510350 5.637259e+01 -145.8750 63.05000
#> 428  108.4375     M        1     7 135.09157827 5.629006e+01 -145.9583 63.05000
#> 429  103.2500     M        1    15 130.87836111 5.621300e+01 -146.0417 63.05000
#> 430  103.8500     M        1     2 126.66468773 5.614139e+01 -146.1250 63.05000
#> 431  118.0000     L        1     0 122.45094766 5.607525e+01 -146.2083 63.05000
#> 432  150.2500     L        1     0 118.23753245 5.601458e+01 -146.2917 63.05000
#> 433  159.2500     L        1     5 114.02367626 5.595936e+01 -146.3750 63.05000
#> 434  139.8333     L        1     1 109.80976963 5.590961e+01 -146.4583 63.05000
#> 435  177.8000     L        1     2 105.59620262 5.586533e+01 -146.5417 63.05000
#> 436  276.1333     L        1     3 101.38221087 5.582651e+01 -146.6250 63.05000
#> 437  134.1875     L        1     0  97.16818394 5.579315e+01 -146.7083 63.05000
#> 438  163.4375     L        1     2  92.95451340 5.576525e+01 -146.7917 63.05000
#> 439  207.7500     L        1     3  88.74043337 5.574282e+01 -146.8750 63.05000
#> 440  117.0500     M        0    NA  84.52633389 5.572585e+01 -146.9583 63.05000
#> 441  110.1875     M        0    NA  80.31260657 5.571435e+01 -147.0417 63.05000
#> 442  109.0625     M        0    NA  76.09848549 5.570831e+01 -147.1250 63.05000
#> 443  119.3750     M        1     0  71.88436072 5.570773e+01 -147.2083 63.05000
#> 444  123.1875     L        1     0  67.67062386 5.571261e+01 -147.2917 63.05000
#> 445  129.1500     L        0    NA  63.45650899 5.572296e+01 -147.3750 63.05000
#> 446  133.3750     L        0    NA  59.24240669 5.573878e+01 -147.4583 63.05000
#> 447  134.1875     L        0    NA  55.02870753 5.576005e+01 -147.5417 63.05000
#> 448  133.9375     L        0    NA  50.81464561 5.578679e+01 -147.6250 63.05000
#> 449  138.3000     L        0    NA  46.60061252 5.581899e+01 -147.7083 63.05000
#> 450  165.5000     L        0    NA  42.38699780 5.585666e+01 -147.7917 63.05000
#> 451  179.3750     L        0    NA  38.17303710 5.589979e+01 -147.8750 63.05000
#> 452  173.8125     L        0    NA  33.95912047 5.594838e+01 -147.9583 63.05000
#> 453  185.0625     L        0    NA  29.74563796 5.600243e+01 -148.0417 63.05000
#> 454  201.5500     L        0    NA  25.53182523 5.606196e+01 -148.1250 63.05000
#> 455  221.3125     M        0    NA  21.31807182 5.612694e+01 -148.2083 63.05000
#> 456  236.1875     M        1     0  17.10476929 5.619739e+01 -148.2917 63.05000
#> 457  252.8125     M        1     0  12.89115180 5.627330e+01 -148.3750 63.05000
#> 458  276.7500     M        1     0   8.67760990 5.635468e+01 -148.4583 63.05000
#> 459  306.8500     M        0    NA   4.46453360 5.644151e+01 -148.5417 63.05000
#> 460  405.1250     M        0    NA   0.25115864 5.653381e+01 -148.6250 63.05000
#> 461  114.6667     M        0    NA 139.38023772 5.265920e+01 -145.8750 63.01667
#> 462  111.1429     M        1    16 135.16189766 5.257660e+01 -145.9583 63.01667
#> 463  127.7500     M        1     3 130.94386615 5.249947e+01 -146.0417 63.01667
#> 464  104.8947     M        1    13 126.72537802 5.242780e+01 -146.1250 63.01667
#> 465  215.5294     M        1     4 122.50682372 5.236161e+01 -146.2083 63.01667
#> 466  204.8750     L        1     8 118.28859375 5.230088e+01 -146.2917 63.01667
#> 467  196.6875     L        1     7 114.06992288 5.224562e+01 -146.3750 63.01667
#> 468  151.8750     L        1     3 109.85120160 5.219583e+01 -146.4583 63.01667
#> 469  135.7000     M        1     0 105.63282040 5.215151e+01 -146.5417 63.01667
#> 470  126.3846     M        1     2 101.41401405 5.211265e+01 -146.6250 63.01667
#> 471  150.1667     M        1     4  97.19517252 5.207926e+01 -146.7083 63.01667
#> 472  176.0833     L        1     4  92.97668784 5.205135e+01 -146.7917 63.01667
#> 473  145.1667     M        1     1  88.75779323 5.202889e+01 -146.8750 63.01667
#> 474  114.8000     M        0    NA  84.53887970 5.201191e+01 -146.9583 63.01667
#> 475  115.3125     L        0    NA  80.32033826 5.200040e+01 -147.0417 63.01667
#> 476  115.6875     L        0    NA  76.10140212 5.199435e+01 -147.1250 63.01667
#> 477  119.6875     M        0    NA  71.88246280 5.199377e+01 -147.2083 63.01667
#> 478  124.5000     M        0    NA  67.66391183 5.199866e+01 -147.2917 63.01667
#> 479  128.1500     M        0    NA  63.44498240 5.200902e+01 -147.3750 63.01667
#> 480  132.0625     M        0    NA  59.22606554 5.202484e+01 -147.4583 63.01667
#> 481  134.7500     L        0    NA  55.00755175 5.204614e+01 -147.5417 63.01667
#> 482  137.1250     L        0    NA  50.78867575 5.207290e+01 -147.6250 63.01667
#> 483  144.7000     L        0    NA  46.56982807 5.210513e+01 -147.7083 63.01667
#> 484  168.5000     L        0    NA  42.35139919 5.214283e+01 -147.7917 63.01667
#> 485  166.9375     L        0    NA  38.13262386 5.218599e+01 -147.8750 63.01667
#> 486  175.4375     L        0    NA  33.91389258 5.223463e+01 -147.9583 63.01667
#> 487  188.0000     L        0    NA  29.69559635 5.228873e+01 -148.0417 63.01667
#> 488  202.0500     L        1     0  25.47696842 5.234830e+01 -148.1250 63.01667
#> 489  220.2500     L        0    NA  21.25840080 5.241334e+01 -148.2083 63.01667
#> 490  238.8125     M        0    NA  17.04028345 5.248384e+01 -148.2917 63.01667
#> 491  263.6250     M        0    NA  12.82185118 5.255982e+01 -148.3750 63.01667
#> 492  291.0625     M        1     2   8.60349395 5.264127e+01 -148.4583 63.01667
#> 493  314.6000     M        1     0   4.38560374 5.272817e+01 -148.5417 63.01667
#> 494  336.7500     M        1     3   0.16741388 5.282056e+01 -148.6250 63.01667
#> 495  180.1250     L        0    NA 139.45534932 4.894582e+01 -145.8750 62.98333
#> 496  140.2857     L        1     2 135.23219588 4.886315e+01 -145.9583 62.98333
#> 497  122.0000     L        1     0 131.00935147 4.878596e+01 -146.0417 62.98333
#> 498  104.6667     M        1    11 126.78605003 4.871423e+01 -146.1250 62.98333
#> 499  148.1053     M        1     9 122.56268245 4.864798e+01 -146.2083 62.98333
#> 500  127.4375     M        1     6 118.33963967 4.858720e+01 -146.2917 62.98333
#> 501  119.0000     M        1     5 114.11615558 4.853189e+01 -146.3750 62.98333
#> 502  154.0000     M        1     0 109.89262109 4.848206e+01 -146.4583 62.98333
#> 503  113.8333     M        1     7 105.66942716 4.843770e+01 -146.5417 62.98333
#> 504  111.0000     M        1    19 101.44580764 4.839881e+01 -146.6250 62.98333
#> 505  112.6875     M        1     4  97.22215296 4.836540e+01 -146.7083 62.98333
#> 506  127.9375     M        1     4  92.99885559 4.833746e+01 -146.7917 62.98333
#> 507  111.6875     M        1     2  88.77514786 4.831499e+01 -146.8750 62.98333
#> 508  117.5000     L        0    NA  84.55142122 4.829799e+01 -146.9583 62.98333
#> 509  117.0833     L        0    NA  80.32806711 4.828646e+01 -147.0417 62.98333
#> 510  116.5833     L        0    NA  76.10431787 4.828041e+01 -147.1250 62.98333
#> 511  122.0833     L        0    NA  71.88056546 4.827983e+01 -147.2083 62.98333
#> 512  128.2500     L        1     0  67.65720182 4.828473e+01 -147.2917 62.98333
#> 513  129.8000     M        0    NA  63.43345929 4.829509e+01 -147.3750 62.98333
#> 514  132.8333     M        0    NA  59.20972932 4.831093e+01 -147.4583 62.98333
#> 515  136.0833     M        1     0  54.98640284 4.833224e+01 -147.5417 62.98333
#> 516  141.5000     L        0    NA  50.76271372 4.835903e+01 -147.6250 62.98333
#> 517  148.2667     L        0    NA  46.53905289 4.839129e+01 -147.7083 62.98333
#> 518  157.1667     L        0    NA  42.31581129 4.842901e+01 -147.7917 62.98333
#> 519  169.3333     L        0    NA  38.09222279 4.847222e+01 -147.8750 62.98333
#> 520  178.7500     L        0    NA  33.86867832 4.852089e+01 -147.9583 62.98333
#> 521  187.6667     L        0    NA  29.64556931 4.857504e+01 -148.0417 62.98333
#> 522  203.2667     L        0    NA  25.42212813 4.863466e+01 -148.1250 62.98333
#> 523  220.3333     L        0    NA  21.19874724 4.869976e+01 -148.2083 62.98333
#> 524  239.5000     L        0    NA  16.97581702 4.877032e+01 -148.2917 62.98333
#> 525  272.8333     L        0    NA  12.75257141 4.884636e+01 -148.3750 62.98333
#> 526  329.2500     M        0    NA   8.52940082 4.892787e+01 -148.4583 62.98333
#> 527  332.1500     M        1     3   4.30669764 4.901485e+01 -148.5417 62.98333
#> 528  347.5000     M        1    18   0.08369434 4.910732e+01 -148.6250 62.98333
#> 529  181.7368     L        0    NA 139.53043829 4.523247e+01 -145.8750 62.95000
#> 530  121.0625     L        1     0 135.30247293 4.514972e+01 -145.9583 62.95000
#> 531  115.3750     L        1     0 131.07481706 4.507246e+01 -146.0417 62.95000
#> 532  101.3750     M        1     9 126.84670376 4.500068e+01 -146.1250 62.95000
#> 533  103.9375     M        1    19 122.61852385 4.493437e+01 -146.2083 62.95000
#> 534  113.5833     M        1     2 118.39067021 4.487354e+01 -146.2917 62.95000
#> 535  118.9167     M        1     0 114.16237434 4.481819e+01 -146.3750 62.95000
#> 536  117.8000     M        1    11 109.93402811 4.476831e+01 -146.4583 62.95000
#> 537  134.6250     L        1     0 105.70602288 4.472392e+01 -146.5417 62.95000
#> 538  140.7500     L        1     0 101.47759165 4.468499e+01 -146.6250 62.95000
#> 539  118.5625     L        1     0  97.24912528 4.465155e+01 -146.7083 62.95000
#> 540  114.9375     L        1     0  93.02101666 4.462359e+01 -146.7917 62.95000
#> 541  115.5000     M        1    11  88.79249726 4.460110e+01 -146.8750 62.95000
#> 542  116.0500     L        0    NA  84.56395845 4.458408e+01 -146.9583 62.95000
#> 543  116.0000     L        0    NA  80.33579313 4.457255e+01 -147.0417 62.95000
#> 544  118.3125     L        0    NA  76.10723275 4.456649e+01 -147.1250 62.95000
#> 545  123.1250     L        0    NA  71.87866868 4.456591e+01 -147.2083 62.95000
#> 546  126.8125     L        0    NA  67.65049383 4.457081e+01 -147.2917 62.95000
#> 547  130.9500     L        0    NA  63.42193965 4.458119e+01 -147.3750 62.95000
#> 548  134.4375     L        0    NA  59.19339802 4.459704e+01 -147.4583 62.95000
#> 549  137.2500     L        0    NA  54.96526082 4.461837e+01 -147.5417 62.95000
#> 550  142.6875     M        1     0  50.73675951 4.464517e+01 -147.6250 62.95000
#> 551  150.4000     M        1     0  46.50828699 4.467746e+01 -147.7083 62.95000
#> 552  158.6250     M        1     0  42.28023413 4.471522e+01 -147.7917 62.95000
#> 553  168.9375     L        0    NA  38.05183389 4.475846e+01 -147.8750 62.95000
#> 554  180.6250     L        0    NA  33.82347768 4.480718e+01 -147.9583 62.95000
#> 555  189.8750     L        0    NA  29.59555683 4.486137e+01 -148.0417 62.95000
#> 556  208.9000     L        0    NA  25.36730436 4.492104e+01 -148.1250 62.95000
#> 557  226.6250     L        0    NA  21.13911114 4.498619e+01 -148.2083 62.95000
#> 558  244.8750     L        0    NA  16.91137002 4.505681e+01 -148.2917 62.95000
#> 559  282.8125     L        0    NA  12.68331252 4.513292e+01 -148.3750 62.95000
#> 560  319.7500     M        1     3   8.45533051 4.521450e+01 -148.4583 62.95000
#> 561  345.0000     M        0    NA   4.22781531 4.530155e+01 -148.5417 62.95000
#> 562  373.8750     L        0    NA   0.00000000 4.539409e+01 -148.6250 62.95000
#> 563  154.7368     L        0    NA 139.60550465 4.151912e+01 -145.8750 62.91667
#> 564  145.3125     L        1     9 135.37272880 4.143631e+01 -145.9583 62.91667
#> 565  139.6875     L        1     6 131.14026293 4.135899e+01 -146.0417 62.91667
#> 566  113.3125     M        1    18 126.90733921 4.128714e+01 -146.1250 62.91667
#> 567  105.3889     M        1     2 122.67434892 4.122077e+01 -146.2083 62.91667
#> 568  133.6111     M        1     0 118.44168537 4.115990e+01 -146.2917 62.91667
#> 569  142.1875     L        1     0 114.20857917 4.110450e+01 -146.3750 62.91667
#> 570  133.6923     L        1     6 109.97542264 4.105458e+01 -146.4583 62.91667
#> 571  146.2500     L        1     0 105.74260757 4.101014e+01 -146.5417 62.91667
#> 572  127.5333     L        1     0 101.50936608 4.097119e+01 -146.6250 62.91667
#> 573  120.0000     L        1     0  97.27608997 4.093772e+01 -146.7083 62.91667
#> 574  117.6875     L        1     0  93.04317106 4.090973e+01 -146.7917 62.91667
#> 575  117.1250     M        1     1  88.80984144 4.088722e+01 -146.8750 62.91667
#> 576  118.5294     L        0    NA  84.57649241 4.087020e+01 -146.9583 62.91667
#> 577  116.5789     L        0    NA  80.34351784 4.085865e+01 -147.0417 62.91667
#> 578  116.3125     M        1     0  76.11014674 4.085259e+01 -147.1250 62.91667
#> 579  121.4375     L        0    NA  71.87677299 4.085201e+01 -147.2083 62.91667
#> 580  125.6875     L        0    NA  67.64378787 4.085691e+01 -147.2917 62.91667
#> 581  133.0500     L        0    NA  63.41042348 4.086730e+01 -147.3750 62.91667
#> 582  141.3125     L        0    NA  59.17707164 4.088316e+01 -147.4583 62.91667
#> 583  148.8125     L        0    NA  54.94412416 4.090451e+01 -147.5417 62.91667
#> 584  150.5625     M        0    NA  50.71081313 4.093134e+01 -147.6250 62.91667
#> 585  157.1000     M        1     0  46.47753037 4.096365e+01 -147.7083 62.91667
#> 586  165.7500     M        1     2  42.24466768 4.100144e+01 -147.7917 62.91667
#> 587  173.5625     M        0    NA  38.01145718 4.104472e+01 -147.8750 62.91667
#> 588  182.8125     L        0    NA  33.77829015 4.109348e+01 -147.9583 62.91667
#> 589  195.4375     L        0    NA  29.54556044 4.114771e+01 -148.0417 62.91667
#> 590  214.5500     L        0    NA  25.31249712 4.120743e+01 -148.1250 62.91667
#> 591  231.8125     L        0    NA  21.07949403 4.127264e+01 -148.2083 62.91667
#> 592  253.4375     L        0    NA  16.84694243 4.134332e+01 -148.2917 62.91667
#> 593  288.5625     L        0    NA  12.61407450 4.141949e+01 -148.3750 62.91667
#> 594  194.0625     L        0    NA 143.91805145 3.789416e+01 -145.7917 62.88334
#> 595  212.0714     L        0    NA 139.68054833 3.780580e+01 -145.8750 62.88334
#> 596  224.5000     L        1     1 135.44296345 3.772291e+01 -145.9583 62.88334
#> 597  226.2857     M        1     5 131.20568902 3.764553e+01 -146.0417 62.88334
#> 598  192.1875     M        1    11 126.96795635 3.757362e+01 -146.1250 62.88334
#> 599   99.6250     L        1     3 122.73015712 3.750720e+01 -146.2083 62.88334
#> 600  129.9500     L        1     0 118.49268511 3.744627e+01 -146.2917 62.88334
#> 601  147.6250     L        0    NA 114.25477005 3.739082e+01 -146.3750 62.88334
#> 602  142.8125     L        0    NA 110.01680466 3.734086e+01 -146.4583 62.88334
#> 603  134.4375     L        0    NA 105.77918120 3.729639e+01 -146.5417 62.88334
#> 604  122.8000     L        0    NA 101.54113090 3.725740e+01 -146.6250 62.88334
#> 605  119.4667     L        0    NA  97.30304549 3.722390e+01 -146.7083 62.88334
#> 606  119.9167     L        0    NA  93.06531875 3.719589e+01 -146.7917 62.88334
#> 607  116.0000     M        1     3  88.82718037 3.717337e+01 -146.8750 62.88334
#> 608  121.5833     L        0    NA  84.58902258 3.715633e+01 -146.9583 62.88334
#> 609  119.6667     M        1     2  80.35123919 3.714477e+01 -147.0417 62.88334
#> 610  120.2500     M        1     0  76.11305986 3.713871e+01 -147.1250 62.88334
#> 611  122.5000     M        0    NA  71.87487685 3.713813e+01 -147.2083 62.88334
#> 612  124.7500     L        0    NA  67.63708393 3.714303e+01 -147.2917 62.88334
#> 613  135.2000     L        1     0  63.39891080 3.715342e+01 -147.3750 62.88334
#> 614  145.3750     L        0    NA  59.16075020 3.716930e+01 -147.4583 62.88334
#> 615  155.8125     L        0    NA  54.92299490 3.719067e+01 -147.5417 62.88334
#> 616  163.6250     L        0    NA  50.68487459 3.721752e+01 -147.6250 62.88334
#> 617  168.1500     M        1     0  46.44678304 3.724986e+01 -147.7083 62.88334
#> 618  177.1250     M        0    NA  42.20911199 3.728768e+01 -147.7917 62.88334
#> 619  186.3750     M        0    NA  37.97109266 3.733099e+01 -147.8750 62.88334
#> 620  195.2500     M        1     0  33.73311730 3.737979e+01 -147.9583 62.88334
#> 621  210.0625     L        0    NA  29.49557814 3.743407e+01 -148.0417 62.88334
#> 622  225.1500     L        0    NA  25.25770643 3.749385e+01 -148.1250 62.88334
#> 623  240.4375     L        0    NA  21.01989391 3.755911e+01 -148.2083 62.88334
#> 624  264.2143     L        0    NA  16.78253431 3.762985e+01 -148.2917 62.88334
#> 625  297.0833     L        0    NA  12.54485738 3.770608e+01 -148.3750 62.88334
#> 626  216.4375     L        0    NA 143.99788461 3.418072e+01 -145.7917 62.85000
#> 627  420.7500     L        0    NA 139.75557360 3.409227e+01 -145.8750 62.85000
#> 628  258.8000     M        1     8 135.51318035 3.400932e+01 -145.9583 62.85000
#> 629  120.8571     M        1     3 131.27109906 3.393187e+01 -146.0417 62.85000
#> 630   97.6667     M        1    25 127.02855860 3.385990e+01 -146.1250 62.85000
#> 631  106.6667     L        1     2 122.78595162 3.379343e+01 -146.2083 62.85000
#> 632  131.8947     L        1    13 118.54367233 3.373245e+01 -146.2917 62.85000
#> 633  142.7500     L        0    NA 114.30094957 3.367695e+01 -146.3750 62.85000
#> 634  145.6250     L        0    NA 110.05817652 3.362695e+01 -146.4583 62.85000
#> 635  142.3750     L        0    NA 105.81574585 3.358245e+01 -146.5417 62.85000
#> 636  129.8824     L        0    NA 101.57288792 3.354343e+01 -146.6250 62.85000
#> 637  120.5263     L        0    NA  97.32999541 3.350990e+01 -146.7083 62.85000
#> 638  120.2500     L        0    NA  93.08746100 3.348186e+01 -146.7917 62.85000
#> 639  115.8750     M        1     0  88.84451503 3.345932e+01 -146.8750 62.85000
#> 640  120.5625     M        1     0  84.60155019 3.344226e+01 -146.9583 62.85000
#> 641  122.2000     M        1     0  80.35895966 3.343070e+01 -147.0417 62.85000
#> 642  124.8750     M        0    NA  76.11597226 3.342463e+01 -147.1250 62.85000
#> 643  128.0000     M        0    NA  71.87298219 3.342405e+01 -147.2083 62.85000
#> 644  132.1667     M        0    NA  67.63038165 3.342896e+01 -147.2917 62.85000
#> 645  138.5333     L        0    NA  63.38740094 3.343936e+01 -147.3750 62.85000
#> 646  148.5833     L        0    NA  59.14443277 3.345525e+01 -147.4583 62.85000
#> 647  160.7500     L        0    NA  54.90186982 3.347663e+01 -147.5417 62.85000
#> 648  172.9167     L        0    NA  50.65894242 3.350351e+01 -147.6250 62.85000
#> 649  180.3333     M        0    NA  46.41604326 3.353587e+01 -147.7083 62.85000
#> 650  187.1667     M        0    NA  42.17356503 3.357373e+01 -147.7917 62.85000
#> 651  200.3333     M        1    40  37.93073805 3.361708e+01 -147.8750 62.85000
#> 652  209.1667     M        0    NA  33.68795503 3.366592e+01 -147.9583 62.85000
#> 653  224.7500     L        0    NA  29.44560862 3.372024e+01 -148.0417 62.85000
#> 654  239.8667     L        0    NA  25.20292920 3.378007e+01 -148.1250 62.85000
#> 655  259.3333     L        0    NA  20.96030893 3.384538e+01 -148.2083 62.85000
#> 656  223.2500     L        0    NA 144.07769359 3.046729e+01 -145.7917 62.81667
#> 657  250.7500     L        1     4 139.83057614 3.037877e+01 -145.8750 62.81667
#> 658  120.1579     M        1     0 135.58337700 3.029575e+01 -145.9583 62.81667
#> 659   97.9412     M        1     4 131.33648927 3.021823e+01 -146.0417 62.81667
#> 660  103.0625     M        1     4 127.08914249 3.014621e+01 -146.1250 62.81667
#> 661  114.6875     L        1     0 122.84172973 3.007967e+01 -146.2083 62.81667
#> 662  128.2308     M        0    NA 118.59464410 3.001864e+01 -146.2917 62.81667
#> 663  146.1333     M        0    NA 114.34711510 2.996310e+01 -146.3750 62.81667
#> 664  150.3333     L        0    NA 110.09953584 2.991306e+01 -146.4583 62.81667
#> 665  150.5714     L        1     0 105.85229942 2.986852e+01 -146.5417 62.81667
#> 666  137.2500     L        0    NA 101.60463532 2.982946e+01 -146.6250 62.81667
#> 667  127.0500     L        0    NA  97.35693614 2.979591e+01 -146.7083 62.81667
#> 668  124.0000     L        0    NA  93.10959655 2.976785e+01 -146.7917 62.81667
#> 669  119.5625     M        0    NA  88.86184445 2.974528e+01 -146.8750 62.81667
#> 670  121.7500     M        0    NA  84.61407348 2.972822e+01 -146.9583 62.81667
#> 671  127.0000     M        0    NA  80.36667729 2.971664e+01 -147.0417 62.81667
#> 672  131.3750     M        0    NA  76.11888377 2.971057e+01 -147.1250 62.81667
#> 673  135.0625     M        0    NA  71.87108709 2.970998e+01 -147.2083 62.81667
#> 674  134.9375     M        0    NA  67.62368139 2.971490e+01 -147.2917 62.81667
#> 675  144.8000     L        0    NA  63.37589457 2.972531e+01 -147.3750 62.81667
#> 676  153.7500     L        0    NA  59.12812028 2.974121e+01 -147.4583 62.81667
#> 677  165.5000     L        0    NA  54.88075165 2.976261e+01 -147.5417 62.81667
#> 678  181.1250     L        0    NA  50.63301811 2.978951e+01 -147.6250 62.81667
#> 679  192.9000     M        0    NA  46.38531281 2.982190e+01 -147.7083 62.81667
#> 680  199.1875     M        0    NA  42.13802884 2.985979e+01 -147.7917 62.81667
#> 681  208.6875     M        0    NA  37.89039568 2.990318e+01 -147.8750 62.81667
#> 682  227.1250     M        0    NA  33.64280645 2.995206e+01 -147.9583 62.81667
#> 683  244.7500     M        0    NA  29.39565476 3.000643e+01 -148.0417 62.81667
#> 684  262.8000     L        1     6  25.14816857 3.006630e+01 -148.1250 62.81667
#> 685  143.0000     L        0    NA 144.15747379 2.675409e+01 -145.7917 62.78333
#> 686  123.3333     L        1     0 139.90555163 2.666550e+01 -145.8750 62.78333
#> 687   99.3750     M        1     7 135.65354782 2.658241e+01 -145.9583 62.78333
#> 688   98.1000     M        1     9 131.40185590 2.650483e+01 -146.0417 62.78333
#> 689  135.1250     L        1     0 127.14970452 2.643274e+01 -146.1250 62.78333
#> 690  122.4375     L        0    NA 122.89748669 2.636615e+01 -146.2083 62.78333
#> 691  119.6875     M        1     3 118.64559748 2.630507e+01 -146.2917 62.78333
#> 692  141.7000     M        0    NA 114.39326398 2.624948e+01 -146.3750 62.78333
#> 693  150.4375     L        0    NA 110.14088023 2.619940e+01 -146.4583 62.78333
#> 694  146.2857     L        0    NA 105.88883980 2.615482e+01 -146.5417 62.78333
#> 695  135.2500     L        0    NA 101.63637126 2.611573e+01 -146.6250 62.78333
#> 696  136.6667     L        0    NA  97.38386767 2.608215e+01 -146.7083 62.78333
#> 697  129.4167     L        0    NA  93.13172410 2.605407e+01 -146.7917 62.78333
#> 698  124.2143     L        0    NA  88.87916761 2.603148e+01 -146.8750 62.78333
#> 699  125.3750     M        1     1  84.62659175 2.601440e+01 -146.9583 62.78333
#> 700  132.5000     M        0    NA  80.37439162 2.600282e+01 -147.0417 62.78333
#> 701  136.8750     M        1     0  76.12179424 2.599674e+01 -147.1250 62.78333
#> 702  146.3750     M        1     0  71.86919319 2.599615e+01 -147.2083 62.78333
#> 703  142.7500     M        0    NA  67.61698355 2.600107e+01 -147.2917 62.78333
#> 704  150.3000     L        0    NA  63.36439236 2.601149e+01 -147.3750 62.78333
#> 705  163.7500     L        0    NA  59.11181369 2.602741e+01 -147.4583 62.78333
#> 706  174.2500     L        0    NA  54.85964161 2.604883e+01 -147.5417 62.78333
#> 707  185.7500     L        0    NA  50.60710316 2.607575e+01 -147.6250 62.78333
#> 708  200.2500     M        0    NA  46.35459344 2.610817e+01 -147.7083 62.78333
#> 709  212.0625     M        1     2  42.10250548 2.614608e+01 -147.7917 62.78333
#> 710  220.6250     M        1     3  37.85006786 2.618951e+01 -147.8750 62.78333
#> 711  243.5625     M        1    10  33.59767416 2.623843e+01 -147.9583 62.78333
#> 712  152.1875     L        0    NA 144.23722978 2.304090e+01 -145.7917 62.75000
#> 713  131.0000     L        1     0 139.98050437 2.295224e+01 -145.8750 62.75000
#> 714  102.6667     M        1     7 135.72369735 2.286908e+01 -145.9583 62.75000
#> 715   99.2667     M        1    35 131.46720270 2.279143e+01 -146.0417 62.75000
#> 716  138.1429     L        1     0 127.21024818 2.271929e+01 -146.1250 62.75000
#> 717  138.9375     L        0    NA 122.95322775 2.265264e+01 -146.2083 62.75000
#> 718  139.1250     L        0    NA 118.69653539 2.259151e+01 -146.2917 62.75000
#> 719  131.2500     M        0    NA 114.43939885 2.253588e+01 -146.3750 62.75000
#> 720  151.1875     M        1     7 110.18221208 2.248575e+01 -146.4583 62.75000
#> 721  142.0625     M        0    NA 105.92536908 2.244114e+01 -146.5417 62.75000
#> 722  136.1250     L        1     0 101.66809757 2.240202e+01 -146.6250 62.75000
#> 723  144.5000     L        0    NA  97.41079101 2.236841e+01 -146.7083 62.75000
#> 724  135.2500     L        0    NA  93.15384494 2.234030e+01 -146.7917 62.75000
#> 725  129.5714     M        0    NA  88.89648551 2.231770e+01 -146.8750 62.75000
#> 726  130.0000     M        0    NA  84.63910724 2.230060e+01 -146.9583 62.75000
#> 727  138.2000     M        0    NA  80.38210462 2.228901e+01 -147.0417 62.75000
#> 728  142.1667     M        0    NA  76.12470382 2.228292e+01 -147.1250 62.75000
#> 729  154.0000     M        0    NA  71.86729985 2.228234e+01 -147.2083 62.75000
#> 730  153.4167     M        0    NA  67.61028774 2.228726e+01 -147.2917 62.75000
#> 731  160.5333     M        1     0  63.35289363 2.229769e+01 -147.3750 62.75000
#> 732  175.0833     L        0    NA  59.09551204 2.231362e+01 -147.4583 62.75000
#> 733  183.3571     L        0    NA  54.83853696 2.233506e+01 -147.5417 62.75000
#> 734  191.6250     L        0    NA  50.58119608 2.236200e+01 -147.6250 62.75000
#> 735  211.1500     L        1    29  46.32388339 2.239445e+01 -147.7083 62.75000
#> 736  138.8421     L        0    NA 144.31696150 1.932773e+01 -145.7917 62.71667
#> 737  113.9375     L        1     0 140.05543431 1.923900e+01 -145.8750 62.71667
#> 738   95.1250     M        1    20 135.79382552 1.915577e+01 -145.9583 62.71667
#> 739  186.0000     M        1     8 131.53252959 1.907806e+01 -146.0417 62.71667
#> 740  147.3750     L        1     3 127.27077341 1.900585e+01 -146.1250 62.71667
#> 741  147.0833     L        0    NA 123.00895081 1.893916e+01 -146.2083 62.71667
#> 742  155.7500     L        0    NA 118.74745779 1.887797e+01 -146.2917 62.71667
#> 743  136.6667     M        0    NA 114.48551967 1.882229e+01 -146.3750 62.71667
#> 744  147.1579     M        0    NA 110.22353133 1.877213e+01 -146.4583 62.71667
#> 745  145.8125     L        0    NA 105.96188724 1.872747e+01 -146.5417 62.71667
#> 746  142.3125     L        0    NA 101.69981422 1.868832e+01 -146.6250 62.71667
#> 747  141.4706     L        0    NA  97.43770615 1.865468e+01 -146.7083 62.71667
#> 748  141.6842     M        1     0  93.17595903 1.862655e+01 -146.7917 62.71667
#> 749  134.8125     M        1     0  88.91379814 1.860393e+01 -146.8750 62.71667
#> 750  134.3125     L        0    NA  84.65161789 1.858682e+01 -146.9583 62.71667
#> 751  143.2941     M        1     0  80.38981426 1.857522e+01 -147.0417 62.71667
#> 752  149.2105     M        0    NA  76.12761252 1.856913e+01 -147.1250 62.71667
#> 753  162.8125     M        1     0  71.86540710 1.856855e+01 -147.2083 62.71667
#> 754  164.6875     M        0    NA  67.60359398 1.857347e+01 -147.2917 62.71667
#> 755  170.0000     M        0    NA  63.34139841 1.858391e+01 -147.3750 62.71667
#> 756  184.4737     M        1     4  59.07921536 1.859985e+01 -147.4583 62.71667
#> 757  196.9286     M        0    NA  54.81743976 1.862131e+01 -147.5417 62.71667
#> 758  210.4167     L        0    NA  50.55529688 1.864827e+01 -147.6250 62.71667
#> 759  124.5294     L        0    NA 144.39666895 1.561458e+01 -145.7917 62.68333
#> 760  123.3125     L        0    NA 140.13034141 1.552577e+01 -145.8750 62.68333
#> 761   99.5625     M        0    NA 135.86393235 1.544248e+01 -145.9583 62.68333
#> 762  175.7500     M        0    NA 131.59783660 1.536470e+01 -146.0417 62.68333
#> 763  143.4000     L        0    NA 127.33128019 1.529244e+01 -146.1250 62.68333
#> 764  149.6875     L        0    NA 123.06465793 1.522568e+01 -146.2083 62.68333
#> 765  159.8750     L        0    NA 118.79836469 1.516445e+01 -146.2917 62.68333
#> 766  157.4286     L        0    NA 114.53162643 1.510873e+01 -146.3750 62.68333
#> 767  150.4667     M        1     0 110.26483800 1.505852e+01 -146.4583 62.68333
#> 768  157.8333     L        0    NA 105.99839427 1.501383e+01 -146.5417 62.68333
#> 769  153.0769     L        0    NA 101.73152119 1.497464e+01 -146.6250 62.68333
#> 770  148.4375     L        0    NA  97.46461309 1.494098e+01 -146.7083 62.68333
#> 771  147.2500     M        0    NA  93.19806639 1.491282e+01 -146.7917 62.68333
#> 772  142.1250     L        0    NA  88.93110549 1.489018e+01 -146.8750 62.68333
#> 773  138.6250     L        0    NA  84.66412575 1.487306e+01 -146.9583 62.68333
#> 774  148.1875     M        0    NA  80.39752257 1.486145e+01 -147.0417 62.68333
#> 775  158.0500     L        0    NA  76.13052033 1.485535e+01 -147.1250 62.68333
#> 776  176.8125     M        0    NA  71.86351492 1.485477e+01 -147.2083 62.68333
#> 777  180.8750     M        1     7  67.59690225 1.485970e+01 -147.2917 62.68333
#> 778  185.2500     M        1     3  63.32990670 1.487014e+01 -147.3750 62.68333
#> 779  203.0000     M        1     0  59.06292364 1.488610e+01 -147.4583 62.68333
#> 780  215.6875     M        1     3  54.79634796 1.490757e+01 -147.5417 62.68333
#> 781  229.4375     M        0    NA  50.52940558 1.493456e+01 -147.6250 62.68333
#> 782  152.0588     L        0    NA 144.47635663 1.190124e+01 -145.7917 62.65000
#> 783  134.9231     L        0    NA 140.20522995 1.181235e+01 -145.8750 62.65000
#> 784  103.0000     M        0    NA 135.93402127 1.172899e+01 -145.9583 62.65000
#> 785  108.2500     M        1     2 131.66312741 1.165115e+01 -146.0417 62.65000
#> 786  136.7000     L        0    NA 127.39177197 1.157883e+01 -146.1250 62.65000
#> 787  195.1250     L        0    NA 123.12035020 1.151202e+01 -146.2083 62.65000
#> 788  203.6875     L        1     0 118.84925895 1.145074e+01 -146.2917 62.65000
#> 789  159.6875     L        0    NA 114.57772176 1.139496e+01 -146.3750 62.65000
#> 790  153.5000     L        0    NA 110.30613442 1.134471e+01 -146.4583 62.65000
#> 791  168.1875     L        0    NA 106.03489225 1.129998e+01 -146.5417 62.65000
#> 792  161.2667     L        0    NA 101.76322030 1.126077e+01 -146.6250 62.65000
#> 793  155.0000     L        0    NA  97.49151387 1.122708e+01 -146.7083 62.65000
#> 794  152.5333     M        1     0  93.22016827 1.119890e+01 -146.7917 62.65000
#> 795  151.3333     L        0    NA  88.94840854 1.117624e+01 -146.8750 62.65000
#> 796  149.6667     L        0    NA  84.67663000 1.115910e+01 -146.9583 62.65000
#> 797  155.0000     M        1     0  80.40522845 1.114748e+01 -147.0417 62.65000
#> 798  169.3000     L        0    NA  76.13342741 1.114138e+01 -147.1250 62.65000
#> 799  190.0625     M        0    NA  71.86162373 1.114080e+01 -147.2083 62.65000
#> 800  199.8750     M        0    NA  67.59021218 1.114573e+01 -147.2917 62.65000
#> 801  203.0625     M        1     6  63.31841783 1.115618e+01 -147.3750 62.65000
#> 802  225.1500     M        1     2  59.04663597 1.117216e+01 -147.4583 62.65000
#> 803  170.1875     L        1     0 144.55601998 8.187910e+00 -145.7917 62.61666
#> 804  142.4000     L        1     8 140.28009561 8.098953e+00 -145.8750 62.61666
#> 805  165.9333     M        1     8 136.00408980 8.015521e+00 -145.9583 62.61666
#> 806   99.5000     M        0    NA 131.72839826 7.937618e+00 -146.0417 62.61666
#> 807  180.6667     L        0    NA 127.45224526 7.865232e+00 -146.1250 62.61666
#> 808  252.2778     L        0    NA 123.17602597 7.798370e+00 -146.2083 62.61666
#> 809  285.7500     M        0    NA 118.90013765 7.737036e+00 -146.2917 62.61666
#> 810  259.1875     L        0    NA 114.62380299 7.681220e+00 -146.3750 62.61666
#> 811  169.2105     M        0    NA 110.34741822 7.630928e+00 -146.4583 62.61666
#> 812  178.4706     L        0    NA 106.07137906 7.586162e+00 -146.5417 62.61666
#> 813  170.0000     L        0    NA 101.79490972 7.546916e+00 -146.6250 62.61666
#> 814  164.6250     M        0    NA  97.51840540 7.513193e+00 -146.7083 62.61666
#> 815  159.7000     L        0    NA  93.24226338 7.484995e+00 -146.7917 62.61666
#> 816  159.5000     L        0    NA  88.96570631 7.462318e+00 -146.8750 62.61666
#> 817  173.3750     L        0    NA  84.68912991 7.445165e+00 -146.9583 62.61666
#> 818  190.5714     L        0    NA  80.41293147 7.433535e+00 -147.0417 62.61666
#> 819  179.9333     M        0    NA  76.13633361 7.427427e+00 -147.1250 62.61666
#> 820  200.7500     L        0    NA  71.85973208 7.426843e+00 -147.2083 62.61666
#> 821  218.0000     M        0    NA  67.58352416 7.431781e+00 -147.2917 62.61666
#> 822  226.0000     M        1    31  63.30693248 7.442243e+00 -147.3750 62.61666
#> 823  154.4375     L        0    NA 144.63565441 4.474810e+00 -145.7917 62.58333
#> 824  198.7000     L        1     5 140.35493410 4.385780e+00 -145.8750 62.58333
#> 825  216.2500     L        0    NA 136.07413239 4.302279e+00 -145.9583 62.58333
#> 826  101.9375     M        0    NA 131.79364543 4.224313e+00 -146.0417 62.58333
#> 827  166.0000     M        1     3 127.51269660 4.151868e+00 -146.1250 62.58333
#> 828  270.2353     M        1     2 123.23168152 4.084951e+00 -146.2083 62.58333
#> 829  367.2500     M        0    NA 118.95099789 4.023567e+00 -146.2917 62.58333
#> 830  422.3333     L        0    NA 114.66986750 3.967705e+00 -146.3750 62.58333
#> 831  217.0714     L        0    NA 110.38868703 3.917371e+00 -146.4583 62.58333
#> 832  182.2000     L        0    NA 106.10785263 3.872569e+00 -146.5417 62.58333
#> 833  176.2500     L        0    NA 101.82658763 3.833290e+00 -146.6250 62.58333
#> 834  171.0625     M        0    NA  97.54528819 3.799540e+00 -146.7083 62.58333
#> 835  171.7500     L        1     0  93.26435047 3.771319e+00 -146.7917 62.58333
#> 836  214.8750     L        1     0  88.98299779 3.748624e+00 -146.8750 62.58333
#> 837  236.8750     L        0    NA  84.70162579 3.731456e+00 -146.9583 62.58333
#> 838  228.5625     L        0    NA  80.42063271 3.719817e+00 -147.0417 62.58333
#> 839  191.8500     L        0    NA  76.13923876 3.713704e+00 -147.1250 62.58333
#> 840  211.8125     L        0    NA  71.85784215 3.713119e+00 -147.2083 62.58333
#> 841  236.5625     L        0    NA  67.57683857 3.718062e+00 -147.2917 62.58333
#> 842  255.2500     L        1     9  63.29545130 3.728532e+00 -147.3750 62.58333
#> 843  252.1667     L        0    NA 144.71526444 7.617278e-01 -145.7917 62.55000
#> 844  256.8000     L        0    NA 140.42974964 6.726257e-01 -145.8750 62.55000
#> 845  129.6316     L        0    NA 136.14415350 5.890564e-01 -145.9583 62.55000
#> 846  128.0625     M        0    NA 131.85887259 5.110266e-01 -146.0417 62.55000
#> 847  128.3750     L        0    NA 127.57312940 4.385220e-01 -146.1250 62.55000
#> 848  216.6500     L        0    NA 123.28732000 3.715500e-01 -146.2083 62.55000
#> 849  310.3750     M        1     0 119.00184252 3.101159e-01 -146.2917 62.55000
#> 850  357.6250     M        1     6 114.71591788 2.542084e-01 -146.3750 62.55000
#> 851  235.3571     L        0    NA 110.42994317 2.038334e-01 -146.4583 62.55000
#> 852  191.2667     L        0    NA 106.14431501 1.589946e-01 -146.5417 62.55000
#> 853  185.5000     L        0    NA 101.85825582 1.196839e-01 -146.6250 62.55000
#> 854  181.0833     M        1     4  97.57216170 8.590565e-02 -146.7083 62.55000
#> 855  187.9333     L        0    NA  93.28643079 5.766205e-02 -146.7917 62.55000
#> 856  267.5500     L        0    NA  89.00028397 3.494797e-02 -146.8750 62.55000
#> 857  303.9375     L        0    NA  84.71411784 1.776622e-02 -146.9583 62.55000
#> 858  235.0000     M        1     0  80.42833056 6.117608e-03 -147.0417 62.55000
#> 859  209.5000     L        0    NA  76.14214301 0.000000e+00 -147.1250 62.55000

Some of the variables of interest include

  • total, which has counts of moose (and is NA for all sites that were not surveyed).
  • strat, a covariate that is either L for Low or M for medium.
  • surveyed, which is a 0 if the site wasn’t sampled and a 1 if the site was sampled.
  • x and y, the spatial coordinates for the centroids of the sites (in a user-defined Trans-Mercator projection).

Fitting the Model and Obtaining Predictions

We can now proceed to use the functions in sptotal in a similar way to how the functions were used for the simulated data. To get a sense of the data, we first give a plot of the raw observed counts:

ggplot(data = AKmoose_df, aes(x = x, y = y)) +
  geom_point(aes(colour = total), size = 4) +
  scale_colour_viridis_c() +
  theme_bw()

where the grey circles are sites that have not been sampled.

slmfit_out_moose <- slmfit(formula = total ~ strat, 
  data = AKmoose_df, xcoordcol = 'x', ycoordcol = 'y',
  CorModel = "Exponential")
summary(slmfit_out_moose)
plot(slmfit_out_moose)

resid_df <- data.frame(residuals = residuals(slmfit_out_moose,
                                             cross.validation = TRUE))
ggplot(data = resid_df, aes(x = residuals)) +
  geom_histogram(colour = "black", fill = "white", bins  = 20) +
  labs(x = "CV Residuals")

pred_moose <- predict(slmfit_out_moose)
pred_moose
plot(pred_moose)

We obtain a predicted total of 1596 animals with 90% lower and upper confidence bounds of 921 and 2271 animals, respectively. Unlike the simulation setting, there is no “true total” we can compare our prediction to, because, in reality, not all sites were sampled!

Allowing Different Covariance Parameters for Strata

Putting strat as a predictor in the model formula means that we are allowing each stratum to have a different mean but are assuming each stratum to have the same variance and covariance. If we want to allow the two strata to have different covariance parameter estimates, we can remove strat from the model formula and add it to the stratacol argument:

slmfit_out_moose_strat <- slmfit(formula = total ~ 1, 
  data = AKmoose_df, xcoordcol = 'x', ycoordcol = 'y',
  stratacol = "strat",
  CorModel = "Exponential")
summary(slmfit_out_moose_strat)
#> $L
#> 
#> Call:
#> total ~ 1
#> 
#> Residuals:
#>     Min      1Q  Median      3Q     Max 
#> -2.8337 -2.8337 -2.8337  0.1663 26.1663 
#> 
#> Coefficients:
#>      Estimate Std. Error t value Pr(>|t|)
#> [1,]    2.834      2.076   1.365    0.176
#> 
#> Covariance Parameters:
#>              Exponential Model
#> Nugget                6.548489
#> Partial Sill         23.421310
#> Range                32.274509
#> 
#> Generalized R-squared: 2.220446e-16 
#> 
#> $M
#> 
#> Call:
#> total ~ 1
#> 
#> Residuals:
#>     Min      1Q  Median      3Q     Max 
#> -4.0571 -4.0571 -2.0571  0.9429 35.9429 
#> 
#> Coefficients:
#>      Estimate Std. Error t value Pr(>|t|)  
#> [1,]    4.057      1.838   2.207    0.029 *
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> 
#> Covariance Parameters:
#>              Exponential Model
#> Nugget                37.62337
#> Partial Sill          12.12722
#> Range                 37.68748
#> 
#> Generalized R-squared: 0

There is now two sets of summary output, one for each stratum. predict() can still be used to obtain an estimate for the total (predict() also gives a predicted total for each stratum):

predict(slmfit_out_moose_strat)
#> 
#> Prediction and Confidence Intervals:
#>       Prediction    SE 90% LB 90% UB
#> L         1133.4 303.2  634.6   1632
#> M          960.8 104.3  789.2   1132
#> Total     2094.2 320.7 1566.8   2622

For this example, our prediction is very different when strata are allowed separate covariance parameters (2094 moose) than when strata are forced to have the same covariance parameters (1596 moose).

To see why this is, we can examine the semi-variograms for each stratum. All functions (e.g. plot(), AIC(), coef(), etc.) that are used on an slmfit() object without stratacol specified can still be used on an slmfit() object with a stratacol specified by running the function in the following way:

plot(slmfit_out_moose_strat[[1]])

plot(slmfit_out_moose_strat[[2]])

We see that the fitted covariance parameters for each strata do look different in this example, as the scale on the semi-variograms changes drastically. Therefore, for this example, it is probably more reasonable to allow the strata to have different covariance parameters and use the stratacol argument.

Sites with Different Areas

Finally, throughout all of the above analyses, we have assumed that the areas of each site were equal. Though this assumption is not accurate for the moose data, due to slightly differing areas based on differing latitudes and longitudes, the assumption approximately holds so that any differences in the prediction that incorporates area is negligible. But, suppose we had sites with very different areas. To showcase how to incorporate site area into the functions in this package, let’s first create a “fake” area variable that has the first 700 sites in the region have an area of 1 square kilometer and has the last 160 sites in the region have an area of 2 square kilometers.

AKmoose_df$fake_area <- c(rep(1, 700), rep(2, 160))

For a spatial model, it makes much more sense to use density as the response variable instead of raw counts if the areas of the sites in the model are drastically different. By supplying an areacol argument to slmfit, the function converts counts to densities, and then gives regression parameters and covariance parameters for the density.

slmfit_out_moose_area <- slmfit(formula = total ~ strat, 
  data = AKmoose_df, xcoordcol = 'x', ycoordcol = 'y',
  CorModel = "Exponential", areacol = 'fake_area')
summary(slmfit_out_moose_area)
#> 
#> Call:
#> total ~ strat
#> 
#> Residuals:
#>     Min      1Q  Median      3Q     Max 
#> -3.3072 -3.3072 -1.0906  0.9094 36.6928 
#> 
#> Coefficients:
#>             Estimate Std. Error t value Pr(>|t|)   
#> (Intercept)   1.0906     0.9138   1.193  0.23401   
#> stratM        2.2166     0.6964   3.183  0.00167 **
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> 
#> Covariance Parameters:
#>              Exponential Model
#> Nugget               20.711292
#> Partial Sill          4.166747
#> Range                23.645337
#> 
#> Generalized R-squared: 0.04479698

The predict function then keeps track of the areacol argument and gives output in the data frame that pertains to both counts and densities:

pred_obj_area <- predict(slmfit_out_moose_area)
head(pred_obj_area$Pred_df[ ,c("total_pred_density", "total_pred_count",
                               "fake_area")])
#>   total_pred_density total_pred_count fake_area
#> 0          0.4029168        0.4029168         1
#> 1          0.3505312        0.3505312         1
#> 2          0.0000000        0.0000000         1
#> 3          0.2841390        0.2841390         1
#> 4          0.2782489        0.2782489         1
#> 5          0.3290719        0.3290719         1
tail(pred_obj_area$Pred_df[ ,c("total_pred_density", "total_pred_count",
                               "fake_area")])
#>     total_pred_density total_pred_count fake_area
#> 854         2.00000000       4.00000000         2
#> 855         0.01496504       0.02993008         2
#> 856         0.02163868       0.04327737         2
#> 857         0.06967022       0.13934044         2
#> 858         0.00000000       0.00000000         2
#> 859         0.49928987       0.99857974         2

Note that, for the first 6 observations, which have an area of 1, the total_pred_density and total_pred_count columns are identical, while, for the last 6 observations, which have an area of 2, the total_pred_density column is half that of the total_pred_count column.

Because we did not specify a column of weights, our prediction in the following output is for the total number of moose.

print(pred_obj_area)
#> Prediction Info:
#>       Prediction    SE 90% LB 90% UB
#> total       1556 393.6    909   2204
#>       Numb. Sites Sampled Total Numb. Sites Total Observed Average Density
#> total                 218               860            742           2.883

If sites have differing areas, the plot() function doesn’t make much sense to use because each site is represented by the same-sized dot. Here, it would be helpful to import the data frame with the predicted counts and densities into a shapefile so that you are able to construct your own graphics that reflect the different-sized sites.

Mean Dissolved Organic Carbon from National Lakes Data

As another example, we took data from the National Aquatic Resource Surveys. With concerns about global warming, the earth’s capacity to store carbon is of great interest, and dissolved organic carbon (DOC) is an estimate of a lake’s ability to store carbon. We will estimate the mean mg/L for DOC from a sample of lakes. If the total lake volume could be calculated (we will not attempt that), then the total dissolved carbon in a population of lakes could be estimated. We will examine DOC in lakes from the 2012 surveys. We combined site data, DOC data, and habitat metrics to create a data set of 1206 lakes in the conterminous United States.

To access the data, type

data(USlakes)

and create a histogram of the log dissolved organic carbon

ggplot(data = USlakes, aes(x = log(DOC_RESULT))) +
  geom_histogram(bins = 20)

Even on the log scale, there appears to be some outliers with very high values, and these may be the result of errors in collection or lab analysis. We will eliminate lakes that have log(DOC) values \(>\) 5 for the purposes of this vignette.

lakes <- USlakes[log(USlakes$DOC_RESULT) < 5, ]

Our new data set has

nrow(lakes)
#> [1] 1204

sites, so we have eliminated 2 sites. To visualize our data more, we make a bubble plot,

plot(USlakes$XCOORD, USlakes$YCOORD, pch = 19, 
  cex = 2 * log(lakes$DOC_RESULT) / max(log(lakes$DOC_RESULT)))

and it appears that there is spatial patterning.

We also have covariates that may help in prediction:

  • ELEVATION: Elevation at lake coordinates (LAT_DD_N83, LON_DD_N83) from NHD Digital Elevation Map layer
  • RVFPUNDWOODY_RIP: riparian zone and vegetation: fraction of understory with nonwoody cover present in the riparian zone
  • FCIBIG_LIT: Fish cover: index of fish cover due to large structures in the littoral zone
  • RVFCGNDBARE_RIP: riparian zone and vegetation: fraction of ground lacking cover in the riparian zone
  • RVFCGNDWOODY_RIP: riparian zone and vegetation: fraction of ground cover by woody vegetation in the riparian zone

In order to explore the association between each predictor and the DOC (but not yet taking into account spatial correlation), we would create scatterplots of DOC vs. each predictor. To save space, we only create one such scatterplot here:

ggplot(data = lakes,
       aes(x = RVFPUNDWOODY_RIP, y = log(DOC_RESULT))) +
  geom_jitter(width = 0.02) +
  geom_smooth(method = "lm", se = TRUE)
#> `geom_smooth()` using formula = 'y ~ x'

It looks like there might be a slight negative relationship between riparian nonwoody-understory cover and DOC, though again we note that this exploratory investigation does not take into account the possible spatial correlation of DOC across sites.

Creating a Subsample Data Set

We have the whole population of lakes, but, with budget cuts, it is likely that this whole population will not always be surveyed in its entirety. So, we will ask the question, “If we sample from this population, can we still get a fairly precise estimate of the mean DOC?”

We will do the same thing that we did with the simulated data, and take a random sample of 500 lakes. Also, because we want the mean, and not a total, we will create a weights column for the lakeobs data set, with each element \(1/N\), where, here, \(N = 1204\).

set.seed(2)
LakeObsID <- sample(1:nrow(lakes), 500)
lakeobs <- lakes
lakeobs$DOC_RESULT <- NA
lakeobs[LakeObsID, 'DOC_RESULT'] <- lakes[LakeObsID, 'DOC_RESULT']
lakeobs$wts <- 1 / nrow(lakeobs)

Fitting the Model and Making Predictions

Even though data are skewed, let’s try it without taking log of response variable. Note that the mean of log-transformed variables is not equal to the log of the mean of set of variables. So if we want a total on the untransformed scale, it would be a mistake to transform the data first, model it, make predictions, sum the predictions, and then exponentiate. It is much simpler to leave the data untransformed and rely on robustness of the method. Let’s see how this works.

slmfitout_exp_lakes <- slmfit(formula = DOC_RESULT ~ ELEVATION +
                                RVFPUNDWOODY_RIP + FCIBIG_LIT +
                                RVFCGNDBARE_RIP + RVFCGNDWOODY_RIP,
                              data = lakeobs, 
                              xcoordcol = 'XCOORD', ycoordcol = 'YCOORD', CorModel = "Exponential")
summary(slmfitout_exp_lakes)
#> 
#> Call:
#> DOC_RESULT ~ ELEVATION + RVFPUNDWOODY_RIP + FCIBIG_LIT + RVFCGNDBARE_RIP + 
#>     RVFCGNDWOODY_RIP
#> 
#> Residuals:
#>      Min       1Q   Median       3Q      Max 
#> -14.4226  -6.1203  -4.2384  -0.6852  88.0354 
#> 
#> Coefficients:
#>                    Estimate Std. Error t value Pr(>|t|)    
#> (Intercept)      18.5585533  2.8978665   6.404  < 2e-16 ***
#> ELEVATION        -0.0015922  0.0009541  -1.669  0.09579 .  
#> RVFPUNDWOODY_RIP -7.7307967  1.2430529  -6.219  < 2e-16 ***
#> FCIBIG_LIT       -4.0559783  1.6831386  -2.410  0.01633 *  
#> RVFCGNDBARE_RIP  -4.4469366  1.6693922  -2.664  0.00798 ** 
#> RVFCGNDWOODY_RIP  1.1457813  1.7641454   0.649  0.51633    
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> 
#> Covariance Parameters:
#>              Exponential Model
#> Nugget                15.09434
#> Partial Sill          91.02786
#> Range             456983.39560
#> 
#> Generalized R-squared: 0.1188291

We see that all covariates are highly significant. There is substantial autocorrelation because the range parameter is very large, and the partial sill is about six times that of the nugget effect. We fit the model again, but this time with the spherical autocorrelation model.

slmfitout_sph_lakes <- slmfit(formula = DOC_RESULT ~ ELEVATION +
                                RVFPUNDWOODY_RIP + FCIBIG_LIT +
                                RVFCGNDBARE_RIP + RVFCGNDWOODY_RIP,
                              data = lakeobs, 
                              xcoordcol = 'XCOORD', ycoordcol = 'YCOORD',
                              CorModel = "Spherical")
summary(slmfitout_sph_lakes)
#> 
#> Call:
#> DOC_RESULT ~ ELEVATION + RVFPUNDWOODY_RIP + FCIBIG_LIT + RVFCGNDBARE_RIP + 
#>     RVFCGNDWOODY_RIP
#> 
#> Residuals:
#>      Min       1Q   Median       3Q      Max 
#> -13.5767  -5.5804  -3.6468  -0.1512  88.6358 
#> 
#> Coefficients:
#>                    Estimate Std. Error t value Pr(>|t|)    
#> (Intercept)      17.7013675  2.0458144   8.652  < 2e-16 ***
#> ELEVATION        -0.0012000  0.0009266  -1.295  0.19589    
#> RVFPUNDWOODY_RIP -7.7512748  1.2399212  -6.251  < 2e-16 ***
#> FCIBIG_LIT       -3.7733882  1.6698727  -2.260  0.02428 *  
#> RVFCGNDBARE_RIP  -4.3820168  1.6562157  -2.646  0.00841 ** 
#> RVFCGNDWOODY_RIP  1.3173261  1.7526027   0.752  0.45263    
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> 
#> Covariance Parameters:
#>              Spherical Model
#> Nugget              15.74324
#> Partial Sill        87.85260
#> Range           761505.47106
#> 
#> Generalized R-squared: 0.1151265

We can use AIC to compare the use of the two autocorrelation models.

AIC(slmfitout_exp_lakes)
#> [1] 3249.515
AIC(slmfitout_sph_lakes)
#> [1] 3248.077

Based on AIC, there is not much difference in fit between the two structures. We will use the exponential covariance structure going forward.

pred_exp_lakes <- predict(slmfitout_exp_lakes,  wtscol = "wts",
                          conf_level = 0.95)
print(pred_exp_lakes)
#> Prediction Info:
#>            Prediction    SE 95% LB 95% UB
#> DOC_RESULT      7.975 0.196  7.591  8.359
#>            Numb. Sites Sampled Total Numb. Sites Total Observed Average Density
#> DOC_RESULT                 500              1204           4111           8.223
mean(lakes$DOC_RESULT)
#> [1] 7.646453

We can see that the prediction, 7.975, is close to the true value, 7.65, and that the confidence interval is quite narrow, and it does contain the true value. If a standard error of 0.196, yielding a coefficient of variation of 0.196/7.975 = 0.0245, is acceptable, then sampling 500 lakes could save money and still provide a useful result on DOC.

Appendix: Statistical Background

An alternative to a sampling-based approach is to assume that the data were generated by a stochastic process and use model-based approaches. It is assumed that the response variable is a realization of a spatial stochastic process. Geostatistical models and methods are used (for a review, see Cressie, 1993). Geostatistics was developed for point samples. If the samples are very small relative to the population size, an infinite population is assumed. In classical geostatistics, the average value over an area can be predicted using methods such as block kriging. Thus it appears that this is closely related to small area estimation, but where samples come from point locations rather than a finite set of sample units. While there is a large literature on geostatistics and block kriging methods, they have been developed for infinite populations. This package is designed for the case where we have a finite collection of plots and we assume that the data were produced by a spatial stochastic process. Detailed developments are given in Ver Hoef (2001, 2008). Comparisons to classical sampling methods can be found in Ver Hoef (2002), and applications in forestry are contained in Ver Hoef and Temesgen (2013) and Temesgen and Ver Hoef (2015).

Citation

To cite this package, type

citation("sptotal")

References

Cressie, N. 1993. Statistics for Spatial Data, Revised Edition John Wiley and Sons, NY.

Temesgen, H. and Ver Hoef, J.M. 2015. Evaluation of the Spatial Linear Model, Random Forest and Gradient Nearest-Neighbour Methods for Imputing Potential Pro- ductivity and Biomass of the Pacific Northwest Forests. Forestry 88(1): 131–142.

Ver Hoef, J.M. 2001. Predicting Finite Populations from Spatially Correlated Data. 2000 Proceedings of the Section on Statistics and the Environment of the American Statistical Association, pgs. 93 – 98.

Ver Hoef, J.M. 2002. Sampling and Geostatistics for Spatial Data. Ecoscience 9: 152–161.

Ver Hoef, J. M. 2008. Spatial Methods for Plot-Based Sampling of Wildlife Populations. Environmental and Ecological Statistics 15: 3-13.

Ver Hoef, J.M. and Temesgen, H. 2013. A Comparison of the Spatial Linear Model to Nearest Neighbor (k-NN) Methods for Forestry Applications. PloS ONE 8(3): e59129.

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