logcondens-package: Estimate a Log-Concave Probability Density from iid...

Description Details Author(s) References Examples

Description

The main function of this package is logConDens: compute the maximum likelihood estimator (MLE) of a log-concave density from one-dimensional i.i.d. observations as well as the kernel smoothed version derived from it. A list of additional functions that can be used to compute quantities relevant in that context can be found below.

Two algorithms are offered to compute the estimate: An active set (logConDens) and an iterative algorithm based on the pool-adjacent-violaters algorithm (icmaLogCon). The latter of these functions is only part of this package for historical reasons: it was the first method that was proposed to estimate a log-concave density, see Rufibach (2007). The more efficient way of computing the estimate is via an active set algorithm. The smoothed versions of the log-concave density and distribution function estimates discussed in Section 3 of Duembgen and Rufibach (2009) are available in the function evaluateLogConDens.

To compute a log-concave density, CDF, and survival function from interval- or right-censored observations use the package logconcens. A log-concave probability mass function can be computed using the package logcondiscr, see Balabdaoui et al (2012) for details. For the computation of a log-concave estimate in any dimension the package LogConDEAD can be used.

Details

Package: logcondens
Type: Package
Version: 2.1.5
Date: 2016-07-11
License: GPL (>=2)

The following additional functions and datasets are provided in the package:

evaluateLogConDens Computes the value of the estimated log-density, density, and distribution function of the MLE and the smoothed estimator at a given x0.

quantilesLogConDens Computes quantiles of the estimated distribution function at a given x0.

intECDF Compute the integrated empirical distribution function of the observations at a given x0.

intF Compute the integral of the distribution function corresponding to the log-concave density estimator at a given x0.

logconTwoSample Compute a permutation test for the difference between two distribution functions.

logConROC Compute ROC curve based on log-concave density estimates within cases and controls.

confIntBootLogConROC_t0 Compute bootstrap confidence intervals at given false positive fractions (= 1 - specificity) for the ROC curve based on log-concave density estimates.

logConCI Compute a confidence interval for the value of the true density at a fixed point, based on the theory developed in Balabdaoui et al (2009). This function was contributed by Mahdis Azadbakhsh and Hanna Jankowski, see Azadbakhsh et al (2012).

isoMean Compute the weighted least squares regression under a monotonicity constraint (the Grenander estimator). This function is used as part of icmaLogCon but is also of independent interest.

The following datasets have been used in several publications to illustrate log-concave density estimation and are therefore included in this package:

reliability Dataset that contains the data analyzed in Duembgen and Rufibach (2009, Figure 2). See the help file for logConDens for the analysis of this data.

brightstar Dataset that contains the data analyzed in Mizera and Koenker (2009, Section 5). The sample consists of measurements of radial and rotational velocities for the stars from the Bright Star Catalog, see Hoffleit and Warren (1991).

pancreas Data from pancreatic cancer serum biomarker study, first published by Wieand et al (1989). Contains data on serum measurements for a cancer antigen (CA-125) and a carbohydrate antigen (CA19.9) both of which are measured on a continuous non-negative scale. The measurements were taken within a case-control study on 90 cases with pancreatic cancer and 51 controls who did not have cancer but pancreatitis.

A print and summary for objects of class dlc, generated by logConDens, are available as well.

Author(s)

Kaspar Rufibach (maintainer), kaspar.rufibach@gmail.com,
http://www.kasparrufibach.ch

Lutz Duembgen, duembgen@stat.unibe.ch,
http://www.imsv.unibe.ch/about_us/staff/prof_dr_duembgen_lutz/index_eng.html

Kaspar Rufibach gratefully acknowledges support by the Swiss National Science Foundation SNF, http://www.snf.ch.

MatLab code with an implementation of the active set algorithm is available on request from Lutz Duembgen.

References

Azadbakhsh, M., Jankowski, H. and Gao, X. (2014). Computing confidence intervals for log-concave densities. Comput. Statist. Data Anal., to appear.

Balabdaoui, F., Jankowski, H., Rufibach, K. and Pavlides, M. (2012). Asymptotics of the discrete log-concave maximum likelihood estimator and related applications. J. R. Stat. Soc. Ser. B Stat. Methodol., 75(4), 769–790.

Baladbaoui, F., Rufibach, K. and Wellner, J. (2009). Limit distribution theory for maximum likelihood estimation of a log-concave density. Ann. Statist., 37(3), 1299–1331.

Duembgen, L., Huesler, A. and Rufibach, K. (2010). Active set and EM algorithms for log-concave densities based on complete and censored data. Technical report 61, IMSV, Univ. of Bern, available at http://arxiv.org/abs/0707.4643.

Duembgen, L. and Rufibach, K. (2009). Maximum likelihood estimation of a log–concave density and its distribution function: basic properties and uniform consistency. Bernoulli, 15(1), 40–68.

Duembgen, L. and Rufibach, K. (2011). logcondens: Computations Related to Univariate Log-Concave Density Estimation. Journal of Statistical Software, 39(6), 1–28. http://www.jstatsoft.org/v39/i06

Hoffleit, D., Warren, W.H. (1991). The Bright Star Catalog. Yale University Observatory, New Heaven.

Mizera, I., Koenker, R. (2010). Quasi-concave density estimation. Ann. Statist., 38(5), 2998–3027.

Rufibach K. (2006). Log-concave Density Estimation and Bump Hunting for i.i.d. Observations. PhD Thesis, University of Bern, Switzerland and Georg-August University of Goettingen, Germany, 2006.
Available at http://www.zb.unibe.ch/download/eldiss/06rufibach_k.pdf.

Rufibach, K. (2007). Computing maximum likelihood estimators of a log-concave density function. J. Stat. Comput. Simul. 77, 561–574.

Rufibach, K. (2012). A smooth ROC curve estimator based on log-concave density estimates. Int. J. Biostat., 8(1), 1–29.

Examples

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## estimate gamma density
set.seed(1977)
x <- rgamma(100, 2, 1)
res <- logConDens(x, smoothed = FALSE, print = TRUE)
summary(res)

## compare performance to ICMA
res2 <- icmaLogCon(x, T1 = 2000, robustif = TRUE, print = TRUE)

res$L
res2$L

## plot resulting functions
par(mfrow = c(2, 2), mar = c(3, 2, 1, 2))
plot(res, which = "density")
plot(res, which = "log-density")
plot(res, which = "CDF")
plot(res$x, res$H, type = 'l'); rug(res$x)

## compute function values at an arbitrary point
x0 <- (res$x[50] + res$x[51]) / 2
evaluateLogConDens(x0, res)

## compute 0.5 quantile of Fhat
quantilesLogConDens(0.5, res)

Example output

Warning messages:
1: In rgl.init(initValue, onlyNULL) : RGL: unable to open X11 display
2: 'rgl_init' failed, running with rgl.useNULL = TRUE 
3: .onUnload failed in unloadNamespace() for 'rgl', details:
  call: fun(...)
  error: object 'rgl_quit' not found 
[1] "iter1 = 1 / L = -2.3991 / max(H) = 0.0092 / #knots = 3"
[1] "iter1 = 2 / L = -2.3966 / max(H) = 0.0012 / #knots = 4"
[1] "iter1 = 3 / L = -2.3964 / max(H) = 6e-04 / #knots = 5"
[1] "iter1 = 4 / L = -2.396 / max(H) = 1e-04 / #knots = 5"
[1] "iter1 = 5 / L = -2.396 / max(H) = 0 / #knots = 5"
[1] "iter1 = 6 / L = -2.396 / max(H) = 0 / #knots = 5"
[1] "iter1 = 7 / L = -2.396 / max(H) = 0 / #knots = 5"

Estimation of a log-concave density from i.i.d. data
 
Number of initial observations: n = 100
 
Number of unique observations (or grid points): m = 100
 
Minimal and maximal weight: 0.01, 0.01
 
log-likelihood: -2.4
 
Maximum likelihood estimate:
Mode: x[24] = 1.1
Value of log-density at mode: -1.1
Value of density at mode: 0.34
 
Number of knots of the MLE: 5
 
Knots of the MLE:
x[1, 24, 69, 97, 100] =
0.22, 1.07, 2.46, 4.16, 4.92

  Iteration LogLikelihood      dirder
1         1     -2.425006 -0.03467376
  Iteration LogLikelihood       dirder
1         2     -2.419353 -0.009992708
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1         3     -2.418137 -0.001772044
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1         4     -2.416945 -0.001574101
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1         5      -2.41645 -0.0006945815
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1         6      -2.41577 -0.0008798964
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1         7     -2.415355 -0.0005995504
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1         8     -2.414757 -0.0008225733
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1         9     -2.414343 -0.0007350699
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1        10     -2.413983 -0.0005205928
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1        11     -2.413456 -0.0007486557
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1        12     -2.413248 -0.0002512967
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1        13     -2.412386 -0.00153559
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1        14     -2.411789 -0.0009290223
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1        15     -2.411372 -0.0005415619
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1        16     -2.411093 -0.0004147231
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1        17     -2.410685 -0.0006527028
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1        18     -2.410475 -0.0002704282
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1        19     -2.409828 -0.001246537
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1        20       -2.4093 -0.0008431235
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1        21     -2.408981 -0.0004023607
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1        22     -2.408776 -0.0002777941
  Iteration LogLikelihood       dirder
1        23     -2.408246 -0.001024056
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1        24     -2.407804 -0.0007142411
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1        25     -2.407297 -0.0007086828
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1        26     -2.407098 -0.0002767254
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1        27     -2.406859 -0.0002946338
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1        28     -2.406711 -0.0001897344
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1        29     -2.406283 -0.0005578257
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1        30     -2.406146 -0.0001807753
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1        31     -2.405938 -0.0002494301
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1        32     -2.405816 -0.0001522329
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1        33     -2.405132 -0.001263769
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1        34     -2.404626 -0.0008944257
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1        35      -2.40432 -0.0004587996
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1        36     -2.404173 -0.0002140603
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1        37     -2.403906 -0.0005106755
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1        38     -2.403688 -0.0003533431
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1        39     -2.403443 -0.000426266
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1        40     -2.403275 -0.0002623039
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1        41     -2.403046 -0.0003587216
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1        42     -2.402924 -0.0001828853
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1        43     -2.402731 -0.0003873662
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1        44     -2.402555 -0.0002897702
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1        45     -2.402366 -0.000253895
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1        46     -2.402299 -9.125916e-05
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1        47     -2.402209 -0.0001091914
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1        48     -2.402154 -6.933331e-05
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1        49     -2.401992 -0.0002037515
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1        50     -2.401945 -6.055584e-05
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1        51     -2.401791 -0.0002572465
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1        52     -2.401696 -0.0001526186
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1        53     -2.401559 -0.0001691131
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1        54     -2.401522 -4.639682e-05
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1        55     -2.401392 -0.0002242298
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1        56     -2.401305 -0.000144678
  Iteration LogLikelihood        dirder
1        57     -2.401077 -0.0002924934
  Iteration LogLikelihood        dirder
1        58     -2.401015 -9.317405e-05
  Iteration LogLikelihood        dirder
1        59     -2.400911 -0.0001594544
  Iteration LogLikelihood        dirder
1        60     -2.400855 -8.673656e-05
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1        61     -2.400669 -0.0002760341
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1        62     -2.400579 -0.0001543224
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1        63     -2.400409 -0.0002028564
  Iteration LogLikelihood        dirder
1        64      -2.40037 -5.752288e-05
  Iteration LogLikelihood        dirder
1        65     -2.400287 -0.0001017498
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1        66     -2.400263 -3.069904e-05
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1        67     -2.400179 -0.0001138783
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1        68     -2.400146 -4.787169e-05
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1        69     -2.400066 -0.0001008084
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1        70      -2.40004 -3.449066e-05
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1        71     -2.399961 -0.0001041606
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1        72     -2.399932 -4.066616e-05
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1        73     -2.399856 -9.894889e-05
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1        74     -2.399829 -3.732802e-05
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1        75     -2.399756 -9.46652e-05
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1        76     -2.399731 -3.368527e-05
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1        77     -2.399659 -9.70821e-05
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1        78     -2.399631 -4.019756e-05
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1        79     -2.399562 -8.550638e-05
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1        80     -2.399541 -2.724274e-05
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1        81     -2.399473 -9.395526e-05
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1        82     -2.399444 -4.118453e-05
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1        83      -2.39938 -7.828527e-05
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1        84     -2.399362 -2.301816e-05
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1        85     -2.399297 -8.955436e-05
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1        86      -2.39927 -4.000173e-05
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1        87     -2.399209 -7.305212e-05
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1        88     -2.399192 -2.075356e-05
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1        89     -2.399131 -8.478724e-05
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1        90     -2.399105 -3.793761e-05
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1        91     -2.399048 -6.879224e-05
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1        92     -2.399033 -1.929787e-05
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1        93     -2.398975 -8.023314e-05
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1        94      -2.39895 -3.584123e-05
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1        95     -2.398896 -6.543386e-05
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1        96     -2.398881 -1.858171e-05
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1        97     -2.398826 -7.595432e-05
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1        98     -2.398803 -3.384621e-05
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1        99     -2.398751 -6.291443e-05
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1       100     -2.398737 -1.859005e-05
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1       101     -2.398685 -7.180877e-05
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1       102     -2.398663 -3.18738e-05
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1       103     -2.398614 -5.995734e-05
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1       104       -2.3986 -1.802069e-05
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1       105      -2.39855 -6.789482e-05
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1       106      -2.39853 -3.003312e-05
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1       107     -2.398483 -5.689598e-05
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1       108      -2.39847 -1.719905e-05
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1       109     -2.398423 -6.42507e-05
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1       110     -2.398404 -2.836877e-05
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1       111      -2.39836 -5.395694e-05
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1       112     -2.398347 -1.636934e-05
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1       113     -2.398303 -6.08241e-05
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1       114     -2.398284 -2.681623e-05
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1       115     -2.398242 -5.117028e-05
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1       116      -2.39823 -1.557183e-05
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1       117     -2.398189 -5.759358e-05
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1       118     -2.398171 -2.535654e-05
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1       119     -2.398132 -4.853489e-05
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1       120      -2.39812 -1.481347e-05
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1       121     -2.398081 -5.454521e-05
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1       122     -2.398064 -2.398114e-05
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1       123     -2.398026 -4.604346e-05
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1       124     -2.398016 -1.409395e-05
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1       125     -2.397978 -5.166749e-05
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1       126     -2.397962 -2.268423e-05
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1       127     -2.397927 -4.36877e-05
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1       128     -2.397916 -1.341147e-05
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1       129     -2.397881 -4.895003e-05
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1       130     -2.397866 -2.146092e-05
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1       131     -2.397832 -4.145959e-05
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1       132     -2.397822 -1.276395e-05
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1       133     -2.397789 -4.638325e-05
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1       134     -2.397775 -2.030677e-05
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1       135     -2.397743 -3.935158e-05
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1       136     -2.397733 -1.214942e-05
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1       137     -2.397701 -4.39582e-05
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1       138     -2.397688 -1.921764e-05
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1       139     -2.397658 -3.735666e-05
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1       140     -2.397649 -1.157259e-05
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1       141     -2.397618 -4.14913e-05
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1       142     -2.397606 -1.79296e-05
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1       143     -2.397577 -3.565492e-05
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1       144     -2.397568 -1.123541e-05
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1       145      -2.39754 -3.892766e-05
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1       146     -2.397528 -1.637917e-05
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1       147     -2.397501 -3.428824e-05
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1       148     -2.397492 -1.119805e-05
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1       149     -2.397465 -3.629958e-05
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1       150     -2.397455 -1.464246e-05
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1       151     -2.397429 -3.320046e-05
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1       152      -2.39742 -1.142675e-05
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1       153     -2.397394 -3.3589e-05
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1       154     -2.397385 -1.273387e-05
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1       155      -2.39736 -3.236944e-05
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1       156     -2.397351 -1.193355e-05
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1       157     -2.397327 -3.077325e-05
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1       158     -2.397319 -1.068002e-05
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1       159     -2.397295 -3.17173e-05
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1       160     -2.397286 -1.266021e-05
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1       161     -2.397264 -2.793923e-05
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1       162     -2.397257 -8.65414e-06
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1       163     -2.397234 -3.094923e-05
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1       164     -2.397225 -1.321002e-05
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1       165     -2.397186 -7.515184e-05
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1       166     -2.397151 -6.828186e-05
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1       167     -2.397112 -4.890823e-05
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1       168     -2.397101 -1.708443e-05
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1       169     -2.397063 -5.505248e-05
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1       170     -2.397046 -3.113078e-05
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1       171     -2.397014 -3.23456e-05
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1       172     -2.397004 -1.586094e-05
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1       173     -2.396998 -7.274649e-06
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1       174     -2.396981 -2.247398e-05
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1       175     -2.396975 -7.911234e-06
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1       176     -2.396958 -2.293149e-05
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1       177     -2.396951 -9.153349e-06
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1       178     -2.396921 -5.815103e-05
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1       179     -2.396894 -5.404495e-05
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1       180     -2.396864 -3.84562e-05
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1       185     -2.396786 -5.03201e-06
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1       190     -2.396706 -2.116696e-05
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1       193     -2.396654 -1.658263e-05
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1       217     -2.396445 -5.348509e-06
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1       219      -2.39644 -4.751901e-06
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1       222      -2.39643 -3.202172e-06
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1       224      -2.39642 -3.933785e-06
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1       225     -2.396407 -2.360552e-05
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1       226     -2.396397 -1.889019e-05
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1       227     -2.396385 -1.137851e-05
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1       229     -2.396371 -2.784595e-06
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1       230     -2.396365 -8.129362e-06
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1       231     -2.396363 -2.946683e-06
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1       232     -2.396357 -8.067059e-06
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1       233     -2.396354 -3.100119e-06
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1       234     -2.396349 -7.463103e-06
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1       236     -2.396341 -7.891011e-06
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1       237     -2.396338 -3.23017e-06
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1       238     -2.396333 -6.960973e-06
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1       239     -2.396331 -2.193353e-06
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1       241     -2.396323 -3.187882e-06
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1       242     -2.396318 -6.626172e-06
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1       243     -2.396316 -2.08727e-06
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1       244     -2.396311 -7.240432e-06
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1       245     -2.396309 -3.034547e-06
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1       246     -2.396304 -6.379498e-06
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1       247     -2.396302 -2.067131e-06
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1       248     -2.396297 -6.859709e-06
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1       249     -2.396295 -2.839043e-06
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1       250      -2.39629 -6.109897e-06
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1       251     -2.396289 -2.008915e-06
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1       252     -2.396284 -6.513285e-06
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1       253     -2.396282 -2.675586e-06
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1       254     -2.396277 -5.838573e-06
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1       255     -2.396276 -1.936311e-06
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1       256     -2.396271 -6.191772e-06
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1       257     -2.396269 -2.531404e-06
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1       258     -2.396265 -5.573121e-06
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1       259     -2.396264 -1.858525e-06
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1       260     -2.396259 -5.890262e-06
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1       261     -2.396257 -2.400305e-06
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1       262     -2.396253 -5.316701e-06
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1       263     -2.396252 -1.779833e-06
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1       264     -2.396248 -5.605841e-06
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1       265     -2.396246 -2.278946e-06
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1       266     -2.396242 -5.070603e-06
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1       267     -2.396241 -1.702352e-06
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1       268     -2.396237 -5.336614e-06
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1       269     -2.396235 -2.165397e-06
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1       270     -2.396231 -4.835224e-06
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1       271      -2.39623 -1.627129e-06
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1       272     -2.396226 -5.081241e-06
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1       273     -2.396225 -2.058471e-06
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1       274     -2.396221 -4.61052e-06
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1       275      -2.39622 -1.554657e-06
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1       276     -2.396216 -4.838706e-06
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1       277     -2.396215 -1.957398e-06
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1       279      -2.39621 -1.48513e-06
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1       280     -2.396207 -4.608186e-06
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1       281     -2.396206 -1.86164e-06
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1       282     -2.396202 -4.191956e-06
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1       283     -2.396201 -1.418586e-06
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1       285     -2.396197 -1.770795e-06
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1       286     -2.396194 -3.997303e-06
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1       287     -2.396193 -1.354978e-06
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1       288     -2.396189 -4.18047e-06
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1       289     -2.396188 -1.684542e-06
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1       290     -2.396185 -3.811828e-06
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1       291     -2.396184 -1.294217e-06
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1       292     -2.396181 -3.98209e-06
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1       293      -2.39618 -1.60261e-06
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1       298      -2.39617 -3.466706e-06
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1       299     -2.396169 -1.180798e-06
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1       300     -2.396166 -3.613682e-06
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1       301     -2.396165 -1.450778e-06
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1       302     -2.396162 -3.306241e-06
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1       303     -2.396161 -1.127907e-06
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1       304     -2.396159 -3.442712e-06
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1       305     -2.396158 -1.38046e-06
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1       306     -2.396155 -3.153325e-06
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1       307     -2.396155 -1.077407e-06
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1       308     -2.396152 -3.279981e-06
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1       309     -2.396151 -1.313619e-06
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1       310     -2.396149 -3.007592e-06
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1       311     -2.396148 -1.02919e-06
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1       312     -2.396146 -3.125083e-06
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1       313     -2.396145 -1.25008e-06
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1       314     -2.396143 -2.868696e-06
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1       315     -2.396142 -9.831491e-07
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1       316      -2.39614 -2.977631e-06
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1       317     -2.396139 -1.189676e-06
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1       318     -2.396137 -2.736308e-06
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1       320     -2.396134 -2.837259e-06
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1       322     -2.396131 -2.610116e-06
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1       323      -2.39613 -8.971986e-07
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1       324     -2.396128 -2.703618e-06
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1       330      -2.39612 -2.375146e-06
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1       331      -2.39612 -8.188077e-07
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1       332     -2.396118 -2.455229e-06
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1       333     -2.396117 -9.76369e-07
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1       334     -2.396115 -2.265819e-06
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1       335     -2.396115 -7.822328e-07
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1       339      -2.39611 -7.472991e-07
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1       340     -2.396108 -2.230018e-06
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1       341     -2.396108 -8.84785e-07
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1       342     -2.396106 -2.062205e-06
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1       343     -2.396106 -7.139316e-07
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1       344     -2.396104 -2.125406e-06
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1       345     -2.396103 -8.423293e-07
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1       346     -2.396102 -1.967446e-06
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1       347     -2.396101 -6.820594e-07
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1       348       -2.3961 -2.025779e-06
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1       349     -2.396099 -8.019499e-07
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1       350     -2.396098 -1.877089e-06
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1       353     -2.396095 -7.635432e-07
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1       355     -2.396093 -6.225322e-07
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1       357     -2.396091 -7.270108e-07
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1       358      -2.39609 -1.708763e-06
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1       359      -2.39609 -5.947509e-07
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1       361     -2.396088 -6.922592e-07
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1       365     -2.396084 -6.591996e-07
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1       369     -2.396081 -6.277481e-07
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1       370      -2.39608 -1.484408e-06
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1       371      -2.39608 -5.186392e-07
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1       689     -2.396014 -1.400516e-08
  Iteration LogLikelihood        dirder
1       690     -2.396014 -3.613292e-08
  Iteration LogLikelihood        dirder
1       691     -2.396014 -1.329162e-08
  Iteration LogLikelihood        dirder
1       692     -2.396014 -3.559747e-08
  Iteration LogLikelihood        dirder
1       693     -2.396014 -1.336667e-08
  Iteration LogLikelihood       dirder
1       694     -2.396014 -3.45004e-08
  Iteration LogLikelihood       dirder
1       695     -2.396014 -1.26941e-08
  Iteration LogLikelihood        dirder
1       696     -2.396014 -3.398284e-08
  Iteration LogLikelihood        dirder
1       697     -2.396014 -1.275745e-08
  Iteration LogLikelihood        dirder
1       698     -2.396014 -3.294171e-08
  Iteration LogLikelihood        dirder
1       699     -2.396014 -1.212341e-08
  Iteration LogLikelihood        dirder
1       700     -2.396014 -3.244166e-08
  Iteration LogLikelihood        dirder
1       701     -2.396014 -1.217613e-08
  Iteration LogLikelihood        dirder
1       702     -2.396014 -3.145351e-08
  Iteration LogLikelihood        dirder
1       703     -2.396014 -1.157834e-08
  Iteration LogLikelihood        dirder
1       704     -2.396014 -3.097056e-08
  Iteration LogLikelihood        dirder
1       705     -2.396014 -1.162143e-08
  Iteration LogLikelihood        dirder
1       706     -2.396014 -3.003261e-08
  Iteration LogLikelihood        dirder
1       707     -2.396014 -1.105775e-08
  Iteration LogLikelihood        dirder
1       708     -2.396014 -2.956636e-08
  Iteration LogLikelihood        dirder
1       709     -2.396014 -1.109212e-08
  Iteration LogLikelihood        dirder
1       710     -2.396014 -2.867595e-08
  Iteration LogLikelihood        dirder
1       711     -2.396014 -1.056052e-08
  Iteration LogLikelihood        dirder
1       712     -2.396014 -2.822599e-08
  Iteration LogLikelihood        dirder
1       713     -2.396014 -1.058704e-08
  Iteration LogLikelihood        dirder
1       714     -2.396014 -2.738063e-08
  Iteration LogLikelihood        dirder
1       715     -2.396014 -1.008563e-08
  Iteration LogLikelihood        dirder
1       716     -2.396014 -2.694654e-08
  Iteration LogLikelihood        dirder
1       717     -2.396014 -1.010506e-08
  Iteration LogLikelihood        dirder
1       718     -2.396014 -2.614388e-08
  Iteration LogLikelihood        dirder
1       719     -2.396014 -9.632063e-09
There were 50 or more warnings (use warnings() to see the first 50)
[1] -2.396013
[1] -2.396014
           xs log-density   density       CDF smooth.density smooth.CDF
[1,] 1.859068    -1.12871 0.3234502 0.4971161       0.323444  0.4965105
      ps quantile
[1,] 0.5 1.867984

logcondens documentation built on May 2, 2019, 6:11 a.m.