Estimate the correlation between a DFA trend and some other timeseries

Fully incorporates the uncertainty from the posterior of the DFA trend

trend_cor(
  rotated_modelfit,
  y,
  trend = 1,
  time_window = seq_len(length(y)),
  trend_samples = 100,
  stan_iter = 300,
  stan_chains = 1,
  ...
)

Arguments

rotated_modelfit

Output from rotate_trends().

y

A numeric vector to correlate with the DFA trend. Must be the same length as the DFA trend.

trend

A number corresponding to which trend to use, defaults to 1.

time_window

Indices indicating a time window slice to use in the correlation. Defaults to using the entire time window. Can be used to walk through the timeseries and test the cross correlations.

trend_samples

The number of samples from the trend posterior to use. A model will be run for each trend sample so this value shouldn't be too large. Defaults to 100.

stan_iter

The number of samples from the posterior with each Stan model run, defaults to 300.

stan_chains

The number of chains for each Stan model run, defaults to 1.

...

Other arguments to pass to sampling

Value

A numeric vector of samples from the correlation coefficient posterior.

Details

Uses a sigma ~ half_t(3, 0, 2) prior on the residual standard deviation and a uniform(-1, 1) prior on the correlation coefficient. Fitted as a linear regression of y ~ x, where y represents the y argument to trend_cor() and x represents the DFA trend, and both y and x have been scaled by subtracting their means and dividing by their standard deviations. Samples are drawn from the posterior of the trend and repeatedly fed through the Stan regression to come up with a combined posterior of the correlation.

Examples

set.seed(1)
s <- sim_dfa(num_trends = 1, num_years = 15)
m <- fit_dfa(y = s$y_sim, num_trends = 1, iter = 50, chains = 1)
#> 
#> SAMPLING FOR MODEL 'dfa' NOW (CHAIN 1).
#> Chain 1: 
#> Chain 1: Gradient evaluation took 3.6e-05 seconds
#> Chain 1: 1000 transitions using 10 leapfrog steps per transition would take 0.36 seconds.
#> Chain 1: Adjust your expectations accordingly!
#> Chain 1: 
#> Chain 1: 
#> Chain 1: WARNING: There aren't enough warmup iterations to fit the
#> Chain 1:          three stages of adaptation as currently configured.
#> Chain 1:          Reducing each adaptation stage to 15%/75%/10% of
#> Chain 1:          the given number of warmup iterations:
#> Chain 1:            init_buffer = 3
#> Chain 1:            adapt_window = 20
#> Chain 1:            term_buffer = 2
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#> Chain 1: 
#> Chain 1:  Elapsed Time: 0.029 seconds (Warm-up)
#> Chain 1:                0.381 seconds (Sampling)
#> Chain 1:                0.41 seconds (Total)
#> Chain 1: 
#> Warning: There were 1 chains where the estimated Bayesian Fraction of Missing Information was low. See
#> https://mc-stan.org/misc/warnings.html#bfmi-low
#> Warning: Examine the pairs() plot to diagnose sampling problems
#> Warning: The largest R-hat is 2.1, indicating chains have not mixed.
#> Running the chains for more iterations may help. See
#> https://mc-stan.org/misc/warnings.html#r-hat
#> Warning: Bulk Effective Samples Size (ESS) is too low, indicating posterior means and medians may be unreliable.
#> Running the chains for more iterations may help. See
#> https://mc-stan.org/misc/warnings.html#bulk-ess
#> Warning: Tail Effective Samples Size (ESS) is too low, indicating posterior variances and tail quantiles may be unreliable.
#> Running the chains for more iterations may help. See
#> https://mc-stan.org/misc/warnings.html#tail-ess
#> Inference for the input samples (1 chains: each with iter = 25; warmup = 12):
#> 
#>                 Q5   Q50   Q95   Mean    SD  Rhat Bulk_ESS Tail_ESS
#> x[1,1]        -1.6  -0.2   0.6   -0.4   0.8  1.47        6       13
#> x[1,2]        -0.7   0.8   1.2    0.4   0.8  1.48        6       13
#> x[1,3]         0.0   1.1   1.9    0.9   0.7  1.87        4       13
#> x[1,4]         0.3   0.9   1.5    0.9   0.5  2.06        4       13
#> x[1,5]        -0.2   0.5   1.4    0.6   0.6  2.06        4       13
#> x[1,6]        -0.5   0.2   0.5    0.1   0.4  0.93       13       13
#> x[1,7]        -0.3   0.4   1.1    0.4   0.5  1.48        7       13
#> x[1,8]        -0.4   0.0   1.2    0.2   0.6  0.92       13       13
#> x[1,9]         0.0   0.4   0.9    0.4   0.3  1.15       10       13
#> x[1,10]       -0.8   0.2   0.7    0.0   0.5  1.45        5       13
#> x[1,11]       -2.0  -1.4  -0.6   -1.3   0.5  1.20        6       13
#> x[1,12]       -2.3  -1.6   0.3   -1.1   1.0  2.06        4       13
#> x[1,13]       -1.5  -1.0   0.8   -0.6   0.9  2.06        4       13
#> x[1,14]       -0.7  -0.4   0.3   -0.3   0.4  0.96       13       13
#> x[1,15]       -0.7   0.4   0.9    0.3   0.6  1.13       10       13
#> Z[1,1]        -4.9  -0.9   7.8    0.0   6.2  2.06       13       13
#> Z[2,1]         0.0   0.6   1.1    0.6   0.4  1.24       13       13
#> Z[3,1]        -1.7  -0.8   1.0   -0.5   0.9  2.06        9       13
#> Z[4,1]        -0.7  -0.1   0.7   -0.1   0.5  1.32       10       13
#> log_lik[1]    -4.5  -0.7  -0.4   -1.6   1.7  2.06        4       13
#> log_lik[2]    -3.7  -0.9  -0.4   -1.4   1.2  1.58        4       13
#> log_lik[3]    -3.7  -0.8  -0.3   -1.4   1.3  2.06        4       13
#> log_lik[4]    -3.7  -1.2  -0.7   -1.6   1.1  1.71        9       13
#> log_lik[5]    -3.8  -1.7  -0.7   -2.0   1.2  2.06        4       13
#> log_lik[6]    -3.7  -1.6  -0.7   -1.8   1.1  1.25        6       13
#> log_lik[7]    -3.7  -1.0  -0.4   -1.5   1.2  1.87        4       13
#> log_lik[8]    -3.7  -1.9  -1.1   -2.1   0.9  1.45        9       13
#> log_lik[9]    -3.7  -1.1  -0.4   -1.6   1.2  2.06        4       13
#> log_lik[10]   -3.7  -1.1  -0.6   -1.5   1.1  1.21        6       13
#> log_lik[11]   -3.7  -1.4  -0.5   -1.5   1.2  2.06        4       13
#> log_lik[12]   -3.7  -0.8  -0.5   -1.3   1.2  1.58        4       13
#> log_lik[13]   -3.9  -0.8  -0.5   -1.4   1.3  2.06        4       13
#> log_lik[14]   -3.8  -2.3  -0.7   -2.3   1.1  1.09       11       13
#> log_lik[15]   -3.7  -0.9  -0.3   -1.4   1.2  1.87        4       13
#> log_lik[16]   -3.7  -0.8  -0.3   -1.3   1.2  1.87        4       13
#> log_lik[17]   -3.7  -0.9  -0.4   -1.4   1.2  1.87        4       13
#> log_lik[18]   -3.7  -0.9  -0.5   -1.7   1.2  1.05        7       13
#> log_lik[19]   -3.7  -0.9  -0.4   -1.4   1.2  1.87        4       13
#> log_lik[20]   -3.7  -1.5  -0.9   -1.8   1.0  1.20       10       13
#> log_lik[21]   -3.8  -0.7  -0.4   -1.5   1.4  2.06        4       13
#> log_lik[22]   -3.7  -0.8  -0.4   -1.3   1.2  1.58        4       13
#> log_lik[23]   -3.7  -0.9  -0.4   -1.4   1.2  1.87        4       13
#> log_lik[24]   -3.7  -0.7  -0.3   -1.2   1.3  2.06        4       13
#> log_lik[25]   -4.1  -0.8  -0.3   -1.5   1.4  2.06        4       13
#> log_lik[26]   -3.7  -0.8  -0.3   -1.3   1.2  2.06        4       13
#> log_lik[27]   -3.7  -0.7  -0.3   -1.3   1.2  2.06        4       13
#> log_lik[28]   -3.7  -0.6  -0.3   -1.2   1.3  1.87        4       13
#> log_lik[29]   -3.7  -1.2  -0.5   -1.6   1.1  1.16        6       13
#> log_lik[30]   -3.7  -0.8  -0.4   -1.4   1.2  1.47        4       13
#> log_lik[31]   -3.7  -1.6  -0.7   -1.8   1.0  1.24        7       13
#> log_lik[32]   -3.7  -0.6  -0.3   -1.3   1.2  1.87        4       13
#> log_lik[33]   -3.8  -0.7  -0.3   -1.3   1.3  1.58        4       13
#> log_lik[34]   -3.7  -1.2  -0.7   -1.6   1.1  1.12        6       13
#> log_lik[35]   -3.7  -0.7  -0.4   -1.3   1.2  1.87        4       13
#> log_lik[36]   -3.7  -0.9  -0.8   -1.5   1.1  1.10        7       13
#> log_lik[37]   -3.7  -0.9  -0.3   -1.4   1.2  1.71        4       13
#> log_lik[38]   -3.7  -0.9  -0.4   -1.4   1.2  1.47        4       13
#> log_lik[39]   -3.7  -0.9  -0.3   -1.4   1.3  1.71        4       13
#> log_lik[40]   -3.7  -1.0  -0.6   -1.4   1.2  2.06        4       13
#> log_lik[41]   -4.6  -0.9  -0.4   -1.6   1.6  1.71        4       13
#> log_lik[42]   -3.7  -2.4  -0.3   -2.2   1.2  1.30        6       13
#> log_lik[43]   -4.3  -1.0  -0.4   -1.9   1.6  1.71        5       13
#> log_lik[44]   -4.8  -3.7  -1.3   -3.5   1.2  1.32        7       13
#> log_lik[45]   -4.0  -0.8  -0.4   -1.8   1.5  1.71        5       13
#> log_lik[46]   -3.7  -1.2  -0.6   -1.5   1.1  1.27        6       13
#> log_lik[47]   -3.7  -0.9  -0.4   -1.6   1.3  1.58        4       13
#> log_lik[48]  -10.6  -4.0  -2.4   -5.3   3.0  1.03       13       13
#> log_lik[49]   -4.0  -0.9  -0.3   -1.6   1.5  2.06        4       13
#> log_lik[50]   -3.7  -1.2  -0.3   -1.5   1.2  2.06        4       13
#> log_lik[51]   -3.7  -0.9  -0.3   -1.5   1.2  1.71        4       13
#> log_lik[52]   -3.7  -1.0  -0.6   -1.5   1.1  1.48        5       13
#> log_lik[53]   -3.8  -0.9  -0.3   -1.4   1.2  2.06        4       13
#> log_lik[54]   -3.7  -0.7  -0.5   -1.3   1.2  2.06        4       13
#> log_lik[55]   -3.7  -0.9  -0.4   -1.3   1.2  2.06        4       13
#> log_lik[56]   -3.7  -2.2  -1.4   -2.4   0.9  1.10        9       13
#> log_lik[57]   -5.0  -1.3  -0.5   -2.0   1.8  1.58        5       13
#> log_lik[58]   -3.7  -2.5  -1.0   -2.2   1.0  0.93       13       13
#> log_lik[59]   -4.1  -1.3  -0.5   -1.8   1.4  1.48        5       13
#> log_lik[60]   -3.7  -2.3  -0.9   -2.1   1.0  1.24        8       13
#> xstar[1,1]    -2.0   0.7   1.9    0.3   1.3  0.97       13       13
#> sigma[1]       0.5   0.7  15.9    3.5   5.7  1.87        4       13
#> lp__        -315.4 -60.3 -46.6 -109.3 109.5  2.06        4       13
#> 
#> For each parameter, Bulk_ESS and Tail_ESS are crude measures of 
#> effective sample size for bulk and tail quantities respectively (an ESS > 100 
#> per chain is considered good), and Rhat is the potential scale reduction 
#> factor on rank normalized split chains (at convergence, Rhat <= 1.05).
r <- rotate_trends(m)
n_years <- ncol(r$trends[, 1, ])
fake_dat <- rnorm(n_years, 0, 1)
correlation <- trend_cor(r, fake_dat, trend_samples = 25)
#> 
#> SAMPLING FOR MODEL 'corr' NOW (CHAIN 1).
#> Chain 1: 
#> Chain 1: Gradient evaluation took 8e-06 seconds
#> Chain 1: 1000 transitions using 10 leapfrog steps per transition would take 0.08 seconds.
#> Chain 1: Adjust your expectations accordingly!
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#> Chain 1: 
#> Warning: The largest R-hat is 1.09, indicating chains have not mixed.
#> Running the chains for more iterations may help. See
#> https://mc-stan.org/misc/warnings.html#r-hat
#> Warning: Bulk Effective Samples Size (ESS) is too low, indicating posterior means and medians may be unreliable.
#> Running the chains for more iterations may help. See
#> https://mc-stan.org/misc/warnings.html#bulk-ess
#> Warning: Tail Effective Samples Size (ESS) is too low, indicating posterior variances and tail quantiles may be unreliable.
#> Running the chains for more iterations may help. See
#> https://mc-stan.org/misc/warnings.html#tail-ess
#> 
#> SAMPLING FOR MODEL 'corr' NOW (CHAIN 1).
#> Chain 1: 
#> Chain 1: Gradient evaluation took 5e-06 seconds
#> Chain 1: 1000 transitions using 10 leapfrog steps per transition would take 0.05 seconds.
#> Chain 1: Adjust your expectations accordingly!
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#> Chain 1: 
#> Warning: Bulk Effective Samples Size (ESS) is too low, indicating posterior means and medians may be unreliable.
#> Running the chains for more iterations may help. See
#> https://mc-stan.org/misc/warnings.html#bulk-ess
#> Warning: Tail Effective Samples Size (ESS) is too low, indicating posterior variances and tail quantiles may be unreliable.
#> Running the chains for more iterations may help. See
#> https://mc-stan.org/misc/warnings.html#tail-ess
#> 
#> SAMPLING FOR MODEL 'corr' NOW (CHAIN 1).
#> Chain 1: 
#> Chain 1: Gradient evaluation took 5e-06 seconds
#> Chain 1: 1000 transitions using 10 leapfrog steps per transition would take 0.05 seconds.
#> Chain 1: Adjust your expectations accordingly!
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#> Chain 1: 
#> Warning: Bulk Effective Samples Size (ESS) is too low, indicating posterior means and medians may be unreliable.
#> Running the chains for more iterations may help. See
#> https://mc-stan.org/misc/warnings.html#bulk-ess
#> Warning: Tail Effective Samples Size (ESS) is too low, indicating posterior variances and tail quantiles may be unreliable.
#> Running the chains for more iterations may help. See
#> https://mc-stan.org/misc/warnings.html#tail-ess
#> 
#> SAMPLING FOR MODEL 'corr' NOW (CHAIN 1).
#> Chain 1: 
#> Chain 1: Gradient evaluation took 5e-06 seconds
#> Chain 1: 1000 transitions using 10 leapfrog steps per transition would take 0.05 seconds.
#> Chain 1: Adjust your expectations accordingly!
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#> Chain 1: 
#> Warning: Bulk Effective Samples Size (ESS) is too low, indicating posterior means and medians may be unreliable.
#> Running the chains for more iterations may help. See
#> https://mc-stan.org/misc/warnings.html#bulk-ess
#> Warning: Tail Effective Samples Size (ESS) is too low, indicating posterior variances and tail quantiles may be unreliable.
#> Running the chains for more iterations may help. See
#> https://mc-stan.org/misc/warnings.html#tail-ess
#> 
#> SAMPLING FOR MODEL 'corr' NOW (CHAIN 1).
#> Chain 1: 
#> Chain 1: Gradient evaluation took 5e-06 seconds
#> Chain 1: 1000 transitions using 10 leapfrog steps per transition would take 0.05 seconds.
#> Chain 1: Adjust your expectations accordingly!
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#> Chain 1: 
#> Warning: Bulk Effective Samples Size (ESS) is too low, indicating posterior means and medians may be unreliable.
#> Running the chains for more iterations may help. See
#> https://mc-stan.org/misc/warnings.html#bulk-ess
#> Warning: Tail Effective Samples Size (ESS) is too low, indicating posterior variances and tail quantiles may be unreliable.
#> Running the chains for more iterations may help. See
#> https://mc-stan.org/misc/warnings.html#tail-ess
#> 
#> SAMPLING FOR MODEL 'corr' NOW (CHAIN 1).
#> Chain 1: 
#> Chain 1: Gradient evaluation took 5e-06 seconds
#> Chain 1: 1000 transitions using 10 leapfrog steps per transition would take 0.05 seconds.
#> Chain 1: Adjust your expectations accordingly!
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#> Running the chains for more iterations may help. See
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#> Warning: Tail Effective Samples Size (ESS) is too low, indicating posterior variances and tail quantiles may be unreliable.
#> Running the chains for more iterations may help. See
#> https://mc-stan.org/misc/warnings.html#tail-ess
#> 
#> SAMPLING FOR MODEL 'corr' NOW (CHAIN 1).
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#> Running the chains for more iterations may help. See
#> https://mc-stan.org/misc/warnings.html#tail-ess
#> 
#> SAMPLING FOR MODEL 'corr' NOW (CHAIN 1).
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#> Running the chains for more iterations may help. See
#> https://mc-stan.org/misc/warnings.html#tail-ess
#> 
#> SAMPLING FOR MODEL 'corr' NOW (CHAIN 1).
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#> Running the chains for more iterations may help. See
#> https://mc-stan.org/misc/warnings.html#tail-ess
#> 
#> SAMPLING FOR MODEL 'corr' NOW (CHAIN 1).
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#> Running the chains for more iterations may help. See
#> https://mc-stan.org/misc/warnings.html#tail-ess
#> 
#> SAMPLING FOR MODEL 'corr' NOW (CHAIN 1).
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#> Running the chains for more iterations may help. See
#> https://mc-stan.org/misc/warnings.html#tail-ess
#> 
#> SAMPLING FOR MODEL 'corr' NOW (CHAIN 1).
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#> Running the chains for more iterations may help. See
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#> SAMPLING FOR MODEL 'corr' NOW (CHAIN 1).
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#> Running the chains for more iterations may help. See
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#> Running the chains for more iterations may help. See
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#> Running the chains for more iterations may help. See
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#> Running the chains for more iterations may help. See
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#> Warning: Bulk Effective Samples Size (ESS) is too low, indicating posterior means and medians may be unreliable.
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#> Running the chains for more iterations may help. See
#> https://mc-stan.org/misc/warnings.html#tail-ess
#> 
#> SAMPLING FOR MODEL 'corr' NOW (CHAIN 1).
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#> Running the chains for more iterations may help. See
#> https://mc-stan.org/misc/warnings.html#tail-ess
#> 
#> SAMPLING FOR MODEL 'corr' NOW (CHAIN 1).
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#> Running the chains for more iterations may help. See
#> https://mc-stan.org/misc/warnings.html#tail-ess
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#> SAMPLING FOR MODEL 'corr' NOW (CHAIN 1).
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#> Running the chains for more iterations may help. See
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#> Running the chains for more iterations may help. See
#> https://mc-stan.org/misc/warnings.html#bulk-ess
#> Warning: Tail Effective Samples Size (ESS) is too low, indicating posterior variances and tail quantiles may be unreliable.
#> Running the chains for more iterations may help. See
#> https://mc-stan.org/misc/warnings.html#tail-ess
#> 
#> SAMPLING FOR MODEL 'corr' NOW (CHAIN 1).
#> Chain 1: 
#> Chain 1: Gradient evaluation took 5e-06 seconds
#> Chain 1: 1000 transitions using 10 leapfrog steps per transition would take 0.05 seconds.
#> Chain 1: Adjust your expectations accordingly!
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#> Chain 1: 
#> Warning: Bulk Effective Samples Size (ESS) is too low, indicating posterior means and medians may be unreliable.
#> Running the chains for more iterations may help. See
#> https://mc-stan.org/misc/warnings.html#bulk-ess
#> Warning: Tail Effective Samples Size (ESS) is too low, indicating posterior variances and tail quantiles may be unreliable.
#> Running the chains for more iterations may help. See
#> https://mc-stan.org/misc/warnings.html#tail-ess
#> 
#> SAMPLING FOR MODEL 'corr' NOW (CHAIN 1).
#> Chain 1: 
#> Chain 1: Gradient evaluation took 5e-06 seconds
#> Chain 1: 1000 transitions using 10 leapfrog steps per transition would take 0.05 seconds.
#> Chain 1: Adjust your expectations accordingly!
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#> Chain 1: 
#> Warning: Bulk Effective Samples Size (ESS) is too low, indicating posterior means and medians may be unreliable.
#> Running the chains for more iterations may help. See
#> https://mc-stan.org/misc/warnings.html#bulk-ess
#> Warning: Tail Effective Samples Size (ESS) is too low, indicating posterior variances and tail quantiles may be unreliable.
#> Running the chains for more iterations may help. See
#> https://mc-stan.org/misc/warnings.html#tail-ess
#> 
#> SAMPLING FOR MODEL 'corr' NOW (CHAIN 1).
#> Chain 1: 
#> Chain 1: Gradient evaluation took 5e-06 seconds
#> Chain 1: 1000 transitions using 10 leapfrog steps per transition would take 0.05 seconds.
#> Chain 1: Adjust your expectations accordingly!
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#> Chain 1: 
#> Warning: The largest R-hat is 1.05, indicating chains have not mixed.
#> Running the chains for more iterations may help. See
#> https://mc-stan.org/misc/warnings.html#r-hat
#> Warning: Bulk Effective Samples Size (ESS) is too low, indicating posterior means and medians may be unreliable.
#> Running the chains for more iterations may help. See
#> https://mc-stan.org/misc/warnings.html#bulk-ess
#> Warning: Tail Effective Samples Size (ESS) is too low, indicating posterior variances and tail quantiles may be unreliable.
#> Running the chains for more iterations may help. See
#> https://mc-stan.org/misc/warnings.html#tail-ess
#> 
#> SAMPLING FOR MODEL 'corr' NOW (CHAIN 1).
#> Chain 1: 
#> Chain 1: Gradient evaluation took 5e-06 seconds
#> Chain 1: 1000 transitions using 10 leapfrog steps per transition would take 0.05 seconds.
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#> Warning: The largest R-hat is 1.07, indicating chains have not mixed.
#> Running the chains for more iterations may help. See
#> https://mc-stan.org/misc/warnings.html#r-hat
#> Warning: Bulk Effective Samples Size (ESS) is too low, indicating posterior means and medians may be unreliable.
#> Running the chains for more iterations may help. See
#> https://mc-stan.org/misc/warnings.html#bulk-ess
#> Warning: Tail Effective Samples Size (ESS) is too low, indicating posterior variances and tail quantiles may be unreliable.
#> Running the chains for more iterations may help. See
#> https://mc-stan.org/misc/warnings.html#tail-ess
#> 
#> SAMPLING FOR MODEL 'corr' NOW (CHAIN 1).
#> Chain 1: 
#> Chain 1: Gradient evaluation took 5e-06 seconds
#> Chain 1: 1000 transitions using 10 leapfrog steps per transition would take 0.05 seconds.
#> Chain 1: Adjust your expectations accordingly!
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#> Chain 1: 
#> Warning: The largest R-hat is 1.06, indicating chains have not mixed.
#> Running the chains for more iterations may help. See
#> https://mc-stan.org/misc/warnings.html#r-hat
#> Warning: Bulk Effective Samples Size (ESS) is too low, indicating posterior means and medians may be unreliable.
#> Running the chains for more iterations may help. See
#> https://mc-stan.org/misc/warnings.html#bulk-ess
#> Warning: Tail Effective Samples Size (ESS) is too low, indicating posterior variances and tail quantiles may be unreliable.
#> Running the chains for more iterations may help. See
#> https://mc-stan.org/misc/warnings.html#tail-ess
hist(correlation)

correlation <- trend_cor(r,
  y = fake_dat, time_window = 5:15,
  trend_samples = 25
)
#> 
#> SAMPLING FOR MODEL 'corr' NOW (CHAIN 1).
#> Chain 1: 
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#> Warning: Bulk Effective Samples Size (ESS) is too low, indicating posterior means and medians may be unreliable.
#> Running the chains for more iterations may help. See
#> https://mc-stan.org/misc/warnings.html#bulk-ess
#> Warning: Tail Effective Samples Size (ESS) is too low, indicating posterior variances and tail quantiles may be unreliable.
#> Running the chains for more iterations may help. See
#> https://mc-stan.org/misc/warnings.html#tail-ess
#> 
#> SAMPLING FOR MODEL 'corr' NOW (CHAIN 1).
#> Chain 1: 
#> Chain 1: Gradient evaluation took 5e-06 seconds
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#> Warning: Bulk Effective Samples Size (ESS) is too low, indicating posterior means and medians may be unreliable.
#> Running the chains for more iterations may help. See
#> https://mc-stan.org/misc/warnings.html#bulk-ess
#> Warning: Tail Effective Samples Size (ESS) is too low, indicating posterior variances and tail quantiles may be unreliable.
#> Running the chains for more iterations may help. See
#> https://mc-stan.org/misc/warnings.html#tail-ess
#> 
#> SAMPLING FOR MODEL 'corr' NOW (CHAIN 1).
#> Chain 1: 
#> Chain 1: Gradient evaluation took 5e-06 seconds
#> Chain 1: 1000 transitions using 10 leapfrog steps per transition would take 0.05 seconds.
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#> Warning: Bulk Effective Samples Size (ESS) is too low, indicating posterior means and medians may be unreliable.
#> Running the chains for more iterations may help. See
#> https://mc-stan.org/misc/warnings.html#bulk-ess
#> Warning: Tail Effective Samples Size (ESS) is too low, indicating posterior variances and tail quantiles may be unreliable.
#> Running the chains for more iterations may help. See
#> https://mc-stan.org/misc/warnings.html#tail-ess
#> 
#> SAMPLING FOR MODEL 'corr' NOW (CHAIN 1).
#> Chain 1: 
#> Chain 1: Gradient evaluation took 6e-06 seconds
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#> Warning: The largest R-hat is 1.08, indicating chains have not mixed.
#> Running the chains for more iterations may help. See
#> https://mc-stan.org/misc/warnings.html#r-hat
#> Warning: Bulk Effective Samples Size (ESS) is too low, indicating posterior means and medians may be unreliable.
#> Running the chains for more iterations may help. See
#> https://mc-stan.org/misc/warnings.html#bulk-ess
#> Warning: Tail Effective Samples Size (ESS) is too low, indicating posterior variances and tail quantiles may be unreliable.
#> Running the chains for more iterations may help. See
#> https://mc-stan.org/misc/warnings.html#tail-ess
#> 
#> SAMPLING FOR MODEL 'corr' NOW (CHAIN 1).
#> Chain 1: 
#> Chain 1: Gradient evaluation took 5e-06 seconds
#> Chain 1: 1000 transitions using 10 leapfrog steps per transition would take 0.05 seconds.
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#> Chain 1: 
#> Warning: Bulk Effective Samples Size (ESS) is too low, indicating posterior means and medians may be unreliable.
#> Running the chains for more iterations may help. See
#> https://mc-stan.org/misc/warnings.html#bulk-ess
#> Warning: Tail Effective Samples Size (ESS) is too low, indicating posterior variances and tail quantiles may be unreliable.
#> Running the chains for more iterations may help. See
#> https://mc-stan.org/misc/warnings.html#tail-ess
#> 
#> SAMPLING FOR MODEL 'corr' NOW (CHAIN 1).
#> Chain 1: 
#> Chain 1: Gradient evaluation took 5e-06 seconds
#> Chain 1: 1000 transitions using 10 leapfrog steps per transition would take 0.05 seconds.
#> Chain 1: Adjust your expectations accordingly!
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#> Chain 1: 
#> Warning: Bulk Effective Samples Size (ESS) is too low, indicating posterior means and medians may be unreliable.
#> Running the chains for more iterations may help. See
#> https://mc-stan.org/misc/warnings.html#bulk-ess
#> Warning: Tail Effective Samples Size (ESS) is too low, indicating posterior variances and tail quantiles may be unreliable.
#> Running the chains for more iterations may help. See
#> https://mc-stan.org/misc/warnings.html#tail-ess
#> 
#> SAMPLING FOR MODEL 'corr' NOW (CHAIN 1).
#> Chain 1: 
#> Chain 1: Gradient evaluation took 6e-06 seconds
#> Chain 1: 1000 transitions using 10 leapfrog steps per transition would take 0.06 seconds.
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#> Chain 1: 
#> Warning: Bulk Effective Samples Size (ESS) is too low, indicating posterior means and medians may be unreliable.
#> Running the chains for more iterations may help. See
#> https://mc-stan.org/misc/warnings.html#bulk-ess
#> Warning: Tail Effective Samples Size (ESS) is too low, indicating posterior variances and tail quantiles may be unreliable.
#> Running the chains for more iterations may help. See
#> https://mc-stan.org/misc/warnings.html#tail-ess
#> 
#> SAMPLING FOR MODEL 'corr' NOW (CHAIN 1).
#> Chain 1: 
#> Chain 1: Gradient evaluation took 6e-06 seconds
#> Chain 1: 1000 transitions using 10 leapfrog steps per transition would take 0.06 seconds.
#> Chain 1: Adjust your expectations accordingly!
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#> Chain 1: 
#> Warning: The largest R-hat is 1.06, indicating chains have not mixed.
#> Running the chains for more iterations may help. See
#> https://mc-stan.org/misc/warnings.html#r-hat
#> Warning: Bulk Effective Samples Size (ESS) is too low, indicating posterior means and medians may be unreliable.
#> Running the chains for more iterations may help. See
#> https://mc-stan.org/misc/warnings.html#bulk-ess
#> Warning: Tail Effective Samples Size (ESS) is too low, indicating posterior variances and tail quantiles may be unreliable.
#> Running the chains for more iterations may help. See
#> https://mc-stan.org/misc/warnings.html#tail-ess
#> 
#> SAMPLING FOR MODEL 'corr' NOW (CHAIN 1).
#> Chain 1: 
#> Chain 1: Gradient evaluation took 6e-06 seconds
#> Chain 1: 1000 transitions using 10 leapfrog steps per transition would take 0.06 seconds.
#> Chain 1: Adjust your expectations accordingly!
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#> Warning: Bulk Effective Samples Size (ESS) is too low, indicating posterior means and medians may be unreliable.
#> Running the chains for more iterations may help. See
#> https://mc-stan.org/misc/warnings.html#bulk-ess
#> Warning: Tail Effective Samples Size (ESS) is too low, indicating posterior variances and tail quantiles may be unreliable.
#> Running the chains for more iterations may help. See
#> https://mc-stan.org/misc/warnings.html#tail-ess
#> 
#> SAMPLING FOR MODEL 'corr' NOW (CHAIN 1).
#> Chain 1: 
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#> Running the chains for more iterations may help. See
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#> Warning: Tail Effective Samples Size (ESS) is too low, indicating posterior variances and tail quantiles may be unreliable.
#> Running the chains for more iterations may help. See
#> https://mc-stan.org/misc/warnings.html#tail-ess
#> 
#> SAMPLING FOR MODEL 'corr' NOW (CHAIN 1).
#> Chain 1: 
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#> Running the chains for more iterations may help. See
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#> Warning: Tail Effective Samples Size (ESS) is too low, indicating posterior variances and tail quantiles may be unreliable.
#> Running the chains for more iterations may help. See
#> https://mc-stan.org/misc/warnings.html#tail-ess
#> 
#> SAMPLING FOR MODEL 'corr' NOW (CHAIN 1).
#> Chain 1: 
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#> Running the chains for more iterations may help. See
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#> Warning: Tail Effective Samples Size (ESS) is too low, indicating posterior variances and tail quantiles may be unreliable.
#> Running the chains for more iterations may help. See
#> https://mc-stan.org/misc/warnings.html#tail-ess
#> 
#> SAMPLING FOR MODEL 'corr' NOW (CHAIN 1).
#> Chain 1: 
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#> Warning: Bulk Effective Samples Size (ESS) is too low, indicating posterior means and medians may be unreliable.
#> Running the chains for more iterations may help. See
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#> Warning: Tail Effective Samples Size (ESS) is too low, indicating posterior variances and tail quantiles may be unreliable.
#> Running the chains for more iterations may help. See
#> https://mc-stan.org/misc/warnings.html#tail-ess
#> 
#> SAMPLING FOR MODEL 'corr' NOW (CHAIN 1).
#> Chain 1: 
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#> Warning: The largest R-hat is 1.09, indicating chains have not mixed.
#> Running the chains for more iterations may help. See
#> https://mc-stan.org/misc/warnings.html#r-hat
#> Warning: Bulk Effective Samples Size (ESS) is too low, indicating posterior means and medians may be unreliable.
#> Running the chains for more iterations may help. See
#> https://mc-stan.org/misc/warnings.html#bulk-ess
#> Warning: Tail Effective Samples Size (ESS) is too low, indicating posterior variances and tail quantiles may be unreliable.
#> Running the chains for more iterations may help. See
#> https://mc-stan.org/misc/warnings.html#tail-ess
#> 
#> SAMPLING FOR MODEL 'corr' NOW (CHAIN 1).
#> Chain 1: 
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#> Warning: Tail Effective Samples Size (ESS) is too low, indicating posterior variances and tail quantiles may be unreliable.
#> Running the chains for more iterations may help. See
#> https://mc-stan.org/misc/warnings.html#tail-ess
#> 
#> SAMPLING FOR MODEL 'corr' NOW (CHAIN 1).
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#> Warning: Tail Effective Samples Size (ESS) is too low, indicating posterior variances and tail quantiles may be unreliable.
#> Running the chains for more iterations may help. See
#> https://mc-stan.org/misc/warnings.html#tail-ess
#> 
#> SAMPLING FOR MODEL 'corr' NOW (CHAIN 1).
#> Chain 1: 
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#> Running the chains for more iterations may help. See
#> https://mc-stan.org/misc/warnings.html#tail-ess
#> 
#> SAMPLING FOR MODEL 'corr' NOW (CHAIN 1).
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#> Running the chains for more iterations may help. See
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#> Running the chains for more iterations may help. See
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#> Running the chains for more iterations may help. See
#> https://mc-stan.org/misc/warnings.html#tail-ess
#> 
#> SAMPLING FOR MODEL 'corr' NOW (CHAIN 1).
#> Chain 1: 
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#> Running the chains for more iterations may help. See
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#> Warning: Tail Effective Samples Size (ESS) is too low, indicating posterior variances and tail quantiles may be unreliable.
#> Running the chains for more iterations may help. See
#> https://mc-stan.org/misc/warnings.html#tail-ess
#> 
#> SAMPLING FOR MODEL 'corr' NOW (CHAIN 1).
#> Chain 1: 
#> Chain 1: Gradient evaluation took 4e-06 seconds
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#> Running the chains for more iterations may help. See
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#> Warning: Bulk Effective Samples Size (ESS) is too low, indicating posterior means and medians may be unreliable.
#> Running the chains for more iterations may help. See
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#> Warning: Tail Effective Samples Size (ESS) is too low, indicating posterior variances and tail quantiles may be unreliable.
#> Running the chains for more iterations may help. See
#> https://mc-stan.org/misc/warnings.html#tail-ess
#> 
#> SAMPLING FOR MODEL 'corr' NOW (CHAIN 1).
#> Chain 1: 
#> Chain 1: Gradient evaluation took 5e-06 seconds
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#> Running the chains for more iterations may help. See
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#> Warning: Tail Effective Samples Size (ESS) is too low, indicating posterior variances and tail quantiles may be unreliable.
#> Running the chains for more iterations may help. See
#> https://mc-stan.org/misc/warnings.html#tail-ess
#> 
#> SAMPLING FOR MODEL 'corr' NOW (CHAIN 1).
#> Chain 1: 
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#> Warning: Tail Effective Samples Size (ESS) is too low, indicating posterior variances and tail quantiles may be unreliable.
#> Running the chains for more iterations may help. See
#> https://mc-stan.org/misc/warnings.html#tail-ess
#> 
#> SAMPLING FOR MODEL 'corr' NOW (CHAIN 1).
#> Chain 1: 
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#> Chain 1: 
#> Warning: Bulk Effective Samples Size (ESS) is too low, indicating posterior means and medians may be unreliable.
#> Running the chains for more iterations may help. See
#> https://mc-stan.org/misc/warnings.html#bulk-ess
#> Warning: Tail Effective Samples Size (ESS) is too low, indicating posterior variances and tail quantiles may be unreliable.
#> Running the chains for more iterations may help. See
#> https://mc-stan.org/misc/warnings.html#tail-ess
#> 
#> SAMPLING FOR MODEL 'corr' NOW (CHAIN 1).
#> Chain 1: 
#> Chain 1: Gradient evaluation took 5e-06 seconds
#> Chain 1: 1000 transitions using 10 leapfrog steps per transition would take 0.05 seconds.
#> Chain 1: Adjust your expectations accordingly!
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#> Chain 1:                0.005 seconds (Total)
#> Chain 1: 
#> Warning: Bulk Effective Samples Size (ESS) is too low, indicating posterior means and medians may be unreliable.
#> Running the chains for more iterations may help. See
#> https://mc-stan.org/misc/warnings.html#bulk-ess
#> Warning: Tail Effective Samples Size (ESS) is too low, indicating posterior variances and tail quantiles may be unreliable.
#> Running the chains for more iterations may help. See
#> https://mc-stan.org/misc/warnings.html#tail-ess
#> 
#> SAMPLING FOR MODEL 'corr' NOW (CHAIN 1).
#> Chain 1: 
#> Chain 1: Gradient evaluation took 5e-06 seconds
#> Chain 1: 1000 transitions using 10 leapfrog steps per transition would take 0.05 seconds.
#> Chain 1: Adjust your expectations accordingly!
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#> Chain 1: 
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#> Chain 1:                0.01 seconds (Total)
#> Chain 1: 
#> Warning: Bulk Effective Samples Size (ESS) is too low, indicating posterior means and medians may be unreliable.
#> Running the chains for more iterations may help. See
#> https://mc-stan.org/misc/warnings.html#bulk-ess
#> Warning: Tail Effective Samples Size (ESS) is too low, indicating posterior variances and tail quantiles may be unreliable.
#> Running the chains for more iterations may help. See
#> https://mc-stan.org/misc/warnings.html#tail-ess
hist(correlation)