Why use a Binning Analysis?
As described in Markus Wallerberger (2019), Vinay Ambegaokar, Matthias Troyer (2010), Tony F. Chan, Gene H. Golub, Randall J. LeVeque (1983), Carsten Bauer (2020), and James Gubernatis, Naoki Kawashima, Philipp Werner (2016), the presence of correlations in a data stream generated in Markov Chain Monte Carlo simulations renders any measures of error severely underestimated.
Theoretical background
The naive approach is to calculate the mean and the variance of the mean, var_of_mean, for a given data stream $X$. For uncorrelated data, this becomes
\[\begin{aligned} \mathtt{mean}(X) &= \frac{1}{M} \sum_{i = 1}^M x_i \\ \mathtt{var\_of\_mean}(X) &= \frac{1}{M(M-1)} \sum_{i = 1}^M (x_i - \mathtt{mean}(X))^2. \end{aligned}\]
Fortunately, in the presence of correlations, the mean doesn't change. Unfortunately, the var_of_mean does. Indeed, one can show that it's given by
\[\mathtt{var\_of\_mean}(X) = \frac{1 + 2\tau_X}{M(M-1)} \sum_{i = 1}^M (x_i - \mathtt{mean}(X))^2,\]
where the as shown in the Figure within the Accumulator type hierarchy. Clearly, the uncorrelated var_of_mean is just increased by a factor of $R_X \equiv 1 + 2\tau_X$, where $\tau_X$ is defined as the integrated autocorrelation_time of the data stream. The binning analysis provides a fast as an O(N) method for calculating the autocorrelation_time. It's also cheap, only requiring O(log N) in RAM. For those who are unfamiliar with the term, normally one would calculate $\tau_X$ by
\[\tau_X = \sum_{i = 1}^M \sum_{i < j} \left[ x_ix_j - \mathtt{mean}(x)^2 \right],\]
which is an O(N^2) summation and can suffer from numerical instabilities.
As one performs pairwise means in each binning level to construct the next highest, and then calculates the var_of_mean for each level, one can see that it rises and then eventually saturates around a particular plateau value. Normalizing by the original var_of_mean calculation assuming the data stream is uncorrelated, one can then calculate $R_X$ in the $\ell^{\rm th}$ binning level as
\[R_X(\ell) = \frac{\mathtt{var\_of\_mean}(X^{(\ell)})}{\mathtt{var\_of\_mean}(X^{(0)})} = \frac{ m^{(\ell)} \mathtt{var}(X^{(\ell)}) }{ \mathtt{var}(X^{(0)}) },\]
where var is the normal variance and $m^{(\ell)}$ is the bin size, or the number of original data values accumulated into a single data point at the $\ell^{\rm th}$ level.
Visualizing a Binning Analysis
The final result of the binning analysis, for sufficiently long data streams, will be to see a sigmoid-like curve. We've wrapped up a fitting and plotting workflow into the plot_binning_analysis function below to demonstrate how it works. Feel free to skip over to the Visualization Examples if you have your own workflow established.
using OnlineLogBinning, Plots, LaTeXStrings
pyplot()
# Return the fitted inflection point
sigmoid_inflection(fit) = fit.param[2] / fit.param[3]
# Define a plotting function with Plots.jl
# This function plots the RxValues, the Sigmoid fit,
# the maximum trustworthy level, and the fitted
# inflection point.
function plot_binning_analysis(bacc)
# Plot the RxValues irrespective of the trusting cutoff
all_levels, all_rxvals = levels_RxValues(bacc, false)
plt = plot( all_levels, all_rxvals;
label = "Binning Analysis",
xlabel = L"Bin Level $\ell$",
ylabel = L"$R_X(\ell)$",
markershape = :circle,
legend = :topleft )
# Fit only the trustworthy levels
levels, rxvals = levels_RxValues(bacc)
fit = fit_RxValues(levels, rxvals)
# Plot the sigmoid fit
plot!(plt, all_levels, sigmoid(all_levels, fit.param);
label = "Sigmoid Fit")
# Plot the maximum trustworthy level
vline!(plt, [ max_trustworthy_level(bacc[level = 0].num_bins) ];
ls = :dash, color = "gray",
label = "Maximum Trustworthy Level")
# Plot the fitted inflection point if it's positive
if sigmoid_inflection(fit) > zero(fit.param[2])
vline!(plt, [ sigmoid_inflection(fit) ];
ls = :dot, color = "red",
label = "Sigmoid Inflection Point")
end
return plt
end[ Info: Installing matplotlib via the Conda matplotlib package...
[ Info: Running `conda install -q -y matplotlib` in root environment
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Preparing transaction: ...working... done
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Executing transaction: ...working... doneVisualization Examples
Now we demonstrate four cases that tend to appear with binning analyses:
- The binning analysis converges to a plateau for a correlated data stream $R_X(\ell \rightarrow \infty) > 0$.
- The binning analysis reveals that the data stream is truly uncorrelated $R_X(\ell \rightarrow \infty) = 0$.
- The binning analysis has not converged because too few data were taken in the data stream to isolate the uncorrelated data.
- The binning analyis has not convered because all of the data in the data stream are more-or-less equally correlated.
The first two cases are ideal. They represent situations where the data stream has enough data to distinguish correlated blocks from one another. In these first two cases, the BinningAnalysisResult will yield plateau_found == true.
The second two cases are not so ideal. Either way they show that the correlations in the data stream are so strong that individual uncorrelated blocks can not be robustly created and more sampling is required. Therefore, their BinningAnalysisResult yields plateau_found == false.
The data shown are pre-simulated signals generated by the TelegraphNoise.jl package (v0.1.0). Random telegraph signals have an analytically-defined autocorrelation time $\tau_X$ related to their average dwell time $T_D$ by $\tau_X = T_D / 2$. Since the signals are random, they are saved previously, but I'll provide each chosen $T_D$ for reproducibility purposes.
Example 1: Clear $R_X$ plateau at a finite value
This first case shows the textbook (James Gubernatis, Naoki Kawashima, Philipp Werner (2016)) example of an $R_X$ plateau found in a binning analysis. The dwell time for this signal was chosen to be $T_D = 16$, meaning $\tau_X = 8$, and the signal generated was $2^{18} = 262144$ in length.
# Read in pre-simulated TelegraphNoise.jl data with a plateau
signal = zeros(Float64, Int(2^18))
read!( joinpath("assets", "telegraph_plateau.bin"), signal )
bacc = BinningAccumulator()
push!(bacc, signal)
plot_binning_analysis(bacc)
result = fit_RxValues(bacc)
result.plateau_found
# output
trueImportantly, note that the fitted inflection point is greater than zero, but less than the maximum trustworthy level. The variations in the calculated $R_X$ values for $\ell$ greater than the maximum trustworthy level are due to strong fluctuations because of small number statistics.
Example 2: Uncorrelated data ~ $R_X(\ell \rightarrow \infty) = 0$
The dwell time for this signal was chosen to be $T_D = 1/2$, meaning $\tau_X = 1/4$, and the signal generated was $2^{14} = 16384$ in length.
# Read in pre-simulated TelegraphNoise.jl data with a plateau
signal = zeros(Float64, Int(2^14))
read!( joinpath("assets", "telegraph_uncorrelated.bin"), signal )
bacc = BinningAccumulator()
push!(bacc, signal)
plot_binning_analysis(bacc)
result = fit_RxValues(bacc)
result.plateau_found
# output
trueNotice that the $R_X$ values decay for increasing $\ell$. This is because the binning analysis cannot detect any smaller blocks of uncorrelated data since the original data stream is indeed correlated.
Example 3: No plateau due to data insufficiency
For insufficiently long data streams, we do not expect a plateau, as shown in the following case:
# Read in pre-simulated TelegraphNoise.jl data without a plateau
signal = zeros(Float64, Int(2^10))
read!( joinpath("assets", "telegraph_no_plateau.bin"), signal )
bacc = BinningAccumulator()
push!(bacc, signal)
plot_binning_analysis(bacc)
result = fit_RxValues(bacc)
result.plateau_found
# output
falseHere, the dwell time was chosen to be $T_D = 256$, meaning $\tau_X = 128$, and the signal generated was $2^{10} = 1024$ in length. Notice, that the binning analysis began to pick up on a set of uncorrelated blocks, but there was not enough simulated data to reveal them before the maximum trustworthy level was reached. In a case like this, one would simply need to run their simulation about 64 times longer reveal a fitted plateau.
Example 4: No plateau due to totally-correlated data
In this final example, we demonstrate what happens for a data stream that has "frozen-in" correlations. By this, we mean one where the autocorrelation time $\tau_X$ far exceeds the data stream size. For this case, $T_D = 2^{16} = 65536$, so $\tau_X = 2^{15} = 32768$, while the signal length is only $2^{12} = 4096$.
# Read in pre-simulated TelegraphNoise.jl data with a plateau
signal = zeros(Float64, Int(2^12))
read!( joinpath("assets", "telegraph_totally_correlated.bin"), signal )
bacc = BinningAccumulator()
push!(bacc, signal)
plot_binning_analysis(bacc)
result = fit_RxValues(bacc)
result.plateau_found
# output
falseIn this case, eventually the binning analysis returns $R_X = 0$, since the blocks start comparing literally identical elements. Notice that this case would return a plateau_found if it were not for the check if any of the $R_X$ values were too small. (In principle this case also would result from a data stream with a defined periodicity, despite having correlations, but this is unavoidable.)
This is the most dangerous case to automate because the variance of such a data stream is naturally very small relative to the mean. As is the case with all Monte Carlo simulations, or statistical data streams, one should be very wary when the error is a mathematical zero, or vanishingly small compared to the calculated mean.