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Ryan Boyden added subsubsection_Dendrograms_We_use_Dendrograms__1.tex
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\subsubsection{Dendrograms}
We use Dendrograms to organize the hierarchical structure of our intensity maps. Figure (Basic Dendrogram) provides a simple 2D depiction of a Dendrogram. To create one, we identify the peak intensity value in a data set, classify it as a "leaf", and catalog other local maxima at smaller intensities, also classified as "leaves". The intensities attached to particular leaves are known as "branches." This technique allows one to visualize local features of intensity levels appearing within our synthetic observations. To account for simulated noise in our data maps, we set a minimum distance between two local maxima, referred to as a "minimum delta." Increasing the minimum delta value decreases the total number of features (necessary?). We refer the reader to Rosolowskly et al. (2008) for a complete description of applying Dendrograms to Molecular Cloud structures.
Koch et al. (2015) defines two Dendogram statistics: the Number of Features Statistic and the Histogram Statistic. To compute the Number of Features Statistic, we generate multiple Dendrograms per run, varying the minimum delta parameter, and count the total number of features associated with each minimum delta. We follow the analysis of Koch et al (2015) (I'm using this alot, any other way to word this? Come back to this) and compute Dendrograms for our two runs using minimum deltas (symbol?) ranging from 10^-2.5 K to 10^0.5 K in 100 logarithmic steps. Figure 12 displays our results. Run W1T1t0.2's plot follows a power-law, while run W2T2t0 appears to diverge from power-law scalings at minimum deltas of 1K. We find that our run with feedback, at all scales, contains a greater amount of structure than our purely turbulent run has.(increasing past min delta creates a power law according to Burkhart et al. (2013), is this a counterexample?
The Histogram Statistic provides another insight on Dendrogram structure. We create series of Dendrograms for our runs using the same range of minimum deltas, but we analyze the intensity "leaves" as PDFs. For each minimum delta, we create a Dendrogram, standardize its intensity leaves and branches, and generate histograms. Figure 13 shows the histograms of our runs plotted over all minimumum deltas. The two runs produce significantly different distributions. We find that run W2T2t0 contains a wider range of probable intensity values than that of run W1T2t0.2, whose histograms are all strongly centered about the mean value. Run W1T2t0.2 also produces a larger span of positive standardized intensity values, but a smaller span of negative ones.
