Stream Graph

Definition

A stream graph, sometimes written as streamgraph, is a stacked plot around a "central axis," resulting in a flowing, organic shape. Stream graphs demonstrate how topics evolve.

Problem Statement

Given a time series \(\{f_1,f_2,\dots,f_n\}\) and a baseline \(g_0\) compute the top layer to \(f_i\) as follows: \[g_i = g_0 + \sum_{j=1}^{i} f_j\]

The question of how to compute g0 remains open. Two simple strategies are

Simple model: \(g_0 \equiv 0\)
Symmetric around the \(x\)-axis: \(g_0 = -\frac{1}{2}\sum_{i=1}^{n} f_i\)

Remark

\(f_i = \{f_i(t_0), f_i(t_1), \dots, f_i(t_n)\}\), thus \(g_i = \{g_i(t_0), g_i(t_1), \dots, g_i(t_n)\}\)

How to compute the baseline

Reference: Stacked Graphs - Geometry and Aesthetic, L. Byron, M. Wattenberg

Simple model \(g_0 \equiv 0\)
Solution 1: minimize silhouette (same as symmetric model)
silhouette: \(f(g_0) = g_0^2 + g_n^2 = g_0^2 + (g_0 + \sum_{i=1}^n f_i)^2\)
\(\frac{df}{dg_0} = 0 \implies g_0 = -\frac{1}{2}\sum_{i=1}^{n} f_i\)
Solution 2: minimize the deviation \(g_i\) from the x-axis (minimizes the wiggle)
deviation: \(f(g_0) = \sum_{i=0}^n g_i^2\)
\(\frac{df}{dg_0}=0 \implies g_0 = -\frac{1}{n+1} \sum_{i=1}^n (n-i+1)f_i\)

Example