User:Lucasvb/An upgrade to the spatial model of voters: Difference between revisions

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 We would expect that if both opinions are already similar, not a lot of convincing is required. We would also expect that the further apart the "opinion units" need to be relocated, the more difficult it is to change someone's opinion.
-In mathematics and engineering, this is a well-studied problem of [https://en.wikipedia.org/wiki/Transportation_theory_(mathematics) optimal transport], and it has found uses everywhere, from artificial intelligence to traffic management.
+In mathematics and engineering, this is a well-studied problem of [[w:Transportation theory (mathematics)|optimal transport]], and it has found uses everywhere, from artificial intelligence to traffic management.
-The intuitive notion of how "difficult" it is to convince someone to believe something else, piece by piece, is captured by the '''''[https://en.wikipedia.org/wiki/Earth_mover%27s_distance earth-mover's distance]''''' ('''''EMD''''') between two distributions. It is, intuitively, the least amount of effort you would need to rearrange one pile of dirt into another pile of dirt.
+The intuitive notion of how "difficult" it is to convince someone to believe something else, piece by piece, is captured by the '''''[[w:earth-mover's distance|earth-mover's distance]]''''' ('''''EMD''''') between two distributions. It is, intuitively, the least amount of effort you would need to rearrange one pile of dirt into another pile of dirt.
 If you replace "dirt" with "opinion unit", you'll immediately arrive at our idea here.
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 Note that different issues are never compared with one another here. Only opinions on the same issue count towards each term.
-It should be clear that that if the distributions are infinitely sharp (i.e. [https://en.wikipedia.org/wiki/Dirac_delta_function Dirac deltas]), the earth-mover's distance is simply the distance between those two sharp peaks. In this way, we recover the old traditional spatial model from our model.
+It should be clear that that if the distributions are infinitely sharp (i.e. [[w:Dirac delta function|Dirac deltas]]), the earth-mover's distance is simply the distance between those two sharp peaks. In this way, we recover the old traditional spatial model from our model.
 === Why the Euclidean distance anyway? ===
-We could have considered other distances (or "metrics") in the same way, but why single out the Eucliean one? Why not use
+We could have considered other distances (or "metrics") in the same way, but why single out the Euclidean one? Why not use
 :<math>d(a,b) = \sum_{i=1}^{N} \text{EMD}(a_i(x),b_i(x))), </math>
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 The following Python code generates the trapezoid distribution for a given opinion, with belief from -1 to +1, and importance from 0 to 1.
+<syntaxhighlight lang="python">
- import numpy as np
+import numpy as np
- from scipy.stats import wasserstein_distance
+from scipy.stats import wasserstein_distance
- L = 5 # bin resolution (number of degrees of agreement/disagreement)
+L = 5 # bin resolution (number of degrees of agreement/disagreement)
- W = 2*L+1 # total number of bins, an odd number so we have a clean zero
+W = 2*L+1 # total number of bins, an odd number so we have a clean zero
- _space = np.linspace(-1,1,W) # array with positions of the bins
+_space = np.linspace(-1,1,W) # array with positions of the bins
- # Earth-mover's distance (or Wasserstein distance) between two opinion distributions
+# Earth-mover's distance (or Wasserstein distance) between two opinion distributions
- def EMD(dist1,dist2):
- 	return wasserstein_distance(_space,_space,dist1,dist2)
+def EMD(dist1,dist2):
+    return wasserstein_distance(_space,_space,dist1,dist2)
- # Generates an opinion distribution as a truncated & bounded trapezoidal distribution
+# Generates an opinion distribution as a truncated & bounded trapezoidal distribution
- #   belief: from -1 to +1
- #   importance: from 0 to 1
+#   belief: from -1 to +1
- def opinion(belief, importance):
+#   importance: from 0 to 1
- 	w = (1 - importance)*(W-1) + 1
+def opinion(belief, importance):
- 	v = (w+1)/2 - L*abs(np.linspace(-1,1,W) - belief*importance)
+    w = (1 - importance)*(W-1) + 1
+    v = (w+1)/2 - L*abs(np.linspace(-1,1,W) - belief*importance)
- 	v[v < 0] = 0
- 	v[v > 1] = 1
+    v[v < 0] = 0
- 	v /= sum(v)
+    v[v > 1] = 1
+    v /= sum(v)
- 	return v
+    return v
- # Visualize distribution with text blocks
+# Visualize distribution with text blocks
- def diststr(op):
+def diststr(op):
- 	return "".join(["_▁▂▃▄▅▆▇█"[int(r/max(op)*7)] for r in op]) # we go up to 7 because the full block looks bad piled up
+    return "".join(["_▁▂▃▄▅▆▇█"[int(r/max(op)*7)] for r in op]) # we go up to 7 because the full block looks bad piled up
- # Show some distributions generated
+# Show some distributions generated
- for w in range(1,W+1):
- 	c = 1 - (w-1)/(W-1)
+for w in range(1,W+1):
- 	for x in range(-L,L+1):
+    c = 1 - (w-1)/(W-1)
-  		op = opinion(x/L,c)
+    for x in range(-L,L+1):
+        op = opinion(x/L,c)
-  		print("(%+0.03f|%0.03f)" % (x/L,c), diststr(op))
+        print("(%+0.03f|%0.03f)" % (x/L,c), diststr(op))
- # Print a few distances
+# Print a few distances
- for i in range(10):
+for i in range(10):
- 	b1, b2 = np.random.rand()*2-1, np.random.rand()*2-1
- 	i1, i2 = np.random.rand(), np.random.rand()
+    b1, b2 = np.random.rand()*2-1, np.random.rand()*2-1
+    i1, i2 = np.random.rand(), np.random.rand()
- 	o1 = opinion(b1, i1)
- 	o2 = opinion(b2, i2)
+    o1 = opinion(b1, i1)
+    o2 = opinion(b2, i2)
- 	print(
+    print(
- 		"(%+0.03f|%0.03f)" % (b1,i1), diststr(o1),
+          "(%+0.03f|%0.03f)" % (b1,i1), diststr(o1),
- 		"vs",
+          "vs",
- 		diststr(o2), "(%+0.03f|%0.03f)" % (b2,i2),
+          diststr(o2), "(%+0.03f|%0.03f)" % (b2,i2),
- 		"= %0.04f" % EMD(o1,o2))
+          "= %0.04f" % EMD(o1,o2))
+</syntaxhighlight>