Python implementation of the sigmoid to map a point value to a rating (where points are milliseconds and the rating is the resulting score).

rating.ipynb

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     "input": "%matplotlib\n\"\"\"\nin reference to http://programmers.stackexchange.com/questions/225487/mapping-values-over-a-curve/225503#225503\n\"\"\"\nimport numpy as np\nimport matplotlib.pyplot as plt\nimport math\n\nMIN_RATING=1\nMAX_RATING=2000\n\nMIN_POINTS=1\nMAX_POINTS=8000\n\nSIGMOID_RANGE=25\n\ndef sigmoid(x):\n    return 1 / (1 + math.exp(-x))\n\ndef map2range(value, low, high, new_low, new_high):\n    '''map a value from one range to another'''\n    return value * 1.0 / (high - low + 1) * (new_high - new_low + 1)\n\ndef map2sigmoid(value, max=SIGMOID_RANGE):\n    '''map a value from its rating range to the sigmoid range'''\n    return map2range(value, MIN_RATING, MAX_RATING, 1.0, max) - (max / 2.0)  \n\ndef map2rating(value, max=SIGMOID_RANGE):\n    '''map a points value to a rating value using the sigmoid function'''\n    rating_range = map2range(value, MIN_POINTS, MAX_POINTS, MIN_RATING, MAX_RATING)\n    sigmoid_range = map2sigmoid(rating_range, max=max)\n    s = sigmoid(sigmoid_range) \n    if s < 0.5:\n        r = math.floor(s * MAX_RATING)\n    else:\n        r = math.ceil(s * MAX_RATING)\n    return r\n\nax = plt.subplot(1,1,1)\nx = np.arange(MIN_RATING,MAX_RATING,1);\ny = [sigmoid(map2sigmoid(y, max=10)) for y in x]\nax.plot(x, y, label=\"range=10\")\ny = [sigmoid(map2sigmoid(y, max=20)) for y in x]\nax.plot(x, y, label=\"range=20\")\ny = [sigmoid(map2sigmoid(y, max=30)) for y in x]\nax.plot(x, y, label=\"range=30\")\n\nhandles, labels = ax.get_legend_handles_labels()\nax.legend(handles, labels, loc=2)\n\nprint \"%5s %7s %7s\" % (\"ms\", \"r=6\", \"r=20\")\nfor points in range(0, 8001, 500):\n    print \"%5d %7.0f %7.0f\" % (points, map2rating(points,max=6), map2rating(points, max=20))\n\n\n\n",
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       "text": "Using matplotlib backend: MacOSX\n   ms     r=6    r=20"
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       "output_type": "stream",
       "stream": "stdout",
       "text": "\n    0      94       0\n  500     135       0\n 1000     190       1\n 1500     265       3\n 2000     364      13\n 2500     490      45\n 3000     641     151\n 3500     814     445\n 4000    1000    1000\n 4500    1186    1555\n 5000    1359    1849\n 5500    1510    1955\n 6000    1636    1987\n 6500    1735    1997\n 7000    1810    1999\n 7500    1865    2000\n 8000    1906    2000\n"
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rating.py

%matplotlib
"""
in reference to http://programmers.stackexchange.com/questions/225487/mapping-values-over-a-curve/225503#225503
"""
import numpy as np
import matplotlib.pyplot as plt
import math

MIN_RATING=1
MAX_RATING=2000

MIN_POINTS=1
MAX_POINTS=8000

SIGMOID_RANGE=25

def sigmoid(x):
    return 1 / (1 + math.exp(-x))

def map2range(value, low, high, new_low, new_high):
    '''map a value from one range to another'''
    return value * 1.0 / (high - low + 1) * (new_high - new_low + 1)

def map2sigmoid(value, max=SIGMOID_RANGE):
    '''map a value from its rating range to the sigmoid range'''
    return map2range(value, MIN_RATING, MAX_RATING, 1.0, max) - (max / 2.0)  

def map2rating(value, max=SIGMOID_RANGE):
    '''map a points value to a rating value using the sigmoid function'''
    rating_range = map2range(value, MIN_POINTS, MAX_POINTS, MIN_RATING, MAX_RATING)
    sigmoid_range = map2sigmoid(rating_range, max=max)
    s = sigmoid(sigmoid_range) 
    if s < 0.5:
        r = math.floor(s * MAX_RATING)
    else:
        r = math.ceil(s * MAX_RATING)
    return r

ax = plt.subplot(1,1,1)
x = np.arange(MIN_RATING,MAX_RATING,1);
y = [sigmoid(map2sigmoid(y, max=10)) for y in x]
ax.plot(x, y, label="range=10")
y = [sigmoid(map2sigmoid(y, max=20)) for y in x]
ax.plot(x, y, label="range=20")
y = [sigmoid(map2sigmoid(y, max=30)) for y in x]
ax.plot(x, y, label="range=30")

handles, labels = ax.get_legend_handles_labels()
ax.legend(handles, labels, loc=2)

print "%5s %7s %7s" % ("ms", "r=6", "r=20")
for points in range(0, 8001, 500):
    print "%5d %7.0f %7.0f" % (points, map2rating(points,max=6), map2rating(points, max=20))