Design꞉ Habit Score

This wiki was written based on version 1.72.2+95. Any subsequent changes are subject to the in-app behavior.

1. Formula

The habit growth curve is based on a Sigmoid function.

1.1. Input mapping

For a day x in [0, days], map it to [-10, 10]:

$$ X = (b_2 - b_1) \frac{x - a_1}{a_2 - a_1} + b_1 $$

Where:

• $a_1 = 0$, $a_2 = \text{days}$
• $b_1 = -10$, $b_2 = 10$

So explicitly:

$$ X = 20 \cdot \frac{x}{\text{days}} - 10 $$

1.2. Core function

$$ Y = \frac{1}{1 + e^{-k (X - x_0)}} $$

Where:

• $k = 0.4$ (growth rate)
• $x_0 = 0$ (center point)

1.3. Output mapping (optional)

If the output should be mapped from an interval $[c_1, c_2]$ to $[d_1, d_2]$:

$$ y = (d_2 - d_1) \frac{y - c_1}{c_2 - c_1} + d_1 $$

Where:

• $[c_1, c_2] = \text{interv}$
• $[d_1, d_2] = _defaultInterv$ (Default: [0, 100])

2. Inverse Formula

This function calculates the day x corresponding to a given habit score y, i.e., the inverse of the habit growth curve.

2.1. Output mapping (inverse of interval mapping)

First, map the output y from an interval $[d_1, d_2]$ back to $[c_1, c_2]$:

$$ Y = (c_2 - c_1) \frac{y - d_1}{d_2 - d_1} + c_1 $$

Where:

• $[c_1, c_2] = \text{interv}$
• $[d_1, d_2] = _defaultInterv$ (Default: [0, 100])

2.2. Inversed Core

The Sigmoid inverse (logit function) is applied:

$$ X = x_0 - \frac{1}{k} \ln \left( \frac{1}{Y} - 1 \right) $$

Where:

• $k = 0.4$ (growth rate)
• $x_0 = 0$ (center point)

2.3. Input mapping (back to days)

Finally, map X from [-10, 10] back to [0, days]:

$$ x = (a_2 - a_1) \frac{X - b_1}{b_2 - b_1} + a_1 $$

Where:

• $[a_1, a_2] = [0, \text{days}]$
• $[b_1, b_2] = [-10, 10]$

So explicitly:

$$ x = \text{days} \cdot \frac{X + 10}{20} $$

3. Habit Growth

• Target Days: Minimum number of days required to complete a habit.
• Daily Goal: Minimum quantity required for single day.
• Incomplete: Daily Goal completely not completed; in positive habits, this is usually 0.
• Partially Completed: Daily Goal not completed; for positive habits, the value is usually with 0 < num < Daily Goal.
• Overachieved: The final number exceeds the value set for the Daily Goal, usually with num > Daily Goal
• Overage Threshold: Maximum value the number can increase after exceeding the Daily Goal

3.1. Growth Score

For a positive habit, when the $DailyNumber ≥ DailyGoal$, an increase of $[1, 1.5]$ is applied relative to the corresponding days of the Target Days, then Formula converts it into the final score.

For a negative habit, simply replace $≥ DailyGoal$ with:

$$ 0 ≤ DailyNumber ≤ DailyGoal $$

3.2. Decline Score

For a positive habit, when the $0 ≤ DailyNumber ≤ DailyGoal$, a reduction of $[100 ÷ TargetDays ÷ 2, 100 ÷ TargetDays]$ in score is applied, which can be used with the Inverse Formula to reverse-calculate the corresponding new total days.

For a negative habit, simply replace $0 ≤ DailyNumber ≤ DailyGoal$ with:

$$ DailyGoal < DailyNumber ≤ OverageThreshold ... $$

When the number exceeds the Overage Threshold, take the maximum value ($100 ÷ TargetDays$).

4. Example