infoeqv 2.0.0

Information Equivalent Designs (Two-Point Reduction)

The Information Equivalent Two-Point Design (or information-preserving reduction) is a concept from optimal experimental design and information theory in statistics, especially in design of experiments (DoE). Here's a deeper explanation of the theory behind your function and where it fits in:

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THEORY: Information Equivalent Designs (Two-Point Reduction)

Objective:

To replace a complex dataset with a simplified two-point design that preserves the same information about a model parameter, usually the mean or location parameter, under a linear or polynomial regression framework.

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💡 Context:

Suppose you have a weighted dataset (x_i, f_i) where:

• x_i: design points (levels of a factor)
• f_i: frequencies or weights (how often that point appears)

Rather than keeping all these points, we want to approximate this design with only two support points, say:

• x_A with weight r
• x_B with weight N - r

Such that the statistical information about the mean or some parameter is unchanged.

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Key Concepts Involved:

1. Information Matrix

In linear regression y = \beta x + \epsilon, the information matrix of the design is:

$$ I = \sum f_i (x_i - \bar{x})^2 $$

We aim to find two new values x_A and x_B, with appropriate frequencies r and N - r, such that the information matrix is the same.

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2. Standardized Design

You normalize the data to make calculations scale-invariant:

$$ d_i = \frac{x_i - \bar{x}}{\max(x) - \bar{x}} $$

This allows focusing only on the relative spread and centrality.

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3. Moments of the Design

• $$\mu_1 = \frac{1}{N} \sum f_i d_i$$ : first central moment (mean)
• $$\mu_2 = \frac{1}{N} \sum f_i d_i^2$$ : second raw moment
• $$$mu_{22} = \mu_2 - \mu_1^2$$ : central second moment (variance)

These determine the shape of the distribution.

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4. Finding Bounds (L and U)

You compute bounds $L$ and $U$ for feasible integer allocations of weight $r$ that will preserve the original information:

$$ L = \frac{N \cdot \mu_{22}}{(1 + \mu_1)^2 + \mu_{22}} \ U = \frac{N \cdot (1 - \mu_1)^2}{(1 - \mu_1)^2 + \mu_{22}} $$

Only if U - L > 1, do feasible two-point equivalents exist.

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5. Finding Equivalent Points

For each valid $$r \in [\lceil L \rceil, \lfloor U \rfloor]$$, calculate:

• Two symmetric points around the mean that match the original design's spread:

$$ x_A = \mu_1 - \sqrt{\frac{N - r}{r} \cdot \mu_{22}} \ x_B = \mu_1 + \sqrt{\frac{r}{N - r} \cdot \mu_{22}} $$

These will become the new support points with weights r and N - r

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Applications:

• Reducing design complexity in regression without losing information.
• Simplifying optimal designs in statistics and engineering.
• Used in experimental planning when cost or resources are limited.
• Sometimes used in machine learning for data reduction or kernel approximation.

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Example in Real Life:

If you’re testing 6 drug doses with varying frequencies and want to reduce the testing to just two key doses that statistically represent the same variability and effect, this method gives you the way to choose those two doses.

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📊 Information Equivalent Design (Two-Point Reduction)

This Python package provides a method for reducing an original frequency distribution to a two-point design with equivalent information. The approach is based on preserving the first and second central moments, ensuring statistical equivalence in terms of spread and location.

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📦 Installation

To install the package, use:

pip install infoeqv

Or, if you have the source code locally:

pip install .

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🧠 Theory

Given a frequency distribution ((x, f)), the goal is to reduce the distribution into a two-point support design that retains its information structure.

We compute:

• ( \mu_1 ): weighted mean of the normalized values
• ( \mu_2 ): second raw moment
• ( \mu_{22} = \mu_2 - \mu_1^2 ): second central moment
• ( N ): total frequency

Then calculate:

[ L = \frac{N \cdot \mu_{22}}{(1 + \mu_1)^2 + \mu_{22}}, \quad U = \frac{N \cdot (1 - \mu_1)^2}{(1 - \mu_1)^2 + \mu_{22}} ]

Only if ( U - L > 1 ), feasible two-point equivalents exist.

Then for each ( r \in [ \lceil L \rceil , \lfloor U \rfloor ] ), calculate:

[ x_A = \mu_1 - \sqrt{ \frac{N - r}{r} \cdot \mu_{22} }, \quad x_B = \mu_1 + \sqrt{ \frac{r}{N - r} \cdot \mu_{22} } ]

These ( x_A ) and ( x_B ) are the new support points with weights ( r ) and ( N - r ) respectively.

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Usage

from infeqvdesign import info_eqv_design

x = [1, 2, 3, 4, 5, 6]
f = [6, 5, 4, 3, 2, 1]

info_eqv_design(x, f)

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Example Output

[[ 5.  16.  1.49  4.93]
 [ 6.  15.  1.33  5.09]
 [ 7.  14.  1.18  5.24]
 [ 8.  13.  1.04  5.38]
 [ 9.  12.  0.91  5.51]
 [10.  11.  0.79  5.63]]

Each row contains:

r	N - r	x_A	x_B
weight 1	weight 2	equivalent point A	equivalent point B

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Requirements

• Python 3.6+
• NumPy

Install dependencies manually with:

pip install numpy

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Author

• Rohit Kumar Behera
• GitHub: @muinrohit

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