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name: grey-relation description: Grey Relational Analysis for MCM/ICM competitions. Use for small-sample correlation analysis and factor importance ranking. Does not require large samples, normal distribution, or linear relationships. Ideal for analyzing which factors most influence outcomes when data is limited.

Grey Relational Analysis (GRA)

Analyze correlation between factors and outcomes using grey relational grade, especially for small samples.

When to Use

• Small sample size: 4-20 data points (unlike correlation which needs >30)
• Factor analysis: Which factors most influence the outcome?
• Non-linear relationships: No assumption of linearity
• Multi-factor ranking: Rank importance of multiple influencing factors
• Incomplete information: "Grey" = between black (unknown) and white (known)

Key difference from correlation coefficient:

• No requirement for large samples
• No assumption of linear relationship
• No requirement for normal distribution

This is NOT forecasting: Use grey-forecaster for prediction, use grey-relation for correlation analysis.

Quick Start

Core Concepts

1. Reference sequence (母序列) $X_0$: The outcome/target variable

   • Example: Student test scores

2. Comparison sequences (子序列) $X_i$: The influencing factors

   • Example: Study hours, sleep hours, exercise hours

3. Grey relational coefficient $\xi_i(k)$: Similarity at each data point $$\xi_i(k) = \frac{\min_i \min_k |x_0(k) - x_i(k)| + \rho \max_i \max_k |x_0(k) - x_i(k)|}{|x_0(k) - x_i(k)| + \rho \max_i \max_k |x_0(k) - x_i(k)|}$$ where $\rho = 0.5$ (resolution coefficient, usually 0.5)

4. Grey relational grade $\gamma_i$: Overall correlation (average of coefficients) $$\gamma_i = \frac{1}{n} \sum_{k=1}^n \xi_i(k)$$

5. Ranking: Higher $\gamma_i$ means stronger correlation

Python Implementation

import numpy as np

# Example: Which factor most affects test scores?
# Columns: [Test Score, Study Hours, Sleep Hours, Exercise Hours]
data = np.array([
    [55, 24, 10, 5],
    [65, 38, 22, 8],
    [75, 40, 18, 6],
    [100, 50, 20, 10]
])

# Step 1: Normalization (dimensionless)
mean = np.mean(data, axis=0)
data_norm = data / mean

# Step 2: Reference sequence (outcome)
Y = data_norm[:, 0]  # Test scores

# Step 3: Comparison sequences (factors)
X = data_norm[:, 1:]  # Study, Sleep, Exercise

# Step 4: Calculate absolute differences |X0 - Xi|
abs_diff = np.abs(X - Y.reshape(-1, 1))

# Step 5: Find min and max differences
a = np.min(abs_diff)  # Two-level minimum
b = np.max(abs_diff)  # Two-level maximum

# Step 6: Resolution coefficient
rho = 0.5

# Step 7: Grey relational coefficient
gamma_matrix = (a + rho * b) / (abs_diff + rho * b)

# Step 8: Grey relational grade (average)
gamma = np.mean(gamma_matrix, axis=0)

# Step 9: Rank factors
factors = ['Study Hours', 'Sleep Hours', 'Exercise Hours']
ranking = sorted(zip(factors, gamma), key=lambda x: x[1], reverse=True)

print("Grey Relational Grade:")
for factor, grade in ranking:
    print(f"{factor}: {grade:.4f}")

print(f"\nMost important factor: {ranking[0][0]}")

MATLAB Implementation

% Data: [Test Score, Study Hours, Sleep Hours, Exercise Hours]
A = [55, 24, 10, 5;
     65, 38, 22, 8;
     75, 40, 18, 6;
     100, 50, 20, 10];

% Step 1: Normalization
Mean = mean(A, 1);
A_norm = A ./ Mean;

% Step 2: Reference sequence
Y = A_norm(:, 1);

% Step 3: Comparison sequences
X = A_norm(:, 2:end);

% Step 4: Absolute differences
absX0_Xi = abs(X - repmat(Y, 1, size(X, 2)));

% Step 5: Min and max
a = min(absX0_Xi(:));
b = max(absX0_Xi(:));

% Step 6: Resolution coefficient
rho = 0.5;

% Step 7: Grey relational coefficient
gamma_matrix = (a + rho * b) ./ (absX0_Xi + rho * b);

% Step 8: Grey relational grade
gamma = mean(gamma_matrix, 1);

% Step 9: Display results
factors = {'Study Hours', 'Sleep Hours', 'Exercise Hours'};
fprintf('Grey Relational Grade:\n');
for i = 1:length(factors)
    fprintf('%s: %.4f\n', factors{i}, gamma(i));
end

[~, idx] = max(gamma);
fprintf('\nMost important factor: %s\n', factors{idx});

Standard Procedure

Step 1: Data Preparation

Organize data into matrix:

• Rows: Samples (observations)
• Columns: [Outcome, Factor1, Factor2, ..., FactorN]

Step 2: Normalization (Dimensionless)

Three common methods:

1. Mean normalization (most common): $$x_i'(k) = \frac{x_i(k)}{\bar{x_i}}$$

2. Min-max normalization: $$x_i'(k) = \frac{x_i(k) - \min x_i}{\max x_i - \min x_i}$$

3. Initial value normalization: $$x_i'(k) = \frac{x_i(k)}{x_i(1)}$$

Step 3: Separate Reference and Comparison

• Reference sequence $X_0$: The outcome variable
• Comparison sequences $X_1, X_2, ..., X_m$: The factors

Step 4: Calculate Differences

$$\Delta_i(k) = |x_0(k) - x_i(k)|$$

Step 5: Find Two-Level Extrema

$$a = \min_i \min_k \Delta_i(k)$$ $$b = \max_i \max_k \Delta_i(k)$$

Step 6: Calculate Grey Relational Coefficient

$$\xi_i(k) = \frac{a + \rho b}{\Delta_i(k) + \rho b}$$

where $\rho \in [0, 1]$ is resolution coefficient (usually 0.5)

Step 7: Calculate Grey Relational Grade

$$\gamma_i = \frac{1}{n} \sum_{k=1}^n \xi_i(k)$$

Step 8: Rank Factors

Sort factors by $\gamma_i$ in descending order.

Examples in Skill

See examples/ folder:

• code1.py / code1.m: Basic grey relational analysis with normalization
• code2.py / code2.m: Alternative implementation
• Inter2Max.m, Mid2Max.m, Min2Max.m: Data preprocessing functions (convert to "larger is better")
• Positivization.m: Positive transformation for indicators

Competition Tips

Problem Recognition (Day 1)

Grey relational analysis indicators:

• "Which factors most influence..."
• "Rank importance of factors"
• Small sample size (< 30 observations)
• No clear linear relationship
• Multiple factors, need to find key drivers

Use GRA when:

• Sample size is small (4-20 points)
• Don't know if relationship is linear
• Want to rank factor importance
• Data doesn't meet assumptions for regression/correlation

Formulation Steps (Day 1-2)

1. Define outcome variable: What are you trying to explain?
2. Identify factors: What might influence the outcome?
3. Collect data: Organize into matrix
4. Normalize: Make data dimensionless
5. Compute GRA: Follow 8-step procedure
6. Interpret: Rank factors by grey relational grade

Implementation Template

import numpy as np

# 1. Prepare data (rows=samples, cols=[outcome, factor1, factor2, ...])
data = np.array([
    [y1, x1_1, x1_2, ...],
    [y2, x2_1, x2_2, ...],
    # ...
])

# 2. Normalize
mean = np.mean(data, axis=0)
data_norm = data / mean

# 3. Separate reference and comparison
Y = data_norm[:, 0]
X = data_norm[:, 1:]

# 4. Calculate differences
abs_diff = np.abs(X - Y.reshape(-1, 1))

# 5. Two-level extrema
a = np.min(abs_diff)
b = np.max(abs_diff)

# 6. Grey relational coefficient
rho = 0.5
gamma_matrix = (a + rho * b) / (abs_diff + rho * b)

# 7. Grey relational grade
gamma = np.mean(gamma_matrix, axis=0)

# 8. Rank and display
factors = ['Factor1', 'Factor2', 'Factor3']
for factor, grade in sorted(zip(factors, gamma), 
                            key=lambda x: x[1], reverse=True):
    print(f"{factor}: {grade:.4f}")

Validation (Day 3)

• Check normalization: All sequences should be dimensionless
• Verify ranking: Does it match intuition/domain knowledge?
• Sensitivity to $\rho$: Try $\rho = 0.3, 0.5, 0.7$ - ranking should be stable
• Compare with correlation: For validation, compute Pearson correlation

Common Pitfalls

1. Confusing with grey forecasting: GRA is for correlation, not prediction

   • Fix: Use grey-forecaster for time series prediction

2. Too large sample: If n > 50, use standard correlation/regression

   • GRA advantage is for small samples

3. Forgot normalization: Different units make comparison meaningless

   • Fix: Always normalize first

4. Wrong reference sequence: Used factor as reference instead of outcome

   • Fix: Reference = outcome variable, comparison = factors

5. Over-interpreting: High $\gamma$ doesn't mean causation

   • GRA shows correlation, not causation

Advanced: Data Preprocessing

Indicator Direction Transformation

Convert all indicators to "larger is better":

1. Cost-type (smaller is better) → Benefit-type: $$x_i'(k) = \max x_i - x_i(k)$$

2. Interval-type (optimal range [a, b]): $$x_i'(k) = \begin{cases} 1 - \frac{a - x_i(k)}{M}, & x_i(k) < a \ 1, & a \leq x_i(k) \leq b \ 1 - \frac{x_i(k) - b}{M}, & x_i(k) > b \end{cases}$$

where $M = \max{a - \min x_i, \max x_i - b}$

Comparison with Other Methods

Method	Sample Size	Assumptions	When to Use
Grey Relational Analysis	Small (4-20)	None	Small sample, non-linear
Pearson Correlation	Large (>30)	Linear, normal	Large sample, linear
Spearman Correlation	Medium (>10)	Monotonic	Non-linear but monotonic
Regression	Large (>30)	Linear, normal	Prediction, causation

References

• references/5-灰色关联分析【微信公众号丨B站：数模加油站】.pdf: Complete tutorial
• Deng, J. (1982) "Control problems of grey systems"

Time Budget

• Data preparation: 15-30 min
• Normalization: 10-15 min
• Implementation: 15-25 min
• Validation: 20-30 min
• Total: 1-1.5 hours

Related Skills

• grey-forecaster: Grey prediction (GM(1,1)) for time series
• pca-analyzer: Alternative for factor analysis (large samples)
• entropy-weight-method: Objective weighting based on data variation
• fuzzy-evaluation: For qualitative factor evaluation