Bray-Curtis Distance

Calculator to compute the Bray-Curtis distance with detailed formulas and examples

Bray-Curtis Distance Calculator

What is calculated?

The Bray-Curtis distance is a measure of dissimilarity between two vectors. It normalizes the Manhattan distance by the sum of both vectors and is frequently used in ecology and bioinformatics.

Input vectors

Values separated by spaces

Same number of values as Vector X

Result
Bray-Curtis distance:
Values range from 0 (identical) to 1 (completely different)

Bray-Curtis Info

Properties

Bray-Curtis distance:

  • Range: [0, 1]
  • 0 = identical vectors
  • 1 = completely different vectors
  • Normalized Manhattan distance

Applications: Widely used in ecology for community comparison and in bioinformatics for gene expression analysis.

Special cases
Identical vectors:
BC([1,2,3], [1,2,3]) = 0
Zero vector:
BC([0,0,0], [1,2,3]) = 1
Proportional vectors:
BC([1,2], [2,4]) = 0
Related distances

→ Manhattan distance
→ Canberra distance
→ Cosine similarity


Formulas for Bray-Curtis distance

Basic formula
\[BC(x,y) = \frac{\sum_{i=1}^n |x_i - y_i|}{\sum_{i=1}^n (x_i + y_i)}\] Standard Bray-Curtis distance
Alternative notation
\[BC(x,y) = \frac{d_{Manhattan}(x,y)}{||x||_1 + ||y||_1}\] Normalized Manhattan distance
Similarity index
\[S_{BC}(x,y) = 1 - BC(x,y)\] Bray-Curtis similarity
Sørensen-Dice relation
\[BC = 1 - \frac{2\sum \min(x_i, y_i)}{\sum (x_i + y_i)}\] Related to Sørensen-Dice
Range
\[0 \leq BC(x,y) \leq 1\] Normalized distance
Symmetry
\[BC(x,y) = BC(y,x)\] Symmetric property

Detailed calculation example

Example: BC([0,3,4,5], [7,6,3,-1])

Given:

  • x = [0, 3, 4, 5]
  • y = [7, 6, 3, -1]

Step 1 - Differences:

\[|0-7| + |3-6| + |4-3| + |5-(-1)|\] \[= 7 + 3 + 1 + 6 = 17\]

Step 2 - Sums:

\[\sum x_i = 0+3+4+5 = 12\] \[\sum y_i = 7+6+3+(-1) = 15\] \[\sum(x_i + y_i) = 27\]

Step 3 - Final result:

\[BC = \frac{17}{27} = 0.6296\]

Interpretation: The vectors are about 63% different (relatively high dissimilarity).

Ecological example

Example: comparing species communities

Site A:

Oak: 20 individuals
Beech: 15 individuals
Spruce: 5 individuals
Pine: 10 individuals

Site B:

Oak: 10 individuals
Beech: 25 individuals
Spruce: 8 individuals
Pine: 7 individuals

Calculation:

\[BC = \frac{|20-10|+|15-25|+|5-8|+|10-7|}{(20+10)+(15+25)+(5+8)+(10+7)} = \frac{26}{100} = 0.26\]

Result: The sites have a relatively similar species composition (BC = 0.26).

Comparison with other distance measures

For vectors [1,2,3] and [2,4,6]
Bray-Curtis
0.000

Proportional vectors = identical

Euclidean
3.742

Accounts for magnitude differences

Manhattan
9.000

Absolute differences

Cosine
0.000

Direction-based

Note: Bray-Curtis and Cosine consider proportional vectors identical, while Euclidean and Manhattan consider magnitude differences.

Mathematical properties

Metric properties
  • Non-negativity: BC(x,y) ≥ 0
  • Symmetry: BC(x,y) = BC(y,x)
  • Identity: BC(x,x) = 0
  • Triangle inequality: Not always satisfied
Special properties
  • Normalization: Range [0,1]
  • Scale-invariant: For proportional vectors
  • Robust: Against outliers in the sum
  • Interpretable: As proportion of dissimilarity
Important notes

Sensitivity: Can be unstable with many zeros

Alternative: Use Canberra distance for sparse data

Practical applications

Ecology
  • Comparing species communities
  • Biodiversity analyses
  • Habitat similarity
  • Vegetation comparisons
Bioinformatics
  • Gene expression comparisons
  • Microbiome analyses
  • Phylogenetic distances
  • Protein sequence comparisons
Data science
  • Document similarity
  • Recommendation systems
  • Clustering algorithms
  • Anomaly detection




Distance functions

Bray Curtis Distance  •  Canberra Distance  •  Chebyshev Distance  •  Cosine Similarity  •  Euclidean  •  Levenshtein Distance  •  Manhattan Distance  •  Matching Distance  •  Minkowski Distance  •  Pearson Correlation Coefficient  •