Sum of Absolute Differences (SAD) Calculator

Online calculator to compute the Sum of Absolute Differences (SAD)

SAD Calculator

The Sum of Absolute Differences

The SAD is an important distance function that measures the deviation between two data series.

Enter Data
First data series (space or semicolon separated)
Second data series (space or semicolon separated)
SAD Results
Deviation:
SAD Properties

Range: SAD is always ≥0 and increases with the deviation between series

SAD ≥0 Additive property L1-norm

SAD Concept

The SAD sums the absolute differences of all corresponding values.
The larger the differences, the higher the SAD.

Values Index

● Series X ● Series Y ⋯ Absolute differences


What is the Sum of Absolute Differences (SAD)?

The Sum of Absolute Differences (SAD) is a fundamental distance measure in statistics:

  • Definition: Sums the absolute values of pairwise differences between two data series
  • Range: Values starting at0, where0 indicates identical series
  • Property: L1-norm or Manhattan distance in multidimensional space
  • Application: Image processing, signal analysis, quality control
  • Interpretation: Direct measure of cumulative deviation
  • Related to: Mean Absolute Error (MAE), Chebyshev distance

Properties of SAD as a Distance Function

The SAD satisfies important properties of a distance function:

Mathematical properties
  • Non-negativity: SAD(x,y) ≥0
  • Identity: SAD(x,x) =0
  • Symmetry: SAD(x,y) = SAD(y,x)
  • Triangle inequality: SAD(x,z) ≤ SAD(x,y) + SAD(y,z)
Practical properties
  • Robustness: Less sensitive to outliers than quadratic measures
  • Linearity: Proportional to the magnitude of deviations
  • Additivity: Sum of individual deviations
  • Scaling: Changes proportionally with the data

Applications of the Sum of Absolute Differences

The SAD is used in many practical areas:

Image Processing & Computer Vision
  • Block-matching in video compression (H.264, MPEG)
  • Motion estimation in video sequences
  • Template matching and object detection
  • Stereo vision and disparity measurement
Signal Processing & Acoustics
  • Audio signal comparison and quality assessment
  • Speech recognition and synthesis
  • Noise reduction and filter design
  • Frequency analysis and spectral comparisons
Statistics & Data Analysis
  • Robust regression and outlier detection
  • Time series analysis and trend comparison
  • Clustering algorithms (k-medoids)
  • Production quality control
Machine Learning & AI
  • Loss function for robust models
  • Feature matching and similarity measurement
  • Anomaly detection in sensor data
  • Optimization of neural networks

Formulas for the Sum of Absolute Differences (SAD)

Basic Formula
\[SAD = \sum_{i=1}^{n} |x_i - y_i|\]

Sum of the absolute values of all pairwise differences

L1-norm Representation
\[SAD = \|x - y\|_1\]

Equivalent representation as the L1-norm of the difference vector

Manhattan Distance
\[d_{Manhattan}(x,y) = \sum_{i=1}^{n} |x_i - y_i|\]

Also known as Manhattan- or Taxicab-distance

Normalized SAD
\[SAD_{norm} = \frac{1}{n} \sum_{i=1}^{n} |x_i - y_i|\]

Mean Absolute Error (MAE)

General Minkowski Distance
\[L_p(x,y) = \left(\sum_{i=1}^{n} |x_i - y_i|^p\right)^{1/p}\]

For p =1 you get SAD (L1-norm), for p =2 the Euclidean distance

Example Calculation for SAD

Given
x = [1,2,3,4,5] y = [3,5,6,7,3]

Calculate: Sum of absolute differences between series x and y

1. Pairwise differences
\[x_1 - y_1 =1 -3 = -2\] \[x_2 - y_2 =2 -5 = -3\] \[x_3 - y_3 =3 -6 = -3\] \[x_4 - y_4 =4 -7 = -3\] \[x_5 - y_5 =5 -3 =2\]

Compute all differences x_i - y_i

2. Absolute values
\[|x_1 - y_1| = |-2| =2\] \[|x_2 - y_2| = |-3| =3\] \[|x_3 - y_3| = |-3| =3\] \[|x_4 - y_4| = |-3| =3\] \[|x_5 - y_5| = |2| =2\]

Take absolute values of all differences

3. Summation
\[SAD =2 +3 +3 +3 +2\] \[SAD =13\]

Sum all absolute differences

4. Interpretation
Total deviation:13
Mean deviation:2.6

The cumulative absolute difference equals13 units

5. Full calculation
\[SAD = |1-3| + |2-5| + |3-6| + |4-7| + |5-3|\] \[SAD =2 +3 +3 +3 +2 =13\]

The sum of absolute differences between the two series equals13

Mathematical Foundations of SAD

The Sum of Absolute Differences (SAD) is a fundamental concept in distance measurement and belongs to the family of Minkowski distances. It represents the L1-norm in multidimensional space and has important properties for robust statistical methods.

Theoretical foundations

SAD is based on the L1-norm and has important mathematical properties:

  • Metric properties: SAD satisfies the axioms of a metric (non-negativity, identity, symmetry, triangle inequality)
  • Convexity: As an L1-norm, SAD is convex, which is advantageous for optimization problems
  • Robustness: Less sensitive to outliers than quadratic distance measures (L2-norm)
  • Continuity: SAD is a continuous function of its arguments
  • Homogeneity: SAD(kx, ky) = |k| × SAD(x, y) for any scalar k

Comparison with other distance measures

SAD is related to various other distance and similarity measures:

Euclidean distance (L2)

The Euclidean distance \(\sqrt{\sum_{i=1}^{n} (x_i - y_i)^2}\) weights large deviations more than the SAD.

Chebyshev distance (L∞)

The maximum norm \(\max_i |x_i - y_i|\) considers only the largest single deviation.

Hamming distance

For binary vectors, Hamming distance counts the number of differing positions.

Minkowski distances

SAD is a special case of the Minkowski distance with parameter p =1.

Applications and variants

SAD appears in various practical forms:

Image processing

In motion estimation and video compression, SAD is used to evaluate block similarity. Algorithms like H.264 use SAD-based techniques for efficient compression.

Optimization

Convexity makes SAD a popular objective in robust regression and Lasso-type methods in statistics.

Clustering

k-medoids algorithms use SAD as a distance measure because it is more robust to outliers than k-means with Euclidean distance.

Time series analysis

SAD offers a robust measure of similarity between different time series.

Advantages and disadvantages

SAD as a distance measure has specific characteristics:

Advantages
  • Robustness: Less sensitive to outliers than quadratic measures
  • Simplicity: Intuitive interpretation and simple computation
  • Convexity: Favorable for optimization procedures
  • Universality: Applicable to various data types
  • Efficiency: Fast computation without complex operations
Disadvantages
  • Sensitivity: Reacts to all deviations, even small ones
  • Dimensionality: May be less informative in high-dimensional data
  • Scaling: Dependent on the unit and scale of the data
  • Equal weighting: Treats all dimensions equally
  • Non-differentiability: Not differentiable at0

Practical considerations

Data preprocessing

Normalization and standardization can significantly improve the usefulness of SAD, especially with differing scales.

Implementation

Efficient implementations use vectorized operations and can be further optimized via parallelization.

Summary

The Sum of Absolute Differences is a versatile and robust distance measure with broad applications in science and engineering. Its mathematical properties make it a valuable tool for similarity analysis, particularly when robustness to outliers is more important than emphasizing large deviations. The choice between SAD and other distance measures should be made in the context of the specific application and desired properties.

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More Statistics Functions

Dice Coefficient  •  Hellinger Distance  •  Jaccard Index  •  MAE - mean absolute error  •  MSE - mean squared error  •  SAD - sum of absolute difference  •  SSD - sum of squared difference  •