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Related Concept Videos

Downsampling01:20

Downsampling

251
When considering a sampled sequence with zero values between sampling instants, one can replace it by taking every N-th value of the sequence. At these integer multiples of N, the original and sampled sequences coincide. This process, known as decimation, involves extracting every N-th sample from a sequence, thereby creating a more efficient sequence.
The Fourier transform of the decimated sequence reveals a combination of scaled and shifted versions of the original spectrum. This...
251
Boundary Conditions: Lossless Lines01:21

Boundary Conditions: Lossless Lines

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Consider a single-phase, two-wire, lossless transmission line terminated by an impedance at the receiving end and a source with Thevenin voltage and impedance at the sending end. The line, with length, has a surge impedance and wave velocity determined by the line's inductance and capacitance.
At the receiving end, the boundary condition states that the voltage equals the product of the receiving-end impedance and current. This relationship is expressed as a function of the incident and...
152
Per-Unit Sequence Models01:26

Per-Unit Sequence Models

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An ideal Y-Y transformer, grounded through neutral impedances, displays per-unit sequence networks akin to those of a single-phase ideal transformer when subjected to balanced positive- or negative-sequence currents. These currents do not produce neutral currents, and their associated voltage drops.
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116
Lossless Lines01:23

Lossless Lines

169
In electrical engineering, a lossless transmission line is characterized by a purely imaginary propagation constant and a resistive characteristic impedance. The ABCD parameters, which describe the relationship between the input and output voltages and currents, indicate an equivalent π circuit with an imaginary series impedance and a shunt admittance. This results in a transmission line that, when the product of the phase constant (beta) and the length of the line is less than pi,...
169
Reducing Line Loss01:18

Reducing Line Loss

193
In a three-phase circuit, line loss is an indicator of energy dissipated as heat due to the resistance of transmission lines. To address this, incorporating transformers into the system—a step-up transformer at the source and a step-down transformer at the load—is a strategic solution. Two three-phase transformers are introduced to improve this.
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Wald-Wolfowitz Runs Test I01:17

Wald-Wolfowitz Runs Test I

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The Wald-Wolfowitz test, also known as the runs test, is a nonparametric statistical test used to assess the randomness of a sequence of two different types of elements (e.g., positive/negative values, successes/failures). It examines whether the order of the elements in a sequence is random or if there is a pattern or trend present. This nonparametric test applies to any ordered data despite the population and sample data distribution, even if a higher sample size is available.
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Empirical Lossless Compression Bound of a Data Sequence.

Lei M Li1,2

  • 1State Key Laboratory of Mathematical Science, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China.

Entropy (Basel, Switzerland)
|August 28, 2025
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Summary
This summary is machine-generated.

This study explores lossless data compression bounds, introducing the Normalized Maximum Likelihood (NML) distribution for optimal minimax compression. The research derives a precise formula for NML code length, applicable to both discrete and continuous data, and validates it with DNA sequence compression. This work advances understanding of data compression limits and practical applications in bioinformatics.

Keywords:
BayesianDNAentropylocal asymptotic normalitylossless compressionnormalized maximum likelihoodpredictive

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Area of Science:

  • Information Theory
  • Data Compression
  • Statistical Inference

Background:

  • Kolmogorov complexity provides an uncomputable theoretical bound for lossless data compression.
  • Shannon's source coding theorem establishes the average compression bound as nH, where n is sequence length and H is entropy.
  • Maximum likelihood estimation (MLE) often underestimates the true compression bound.

Purpose of the Study:

  • To derive and analyze the lossless compression bound for individual data sequences.
  • To investigate the optimality of the Normalized Maximum Likelihood (NML) distribution for data compression.
  • To extend compression bound calculations to both discrete and continuous data, with applications in bioinformatics.

Main Methods:

  • Utilizing the Normalized Maximum Likelihood (NML) distribution, which is shown to be optimal in a minimax sense.
  • Applying local asymptotic normality to derive the asymptotic code length for NML.
  • Developing a Bayesian approach to predict optimal code length, leading to a mixture code.
  • Calculating compression bounds for protein-encoding DNA sequences using different parsing models.

Main Results:

  • The NML code length is analytically derived as nH(θ^n) + (d/2)log(n/2π) + log∫|I(θ)|^(1/2)dθ + o(1).
  • A mixture code, derived via Bayesian prediction, has a length of nH(θ^n) + (d/2)log(n/2π) + log|I(θ^n)|^(1/2)w(θ^n) + o(1).
  • Compression of DNA sequences is maximized when parsing aligns with amino acid codons, demonstrating practical application.
  • Empirical compression bounds improve with increasing dictionary size.

Conclusions:

  • The NML distribution provides an optimal, computable approach to data compression bounds.
  • The derived asymptotic formulas offer accurate estimations for both discrete and continuous data compression.
  • Parsing strategies significantly impact compression efficiency, particularly for biological sequences.
  • The study provides a robust framework for understanding and calculating theoretical compression limits in various data types.