Frequency analysis | EPFL Graph Search

In cryptanalysis, frequency analysis (also known as counting letters) is the study of the frequency of letters or groups of letters in a ciphertext. The method is used as an aid to breaking classical ciphers. Frequency analysis is based on the fact that, in any given stretch of written language, certain letters and combinations of letters occur with varying frequencies. Moreover, there is a characteristic distribution of letters that is roughly the same for almost all samples of that language. For instance, given a section of English language, , , and are the most common, while , , and are rare. Likewise, , , , and are the most common pairs of letters (termed bigrams or digraphs), and , , , and are the most common repeats. The nonsense phrase "ETAOIN SHRDLU" represents the 12 most frequent letters in typical English language text. In some ciphers, such properties of the natural language plaintext are preserved in the ciphertext, and these patterns have the potential to be exp

Predictive Hydrology: A Frequency Analysis Approach

Anne-Catherine Favre

The unusual frequency of hydro-meteorological events in recent decades, often with catastrophic consequences for society and the environment, require new methods for designing water management projects and the structures meant to protect us from natural hazards. These methods and techniques are often based on the statistical modeling techniques of frequency analysis. Predictive Hydrology: A Frequency Analysis Approach is the first book to address both the theoretical concepts and the methodological approaches used in frequency hydrology—spelling out the fundamental methods to consider, providing concise instruction on the techniques that are involved, and including examples and critiques based on practical applications. It explores some of the recent research developments in the field. Published originally in French, this English translation targets students in civil engineering, environmental sciences and technology, hydrology, geography, geology and ecology. This book will also serve as a useful reference not only for teachers and researchers, but for engineering practitioners, who are constantly faced with the problems of handling data, but often find themselves without the appropriate analytical tools.

Science Publishers, CRC Press2012

Interpreting deep learning models for epileptic seizure detection on EEG signals

David Atienza Alonso, Leila Cammoun, Valentin Alexandre Guy Gabeff, Tomas Teijeiro Campo, Marina Zapater Sancho

While Deep Learning (DL) is often considered the state-of-the art for Artificial Intelligence-based medical decision support, it remains sparsely implemented in clinical practice and poorly trusted by clinicians due to insufficient interpretability of neural network models. We have tackled this issue by developing interpretable DL models in the context of online detection of epileptic seizure, based on EEG signal. This has conditioned the preparation of the input signals, the network architecture, and the post-processing of the output in line with the domain knowledge. Specifically, we focused the discussion on three main aspects: 1) how to aggregate the classification results on signal segments provided by the DL model into a larger time scale, at the seizure-level; 2) what are the relevant frequency patterns learned in the first convolutional layer of different models, and their relation with the delta, theta, alpha, beta and gamma frequency bands on which the visual interpretation of EEG is based; and 3) the identification of the signal waveforms with larger contribution towards the ictal class, according to the activation differences highlighted using the DeepLIFT method. Results show that the kernel size in the first layer determines the interpretability of the extracted features and the sensitivity of the trained models, even though the final performance is very similar after post-processing. Also, we found that amplitude is the main feature leading to an ictal prediction, suggesting that a larger patient population would be required to learn more complex frequency patterns. Still, our methodology was successfully able to generalize patient inter-variability for the majority of the studied population with a classification F1-score of 0.873 and detecting 90% of the seizures.

2021

Frequency analysis of classical Preisach model

Marten Sjöström

The paper contains an analysis of the classical Preisach model from a frequency point of view. A Fourier analysis retrieves the necessary and sufficient conditions on the Preisach weighting function to have an output with half-wave symmetry, i.e. only odd harmonics to be present. For a wide class of weighting functions, it is shown that all odd frequencies are present. As a consequence, a filtering method is suggested that enhances time-series data from sinusoidal input, which can be used for Preisach model identification.

1999

Interpreting deep learning models for epileptic seizure detection on EEG signals

David Atienza Alonso, Leila Cammoun, Valentin Alexandre Guy Gabeff, Tomas Teijeiro Campo, Marina Zapater Sancho

2021