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What is a temporal feature, what features make something temporal in nature? Is this problem agnostic? How does it change from different fields of study?

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In general, the expression "temporal feature" might refer to any feature that is associated with or changes over time.

However, in the context of signal processing, a temporal feature might refer to any feature of the data before being transformed to the Fourier, frequency or spectral domain, using the Fourier transform. In this context, the domain of the untransformed data is often called "time domain" (as opposed to the "frequency" or "spectral" domain, which is the domain of the transformed data), even though it might not be strictly associated with or defined as a function of time. For example, in image processing, an image can be interpreted as a 2D signal. The domain of an image can be referred to as the "time domain", even though it is usually and more correctly referred to as the "spatial domain" (given that an image can be thought of as a function from a pixel, which is defined by two numbers $x$ and $y$, to a value, e.g. a grayscale value). You can transform this image, using the Fourier transform, to the spectral domain. In that case, the domain of the result of the transformation can be referred to as the "spectral domain".

In the paper Learning Temporal Features Using a Deep Neural Network and its Application to Music Genre Classification, the authors define spectral and temporal features

Extracting features from audio that are relevant to the task at hand is a very important step in many music information retrieval (MIR) applications, and the choice of features has a huge impact on the performance. For the past decades, numerous features have been introduced and successfully applied to many different kinds of MIR systems. These audio features can be broadly categorized into two groups: 1) spectral and 2) temporal features.

Spectral features (SFs) represent the spectral characteristics of music in a relatively short period of time. In a musical sense, it can be said to reveal the timbre or tonal characteristics of music. Some of popular SFs include: spectral centroid, spectral spread, spectral flux, spectral flatness measure, mel-frequency cepstral coefficients (MFCCs) and chroma.

On the other hand, temporal features (TFs) describe the relatively long-term dynamics of a music signal over time such as temporal transition or rhythmic characteristics. These include zero-crossing rate (ZCR), temporal envelope, tempo histogram, and so on.

The two groups are not mutually exclusive, however, and many MIR applications use a combination of many different features.

These definitions are given in the context of signal processing (as mentioned above).

To conclude, the meaning of "temporal feature" might change depending on the context. Hence, you should interpret it given the context, but it is almost always associated with time (in some way).

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