Which ONE of the following fractions is used to define the inverse document frequency (idf) of the term $t$?
Understanding Inverse Document Frequency (IDF)
The Inverse Document Frequency (IDF) measures how important a term is in a collection of documents. It diminishes the weight of terms that occur very frequently across documents, such as common words ('the', 'is', 'a'), and increases the weight of terms that occur rarely.
IDF Formula Derivation
Let:
- $N$ be the total number of documents in the collection.
- $d$ be the number of documents where the specific term $t$ appears (document frequency, often denoted as $df_t$).
The core component of the IDF calculation involves the ratio of the total documents to the documents containing the term. While IDF is typically calculated using a logarithmic function (e.g., $\log(\frac{N}{d})$ or $\log(\frac{N}{d+1})$ to avoid division by zero if a term doesn't appear), the question asks for the specific fraction used in its definition, representing the inverse relationship between document frequency and importance.
This fraction represents how many documents are 'missed' by the term. A higher value indicates a rarer term.
Selecting the Correct Fraction
Based on the definition and common formulas for IDF, the fraction representing the relationship between total documents ($N$) and the documents containing the term ($d$) is:
$ \frac{N}{d} $
Comparing this with the given options:
- Option 1: $\frac{N}{d}$ - Matches the required fraction.
- Option 2: $\frac{1}{N+d}$ - Incorrect structure.
- Option 3: $\frac{1}{N-d}$ - Incorrect structure.
- Option 4: $\frac{1}{Nd}$ - Incorrect structure.
Therefore, the fraction used to define the inverse document frequency (idf) of the term $t$ is $\frac{N}{d}$.
Important Questions from Numerical Computation
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If the in-situ density of coal is 1320 kg/m$^3$ and the density of blasted coal is 952 kg/m$^3$, the swell factor is _____________ (rounded off to 3 decimal places)
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The Fourier transform and its inverse transform are respectively defined as $\tilde{f}(\omega) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{+\infty} f(x)e^{i\omega x}dx$ and $f(x) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{+\infty} \tilde{f}(\omega)e^{-i\omega x}d\omega$. Consider two functions $f$ and $g$. Another function $f * g$ is defined as
$(f * g)(x) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{+\infty} f(y)g(x - y)dy$
Which of the following relation is/are true?
Note: Tilde ($\sim$) denotes the Fourier transform.
