disp_DA_tdm: Calculate the dispersion measure D_{A} for a term-document...
In tlda: Tools for Language Data Analysis
disp_DA_tdm R Documentation
Calculate the dispersion measure D_{A} for a term-document matrix
Description
This function calculates the dispersion measure D_{A}. It offers two different computational procedures, the basic version as well as a computational shortcut. It also allows the user to choose the directionality of scaling, i.e. whether higher values denote a more even or a less even distribution. It also provides the option of calculating frequency-adjusted dispersion scores.
Usage
disp_DA_tdm(
tdm,
row_partsize = "first",
directionality = "conventional",
procedure = "basic",
freq_adjust = FALSE,
freq_adjust_method = "even",
unit_interval = TRUE,
digits = NULL,
verbose = TRUE,
print_scores = TRUE
)
Arguments
tdm
A term-document matrix, where rows represent items and columns represent corpus parts; must also contain a row giving the size of the corpus parts (first or last row in the term-document matrix)
row_partsize
Character string indicating which row in the term-document matrix contains the size of the corpus parts. Possible values are "first" (default) and "last"
directionality
Character string indicating the directionality of scaling. See details below. Possible values are "conventional" (default) and "gries"
procedure
Character string indicating which procedure to use for the calculation of D_{A}. See details below. Possible values are 'basic' (default), 'shortcut'.
freq_adjust
Logical. Whether dispersion score should be adjusted for frequency (i.e. whether frequency should be 'partialed out'); default is FALSE
freq_adjust_method
Character string indicating which method to use for devising dispersion extremes. See details below. Possible values are "even" (default) and "pervasive"
unit_interval
Logical. Whether frequency-adjusted scores that exceed the limits of the unit interval should be replaced by 0 and 1; default is TRUE
digits
Rounding: Integer value specifying the number of decimal places to retain (default: no rounding)
verbose
Logical. Whether additional information (on directionality, formulas, frequency adjustment) should be printed; default is TRUE
print_scores
Logical. Whether the dispersion scores should be printed to the console; default is TRUE
Details
This function takes as input a term-document matrix and returns, for each item (i.e. each row) the dispersion measure D_{A}. The rows in the matrix represent the items, and the columns the corpus parts. Importantly, the term-document matrix must include an additional row that records the size of the corpus parts. For a proper term-document matrix, which includes all items that appear in the corpus, this can be added as a column margin, which sums the frequencies in each column. If the matrix only includes a selection of items drawn from the corpus, this information cannot be derived from the matrix and must be provided as a separate row.
Directionality: D_{A} ranges from 0 to 1. The conventional scaling of dispersion measures (see Juilland & Chang-Rodriguez 1964; Carroll 1970; Rosengren 1971) assigns higher values to more even/dispersed/balanced distributions of subfrequencies across corpus parts. This is the default. Gries (2008) uses the reverse scaling, with higher values denoting a more uneven/bursty/concentrated distribution; use directionality = 'gries' to choose this option.
Procedure: Irrespective of the directionality of scaling, two computational procedures for D_{A} exist (see below for details). Both appear in Wilcox (1973), where the measure is referred to as "MDA". The basic version (represented by the value basic) carries out the full set of computations required by the composition of the formula. As the number of corpus parts grows, this can become computationally very expensive. Wilcox (1973) also gives a "computational" procedure, which is a shortcut that is much quicker and closely approximates the scores produced by the basic formula. This version is represented by the value shortcut.
Frequency adjustment: Dispersion scores can be adjusted for frequency using the min-max transformation proposed by Gries (2022, 2024). The frequency-adjusted score for an item considers the lowest and highest possible level of dispersion it can obtain given its overall corpus frequency as well as the number (and size) of corpus parts. The unadjusted score is then expressed relative to these endpoints, where the dispersion minimum is set to 0, and the dispersion maximum to 1 (expressed in terms of conventional scaling). The frequency-adjusted score falls between these bounds and expresses how close the observed distribution is to the theoretical maximum and minimum. This adjustment therefore requires a maximally and a minimally dispersed distribution of the item across the parts. These hypothetical extremes can be built in different ways. The method used by Gries (2022, 2024) uses a computationally expensive procedure that finds the distribution that produces the highest value on the dispersion measure of interest. The current function constructs extreme distributions in a different way, based on the distributional features pervasiveness (pervasive) or evenness (even). You can choose between these with the argument freq_adjust_method; the default is even. For details and explanations, see vignette("frequency-adjustment").
To obtain the lowest possible level of dispersion, the occurrences are either allocated to as few corpus parts as possible (pervasive), or they are assigned to the smallest corpus part(s) (even).
To obtain the highest possible level of dispersion, the occurrences are either spread as broadly across corpus parts as possible (pervasive), or they are allocated to corpus parts in proportion to their size (even). The choice between these methods is particularly relevant if corpus parts differ considerably in size. See documentation for find_max_disp().
In the formulas given below, the following notation is used:
-
k the number of corpus parts
-
R_i the normalized subfrequency in part i, i.e. the number of occurrences of the item divided by the size of the part
-
r_i a proportional quantity; the normalized subfrequency in part i (R_i) divided by the sum of all normalized subfrequencies
The value basic implements the basic computational procedure (see Wilcox 1973: 329, 343; Burch et al. 2017: 194; Egbert et al. 2020: 98). The basic version can be applied to absolute frequencies and normalized frequencies. For dispersion analysis, absolute frequencies only make sense if the corpus parts are identical in size. Wilcox (1973: 343, 'MDA', column 1 and 2) gives both variants of the basic version. The first use of D_{A} for corpus-linguistic dispersion analysis appears in Burch et al. (2017: 194), a paper that deals with equal-sized parts and therefore uses the variant for absolute frequencies. Egbert et al. (2020: 98) rely on the variant using normalized frequencies. Since this variant of the basic version of D_{A} works irrespective of the length of the corpus parts (equal or variable), we will only give this version of the formula. Note that while the formula represents conventional scaling (0 = uneven, 1 = even), in the current function the directionality is controlled separately using the argument directionality.
1 - \frac{\sum_{i = 1}^{k-1} \sum_{j = i+1}^{k} |R_i - R_j|}{\frac{k(k-1)}{2}} \times \frac{1}{2\frac{\sum_i^k R_i}{k}} (Egbert et al. 2020: 98)
The function uses a different version of the same formula, which relies on the proportional r_i values instead of the normalized subfrequencies R_i. This version yields the identical result; the r_i quantities are also the key to using the computational shortcut given in Wilcox (1973: 343). This is the basic formula for D_{A} using r_i instead of R_i values:
1 - \frac{\sum_{i = 1}^{k-1} \sum_{j = i+1}^{k} |r_i - r_j|}{k-1} (Wilcox 1973: 343; see also Soenning 2022)
The value shortcut implements the computational shortcut given in Wilcox (1973: 343). Critically, the proportional quantities r_i must first be sorted in decreasing order. Only after this rearrangement can the shortcut procedure be applied. We will refer to this rearranged version of r_i as r_i^{sorted}:
\frac{2\left(\sum_{i = 1}^{k} (i \times r_i^{sorted}) - 1\right)}{k-1} (Wilcox 1973: 343)
The value shortcut_mod adds a minor modification to the computational shortcut to ensure D_{A} does not exceed 1 (on the conventional dispersion scale):
\frac{2\left(\sum_{i = 1}^{k} (i \times r_i^{sorted}) - 1\right)}{k-1} \times \frac{k}{k - 1}
Value
A numeric vector the same length as the number of items in the term-document matrix
Author(s)
Lukas Soenning
References
Burch, Brent, Jesse Egbert & Douglas Biber. 2017. Measuring and interpreting lexical dispersion in corpus linguistics. Journal of Research Design and Statistics in Linguistics and Communication Science 3(2). 189–216. \Sexpr[results=rd]{tools:::Rd_expr_doi("doi:10.1558/jrds.33066")}
Carroll, John B. 1970. An alternative to Juilland’s usage coefficient for lexical frequencies and a proposal for a standard frequency index. Computer Studies in the Humanities and Verbal Behaviour 3(2). 61–65. \Sexpr[results=rd]{tools:::Rd_expr_doi("doi:10.1002/j.2333-8504.1970.tb00778.x")}
Egbert, Jesse, Brent Burch & Douglas Biber. 2020. Lexical dispersion and corpus design. International Journal of Corpus Linguistics 25(1). 89–115. \Sexpr[results=rd]{tools:::Rd_expr_doi("doi:10.1075/ijcl.18010.egb")}
Gries, Stefan Th. 2022. What do (most of) our dispersion measures measure (most)? Dispersion? Journal of Second Language Studies 5(2). 171–205. \Sexpr[results=rd]{tools:::Rd_expr_doi("doi:10.1075/jsls.21029.gri")}
Gries, Stefan Th. 2024. Frequency, dispersion, association, and keyness: Revising and tupleizing corpus-linguistic measures. Amsterdam: Benjamins. \Sexpr[results=rd]{tools:::Rd_expr_doi("doi:10.1075/scl.115")}
Gries, Stefan Th. 2008. Dispersions and adjusted frequencies in corpora. International Journal of Corpus Linguistics 13(4). 403–437. \Sexpr[results=rd]{tools:::Rd_expr_doi("doi:10.1075/ijcl.13.4.02gri")}
Juilland, Alphonse G. & Eugenio Chang-Rodríguez. 1964. Frequency dictionary of Spanish words. The Hague: Mouton de Gruyter. \Sexpr[results=rd]{tools:::Rd_expr_doi("doi:10.1515/9783112415467")}
Rosengren, Inger. 1971. The quantitative concept of language and its relation to the structure of frequency dictionaries. Études de linguistique appliquée (Nouvelle Série) 1. 103–127.
Soenning, Lukas. 2022. Evaluation of text-level measures of lexical dispersion: Robustness and consistency. PsyArXiv preprint. https://osf.io/preprints/psyarxiv/h9mvs/
Wilcox, Allen R. 1973. Indices of qualitative variation and political measurement. The Western Political Quarterly 26 (2). 325–343. \Sexpr[results=rd]{tools:::Rd_expr_doi("doi:10.2307/446831")}
Examples
disp_DA_tdm(
tdm = biber150_spokenBNC2014[1:20,],
row_partsize = "first",
procedure = "basic",
directionality = "conventional",
freq_adjust = FALSE)
tlda documentation built on June 8, 2025, 11:41 a.m.
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| disp_DA_tdm | R Documentation |
Calculate the dispersion measure D_{A} for a term-document matrix
Description
This function calculates the dispersion measure D_{A}. It offers two different computational procedures, the basic version as well as a computational shortcut. It also allows the user to choose the directionality of scaling, i.e. whether higher values denote a more even or a less even distribution. It also provides the option of calculating frequency-adjusted dispersion scores.
Usage
disp_DA_tdm(
tdm,
row_partsize = "first",
directionality = "conventional",
procedure = "basic",
freq_adjust = FALSE,
freq_adjust_method = "even",
unit_interval = TRUE,
digits = NULL,
verbose = TRUE,
print_scores = TRUE
)
Arguments
tdm |
A term-document matrix, where rows represent items and columns represent corpus parts; must also contain a row giving the size of the corpus parts (first or last row in the term-document matrix) |
row_partsize |
Character string indicating which row in the term-document matrix contains the size of the corpus parts. Possible values are |
directionality |
Character string indicating the directionality of scaling. See details below. Possible values are |
procedure |
Character string indicating which procedure to use for the calculation of |
freq_adjust |
Logical. Whether dispersion score should be adjusted for frequency (i.e. whether frequency should be 'partialed out'); default is |
freq_adjust_method |
Character string indicating which method to use for devising dispersion extremes. See details below. Possible values are |
unit_interval |
Logical. Whether frequency-adjusted scores that exceed the limits of the unit interval should be replaced by 0 and 1; default is |
digits |
Rounding: Integer value specifying the number of decimal places to retain (default: no rounding) |
verbose |
Logical. Whether additional information (on directionality, formulas, frequency adjustment) should be printed; default is |
print_scores |
Logical. Whether the dispersion scores should be printed to the console; default is |
Details
This function takes as input a term-document matrix and returns, for each item (i.e. each row) the dispersion measure D_{A}. The rows in the matrix represent the items, and the columns the corpus parts. Importantly, the term-document matrix must include an additional row that records the size of the corpus parts. For a proper term-document matrix, which includes all items that appear in the corpus, this can be added as a column margin, which sums the frequencies in each column. If the matrix only includes a selection of items drawn from the corpus, this information cannot be derived from the matrix and must be provided as a separate row.
Directionality:
D_{A}ranges from 0 to 1. The conventional scaling of dispersion measures (see Juilland & Chang-Rodriguez 1964; Carroll 1970; Rosengren 1971) assigns higher values to more even/dispersed/balanced distributions of subfrequencies across corpus parts. This is the default. Gries (2008) uses the reverse scaling, with higher values denoting a more uneven/bursty/concentrated distribution; usedirectionality = 'gries'to choose this option.Procedure: Irrespective of the directionality of scaling, two computational procedures for
D_{A}exist (see below for details). Both appear in Wilcox (1973), where the measure is referred to as "MDA". The basic version (represented by the valuebasic) carries out the full set of computations required by the composition of the formula. As the number of corpus parts grows, this can become computationally very expensive. Wilcox (1973) also gives a "computational" procedure, which is a shortcut that is much quicker and closely approximates the scores produced by the basic formula. This version is represented by the valueshortcut.Frequency adjustment: Dispersion scores can be adjusted for frequency using the min-max transformation proposed by Gries (2022, 2024). The frequency-adjusted score for an item considers the lowest and highest possible level of dispersion it can obtain given its overall corpus frequency as well as the number (and size) of corpus parts. The unadjusted score is then expressed relative to these endpoints, where the dispersion minimum is set to 0, and the dispersion maximum to 1 (expressed in terms of conventional scaling). The frequency-adjusted score falls between these bounds and expresses how close the observed distribution is to the theoretical maximum and minimum. This adjustment therefore requires a maximally and a minimally dispersed distribution of the item across the parts. These hypothetical extremes can be built in different ways. The method used by Gries (2022, 2024) uses a computationally expensive procedure that finds the distribution that produces the highest value on the dispersion measure of interest. The current function constructs extreme distributions in a different way, based on the distributional features pervasiveness (
pervasive) or evenness (even). You can choose between these with the argumentfreq_adjust_method; the default iseven. For details and explanations, seevignette("frequency-adjustment").To obtain the lowest possible level of dispersion, the occurrences are either allocated to as few corpus parts as possible (
pervasive), or they are assigned to the smallest corpus part(s) (even).To obtain the highest possible level of dispersion, the occurrences are either spread as broadly across corpus parts as possible (
pervasive), or they are allocated to corpus parts in proportion to their size (even). The choice between these methods is particularly relevant if corpus parts differ considerably in size. See documentation forfind_max_disp().
In the formulas given below, the following notation is used:
-
kthe number of corpus parts -
R_ithe normalized subfrequency in parti, i.e. the number of occurrences of the item divided by the size of the part -
r_ia proportional quantity; the normalized subfrequency in parti(R_i) divided by the sum of all normalized subfrequencies
The value basic implements the basic computational procedure (see Wilcox 1973: 329, 343; Burch et al. 2017: 194; Egbert et al. 2020: 98). The basic version can be applied to absolute frequencies and normalized frequencies. For dispersion analysis, absolute frequencies only make sense if the corpus parts are identical in size. Wilcox (1973: 343, 'MDA', column 1 and 2) gives both variants of the basic version. The first use of D_{A} for corpus-linguistic dispersion analysis appears in Burch et al. (2017: 194), a paper that deals with equal-sized parts and therefore uses the variant for absolute frequencies. Egbert et al. (2020: 98) rely on the variant using normalized frequencies. Since this variant of the basic version of D_{A} works irrespective of the length of the corpus parts (equal or variable), we will only give this version of the formula. Note that while the formula represents conventional scaling (0 = uneven, 1 = even), in the current function the directionality is controlled separately using the argument directionality.
1 - \frac{\sum_{i = 1}^{k-1} \sum_{j = i+1}^{k} |R_i - R_j|}{\frac{k(k-1)}{2}} \times \frac{1}{2\frac{\sum_i^k R_i}{k}} (Egbert et al. 2020: 98)
The function uses a different version of the same formula, which relies on the proportional r_i values instead of the normalized subfrequencies R_i. This version yields the identical result; the r_i quantities are also the key to using the computational shortcut given in Wilcox (1973: 343). This is the basic formula for D_{A} using r_i instead of R_i values:
1 - \frac{\sum_{i = 1}^{k-1} \sum_{j = i+1}^{k} |r_i - r_j|}{k-1} (Wilcox 1973: 343; see also Soenning 2022)
The value shortcut implements the computational shortcut given in Wilcox (1973: 343). Critically, the proportional quantities r_i must first be sorted in decreasing order. Only after this rearrangement can the shortcut procedure be applied. We will refer to this rearranged version of r_i as r_i^{sorted}:
\frac{2\left(\sum_{i = 1}^{k} (i \times r_i^{sorted}) - 1\right)}{k-1} (Wilcox 1973: 343)
The value shortcut_mod adds a minor modification to the computational shortcut to ensure D_{A} does not exceed 1 (on the conventional dispersion scale):
\frac{2\left(\sum_{i = 1}^{k} (i \times r_i^{sorted}) - 1\right)}{k-1} \times \frac{k}{k - 1}
Value
A numeric vector the same length as the number of items in the term-document matrix
Author(s)
Lukas Soenning
References
Burch, Brent, Jesse Egbert & Douglas Biber. 2017. Measuring and interpreting lexical dispersion in corpus linguistics. Journal of Research Design and Statistics in Linguistics and Communication Science 3(2). 189–216. \Sexpr[results=rd]{tools:::Rd_expr_doi("doi:10.1558/jrds.33066")}
Carroll, John B. 1970. An alternative to Juilland’s usage coefficient for lexical frequencies and a proposal for a standard frequency index. Computer Studies in the Humanities and Verbal Behaviour 3(2). 61–65. \Sexpr[results=rd]{tools:::Rd_expr_doi("doi:10.1002/j.2333-8504.1970.tb00778.x")}
Egbert, Jesse, Brent Burch & Douglas Biber. 2020. Lexical dispersion and corpus design. International Journal of Corpus Linguistics 25(1). 89–115. \Sexpr[results=rd]{tools:::Rd_expr_doi("doi:10.1075/ijcl.18010.egb")}
Gries, Stefan Th. 2022. What do (most of) our dispersion measures measure (most)? Dispersion? Journal of Second Language Studies 5(2). 171–205. \Sexpr[results=rd]{tools:::Rd_expr_doi("doi:10.1075/jsls.21029.gri")}
Gries, Stefan Th. 2024. Frequency, dispersion, association, and keyness: Revising and tupleizing corpus-linguistic measures. Amsterdam: Benjamins. \Sexpr[results=rd]{tools:::Rd_expr_doi("doi:10.1075/scl.115")}
Gries, Stefan Th. 2008. Dispersions and adjusted frequencies in corpora. International Journal of Corpus Linguistics 13(4). 403–437. \Sexpr[results=rd]{tools:::Rd_expr_doi("doi:10.1075/ijcl.13.4.02gri")}
Juilland, Alphonse G. & Eugenio Chang-Rodríguez. 1964. Frequency dictionary of Spanish words. The Hague: Mouton de Gruyter. \Sexpr[results=rd]{tools:::Rd_expr_doi("doi:10.1515/9783112415467")}
Rosengren, Inger. 1971. The quantitative concept of language and its relation to the structure of frequency dictionaries. Études de linguistique appliquée (Nouvelle Série) 1. 103–127.
Soenning, Lukas. 2022. Evaluation of text-level measures of lexical dispersion: Robustness and consistency. PsyArXiv preprint. https://osf.io/preprints/psyarxiv/h9mvs/
Wilcox, Allen R. 1973. Indices of qualitative variation and political measurement. The Western Political Quarterly 26 (2). 325–343. \Sexpr[results=rd]{tools:::Rd_expr_doi("doi:10.2307/446831")}
Examples
disp_DA_tdm(
tdm = biber150_spokenBNC2014[1:20,],
row_partsize = "first",
procedure = "basic",
directionality = "conventional",
freq_adjust = FALSE)
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