# Calibrate raw data to crisp or fuzzy sets

## Description

This function transforms (calibrates) the raw data to either crisp or fuzzy sets values, using the direct method of calibration.

## Usage

    calibrate(x, type = "fuzzy", method = "direct", thresholds = NA,
              logistic = TRUE, idm = 0.95, ecdf = FALSE, below = 1, above = 1, ...)

## Arguments

`x` |  |  |  | A numerical causal condition.
`type` |  |  |  | Calibration type, either `"crisp"` or `"fuzzy"`.
`method` |  |  |  | Calibration method, either `"direct"`, `"indirect"`, or `"TFR"`.
`thresholds` |  |  |  | A vector of (named) thresholds.
`logistic` |  |  |  | Calibrate to fuzzy sets using the logistic function.
`idm` |  |  |  | The set inclusion degree of membership for the logistic function.
`ecdf` |  |  |  | Calibrate to fuzzy sets using the empirical cumulative distribution function of the raw data.
`below` |  |  |  | Numeric (non-negative), determines the shape below crossover.
`above` |  |  |  | Numeric (non-negative), determines the shape above crossover.
`...` |  |  |  | Additional parameters, mainly for backwards compatibility.

## Details

Calibration is a transformational process from raw numerical data (interval or ratio level of measurement) to set membership scores, based on a certain number of qualitative anchors.

When `type = "crisp"`, the process is similar to recoding the original values to a number of categories defined by the number of thresholds. For one threshold, the calibration produces two categories (intervals): 0 if below, 1 if above. For two thresholds, the calibration produces three categories: 0 if below the first threshold, 1 if in the interval between the thresholds and 2 if above the second threshold etc.

When `type = "fuzzy"`, calibration produces fuzzy set membership scores, using three anchors for the increasing or decreasing s-shaped distributions (including the logistic function), and six anchors for the increasing or decreasing bell-shaped distributions.

The argument `thresholds` can be specified either as a simple numeric vector, or as a named numeric vector. If used as a named vector, for the first category of s-shaped distributions, the names of the thresholds should be:

`"e"` | for the full set exclusion
`"c"` | for the set crossover
`"i"` | for the full set inclusion

For the second category of bell-shaped distributions, the names of the thresholds should be:

`"e1"` | for the first (left) threshold for full set exclusion
`"c1"` | for the first (left) threshold for set crossover
`"i1"` | for the first (left) threshold for full set inclusion
`"i2"` | for the second (right) threshold for full set inclusion
`"c2"` | for the second (right) threshold for set crossover
`"e2"` | for the second (right) threshold for full set exclusion

If used as a simple numerical vector, the order of the values matter.

If `e` $<$ `c` $<$ `i`, then the membership function is increasing from `e` to `i`. If `i` $<$ `c` $<$ `e`, then the membership function is decreasing from `i` to `e`.

Same for the bell-shaped distribution, if `e1` $<$ `c1` $<$ `i1` $\le$ `i2` $<$ `c2` $<$ `e2`, then the membership function is first increasing from `e1` to `i1`, then flat between `i1` and `i2`, and then decreasing from `i2` to `e2`. In contrast, if `i1` $<$ `c1` $<$ `e1` $\le$ `e2` $<$ `c2` $<$ `i1`, then the membership function is first decreasing from `i1` to `e1`, then flat between `e1` and `e2`, and finally increasing from `e2` to `i2`.

When `logistic = TRUE` (the default), the argument `idm` specifies the inclusion degree of membership for the logistic function. If `logistic = FALSE`, the function returns linear s-shaped or bell-shaped distributions (curved using the arguments `below` and `above`), unless activating the argument `ecdf`.

If there is no prior knowledge on the shape of the distribution, the argument `ecdf` asks the computer to determine the underlying distribution of the empirical, observed points, and the calibrated measures are found along that distribution.

Both `logistic` and `ecdf` arguments can be used only for s-shaped distributions (using 3 thresholds), and they are mutually exclusive.

The parameters `below` and `above` (active only when both `logistic = TRUE` and `ecdf` are deactivated, establish the degree of concentration and dilation (convex or concave shape) between the threshold and crossover:

`0 < below < 1` | dilates in a concave shape below the crossover
`below = 1` | produces a linear shape (neither convex, nor concave)
`below > 1` | concentrates in a convex shape below the crossover
`0 < above < 1` | dilates in a concave shape above the crossover
`above = 1` | produces a linear shape (neither convex, nor concave)
`above > 1` | concentrates in a convex shape above the crossover

Usually, `below` and `above` have equal values, unless specific reasons exist to make them different.

For the `type = "fuzzy"` it is also possible to use the `"indirect"` method to calibrate the data, using a procedure first introduced by Ragin (2008). The indirect method assumes a vector of thresholds to cut the original data into equal intervals, then it applies a (quasi)binomial logistic regression with a fractional polynomial equation.

The results are also fuzzy between 0 and 1, but the method is entirely different: it has no anchors (specific to the direct method), and it doesn't need to specify a calibration function to calculate the scores with.

The third method applied to fuzzy calibrations is called `type = "TFR"` and calibrates categorical data (such as Likert type response scales) to fuzzy values using the Totally Fuzzy and Relative method (Chelli and Lemmi, 1995).

## Value

A numeric vector of set membership scores, either crisp (starting from 0 with increments of 1), or fuzzy numeric values between 0 and 1.

## References

Thiem, A.; Dusa, A. (2013) Qualitative Comparative Analysis with R: A User's Guide. New York: Springer.

Thiem, A. (2014) “Membership Function Sensitivity of Descriptive Statistics in Fuzzy-Set Relations.” International Journal of Social Research Methodology vol.17, no.6, pp.625-642.

## Examples

    # generate heights for 100 people
    # with an average of 175cm and a standard deviation of 10cm
    set.seed(12345)
    x <- rnorm(n = 100, mean = 175, sd = 10)

    cx <- calibrate(x, type = "crisp", thresholds = 175)
    plot(x, cx, main="Binary crisp set using 1 threshold",
         xlab = "Raw data", ylab = "Calibrated data", yaxt="n")
    axis(2, at = 0:1)

    cx <- calibrate(x, type = "crisp", thresholds = c(170, 180))
    plot(x, cx, main="3 value crisp set using 2 thresholds",
         xlab = "Raw data", ylab = "Calibrated data", yaxt="n")
    axis(2, at = 0:2)

    # calibrate to an increasing, s-shaped fuzzy-set
    cx <- calibrate(x, thresholds = "e=165, c=175, i=185")
    plot(x, cx, main = "Membership scores in the set of tall people",
         xlab = "Raw data", ylab = "Calibrated data")

    # calibrate to a decreasing, s-shaped fuzzy-set
    cx <- calibrate(x, thresholds = "i=165, c=175, e=185")
    plot(x, cx, main = "Membership scores in the set of short people",
         xlab = "Raw data", ylab = "Calibrated data")

    # when not using the logistic function, linear increase
    cx <- calibrate(x, thresholds = "e=165, c=175, i=185", logistic = FALSE)
    plot(x, cx, main = "Membership scores in the set of tall people",
         xlab = "Raw data", ylab = "Calibrated data")

    # tweaking the parameters "below" and "above" the crossover,
    # at value 3.5 approximates a logistic distribution, when e=155 and i=195
    cx <- calibrate(x, thresholds = "e=155, c=175, i=195",
          logistic = FALSE, below = 3.5, above = 3.5)
    plot(x, cx, main = "Membership scores in the set of tall people",
         xlab = "Raw data", ylab = "Calibrated data")

    # calibrate to a bell-shaped fuzzy set
    cx <- calibrate(x, thresholds = "e1=155, c1=165, i1=175, i2=175, c2=185, e2=195",
          below = 3, above = 3)
    plot(x, cx, main = "Membership scores in the set of average height",
         xlab = "Raw data", ylab = "Calibrated data")

    # calibrate to an inverse bell-shaped fuzzy set
    cx <- calibrate(x, thresholds = "i1=155, c1=165, e1=175, e2=175, c2=185, i2=195",
          below = 3, above = 3)
    plot(x, cx, main = "Membership scores in the set of non-average height",
         xlab = "Raw data", ylab = "Calibrated data")

    # the default values of "below" and "above" will produce a triangular shape
    cx <- calibrate(x, thresholds = "e1=155, c1=165, i1=175, i2=175, c2=185, e2=195")
    plot(x, cx, main = "Membership scores in the set of average height",
         xlab = "Raw data", ylab = "Calibrated data")

    # different thresholds to produce a linear trapezoidal shape
    cx <- calibrate(x, thresholds = "e1=156, c1=164, i1=172, i2=179, c2=187, e2=195")
    plot(x, cx, main = "Membership scores in the set of average height",
         xlab = "Raw data", ylab = "Calibrated data")

    # larger values of above and below will increase membership in or out of the set
    cx <- calibrate(x, thresholds = "e1=155, c1=165, i1=175, i2=175, c2=185, e2=195",
          below = 10, above = 10)
    plot(x, cx, main = "Membership scores in the set of average height",
         xlab = "Raw data", ylab = "Calibrated data")

    # while extremely large values will produce virtually crisp results
    cx <- calibrate(x, thresholds = "e1=155, c1=165, i1=175, i2=175, c2=185, e2=195",
          below = 10000, above = 10000)
    plot(x, cx, main = "Binary crisp scores in the set of average height",
         xlab = "Raw data", ylab = "Calibrated data", yaxt="n")
    axis(2, at = 0:1)
    abline(v = c(165, 185), col = "red", lty = 2)

    # check if crisp
    cx

      [1] 1 1 1 1 1 0 1 1 1 1 1 0 1 1 1 1 1 1 0 1 1 0 1 0 0 0 1 1 1 1 1 0 0 0 1 1 1 0 0 1 0
     [42] 0 0 1 1 0 0 1 1 0 1 0 1 1 1 1 1 1 0 0 1 0 1 0 1 0 1 0 1 0 1 1 0 0 0 1 1 1 1 0 1 0
     [83] 1 0 1 0 1 1 1 1 1 1 0 1 1 1 1 0 1 1

    # using the empirical cumulative distribution function
    # require manually setting logistic to FALSE
    cx <- calibrate(x, thresholds = "e=155, c=175, i=195", logistic = FALSE,
          ecdf = TRUE)
    plot(x, cx, main = "Membership scores in the set of tall people",
         xlab = "Raw data", ylab = "Calibrated data")

    ## the indirect method, per capita income data from Ragin (2008)
    inc <- c(40110, 34400, 25200, 24920, 20060, 17090, 15320, 13680, 11720,
             11290, 10940, 9800, 7470, 4670, 4100, 4070, 3740, 3690, 3590,
             2980, 1000, 650, 450, 110)

    cinc <- calibrate(inc, method = "indirect",
            thresholds = "1000, 4000, 5000, 10000, 20000")

    plot(inc, cinc, main = "Membership scores in the set of high income",
         xlab = "Raw data", ylab = "Calibrated data")

    set.seed(12345)
    values <- sample(1:7, 100, replace = TRUE)

    TFR <- calibrate(values, method = "TFR")

    table(round(TFR, 3))

        0 0.151 0.314 0.477 0.605 0.814     1
       14    13    14    14    11    18    16

## Author

Adrian Dusa