QCApro (version 1.1-1)

calibrate: Calibrate Raw Data into Configurational Data

Description

This function generates configurational data from raw data (base variables) and some specified threshold(s). The calibration of bivalent fuzzy-set factors is possible for positive and negative end-point and mid-point concepts, using the method of transformational assignment.

Usage

calibrate(x, type = "crisp", thresholds = NA, include = TRUE,  logistic = FALSE, idm = 0.95, ecdf = FALSE, p = 1, q = 1)

Arguments

x

An interval or ratio-scaled base variable.

type

The calibration type, either "crisp" or "fuzzy".

thresholds

A vector of thresholds.

include

Logical, include threshold(s) (type = "crisp" only).

logistic

Calibrate to fuzzy-set variable using the logistic function.

idm

The set inclusion degree of membership for the logistic function.

ecdf

Calibrate to fuzzy-set variable using the empirical cumulative distribution function of the base variable.

p

Parameter: if $p > 1$ concentration, if $0 < p < 1$ dilation below crossover.

q

Parameter: if $q > 1$ dilation, if $0 < q < 1$ concentration above crossover.

Value

A numeric vector of set membership scores between 0 and 1 for bivalent crisp-set factors and bivalent fuzzy-set variables, or a numeric vector of levels for multivalent crisp-set factors (beginning with 0 at increments of 1).

Contributors

Dusa, Adrian

: programming

Details

Calibration is the process by which configurational data is produced, that is, by which set membership scores are assigned to cases. With interval and ratio-scaled base variables, calibration can be based on transformational assignments using (piecewise-defined) membership functions.

For type = "crisp", one threshold produces a factor with two levels: 0 and 1. More thresholds produce factors with multiple levels. For example, two thresholds produce three levels: 0, 1 and 2.

For type = "fuzzy", this function can generate bivalent fuzzy-set variables by linear, s-shaped, inverted s-shaped and logistic transformation for end-point concepts. It can generate bivalent fuzzy-set variables by trapezoidal, triangular and bell-shaped transformation for mid-point concepts (Bojadziev and Bojadziev 2007; Clark et al. 2008; Thiem 2014; Thiem and Dusa 2013).

For calibrating bivalent fuzzy-set variables based on end-point concepts, thresholds should be specified as a numeric vector c(thEX, thCR, thIN), where thEX is the threshold for full exclusion, thCR the threshold for the crossover, and thIN the threshold for full inclusion.

If thEX $<$ thCR $<$ thIN, then the membership function is increasing from thEX to thIN. If thIN $<$ thCR $<$ thEX, then the membership function is decreasing from thIN to thEX.

For calibrating bivalent fuzzy-set variables based on mid-point concepts, thresholds should be specified as a numeric vector c(thEX1, thCR1, thIN1, thIN2, thCR2, thEX2), where thEX1 is the first (left) threshold for full exclusion, thCR1 the first (left) threshold for the crossover, thIN1 the first (left) threshold for full inclusion, thIN2 the second (right) threshold for full inclusion, thCR2 the second (right) threshold for crossover, and thEX2 the second (right) threshold for full exclusion.

If thEX1 $<$ thCR1 $<$ thIN1 $\le$ thIN2 $<$ thCR2 $<$ thEX2, then the membership function is first increasing from thEX1 to thIN1, then flat between thIN1 and thIN2, and finally decreasing from thIN2 to thEX2. In contrast, if thIN1 $<$ thCR1 $<$ thEX1 $\le$ thEX2 $<$ thCR2 $<$ thIN2, then the membership function is first decreasing from thIN1 to thEX1, then flat between thEX1 and thEX2, and finally increasing from thEX2 to thIN2.

The parameters p and q control the degree of concentration and dilation. They should be left at their default values unless good reasons for changing them exist.

If logistic = TRUE, the argument idm specifies the inclusion degree of membership.

If ecdf = TRUE, calibration is based on the empirical cumulative distribution function of x.

References

Bojadziev, George, and Maria Bojadziev. 2007. Fuzzy Logic for Business, Finance, and Management. 2nd ed. Hackensack, NJ: World Scientific. Link.

Clark, Terry D., Jennifer M. Larson, John N. Mordeson, Joshua D. Potter, and Mark J. Wierman. 2008. Applying Fuzzy Mathematics to Formal Models in Comparative Politics. Berlin: Springer. Link.

Thiem, Alrik. 2014. “Membership Function Sensitivity of Descriptive Statistics in Fuzzy-Set Relations.” International Journal of Social Research Methodology 17 (6):625-42. DOI: 10.1080/13645579.2013.806118.

Thiem, Alrik, and Adrian Dusa. 2013. Qualitative Comparative Analysis with R: A User's Guide. New York: Springer. Link.

Examples

# base variable; random draw from standard normal distribution
set.seed(30)
x <- rnorm(30)

# calibration thresholds
th <- quantile(x, seq(from = 0.05, to = 0.95, length = 6))

# calibration of bivalent crisp-set factor
calibrate(x, thresholds = th[3])

# calibration of trivalent crisp-set factor
calibrate(x, thresholds = c(th[2], th[4]))

# fuzzy-set calibration
# 1. positive end-point concept, linear
# 2. positive and corresponding negative end-point concept, logistic
# 3. positive end-point concept, ECDF
# 4. negative end-point concept, s-shaped (quadratic)
# 5. negative end-point concept, inverted s-shaped (root)
# 6. positive mid-point concept, triangular
# 7. positive mid-point concept, trapezoidal
# 8. negative mid-point concept, bell-shaped

yl <- "Set Membership"
xl <- "Base Variable Value"

par(mfrow = c(2,4), cex.main = 1)

plot(x, calibrate(x, type = "fuzzy", thresholds = c(th[1], (th[3]+th[4])/2, 
   th[6])), xlab = xl, ylab = yl, 
   main = "1. positive end-point concept,\nlinear")

plot(x, calibrate(x, type = "fuzzy", thresholds = c(th[1], (th[3]+th[4])/2, 
  th[6]), logistic = TRUE, idm = 0.99), xlab = xl, ylab = yl, 
  main = "2. positive and corresponding negative\nend-point concept, logistic")
  points(x, calibrate(x, type = "fuzzy", thresholds = c(th[6], (th[3]+th[4])/2, 
    th[1]), logistic = TRUE, idm = 0.99))

plot(x, calibrate(x, type = "fuzzy", thresholds = c(th[1], (th[3]+th[4])/2, 
  th[6]), ecdf = TRUE), xlab = xl, ylab = yl, 
  main = "3. positive end-point concept,\nECDF")

plot(x, calibrate(x, type = "fuzzy", thresholds = c(th[6], (th[3]+th[4])/2, 
  th[1]), p = 2, q = 2), xlab = xl, ylab = yl, 
  main = "4. negative end-point concept,\ns-shaped (quadratic)")

plot(x, calibrate(x, type = "fuzzy", thresholds = c(th[6], (th[3]+th[4])/2, 
  th[1]), p = 0.5, q = 0.5), xlab = xl, ylab = yl, 
  main = "5. negative end-point concept,\ninverted s-shaped (root)")

plot(x, calibrate(x, type = "fuzzy", thresholds = th[c(1,2,3,3,4,5)]),
  xlab = xl, ylab = yl, main = "6. positive mid-point concept,\ntriangular")

plot(x, calibrate(x, type = "fuzzy", thresholds = th[c(1,2,3,4,5,6)]),
  xlab = xl, ylab = yl, main = "7. positive mid-point concept,\ntrapezoidal")

plot(x, calibrate(x, type = "fuzzy", thresholds = th[c(3,2,1,5,4,3)],
  p = 3, q = 3), xlab = xl, ylab = yl, 
  main = "8. negative mid-point concept,\nbell-shaped")