z-Transformation 20. Jun 2018 z-Transformation (= z-scores) The so-called z-transformation is often applied to normalise (= standardise)data to be able to compare ‘values’ between different speakers or data-ranges. The idea behind this transformation is to subtract the arithmetic mean of some data (e.g. of a speaker or a segment) from a particular value. This shifts this value into a range of negative and positive values around the mean. And then divide this difference by the standard deviation of the data (speaker or segment). This will normalize the data to ±1 for a value that is ± the standard deviation away from the mean. As an outcome, the z-transformed data form a distribution around ‘0’ with the standard deviation of ‘1’, independent of the original data range. Formula: z=xi−mean(x) st.dev(x) ? A typical formula in JMP looks then like (in this example, F1-values normalised for each speaker): F1 - Col Mean (F1, Speaker) Col Std Dev (F1, Speaker) ? In case the data should be normalised in relation to all data (and not speaker specific, JMP provides the function Col Standardize: Col Standardize (F1) Transformation of pitch data Using semitones or ERB values allows the comparison of speaker with very different pitch ranges and is the preferred method. Using z-scores additionally normalises each segment by transforming the pitch values to a distribution around the a mean of ‘0’ and the standard deviation of ‘1’ by the formula (in this example, for a phrase): z=F0(at time point) - F0mean(of phrase) F0stdev(of phrase) ? . Using z-score transformed semitone values to compare pitch contours is an often applied method but can lead to misleading results if a contour is rather flat (cf. https://sites.google.com/site/ tonemodelling/anaposts/z-transformdoesnotworkforpitchcontoursflat). The normalisation by subtracting the mean only seems to be a more appropriate method, but I suggest to apply this method only to ERB or semitone scaled data (and not to Hertz values directly): z = F0(at time point) - F0mean(of segment) ?
z-Transformation 20. Jun 2018 Transformation of Formant data (vowel space) For formant data of vowels, there have been several normalisation methods proposed. A good overview article is Fabricius, Anne H.; Watt,Dominic & Johnson,Daniel Ezra (2009). A comparison of three speaker- intrinsic vowel formant frequency normalization algorithms for sociophonetics. Language Variation and Change21, 413–435. There is also an online version to apply these methods <http://lingtools.uoregon.edu/norm/norm1.php> and an R-function <http://lingtools.uoregon.edu/norm/package/html/norm.wattfabricius.html>