"Rejoinder for “A Spatial Modeling Approach for Linguistic Object Data: Analyzing Dialect Sound Variations Across Great Britain”." Journal of the American Statistical Association, 114(527), pp. 1103–110

Aston, John A. D.

Coleman, John S.

Pigoli, Davide

Tavakoli, Shahin

English

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     warwick.ac.uk/lib-publications      Manuscript version: Author’s Accepted Manuscript The version presented in WRAP is the author’s accepted manuscript and may differ from the published version or Version of Record.  Persistent WRAP URL: http://wrap.warwick.ac.uk/126480                                                         How to cite: Please refer to published version for the most recent bibliographic citation information.  If a published version is known of, the repository item page linked to above, will contain details on accessing it.  Copyright and reuse: The Warwick Research Archive Portal (WRAP) makes this work by researchers of the University of Warwick available open access under the following conditions.   Copyright © and all moral rights to the version of the paper presented here belong to the individual author(s) and/or other copyright owners. To the extent reasonable and practicable the material made available in WRAP has been checked for eligibility before being made available.  Copies of full items can be used for personal research or study, educational, or not-for-profit purposes without prior permission or charge. Provided that the authors, title and full bibliographic details are credited, a hyperlink and/or URL is given for the original metadata page and the content is not changed in any way.  Publisher’s statement: Please refer to the repository item page, publisher’s statement section, for further information.  For more information, please contact the WRAP Team at: wrap@warwick.ac.uk.  Rejoinder for A Spatial Modeling Approach forLinguistic Object Data: Analysing dialect soundvariations across Great BritainShahin Tavakoli,Department of Statistics, University of Warwick, UK,Davide Pigoli,Department of Mathematics, King's College London, UK,John A. D. Aston∗†Statistical Laboratory, University of Cambridge, UK,andJohn S. ColemanPhonetics Laboratory, University of Oxford, UKAugust 9, 2019We thank the discussants for their thoughtful comments and suggestions for future re-search directions. They raise a number of interesting questions, which we address here. Forease of reference, we will write PM when referring to the discussion of Petersen & Müller,and DPS when referring to the discussion of Dryden, Preston & Severn (and Marron whenreferring to the discussion from Marron).1. Choice of the metric: DPS suggest that the choice of the metric on the space ofcovariance matrices can give an extra level of tuning for the practitioner (PM alsopoint this out), and propose to look at the family of power Euclidean metric betweencovariances. Denoting by dα(A,B) = |||Aα −Bα|||, DPS come up with a nice expressionfor the dα-covariance, for αs of the form α = k/(2m) for integers k,m (dyadic rationals).DPS propose to look at the case α = 3/4, which could provide a reasonable compromise∗The authors gratefully acknowledge support from EPSRC grant EP/K021672/2†Address for correspondence: Professor John Aston, Statistical laboratory, Department of Pure Math-ematics and Mathematical Statistics, University of Cambridge, CB3 0WB Cambridge, United Kingdom.Email: j.aston@statslab.cam.ac.uk1between our (popular) choice α = 1/2 and α = 1 (which gives the usual covariancematrix). Choosing a good α is a difficult question, and depends on the applicationthat is being considered. Petersen & Müller (2016) propose methods for choosingα in an adaptive way, by using a criterion-based approach. The criteria they giveas examples are based on desired qualities of the interpolated (dE-)covariance matrix(such as eigenvalue decay), and it is therefore not straightforward to extend theirmethods to the case of dα-covariance matrices, for which eigenvalues do not have aclear interpretation through a KarhunenLoève expansion.2. Local linear smoothing: PM suggest the use of local linear regression (local FréchetRegression in the terminology of Petersen et al. 2019), which should give smaller biasnear the boundaries. We agree that local linear smoothers are better than local con-stant smoother: however the local linear smoother proposed by PM assumes that themetric on the predictor space (Great Britain in our case, denoted by E) is the Euclideanmetric, since z → β0 + β1(z − x) is a line only under the Euclidean metric. In ourcase our distance on Great Britain is the shortest inland distance, which we approx-imate by the graph distance dg. While this distance is equal (approximately) to theEuclidean distance for two points for which the (Euclidean) straight line connectingthem lies within Great Britain, this is not true for points near the boundary (and inparticular, the Euclidean metric is not adequate, not because Great Britain is not big,but because it is non-convex). Hence the method proposed is not directly applicable.Since only distances in the predictor space are available, we would need to use methodsto perform local linear smoothing when only pairwise distances between the predictorsare available. Baíllo & Grané (2009), Boj et al. (2010, 2016) propose such methods,however they only deal with scalar responses, and they would need to be extended todeal with a non-Euclidean metric in the response space. This is indeed an interestingavenue for future research.3. 1 step procedure: The comment of PM regarding the one-step estimation of thed-covariances is interesting. To obtain such a one-step estimation, one however needsto have an estimate µˆ(x) of the mean field at x ∈ E , which in the case of singleobservation at each sampling point requires an initial smoothing step to compute (weomit the time index t for ease of notation). Since we have multiple observations ateach sampling points in E , we can obtain an unbiased estimate of µ(Xl) by taking thesample average at Xl ∈ E , Y t. Then, the method advocated in our paper results innon-parametric smoothing of (Xl, Ω˘l), l = 1, . . . , L. The NadarayaWatson estimatorat x ∈ E is Ωˆ(x) =[∑Ll=1wl(x)√Ω˘l]2, wherewl(x) = w˜l(x)/L∑l′=1w˜l′(x) & w˜l(x) = Kh (dg(x,Xl)) . (0.1)If one uses the pointwise dS-covariances Ω˘lj =√(Yli − Y l)(Yli − Y l)T for the one-stepprocedure of PM, using a NadarayaWatson estimator, the resulting estimator Ωˆ∗(x)2of Ω(x) would actually also result in a convex sum over the sample dS-covariances Ω˘l,namely Ωˆ∗(x) =[∑Ll=1w∗l (x)√Ω˘l]2, which we call the PM estimator, wherew∗l (x) =nl∑Lk=1wk(x)nkwl(x).Compare this with (0.1). We see that the weights in the PM estimator are scaled bya factor nl/(∑Lk=1wk(x)nk), the ratio between the number of observations at Xl, andthe NadarayaWatson estimator of the number of observations at x. Hence, in thePM estimator, the locations that have more observations (respectively less observa-tions) than the estimated number of observations at x will have their weight increased(respectively decreased) compared to our estimator. Whether this is in general desiredor not would require some theoretical investigation, and is left for future research. Inour case, this does not seem desirable, due to the observational nature of the dataset.4. Sharp Changes: PM mention the possibility of using a model that assumes that themean and dS-covariance field are piecewise constant. This is a very nice suggestion,that was hinted at by some colleagues. It is well-known by English speakers that certainregions of Great Britain have distinct accents, and therefore it would make sense to tryto find sharp transitions (if they exist) between distinct dialect regions This opens thedoor to detecting and estimating the boundaries where the sharp changes occur, andtesting whether the mean and dS-covariance fields are constant within these boundaries,and the suggestions made by PM give some initial ideas for investigating this.5. Cross-validation scale: Marron points out that in our cross-validation plots, thetuning parameter h should be displayed on the log scale, since it is a scaling parameter.It is true that this would have been preferable, but we do not believe that it wouldchange the results of the cross-validation.6. Sonic interpretations of the principal component (PC) loadings: Marronraises an excellent question when asking about the possibility of drawing some sim-ple sonic interpretations of the PC loadings. We provide as supplementary materialssounds that exhibit the effect of the PC loadings. If we denote by µ the MFCC ofthe vowel sound of a good quality recording of the word class, and let ϕ be the PCloading (MFCC) which we would like to interpret. Letting MFCCToSound denote thefunction that maps MFCCs to (playable) sound waves, we can listen to the effect ofthe PC loading ϕ by listening to MFCCToSound(µ + λϕ) for various values of λ ∈ R.Negative values of λ will correspond to low PC scores (in yellow on the left-hand sidesubplot of Figure 4 of the paper) whereas positive values of λ will correspond to highPC scores (in red in the same Figure). In order to make the interpretation easier, wereplace the vowel sound of the recording used to get µ by the generated vowel soundsMFCCToSound(µ+ λϕ), and play the entire word sound. These are sequentially playedforλ ∈ {−10,−7.78,−5.56,−3.33,−1.11, 1.11, 3.33, 5.56, 7.78, 10},for PCs 1 and 2, andλ ∈ {−4,−3.11,−2.22,−1.33,−0.44, 0.44, 1.33, 2.22, 3.11, 4},3for PC 3. The sounds can be found in the supplementary materials, in the filesounds/effect-pcX.wav, where X= 1, 2, 3. Listening to these files, we can hear thatnegative λs for PCs 1, 2 and 3 seem to correspond to the short and open from vowel [a](as in pat), whereas positive λs seem to correspond to the long back vowel [A] (aah),similar to the vowel in part. Interpreting the left-hand side color maps in Figure 4 ofthe paper reveals that we indeed capture with the PC 1 and PC 2 scores the expected[a]-[A] differences, while the interpretation of the map of PC 3 scores is less straightfor-ward. The sonic interpretation of the PC loadings requires however carefully listeningto these generated sounds (with high-fidelity headphones or loudspeakers), and is alsoconfounded by the rather low signal-to-noise ratio in the sound tokens used for theanalysis. Furthermore, the range of values for λ considered for these sound reconstruc-tions is greater than the ones found in the projected mean MFCC field: this reflectsthe great amount of noise in our non-parametric regression (indeed, the PC scores ofthe original vowel sound MFCCs are [−8.09, 13.27] for PC 1, [−8.69, 8.59] for PC 2,and [−5.39, 6.00] for PC 3). The noise is due to possible confounding effect of sex orage on the MFCCs, the microphone and room reverberation effect, in addition to thelow signal-to-noise in the recordings.We would like to thank again the discussants for their critical comments. We also extendour thanks to the associate editor and the three referees that all provided valuable feedbackon earlier versions of the paper.ReferencesBaíllo, A. & Grané, A. (2009), `Local linear regression for functional predictor and scalarresponse', Journal of Multivariate Analysis 100(1), 102111.Boj, E., Caballé, A., Delicado, P., Esteve, A. & Fortiana, J. (2016), `Global and localdistance-based generalized linear models', Test 25(1), 170195.Boj, E., Delicado, P. & Fortiana, J. (2010), `Distance-based local linear regression for func-tional predictors', Computational Statistics & Data Analysis 54(2), 429437.Petersen, A. & Müller, H.-G. (2016), `Fréchet integration and adaptive metric selection forinterpretable covariances of multivariate functional data', Biometrika 103(1), 103120.Petersen, A., Müller, H.-G. et al. (2019), `Fréchet regression for random objects with eu-clidean predictors', The Annals of Statistics 47(2), 691719.4

Rejoinder for “A spatial modeling approach for linguistic object data : analysing dialect sound variations across Great Britain”

http://wrap.warwick.ac.uk/126480/1/WRAP-rejoinder-spatial-modeling-approach-linguistic-object-data-Tavakoli-2019.pdf

Rejoinder for “A spatial modeling approach for linguistic object data : analysing dialect sound variations across Great Britain”

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