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Higher Order Cepstral Moment Normalization (HOCMN) for Robust Speech Recognition Speaker: Chang-wen Hsu Advisor: Lin-shan Lee 2007/02/08

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Outline Introduction CMS/CMVN/HEQ Higher Order Cepstral Moment Normalization (HOCMN) Even order HOCMN Odd order HOCMN Cascade system Fundamental principles Experimental Results Conclusions

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Introduction Feature normalization in cepstral domain is widely used in robust speech recognition: CMS: normalizing the first moment CMVN: normalizing the first and second moments Cepstrum Third-order Normalization (CTN): normalizing the first three moments (Electronics Letters, 1999) HEQ: normalizing the full distribution (all order moments) How about normalizing a few higher order moments only? Higher order moments are more dominated by higher value samples Normalizing only a few higher order moments may be good enough, while avoiding over-normalization

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Introduction Cepstral Normalization CMS: CMVN: Time progressively

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Introduction Histogram Equalization

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Higher Order Cepstral Moment Normalization If the distribution of the cepstral coefficients can be assumed to be quasi-Gaussian: Odd order moments can be normalized to zero Even order moments can be normalized to some specific values Define notation: X(n): a certain cepstral coefficient of the n-th frame X [k] (n): with the k-th moment normalized X [k,l] (n): with both the k-th and l-th moments normalized X [k,l,m] (n): with the k-th, l-th and m-th moments normalized HOCMN [k,l,m] : an operator normalizing the k-th, l-th and m-th moments For example

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Cepstral Moment Normalization Moment estimation: Time average of MFCC parameters Purpose: For odd order L For even order N

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Even order HOCMN Only the moment for a single even order N can be normalized and CMS can always be performed in advance Therefore, the new feature coefficients can be expressed as Let the desired value of the N-th moment of the new feature coefficient be, that is

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Even order HOCMN Aurora 2, clean condition training, word accuracy averaged over 0~20dB and all types of noise (sets A,B,C) CMVN=HOCMN [1,2]

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l Acc. [1,100] Even order HOCMN Evaluation of the expectation value for the moments Sample average over a reference interval Full utterance Moving window of l frames …… X(n-3) X(n-2) X(n-1) X(n) X(n+1) X(n+2) X(n+3) …… l to be normalized l =86 is best

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Experimental results CMVN (l=86) CMVN (full-utterance) Aurora 2, clean condition training, word accuracy averaged over 0~20dB and all types of noise (sets A,B,C)

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Odd order HOCMN (1/3) Besides the first moment (CMS), only another single moment of odd order L can be normalized in addition The L-th HOCMN can be obtained from the (L-1)-th HOCMN (which is for an even number as discussed previously) Then, the new feature coefficients can be expressed as a and c are to be solved

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Odd order HOCMN (2/3) To solve a and c The first moment is set to zero The N-th moment is set to zero After some mathematics and approximation

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Odd order HOCMN (3/3) Because the formula for a above is only an approximation, a recursive solution can be obtained in about two iterations

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Cascade system Cascading an odd order operator HOCMN [1,L] (L is an odd number) and an even order operator HOCMN [1,N] (N is an even number) can obtain an operator HOCMN [1,L,N]

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Experimental results CN CTN=HOCMN [1,2,3] CN (l=86) Aurora 2, clean condition training, word accuracy averaged over 0~20dB and all types of noise (sets A,B,C) CMVN CTN=HOCMN [1,2,3] CMVN (l=86)

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Skewness and Kurtosis Skewness Third moment about the mean and normalized to the standard deviation Pdf departure from symmetric Positive/negative indicate skew to right/left Zero indicate symmetric Kurtosis Fourth moment about the mean and normalized to the standard deviation Peaked or flat with tails of large size as compared to standard Gaussian 3 is the fourth moment of N(0,1) Positive/negative indicate flatter/more peaked

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Skewness and Kurtosis 1st-moment always normalized Define: Generalized skewness of odd order L L are not necessary 3 Similar meaning as skewness (skew to right or left) except in the sense of L – th moment Define: Generalized kurtosis of even order N N are not necessary 4 Similar meaning as kurtosis (peaked or flat) except in the sense of N – th moment

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Skewness and Kurtosis Normalizing odd order moment is to constrain the pdf to be symmetric about the origin Except in the sense of L-th moment Normalizing even order moment is to constrain the pdf to be equally flat with tails of equal size Except in the sense of N-th moment

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The order of normalized moments are not necessary integers Generalized moment Type 1: Reduced to odd order moment when u is an odd integer L (ex: L=1 or 3) Type 2: Reduced to even order moment when u is an even integer N (ex: N=2 or 4) HOCMN with non-integer moment orders Generalized Moments

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Experimental Setup Aurora2 database Training: Clean condition training Testing: Set A, B and C Development: All from clean training data 39-dimension feature coefficients C0~C12 MFCC, Δ, Δ 2 Normalization performed on C0~C12

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Experimental Results Higher order moments can derive more robust features Normalizing only three orders of moments are better than full distribution

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Experimental Results

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Experimental Results

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PDF Analysis HEQ Over fitting to Gaussian Loss original statistics HOCMN Fitting the generalized skewness and kurtosis Retain more speech nature HEQ HOCMN Original C0 & C1

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Distance Analysis Distance definition: HOCMN can derive smaller distance between clean and noisy speech distance reduction has similar trend as error rate reduction

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Experimental Results Slight improvement for HOCMN with non-integer order moments Especially for lower SNR values Other robust techniques can be combined with it

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Experimental Results

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Experimental Results For multi-condition training: HOCMN performs better than CMVN for all SNR values Better than HEQ for higher SNR values

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Conclusions We proposed a unified framework for higher moment order cepstral normalization Normalization of higher moment order gives more robust features Parameter set can be appropriately selected by development set Skewness/kurtosis/distance analysis can further demonstrate the concepts of the normalization techniques

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