Method for noise adaptation in automatic speech recognition using transformed matrices

Abstract

The improved noise adaptation technique employs a linear or non-linear transformation to the set of Jacobian matrices corresponding to an initial noise condition. An α-adaptation parameter or artificial intelligence operation is employed in a linear or non-linear way to increase the adaptation bias added to the speech models. This corrects shortcomings of conventional Jacobian adaptation, which tend to underestimate the effect of noise. The improved adaptation technique is further enhanced by a reduced dimensionality, principal component analysis technique that reduces the computational burden, making the adaptation technique beneficial in embedded recognition systems.

Description

BACKGROUND AND SUMMARY OF THE INVENTION

The present invention relates generally to automatic speech recognition systems. More particularly, the invention relates to techniques for adapting the recognizer to perform better in the presence of noise.

Current automatic speech recognition systems perform reasonably well in laboratory conditions, but degrade rapidly when used in real world applications. One of the important factors influencing recognizer performance in real world applications is the presence of environmental noise that corrupts the speech signal. A number of methods, such as spectral subtraction or parallel model combination, have been developed to address the noise problem. However, these solutions are either too limited or too computationally expensive.

Recently, a Jacobian adaptation method has been proposed to deal with additive noise, where the noise changes from noise A to noise B. For example, U.S. Pat. No. 6,026,359 to Yamaguchi describes such a scheme for model adaptation in pattern recognition, based on storing Jacobian matrices of a Taylor expansion that expresses model parameters. However, for this method to perform well it is necessary to have noise A and noise B close to one another in terms of character and level. For example, the Jacobian adaptation technique is likely to work well where noise A is measured within the passenger compartment of a given vehicle travelling on a smooth road at 30 miles an hour, and where Noise B is of a similar character, such as the noise measured inside the same vehicle on the same road travelling at 45 miles per hour.

The known Jacobian adaptation technique begins to fail where noise A and B lie farther apart from one another, such as where noise A is measured inside the vehicle described above at 30 miles per hour and noise B is measured in the vehicle with windows down or at 60 miles per hour.

This shortcoming of the proposed Jacobian noise adaptation method limits its usefulness in many practical applications because it is often difficult to anticipate at training time the noise that may be present at testing time (when the system is in use). Also, improvements in Jacobian noise adaptation techniques are limited in many applications because the computational expense (processing time and/or memory requirements) needed makes them impractical.

The present invention addresses the foregoing shortcoming. Instead of using Jacobian matrices, the invention uses a transformed matrices which resembles the form of a Jacobian matrix but comprises different values. The transformed matrices compensate for the fact that the respective noises at training time and at recognition time may be far apart. The presently preferred embodiment of the inventive method effects a linear or non-linear transformation of the Jacobian matrices using an α-adaptation parameter to develop the transformed matrices. The transformation process can alternatively be effected through other linear or non-linear transformation means, such as using a neural network or other artificial intelligence mechanism. To speed computation, the resulting transformed matrices may be reduced through a dimensionality reduction technique such as principal component analysis.

For a more complete understanding of the invention, its objects and advantages, refer to the following specification and to the accompanying drawings.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 diagrammatically represents different noise conditions, useful in understanding the invention;

FIG. 2 is a data flow diagram of both training and recognition phases, illustrating a presently preferred implementation of the improved Transformed Matrix adaptation;

FIG. 3 is a log-spectral plot comparing conventional Jacobian Adaptation with Parallel Model Combination (PMC) adaptation;

FIGS. 4 and 5 are α-adaptation parameter curves, showing the effect of different α values upon recognition accuracy.

DESCRIPTION OF THE PREFERRED EMBODIMENT

FIG. 1 illustrates the problem that the present invention is designed to solve. As shown at 10 , assume, for example, that the automatic speech recognition system must work in a noisy environment, such as within the passenger compartment of a moving vehicle. The noise level measured within the passenger compartment typically increases from noise A to noise A′ as the vehicle speeds up. Although the noise level may increase from A to A′, the character or quality of the noise remains largely the same. In a moving vehicle, for example, the noise spectrum typically changes in a predictable way as the vehicle speeds up. Wind noise increases in amplitude but retains its largely random white noise or pink noise character. Road surface noise (the sound of the tires rolling upon the road surface) increases in frequency in proportion to the increase in speed.

Unfortunately, in many real-world applications, the character and quality of the ambient noise cannot be as readily predicted as the conditions at 10 in FIG. 1 would imply. Consider the portable cellular telephone, for example. The cellular phone may be used inside a moving vehicle where it experiences the range of noises illustrated at 10 ; or it may be used on a street corner where completely different traffic sounds abound; or it may be used in a shopping mall with yet an entirely different noise quality. This wide diversity in different noise qualities is illustrated in FIG. 1 at 12 , where three different noise patterns have been diagrammatically depicted as noise A, noise B and noise C. The unpredictability of noise quality has heretofore presented a significant challenge for automatic speech recognition systems that must perform in these varied noisy environments.

FIG. 2 illustrates an exemplary embodiment of the invention in a speech recognition application. The speech recognition application employs a model-based recognizer. The models are developed during training and are then later used during recognition. In FIG. 2 , the training phase of the system is illustrated generally at 20 and the recognition phase at 40 . Input speech is provided during the training phase under noise condition A, as illustrated at 22 . The input speech is used to train speech models as indicated at step 24 , with exemplary speech models diagrammatically represented at 26 . In the typical input speech signal there will be times during which no speech is present, such as prior to the beginning of speech or after the end of speech. These non-speech portions may be used to record data indicative of the reference noise N a that is associated with noise condition A. In FIG. 2 , the reference noise N a is stored at block 28 . If desired, the noise may be modeled (background model) using the same training operation used to construct the speech models 26 .

After training the speech models, a set of Transformed Matrices is calculated at step 30 and stored at 32 . These matrices are used during recognition to adapt the speech models so that they will perform better under the noise conditions existing during recognition. The basic Jacobian adaptation process assumes that the quality of the noise during recognition time is approximately the same as during training time. Otherwise, classic Jacobian adaptation may produce less than optimal results.

The improved adaptation technique is based on the use of a set of Transformed Matrices generated for the initial noise condition N a . The Transformed Matrices are computed by applying a linear or non-linear transformation to a set of the Jacobian matrices developed for the initial noise condition N a . The presently preferred embodiments perform the transformation (both linear and non-linear) by applying an α-adaptation factor as presented in the next section. While the α-adaptation factor is presently preferred, a neural network or other artificial intelligence component may be used to effect the transformation.

The manipulation of matrices can be a highly computationally expensive process. A considerable cost factor is the memory space needed to store all of the matrices. In a typical embodiment, the speech models for each entry in the lexicon may employ multiple Hidden Markov Model states, with multiple Gaussian densities associated with each state. There would thus be one matrix for each of the Gaussians in each of the states. This could result in several hundred matrices needing to be stored.

The preferred embodiment performs a matrix decomposition step 36 to generate a reduced-complexity set of Jacobian matrices 38 . As will be more fully discussed below, the presently preferred decomposition technique uses principal component analysis (PCA) to construct the reduced-complexity Transformed Matrices.

At recognition time, input speech from the user is provided at step 42 . The input speech is associated with a noise condition B (also referred to as the target noise N b ) as illustrated at 44 . As previously discussed, if the noise condition B is different in quality from the noise condition A used at training time, the traditional Jacobian adaptation technique may produce less than optimal results. However, we have found that the α-adaptation process (performed when the Jacobian matrices are defined during training) greatly improves recognition performance under adverse noise conditions. Results of our tests are provided in the example presented below.

The target noise N b , shown at 44 in FIG. 2 , is extracted from the input speech 42 and then used to calculate the difference from the reference noise N a as indicated at 46 . New adapted speech models are then calculated using this noise difference and the reduced Transformed Matrices developed during training, as illustrated at 48 . The resulting adapted speech models 50 are then used at 52 to perform speech recognition on the input speech 42 to provide the recognition output 54 .

Alpha Adaptation

To better appreciate how our Transformed Matrices work, it is useful to understand conventional Jacobian adaptation. Conventional Jacobian adaptation is related to another form of adaptation known as parallel model combination (PMC). Traditionally, Jacobian adaptation is used as an approximation for PMC, in order to reduce the computational burden that PMC requires. PMC is highly computationally expensive because, for each density of the speech model, mean vectors must be transformed into the spectral domain. Then, after adding the mean vector to the target noise, the resulting vector must be transformed back into the cepstral domain. This double transformation, which makes use of a matrix multiplication and of two non-linear functions, is usually too time consuming for embedded systems.

Traditionally, Jacobian adaptation is used as an approximation of PMC in the cepstral domain. For comparison purposes, Equation 1, reproduced below, describes the PMC calculation, where capital F represents the matrix of Discrete Cosign Transform (DCT). Equation 2 represents the traditional Jacobian adaptation calculation that is used as an approximation of the more computationally costly PMC calculation.

Equation 1

C ( S+N ) =F· log(exp( F −1 ·C ( S )))+exp( F −1 ·C ( N ))

Equation 2

$$ Δ ⁢   ⁢ C ⁡ ( S + N ) = ∂ C ⁡ ( S + N ) ∂ C ⁡ ( N ) · Δ ⁢   ⁢ C ⁡ ( N ) = F · N S + N · F - 1 · Δ ⁢   ⁢ C ⁡ ( N ) EQUATION  2

The problem with the Jacobian adaptation approximation is that it holds only where the target noise (experienced during system use) is similar in quality to the reference noise (present during system training). The nature of the problem is illustrated in FIG. 3 , which plots the evaluation, in the log-spectral domain, of the noisy speech parameters when the noise increases. Specifically, the plots show in the log-spectral domain how PMC adaptation and conventional Jacobian adaptation compare. In FIG. 3 the shaded region on the left corresponds to the condition where the speech signal is far more powerful than the noise, whereas the region on the right corresponds to conditions where the noise is more powerful than the speech signal. If both the training and testing environments are in the same of these two regions, then Jacobian Adaptation and PMC perform similarly. However, if one of these two environments is in the middle region, or if it is in another region than the other environment, then Jacobian Adaptation differs from PMC, and actually always underestimates the adaptation of the models.

We have discovered that the conventional Jacobian adaptation can be greatly improved through a linear or non-linear transformation of the Jacobian matrices. To effect the linear or non-linear transformation, the preferred embodiments employ a parameter that we call an α-adaptation parameter. Equation 3 below illustrates the presently preferred use of the α-adaptation parameter to effect a non-linear transformation. Equation 4 shows an alternate use of the parameter to effect a linear transformation. As noted above, while the use of an α-adaptation parameter to effect the transformation is presently preferred, other transformation techniques are also possible. For example a neural network or other artificial intelligence component may be used to transform Jacobian matrices for the initial noise condition. Another transformation technique involves applying a first α-adaptation parameter or factor to the input speech and a second α-adaptation parameter or factor to the noise. Other variations are also possible. $$ Δ ⁢   ⁢ C ⁡ ( S + N ) = F · α ⁢   ⁢ N S + α ⁢   ⁢ N · F - 1 · Δ ⁢   ⁢ C ⁡ ( N ) EQUATION  3

Equation 4

$$ JA :   ⁢ Δ ⁢   ⁢ C ⁡ ( S + N ) = ∂ C ⁡ ( S + N ) ∂ C ⁡ ( N ) · αΔ ⁢   ⁢ C ⁡ ( N ) EQUATION  4

Referring to Equation 3, the α-adaptation parameter functions as follows. If the reference noise is close to zero, and if α is not too large, then both tangents (computed respectively at x-coordinate N and αN) are horizontal. If the reference noise is very important, then both tangents will correspond to the line y=x. If the reference noise belongs to the central region of FIG. 3 , then the new slope of the tangent will be greater than the conventional Jacobian adaptation curve would have produced.

Use of the α-adaptation parameter in Equation 3 results in a non-linear transformation of the matrices. Both numerator and denominator are multiplied by the parameter, hence producing a non-linear transformative effect. In Equation 4 the α-adaptation parameter is multiplied against the resulting numerator/denominator quotient, hence producing a linear transformative effect.

In both cases, the main effect of the α-adaptation parameter is to increase the adaptation bias added to the speech models. This is useful because it corrects the shortcoming of conventional Jacobian adaptation to underestimate the effect of noise. In a later section of this document we present our experimental results, showing the improvements which are possible using the α-adaptation parameter.

Selection of the α-Adaptation Parameter

Theoretically, the optimal value of the α-adaptation parameter is dependent on the environment: the value for a should be greater where the mismatch between target noise and reference noise is greater. However, we have discovered that the α-adaptation parameter is far more stable than theory would have predicted. When used to generate Transformed Matrices as a replacement for conventional Jacobian adaptation, the variation in speech recognition accuracy is low for small values of α, increases for medium values of α and becomes low again when α increases beyond a certain point. This phenomenon is due to the shape of the curve in FIG. 3 . Specifically, whatever the value of α is, the slope of the tangent will only vary between 0 and 1.

To clarify this point, we have realized a set of experiments for digits recognition in adverse environments. Twelve context-independent models of digits were constructed: the numbers from 1 to 9 plus models for “o” and “zero”, plus a model for silence. The silence was modeled by a Hidden Markov Model (HMM) with five states. The remaining models used fifteen states. Each state of all the HMMs employs four Gaussian densities. The training set used to train the models comprised 3803 sequences of digits, pronounced by 80 speakers. The training set was recorded under laboratory conditions without noise. FIGS. 4 and 5 represent the variation of accuracy when α is varied in a range from 1 to 4. The data was generated based on six different acoustic environments:

The validation corpus, which is recorded in clean conditions.

The same corpus, with added carnoise with a SNR of 10 dB.

The same corpus, with added car noise with a SNR of 0 dB.

The same corpus, with added white noise with a SNR of 15 dB.

The test corpus, recorded in a car at 30 mph.

Another test corpus, recorded in a car at 60 mph.

Referring to FIGS. 4 and 5 , it can be seen that whatever the acoustic environment is, the variation of accuracy for different values of α is very low in the range of α=2.4 to α=3.6. This shows that α has a stable range that may be suitably exploited in a practical embodiment of the invention. While we presently prefer an α-adaptation parameter between about 2.4 to 3.6, it will be understood that this is merely representative of one possible stable range. In general, other values of α may be used with beneficial results. Stated differently, the decrease of accuracy between the true “optimal” value of α and any other value of α that may be chosen in a considered range (e.g. 2.4-3.6) is very low. Our data shows that the decrease in accuracy from the “optimal” point is less than three percent. This makes our improved Jacobian adaptation a very robust method.

Dimensionality Reduction for Reducing Computational Expense

As noted above, although Jacobian adaptation is less computationally expensive than PMC, it still places a fairly taxing burden on the recognition system, particularly for embedded systems.

Indeed, we have seen that each Tranformed Matrix can be expressed by the following Equation 5:

Equation 5

$$ ∂ C ⁡ ( S + N ) ∂ C ⁡ ( N ) · = F · S + α ⁢   ⁢ N α ⁢   ⁢ N · F - 1 EQUATION  5

where $$ S + α ⁢   ⁢ N α ⁢   ⁢ N

is a diagonal matrix with dimensions NFiltXNFilt, where Nfilt is the number of filters used in the spectral filter-bank.

Thus, each Transformed Matrix can be expressed as the weighted sum of Nfilt canonical matrices, which are in fact a base of the space to which Jacobian matrices belong. These canonical matrices are defined by:

J i =F·diag ( i ) ·F −1

where diag(i) refers to a diagonal Nfilt×Nfilt matrix with 0 everywhere but 1 at position i.

Each Transformed Matrix can thus be expressed as:

Equation 6

$$ ∂ C ⁡ ( S + N ) ∂ C ⁡ ( N ) = ∑ i = 1 Nfilt ⁢   ⁢ γ i · J i EQUATION  6

Thus, instead of storing Nd matrices (where Nd is the total number of densities in all the speech models), it is enough to store Nfilt canonical matrices, plus Nd times Nfilt coefficients γ i . This considerably decreases the storage requirements.

However, this solution can be further improved because it increases the time-complexity of the algorithm: indeed, when all the Transformed Matrices are stored, Equation 2 can be applied directly to all the densities, which costs Nd matrix multiplication.

If the second solution is chosen, the right part of Equation 2 becomes: $$ ∂ C ⁢ ( S + N ) ∂ C ⁢ ( N ) · Δ ⁢   ⁢ C ⁢ ( N ) = ( ∑ i = 1 Nfilt ⁢   ⁢ γ i · J l ) · Δ ⁢   ⁢ C ⁢ ( N ) = ∑ i = 1 Nfilt ⁢   ⁢ γ i · ( J i · Δ ⁢   ⁢ C ⁢ ( N ) )

In this equation, the cost is Nfilt matrix additions, and Nfilt matrix multiplication by a scalar: this must be repeated for each density. The total cost is thus 2·Nd·Nfilt matrix operations.

If we do not want to use extra computational time, the number of canonical matrices has to be reduced.

The presently preferred technique to reduce the dimension of a space is to realize a Principal Component Analysis on the set of elements belonging to this space. We have thus first computed all the vectors $$ S + α ⁢   ⁢ N α ⁢   ⁢ N

and realized a single value decomposition on this set of vectors. The resulting canonical vectors have been used to compute the Nfilt canonical Jacobian matrices $$ F · S + α ⁢   ⁢ N α ⁢   ⁢ N · F - 1

sorted with descending order of their eigen-values.

Using principal component analysis, as described above, can yield considerable improvement in reducing computational burden. Experiments have shown that it is possible to reduce the number of useful canonical matrices down to five matrices. Even further reduction may be possible. Reducing the number of matrices decreases the space requirements as well as the computation time needed to perform the adaptation. To better understand the improvement achieved through dimensionality reduction (principal component analysis) Table I compares the Transformed Matrix adaptation process both with and without employing principal component analysis.

	TABLE I
						Comp.
	dim.	alpha	Clean	30 mph	60 mph	time
Without	32	2	98.74%	95.29%	90.21%	2439.28
PCA						us
With PCA	7	2.4	98.56%	95.70%	90.63%	1932.42
						us

In the above Table I the first column identifies the number of dimensions, that is the number of canonical matrices. The next column identifies the α-adaptation value used. The remaining columns give the percentage of recognition accuracy and the associated computation time required (cumulative time in, microseconds, of the adaptation over the whole database) for the following three environmental conditions: clean (no noise), vehicle at 30 miles per hour and vehicle at 60 miles per hour.

Experimental Results

The noise adaptation technique described in the foregoing were tested under various noise conditions. The results of our tests are reproduced in this section. To test the adaptation system a speech recognizer for a car navigation system was employed. Of course, the adaptation techniques described herein are not restricted to car navigation or any other recognition task. Car navigation was selected for our tests because noise conditions within a moving vehicle can vary quite widely over different vehicle speeds. Thus a test of the adaptation system in a vehicular environment was selected as a good measure of the adaptation system's capabilities.

The experimental setup of these experiments is the same as previously described. Three testing sets were constructed: (1) comprising a validation set, composed of 462 sequences of digits pronounced by 20 speakers (different than in the training set), recorded in the same conditions as used in the training set; (2) composed of 947 sequences of digits pronounced by different speakers and recorded in a car at 30 miles per hour; (3) composed of 475 sequences of five digits pronounced by the same speakers, but recorded in the car at 60 miles per hour.

Recognition was performed using a simple loop grammar, with equal transition probabilities for all the numbers (“o” and “zero” models the same number) and silence. Accuracy was computed on ten numbers, after removing the silences in the recognized sentences.

For these first experiments, the signal was coded into a sequence of vectors of nine PLP coefficients (including the residual error) plus nine delta coefficients. Adaptation, if performed, was applied to only the means of the first nine static coefficients. For adaptation, the target noise was computed using 30 first frames of each sentence.

The results reproduced in Table II below, compare the performance of the Hidden Markov Models (HMM) without adaptation with the results obtained using parallel model combination (PMC) and traditional Jacobian adaptation (JA). Table II thus shows how both parallel model combination and Jacobian adaptation improve recognition performance in the presence of noise. Table II does not, however, show the performance of the improved Transformed Matrix adaptation using α-adaptation. This table is presented to serve as a baseline against which the improved Transformed Matrix adaptation technique may be better understood.

	TABLE II
		Clean
	system	validation	30 mph	60 mph
	HMM (no	98.84%	56.27%	35.83%
	adaptation)
	PMC	95.78%	91.72%	89.60%
	JA	98.66%	83.76%	70.02%

	TABLE III
	Adaptation	alpha	clean	30 mph	60 mph
	No		98.84%	56.27%	35.83%
	PMC		95.78%	91.72%	89.60%
	JA		98.66%	83.76%	70.02%
	α-PMC	1.3	96.03%	91.67%	89.81%
	α-TM	3	98.73%	95.24%	89.81%

Table III shows the comparative performance of both parallel model combination and Jacobian adaptation, with and without the alpha factor. In Table III the Transformed Matrix adaptation with α-adaptation is designated “α-TM.” For comparison purposes, the alpha factor was also applied in the parallel model combination technique as shown in the row designated “α-PMC.”

Comparing the results in Table III, note that the improved Transformed Matrix adaptation (α-TM) performs significantly better than standard Jacobian adaptation (JA) in the presence of noise. While the alpha factor did not substantially degrade the performance of PMC adaptation, it did not provide significant improvement either.

The results of our experiments show that the improved Transformed Matrix adaptation technique, employing the α-adaptation factor, gives considerably better results that standard Jacobian adaptation. Moreover, because Transformed Matrix adaptation is inherently less computationally expensive than PMC, it becomes an ideal candidate for embedded recognition systems that do not have a lot of processing power or memory. Such applications include, for example, cellular telephone recognition systems and vehicular navigation systems and other consumer products.

In addition, still further improvements in system performance can be had through use of the dimensionality reduction techniques described herein. When combined with Transformed Matrix adaptation, the result is a compact, efficient and robust adaptation system that will serve well in many recognition applications.

While the invention has been described in its presently preferred embodiments, it will be understood that the invention is capable of modification without departing from the spirit of the invention as set forth in the appended claims.

Claims

1. A method of performing noise adaptation in a speech recognition system, comprising:constructing a set of speech models under a first noise condition; providing a set of Jacobian matrices for said speech models under said first noise condition, transforming said Jacobian matrices to define a set of transformed matrices and storing said transformed matrices for use in speech recognition, said transforming step is performed by applying an alpha adaptation factor to said Jacobian matrices; providing input speech under a second noise condition; determining a first change in noise condition based on said first and second noise conditions; and using said first change in noise condition and said transformed matrices to adapt said set of speech models.

2. The method of claim 1 wherein said step of transforming said Jacobian matrices is performed using a linear transformation.

3. The method of claim 1 wherein said step of transforming said Jacobian matrices is performed using a non-linear transformation.

4. The method of claim 1 wherein said alpha adaptation factor comprising a weighting factor having a value substantially associated with the numerical range of 2.4 to 3.6.

5. The method of claim 1 further comprising decomposing said transformed matrices using a dimensionality reduction process.

6. The method of claim 5 wherein said dimensionality reduction process employs principal component analysis.

7. A method of constructing adaptation matrices for noise adaptation in a speech recognition system developed during training time and used at use time, comprising:constructing a set of speech models under a first noise condition associated with said training time; calculating a set of Jacobian matrices for said speech models under said first noise condition; transforming said set of Jacobian matrices using a predetermined transformation operation to compensate for differences between the noise at training time and the noise at use time, said transformation operation comprises applying an alpha adaptation factor to said Jacobian matrices; and storing said transformed set of matrices as adaptation matrices for use by said speech recognition system during use time.

8. The method of claim 7 wherein said transforming operation comprises performing a linear transformation upon said Jacobian matrices.

9. The method of claim 7 wherein said transforming operation comprises performing a non-linear transformation upon said Jacobian matrices.

10. The method of claim 7 wherein said alpha adaptation factor comprising a weighting factor having a value substantially associated with the numerical range of 2.4 to 3.6.

11. The method of claim 7 further comprising decomposing said adaptation matrices using a dimensionality reduction process.

12. The method of claim 11 wherein said dimensionality reduction process employs principal component analysis.

13. A speech recognizer comprising:a set of speech models trained under a first noise condition; an adaptation system that adapts said speech models in accordance with a second noise condition; a memory associated with said adaptation system containing a stored set of adaptation matrices corresponding to said speech models; and wherein adaptation matrices comprise transformed matrices constructed by applying a linear or non-linear transformation to a set of Jacobian matrices for said first noise condition, said transformation of said Jacobian matrices employs an alpha adaptation factor.

14. The recognizer of claim 13 wherein said alpha adaptation factor applied to said Jacobian matrices as a weighting factor having a value substantially associated with the numerical range of 2.4 to 3.6.

15. The recognizer of claim 13 wherein said set of adaptation matrices are decomposed through dimensionality reduction.

16. The recognizer of claim 13 wherein said set of adaptation matrices are decomposed by principal component analysis.