EE513 Audio Signals and Systems Statistical Pattern Classification Kevin D. Donohue Electrical and Computer Engineering University of Kentucky
EE513Audio Signals and Systems
Statistical Pattern ClassificationKevin D. Donohue
Electrical and Computer EngineeringUniversity of Kentucky
Interpretation of Auditory ScenesHuman perception and cognition greatly exceeds any computer-based system for abstracting sounds into objects and creating meaningful auditory scenes. This perception of objects (not just detecting acoustic energy) allows for interpretation of situations leading to an appropriate response or further analyses.
Sensory organs (ears) separate acoustic energy into frequency bands and convert band energy into neural firingsThe auditory cortex receives the neural responses and abstracts an auditory scene.
Auditory ScenePerception derives a useful representation of reality from sensory input.Auditory Stream refers to a perceptual unit associated with a single happening (A.S. Bregman, 1990) .
Acoustic to Neural
Conversion
Organize into Auditory Streams
Representation of Reality
Computer InterpretationIn order for a computer algorithm to interpret a scene
Acoustic signals must be converted to numbers using meaningful models.Sets of numbers (or patterns) are mapped into events (perceptions).Events are analyzed with other events in relation to the goal of the algorithm and mapped into a situation (cognition or deriving meaning).Situation is mapped into an action/response.
Numbers extracted from the acoustic signal for the purpose of classification (determination of event) are referred to as features.
Time -based features are extracted from signal transforms such as:EnvelopeCorrelations
Frequency-based features are extracted from signal transforms such as:Spectrum (Cepstrum)Power Spectral Density
Feature Selection Example Consider a problem of discriminating between the spoken words yes and no based on 2 features:1. The estimate of first formant frequency g1 (resonance of the
spectral envelope)2. The ratio in dB of the amplitude of the second formant frequency
over the third formant frequency g2.
A fictitious experiment was performed and these 2 features were computed for 25 recordings of people saying these words. The feature were plotted for each class to develop an algorithm to classify these samples correctly.
Feature PlotDefine a feature vector.
Plot G, given a yes was spoken, with green o’s, and given a nowas spoken, be wiht red x’s.
⎥⎦⎤
⎢⎣⎡=
2
1ggG
Minimum Distance ApproachCreate representative vector for yes and no features
For a new sample with estimated features, use decision rule:
Results in 3 incorrect decisions.
∑=
=25
1)|(
251
nyes yesnGμ
∑=
=25
1)|(
251
nno nonGμ
yesno
yes
noμGμG −≥
Normalization With STDThe frequency features had larger values than the amplitude ratios, and therefore had more influence in the decision process.
Remove scale differences by normalizing each feature by its standard deviation over all classes.
Now 4 errors result (why would it change?)
( ) ( ) ⎟⎟⎠
⎞⎜⎜⎝
⎛−+−= ∑ ∑
= =
25
1
25
1
2|
2| )|()|(25
1
n nnoiiyesiii μnongμyesngσ
Minimum Distance ClassifierConsider feature vector x with the potential to be classified as belonging to K exclusive classes. Classification decision will be based on the distance of the feature vector to one of the template vectors representing each of the K classes. The decision rule is for a given observation x and set of template vectors zk for each class, decide on class k such that:
[ ])()(minarg kTkkk
D zxzx −−=
Minimum Distance ClassifierIf some features need to be weighted more than others in the decision process, as well as exploiting correlation between the features, the distance for each feature can be weighted to result in the weighted minimum distance classifier:
where W is a square matrix of weights with dimension equal to length of x. If W is a diagonal matrix, it simply scales each of the features in the decision process. Off diagonal terms scale the correlation between features. If W is the inverse of the covariance matrix of the features in x, and zk is the mean feature vector for each class, then the above distances are referred to as the Mahanalobis distance.
[ ])()(minarg kTkkk
D zxWzx −−=
[ ] ( )( )[ ]1
1E1 E
−
=⎟⎠⎞
⎜⎝⎛
∑ −−==K
k
Tkkk kK
k zxzxWxz
Correlation ReceiverIt can be shown that selecting the class based on the minimum distance between the observation vector and the template vector is equivalent to finding the maximum correlation between the observation vector and the template:
or
where the template vectors have been normalized such that
[ ] [ ]kTkk
kT
kkk
CD zxzxzx ==−−= maxargminarg )()(
kPkTk allfor constant) a is (P =zz
[ ] [ ]kTkk
kT
kkk
CD WzxzxWzx ==−−= maxargminarg )()(
DefinitionsRandom variable (RV) is a function that maps events (sets) into a discrete set of real numbers for a discrete RV, or a continuous set of real numbers for a continuous RV.
Random process (RP) is a series of RVs indexed by a countable set for a discrete RP, or by a non-countable set for continuous RP.
Definitions: PDF First Order
The likelihood of RV values is described through the probability density function (pdf).
[ ] ∫=
Definitions: Joint PDF
The probabilities describing more than one RV is described by a joint pdf.
( ) ( )[ ] ∫ ∫=
Definitions: Conditional PDFThe probabilities describing a RV given that the another event has already occurred is described by a conditional pdf.
Closely related to this is Bayes’ rule:)(
),()|(| ypyxpyxp
Y
XYYX =
)()()|(
)|(
)()|(),()()|(
||
||
xpypyxp
xyp
xpxypyxpypyxp
X
YYXXY
XXYXYYYX
=
==
Examples: Gaussian PDFA first order Gaussian RV pdf (scalar x) with mean µ and standard deviation σ is given by:
A higher order joint Gaussian pdf (column vector x) with mean vector m and covariance matrix ∑ is given by:
⎟⎟⎠
⎞⎜⎜⎝
⎛ −−= 2
2
2 2)(exp
21)(
σμ
πσxxpX
( )[ ][ ][ ]T
Tn
Tn
xxx
p
))((EE
,,
)()(21exp
21)(
21
12/12/X
mxmxxm
x
mxmxx
−−=∑
==
⎟⎠⎞
⎜⎝⎛ −∑−−
∑= −
L
π
Example UncorrelatedProve that for an Nth order sequence of uncorrelated Gaussian zero-mean RVs the joint PDF can be written as:
Note that for Gaussian RVs uncorrelated implies statistical independence.Assume variances are equal for all elements. What would the autocorrelation of this sequence look like?How would the above analysis change if RVs were not zero mean?
∏=
⎟⎟⎠
⎞⎜⎜⎝
⎛−=
N
i i
i
i
Xxp
12
2
2 2)(exp
21)(
σπσx
Class PDFsWhen features are modeled as RVs, their pdfs can be used to derive distance measures for the classifier, and an optimal decision rule that minimizes classification error can be designed.Consider K classes individually denoted by ωk. Feature values associated with each class can be described by:a posteriori probability (likelihood the class after observation/data)
a priori probability (likelihood the class before observation/data)
Likelihood function (likelihood observation/data given a class)
)( xkkp ω
)( kp ωxx
)( kkp ω
Class PDFsThe likelihood function can be estimated through empirical studies. Consider 3 speakers whose 3rd formant frequency is distributed by:
Classifier probabilities can be obtained from Bayes’ rule
)()()(
)(xppxp
xpx
kkkxkk
ωωω =
Decision Thresholds
)( 1ωxpx
)( 2ωxpx
)( 3ωxpx
Maximum a posteriori Decision RuleFor K classes and observed feature vector x, the maximum a posteriori (MAP) decision rule states:
or by applying Bayes’ rule:
For the binary case this reduces to the (log) likelihood ratio
ijppω jkiki ≠∀> )()( if Decide xx ωω
ijp
pppω
ik
jkjii ≠∀> )(
)()()( if Decide
ωωω
ωx
x xx
( ) ( ) ⎟⎟⎠
⎞⎜⎜⎝
⎛
>
<
)()(
ln)(ln)(ln )()(
)()(
ik
jkji
ik
jk
j
i
pp
pppp
pp
i
j
i
j
ωω
ωωωω
ωω
ω
ω
ω
ω
xxxx
xxx
x
ExampleConsider a 2 class problem with Gaussian distributed feature vectors
Derive the log likelihood ratio and describe how the classifier uses distance information to discriminate between the classes.
[ ][ ] [ ][ ][ ]2222
1111
2211
21
))((E
))((E
E E,,
ω
ω
ωω
T
T
TNxxx
mxmx
mxmx
xmxmx
−−=∑
−−=∑
==
= L
HomeworkConsider a 2 features for use in a binary classification problem. The features are Gaussian distributed are form feature vectorx = [x1, x2]T. Derive the log likelihood ratio and corresponding classifier for the 3 different cases listed below:• 1) 2)
3) 4)
Comment how each classifier computes “distance” and uses it in the classification process.
[ ]
⎥⎦
⎤⎢⎣
⎡=∑⎥
⎦
⎤⎢⎣
⎡=∑
−=−=
==
2.0008.0
,2.10
06.01,1 ]1,1[
5.0)()(
21
21
21TT
kk pp
mm
ωω[ ]
⎥⎦
⎤⎢⎣
⎡−
−=∑=∑
−=−=
==
5.02.02.05.0
1,1 ]1,1[
5.0)()(
21
21
21TT
kk pp
mm
ωω
[ ]
⎥⎦
⎤⎢⎣
⎡=∑⎥
⎦
⎤⎢⎣
⎡=∑
==
==
5.0005.0
,1.00
01.00,0
5.0)()(
21
21
21T
kk pp
mm
ωω
[ ]
⎥⎦
⎤⎢⎣
⎡=∑⎥
⎦
⎤⎢⎣
⎡=∑
−=−=
==
2.0008.0
,2.10
06.01,1 ]1,1[
8.0)( 2.0)(
21
21
21TT
kk pp
mm
ωω
Classification Error
Classification error is the percentage of decision statistics that occur on the wrong side of the threshold, scaled by the percentage of times such an event occurs.
1T
)( 1ωλλp
)( 2ωλλp
)( 3ωλλp
2T
∫∫∫∫∞−
∞
∞−
∞
+⎟⎟⎠
⎞⎜⎜⎝
⎛++=
2
2
1
1
)()()()()()()( 3322211T
kT
T
kT
ke dppdpdppdppp λωλωλωλλωλωλωλω λλλλ
Homework
For the previous example, write an expression for probability of a correct classification by changing the integrals and limits (i.e. do not simply write pc=1-pe)
Approximating a Bayes ClassifierIf density functions are not known:
Determine template vectors that minimize distances to feature vectors in each class for training data (vector quantization).
Assume form of density function and estimate parameters (directly or iteratively) from the data (parametric or expectation maximization).
Learn posterior probabilities directly from training data and interpolate on test data (neural networks).
EE513�Audio Signals and SystemsInterpretation of Auditory ScenesAuditory SceneComputer InterpretationFeature Selection Example Feature PlotMinimum Distance ApproachNormalization With STDMinimum Distance ClassifierMinimum Distance ClassifierCorrelation ReceiverDefinitionsDefinitions: PDF First OrderDefinitions: Joint PDFDefinitions: Conditional PDFExamples: Gaussian PDFExample UncorrelatedClass PDFsClass PDFsMaximum a posteriori Decision RuleExampleHomeworkClassification ErrorHomeworkApproximating a Bayes Classifier