Lecture 3 Identification

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    Lecture #3 Identification

    ECE 842

    Ali Keyhani

    Identification is the determination on the basis of input and output, of a process (model), within a specified

    class of processes (models), to which the process under test is equivalent.

    y(.)

    u(.)

    w(.)

    y(.)

    u(.)

    w(.)

    Process

    Equivalent

    Model

    The equivalent model is a model which has captured the underlying mechanism which has generated

    the original process.

    Three Step Modeling Procedure for Process {y(.)} excited by white noise

    1. Model IdentificationMeans the determination of n and m for process.2. Parameter Estimation; The determination of { ^ )(ka } from { Y(j), J < k }3. Diagnostic Checking

    Checking the {w(.)} sequence to establish that the noise indeed is white.

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    Classical Approach to Identification and Modeling

    Given

    Build The Model OverThis Portion of PastHistory

    Test The Model OverThis Portion of PastHistory

    A 3 Step Procedure for Modeling IS:

    Identify A CandidateModel

    Estimate The Parameter ofThe Identified Model

    Diagnostic CheckingIs The Noise

    White?

    Go To IdentifyAnother Model

    Done

    No

    Yes

    k

    Y(k)

    Covariance: A partial indication of the degree to which one variable is related to another is given by the

    covariance, which is the expectation of the product of the deviation of two random variables from their

    means.

    {Y(.)} random process Y(.)

    {x(.)} random process x(.)

    Cov {x(k) Y(k)} = E [(x(k) ux) (Y(k) uy)]

    = E [x(k) Y(k)] ux uy

    Where

    ux = E [x(k)]

    uy = E [Y(k)]

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    Cross Correlation Coefficient

    yx

    yxuukYkxE

    =

    )]()([

    Variance of x ]))([( 22 xx ukxE =

    Variance of y ]))([( 22yy

    ukYE =

    Notes:

    1) The correlation coefficient is a measure of the degree of linear dependence between x and y.2) If x and y are independent, = 0 (the inverse is not true).3) If Y is a linear function of x, = 1

    Zero Mean Process: Given process {Y(.)}, if the mean of the process is estimated and then subtracted

    from each Y(k), we obtain a zero mean process.

    Ex.

    N

    kkY 11 )}({ =

    = )(1

    11

    ^

    kYN

    u y

    1

    ^

    1 )1()1( yuYY =

    1

    ^

    1

    1

    1

    ^

    1

    )()(

    .

    )}({.

    .

    )2()2(

    y

    N

    k

    y

    ukYkY

    processmeanzeroaiskY

    uYY

    =

    =

    =

    For zero mean processes {x(.)} and {Y(.)}

    Cov { x(k) Y(k) } = E [ x(k) Y(k) ] = xy

    xy is called cross covariance or correlation between x and Y variables

    The basic cross correlation or covariance equation

    E [ x(k) Y(k+j) ] = xy (j)

    E [ Y(k) x(k+j) ] = yx (-j); i.e. xy (j) =yx (-j)

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    For j = 0

    E [ x(k) Y(k) ] = xy =xy (0)

    The sample estimate of cross correlation for covariance,

    mjjkYkxjN

    jjN

    k

    xy .,.........1,0)()(1

    )(1

    ^

    =+

    =

    =

    mjjkxkYjN

    jjN

    k

    yx .,.........1,0)()(1

    )(1

    ^

    =+

    =

    =

    Sample cross correlation coefficient of zero mean {Y(.)} and {x(.)} processes.

    yx

    xy

    kYkxE^^

    ^^ )]()([)0(

    =

    =

    ==N

    k

    xykYkx

    NkYkxE

    1

    ^^

    )()(1

    )]()([)0(

    yx

    xy

    xy ^^

    ^

    ^ )0()0(

    =

    =

    ==N

    k

    xxkx

    N 1

    22^

    0

    ^

    ))((1

    1

    =

    ==N

    k

    yykY

    N 1

    22^

    0

    ^

    ))((1

    1

    J Period apart cross correlation for process {x(.)} and {Y(.)} with zero mean.

    xy (j) = E [ x(k) Y(k+j) ] = xy (j)

    Cross correlation coeff.

    yx

    xy

    xy

    jj

    )()( =

    Same cross correlation coeff.

    yx

    xyxy ^^

    ^

    ^

    )0()0(

    =

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    Historically, the correlation function is defined without restriction on mean and without normalization:

    Correlation function mjjkYkxjN

    jRN

    k

    xy .,.........1,0)()(1

    )(1

    1

    ^

    =+

    =

    =

    )]()([)(^^

    jkYkxEjRxy +=

    The covariance function has the mean value removed

    mjujkYukxjN

    jCjjN

    k

    yxxyxy .,.........1,0])(][)([1

    )()(1

    ^

    =+

    ==

    =

    In practice, the mean value usually has been subtracted from the data.

    The correlation coefficient function

    yx

    xy

    xy

    jj

    )()( = , )0(xx = , )0(yy =

    j Period Apart, auto-covariance for process {Y(.)} with zero mean E [Y(k)] = 0

    (j) = e [ (Y(k) Y(k+j) ]

    Autocorrelation coefficient

    0

    )()(

    jj =

    Sample correlation coefficient

    0

    ^

    ^

    0

    ^

    ^^ )()]()([

    )(

    jjkYkYE

    j =+

    =

    Where

    ..........2,1,0)()(1

    )]()([)(1

    ^^

    =+

    =+=

    =

    jjkYkYjN

    jkYjYEjjN

    k

    yy

    =

    =N

    k

    kYN 1

    2

    0

    ^

    ))((1

    1

    Note that j period apart autocorrelation is a partial indication of the degree to which present is dependent to

    the past. i.e.

    1) The autocorrelation is a measure of the degree of linear dependence between the present and thepast.

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    2) If Y(k) and Y(k+j) are independent, then 0)(^ =yyy .3) 20)0( =YYR i.e the mean-square value of the random process can always be obtained by setting the

    j = 0.

    4) )()( jRjR YYYY = 5) )0()( YYYY RjR the largest value of the autocorrelation always occurs at j = 0.