# Chemometrics from basick to wavelet transform - Chau F.T

ISBN 0-471-20242-8

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-12 1 265

-11 -345 95

-10 -1,122 -38 11 628

-9 -1 255 -95 -6 460 340

-8 -915 -95 -13 005 255 195

-7 -255 -55 -11 220 -420 -195 2 145

-6 590 10 -3 940 290 -260 -2 860 110

-5 1 503 87 6 378 18 -117 -2 937 -198 18

-4 2 385 165 17 655 405 135 165 -135 -45 15

-3 3 ,155 235 28 190 790 415 3 755 110 -10 -55 5

-2 3 750 290 36 660 1 110 660 7 500 390 60 30 -30

-1 4 , 125 325 42 120 1 320 825 10 125 600 120 135 75

0 4 253 -339 44 003 1 393 883 11 063 677 143 179 131

1 4 , 125 325 42 120 1 320 825 10 125 600 120 135 75

2 3 750 290 36 660 1 110 660 7 500 390 60 30 -30

3 3 ,155 235 28 190 790 415 3 755 110 -10 -55 5

4 2 385 165 17 655 405 135 165 -135 -45 15

5 1 503 87 6 378 18 -117 -2 937 -198 18

6 590 10 -3 940 290 -260 -2 860 110

7 -255 -55 -11 220 -420 -195 2 145

8 -915 -95 -13 005 255 195

9 -1 255 -95 -6 460 340

10 -1,122 -38 11 628

11 -345 95

12 1 265

30,015 6 555 260 015 7 429 4 ,199 46 189 2 ,431 429 429 231

2.1.3. Kalman Filtering

Kalman filtering is a kind of optimal linear recursive estimation method. Its operation speed is very high , and relatively small memory space is required for computation. Kalman filtering has been extensively used in engineering , especially in space technology. Recursive operation is the key feature of the method. Here we will first introduce what recursive operation is before discussing Kalman filtering in detail.

The basic idea of recursive operation is its efficient use of the results obtained previously and also the newly acquired information so as to avoid unnecessary repeated calculation. Let us first have a look at the basic feature of the recursive operation through a simple example. The mean

digital smoothing and filtering methods

33

signal point x 10-3 Smoothing with window size=13

"05

?=

D)

05

signal point

signal point x 10-3 Smoothing with window size=17

signal point

Figure 2.3. Smoothing results obtained by the Savitsky-Golay filter with different window sizes. They are depicted by four plots with the original curve (solid line), the raw noisy signals (cross line), and the smoothed curve (dashed line) with window size of 7 (a), 11 (b), 13 (c), and 17 (d).

value is usually evaluated using the following formula

^Xj

x =

(2.11)

where ?xi denotes the sum of n observations, say x, (i = 1,... , n). When one measures a new Xj (i = n + 1), one has to calculate the mean again using Equation (2.11). Hence, all the n observations obtained before should be stored in the computer for future use. However, for recursive operation, a new mean can be evaluated through the following formula without using all the observations:

X n+1 — xn +

Xn+1 — x n

n + 1

(2.12)

Comparing this formula with Equation (2.11), one can obviously see that the recursive operation is faster and more efficient, and this is the attractive feature of Kalman filtering.

n

34

one-dimensional signal processing techniques in chemistry

Kalman filter is based on a dynamic system model

x(k) = F(k, k - 1)x(k - 1) + w(k) (2.13)

and a measurement model

y (k) = h(k)fx(k - 1) + e (k) (2.14)

where x(k), y(k), and h(k) denote the state vector, the measurement, and the measurement function vector, respectively. The variable k represents a measurement point that can be time, wavelength, or other. It should be noted that F(k, k - 1) is the system transition matrix which represents how the system transits from state (k - 1) to state k. Very often, it is an identity matrix for smoothing purposes. w(k) denotes the dynamic system noise, and could be a zero vector approximately because the smoothing filter can be regarded as a static model. e(k) is the measurement noise, which can be a stochastic variable with zero mean and constant variance obeying the Gaussian distribution.

The core recursive state estimate update in Kalman filtering is given by the following equation

x(k) = x(k - 1) + g(k)[y(k) - h(k)fx(k - 1)] (2.15)

where the vector g(k) is called Kalman gain. Comparing this equation with Equation (2.12), one can easily see the similarity between the two. The Kalman gain, g(k), corresponds to 1/(n + 1) in Equation (2.12) and is used to adjust the difference between the state vectors x(k) and x(k - 1) through the term of measurement difference, of [y(k) - h(k)fx(k - 1)]. Through Equation (2.15), one can also see that the state estimate update is just based on the newly measured y(k) and the state vector x(k - 1) obtained before. Equation (2.15) makes the efficient usage of recursive operation possible.

The Kalman gain can be determined by the following formula

g(k) = P(k - 1)h(k)[h(k)fP(k - 1)h(k) + r(k)]-1 (2.16)

where r(k) represents the variance of the measurement noise e(k). P(k - 1) is the covariance matrix of the system estimated from the (k - 1) observations obtained before through

P(k) = [I - g(k - 1)h(k)f]P(k - 1)[I - g(k - 1)h(k)f]

+ g(k - 1)r(k)g(k - 1)f (2.17)

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