Quantum Computing Based Machine Learning Method and Its Application in Radar Emitter Signal Recognition Gexiang Zhang1,2,*, Laizhao Hu1; and Weidong Jin2 1
National EW Laboratory, Chengdu, 610036 Sichuan, China
[email protected] 2 School of Electrical Engineering, Southwest Jiaotong University, Chengdu 610031, Sichuan, China
Abstract. Feature selection plays a central role in data analysis and is also a crucial step in machine learning, data mining and pattern recognition. Feature selection algorithm focuses mainly on the design of a criterion function and the selection of a search strategy. In this paper, a novel feature selection approach (NFSA) based on quantum genetic algorithm (QGA) and a good evaluation criterion is proposed to select the optimal feature subset from a large number of features extracted from radar emitter signals (RESs). The criterion function is given firstly. Then, detailed algorithm of QGA is described and its performances are analyzed. Finally, the best feature subset is selected from the original feature set (OFS) composed of 16 features of RESs. Experimental results show that the proposed approach reduces greatly the dimensions of OFS and heightens accurate recognition rate of RESs, which indicates that NFSA is feasible and effective.
1
Introduction
Feature selection is the process of extracting the most discriminatory information and removing the irrelevant and redundant information from a large number of measurable attributes. [1] Good features can enhance within-class pattern similarity and between-class pattern dissimilarity. [2] The minimum number of relevant and significant features can simplify the design of classifiers instead of degrading the performances of algorithms devoted to feature extraction and classification. So feature selection plays a central role in data analysis and is a crucial step in pattern recognition, machine learning and data mining. [1–5]1 Feature selection algorithms presented in the literatures can be classified into two categories based on whether or not feature selection is performed independently of the learning algorithm used to construct the classifier. The feature selection algorithms accomplished independently from the performance of a specific learning algorithm are referred to as the filter selection approach. Conversely, the feature *
Student Member IEEE
1
This work was supported by the National Defence Foundation (No.51435030101ZS0502; No. 00JSOS.2.1.ZS0501), by the National Natural Science Foundation of China (No.69574026), and was also supported by the Doctoral Innovation Foundation of SWJTU
V. Torra and Y. Narukawa (Eds.): MDAI 2004, LNAI 3131, pp. 92-103, 2004. © Springer-Verlag Berlin Heidelberg 2004
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selection algorithms directly related to the performance of the learning algorithms are regarded as wrapper selection approach. [4] Radar emitter signal recognition is a typical kind of pattern recognition. Lots of features extracted from radar emitter pulse signals with different intra-pulse modulation laws not only consume much time, but also introduce useless information to interfere with useful features because feature extraction is always a subjective process. Feature subset selection in radar emitter signal recognition can be considered as a global combinatorial optimization problem. [5] It is difficult to select the optimal m features from C nm paths which cover all combinations of the n features. Though, genetic algorithm (GA) is a good search technique developed rapidly in recent years for the combinatorial optimization problem. But conventional genetic algorithms (CGA) often have slow convergent speed and premature phenomenon in applications and have weak capability of balancing exploration and exploitation, that's to say, the characteristics of population diversity and selective pressure are not easy to be implemented, simultaneously. [6-7] Based on the principles of quantum computing [8-9], Genetic quantum algorithm (GQA) [10] was presented to solve combinatorial optimization problem and the results demonstrate that GQA is superior to CGA greatly. However, there are several shortcomings, such as non-determinability of lookup table of updating quantum gates, requiring prior knowledge of the best solution and premature phenomenon in GQA. In this paper, a novel feature selection approach (NFSA) based on quantum genetic algorithm and a good evaluation criterion is proposed. Because the main parts of feature selection methods are evaluation criterion of the optimal feature subset and automatic search algorithm. So a valid evaluation criterion is proposed to select the optimal feature subset from the original feature set firstly. Then, a novel quantum genetic algorithm (NQGA) is presented based on the concepts and theories of quantum computing. In NQGA, a novel update strategy of rotation angles of quantum gates, immigration operation and catastrophe operations are introduced to enhance search capability and to avoid premature convergence. The performances of NQGA are compared with GQA. After neural network classifiers are designed, the optimal feature subset is selected from the original feature set composed of 16 features of radar emitter signals using NFSA. Experimental results show that the proposed feature selection approach reduces greatly the dimensions of original feature set and heightens accurate recognition rate of radar emitter signals, which indicates that the introduced approach is feasible and effective. This paper is organized as follows. Section 2 gives the criterion function for evaluating the best feature subset. Section 3 describes the algorithm of NQGA in detail. Neural network classifiers are designed and the simulation experiments of feature selection and radar emitter signal recognition are made and experimental results are analyzed in section 4. Concluding remarks are listed in Section 5.
2
Evaluation Criterion
To facilitate the selection process, the quality of any feature has to be assessed via some well-designed criterion functions, and it is more important to design the feature
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selection criterion for a set of features because consideration of individual feature does not reveal the redundancy in the input data. Suppose that the maximum within-class clustering of the i -th class is represented with Cii . We define Cii as
⎧ ⎪⎡ 1 Cii = max ⎨ ⎢ q ⎪ ⎢⎣ M i ⎩
M iq
∑ k =1
1 ⎫ p ⎤ ⎪ q q xik − E ( X i ) ⎥ ⎬ ⎥⎦ ⎪ ⎭ p
(1)
q = 1, 2,L , N , N is the number of features, M i q is the number of samples
where
q
of the q -th feature of the i -th class, xik is the k -th sample value of the q -th q q q q q feature of the i -th class, X i = ⎡ xi1 , xi 2 ,L , xiM q ⎤ , E ( X i ) is the expectation
⎣
q
of X i . p (
i
⎦
p ≥ 1 ) is an integer. Similarly, the maximum within-class clustering
C jj of the j -th class is ⎧ ⎪⎡ 1 C jj = max ⎨ ⎢ q ⎪ ⎣⎢ M j ⎩
M jq
∑ k =1
1 ⎫ p ⎤ ⎪ q q x jk − E ( X j ) ⎥ ⎬ ⎦⎥ ⎪ ⎭ p
(2)
q
where q , p and N is the same as Eq.(12), M j is the number of samples of the
q -th feature of the j -th class, x jk q is the k -th sample value of the q -th feature of
j -th class, X j q = ⎡⎢ x j1q , x j 2 q ,L , x jM q q ⎤⎥ , E ( X j q ) is the expectation of X j q . j ⎣ ⎦ The minimum distance Dij between the i -th class and the j -th class is
the
{
Dij = min E ( X i q ) − E ( X j q )
}
Thus, the between-class separability Sij between the i -th class and the
(3)
j -th class
is defined as
Sij =
Dij Cii + C jj
(4)
Assume that there are totally H classes to be recognized, the criterion function of QAFS is represented with
Quantum Computing Based Machine Learning Method and Its Application
f = Obviously, the bigger
3
H −1 H 2 ∑ ∑ Sij H ( H − 1) i =1 j =i +1
95
(5)
f is, the better the selected feature subset is.
Search Strategy
In CGA, crossover and mutation operations are used to maintain the diversity of population, while the evolutionary operation that quantum gates operate on the probability amplitudes of basic quantum states is applied to maintain the diversity of population in GQA. So the technique of updating quantum gates is a key problem of GQA. In reference [10], quantum logic gates are updated by comparing binary bits, fitness and probability amplitudes of the current solution with those of the best solution in last generation. The look-up table in GQA is very complicated and the values of the angles of quantum gates are decided difficultly. It was pointed out in reference [10] that the method of updating quantum gates was suitable to solve the optimization problems such as knapsack problem. The reason is that the update strategy is based on knowing about the criterion of the optimal solution of optimization problems beforehand. For example, in knapsack problem, the criterion of the optimal solution is that the number of “1” should be as bigger as possible within constraint conditions because more “1” means bigger fitness of chromosome. However, the criterions of the optimal solutions cannot be known in many other optimization problems and in practical applications. What's more, premature phenomenon appears easily in GQA because all solutions have the same evolution direction. So, a novel quantum genetic algorithm is proposed to overcome these shortcomings. 3.1 Chromosome Representation Quantum bit (qubit) chromosome representation has good characteristics of representing any linear superposition of solutions and a qubit may be in the ‘1' state, in the ‘0' state, or in any superposition of the two, which is not in binary, numeric, or symbol representation. So we also adopt the representation in NQGA. The following description introduces the representation briefly. In quantum computing, the state of a qubit can be represented as
ψ =α 0 + β 1
(6)
where α and β are probability amplitudes of the corresponding states. Normalization of the state to unity guarantees 2
2
α + β =1
(7)
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where
α
2
gives the probability that the qubit will be found in ‘0' state and
the probability that the qubit will be found in ‘1' state. A system with contain information of can be represented as
β
2
gives
m qubits can
m
2 states and any linear superposition of all possible states 2m
| ψ i = ∑ Ck | S k 〉
(8)
k =1
where
Ck specifies the probability amplitude of the corresponding states Sk and 2
subjects to the normalization condition C1 + C2 probability amplitudes of
2
+ L + C2 n
2
= 1 . Thus, the
m qubits are represented as ⎡α α L α m ⎤ Pm = ⎢ 1 2 ⎥ ⎣ β1 β 2 L β m ⎦
where
2
(9)
2
α i + β i = 1, i = 1, 2,L , m .
3.2 Evolutionary Strategy Before evolutionary algorithm of NQGA is described in detail, two definitions and their interpretation are given firstly to understand easily the introduced algorithm. Definition 1: The probability amplitude of one qubit is defined with a pair of real number, ( α , β ), as
[α where
α and β
β]
T
(10)
satisfy equation (6) and (7).
Definition 2: The phase of a qubit is defined with an angle ζ as
ζ = arctan( β / α ) and the product
α and β
(11)
is represented with the symbol d , i.e.
d =α ⋅β
(12)
where d stands for the quadrant of qubit phase ζ . If d is positive, the phase ζ lies
in the first or third quadrant, otherwise, the phase ζ lies in the second or fourth quadrant. So the phase of the i th qubit in Eq.(4) is
ζ i = arctan( β i / α i )
(13)
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The detailed algorithm of NQGA is as follows. Step 1: Choose the population size n and the number m of qubits. Generate an initial population Ps containing n individuals Ps = {P1 , P2 ,L , Pn } , where Pj
j = 1, 2,L , n is the j th individual of population and Pj is ⎡α j1 α j 2 L α jm ⎤ Pj = ⎢ ⎥ ⎢⎣ β j1 β j 2 L β jm ⎥⎦ where the values of all
α ji , β ji (i = 1, 2,L , m)
(14) are 1/
2
which indicates the
quantum superposition state is composed of all basic quantum states by the same probability at the beginning of search process. Evolutionary generation g is set 0. Step 2: According to probability amplitudes of all individuals in population, construct observation states R of basic quantum states, R = {a1 , a2 ,L , an } , where
a j ( j = 1, 2,L , n) is observation state of j th individual and a j is a binary string, i.e. a j = b1b2 L bm , where
bk (k = 1, 2,L , m) is a binary bit composed of “1” or
“0”. Step 3: All individuals in observation states R are evaluated by using fitness function represented with equation (5). Step 4: The best solution sc in current generation is maintained. If sc is more than the best solution
so in evolutionary process, so is replaced by sc and so is
maintained. If satisfactory solution is obtained or the maximum generation arrives, the algorithm ends, otherwise, the algorithm continues. Step 5: Quantum rotation gate G is chosen as quantum logic algorithm in NQGA and G is represented as
⎡cos θ G=⎢ ⎣sin θ where
θ
is rotation angle of G and
− sin θ ⎤ cos θ ⎥⎦
θ = k ⋅ h(α , β ) . k
(15) is a coefficient and the
value of k has an effect on the speed of convergence. The value of k must be chosen reasonably. If k is too big, search grid of the algorithm is large and the solutions may diverge or have a premature convergence to a local optimum, and if it is too little, search grid of the algorithm is also little and the algorithm may be in a stagnant state. So k is defined as a variable. In CGA, adjustable coefficients are often relative to the maximal fitness and average fitness in current generation, while the strategy cannot be used in NQGA because it will affect the good characteristic of short computing time. Here, taking advantage of rapid convergence of NQGA, k is defined as a variable that is relative to evolutionary generations. Thus, search grid of
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Gexiang Zhang et al. − t / max t
NQGA can be adjusted adaptively. For example k = 0.5e , where t is evolutionary generation and maxt is a constant determined by the complexity of optimization problem. The function h(α i , β i ) determines the search direction of convergence to a global optimum. The below lookup table (Table 1) can be used as a strategy to make the algorithm converge. Thus, the changing value of rotation angle of quantum rotation gate is determined by comparing the quantum phase of the current solution with the quantum phase of the best solution, which is called quantum phase comparison approach. Table 1. Look-up table of function h(α , β )
d1 > 0 True True False False Table 1:
h(α , β )
d2 > 0
| ζ 1 |>| ζ 2 |
| ζ 1 |
