This article represents results of research of potential functions method in case of discrete signal with additive noise. Noise in transmission channel has Gaussian distribution, i.e. it is a white noise. Besides, noises in the channels have a static interrelation. The main aim of the paper is researching of applicability of the method of potential functions for recognition of the binary signal from it mix with additive noise. Number of researches have been conducted to assess the possibility of using the described method. The results were presented in a series of graphics. Tests were made for different numbers of parameters of the model, which had described in this paper. There was done comparative analysis of working of potential functions and k-nearest neighbors algorithm, which had considered in previous author's works. The results of this research can be used as the base for further research and can be applicable in organization of transport channels.
The paper is the logical continuation of [
In this paper we propose potential functions for solving the challenge. Signal is transmitted on two channels, noises in the channels are statistically connected. One channel is contained noise only, the second – noise and useful signal. If the second channel doesn’t exist, we can introduce it in some cases.
MATHEMATICAL REPRESENTATION OF THE
CHANNELS MODEL
There are two classes of objects, w0 and w1 (“0” and “1”, respectively). Objects of w0 have a negative potential, and objects of w1 - a positive one. The value of a potential is computed by potential of the point charge, which is determined as follows: g(x) = ∑ qi K(x, xi) (1) where K(x, xi) – potential function, it is inversely proportional to value || x - xi ||; xi – i-th element of the training sample.
qi – charge of the i-the elements «ones» and «zeros» q1 = +1, q0 = -1.
For every new element is computed it's potential relatively received earlier: if g(x) > 0, then x w1 ; if g(x) < 0, then x w0; case g(x) = 0 is impossible from the point of view of the theory of probability, as K(x, xi) is continuous random value.
There are most commonly used potential function, which has peak at x=xi and decreases monotonically to null during || x - xi || → . It’s convenient to present potential function as function of distance between entry and other elements of the set.
In this paper the potential function K(x, xi) is:
K(x, xi) = exp( -a || x - xi ||2) The parameter a > 0 – constant.
Evaluation of the successful recognition of the binary signal in the channel looks like:
P* = m / N, where (2) (3) m – an amount of successfully determined classes of the message;
N – the length of the test set.
There are using model of the channels y1, y2, which described, as y1 = a * / 2 + + ; y2 = b * / 2 + + h * , where , , , – random variables, takes the values 1 and -1 with probability P = 1/2; N(0, 2) - the noise in the first channel; N(0, 2) – the noise in the second channel, and are statistically interrelated, r, it is correlation coefficient of them; N(0, 2) – common noise in the channels, h – constant; and – random values, which are independent from and , and each other; and – natural noises, attending in the channels; – noise, appearing in the channels with some technical conditions; a = |m11 - m-11| – discrete signal in the channel y1; b = |m12 - m-12| – discrete signal in the channel y2;
In this work N(m, 2) – normal distribution law with parameters (m, 2)
Conditional probability densities P(y1 | = 1) и P(y1 | = -1) have a normal law with parameters N(m1, 2+2 ) и N(m-1, 2+2). For 2 that parameters are defined similarly (so, normal law has parameters N(m±1, 2+2 )).
III.
For investigations of method of potential functions there had been conducted the set of experiments, which allowed to valuate recognition probability of the signal in different cases.
1 P , ion 0,95 t i gn 0,9 o c re 0,85 l a ign 0,8 s fo 0,75 y t i ilb 0,7 a b ro 0,65 P 0,6 0,55 -1,0 -0,8 -0,6 -0,4 -0,2 0,0 0,2 0,4 0,6
Correlation coefficient of noises ε and , rε, 0,8 1,0 P, b=0 P, b=3
P, b=1 P, b=5
Correlation coefficient of the noises and took values from -1 to 1 in 0.1 steps. Standard deviation of more 10 times than standard deviations of and . Signal-to-noise ratio [further – ratio] is considerably less, than 1. Matlab 2011a was used for experiments.
The results of the algorithm work in case, where isn’t common noise (it’s dispersion is equal to zero), are shown if Figure 1. In that case parameter b are changing.
The behavior of the system is predictable in this case, the probability of recognition of the signal grows in the power increasing. Mention must be made of the high level of recognition in case b = 1.
As demonstration of the algorithm working are presented results of working for two parameter sets: (a=1, b=1; a=2, b=1). Experiments were made for different values of h: h=0, h=|0,5|, h=|1|, h=|1,5, h=|5|.
Results of case (a = 1, b = 1) are shown in Fig.2 and Fig.3. It’s demonstrates, that the quality of recognition depends on parameter h. Fig.3 shows, that the probability of recognition strives to 1 for correlation coefficient with value -1. This is due to the fact that ratio of the model of the signals is much more, than 1. In some cases in ratio signal grow and noise decreased. When it’s happening, you can notice the situation like in the Fig.3, Fig.4, Fig.5.
Results for a = 1, b = 2 are shown in Fig.4 and Fig.5. Fig.4 shows, that signal recognition significantly improves with h = 1. As a whole, we note, that recognition probability with b=2 larger, than with b=1. You can see, that we found some excludes: when correlation coefficient is going to -1, P→1. It is going because mathematic expression of ratio grows very fast in some point: where fraction signal/noise strives to infinity.
0,59 P 0,58 , n tio 0,57 i n g co 0,56 e r l an 0,55 g i s fo 0,54 y t i ilb 0,53 a b roP 0,52 0,51 0,5 -1,0 -0,8 -0,6 -0,4 -0,2 0,0 0,2 0,4 0,6 0,8 1,0 0,95 P ,no 0,9 i t in0,85 g eco 0,8 r l an0,75 g ifs 0,7 o ty0,65 i l i ab 0,6 b roP0,55
0,5 0,95 0,9 P ,ino0,85 t ign 0,8 o c re0,75 l a ign 0,7 s fo0,65 y t iilb 0,6 a ob0,55 r P 0,5
1 P0,95 , ion 0,9 t i n og 0,85 c e lra 0,8 n isg 0,75 f toy 0,7 i l i ab 0,65 b o rP 0,6 0,55 0,5 -1,0 -0,8 -0,6 -0,4 -0,2 0,0 0,2 0,4 0,6
Correlation coefficient of noises ε and , rε, h=0 h=0,5 h=1 h=1.5 h=5 0,8 1,0 Fig. 4 Signal recognition probability for a = 1, b = 2, h ≥ 0 -1,0 -0,8 -0,6 -0,4 -0,2 0,0 0,2 0,4 0,6 0,8 1,0
Correlation coefficient of noises ε and , rε, h=0 h=-0,5 h=-1 h=-1.5 h=-5 Fig. 5 Signal recognition probability for a = 1, b = 2, h ≤ 0 -1,0 -0,8 -0,6 -0,4 -0,2 0,0 0,2 0,4 0,6 0,8 1,0
Correlation coefficient of noises ε and , rε, h=0 h=-1.5 h=-0,5 h=-5 h=-1 Fig. 3 Signal recognition probability for a=1, b=1, h ≤ 0 -1,0 -0,8 -0,6 -0,4 -0,2 0,0 0,2 0,4 0,6 0,8 1,0 IV. COMPARISION WITH K-NEAREST NEIGHBORS
ALGORITHM
Previous work of authors [
A description of k-nearest neighbors is presented further (in
the context of the work). There are two classes of elements: w0
and w1. Elements with label of the class w0 are “0”, and with
label of w1 are “1”, respectively. There are generated sample,
which size is large enough (in the work the sample’s size is
around 10000 [
A new classified two-dimensional object x comes to the input of the qualifier. The distance || x - xi || between x and elements of the training set is computed. k nearest are selected. x will get the label, which had the most of the k choosed elements. k is always ood in order to avoid collisions. -1,0 -0,8 -0,6 -0,4 -0,2 0,0 0,2 0,4 0,6 0,8 1,0
Correlation coefficient of noises ε and , rε,
PF, h=0 PF, h=1 PF, h=5 0,45 -1,0 -0,8 -0,6 -0,4 -0,2 0,0 0,2 0,4 0,6 0,8 1,0
Correlation coefficient of noises ε and , rε,
So, we compared the method of potential functions and k nearest neighbors’ algorithm. As earlier, results are presented as graphs. Size of the training set also are 10 000. On the x-axis you have got values of the correlation coefficient rε, to within 0.1. On the y-axis – the probability of the signal recognition. PF means Potential Functions and K – k nearest neighbors.
We can see, that on the Fig.6, Fig.7, Fig.8 potential function method has a little win in front of the k-nearest method.
V.
In the paper are presented the set of the experiments results,
which demonstrate probabilities of using potential function for
signal recognition. We considered behavior of the probability of
the signal recognition in mathematical model (2) and made the
following conclusions. The method can be used in case of two
channels. There are opportunity to use it in more channels [7],
but it requires additional researches. Besides, we showed there
are points, where the recognition probability grows to 1. The
reason has been at analytical function of ratio (it had showed in
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