A coding technique is described which improves error performance of synchronous data links without sacrificing data rate or requiring more bandwidth. This is achieved by channel coding with expanded sets of multilevel/phase signals in a manner which increases free Euclidean distance. Soft maximum--likelihood (ML) decoding using the Viterbi algorithm is assumed. Following a discussion of channel capacity, simple hand-designed trellis codes are presented for 8 phase-shift keying (PSK) and 16 quadrature amplitude-shift keying (QASK) modulation. These simple codes achieve coding gains in the order of 3-4 dB. It is then shown that the codes can be interpreted as binary convolutional codes with a mapping of coded bits into channel signals, which we call "mapping by set partitioning." Based on a new distance measure between binary code sequences which efficiently lower-bounds the Euclidean distance between the corresponding channel signal sequences, a search procedure for more powerful codes is developed. Codes with coding gains up to 6 dB are obtained for a variety of multilevel/phase modulation schemes. Simulation results are presented and an example of carrier-phase tracking is discussed.

Channel coding with multilevel/phase signals

In the present study k-Nearest Neighbor classification method, have been studied for economic forecasting. Due to the effects of companies’ financial distress on stakeholders, financial distress prediction models have been one of the most attractive areas in financial research. In recent years, after the global financial crisis, the number of bankrupt companies has risen. Since companies' financial distress is the first stage of bankruptcy, using financial ratios for predicting financial distress have attracted too much attention of the academics as well as economic and financial institutions. Although in recent years studies on predicting companies’ financial distress in Iran have been increased, most efforts have exploited traditional statistical methods; and just a few studies have used nonparametric methods. Recent studies demonstrate this method is more capable than other methods.

Application of K-Nearest Neighbor (KNN) Approach for Predicting Economic Events: Theoretical Background

The $k$th-nearest neighbor rule is arguably the simplest and most intuitively appealing nonparametric classification procedure. However, application of this method is inhibited by lack of knowledge about its properties, in particular, about the manner in which it is influenced by the value of $k$; and by the absence of techniques for empirical choice of $k$. In the present paper we detail the way in which the value of $k$ determines the misclassification error. We consider two models, Poisson and Binomial, for the training samples. Under the first model, data are recorded in a Poisson stream and are "assigned" to one or other of the two populations in accordance with the prior probabilities. In particular, the total number of data in both training samples is a Poisson-distributed random variable. Under the Binomial model, however, the total number of data in the training samples is fixed, although again each data value is assigned in a random way. Although the values of risk and regret associated with the Poisson and Binomial models are different, they are asymptotically equivalent to first order, and also to the risks associated with kernel-based classifiers that are tailored to the case of two derivatives. These properties motivate new methods for choosing the value of $k$.

Choice of neighbor order in nearest-neighbor classification

This paper studies minimax aspects of nonparametric classification. We first study minimax estimation of the conditional probability of a class label, given the feature variable. This function, say f, is assumed to be in a general nonparametric class. We show the minimax rate of convergence under square L/sub 2/ loss is determined by the massiveness of the class as measured by metric entropy. The second part of the paper studies minimax classification. The loss of interest is the difference between the probability of misclassification of a classifier and that of the Bayes decision. As is well known, an upper bound on risk for estimating f gives an upper bound on the risk for classification, but the rate is known to be suboptimal for the class of monotone functions. This suggests that one does not have to estimate f well in order to classify well. However, we show that the two problems are in fact of the same difficulty in terms of rates of convergence under a sufficient condition, which is satisfied by many function classes including Besov (Sobolev), Lipschitz, and bounded variation. This is somewhat surprising in view of a result of Devroye, Gorfi, and Lugosi (see A Probabilistic Theory of Pattern Recognition, New York: Springer-Verlag, 1996).

http://ssg.mit.edu/cal/abs/2000_spring/np_dens/classification/yang_it_11_99.pdf

Minimax nonparametric classification .I. Rates of convergence

Nearest neighbor methods are a popular class of nonparametric estimators with several desirable properties, such as adaptivity to different distance scales in different regions of space. Prior work on convergence rates for nearest neighbor classification has not fully reflected these subtle properties. We analyze the behavior of these estimators in metric spaces and provide finite-sample, distribution-dependent rates of convergence under minimal assumptions. As a by-product, we are able to establish the universal consistency of nearest neighbor in a broader range of data spaces than was previously known. We illustrate our upper and lower bounds by introducing smoothness classes that are customized for nearest neighbor classification.

Rates of Convergence for Nearest Neighbor Classification

Multi- h phase coded signals are constant-envelope sinusoids whose phase varies in a piecewise linear manner driven by cyclically chosen slopes. The transmission of these over a Gaussian noise channel with Viterbi decoder is simulated. Decoded error events are of two types normal and long, but in either case only a few bit errors are caused; events are empirically independent, they do not propagate, and their frequency is that predicted by signal space theory. Power gain compared to MSK modulation varies from 1.5 dB at bit error rate 0.05 to 4 dB at 0.0001 .

J. Anderson

Papers

Simulated error performance of multi-h phase codes (Corresp.)