Maximum Likelihood Probability Research Articles

Dynamic Hand Gesture Recognition using Hidden Markov Model by Microsoft Kinect Sensor

Hand gesture recognition is one of the leading applications of human computer interaction. With diversity of applications of hand gesture recognition, sign language interpretation is the most demanding application. In this paper, dynamic hand gesture recognition for few subset of Indian sign language recognition was considered. The use of depth camera such as Kinect sensor gave skeleton information of signer body. After detailed study of dynamic ISL vocabulary with reference to skeleton joint information, angle has identified as a feature with reference to two moving hand. Here, real time video has been captured and gesture was recognized using Hidden Markov Model (HMM). Ten state HMM model was designed and normalized angle feature of dynamic sign was being observed. Maximum likelihood probability symbol was considered as a recognized gesture. Algorithm has been tested on ISL 20 dynamic signs of total 800 training set of four persons and achieved 89.25% average accuracy. General Terms Human Computer Interaction, Pattern Recognition, Machine Learning

Targeted Maximum Likelihood Estimation for Pharmacoepidemiologic Research.

Targeted maximum likelihood estimation has been proposed for estimating marginal causal effects, and is robust to misspecification of either the treatment or outcome model. However, due perhaps to its novelty, targeted maximum likelihood estimation has not been widely used in pharmacoepidemiology. The objective of this study was to demonstrate targeted maximum likelihood estimation in a pharmacoepidemiological study with a high-dimensional covariate space, to incorporate the use of high-dimensional propensity scores into this method, and to compare the results to those of inverse probability weighting. We implemented the targeted maximum likelihood estimation procedure in a single-point exposure study of the use of statins and the 1-year risk of all-cause mortality postmyocardial infarction using data from the UK Clinical Practice Research Datalink. A range of known potential confounders were considered, and empirical covariates were selected using the high-dimensional propensity scores algorithm. We estimated odds ratios using targeted maximum likelihood estimation and inverse probability weighting with a variety of covariate selection strategies. Through a real example, we demonstrated the double robustness of targeted maximum likelihood estimation. We showed that results with this method and inverse probability weighting differed when a large number of covariates were included in the treatment model. Targeted maximum likelihood can be used in high-dimensional covariate settings. In high-dimensional covariate settings, differences in results between targeted maximum likelihood and inverse probability weighted estimation are likely due to sensitivity to (near) positivity violations. Further investigations are needed to gain better understanding of the advantages and limitations of this method in pharmacoepidemiological studies.

FEH Local: Improving flood estimates using historical data

The traditional approach to design flood estimation (for example, to derive the 100-year flood) is to apply a statistical model to time series of peak river flow measured by gauging stations. Such records are typically not very long, for example in the UK only about 10% of the stations have records that are more than 50 years in length. Along-explored way to augment the data available from a gauging station is to derive information about historical flood events and paleo-floods, which can be obtained from careful exploration of archives, old newspapers, flood marks or other signs of past flooding that are still discernible in the catchment, and the history of settlements. The inclusion of historical data in flood frequency estimation has been shown to substantially reduce the uncertainty around the estimated design events and is likely to provide insight into the rarest events which might have pre-dated the relatively short systematic records. Among other things, the FEH Local project funded by the Environment Agency aims to develop methods to easily incorporate historical information into the standard method of statistical flood frequency estimation in the UK. Different statistical estimation procedures are explored, namely maximum likelihood and partial probability weighted moments, and the strengths and weaknesses of each method are investigated. The project assesses the usefulness of historical data and aims to provide practitioners with useful guidelines to indicate in what circumstances the inclusion of historical data is likely to be beneficial in terms of reducing both the bias and the variability of the estimated flood frequency curves. The guidelines are based on the results of a large Monte Carlo simulation study, in which different estimation procedures and different data availability scenarios are studied. The study provides some indication of the situations under which different estimation procedures might give a better performance.

Unexpected diversity and new species in the sponge-Parazoanthidae association in southern Japan

Currently the genera Parazoanthus (family Parazoanthidae) and Epizoanthus (family Epizoanthidae) are the only sponge-associated zoantharians (Cnidaria, Anthozoa). The Parazoanthidae-sponge associations are widely distributed in tropical and subtropical waters from the intertidal to the deep sea in the Atlantic and Indo-Pacific Oceans. However, the taxonomic identification of both parties is often confused due to variable morphology and wide ecological ranges. In particular, Parazoanthidae species diversity remains poorly understood in the Indo-Pacific. In the present study, the diversity of the sponge-zoanthid association in the Indo-Pacific was investigated with 71 Parazoanthidae specimens collected from 29 different locations in Japan (n=22), Australia (n=6) and Florida, USA (n=1). For all specimens morphological analyses were performed and total DNA was extracted and amplified for four DNA markers (COI-mtDNA, mt 16S-rDNA, ITS-rDNA and ALG11-nuDNA). The combined data demonstrate that the specimens of this study are clearly different from those of all described Parazoanthus species, and lead us to erect Umimayanthus gen. n., within family Parazoanthidae, containing the three newly described species U. chanpuru sp. n., U. miyabi sp. n., U. nakama sp. n. The new genus also includes the previously described species U. parasiticus (Duchassaing and Michelotti, 1860; comb. nov.), previously belonging to the genus Parazoanthus. Neighbor joining, maximum likelihood and Bayesian posterior probability phylogenetic trees clearly demonstrate the monophyly of Umimayanthus gen. n. to the exclusion of all outgroup sequences. The phylogenetic results were also compared to morphological features, and polyp sizes, amount of sand content in tissues, types of connections between polyps, and cnidae data, in particular holotrichs-1, were useful in distinguishing the different species within this new genus. This new genus can be distinguished from all other Zoantharia by a unique and conserved 9bp insertion and a 14bp deletion in the mt 16S-rDNA region. Additionally, compared to Parazoanthus sensu stricto (i.e. P. axinellae [Schmidt, 1862]), Umimayanthus spp. are only found associated to sponges, and have a coenenchyme much less developed than Parazoanthus sensu stricto. Each new species can be distinguished from other congeners by a unique DNA sequence, numbers of tentacle, maximum sizes of holotrichs, associated sponge morphology, and colony morphology. The identification of the host sponge species is the next logical step in this research as this may also aid in the distinction of Umimayanthus species.

Who are the Children Going to School in Urban Punjab (Pakistan)?

Purpose – The purpose of this paper is to analyze the demand side determinants of schooling of Pakistani urban children and the factors affecting boys and girls’ schooling separately.Design/methodology/approach – This is an empirical study using the non-linear maximum likelihood probability (probit) function on primary data.Findings – Besides other variables it has been observed that the poverty remains an important determinant of school participation. Poor households keep their children out of school due to their inability to afford the cost of schooling.Research limitations/implications – On the basis of this study a socioeconomic policy can be formulated for a developing country like Pakistan.Practical implications – A development policy can be formulated on the basis of this research for the enhancement of human resource development for a developing and an orthodox economy like Pakistan.Originality/value – The paper is beneficial to the researchers, policy makers, and social scientists for the enhancement of the level of social welfare through its findings

Recognition of Acoustic Emission Signal based on the Algorithms of TDNN and GMM

Friction fault diagnosis of rotating machinery based on acoustic emission ( AE) technique is a research hotspot in recent years. The rotating machinery will produce multi-source noise during the operation process, so how to correctly identify the friction acoustic emission signals has become a key factor for accurate diagnosis of the fault. In this paper, it proposes a Gaussian mixed model (GMM) based on an embedded time delay neural network (TDNN) to identify friction acoustic emission signals. It comprehensively utilizes the advantages of the learning ability of time delay neural network abo ut data structure and data distribution presentation capability of Gaussian mixture model. Time delay neural network fully exploits the time-ordered of eigenvector set, makes the maximum likelihood probability more reasonable which needs to assume that the variables are independent of each other through the transformation of the time delay network and uses them for the training as a whole with the criteria of maximum likelihood (ML) probability. During the training process, the parameters of Gaussian mixture model and neural network update alternately. The average amplitude, maximum amplitude, amplitude dynamic range, the Hurst exponent and approximate entropy (ApEn) of friction acoustic emission signals are selected as the characteristic parameters of fault recognition and these five parameters constitutes the input parameters vector of the identification model. Through the verificatio n the AE signals of different friction states collected on the rotor test bed, the experimental results show that the identification method o f rotor friction acoustic emission signals of Gaussian mixed model based on embedded time delay neural network is an effective mean of identification with high recognition efficiency.

PTree: pattern-based, stochastic search for maximum parsimony phylogenies.

Phylogenetic reconstruction is vital to analyzing the evolutionary relationship of genes within and across populations of different species. Nowadays, with next generation sequencing technologies producing sets comprising thousands of sequences, robust identification of the tree topology, which is optimal according to standard criteria such as maximum parsimony, maximum likelihood or posterior probability, with phylogenetic inference methods is a computationally very demanding task. Here, we describe a stochastic search method for a maximum parsimony tree, implemented in a software package we named PTree. Our method is based on a new pattern-based technique that enables us to infer intermediate sequences efficiently where the incorporation of these sequences in the current tree topology yields a phylogenetic tree with a lower cost. Evaluation across multiple datasets showed that our method is comparable to the algorithms implemented in PAUP* or TNT, which are widely used by the bioinformatics community, in terms of topological accuracy and runtime. We show that our method can process large-scale datasets of 1,000–8,000 sequences. We believe that our novel pattern-based method enriches the current set of tools and methods for phylogenetic tree inference. The software is available under: http://algbio.cs.uni-duesseldorf.de/webapps/wa-download/.

Power Quality Disturbance Classification Using S-transform and Hidden Markov Model

In this article, a combination method based on S-transform and the hidden Markov model is presented for power quality disturbance classification. S-transform is a new method that has been used in signal processing and power quality disturbance classification applications. S-transform not only has the advantages of both wavelet transform and fast Fourier transform but also characteristics superior than both mentioned transform methods. Besides, the hidden Markov model is a powerful and effective method on signal processing applications. This method computes the maximum likelihood probability between training and testing data signals for identification. The proposed method makes the classification more simple and accurate. The identification procedure has two major steps. First, features are extracted by S-transform, and in the second step, classification is done using a decision tree structure of hidden Markov models. Simulation results by simulated and experimental test signals reveal the robustness of proposed method and the possibility of using it for real applications.

강수지표를 이용한 우리나라 강수량의 시간적인 특성 변화

본 연구에서는 과거에서부터 기록된 우리나라 65개의 관측소들을 대상으로 강수량 데이터를사용하여 강수빈도와 극한 지표들을 산출해 냄으로써, 지표에 따라 지역별 분포를 살펴보고 시공간적인 증 감율에 대한 결과를 제시하였다. 그 결과를 시계열로 도시함으로 쉽게 파악할 수 있지만경향이 뚜렷하게 나타나지 않는 경우에는 통계적 방법인 Mann-Kendall 경향성 분석을 실시하여 이에 대해 파악하였다. 더불어 FARD를 이용하여 100년과 200년의 빈도확률강수량을 산정하고 이를 비교하였다. 강우특성을 크게 Amount, Extreme, Frequency로 분류하고 각각의 지수를 강우량 지수(Rainfall Index for Amount), 강우극치 지수(Rainfall Index for Extremes), 강우빈도 지수(Rainfall Index for Frequency)로 정의하고 RIA, RIE, RIF를 산술평균하여 홍수에 대한 종합적인 강우 위험도를 나타내는 평균 강우지수(Average Rainfall Index, ARI)를 산정하였다. 그 결과 과거에 비해 최근 10년간 강우량의 정량적인 평가 특성은 양적으로 22.3%, 극치사상 발생측면은 26.2%, 빈도측면은 5.1% 증가하였음을 확인 할 수 있었다. 본연구의 결과는 최근 우리나라 기후변화 경향을 구체적으로 제시하고 있으므로 기후변화 대비 적응 연구의 기초 자료로 활용될 수 있다. This study suggests the results of temporal and spatial variations for rainfall data in the Korean Peninsula. We got the index of the rainfall amount, frequency and extreme indices from 65 weather stations. The results could be easily understood by drawing the graph, and the Mann-Kendall trend analysis was also used to determine the tendency (up & downward/no trend) of rainfall and temperature where the trend could not be clear. Moreover, by using the FARD, frequency probability rainfalls could be calculated for 100 and 200 years and then compared each other value through the moment method, maximum likelihood method and probability weighted moments. The Average Rainfall Index (ARI) which is meant comprehensive rainfalls risk for the flood could be obtained from calculating an arithmetic mean of the RI for Amount (RIA), RI for Extreme (RIE), and RI for Frequency (RIF) and as well as the characteristics of rainfalls have been mainly classified into Amount, Extremes, and Frequency. As a result, these each Average Rainfall Indices could be increased respectively into 22.3%, 26.2%, and 5.1% for a recent decade. Since this study showed the recent climate change trend in detail, it will be useful data for the research of climate change adaptation.

Polyandry Is a Common Event in Wild Populations of the Tsetse Fly Glossina fuscipes fuscipes and May Impact Population Reduction Measures

Background Glossina fuscipes fuscipes is the main vector of human and animal trypanosomiasis in Africa, particularly in Uganda. Attempts to control/eradicate this species using biological methods require knowledge of its reproductive biology. An important aspect is the number of times a female mates in the wild as this influences the effective population size and may constitute a critical factor in determining the success of control methods. To date, polyandry in G.f. fuscipes has not been investigated in the laboratory or in the wild. Interest in assessing the presence of remating in Ugandan populations is driven by the fact that eradication of this species is at the planning stage in this country.Methodology/Principal FindingsTwo well established populations, Kabukanga in the West and Buvuma Island in Lake Victoria, were sampled to assess the presence and frequency of female remating. Six informative microsatellite loci were used to estimate the number of matings per female by genotyping sperm preserved in the female spermathecae. The direct count of the minimum number of males that transferred sperm to the spermathecae was compared to Maximum Likelihood and Bayesian probability estimates. The three estimates provided evidence that remating is common in the populations but the frequency is substantially different: 57% in Kabukanga and 33% in Buvuma.Conclusions/SignificanceThe presence of remating, with females maintaining sperm from different mates, may constitute a critical factor in cases of re-infestation of cleared areas and/or of residual populations. Remating may enhance the reproductive potential of re-invading propagules in terms of their effective population size. We suggest that population age structure may influence remating frequency. Considering the seasonal demographic changes that this fly undergoes during the dry and wet seasons, control programmes based on SIT should release large numbers of sterile males, even in residual surviving target populations, in the dry season.

Modem theory: an introduction to telecommunications

At the heart of any modern communication system is the modem, connecting the data source to the communication channel. This first course in the mathematical theory of modem design introduces the theory of digital modulation and coding that underpins the design of digital telecommunications systems. A detailed treatment of core subjects is provided, including baseband and passband modulation and demodulation, equalization, and sequence estimation. The modulation waveforms for communication channels and digital recording channels are treated in a common setting and with unified terminology. A variety of more advanced topics is also covered, such as trellis codes, turbo codes, the Viterbi algorithm, block codes, maximum likelihood and maximum posterior probability, iterative demodulation, and jamming. Numerous end-of-chapter exercises are also included to test the reader's understanding throughout. This insightful book is ideal for senior undergraduate students studying digital communications and is also a useful reference for practising engineers.

Labor force participation of married women in Punjab (Pakistan)

PurposeThe purpose of this paper is to highlight the factors that influence the decision of married women (in the age group of 16‐60 years) to participate in labor force activities.Design/methodology/approachThis is an empirical study employing the non‐linear maximum likelihood probability (probit) function on primary data (3,911 observations).FindingsBesides other variables it has been observed that poverty remains an important determinant of female labor participation.Research limitations/implicationsOn the basis of this paper, a socio‐economic policy can be formulated for a developing country like Pakistan.Practical implicationsA development policy (especially considering the gender aspects) can be formulated on the basis of this research for the enhancement of human resource development for a developing and an orthodox economy like Pakistan.Originality/valueThis paper is beneficial to researchers, policy makers, and social scientists for the enhancement of the level of social welfare and equity through its findings.

Who are the children going to school in Urban Punjab (Pakistan)?

PurposeThe purpose of this paper is to analyze the demand side determinants of schooling of Pakistani urban children and the factors affecting boys and girls' schooling separately.Design/methodology/approachThis is an empirical study using the non‐linear maximum likelihood probability (probit) function on primary data.FindingsBesides other variables it has been observed that the poverty remains an important determinant of school participation. Poor households keep their children out of school due to their inability to afford the cost of schooling.Research limitations/implicationsOn the basis of this study a socio‐economic policy can be formulated for a developing country like Pakistan.Practical implicationsA development policy can be formulated on the basis of this research for the enhancement of human resource development for a developing and an orthodox economy like Pakistan.Originality/valueThe paper is beneficial to the researchers, policy makers, and social scientists for the enhancement of the level of social welfare through its findings.

Labor Force Participation of Married Women in Punjab (Pakistan)

Purpose – The purpose of this paper is to highlight the factors that influence the decision of married women (in the age group of 16-60 years) to participate in labor force activities.Design/methodology/approach – This is an empirical study employing the non-linear maximum likelihood probability (probit) function on primary data (3,911 observations).Findings – Besides other variables it has been observed that poverty remains an important determinant of female labor participation.Research limitations/implications – On the basis of this paper, a socio-economic policy can be formulated for a developing country like Pakistan.Practical implications – A development policy (especially considering the gender aspects) can be formulated on the basis of this research for the enhancement of human resource development for a developing and an orthodox economy like Pakistan.Originality/value – This paper is beneficial to researchers, policy makers, and social scientists for the enhancement of the level of social welfare and equity through its findings.

Improvement of Gallager Upper Bound and its Variations for Discrete Channels

A new tight upper bound on the maximum-likelihood (ML) word and bit-error decoding probabilities for specific codes over discrete channels is presented. It constitutes an enhanced version of the Gallager upper bound and its variations resulting from the Duman-Salehi second bounding technique. An efficient technique is developed that, in the case of symmetric channels, overcomes the difficulties associated with the direct computation of the proposed bound. Surprisingly, apart from the distance and input-output weight enumerating functions (IOWEFs), the bound depends also on the coset weight distribution of the code.

A Novel Discriminative Approach Based on Hidden Markov Models and Wavelet Transform to Transformer Protection

In this paper we present a combinatorial scheme based on hidden Markov models (HMM) and wavelet transform (WT) to discriminate between magnetizing inrush currents and internal faults in power transformers. HMMs are powerful tools for transient classification which compute the maximum likelihood probability between training and testing data signals for identification. The WT is employed to extract certain features which reduce the computation burden of HMMs and enhance detection accuracy. The newly extracted feature efficiently discriminates between faults by different trends. The k-means clustering technique is applied to reduce the training procedure time investment. Since the discrimination method is based on the probabilistic characteristics of the signals without application of any deterministic index, more reliable and accurate classification is achieved. This method is independent of the selection thresholds. Based on the proposed algorithm a highspeed relay response (a quarter of a cycle) can be achieved. The suitable performance of this method is demonstrated by simulation of different faults and switching conditions on a power transformer using PSCAD/EMTDC software.

Consistent Estimation of Divergence Times in Phylogenetic Trees with Local Molecular Clocks

This PhD thesis consists of an introduction and five papers, dealing with statistical methods in phylogenetics.A phylogenetic tree describes the evolutionary relationships among species assuming that they share a common ancestor and that evolution takes place in a tree like manner. Our aim is to reconstruct the evolutionary relationships from aligned DNA sequences.In the first two papers we investigate two measures of confidence for likelihood based methods, bootstrap frequencies with Maximum Likelihood (ML) and Bayesian posterior probabilities. We show that an earlier claimed approximate equivalence between them holds under certain conditions, but not in the current implementations of the two methods.In the following two papers the divergence times of the internal nodes are considered. The ML estimate of the divergence time of the root is improved if longer sequences are analyzed or if more taxa are added. We show that the gain in precision is faster with longer sequences than with more taxa. We also show that the algorithm of the software package PATHd8 may give biased estimates if the global molecular clock is violated. A change of the algorithm to obtain unbiased estimates is therefore suggested.The last paper deals with non-informative priors when using the Bayesian approach in phylogenetics. The term is not uniquely defined in the literature. We adopt the idea of data translated likelihoods and derive the so called Jeffreys' prior for branch lengths using Jukes Cantor model of evolution.

Diversification of the arboreal mice of the genus Habromys (Rodentia: Cricetidae: Neotominae) in the Mesoamerican highlands

The arboreal mice of the genus Habromys (Rodentia: Cricetidae: Neotominae) are among the most poorly known Neotropical rodents. We investigated species-level phylogenetic relationships among the seven described Habromys species using 1331 aligned bases from the mitochondrial ND3 and ND4 regions. Sequences were obtained from 30 specimens of the seven known species of Habromys and we performed maximum parsimony, maximum likelihood, and Bayesian probabilities analyses. The monophyly of the genus Habromys within the Neotomines was verified. The northernmost H. simulatus is sister to the remaining species of the group; within the latter, the southernmost clade (Oaxaca to Central America) is sister to the Transmexican Volcanic Belt clade. Four major clades are clearly distinguished: H. simulatus from the Sierra Madre Oriental and the closely associated Sierra Mazateca; H. delicatulus and H. schmidlyi from the Transmexican Volcanic Belt; H. lepturus, H. chinanteco, and H. ixtlani from the northern Oaxacan highlands; and H. lophurus from Nuclear Central America. Within species, the analyses suggest that H. simulatus and H. lophurus are each composed by two different taxa.

Parasites and the island syndrome: the colonization of the western Mediterranean islands by Heligmosomoides polygyrus (Dujardin, 1845)

AbstractAim Populations of free‐living vertebrates on islands frequently differ from their mainland counterparts by a series of changes in morphometric, life‐history, behavioural, physiological and genetic traits, collectively referred to as the ‘island syndrome’. It is not known, however, whether the ‘island syndrome’ also affects parasitic organisms. The present study establishes the colonization pattern of the Mediterranean islands by the nematode Heligmosomoides polygyrus, a direct and specific parasite of rodent hosts of the Apodemus genus, and evaluates the effects of island colonization by this species on two components of the island syndrome: the loss of genetic diversity and the enlargement of the ecological niche.Location Heligmosomoides polygyrus was sampled on seven western Mediterranean islands − Corsica, Crete, Elba, Majorca, Minorca, Sardinia and Sicily − as well as in 20 continental locations covering the Mediterranean basin.Methods The mitochondrial cytochrome b gene (690 base pairs) was sequenced in 166 adult H. polygyrus individuals sampled in the 27 continental and island locations. Phylogenetic reconstructions in distance, parsimony, maximum likelihood and Bayesian posterior probabilities were carried out on the whole cytochrome b gene data set. The levels of nucleotide, haplotype and genetic divergence (Kimura two‐parameter distance estimator) diversities were estimated in each island population and in the various continental lineages.Results Phylogenetic reconstructions show that the mainland origins of H. polygyrus were continental Spain for the Balearic Islands (Majorca, Minorca), northern Italy for the Tyrrhenian Islands (Corsica, Sardinia, Elba), southern Italy for Sicily, and the Balkan region for Crete. A comparison of island H. polygyrus populations with their mainland source populations revealed two characteristic components of the island syndrome in this parasite. First, island H. polygyrus populations display a significant loss of genetic diversity, which is related (r2 = 0.73) to the distance separating the island from the mainland source region. Second, H. polygyrus exhibits a niche enlargement following insularization. Indeed, H. polygyrus in Corsica is present in both A. sylvaticus and Mus musculus domesticus, while mainland H. polygyrus populations are present exclusively in Apodemus hosts.Main conclusions Our results show that H. polygyrus has undergone a loss of genetic diversity and a niche (host) enlargement following colonization of the western Mediterranean islands. To our knowledge, this study provides the first evidence for components of the ‘island syndrome’ in a parasitic nematode species.

On the MDL principle for i.i.d. sources with large alphabets

Average case universal compression of independent and identically distributed (i.i.d.) sources is investigated, where the source alphabet is large, and may be sublinear in size or even larger than the compressed data sequence length n. In particular, the well-known results, including Rissanen's strongest sense lower bound, for fixed-size alphabets are extended to the case where the alphabet size k is allowed to grow with n. It is shown that as long as k=o(n), instead of the coding cost in the fixed-size alphabet case of 0.5logn extra code bits for each one of the k-1 unknown probability parameters, the cost is now 0.5log(n/k) code bits for each unknown parameter. This result is shown to be the lower bound in the minimax and maximin senses, as well as for almost every source in the class. Achievability of this bound is demonstrated with two-part codes based on quantization of the maximum-likelihood (ML) probability parameters, as well as by using the well-known Krichevsky-Trofimov (KT) low-complexity sequential probability estimates. For very large alphabets, kGtn, it is shown that an average minimax and maximin bound on the redundancy is essentially (to first order) log(k/n) bits per symbol. This bound is shown to be achievable both with two-part codes and with a sequential modification of the KT estimates. For k=Theta(n), the redundancy is Theta(1) bits per symbol. Finally, sequential codes are designed for coding sequences in which only m<min{k,n} alphabet symbols occur