Masters Theses Department of Mathematics and Statistics
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Item Modelling transmission dynamics of typhoid fever with fear of infection and vaccination in Kenya(UoEm, 2025-06-03) Wangui, Jackline WanjikuDespite the great advancements in healthcare systems and sanitary improvements globally, sub-Saharan Africa including Kenya bears a significant burden of infectious diseases, among which typhoid fever continues to exert a notable toll. In this study, we developed a deterministic mathematical model to examine the interplay between human responses driven by the psychological factor of fear of infection, vaccination efforts, and the dynamics of human-to-human and environmental transmission of typhoid fever. The mathematical model was analyzed using theories of first-order ordinary differential equations to establish the existence of equilibrium points and their conditions for local and global stability. The reproduction number, R0 , was established and distinct pathways for the transmission of infection were identified, shedding light on the crucial interactions among key population groups fueling the spread of typhoid fever disease. The model results suggest that, typhoid fever infection is heightened by both direct and indirect contact with infected individuals and contaminated environments. Additionally, lack or limited awareness contributes to decreased fear of infection and reluctance towards vaccination, further exacerbating the situation. Moreover, an increase in environmental transmission is observed due to elevated discharge rates from infected individuals. This study contributes valuable insights into the design of effective mitigation strategies aimed at combating typhoid fever in resource-limited settingsItem The relationship between constructive simulation, convectional method and learning outcomes of christian religious education. The case of Embu county, Kenya(UoEm, 2025-06-03) Mugo, Annis MuthoniABSTRACT Learning outcomes in Christian Religious Education are attributed to the teaching and learning approaches utilized by teachers. In Kenya, conventional methods are prevalent in classrooms, as teachers often prefer methods that alleviate their workload. Nevertheless, the implementation of learner-centred methods such as constructive simulation enhances learning outcomes. Although numerous researchers have advocated for the constructive simulation method as a means to promote interactive education across various subjects, there remains scarcity of research in the context of Christian Religious Education. Therefore, this study established the relationship between constructive simulation, conventional method and learning outcomes of CRE. This was done by evaluating the dissimilarity in learning outcomes and conceptions of CRE learners instructed by constructive simulation, and those instructed using a conventional approach. This research adopted a mixed method approach where, a Quasi- experimental research design with groups under treatment and control to collect quantitative data, while the descriptive research design was utilized to collect qualitative data, guided by Vygotsky's social cognitive theory and John Dewey’s philosophy of reflective practice. In total, 108 form two CRE learners in sub-county secondary schools were purposively selected for the study. Data were collected using the Learner Attainment test in CRE as the assessment tool, student questionnaires, and lesson observation schedules. The research utilised correlation analysis, t-test and regression statistical models to establish the effectiveness of the two teaching methods, as well as the relationship between learners’ conceptions and their learning outcomes in CRE. Results from the t-test revealed that, constructive simulation (t (49) = − 9.76, n = 50, p < 0.05) significantly outperformed the conventional method of teaching (t (39) = 2.700, n = 40, p < 0.324), demonstrating that constructive simulation was more effective in enhancing learning outcomes in CRE. The correlation and regression analysis indicated a positive link between the two assessment tests of learners taught using constructive simulation (r = .510, p < 0.01) and conventional methods (r = .673, p < 0.01). Furthermore, learners’ conceptions of constructive simulation (β = 0.252, p < 0.00) and conventional method (β = 0.065, p < 0.01) had a positive relationship with learning outcomes of CRE. The study therefore ascertained the relationship between constructive simulation and learning outcome as operative. Moreover, learners’ conceptions of teaching methods had an impact on their learning outcome. From the study findings, constructive simulation was more effective in enhancing learning outcomes as opposed to the conventional method. These findings are pertinent for a diverse range of stakeholders, including educators, policymakers, and scholars advocating for evidence-based approaches like constructive simulation in education. The results, therefore, suggest when designing curricula and formulating educational policies, educators and policymakers should incorporate constructive simulation as a learner-centred method.Item Forecasting Kenya's public debt using time series analysis(UoEm, 2025-06-03) Obwoge, Frankline KeraroABSTRACT Public debt management and forecasting remain challenging for developing economies, including Kenya, where accurate predictions are essential for sustainable fiscal planning. This study aimed to analyze and forecast Kenya's public debt using two time series forecasting approaches: the Autoregressive Integrated Moving Average model and the Holt exponential smoothing model. The study sought to evaluate the performance of these models to determine the most efficient forecasting method for Kenya's debt forecasting. The research employed a cross-sectional study design, utilizing public debt data from the Central Bank of Kenya spanning January 2010 to December 2023. The methodology involved initial data preprocessing, stationarity testing, and pattern analysis, followed by dividing the data into training and testing sets. Both models were fitted to the training data, with parameters optimized through minimization of the Akaike Information Criterion and smoothing parameters. Results revealed that the Autoregressive Integrated Moving Average model demonstrated superior performance in forecasting domestic debt, with a Root Mean Square Error of 0.02649721 compared to 0.0311399 for the Holt exponential smoothing model. For external debt forecasting, the Holt exponential smoothing model showed marginally better results. In forecasting total public debt, the Autoregressive Integrated Moving Average model again proved more accurate, with a Root Mean Square Error of 0.05710133 compared to 0.06144849 for the Holt model. Based on these findings, the study recommends using the Autoregressive Integrated Moving Average model for forecasting domestic and total public debt in Kenya, while the Holt exponential smoothing method for external debt forecasting. Regular reassessment of model performance is encouraged to maintain accuracy as debt patterns evolve. Future research should consider incorporating multiple economic variables, exploring advanced time series models, and integrating debt sustainability frameworks to enhance forecasting accuracy.Item Mathematical Modelling of Host-Pest Interaction in the Presence of Insecticides and Resistance(UoEm, 2024-09) Gatwiri, Moreen BrendaSeveral pest management programs have been developed to control the rising agricultural pest populations. However, the challenge of rapid evolution and pest resistance towards the control measures continues to cause high production losses to maize farmers in Africa. Few models have attempted to address the issue of Fall Armyworm (FAW) but have not incorporated the effect of insecticides resistance. The knowledge on the effect of insecticides resistance is still scanty. Models with resistance would help predict the dynamics of FAW population thus mitigate loses. The main objectives of this work were to develop, analyse, and numerically simulate a susceptible- infected deterministic mathematical model expressing the FAW-maize interaction and population dynamics under insecticidal sprays and resistance FAW larvae. Three model steady states are established and their local stability conducted using either the eigenvalue or the Routh- Hurwitz stability criterions and the global stability analyzed using Castillo Chavez, Perron eigen vector, and the Lyapunov methods. An expression for the Basic reproduction number, 𝑅0, and the sensitivity analysis of its parameter values is provided. Numerical analysis is conducted to various model parameter values. The results established all the model steady states to be locally and globally asymptotically stable at 𝑅0≤1. Also, resistance 𝜔 increased the infection rates by increasing the FAW larvae survival rate 𝜆 and reducing the insecticidal efficacy 𝛿𝑅 and 𝛿𝑁. This work informs the agriculture sector and policy makers on pest control with the best ways to use insecticides to minimize pest resistance and enhance efficacy in production. Pest control measures should be modified to lower the FAW survival rate and all model parameters contributing to resistance formation by FAW larvae in order to minimize FAW- host interaction thus reducing crop damage.Item Modeling exchange rate fluctuation on tourism demand(Abraham Kipkemei Maiyo, 2024-08)This research examined the effect of exchange rate fluctuations on tourism demand in Kenya using Vector Autoregressive (VAR) and Bayesian Structural Time Series (BSTS) models. Secondary data from 2010 to 2023 was sourced from the Ministry of Tourism, Wildlife & Heritage, and the Central Bank of Kenya. The VAR model revealed significant causal effects from exchange rate fluctuations, particularly the US Dollar and Euro, on regional tourism proxied by the Ugandan Shilling, with no reverse causality detected. Exogenous exchange rate shocks accounted for a substantial portion of forecast uncertainties in tourism demand. The BSTS model effectively captured trend, seasonality, and inherent uncertainty in tourism demand forecasting, with residual diagnostics confirming model validity. Forecasts demonstrated a downward trend in tourism demand over time. Comparative analysis showed the BSTS approach outperformed the VAR model, with a significantly lower Root Mean Squared Error (RMSE) of 0.0635 compared to 0.9875 for VAR and a higher forecasting efficiency ratio of 15.55. The findings indicate that major currency exchange rate fluctuations significantly affect Kenya's tourism flow. Recommendations include adopting policies for controlling exchange rate risk, incorporating the BSTS model into forecasting frameworks, monitoring economic shifts and consumer preferences, and considering external factors in modeling. By adopting these recommendations and stochastic modeling approaches, policymakers and industry players can make informed decisions on exchange rate risks, pricing strategies, and marketing to boost regional tourism.Item In-host Density-dependent Model of High-risk HPV Virions, Basal Cells, and Lymphocytes T-cells Incorporating Functional Responses(ELOSY MAKENA, 2024-08)Cervical cancer is one of the most common types of cancer and it is caused mostly by high-risk Human Papillomavirus (HPV) and continues to spread at an alarming rate. While HPV impacts have been investigated before, there are currently only a scanty number of mathematical models that account for HPV’s dynamic role in cervical cancer. The objectives were to develop an in-host density-dependent deterministic model for the dynamics implications of basal cells, virions, and lymphocytes incorporating immunity and functional responses. Analyze the model using techniques of epidemiological models such as basic reproduction number and simulate the model using Matlab ODE solver. Six compartments are considered in the model that is; Susceptible cells (S), Infected cells (I), Precancerous cells (P), Cancerous cells (C), Virions (V), and Lymphocytes (L). Next generation matrix (NGM), survival function, and characteristic polynomial method were used to determine the basic reproduction number denoted as 𝑅𝑅0. The findings from this research indicated that the Disease-Free Equilibrium point is locally asymptotically stable whenever 𝑅𝑅0 ∗ < 1 and globally asymptotically stable if 𝑅𝑅0 ∗ ≤ 1 and the Endemic Equilibrium is globally asymptotically stable if 𝑅𝑅0 ∗ > 1. The results obtained show that the progression rate of precancerous cells to cancerous cells (𝜃𝜃) has the most direct impact on the model. The model was able to estimate the longevity of a patient as 10 days when (𝜃𝜃) increases by 8%. The findings of this research will help healthcare providers, public authorities, and non-governmental health groups in creating effective prevention strategies to slow the development of cervical cancer. More research should be done to determine the exact number of cancerous cells that can lead to the death of a cervical cancer patient since this paper estimated a proportion of 75%. Keywords: In-host model, functional responses, stability analysis, simulation and reproduction number.Item Mathematical Modelling of Drug Abuse, Unemployment and Mental Stress on Population Dynamics of Mental Ilness(UoEm, 2024-01) Musyoka, Albanus MuambiThere has been a rise in the number of reported cases of mental illness in both High Income Countries (HICs) and Low and Middle Income Countries (LMICs). Non-communicable Diseases (NCDs) seldom make use of mathematical modeling. This research suggests eight first-order differential equations to form the basis of a mathematical model for psychiatric disorders. There are eight distinct categories created to reflect the public at large: the vulnerable, the working and jobless, drug addicts, the emotionally distraught, and the mentally ill. Theoretically, the well-posedness of the model equations is established by examining the positive, bounded, existing, unique solutions and the local and global stability. The eigenvalue approach was used to investigate local stability, and a Lyapunov function was created to analyze global behavior. In order to back up the analytical results, we performed a numerical investigation of the dynamical behavior of the model's equations using the fourth-order Runge-Kutta technique with the use of the MATLAB software package. To better understand the impact of environmental factors on mental disease, researchers have experimented with changing a number of variables related to mental stress, unemployment and drug addiction among certain groups. Based on the findings, the prevalence of mental illness skyrocketed anytime variables related to psychological strain or substance (drug) addiction rose in severity. In conclusion, lowering the growing rates of mental illness may be accomplished through increasing options for employment, improving working conditions, and fostering a welcoming workplace.Item Probabilistic Analysis of Covid-19 Transmission In Kenya using Markov Chain(UoEm, 2023) Mungania, Joseph MugambiThe COVID-19 pandemic has had a profound impact on global health and has highlighted the importance of understanding the transmission dynamics of infectious diseases. This study aimed to construct a COVID-19 transmission matrix in Kenya using the Markov chain and to examine the equilibrium distribution and steady states for COVID-19 transmission in Kenya. The study utilized data from the Ministry of Health in Kenya and other sources to estimate the transition probabilities used in the Markov chain model. The results showed that the transmission of COVID 19 in Kenya is primarily driven by human mobility and the spread of the virus from infected individuals to susceptible individuals. The equilibrium distribution indicated that the steady state for COVID-19 transmission in Kenya is heavily dependent on the control measures that are in place. The steady states for COVID-19 transmission in Kenya were estimated to be lower for scenarios with more stringent control measures in place. The results of the study showed that the COVID-19 transmission matrix in Kenya is dynamic and influenced by a range of factors, including human behavior, the availability of effective interventions, and the emergence of new variants of the virus. The equilibrium distribution of COVID-19 transmission in Kenya was found to be influenced by the presence of comorbidities, the availability of effective treatments, and the degree of community transmission. The steady states for COVID-19 transmission in Kenya were found to be influenced by the effectiveness of interventions, including the use of masks, social distancing measures, and the availability of vaccines. The results of this study provide important insights into the transmission dynamics of COVID-19 in Kenya, and can inform the development of more effective strategies for controlling its spread. In conclusion, the results of this study demonstrate the utility of Markov chain models for the probabilistic analysis of COVID-19 transmission. The findings of this study highlight the need for continued monitoring of COVID 19 transmission in Kenya, and for the development of effective interventions to control its spread. In conclusion, the probabilistic analysis of COVID-19 transmission in Kenya conducted in this study is an important step towards understanding the transmission dynamics of the virus and towards developing effective control measures. Further research is needed to improve the accuracy of the model and to understand the complex dynamics of COVID-19 transmission in Kenya and other populations.Item Mathematical Modelling and Simulation of Competition for Students’ Population Via Influence And Economic Factors With Holling Type Ii Response(UoEm, 2023-08) Odhiambo, Brian OtienoThe increase in Kenyan population attracted the establishment of more schools, both public schools and private schools. This was due to the need to cater for the increasing number of students being enrolled in schools. Moreover, the dynamics of students’ population both in public schools and private schools have created the changes in the schools’ population. This occurs through transfer from one category of school to the other, through completion of the learning period and through drop out due to unknown reasons. This subjected both the public schools and private schools to compete in order to maintain a good number of students under their custody. In this work, a modified Lotka-Volterra model of schools and non-enrolled entities population in the education system is studied. Private schools and non-enrolled entities play the role of a predator in public schools. Again, public schools and non-enrolled entities play the role of predators in private schools. This study uses integrated Holling type II functional response to analyze the model. Establishment of equilibrium points and their stability are determined using the Routh-Hurwitz criterion and eigenvalue method. Global stability has been done for the positive equilibrium point. Hopf bifurcation is also done around the positive equilibrium point. Data obtained from the Ministry of Education and the sources cited were used to estimate the model parameters. Finally, graphical illustration of various parameter is derived to show their effect on schools when they are varied. The study revealed that the increase in transfer rate from private to non-enrolled, transfer rate from public to non-enrolled and the non-enrolled entity predation on public schools greatly affects the schools’ population as they are the ones leading to predation in school. Therefore, proper strategies should be developed to focus on reducing the parameters that affects the schools’ population adversely to avoid leading schools’ population to extinct.Item Modelling Conditions for Quality Egg Storage Using Randomized Complete Block Design(University of Embu, 2021-09) Gogo, Jacqueline AkeloRearing chicken has contributed positively to global nutrition, especially egg production. This practice attracts both large and small-scale poultry keeping within the world’s economy. Egg storage has been a problem due to ineffective methods subjecting many farmers and egg retailers to losses. These methods include various models involving statistical analysis of the storage conditions on the egg quality. However, they do not provide sufficient information. Therefore, confusion persists between the use of fixed and mixed-effect models. The confusion is because some studies analyse randomized complete block design as fixed while others as mixed effect models. Apparent deficiencies of the evidenced information from the randomized complete block design model prompted this study. The quality of the eggs was determined by the physical characterization and changes of both external and internal properties under different temperature conditions and storage duration. The study evaluated the effect of storage temperature at three levels (5 o C, 19.5 o o ) on egg quality using fixed and mixed-effects models. This study used a total of 618 fresh and unfertilized eggs from the ISA (Institut de Sélection Animale) brown layers. Restricted maximum likelihood and analysis of variance methods were used to determine the efficiency of fixed and mixed effect models. Results showed that the physical components of the egg were significantly affected at 5 C and 30 C) and time at four levels (2 nd th nd , 12 , 22 nd and 32 o o C, 19.5 o P ( 0.05). C and 30 C The effect was more adverse on eggs stored o at 30 C for 32 days. However, storage temperatures of 5 o o C and 19.5 C led to an extensive reduction in the Haugh unit, yolk index, and egg white. Contrariwise, it increased the weight loss, the albumen diameter under storage for 2 nd th nd , 12 , 22 nd , and 32 -time intervals. This study recommends a temperature of 5 o C for egg quality preservation. The eggs should be reserved in fridge-freezers for 32 days, at 19.5 o C for fourteen days, and at 30 o C for seven days maximal. The fixed-effect models exhibited smaller components in diameter and height of albumen, yolk index, weight loss, and Haugh unit. This overlapped instances where the fixed-effect models were significantly the same as the mixed-effect models. This study proposes that the fixed effect model is the most appropriate for randomized completely block design experiments. This study obtained G-optimal efficiency of 68% to predict the optimal levels of egg storage for quality maintenance. This study strongly recommends further studies to consider optimization using other classes of storage conditions.Item Socio-economic determinants of low birth weight in Kenya: Logistic regression analysis(University of Nairobi, 2010) Atitwa, EdwinBabies born with Low-birth weight are at increased risk for serious health problems which are accompanied by disabilities and even death. Hence this study aims to determine socioeconomic factors that lead to low birth weight of children in Kenya. Data used was from Kdhs 2003 and the significant effect of socio-economic determinants on low birth weight was examined using logistic regression analysis data is categorical and continuous in nature, where predictor variables being socio-economic determinants and birth weight being dependent variable. Results indicate that out of ten socio-economic factors involved in the study, six revealed some significant effects on the children with low birth weight, four determinants namely , sex of the child, antenatal visit for pregnancy, antenatal care attendance and time wanted pregnancy has no significant impact to the infant birth weight holding other factors constant. The six major socio-economic determinants which can alter the weight at birth for babies born in Kenya were Religion, Educational attainment, Age of household head, Smokes nothing, Current marital status, and Size of child at birth. Therefore Socio-economic determinants have a significant effect on Low birth weight which suggests a strong negative associated with infant survival in Kenya independent of other risk factors. Children can be ensured a healthy start in life if women start pregnancy healthy and well nourished, and go through pregnancy and childbirth safely. Also expectant mothers should keenly focus on the socio-economic determinants by avoiding some like smoking, acquire education so that they can understand how to take care of themselves and not always to depend on the doctor's advice, give enough time and prepare for the pregnancyItem Crame´R-Rao Bound of Direction Finding using Uniform Circular Array And 2-Circle Concentric Uniform Array(university of Embu, 2019-09) MUSYOKA, DAVIDSource direction-of-arrival estimation problem has received much attention in recent years following its significant role in array-signal processing and wide range of applications such as radar, wireless communication, sonar, seismology among others. Direction finding has been solved by several techniques such as Maximum likelihood estimator, MUltiple Signal Classification, Estimation of Parameters via Rotational Invariance Technique and Cram´er- Rao bound using array of sensors in both uniformly-spaced and non-uniformly-spaced. The sensors have further been arranged in different geometric patterns ranging from onedimension to three-dimensional. However, little effort has been made in direction finding using concentric planar arrays with fixed centers at the Cartesian origin. In this study, a new planar sensor-array geometry (the 2-circle concentric uniform array geometry) centered at the Cartesian origin, that maximizes the array’s spatial aperture mainly for bivariate azimuth-polar resolution of direction-of-arrival estimation problem was proposed. The proposed geometry provides almost invariant azimuth angle coverage and offers the advantage of full rotational symmetry (circular invariance) while maintaining an inter-sensor spacing not exceeding half wavelength (for non-ambiguity with respect to the Cartesian direction cosines) among other advantages. The study adopted Cram´er-Rao bound technique of direction finding via a uniform circular array (single ring array) and the proposed geometry to estimate the bivariate azimuth-polar angles-of-arrival. Both the array manifolds and the Cram´er-Rao bounds for the uniform circular array and that of the proposed array grid were derived. Further, a better-accurate performance in direction finding of the proposed array grid over that of the single ring array grid was analytically verified under different constraints of investigation. It was found that the proposed sensor-array geometry has better estimation accuracy than a single ring array and the 2-circle concentric uniform array geometry would have the best estimation accuracy for minimal number of sensors hence reducing the hardware cost. The study therefore recommends that the 2-circle concentric uniform array geometry should be used for direction finding with minimal number of sensors and with an inter-sensor spacing not exceeding half a wavelength as opposed to a uniform circular array geometry. xItem Cramér-Rao Bound of Direction-Finding using uniform Hexagonal Array(University of Embu, 2019-09) Ndiritu, Grace W.Direction-of-arrival (DOA) estimation is an important branch in the field of array signal processing. It can be applied in such fields as wireless communication, sonar, radar, biomedicine, and radio detection. This fact together with the development of the geometries used in the past years is the principal motivation of this research project. Although various studies have focused on the uniform hexagonal array for direction finding, there is scanty use of the uniform hexagonal array in conjunction with Cram´er-Rao bound for direction finding. In this research project, the direction-of-arrival estimation of Cram´er-Rao bound based on the uniform hexagonal array was studied. The proposed approach concentrated on deriving the array manifold vector for the uniform hexagonal array and Cram´er-Rao bound of the uniform hexagonal array. The Cram´er-Rao bound based on the uniform hexagonal array were compared with Cram´er-Rao bound based on the uniform circular array. The array manifold vector and Cram´er-Rao bound for the uniform hexagonal array were derived. The Cram´er-Rao bound based on the uniform hexagonal array was compared with Cram´er-Rao bound of uniform circular array. The conclusions are as follows, the Cram´er-Rao bound of uniform hexagonal array decreases with an increase in the number of sensors, whereas that of circular array reduces with increase in the number of sensors. The comparison between the uniform hexagonal array and uniform circular array shows that the Cram´er-Rao bound of the uniform hexagonal array was slightly higher as compared to the Cram´er-Rao bound of the uniform circular array. Thus, uniform circular array is a better approximator as compared to uniform hexagonal array. Graphical representation validated the analytical result.Item Spatial Analysis and Mapping of Infant Mortality in Kenya On The Basis Of Demographic and Health Survey Data(University of Nairobi, 2010) Mutwiri, Robert M.This study set out to examine and map the spatial variation of infant mortality in Kenya. We used data from Demographic and Health Survey (DHS) database to explore spatial variation. Generalized linear mixed inodel(GLMM) with Enumeration Areas (EA) specific random effects was used to assess the effects of geographical heterogeneity and other covariates. The model based Geostatistical methods were used to quantify the spatial variations of the observations using the variograms and fitted the exponential and matern parametric models to the sample variograms. Then utilizing the fitted variogram function, Trans-gaussian kriging was performed infant mortality rates based on both models and produced smooth maps. Generalized linear mixed model (GLMM) showed significant geographical heterogeneity in infant mortality. However, moran's I statistic showed spatial autocorrelation unaccounted for by GLMM. Modeling the correlation between people as a decreasing function of the spatial distance between them, Geostatistical models gave information no only on the magnitude but also on the scale of spatial variation. The socioeconomic status and infant mortality varied significantly across districts in Kenya. EA indicators better explained spatial variation of mortality when measured across a continuous space rather than within administrative areas. The resulting map broadly agreed with the the previous studies on the variation of risk in the country, and further showed marked variation even at local level. High risk areas were in Nyanza regions, while low risk areas are in Central of the country. The maps provided an initial description of the geographic variation of IMR in Kenya, and might help in the choice and design of interventions, which is crucial for reducing the child mortality by two thirds by 2015.Item Review of Methods of Estimating Parameters In Nonlinear Mixed-Effects (Nlme) Models(University of Nairobi, 2007-07) Mbunzi, Stephen M.This study is a critical review of theoreticalissues that underline the linear mixed effects (LME) and nonlinear mixed effects (NLME) models. These two areas are revisited under maximum likelihood and restricted maximumlikelihood estimation frameworks. We also review methods of estimating parameters in both linear and nonlinear mixed effects models. In the case of LME, we consider different ways of developing the likelihood estimators, key among these methods are the “pseudo-data” approach, orthogonal triangular decomposition method and the use of penalized least squares problem. For NLME, we intended to investigate the computational efficiency and accuracy of computational methods, like the b-splines, that could be used to approximate the log-likelihood function in non-linear mixed effects models. This was not achieved in this study but can be an interesting area for further research work. We critically review the four methods of estimating parameters by Pinheiro and Bates (1995) through proving a number of lemmas. Our proves led us to same stated results by different researchers in different papers. This is a key issue in the investigation of other expansion methods and comparing their computational efficiency and accuracy with these existing ones. We conclude by giving an insight into linear mixed effects models by analyzing a data set from livestock where we examine incorporation of random effects to study variations among rams (sires) and ewes (dams) and their influences on lamb weaning weight. Factors like year of birth of the lamb, sex of lamb, age at weaning, age of dam, ewe breed and ram breed are found to influence the weaning weight differently. With the random terms (ewes and rams) specified in the model the estimate of the residual among lamb variance is found to reduce due to taking into account the variations among rams and ewes within breeds. It was our intention to obtain heritability estimates which determine the proportion of the variation among offspring that have been handed down from parents out of these random estimates.Item Connecting People using Latent Semantic Analysis for Knowledge Sharing(2010-01) Mugo, David M.A shift from technology-oriented knowledge management to people-oriented knowledge management is indispensable. To achieve this, organizations must understand the nature of knowledge. In this work, knowledge has been found to be both a process and a collection of artifacts. This makes knowledge and the knower to be two inseparable entities. Consequently, the appropriate way to share both the explicit and the implicit knowledge components is through people-with-people connection. However, from existing barriers like location and time differences among others, people-with-documents connection is proposed as an intermediate step. The investigation of latent semantic analysis (LSA) in achieving people-with-documents connection has revealed decreased precision performance at higher recall performance. A solution to include annotations in the technique has been proposed to refine knowledge representation into the LSA technique. Annotation process based on domain ontologies has been proposed to compliment the LSA knowledge mining process from documents with domain knowledge represented by ontologiesItem An Investigation of the Latent Semantic Analysis Technique for Document Retrieval(2014) Mugo, David M.Latent semantic analysis (LSA) application in information retrieval promises to offer better performance by overcoming some limitations that plagues traditional termmatching techniques. These term-matching techniques have always relied on matching query terms with document terms to retrieve the documents having terms matching the query terms. However, by use of these traditional retrieval techniques, users’ needs have not been adequately served. While users want to search through information based on conceptual content, natural languages have limited the expression of these concepts. They present synonymy problem (a situation where several words may have the same meaning) and polysemy problem (a situation where a word may have several meanings). Due to these natural language problems, individual words contained in users’ queries, may not explicitly specify the intended user’s concept, which may result in the retrieval of some irrelevant documents. LSA seems to be a promising technique in overcoming these natural language problems especially synonymy problem. It deals with exploiting the global relationships between terms and documents and then mapping these documents and terms in a proximity space, where terms and documents that are closely related are mapped close to each other in this space. Queries are then mapped to this space with documents being retrieved based on similarity measures. In this report, LSA performance in documents retrieval is investigated and compared with traditional term-matching techniques.