Maternal Health Services: Organized services to provide health care to expectant and nursing mothers.Maternal Welfare: Organized efforts by communities or organizations to improve the health and well-being of the mother.Maternal Mortality: Maternal deaths resulting from complications of pregnancy and childbirth in a given population.Midwifery: The practice of assisting women in childbirth.Health Services Accessibility: The degree to which individuals are inhibited or facilitated in their ability to gain entry to and to receive care and services from the health care system. Factors influencing this ability include geographic, architectural, transportational, and financial considerations, among others.Delivery, Obstetric: Delivery of the FETUS and PLACENTA under the care of an obstetrician or a health worker. Obstetric deliveries may involve physical, psychological, medical, or surgical interventions.Pregnancy: The status during which female mammals carry their developing young (EMBRYOS or FETUSES) in utero before birth, beginning from FERTILIZATION to BIRTH.Prenatal Care: Care provided the pregnant woman in order to prevent complications, and decrease the incidence of maternal and prenatal mortality.Power (Psychology): The exertion of a strong influence or control over others in a variety of settings--administrative, social, academic, etc.Patient Acceptance of Health Care: The seeking and acceptance by patients of health service.Developing Countries: Countries in the process of change with economic growth, that is, an increase in production, per capita consumption, and income. The process of economic growth involves better utilization of natural and human resources, which results in a change in the social, political, and economic structures.Health Services: Services for the diagnosis and treatment of disease and the maintenance of health.Mental Health Services: Organized services to provide mental health care.Socioeconomic Factors: Social and economic factors that characterize the individual or group within the social structure.Health Services Needs and Demand: Health services required by a population or community as well as the health services that the population or community is able and willing to pay for.Health Services Research: The integration of epidemiologic, sociological, economic, and other analytic sciences in the study of health services. Health services research is usually concerned with relationships between need, demand, supply, use, and outcome of health services. The aim of the research is evaluation, particularly in terms of structure, process, output, and outcome. (From Last, Dictionary of Epidemiology, 2d ed)Delivery of Health Care: The concept concerned with all aspects of providing and distributing health services to a patient population.Reproductive Health Services: Health care services related to human REPRODUCTION and diseases of the reproductive system. Services are provided to both sexes and usually by physicians in the medical or the surgical specialties such as REPRODUCTIVE MEDICINE; ANDROLOGY; GYNECOLOGY; OBSTETRICS; and PERINATOLOGY.Health Status: The level of health of the individual, group, or population as subjectively assessed by the individual or by more objective measures.Rural Health Services: Health services, public or private, in rural areas. The services include the promotion of health and the delivery of health care.Health Policy: Decisions, usually developed by government policymakers, for determining present and future objectives pertaining to the health care system.Public Health: Branch of medicine concerned with the prevention and control of disease and disability, and the promotion of physical and mental health of the population on the international, national, state, or municipal level.Community Health Services: Diagnostic, therapeutic and preventive health services provided for individuals in the community.Child Health Services: Organized services to provide health care for children.Health Care Surveys: Statistical measures of utilization and other aspects of the provision of health care services including hospitalization and ambulatory care.Quality of Health Care: The levels of excellence which characterize the health service or health care provided based on accepted standards of quality.Health Care Reform: Innovation and improvement of the health care system by reappraisal, amendment of services, and removal of faults and abuses in providing and distributing health services to patients. It includes a re-alignment of health services and health insurance to maximum demographic elements (the unemployed, indigent, uninsured, elderly, inner cities, rural areas) with reference to coverage, hospitalization, pricing and cost containment, insurers' and employers' costs, pre-existing medical conditions, prescribed drugs, equipment, and services.Health Surveys: A systematic collection of factual data pertaining to health and disease in a human population within a given geographic area.Health Promotion: Encouraging consumer behaviors most likely to optimize health potentials (physical and psychosocial) through health information, preventive programs, and access to medical care.Community Mental Health Services: Diagnostic, therapeutic and preventive mental health services provided for individuals in the community.Maternal-Child Health Centers: Facilities which administer the delivery of health care services to mothers and children.Primary Health Care: Care which provides integrated, accessible health care services by clinicians who are accountable for addressing a large majority of personal health care needs, developing a sustained partnership with patients, and practicing in the context of family and community. (JAMA 1995;273(3):192)Family Planning Services: Health care programs or services designed to assist individuals in the planning of family size. Various methods of CONTRACEPTION can be used to control the number and timing of childbirths.Human Rights Abuses: Deliberate maltreatment of groups of humans beings including violations of generally-accepted fundamental rights as stated by the Universal Declaration of Human Rights, adopted and proclaimed by the United Nations General Assembly resolution 217 A (III) of 10 December 1948.Adolescent Health Services: Organized services to provide health care to adolescents, ages ranging from 13 through 18 years.Health Expenditures: The amounts spent by individuals, groups, nations, or private or public organizations for total health care and/or its various components. These amounts may or may not be equivalent to the actual costs (HEALTH CARE COSTS) and may or may not be shared among the patient, insurers, and/or employers.Preventive Health Services: Services designed for HEALTH PROMOTION and prevention of disease.Health Planning: Planning for needed health and/or welfare services and facilities.Health Services for the Aged: Services for the diagnosis and treatment of diseases in the aged and the maintenance of health in the elderly.Health Services Administration: The organization and administration of health services dedicated to the delivery of health care.Mental Health: The state wherein the person is well adjusted.Occupational Health Services: Health services for employees, usually provided by the employer at the place of work.Healthcare Disparities: Differences in access to or availability of medical facilities and services.State Medicine: A system of medical care regulated, controlled and financed by the government, in which the government assumes responsibility for the health needs of the population.Attitude to Health: Public attitudes toward health, disease, and the medical care system.National Health Programs: Components of a national health care system which administer specific services, e.g., national health insurance.Insurance, Health: Insurance providing coverage of medical, surgical, or hospital care in general or for which there is no specific heading.Postnatal Care: The care provided to women and their NEWBORNS for the first few months following CHILDBIRTH.Health Priorities: Preferentially rated health-related activities or functions to be used in establishing health planning goals. This may refer specifically to PL93-641.Public Health Administration: Management of public health organizations or agencies.Women's Health Services: Organized services to provide health care to women. It excludes maternal care services for which MATERNAL HEALTH SERVICES is available.World Health: The concept pertaining to the health status of inhabitants of the world.Urban Health Services: Health services, public or private, in urban areas. The services include the promotion of health and the delivery of health care.United States Public Health Service: A constituent organization of the DEPARTMENT OF HEALTH AND HUMAN SERVICES concerned with protecting and improving the health of the nation.Rural Health: The status of health in rural populations.Health Services, Indigenous: Health care provided to specific cultural or tribal peoples which incorporates local customs, beliefs, and taboos.Health Knowledge, Attitudes, Practice: Knowledge, attitudes, and associated behaviors which pertain to health-related topics such as PATHOLOGIC PROCESSES or diseases, their prevention, and treatment. This term refers to non-health workers and health workers (HEALTH PERSONNEL).Cross-Sectional Studies: Studies in which the presence or absence of disease or other health-related variables are determined in each member of the study population or in a representative sample at one particular time. This contrasts with LONGITUDINAL STUDIES which are followed over a period of time.Health: The state of the organism when it functions optimally without evidence of disease.Rural Population: The inhabitants of rural areas or of small towns classified as rural.Health Personnel: Men and women working in the provision of health services, whether as individual practitioners or employees of health institutions and programs, whether or not professionally trained, and whether or not subject to public regulation. (From A Discursive Dictionary of Health Care, 1976)Qualitative Research: Any type of research that employs nonnumeric information to explore individual or group characteristics, producing findings not arrived at by statistical procedures or other quantitative means. (Qualitative Inquiry: A Dictionary of Terms Thousand Oaks, CA: Sage Publications, 1997)Health Facilities: Institutions which provide medical or health-related services.Home Care Services: Community health and NURSING SERVICES providing coordinated multiple services to the patient at the patient's homes. These home-care services are provided by a visiting nurse, home health agencies, HOSPITALS, or organized community groups using professional staff for care delivery. It differs from HOME NURSING which is provided by non-professionals.Delivery of Health Care, Integrated: A health care system which combines physicians, hospitals, and other medical services with a health plan to provide the complete spectrum of medical care for its customers. In a fully integrated system, the three key elements - physicians, hospital, and health plan membership - are in balance in terms of matching medical resources with the needs of purchasers and patients. (Coddington et al., Integrated Health Care: Reorganizing the Physician, Hospital and Health Plan Relationship, 1994, p7)Financing, Government: Federal, state, or local government organized methods of financial assistance.Questionnaires: Predetermined sets of questions used to collect data - clinical data, social status, occupational group, etc. The term is often applied to a self-completed survey instrument.Interviews as Topic: Conversations with an individual or individuals held in order to obtain information about their background and other personal biographical data, their attitudes and opinions, etc. It includes school admission or job interviews.Great BritainHealth Behavior: Behaviors expressed by individuals to protect, maintain or promote their health status. For example, proper diet, and appropriate exercise are activities perceived to influence health status. Life style is closely associated with health behavior and factors influencing life style are socioeconomic, educational, and cultural.United StatesInfant, Newborn: An infant during the first month after birth.United States Indian Health Service: A division of the UNITED STATES PUBLIC HEALTH SERVICE that is responsible for the public health and the provision of medical services to NATIVE AMERICANS in the United States, primarily those residing on reservation lands.Health Care Rationing: Planning for the equitable allocation, apportionment, or distribution of available health resources.Health Education: Education that increases the awareness and favorably influences the attitudes and knowledge relating to the improvement of health on a personal or community basis.Public Sector: The area of a nation's economy that is tax-supported and under government control.Women's Health: The concept covering the physical and mental conditions of women.BangladeshPoverty: A situation in which the level of living of an individual, family, or group is below the standard of the community. It is often related to a specific income level.Pregnancy Complications: Conditions or pathological processes associated with pregnancy. They can occur during or after pregnancy, and range from minor discomforts to serious diseases that require medical interventions. They include diseases in pregnant females, and pregnancies in females with diseases.Health Plan Implementation: Those actions designed to carry out recommendations pertaining to health plans or programs.IndiaObstetrics: A medical-surgical specialty concerned with management and care of women during pregnancy, parturition, and the puerperium.Program Evaluation: Studies designed to assess the efficacy of programs. They may include the evaluation of cost-effectiveness, the extent to which objectives are met, or impact.Politics: Activities concerned with governmental policies, functions, etc.Attitude of Health Personnel: Attitudes of personnel toward their patients, other professionals, toward the medical care system, etc.Mental Disorders: Psychiatric illness or diseases manifested by breakdowns in the adaptational process expressed primarily as abnormalities of thought, feeling, and behavior producing either distress or impairment of function.Oral Health: The optimal state of the mouth and normal functioning of the organs of the mouth without evidence of disease.Regional Health Planning: Planning for health resources at a regional or multi-state level.Public Health Practice: The activities and endeavors of the public health services in a community on any level.Dental Health Services: Services designed to promote, maintain, or restore dental health.Occupational Health: The promotion and maintenance of physical and mental health in the work environment.Home Childbirth: Childbirth taking place in the home.Health Care Sector: Economic sector concerned with the provision, distribution, and consumption of health care services and related products.Health Status Disparities: Variation in rates of disease occurrence and disabilities between population groups defined by socioeconomic characteristics such as age, ethnicity, economic resources, or gender and populations identified geographically or similar measures.Outcome Assessment (Health Care): Research aimed at assessing the quality and effectiveness of health care as measured by the attainment of a specified end result or outcome. Measures include parameters such as improved health, lowered morbidity or mortality, and improvement of abnormal states (such as elevated blood pressure).EnglandHealth Resources: Available manpower, facilities, revenue, equipment, and supplies to produce requisite health care and services.Quality Assurance, Health Care: Activities and programs intended to assure or improve the quality of care in either a defined medical setting or a program. The concept includes the assessment or evaluation of the quality of care; identification of problems or shortcomings in the delivery of care; designing activities to overcome these deficiencies; and follow-up monitoring to ensure effectiveness of corrective steps.Vulnerable Populations: Groups of persons whose range of options is severely limited, who are frequently subjected to COERCION in their DECISION MAKING, or who may be compromised in their ability to give INFORMED CONSENT.Infant Welfare: Organized efforts by communities or organizations to improve the health and well-being of infants.Ethiopia: An independent state in eastern Africa. Ethiopia is located in the Horn of Africa and is bordered on the north and northeast by Eritrea, on the east by Djibouti and Somalia, on the south by Kenya, and on the west and southwest by Sudan. Its capital is Addis Ababa.Contract Services: Outside services provided to an institution under a formal financial agreement.Men: Human males as cultural, psychological, sociological, political, and economic entities.Health Manpower: The availability of HEALTH PERSONNEL. It includes the demand and recruitment of both professional and allied health personnel, their present and future supply and distribution, and their assignment and utilization.Lesotho: A kingdom in southern Africa, within the republic of SOUTH AFRICA. Its capital is Maseru.United States Dept. of Health and Human Services: A cabinet department in the Executive Branch of the United States Government concerned with administering those agencies and offices having programs pertaining to health and human services.United Nations: An international organization whose members include most of the sovereign nations of the world with headquarters in New York City. The primary objectives of the organization are to maintain peace and security and to achieve international cooperation in solving international economic, social, cultural, or humanitarian problems.Personal Health Services: Health care provided to individuals.Catchment Area (Health): A geographic area defined and served by a health program or institution.Community Health Centers: Facilities which administer the delivery of health care services to people living in a community or neighborhood.Social Class: A stratum of people with similar position and prestige; includes social stratification. Social class is measured by criteria such as education, occupation, and income.Urban Population: The inhabitants of a city or town, including metropolitan areas and suburban areas.Community Health Planning: Planning that has the goals of improving health, improving accessibility to health services, and promoting efficiency in the provision of services and resources on a comprehensive basis for a whole community. (From Facts on File Dictionary of Health Care Management, 1988, p299)Obstetric Labor Complications: Medical problems associated with OBSTETRIC LABOR, such as BREECH PRESENTATION; PREMATURE OBSTETRIC LABOR; HEMORRHAGE; or others. These complications can affect the well-being of the mother, the FETUS, or both.Logistic Models: Statistical models which describe the relationship between a qualitative dependent variable (that is, one which can take only certain discrete values, such as the presence or absence of a disease) and an independent variable. A common application is in epidemiology for estimating an individual's risk (probability of a disease) as a function of a given risk factor.Urban Health: The status of health in urban populations.NepalEnvironmental Health: The science of controlling or modifying those conditions, influences, or forces surrounding man which relate to promoting, establishing, and maintaining health.School Health Services: Preventive health services provided for students. It excludes college or university students.Student Health Services: Health services for college and university students usually provided by the educational institution.Marketing of Health Services: Application of marketing principles and techniques to maximize the use of health care resources.Educational Status: Educational attainment or level of education of individuals.Mothers: Female parents, human or animal.Community Networks: Organizations and individuals cooperating together toward a common goal at the local or grassroots level.Pregnant Women: Human females who are pregnant, as cultural, psychological, or sociological entities.Puerperal Infection: An infection occurring in PUERPERIUM, the period of 6-8 weeks after giving birth.Referral and Consultation: The practice of sending a patient to another program or practitioner for services or advice which the referring source is not prepared to provide.Prevalence: The total number of cases of a given disease in a specified population at a designated time. It is differentiated from INCIDENCE, which refers to the number of new cases in the population at a given time.Malawi: A republic in southern Africa east of ZAMBIA and MOZAMBIQUE. Its capital is Lilongwe. It was formerly called Nyasaland.PakistanNeeds Assessment: Systematic identification of a population's needs or the assessment of individuals to determine the proper level of services needed.Perinatal Care: The care of women and a fetus or newborn given before, during, and after delivery from the 28th week of gestation through the 7th day after delivery.Private Sector: That distinct portion of the institutional, industrial, or economic structure of a country that is controlled or owned by non-governmental, private interests.Emergency Medical Services: Services specifically designed, staffed, and equipped for the emergency care of patients.World Health Organization: A specialized agency of the United Nations designed as a coordinating authority on international health work; its aim is to promote the attainment of the highest possible level of health by all peoples.Focus Groups: A method of data collection and a QUALITATIVE RESEARCH tool in which a small group of individuals are brought together and allowed to interact in a discussion of their opinions about topics, issues, or questions.Residence Characteristics: Elements of residence that characterize a population. They are applicable in determining need for and utilization of health services.Transients and Migrants: People who frequently change their place of residence.Patient Satisfaction: The degree to which the individual regards the health care service or product or the manner in which it is delivered by the provider as useful, effective, or beneficial.Community Health Workers: Persons trained to assist professional health personnel in communicating with residents in the community concerning needs and availability of health services.


text:
Nimona Animated Movie Gets Release Date
By Tim Karan/June 30, 2017 3:05 pm EST
Nimona will make the leap from Tumblr to the big screen.
According to Exhibitor Relations, a movie adaptation of the popular web comic-turned-graphic novel will be released by 20th Century Fox Animation on February 14, 2020.
Noelle Stevenson originally came up with the story while she was a student at the Maryland Institute College of Art, and she posted Nimona to Tumblr. It became a graphic novel in 2015, and Fox acquired the rights that year.
Based on Stevenson's experience with cosplay, the story follows a young shapeshifter who is the sidekick of supervillain Lord Ballister Blackheart. Blackheart is a knight who became a mad scientist. Together, the pair fight against Blackheart's nemesis, Sir Ambrosius Goldenloin.
Patrick Osborne, director of the Oscar-winning short Feast, will helm the movie based on a script by Marc Haimes (Kubo and the Two Strings).
Stevenson also worked on the Lumberjanes comic series, which Fox is also developing into a film.
While wait to hear about the cast, take a look at the surprising faces behind some famous cartoon characters.

redpajama_set_name: RedPajamaCommonCrawl


text:
C. M. Kaliko Baker
Hope Polopeka
C. M. Kaliko Baker, PhD, has been a Kumu 'Ōlelo Hawai'i (Hawaiian language instructor) in Kawaihuelani Center for Hawaiian Language at the University of Hawai'i at Mānoa since 1996 and currently holds an Assistant Professor Position. His research and scholarly efforts primarily focus on Hawaiian, particularly discourse grammar, traditional Hawaiian narratives, and the revitalization of Hawaiian. He completed his PhD in Linguistics in 2012 entitled A-class Genitive Subject Effect: A pragmatic and discourse grammar approach to a- and o-class genitive subject selection in Hawaiian, in which he explored genitive class selection in Hawaiian. Since obtaining his PhD, Kaliko has been writing and researching primarily in the space of Kanaka Maoli identity, particularly how ʻŌlelo Hawaiʻi is central to indexicalizing and presenting who we are as Kānaka Maoli. He co-authors Hawaiian-medium plays with Tammy Haili'ōpua Baker and also serves as dramaturge under Ka Hālau Hanakeaka. 2022 will bring a book that Kaliko and Hailiʻōpua edited entitled Moʻolelo: The foundation of Hawaiian Knowledge which includes essays from top scholars in the field of Hawaiian knowledge. Currently, he is the president of a non-profit organization, Halele'a Arts Foundation, which strives to promote Hawaiian-medium theatrical and other media projects.
Dr. Baker is a long time Protect Kaho'olawe 'Ohana member, the grassroots organization that stopped the U.S. Navy's bombing and desecration of Kaho'olawe. Since 2003, he has served as the head Moʻo Lono leading the Makahiki ceremonies while moving the practice forward through the incorporation of ʻŌlelo Hawaiʻi and literary fluency into the ceremonies. Recently, he has been leading huakaʻi ʻōlelo Hawaiʻi under the PKO to Kahoʻolawe during UH's Spring Break during which the UH Mānoa and Hilo campuses connect and work on Kahoʻolawe using ʻōlelo Hawaiʻi as their means of communication.
As a lifelong resident of Koʻolaupoko, Oʻahu-a-Lua, Dr. Baker keeps his hands in his community by taking his students to Ke Kula ʻo S.M. Kamakau in Haʻikū and volunteering at Ka Loko Iʻa o Heʻeia, amongst other things.
Leka Uila: cbaker@hawaii.edu


text:
Arteta tipped to bin off a certain type of player at Arsenal
By Harry Sherlock
Arsenal have endured a difficult start to the season.
The Gunners have been inconsistent throughout the beginning of 2020/21, roundly failing to string together a run of form.
Arteta's side won their opening two games of the season, beating Fulham and West Ham United, but they were then thumped 3-1 by Liverpool, prior to a 2-1 win over Sheffield United.
Following the victory, they suffered back-to-back defeats, losing to Manchester City and Leicester City. A 1-0 win over Manchester United seemed to lift some of the gloom but they were then thrashed 3-0 by Aston Villa before the international break.
This weekend, they take on Leeds United and a defeat to Marcelo Bielsa's side would see the 15th-placed Whites overtake the Gunners, who have taken 12 points from eight games.
The fixtures only get tougher too; following their game against Leeds, the Gunners take on Wolves and Tottenham Hotspur.
Thoughts are already shifting towards the January transfer window and perhaps even to the summer as the Gunners look to strengthen the squad Arteta is currently working with.
And Kevin Campbell, the ex-Arsenal striker, believes that the Spaniard must purge the squad of any players with a "loser's mentality" if they are to have any hope of becoming genuine title contenders.
Speaking to Football Insider, he said: "It all depends on the recruitment and the mindset of the team.
"There has been some players who have come in and made a difference but there are still players within that squad who have a loser's mentality. That is the hardest thing to change.
"In a cup competition it is easy but when it is the league, week in, week out. It is tough to have that strong mentality but it makes a hell of a difference.
"Mikel Arteta knows what is required because he has been at Manchester City. He knows what is in the dressing room and some of those players have not got it.
"It is down to Arteta to get that across but I can see certain players will not be at the club beyond the summer."
GIVEMESPORT'S Harry Sherlock says…
Arsenal are miles away from both Liverpool and Manchester City.
The defence is porous – they have a negative goal difference this season – and their attack, thus far this season, is blunt.
In eight games, they have scored nine goals; only Wolves, Fulham, West Bromwich Albion, Burnley, and Sheffield United have scored fewer. The final four 17th, 18th, 19th, and 20th respectively.
In a way, then, Campbell is right; the Gunners need to get rid of the players who offer precious little to the squad but they also need a sustained period of investment.
An entire restructure of the squad is required if the club are to have any chance of matching those at the very top of the table.
One transfer window isn't going to be enough; this could take two or three seasons.
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Sankhuwasabha, Nov. 21: President Bidya Devi Bhandari inaugurated the Makalu Diamond Jubilee celebrations commemorating the first ascent on Mt Makalu amidst a special programme here today.
On the occasion, the President said Sankhuwasabha, home to Mt Makalu which is one of the eight highest peaks in the world, and popular as the 'capital of rhododendron, can be developed as a major tourist destination.
Seeking the attention of the bodies concerned towards promoting tourism in a planned means in the district and eventually lifting the local lifestyle, the President said, "Peace and stability are the current need of the country".
The Head-of-the-State insisted on unity among the people of mountain, hill and Tarai to see the overall development of the country and to lead it towards prosperity.
Minister for Tourism and Civil Aviation Jeevan Bahadur Shahi, speaking of the potentiality of huge income generation through the means of tourism here, said the arrangements will be made for immediately allocating the amount to be collected through royalty of the Makalu climbing to the District Development Committee so that it can be used in the development endeavours in the district.
Lawmaker Taraman Gurung and CPN (UML)'s central member Rajendra Gautam said Sakhuwasabha has a bright future in terms of tourism, seeking the government assistance towards that end.
Main Celebrations Committee coordinator Minga Sherpa hoped of getting support from all quarters for the entire development of the Makalu region.
Tourists from Japan and the US also joined the celebrations. An agro festival has been organised as a part of the celebrations.
The Chichila Bazar that lies to the north of the district headquarters Khandbari carries abundant potentiality of tourism development.


text:
\section{Introduction}

Bayesian state estimation has a wide range of applications from positioning \cite{6746180} and tracking \cite{Bar-Shalom} to brain imaging \cite{hiltunen2011state} and modeling spread of infectious diseases \cite{opac-b1124487}. The number of state variables varies depending on the application e.g.\ in some positioning applications only 2 position variables are estimated, but in others the state may contain also variables for the locations of thousands of landmarks.

In general, the Bayesian state estimate cannot be computed in closed form. Under certain conditions closed form solutions exist. For example, when measurement and state transition functions are linear and associated noises are Gaussian, the Bayesian state estimate can be computed in closed form by the \ac{KF} algorithm \cite{kalman}.

When the conditions are not met an approximation has to be used. \acp{KFE} are based on approximate linear-Gaussian models. In the literature, there are several different types of \acp{KFE}, with different demands on computational resources. The computational complexity of the \ac{KFE} increases when the number of estimated state variables or dimensionality of the measurement vector increases; in some \acp{KFE} even exponentially \cite{ITO}. In many applications the computational resources are severely limited e.g.\ when doing tracking using wearable devices.

In this survey, we study different methods to optimize the computation of the state estimation with \acp{KFE} by exploiting the structure of the state transition and measurement models. We assume that the reader is familiar with the basics of \acp{KFE}. The presented algorithms are such that the result is exact when applied to a situation where the \ac{KF} produces the exact result. We give the algorithms in a general form so that they can be applied to as wide a range of problems as possible, but still in a form that they are easy to implement. Some of the surveyed algorithms in the original sources are given only for a certain \ac{KFE}. The general notation used in this survey allows the optimizations to be implemented for different \acp{KFE}. In the literature, there is not a similar unified presentation of different optimization methods. In addition to surveying the algorithms in the literature, we give some generalizations of the algorithms and new ways of applying the optimizations with the \acp{KFE}.

A drawback of \acp{KFE} is that the estimate is approximated as a Gaussian, which makes the estimate inaccurate when the true posterior is far from normal. \acp{GMF} (a.k.a. Gaussian sum filters)  use a sum of normal densities to estimate the probability distributions and can approximate any probability density function~\cite{Sorenson_Alspach_1971}. Because \acp{GMF} use several \acp{KFE} the computational load is larger than with algorithms that use only one Gaussian. The optimizations in this survey can be applied also in the \acp{GMF} for state propagation and update of individual components.

For implementations, we assume that a library for basic linear algebra is available. In many practical applications, the algorithms can be optimized further by taking the application specific structure of the matrices into account. Often the matrices are sparse, and the sparsity can be exploited for optimization either by hand or by implementing the algorithms with sparsity-optimized subroutines. These can be found, for example, in Sparse Basic Linear Algebra Subprograms\footnote{\url{http://math.nist.gov/spblas/}} or in Matlab\footnote{\url{http://www.mathworks.com/products/matlab/}}.

The computation time of algorithms can also be reduced by using parallel computing. The matrix operations can be effectively parallelized and the parallelization can be done in the linear algebra library that handles the matrix operations. Thus, we do not consider parallelization in this survey.

The remainder of this survey is organized as follows. In the next section we present the common notations that are used throughout the paper. In Section~\ref{sec:background} the background of \acp{KFE} is presented. Section~\ref{sec:optimizations} contains  linearity based optimizations of the \acp{KFE}. Section~\ref{sec:unobserved} presents optimizations based on unobserved and static state variables. In Section~\ref{sec:inno} the solution of the set of linear equations containing the inverse of the innovation covariance is optimized. Section~\ref{sec:examples} gives example applications of the optimization methods. Section~\ref{sec:conclusions} concludes the article.

\section{Notations}
\label{sec:notations}
In the following list there are the variables that are used throughout the paper. In different sections there are algorithm specific notations that are explained as they occur.\vspace{-0.5\topsep}
\begin{itemize}
\item Scalars
\begin{description}
 	\item[$j$] number of iterations
	\item[$m$] dimension of measurement
 	\item[$n$] dimension of state (or augmented state)
 \end{description}
\item Random variables
\begin{description}
	\item[$x$] state
	\item[$z$] augmented state
	\item[$\varepsilon$] measurement noise
	\item[$\varepsilon^\state$] state transition noise
	\item[$\measnoise$] measurement noise
\end{description}
\item Subscripts
\begin{description}
 	\item[$i$] iteration index
 	\item[$t$] time index
\end{description}

\item Functions
\begin{description}
	\item[$f(\cdot)$] state transition function
	\item[$g(\cdot)$] non-specific function
	\item[$G(\cdot)$] Matrix valued function
	\item[$h(\cdot)$] measurement function
\end{description}

\item Expected values
\begin{description}
	\item[$\E x$] expected value of x
	\item[$P_{ g(x)x}$] covariance of $g(x)$ and $x$
 	\item[$\mu_x$] mean of $x$
\end{description}

\item Other variables
\begin{description}
	\item[$I$] identity matrix
 	\item[$J$] matrix that defines statistically linearized relationship between state and measurement, which is the Jacobian matrix in linear systems
 	\item[$K$] Kalman gain
 	\item[$S$] innovation covariance
	\item[$\mathbf{0}$] zero matrix

\end{description}

\item Acronyms
\begin{acronym}

\acro{CKF}{Cubature Kalman Filter}
\acro{DD}{Divided Difference}
\acro{EKF}{Extended Kalman Filter}
\acro{EKF2}{Second Order Extended Kalman Filter}
\acro{GF}{Gaussian Filter}
\acro{GMF}{Gaussian Mixture Filter}
\acro{KF}{Kalman Filter}
\acro{KFE}{Kalman Filter Extension}
\acro{PDR}{Pedestrian Dead Reckoning}
\acro{QKF}{Gauss-Hermite Quadrature Kalman filter}
\acro{RBPF}{Rao-Blackwellized Particle Filter}
\acro{RUF}{Recursive Update Filter}
\acro{SLAM}{Simultaneous Localization and Mapping}
\acro{SOKF2}{Second Order Polynomial Kalman Filter}
\acro{S2KF}[S\textsuperscript{2}KF]{Smart Sampling Kalman Filter}
\acro{TDoA}{Time Difference of Arrival}
\acro{UKF}{Unscented Kalman Filter}
\acro{URUF}{Unscented Recursive Update Filter}
\end{acronym}
\end{itemize}

\section{Background}
\label{sec:background}
In discrete-time Bayesian filtering, the state of a dynamical system is estimated based on noisy measurements.  The state model describes how the $n$-dimensional state $x$ is propagated in time. A state model can be expressed as
\begin{equation}
	x_t = f_{t}(x_{t-1}, \varepsilon^\state_t),
\end{equation}
where $f(\cdot)$ is the state transition function, $x_{t-1}$ is the state of the previous time step, and $\varepsilon^\state_t$ is the state transition noise. The state's distribution is updated using measurements of the form
\begin{equation}
	y_t = h_{t}(x_{t}, \measnoise_t),
\end{equation}
where $y_t$ is the realized measurement, $h(\cdot)$ is the measurement function, and $\measnoise$ is the measurement noise.

To shorten notations we use augmented state $ z = \left[ \smatr{x\\ \varepsilon} \right]  $, where applicable. In the augmented state $\varepsilon$ is either state transition or measurement noise. The state and measurement noises are assumed independent and white. We also assume that all the interesting variables are in the state. If the state or measurement noises are not white, they can be modeled using more  variables in the state. Also we omit the time index $t$ when its absence should not cause confusion.

In general, a \ac{KFE} can be divided into two parts:
\begin{enumerate}
\item Prediction:
\begin{align}
	\mu_t^- &= \mu_{f(z_{t-1})} \\
	P_t^- &= P_{{f(z_{t-1})}{f(z_{t-1})}},
\end{align}
where $\mu_t^-$ is the prior mean computed using the state transition function and posterior of the previous time step and $P_t^-$ is the prior covariance.
\item Update:
\begin{align}
	y_t^- &= \mu_{h(z_t^-)} \\
	S_t & =   P_{h(z_t^-)h(z_t^-)} \label{equ:innocov}\\
	K_t &= P_{x_t^-h(z_t^-)}S_t ^{-1} \label{equ:Kalmangain} \\
	\mu_t &= \mu_t^-  + K(y_t - y_t^-)  \\
	P_t & = P_t^- -K_tS_tK_t^T,
\end{align}
	where $y_t^-$ is the predicted mean, $z_t^- \sim \N(\mu_t^-, P_t^- )$, $S_t$ is the innovation covariance, $K_t$ is the Kalman gain, $\mu_t$ is the updated state, and $P_t$ is the updated covariance.
\end{enumerate}
The mean and covariance matrices associated to measurement and state transition functions can be written using expectations:
\begin{align}
	\mu_{g(z)} & =\E g(z) \label{equ:E1}\\
	P_{g(z)g(z)} & =  \E  (g(z)-\mu_{g(z)})(g(z)-\mu_{g(z)})^T  \label{equ:E2}\\
	P_{zg(z)} & = \E (z- \mu_z  ) \left( g(z)-\mu_{g(z)} \right)^T \label{equ:E3}.
\end{align}
In \eqref{equ:Kalmangain}, $P_{x_t^-h(z_t^-)}$ refers to the rows that correspond to the state variables of  $P_{z_t^-h(z_t^-)}$.  When a function is linear, that is, of the form
\begin{equation}
	g(z) = J z,
\end{equation}
 the expectations have analytic form:
\begin{align}
	\mu_{g(z)} & = J\mu_z \\
	P_{g(z)g(z)} & =  JP_{zz}J^T\\
	P_{zg(z)} & = P_{zz}J^T \label{equ:PPJ}
\end{align}
and the algorithm is the \ac{KF}.

There are extensions that approximate the expectations \eqref{equ:E1}-\eqref{equ:E3} using analytic differentiation (or integration) of the functions. For example, the \ac{EKF} uses the first order Taylor expansion and the \ac{EKF2} does the linearization based on the second order Taylor expansion \cite{jazwinski}. There are also algorithms that use numerical differentiation to approximate  \eqref{equ:E1}-\eqref{equ:E3}. In \ac{DD} filters~\cite{norgaard} the computation is based on numerical differentiation of the functions to obtain a linear model. \ac{DD} filters use $2n+1$ function evaluations. The \ac{SOKF2}~\cite{ITO} uses $ \frac{ n^2}{2} + \frac{3}{2}n + 1$ points to fit a second order polynomial to the function and then computes the analytic covariance matrices for the polynomial.

One very commonly used and large group of \acp{KFE} is algorithms that can be written in form:
\begin{align}
	\mu_{g(z)} &\approx  \sum w_i^s g(\chi_i) \label{equ:UKF1}\\
	P_{g(z)g(z)} & \approx  \sum w_i^c (g(\chi_i) - \mu_{g(z)}) (g(\chi_i) - \mu_{g(z)})^T \label{equ:UKF2}\\
	P_{zg(z)} &\approx  \sum w_i^c (\chi_i - z) (g(\chi_i) - \mu_g(z))^T, \label{equ:UKF3}
\end{align}
where $\chi_i$ are so-called sigma-points that are chosen according to the prior distribution and $w_i^s$ and $w_i^c$ are associated weights. Examples of this kind of filters are \ac{UKF}~\cite{UKF}, different \acp{CKF}~\cite{cubature}, and  \ac{QKF}~\cite{ITO}.

The selection and number of sigma-points and weights depend on the algorithm used. The \ac{UKF} is used usually with $2n +1$ sigma-points. The \ac{GF} uses $kn+1$ sigma-points, where $k > 1$ is an integer parameter \cite{huber2008gaussian}. \acp{CKF} are developed for different orders and they use $\mathcal{O}(n^m)$ sigma-points, where $m$ is the order of the cubature rule. The number of sigma-points in \acp{QKF} increases exponentially, as the number of sigma-points is $\alpha^n$, where $\alpha > 1$~\cite{ITO}. There are also algorithms that allow an arbitrary number of points, for example \ac{S2KF}~\cite{S2KF}, which uses at least $n+1$ sigma-points.

Algorithms that use \eqref{equ:UKF1}-\eqref{equ:UKF2} do not compute  $J$ explicitly. In some optimizations  $J$ is required and can be computed using statistical linearization~\cite{sarkka2013bayesian}
\begin{equation}
	 J  = P_{g(z)z} P_{zz}^{-1}. \label{equ:virtJac}
\end{equation}
This matrix defines the relationship between $P_{zz}$ and $P_{g(z)z}$. Because this computation requires solving a set of linear equations it should be avoided when the dimension of $z$ is large.

The computational requirements of the algorithms can be reduced in several ways. One way is to compute the expected values numerically only for state variables that are transformed by a nonlinear function and compute the expectations for linear parts analytically. The improving of implementation efficiency using the linear \ac{KF} update for the linear state variables is more familiar in particle filtering. Specifically \ac{RBPF} solves the estimates of conditionally linear variables using a \ac{KF} and rest of the variables using particles \cite{Doucet:2000:RPF:647234.720075}. The dimension of the output of the nonlinear part can also be reduced to reduce the computational burden. In \eqref{equ:UKF2} the $m\times m$ covariance matrix is updated for every sigma point. Thus, halving the dimension $m$ of $g(\cdot)$  reduces the operations applied to the covariance matrix to one fourth. This kind of update optimizations are considered in Section~\ref{sec:optimizations}.

When some state variables are unobservable i.e.\ they do not contribute to the output of the measurement function and are static i.e.\ their mean and covariance does not change in the state propagation, they do not need to be updated at every time step. Computing their estimate only when they are observed or their mean or covariance changes can also be used to reduce the computational burden. These optimizations are presented in Section~\ref{sec:unobserved}. It is also possible to compute the Kalman gain \eqref{equ:Kalmangain} faster by exploiting the structure of the innovation covariance \eqref{equ:innocov} as shown in Section~\ref{sec:inno}.

\section{Optimizations Based on the Linearity in Nonlinear functions}

\label{sec:optimizations}

\subsection{Partially Linear Functions}
\label{sec:partlin}
In~\cite{1257248}, algorithms for treating different setups of \ac{UKF} were presented. The algorithms considered different implementations when the state transition model or measurement model has one of the following three forms
\begin{align}
	g(x,\varepsilon)&=\Mfunx+\varepsilon \\
	g(x,\varepsilon) & =g_1(x)+\varepsilon \label{equ:additivenoise}\\
	g(x,\varepsilon) & =g_2(x,\varepsilon).
\end{align}
The algorithms improve the computation efficiency when either or both of the state and measurement models belong to the first or second group. If both are in the first group the updates can be computed exactly using the \ac{KF}. If the function belongs to the second group then \eqref{equ:E1}-\eqref{equ:E3}
become
\begin{align}
	\mu_{g(z)} & = \left(\E g(x)\right) + \mu_\varepsilon \label{equ:U1}\\
	P_{g(z)g(z)} & =  \left(\E  (g(x)-\mu_{g(x)})(g(x)-\mu_{g(x)})^T\right)  + P_{\varepsilon\noise} \label{equ:U2}\\
	P_{xg(z)} & = \E (x- \mu_x  ) \left( g(x)-\mu_{g(x)} \right)^T \label{equ:U3}
\end{align}
and the approximation can be computed for the dimension of the state instead of the dimension of the augmented state. This optimization is widely used among different filters and is often considered to be the standard model.

In~\cite{1660584} the function is split into a nonlinear $g_1(\cdot)$ part that depends only on a part of the  state $z_n$  and linear parts
\begin{equation}
	h(z) = \matr{ g_1(z_n ) \\ H_1 z}. \label{equ:lnonl}
\end{equation}
The \ac{UKF} is used for computing the expected values of the nonlinear part and the correlation between nonlinear and linear parts.

We generalize the above models to functions of form:
\begin{equation}
	g(z) = A g_n\left(Tz \right) + Hz, \label{equ:fullequ}
\end{equation}
where $\rank T$ equals the number of rows in $T$.  To reduce the required resources in computation $T$  is chosen to have minimal number of rows and $g(\cdot)$ to have minimal number of elements. Expectations \eqref{equ:E1}-\eqref{equ:E3} are computed for $Tz$, which should have a smaller dimension than $z$.

For computing the update, an algorithm that approximates expectations \eqref{equ:E1}-\eqref{equ:E3} is required. These are usually computed in the update stage of a \ac{KFE}. The cross correlation matrix $P_{zg(z)}$ \eqref{equ:UKF3} is not needed in the normal state propagation, but the algorithm from the update stage can be used for this.

The transformed augmented state is denoted
\begin{equation}
\tilde z = Tz \sim \N(\mu_{\tilde z}=T\mu_z,P_{\tilde z\tilde z}= TP_{zz} T^T   ). \label{equ:transt}
\end{equation}
When expectations
$\mu_{g_n(\tilde z)}$ \eqref{equ:E1}, $P_{g_n(\tilde z)g_n(\tilde z)}$ \eqref{equ:E2}, and $P_{\tilde zg_n(\tilde z)}$
\eqref{equ:E3} are known, the expectations for \eqref{equ:fullequ} are
\begin{align}
	\mu_{g(z)} & = A \mu_{g_N(Tz)} + H\mu_z \label{equ:mom1} \\
	P_{g(z)g(z)} &= \matr{A & H} \matr{ P_{g_n(\tilde z)g_n(\tilde z)} & P_{zg(\tilde z)}^T \\ P_{zg_n(\tilde z)} & P_{zz} } \matr{ A^T \\  H^T } \label{equ:isoP}\\
	P_{z g(z)} &= \matr{ P_{zg_n(\tilde z)} & P_{zz}  }\matr{A^T \\ H^T}, \label{equ:mom3}
\end{align}
where
\begin{align}
	P_{zg(\tilde z)} & = P_{zz}T^T\left(T P_{zz} T^T \right)^{-1} P_{\tilde zg(\tilde z)}  . \label{equ:replace}
\end{align}
This is based on the fact that the cross term $P_{\tilde zg_n(\tilde z)}$ describes the linear dependency of function $g_n(\tilde z)$ and $\tilde z$ as in \eqref{equ:virtJac}:
\begin{equation}
P_{\tilde zg_n(\tilde z)} = P_{\tilde z\tilde z}J^T  = TP_{zz}T^TJ^T \label{equ:Pz}.
\end{equation}
and the term $P_{zg_n(\tilde z)}$ is
\begin{equation}
P_{ zg_n(\tilde z)} = P_{zz} T^T J^T. \label{equ:pzg}
\end{equation}
Solving $J$ from \eqref{equ:Pz} and substituting it into \eqref{equ:pzg} we get \eqref{equ:replace}.

In the update, the matrix with cross terms is required for state variables only. This can be extracted by
\begin{equation}
	P_{xh(z)} = \matr{I & 0}P_{zh(z)}
\end{equation}
when the state variables appear first in the augmented state.

\begin{algorithm}[tbp!]
\SetAlgoNoLine
\KwIn{
	  $x_{t-1} \sim \N(\mu_{x_{t-1}}, P_{x_{t-1}x_{t-1}})$ \tcp{State estimate from previous time step}
          $\noise_q \sim \N(\mu_{\noise_q},P_{\noise_q \noise_q}  )$,  \tcp{State transition noise}}
	 $P_{x_{t-1} \noise_q}$ \tcp{cross covariance between state and state transition noise}
	 $f(x,\noise_q)=f(z)=A g\left(Tz \right) + Hz$,  \tcp{State transition function of given form}

  \KwOut{
	$x \sim \N( \mu_{x^-}, P_{x^-x^-} )$ \tcp{Propagated state}
}
$\mu_{z} = \matr{ \mu_{x_{t-1}} \\ \mu_{\noise_q}}$ \tcp{Augmented mean}
$P_{zz} = \matr{ P_{x_{t-1}x_{t-1}} & P_{x_{t-1} \noise_q} \\ P_{x_{t-1} \noise_q}^T & P_{\noise_q \noise_q}}$ \tcp{Augmented covariance}

$\mu_{\tilde z} = T\mu_z$ \tcp{Transformed mean}
 $P_{\tilde z\tilde z} = TP_{zz}T^T$ \tcp{Transformed covariance}
Compute $\mu_{g(\tilde z)}$,$P_{g(\tilde z)g(\tilde z)}$, and $P_{\tilde zg(\tilde z)}$ using a \ac{KFE} \\
$P_{zg(\tilde z)}  = P_{zz}T^T\left(T P_{zz} T^T \right)^{-1} P_{\tilde zg(\tilde z)}$ \\
$\mu_{x^-}  = A \mu_{g(Tz)} + H\mu_z$  \tcp{Predicted mean of state}
$P_{x^-x^-} = \matr{A & H} \matr{ P_{g(\tilde z)g(\tilde z)} & P_{zg(\tilde z)}^T \\ P_{zg(\tilde z)} & P_{zz} } \matr{ A^T \\  H^T }$  \tcp{Predicted covariance}
\caption{State transition using a partially linear measurement}
\label{algo:partiallinearpred}
\end{algorithm}
\begin{algorithm}[tbp!]
\SetAlgoNoLine
\KwIn{
	  $x^- \sim \N(\mu_{x^-}, P_{x^-x^-})$ \tcp{Prior state}
          $\measnoise \sim \N(\mu_{\measnoise},P_{\measnoise \measnoise}  )$,  \tcp{Measurement noise}}
        	 $P_{x^- \measnoise}$ \tcp{cross covariance between state and measurement noise}
          $h(x,\measnoise)=h(z)=A g\left(Tz \right) + Hz$,  \tcp{Measurement function of given form}
\KwOut{
	$x \sim \N( \mu_x, P_{xx} )$ \tcp{Posterior estimate}
}
$\mu_z = \matr{\mu_{x^-} \\ \mu_{\measnoise}}$  \tcp{Mean of the augmented state}
$P_{zz} = \matr{P_{x^-x^-} & P_{x^-\measnoise} \\ P_{x^-\measnoise}^T & P_{\measnoise\measnoise}}$ \tcp{Covariance matrix of the augmented state}
$\mu_{\tilde z} = T\mu_z$ \tcp{Transformed mean}
 $P_{\tilde z\tilde z} = TP_{zz}T^T$ \tcp{Transformed covariance}
Compute $\mu_{g(\tilde z)}$,$P_{g(\tilde z)g(\tilde z)}$, and $P_{\tilde zg(\tilde z)}$ using a \ac{KFE} \\
$P_{zg(\tilde z)}  = P_{zz}T^T\left(T P_{zz} T^T \right)^{-1} P_{\tilde zg(\tilde z)}$ \\
$\mu_{h(z)}  = A \mu_{g(Tz)} + H\mu_z$  \tcp{Predicted mean of measurement}
$P_{h(z)h(z)} = \matr{A & H} \matr{ P_{g(\tilde z)g(\tilde z)} & P_{zg(\tilde z)}^T \\ P_{zg(\tilde z)} & P_{zz} } \matr{ A^T \\  H^T }$  \tcp{Innovation covariance}
$P_{x h(z)} = \matr{ P_{zg(\tilde z)} & P_{zz}  }_{[1:n,:]}\matr{A^T \\ H^T}$ \tcp{State-measurement cross correlation}
$K=P_{x h(z)} P_{h(z)h(z)}^{-1}$ \tcp{Kalman Gain}
$\mu_x = K (y-\mu_{h(z)})$ \tcp{Posterior mean}
$P_{x\state} = P_{x^-x^-} - KP_{h(z)h(z)} K^T$ \tcp{Posterior covariance}
\caption{Update using a partially linear measurement}
\label{algo:partiallinearup}
\end{algorithm}

The algorithm for state transition is given in Algorithm~\ref{algo:partiallinearpred} and an algorithm for state update is given in  Algorithm~\ref{algo:partiallinearup}.  Use of these algorithms is beneficial when the dimension of $\tilde z$ is smaller than $z$ and the matrix inverse in \eqref{equ:replace} is applied in a small dimension. Algorithms~\ref{algo:partiallinearpred} and~\ref{algo:partiallinearup} are given in a general form and for applications they can be optimized further by considering the structures of matrices. Examples of exploiting the partially linear state are given in sections~\ref{sec:pdr} and~\ref{sec:tdoa}.

\subsection{Conditionally Linear Measurements}
\label{sec:condlin}
In~\cite{4217985}, a situation where a function can be divided into nonlinear $z_n$ and conditionally linear $z_l$ parts
\begin{equation}
	g(z) = g_n(z_n )+ G_n( z_n )z_l \label{equ:condl}
\end{equation}
is considered. In the original article, the distributions of this nonlinear function is approximated using a modification of the \ac{UKF}. The number of used sigma-points depends on the dimension of  $z_n$ instead of the dimension of the full state.

This algorithm is converted to be used with the \ac{GF} in~\cite{5203794}.  Here we present the algorithm in a general form that allows it to be also with any \ac{KFE} that uses weighted points to compute the expectations, as in \eqref{equ:U1}-\eqref{equ:U3}, but it cannot be used with other types of filters e.g.\ \ac{DD} or \ac{SOKF2}.

In the algorithm the sigma-points are computed for the nonlinear part only and then used for computing the conditional probabilities of the conditionally linear part:
\begin{align}
	\mu_{z_l | \chi_i}  & = \mu_{z_l} +  P_{z_lz_n}P^{-1}_{z_nz_n}(\chi_i - \mu_{z_n}) \label{equ:condmu}\\
	P_{z_lz_l | \chi_i } & = P_{z_lz_l} - P_{z_lz_n}P^{-1}_{z_nz_n}P_{z_nz_l}. \label{equ:condP}
\end{align}
The matrices in the equations above are independent of the sigma-point $\chi_i$. Thus, $P_{z_lz_n}P^{-1}_{z_nz_n}$ and $P_{z_lz_l | \chi_i }$ need to be computed only once. According to~\cite{4217985},  the expectations for a function of form \eqref{equ:condl} can be approximated as
\begin{align}
{\mathcal Y}_i = &  g_n (\chi_i)    +G_n (\chi_i)  \mu_{z_l | \chi_i} \\
\mu_{g(z)}  = &\sum w_i {\mathcal Y}_i \ \\
P_{g(z)g(z)}  =&  \sum w_i \left(\left(  {\mathcal Y}_i - \mu_{g(z)}  \right)\left({\mathcal Y}_i  - \mu_{g(z)}   \right)^T + G(\chi_i)  P_{z_l z_l | \chi_i}  G (\chi_i)^T\right) \\
P_{zg(z)} = &\sum w_i\left( \left( \matr{\chi_i \\ \mu_{z_l | \chi_i}} - \mu_{z} \right)\left( {\mathcal Y}_i  - \mu_{g(z)} \right)^T  + \matr{0 \\P_{z_l z_l | z_n}G(\chi_i)^T }\right).
\end{align}
These formulas can be used to compute the expectations for the nonlinear part $g(\tilde z)$ in algorithms~\ref{algo:partiallinearpred} and~\ref{algo:partiallinearup}, if there are conditionally linear state variables. This is done in the example in Section~\ref{sec:pdr}.

\section{Optimizations Based on Unobserved and Static State Variables}
\label{sec:unobserved}
\subsection{Part of State Is Unobserved and Static}
\label{sec:unobserved1}
A state variable is considered static when its distribution does not change in the state transition and unobserved when it does not have an effect on the measurement function. This section considers the situation where some of the state variables are unobserved and static. This situation has been considered in \cite{davison1998mobile,938382}. The presented algorithm is based on the latter, which has more compact formulas. The noise variables are, by definition, independent of other noise variables at other times. Thus, they cannot be static and the augmented notation is not used in this section.

The state is divided into two parts, active $x_a$ and inactive $x_u$. The inactive part contains variables that are static and unobserved and the active part contains the remaining state variables.

In~\cite{938382}, the algorithm that does updates with \ac{EKF} stores the following auxiliary variables:
\begin{align}
	\psi_t &= \psi_{t-1}+\Phi_{t-1}^TJ_t^TP_{h(x_a)h(x_a),t}^{-1}J_t\Phi_{t-1} \label{equ:hti2}\\
	c_t &= c_{t-1}+\Phi_{t-1}^TJ_t^TP_{h(x_a)h(x_a),t}^{-1}(y-J_t\mu_t^-) \label{equ:hti3}\\
	\Phi_t &= (I - K_tJ_t) \Phi_{t-1}^T \label{equ:hti4}.
\end{align}
Initial values for these variables are $\psi_{t_0}=\mathbf{0}$, $\Phi_{t_0}=I$, and $c_{t_0}=\mathbf{0}$.
When needed the full estimate can be extracted by
\begin{align}
	\mu_{u,t} & = \mu_{u,t_0} +\Pa{\inacta}{t_0}c_{t}\\
	\Pa{u\inact}{t} &=  \Pa{u\inact}{t_0}-\Pa{au}{t_0} \psi_{t} \Pa{\inacta}{t_0} \\
	\Pa{au}{t} &=  \Phi_{t}^T \Pa{au}{t_0} \\
	\Pa{\inacta}{t} &=    \Pa{\inacta}{t}\Phi_{t}.
\end{align}

The auxiliary variables \eqref{equ:hti2}-\eqref{equ:hti4} contain $2n_a^2+n_a$ elements that have to be stored (the memory requirement can be reduced by noting that $\psi_t$ is symmetric), but only the matrix of the latest time step is needed. When $n_a \ll n_u$ the amount of additional memory required is negligible compared to the total used memory. It is worthwhile to note that the only term that contains a matrix inverse $J_t^TP_{h(x_a)h(x_a),t}^{-1}$ appears in equations \eqref{equ:hti2} and \eqref{equ:hti3} and also in the computation of the Kalman gain for \ac{EKF}. To speed up the algorithm $J_t^TP_{h(x_a)h(x_a),t}^{-1}$  is computed only once.

Using \eqref{equ:virtJac} we can rewrite \eqref{equ:hti2}-\eqref{equ:hti4} in a more general form
\begin{align}
	\psi_t &= \psi_{t-1}+\Phi_{t-1}^TP_{a\act}^{-1}P_{a h(x)}P_{h(x_a)h(x_a),t}^{-1}P_{ h(x)a}P_{a\act}^{-1}\Phi_{t-1} \\
	c_t &= c_{t-1}+\Phi_{t-1}^TP_{a\act}^{-1}P_{a h(x)}P_{h(x_a)h(x_a),t}^{-1}(y-\mu_t^-) \\
	\Phi_t &= (I - K_tP_{a h(x)}^TP_{a\act}^{-1})\Phi_{t-1}.
\end{align}
This formulation can be used with a wider set of \acp{KFE} than previous, but it contains the inverse of the active state $P_{a\act}^{-1}$. The computation of this inverse is feasible when the active state is small. The full algorithm is given in Algorithm~\ref{algo:staticvariables}.

\begin{algorithm}[tbp]
\SetAlgoNoLine
\KwIn{
	  $x_{t_0} = \matr{ x_{a,t_0} \\ x_{u,t_0}} \sim \N\left( \matr{ \mu_{a,t_0}\\ \mu_{u,t_0}}  , \matr{ P_{a\act,t_0} &  P_{au,t_0} \\ P_{\inacta,t_0} & P_{u\inact,t_0}} \right)  $ \tcp{Initial state}
 $x_{t} = h_t(x_{a,t},\measnoise_t)$,  \tcp{Measurement model}
                    $x_{t+1} = \matr{x_{a,t+1}\\ x_{u,t+1}} = \matr {f_{t+1}(x_{a,t},\varepsilon^\state) \\ x_{u,t}}$  \tcp{State transition model}
          }
          \KwOut{
	  $x_{t_{0+\Delta}} = \matr{ x_{a,t_{0+\Delta}} \\ x_{u,t_{0+\Delta}}} \sim \N\left( \matr{ \mu_{a,t_{0+\Delta}}\\ \mu_{u,t_{0+\Delta}}}  , \matr{ P_{a\act,t_{0+\Delta}} &  P_{au,t_{0+\Delta}} \\ P_{\inacta,t_{0+\Delta}} & P_{u\inact,t_{0+\Delta}}} \right)  $ \tcp{State at $t_{0+\Delta}$}

	 Initialize auxiliary variables:\\
	 $\psi_{t_0}=\mathbf{0}$\\
	  $\Phi_{t_0}=I$\\
	  $c_{t_0}=\mathbf{0}$ \\
	\Indm
\For{$t=t_0+1$ to  $t_{0+\Delta}$}
{

	Prediction of the active part: \\
	\Indp
	$ \mu_{a,t}^- = \mu_{f_t(x_{a,t-1},\varepsilon^\state_t)}$  \tcp{Predicted mean of active part of the state}
	$ P_{a\act,t}^- = P_{ f(x_{a,t-1}\varepsilon^\state_t), f(x_{a,t-1}, \varepsilon^\state_t)   }$ \tcp{Predicted covariance of the active part}
	\Indm
	Update of the active part: \\
	\Indp
	$y_{t}^- = \mu_{h_t( x_{a,t}^-, \measnoise_t)}$ \\
	$S_{t}  =   P_{h_t( x_{a,t}^-, \measnoise_t) h_t( x_{a,t}^-, \measnoise_t) }$ \\
	$K_{t} = P_{x_ah_t( x_{a,t}^-, \measnoise_t)}S_{i}^{-1}$ \\
	$\mu_{a,t} = \mu_{a,t}^-  + K( y_{t} - y_{t}^-)$  \\
	$P_{a\act,t}  = P_{a\act,t}^- -K_{t}S_{t}K_{t}^T$\\
	\Indm Update auxiliary variables: \\
	\Indp
	$\psi_t = \psi_{t-1}+\Phi_{t-1}^TP_{a\act,t}^{-1}P_{a h(x),t}P_{h_t(x_{a,t})h_t(x_a),t}^{-1}P_{ h(x_{a,t})a_t}P_{a\act,t}^{-1}\Phi_{t-1} $ \\
	$c_t = c_{t-1}+\Phi_{t-1}^TP_{a\act,t}^{-1}P_{a h(x),t}P_{h(x_a)h(x_a),t}^{-1}( y_{t} - y_{t}^-)$\\
	$\Phi_t = (I - K_tP_{a h(x),t}^TP_{a\act,t}^{-1})\Phi_{t-1}$
}
Compute the full mean and covariance: \\
\Indp
$	\mu_{x,t_{0+\Delta}}  = \matr{ \mu_{a,t_0+\Delta} \\ \mu_{u,t_0} +\Pa{\inacta}{t_0}c_{t_{0+\Delta}}} $ \tcp{Posterior mean}
$	P_{xx,t_{0+\Delta}} = \matr{
                 P_{a\act,t_0+\Delta}  & \Phi_{t_{0+\Delta}}^T \Pa{au}{t_0} \\
                  \left(\Phi_{t_{0+\Delta}}^T \Pa{au}{t_0}\right)^T & \Pa{u\inact}{t_0}-\Pa{au}{t_0} \psi_{t_{0+\Delta}} \Pa{\inacta}{t_0}} $  \tcp{Posterior covariance}
}

\caption{State estimation when a part of the state is unobserved and static.}
\label{algo:staticvariables}
\end{algorithm}

\subsection{Part of State is Unobserved and Whole State Is Static}
\label{sec:unobserved2}
In \cite{LefebvreBS05}, an update algorithm for the situation when all state variables are static is presented. In this algorithm the update is done only for observed variables at every time step. When the full state estimate is required at time index $t_{0}+\Delta$ it can be extracted by:
\begin{align}
	L &= \Pa{\inacta}{t_0}\Pa{a\act}{t_0}^{-1} \label{equ:unobstart}\\
	\mu_{u,t_{0}+\Delta} &= \mu_{u,t_0} + L\left( \mu_{a,t_{0}+\Delta} - \mu_{a,t_0}\right)\\
	\Pa{\inacta}{t_{0}+\Delta} &= L\Pa{a\act}{t_0+\Delta} \\
	\Pa{u\inact}{t_{0}+\Delta} &= \Pa{u\inact}{t_0} -L\left( \Pa{a\act}{t_{0}} -\Pa{a\act}{t_{0}+\Delta} \right) L^T.\label{equ:unobend}
\end{align}
This algorithm does not require any extra variables to be stored during the estimation process. An example of the use of this optimization is given in Section~\ref{sec:iteKFE}.

\section{Optimizations Related to the Inverse of the Innovation Covariance}
\label{sec:inno}
\subsection{Block Diagonal Measurement Covariance}
\label{sec:Innopti}
The formula for the Kalman gain \eqref{equ:Kalmangain} contains the inverse of the innovation covariance~$S$. When the dimension of the innovation covariance (i.e.\ the number of measurements) is large the computation of the inverse or solving the set of linear equations is a computationally expensive operation.

When measurements are linear and the measurement covariance is diagonal, the Kalman update can be applied one measurement element at a time and the partially updated state used as a prior for the next measurement \cite{stengel1986stochastic}. This kind of update can be generalized also to block diagonal measurement covariances and the update can be done by applying one block at a time. This reduces the computation time when the state dimension is small, the measurement dimension is large, and the measurements are independent.

When some measurements are independent, the \ac{KFE} has superlinear complexity, and different measurement elements contain different state terms, updating state using only a part of measurements at once can be effectively combined with the algorithms presented in Section~\ref{sec:optimizations}. The block update does not change the estimate when the measurement model is linear, but when the model is nonlinear the linearization is redone in the partially updated state. This may alter the output of the algorithm. In~\cite{PUKF,Raitoharju2016} application of measurements one element at a time is used to improve the estimation accuracy by applying the most linear measurements first. The algorithm for updating by blocks is given in Algorithm~\ref{algo:blocup}.
\begin{algorithm}[tbp]
\SetAlgoNoLine
\KwIn{
	  $x_0 \sim \N(\mu_{x,0}, P_{xx,0})$ \tcp{Prior state}
          $\measnoise_i \sim \N(\mu_{\measnoise_i},P_{\measnoise_i \measnoise_i}  )$, $1\leq i \leq n$ \tcp{Measurement noises}}
          $h_i(x,\measnoise_i)$, $1\leq i \leq n$ \tcp{Measurement functions}
\KwOut{
	$x_n \sim \N( \mu_{x,n},, P_{xx,n} )$ \tcp{Posterior estimate}
}
\For{i=1 to n}
{
	$y_{i}^- = \mu_{h_i( x_{i-1}, \measnoise_i )}$ \\
	$S_{i}  =   P_{h_i(z_{t,i}^-)h(x_{i-1}, \measnoise_i)}$ \\
	$K_{i} = P_{xh_i(z_{t,i}^-)}S_{i}^{-1}$ \\
	$\mu_{x,i} = \mu_{x,i-1}  + K( y_{i} - y_{i}^-)$  \\
	$P_{xx,i}  = P_{xx,i}^- -K_{i}S_{i}K_{i}^T$
}

\caption{Block Update \ac{KF}}
\label{algo:blocup}
\end{algorithm}

\subsection{Applying the Matrix Inversion Lemma to Innovation Covariance}
\label{sec:innopti2}
When the measurements are not independent the block update formula cannot be used. In some situations the innovation covariance can be written in the form
\begin{equation}
	S = P_s + UP_vU^T, \label{equ:innoSSS}
\end{equation}
where $P_s$ is easy to invert, $P_v$ has a small dimension and $U$ is a transformation matrix.

Using matrix inversion lemma the inverse of \eqref{equ:innoSSS} is
\begin{equation}
	S^{-1} = P_s^{-1} - P_s^{-1} U\left( P_v^{-1} + UP_s^{-1}U^T  \right)^{-1} U^T P_s^{-1}. \label{equ:woodbury}
\end{equation}
This formula is worth using if the inverse of $P_s$ is easy to compute or can be computed offline and the dimension of $P_v$ is smaller than the dimension of  $P_s$.  An example of its use is given in Section~\ref{sec:tdoa}.

\section{Example Applications}
\label{sec:examples}
\subsection{Pedestrian Dead Reckoning}
\label{sec:pdr}
In \ac{PDR}, pedestrian's steps and heading changes are measured with accelerometers and gyroscopes. The change of position is based on these. In \cite{nurminen2013a}, a state consists of two-dimensional position $r$, heading~$\theta$, step length $l$, and possibly a floor index. The state transition is computed with a particle filter. Here we show how the state model without floor index and with normal  noise can be optimized for use with \acp{KFE}. We use in this example algorithms presented in Sections~\ref{sec:partlin} and~\ref{sec:condlin}.

The nonlinear state transition model is
\begin{equation}
	x_{t+1} = \matr{r_{1,t+1} \\ r_{2,t+1} \\\theta_{t+1} \\ l_{t+1}  } =
	\matr{r_{1,t}  + \left(l_{t}+\varepsilon_{l} \right) \cos \left(\theta_t + \varepsilon_\theta \right)+ \varepsilon_{r,1} \\
	          r_{2,t}  + \left(l_{t}+\varepsilon_{l} \right) \sin \left(\theta_t + \varepsilon_\theta \right) + \varepsilon_{r,2} \\
	          \theta_{t} + \varepsilon_\theta \\
	          l_t + \varepsilon_l
	          }, \label{equ:fullpdr}
\end{equation}
where  $\varepsilon_\theta \sim \N(\Delta_t, \sigma^2_\theta)$ is the noisy heading change, with $\Delta_t$ measured from gyroscopes, $l$ is the footstep length and $r$ is the position of the pedestrian. The augmented state is $z= \matr{r_1 & r_2 & \theta & l & \varepsilon_{r,1} & \varepsilon_{r,2} & \varepsilon_\theta & \varepsilon_l}^T$. The state transition model can be written  in the form of \eqref{equ:fullequ}:
\begin{align}
	\tilde z & = \matr{\theta_t + \varepsilon_\theta \\ l_{t}+\varepsilon_{l}   } \\
	T &= \matr{\mathbf{0}_{2 \times 2} & I_{2 \times 2} & \mathbf{0}_{2 \times 2} & I_{2 \times 2}  } \\
	         	g(\tilde{z}) & = \matr{ \tilde{z}_2 \cos \tilde{z}_1 \\  \tilde{z}_2 \sin \tilde{z}_1 } \label{equ:reducedpdr} \\
		A &= \matr{I_{2 \times 2} \\ \mathbf{0}_{2 \times 2} }\\
	H &= \matr{I_{4 \times 4} & I_{4 \times 4} }
\end{align}
In this formulation the reduced state $\tilde z$ is two-dimensional and the nonlinear part of the state transition function is also two-dimensional.

If this problem is to be solved with a \ac{KFE} that uses sigma-points, we can optimize the implementation further. This can be done because $\tilde{z}_2$ in \eqref{equ:reducedpdr} is conditionally linear, given $\tilde{z}_1$. The parts of function \eqref{equ:condl}  are
\begin{align}
	z_n &= \tilde{z}_1 \\
	z_l &= \tilde{z}_2 \\
	g_n(\tilde{z}_1) &= \matr{ 0\\0 } \\
	G_n(\tilde{z}_1) &=\matr{ \sin \tilde{z}_1\\\cos \tilde{z}_1 } .
\end{align}
Using this the dimension of the nonlinear part of the state is reduced to 1 and the sigma-points are generated for only one dimension.

Figure~\ref{fig:stepex} shows mean estimates of the first position variable ($r_1$) after one propagation step in a test. \begin{figure}[!b]
\centering
\includegraphics[width=\columnwidth]{stepex.eps}
	\caption{Estimated mean of $r_1$ with different number of function evaluation points using \ac{S2KF}}
	\label{fig:stepex}
\end{figure}
The estimates are computed using \ac{S2KF} with varying number of sigma-points. Implementation of the sigma-point generator and \ac{S2KF} can be found in \cite{nonlinearestimationtoolbox}.  \ac{S2KF} is applied to original state \eqref{equ:fullpdr} and to above presented reduced states with 2 and 1 nonlinear variables. The propagation of the state with the full state model cannot be computed with fewer than 9 sigma-points. The state computed using the model that uses conditional linearity in addition to the linearity has the smoothest convergence to an estimate and the estimate with full state estimate has the largest variation.  This shows that the use of optimizations can also improve the estimation accuracy.

\subsection{Source Tracking Using a Microphone Array \ac{TDoA}}
\label{sec:tdoa}
In tracking of a sound source using a microphone array the received acoustic signals of two microphones are compared so that the \ac{TDoA} of the acoustic signal is measured. In ideal situation the \ac{TDoA} is
\begin{align}
h_{i,j}(r_0) & = \frac{\norm{r_i - r_0 } -  \norm{r_j - r_0 }}{v},
\end{align}
where $r_0$ is the location of the sound source, $r_i$ is the location of $i$th microphone and $v$ is the speed of sound. When there are $m$ microphones in the array there are at most $ \frac{m(m-1)}{2}$ \ac{TDoA} measurements available  \cite{oualil2012tdoa}.

In practice,  measurements contain noise. Here we consider model with noises
\begin{align}
h_{i,j}(x) & = {\norm{r_i - x } -  \norm{r_j - x } + \varepsilon_i - \varepsilon_j + \varepsilon_{i,j}},
\end{align}
where $\varepsilon_i$ is the noise corresponding to $i$th microphone and $\varepsilon_{i,j}$ is error corresponding to the correlation of the measured signals from the microphone pair and the speed of sound is multiplied out from the equation.

When $\varepsilon_{i,j}=0$ for all microphone pairs the measurement noises are not independent for all possible pairs and the measurements can be modeled with $m-1$ measurements of form
\begin{align}
h_i(x) & =   \norm{r_i - x } -   \norm{r_m - x } + \varepsilon_i - \varepsilon_m.
\end{align}
This kind of assumption is done for example in \cite{bechler2004system}. In practice, this assumption does not hold and the selection of microphone pairs is a tradeoff, where pairs close to each other have smaller correlation error $\varepsilon_{i,j}$, but worse geometry than pairs far away from each other~\cite{silverman1998huge}.
Here we consider the situation where all possible microphone pairs are used and errors $\varepsilon_i$ and $\varepsilon_{i,j}$ are modeled  as Gaussians. The augmented state model is
\begin{equation}
z=\matr{x^T & \varepsilon_{1,1} & \varepsilon_{1,2} & \ldots & \varepsilon_{m,m} & \varepsilon_{1} & \ldots &\varepsilon_{m} }^T,
\end{equation}
where $x$ contains 3 position and velocity variables.

Using \eqref{equ:U1}-\eqref{equ:U3} the nonlinear part of the estimation can be done only for the state variables and  sigma-points would be required only for 6 dimensional state instead of using sigma-points also for $\frac{m(m+1)}{2}$ noise terms. The measurement model can be written in compact form using \eqref{equ:fullequ}:
\begin{align}
\tilde z &= x_{1:3} \\
T & = \matr{I_{3\times3} & \mathbf{0_{3\times3}}} \\
g(\tilde z) & = \matr {\norm{r_1 - \tilde z } \\ \norm{r_2 - \tilde z } \\ \vdots \\ \norm{r_m - \tilde z }} \\
A & = \matr{ \mathbf{1}_{m-1\times 1} & \multicolumn{2}{c}{-I_{m-1\times m-1}} \\
 	       \mathbf{0}_{m-2\times 1}  & \mathbf{1}_{m-2\times 1 }  & -I_{m-2 \times m-2} \\
	       \mathbf{0}_{m-3\times 2}  & \mathbf{1}_{m-3\times 1 }  & -I_{m-3 \times m-3} \\
	       \multicolumn{3}{c}{\vdots} \\
	       \mathbf{0}_{1\times m-2}  & 1  & -1} \\
H&=	 \matr{ \mathbf{0}_{  \frac{m(m-1)}{2}\times 6  } & A & I_{ \frac{m(m-1)}{2} \times  \frac{m(m-1)}{2}}}.
\end{align}
Using these the dimension of the nonlinear function is reduced from $\frac{m(m-1)}{2}$ to $m$. This reduces the number of updated elements for each sigma-point in \eqref{equ:UKF2} from $\left(\frac{m(m-1)}{2}\right)^2$ to $m^2$. $A$ and $H$ matrices are sparse and an application of sparse linear algebra codes would further enhance the performance of the algorithm.

The dimension of the innovation covariance $S$ is $\left(\frac{m(m-1)}{2}\right)^2 \times \left(\frac{m(m-1)}{2}\right)^2$. The computation of the inverse of the innovation covariance for Kalman gain \eqref{equ:Kalmangain} can be optimized using the inversion formula \eqref{equ:woodbury}. The noise terms are assumed independent and, thus, the noise covariance matrix $R$ is diagonal. We split noises into two parts, whose covariance matrices are $D_1$ and $D_2$. $D_1$ corresponds to microphone specific terms $\measnoise_i$ and $D_2$ to microphone pair specific terms $\measnoise_{i,j}$. Because $\tilde z$ contains only state variables and they do not contribute to the linear part ($AP_{zg(\tilde z)^T}H^T=\mathbf{0}$), the innovation covariance can be written as
\begin{equation}
	S = D_2 + A( P_{g(\tilde z)g(\tilde z)} + D_1) A^T
\end{equation}
and its inverse is
\begin{equation}
\begin{aligned}
	S^{-1} &=  D_2^{-1} + D_2^{-1}A\left[( P_{g(\tilde z)g(\tilde z)} + D_1 )^{-1} + A  D_2^{-1}A^T  \right]^{-1} A^T D_2^{-1}.
	\end{aligned}
\end{equation}
Because $D_2^{-1}$ is diagonal its inverse is diagonal with the reciprocal of the diagonal elements of $D_2$ on its diagonal. Other inverses that appear in this formula are applied to $m\times m$ matrices instead of $\left(\frac{m(m-1)}{2}\right)^2 \times \left(\frac{m(m-1)}{2}\right)^2$ matrices.

\subsection{Optimization of Iterative \acp{KFE} }
\label{sec:iteKFE}
Optimizations presented in sections~\ref{sec:unobserved1} and \ref{sec:unobserved2} are most widely used in the field of \ac{SLAM}. In \ac{SLAM}, the state contains a dynamic part that represents the agent (robot) that is mapping static landmarks in the environment and localizing itself. The landmarks are the static part of the state.

We propose that these optimizations can be applied also to \acp{KFE} that do the update iteratively e.g.\ \ac{RUF} \cite{RUKF} and its extension for sigma-point filters \cite{RUFEXT} or posterior linearization filters \cite{IPLF}. In~\ac{RUF}, the state is updated by applying the update in parts that have smaller impact than the original update. After each part of the update the linearization is recomputed. These iterative filters are developed assuming additive Gaussian noise. Instead of making the updates for full state the update could be done iteratively only for the observed variables. Because the iterative update is actually one update, the unobserved state variables are static during the partial updates.

\begin{figure}[tb!]
\centering
\includegraphics[width=0.8\columnwidth,clip=true, trim=0cm 1.1cm 0cm 0.7cm]{rangex.eps}
\caption{Posterior means computed with \ac{RUF} with varying number of iterations.}
\label{fig:rangex}
\end{figure}

In this example we consider a state model with 3D position, velocity, and acceleration i.e.\ 9 state variables. Measurements used are range measurements from 3 base stations. The measurement model for range from the $i$th base station is
\begin{equation}
	y_i =  \left\| x_{[1:3]}-  r_i \right\| + \measnoise_i, \label{equ:rangemeas}
\end{equation}
where $r_i$ is the location of the $i$th base station. In our example, the prior has a large variance and its mean is located so that the linearization about the mean is not accurate. Figure~\ref{fig:rangex} shows the example situation and the posterior means which are computed with different number of iterations of \ac{RUF}.
The estimate with one iteration ($T=1$) is identical to \ac{EKF} estimate and is not close to the true location. The estimate with 10 iterations is close to the true location. The computation of 10 iterations with \ac{RUF} involves the updates to the $9\times 9$ state covariance matrices 10 times. Because the measurement model \eqref{equ:rangemeas} depends only on 3 first state variables the active state can be selected to consist of only those and the iterative update is computed only for the position variables. The full covariance update is computed after all iterations are applied. In this update scheme, the $3\times 3$ covariance matrix is updated $T$ times and the $9\times9$ matrix is updated only once using \eqref{equ:unobstart}-\eqref{equ:unobend}.

\section{Conclusions}
\label{sec:conclusions}

In this survey, we collected different optimizations for \acp{KFE} that exploit the structure of the state and measurement models to reduce the computational load. Table~\ref{tbl:optitaulu} summarizes the structures of models that are exploited in different optimizations. These structures are present in various nonlinear estimation problems.

\begin{table*}[tb]
\caption{Summary of presented optimizations} \label{tbl:optitaulu}
\footnotesize
\centering
{\begin{tabular}{c|c}
Section & Exploited structure \\ \hline
Section~\ref{sec:partlin} & Partially linear functions ${f(z) = A g(Tz) + Hz}$ \\
Section~\ref{sec:condlin} & Conditionally linear functions ${f(z) = g_n(z_n) + G(z_n)z_l}$ \\
Section~\ref{sec:unobserved1} & Part of state is unobserved and static \\
Section~\ref{sec:unobserved2} & Part of state is unobserved and whole state is static \\
Section~\ref{sec:Innopti} & Measurement covariance is block diagonal \\
Section~\ref{sec:innopti2} &Innovation covariance of form ${S = D_2 + A( P_{g(\tilde z)g(\tilde z)} + D_1) A^T}$
 \\
\end{tabular}
}
\end{table*}

\begin{table*}[tb]
\caption{Summary of properties of selected \acp{KFE}} \label{tbl:Kalmantaulu}
\footnotesize
\centering
	{\begin{tabular}{c|cccc}
	 Algorithm & \rotatebox{90}{\parbox{4.4cm}{Order of measurement function evaluations. $\alpha$ = algorithm specific parameter ($\alpha>0$), $j$ = number of iterations}}  & \rotatebox{90}{\parbox{4.4cm}{Computation of $J$ \\ A=analytically, I=using \eqref{equ:virtJac}, N=numerically inside algorithm }} & \rotatebox{90}{Expectations of form \eqref{equ:UKF1}-\eqref{equ:UKF3}} &\rotatebox{90}{Iterative algorithm}  \\ \hline
	 \acf{EKF} \cite{jazwinski}& 1 & A & \\
	 \acf{EKF2} \cite{jazwinski} & 1 & A & \\
	 \acf{UKF} \cite{UKF}& n  &  I & X &\\
	 \acf{CKF} \cite{cubature}& $n^\alpha$ & I & X &  \\
	 \acf{S2KF} \cite{S2KF}& $n+\alpha$ & I & X &  \\
	 \acf{GF} \cite{huber2008gaussian}& $\alphan$ & I & X &  \\
	 \acf{QKF} \cite{ITO}& $\alpha^n$ & I & X &  \\
	 \acf{DD} \cite{norgaard}& $n$ &N & \\
	 \acf{SOKF2} \cite{ITO}& $n^2$ & N& \\
	  \acf{RUF} \cite{RUKF} & $j$ & A & &X \\
	  \acf{URUF} \cite{RUFEXT} & $j n$ & I & X &X \\
	\end{tabular}}
\end{table*}
Table~\ref{tbl:Kalmantaulu} shows some properties of selected \acp{KFE}.  Different optimizations give different amount of speed increase with different \acp{KFE}. Optimizations presented in sections \ref{sec:partlin} and \ref{sec:condlin} are the most useful with \acp{KFE} that have a high number of function evaluations as a function of state dimension. The exploitation of conditionally linear part of functions in Section~\ref{sec:condlin} requires that the expectations are approximated with equations of form \eqref{equ:UKF1}-\eqref{equ:UKF3}. The optimization that exploits the partially unobserved and static state in Section~\ref{sec:unobserved1} is better suited for algorithms that compute the $J$ within the algorithm (A or N in Table~\ref{tbl:Kalmantaulu}).

The main contribution of this survey is collecting the different code optimization methods into a single reference and using a general notation that allows to use them with various \acp{KFE}.
In addition to giving a unifying source for code optimization methods, we make the following contributions:
\begin{enumerate}
	\item We point out the possibility to use a linear transformation to find the minimal nonlinear subspace (Section~\ref{sec:partlin}).
	\item We introduce an algorithm for systems with conditionally linear states (Section~\ref{sec:condlin}); it can be used to compute the moments for nonlinear part and solve others using the linearity (Section~\ref{sec:partlin}) as is done in the example in Section~\ref{sec:pdr}.
	\item We presented an example showing that in addition to gaining increase in speed of computation the optimization algorithms may also lead to better estimation accuracy (Section~\ref{sec:pdr}).
	\item We show how to use the matrix inversion lemma to reduce the computational complexity in \ac{TDoA}-based positioning (Section~\ref{sec:tdoa}).
	\item  We presented an example showing that the optimization that exploits static state and partly unobserved variables can applied with iterative \acp{KFE} (Section~\ref{sec:iteKFE}).
\end{enumerate}

In future, the optimizations in this paper could be developed to work with filters that use square root form of the covariance matrix. The square root form provides better numerical properties for filtering. Efficient algorithms for \ac{KF} have been long known \cite{thornton,1977fmd} and there are efficient implementations of the sigma-point filters \cite{SQKF,940586} for situations where noise is additive and independent of the state. However, most of the optimizations presented in this survey have not been considered for square-root form in literature.

\balance
\bibliographystyle{elsarticle-harv}

redpajama_set_name: RedPajamaArXiv


The field of exoplanet detection has seen many new developments since the discovery of the first exoplanet. Observational surveys by the NASA Kepler Mission and several other instrument have led to the confirmation of over 1900 exoplanets, and several thousands of exoplanet potential candidates. All this progress, however, has yet to provide the first confirmed exomoon. Since all previous attempts to discover exomoons have failed, a novel method to detect them is proposed in this dissertation, which describes development of the method and its applications to select the best exomoon candidates for observational searches. The main goal of these searches is to verify the validity and effectiveness of the method, and discover the first exomoon by using the world largest and most suitable radio telescopes. The discovery of first exomoon would begin a new era of exploratory research in exoplanetary systems. The idea that exomoons can be discovered with radio telescopes was proposed by Noyola, Satyal and Musielak et al. (2014), who suggested that the interaction between Io and the Jovian magnetosphere could also be found in exoplanet-exomoon pairs, and the resulting radio emissions could be used to directly detect these systems. The main results of the original study obtained for single prograde exomoons are also described in this dissertation, which in addition extends the previous study to multiple exomoon systems, as well as retrograde orbits. The main objective of these studies is to identify the best exomoon candidates for detection by chosen radio telescopes. One such candidates, Epsilon Eridani b, was selected and observed by the Giant Metre Radio Telescope (GMRT) in India. The preliminary results of these observations do not show any expected radio emission from the chosen systems. Thus, implementation of several important improvements to the method is discussed in details in this dissertation.


Bermondsey Central Hall Methodist Church serves a diverse congregation of about 320 and houses the South London Mission, which has been supporting the community since 1889. The modern London Shard skyscraper rises behind the church.
Editor's note: Part of an occasional series of stories, "British Methodism: Staying relevant in Wesley's homeland," about today's Methodist Church in Britain.
Baffoe was an active part of the Methodist congregation. But he saw lots of people "fall through the net" in those days, including some of his friends.
Because of that, the 35-year-old takes satisfaction in being able to offer hope to the community through his role as faith and community development officer for Bermondsey's South London Mission. Both the church and the mission, he says, provide the "good news" and spirit of Jesus in the community.
That's exactly the kind of spirit that Wesleyan Methodists wanted to invoke, back in the late 19th century, when the church realized there were poorer classes of people they simply were not reaching.
So — in what had been one of the worst districts of London — the Methodists opened a grand hall, complete with a 2,000-seat sanctuary, in September 1900.
A group of African women affectionately known as "the aunties" share prayer books during a praise service at Bermondsey Central Hall Methodist Church. The women form the backbone of the congregation according to their pastor, the Rev. Janet Corlett. From left are: Frances Dakowa, Omo Harris, Kayode Cline-Thomas and Joyce Davies.
Peter Baffoe grew up in the area around Bermondsey Central Hall and remembers a time when it was considered unsafe. He now serves as faith and community development officer for Bermondsey's South London Mission.
Behind its Victorian façade, the building is re-configured now, but still serves as a base for the church's South London Mission, which was established in 1889. It's also home to a growing congregation.
Historically a white working-class area, the north part of the borough now includes first-generation West Africans, second- and third-generation Caribbean natives, Eastern Europeans and an emerging Spanish-Latin American community. It also includes both expensive apartments and social housing for low-income families.
The majority white and elderly Bermondsey congregation of the late 1970s is now majority African of all ages. Of its 320 members, half are from Sierra Leone, reflecting a large migration to England following that country's civil war from 1991 to 2002.
Another 30 percent are from Ghana. The remaining 20 percent include Nigerians and people from Zimbabwe, Zambia, Kenya and South Africa, according to the Rev. Janet Corlett, Bermondsey's minister.
The Rev. Janet Corlett (right) takes part in a prayer and praise service at Bermondsey Central Hall Methodist Church.
"Of course, now we have a growing number of second-generation members who are well and truly children of London but whose parents are from elsewhere," she said.
In a way, she, too, is an outsider, having grown up on the Isle of Man, a self-governing island in the Irish Sea between England and Ireland, and having studied or worked in places as diverse as the Philippines, India, Italy, Zimbabwe and Honduras.
On a busy Wednesday in early December, the church's doors were wide open and Bermondsey Central Hall was full of activity.
A morning prayer and praise service included about 10 of "the aunties." These older African women form the backbone of the congregation — women like Remi Solanke, who was raised Methodist and moved to the neighborhood in 1992. The retired civil servant has lived in London more than 40 years and has children and grandchildren here but goes back to her native Nigeria every other year to visit.
The aunties can be there when Bermondsey's working members cannot, Corlett said. They turn up regularly for Bible class, run the prayer and praise service, take the lead for the over-60s exercise class, volunteer at the night-shelter and make sandwiches for the youth brigade.
Eric Bediako (left) leads a Bible study with Judith Alpha (center) and Joyce Davies at Bermondsey Central Hall Methodist Church.
Daphne Sawyer lives at Cluny Place House, a hostel run by the Bermondsey South London Mission that offers short-term support and a safe space for people having trouble finding an affordable rental.
After the prayer service, the group moved to a corner of the building where the "Pop-In" lounge offers hospitality. Open on weekdays from 10 a.m. to 2 p.m., the Pop-In also serves as a neighborhood welcome spot for the South London Mission.
To address housing needs, the mission runs a hostel called Cluny Place House — similar to a student residence hall — that offers short-term support and a safe space for people, especially singles, having trouble finding an affordable rental. It also generates income for the mission.
Particularly affected are those who don't qualify for social housing, those who, "despite the fact they're working a good job, can't afford high London prices," said Peter Baffoe.
Daphne Sawyer, a resident at Cluny Place since 2009, fits that description. Both she and her husband, Edward, are university graduates but, she noted, West Africans often don't get jobs they are qualified for.
Sawyer was living in East London when she lost her job and was evicted from her previous residence. Her husband had worked for a teaching agency as a substitute teacher but those jobs dried up. He currently is employed as a painter at a hospital.
The Rev. Janet Corlett (left) welcomes Frances Dakowa for a prayer and praise service at Bermondsey Central Hall Methodist Church.
Winnie Baffoe heads the "Mummies Republic" program for young mothers at Methodist-run South London Mission.
The mission also connects with young mothers through "Mummies Republic." Winnie Baffoe — employed by the church as a "Fresh Expressions" pioneer worker and married to Peter Baffoe — developed the program as part of her "T-Rooms" project for Fresh Expressions, an initiative for church revitalization.
She works with FareShare food bank, a South London Mission partner, and the London Borough of Southwark's mental health team and heath visitors program. Some of the mothers take part in "Talking Therapies," a trial program at the mission being monitored by the National Health Service to see if it could work at other faith-based institutions.
16 young people share their experiences before being confirmed at Bermondsey Central Hall Methodist Church, Pentecost, 2017.
Winnie Baffoe talks about helpimg createT-Rooms, a quiet place for families and inidividuals to commune with others and with God.
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Read more from our series on the church in England, Brtitsh Methodism: Staying Relevant in Wesley's Homeland.
A focus on young people has been a common thread throughout the church's and mission's history.
In 2011, Corlett started the Boys and Girls Brigade, a youth program particularly strong in the Methodist, Baptist and Presbyterian traditions, at Bermondsey Central Hall. More than 30 children aged 4 to 11 take part every Wednesday. "It has become a key part of our ministry to young people and families in the neighborhood," she said.
There is a youth club and other activities for older teens and on Saturdays, Peter Baffoe said, they "hire a pitch" at a local school for football.
Last year, the church had 16 teenagers — 13 boys and 3 girls — in its confirmation class. Thirteen of the confirmands had been baptized at Bermondsey, "so we had, in a way, fulfilled our baptismal promise as a congregation," Corlett noted.
One of the active young people is Winston Shaw, 17, of East London, who was taking advantage of the empty sanctuary that Wednesday afternoon to practice the organ.
"I try to repay my love for God through music," explained Shaw, who will be attending university during the next academic year and plans to go into teaching. Later that day, he would volunteer with the Boys and Girls Brigade.
"The thing about this church congregation is we're family," Shaw said. "I think it's the love that binds us together."
Winston Shaw, 17, stops by after school to practice the organ at Bermondsey Central Hall Methodist Church. "I try to repay my love for God through music," Shaw said.
Bloom is assistant news editor of United Methodist News Service and DuBose is a photographer for United Methodist Communications. Contact them at 615-742-5470 or newsdesk@umcom.org. To read more United Methodist news, subscribe to the free Daily or Weekly Digests.