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125-397: In statistics and control theory , Kalman filtering (also known as linear quadratic estimation ) is an algorithm that uses a series of measurements observed over time, including statistical noise and other inaccuracies, to produce estimates of unknown variables that tend to be more accurate than those based on a single measurement, by estimating a joint probability distribution over

250-429: A Markov chain built on linear operators perturbed by errors that may include Gaussian noise . The state of the target system refers to the ground truth (yet hidden) system configuration of interest, which is represented as a vector of real numbers . At each discrete time increment, a linear operator is applied to the state to generate the new state, with some noise mixed in, and optionally some information from

375-474: A distribution (sample or population): central tendency (or location ) seeks to characterize the distribution's central or typical value, while dispersion (or variability ) characterizes the extent to which members of the distribution depart from its center and each other. Inferences made using mathematical statistics employ the framework of probability theory , which deals with the analysis of random phenomena. A standard statistical procedure involves

500-469: A population , for example by testing hypotheses and deriving estimates. It is assumed that the observed data set is sampled from a larger population. Inferential statistics can be contrasted with descriptive statistics . Descriptive statistics is solely concerned with properties of the observed data, and it does not rest on the assumption that the data come from a larger population. Consider independent identically distributed (IID) random variables with

625-461: A Kalman Filter is often difficult due to the difficulty of getting a good estimate of the noise covariance matrices Q k and R k . Extensive research has been done to estimate these covariances from data. One practical method of doing this is the autocovariance least-squares (ALS) technique that uses the time-lagged autocovariances of routine operating data to estimate the covariances. The GNU Octave and Matlab code used to calculate

750-418: A decade earlier in 1795. The modern field of statistics emerged in the late 19th and early 20th century in three stages. The first wave, at the turn of the century, was led by the work of Francis Galton and Karl Pearson , who transformed statistics into a rigorous mathematical discipline used for analysis, not just in science, but in industry and politics as well. Galton's contributions included introducing

875-458: A given probability distribution : standard statistical inference and estimation theory defines a random sample as the random vector given by the column vector of these IID variables. The population being examined is described by a probability distribution that may have unknown parameters. A statistic is a random variable that is a function of the random sample, but not a function of unknown parameters . The probability distribution of

1000-484: A given probability of containing the true value is to use a credible interval from Bayesian statistics : this approach depends on a different way of interpreting what is meant by "probability" , that is as a Bayesian probability . In principle confidence intervals can be symmetrical or asymmetrical. An interval can be asymmetrical because it works as lower or upper bound for a parameter (left-sided interval or right sided interval), but it can also be asymmetrical because

1125-471: A given situation and carry the computation, several methods have been proposed: the method of moments , the maximum likelihood method, the least squares method and the more recent method of estimating equations . Interpretation of statistical information can often involve the development of a null hypothesis which is usually (but not necessarily) that no relationship exists among variables or that no change occurred over time. The best illustration for

1250-498: A join-tree or Markov tree . Additional methods include belief filtering which use Bayes or evidential updates to the state equations. A wide variety of Kalman filters exists by now: Kalman's original formulation - now termed the "simple" Kalman filter, the Kalman–Bucy filter , Schmidt's "extended" filter, the information filter , and a variety of "square-root" filters that were developed by Bierman, Thornton, and many others. Perhaps

1375-618: A linear interpolation, x = ( 1 − t ) ( a ) + t ( b ) {\displaystyle x=(1-t)(a)+t(b)} for t {\displaystyle t} between [0,1]. In our case: This expression also resembles the alpha beta filter update step. If the model is accurate, and the values for x ^ 0 ∣ 0 {\displaystyle {\hat {\mathbf {x} }}_{0\mid 0}} and P 0 ∣ 0 {\displaystyle \mathbf {P} _{0\mid 0}} accurately reflect

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1500-555: A mathematical discipline only took shape at the very end of the 17th century, particularly in Jacob Bernoulli 's posthumous work Ars Conjectandi . This was the first book where the realm of games of chance and the realm of the probable (which concerned opinion, evidence, and argument) were combined and submitted to mathematical analysis. The method of least squares was first described by Adrien-Marie Legendre in 1805, though Carl Friedrich Gauss presumably made use of it

1625-1033: A meaningful order to those values, and permit any order-preserving transformation. Interval measurements have meaningful distances between measurements defined, but the zero value is arbitrary (as in the case with longitude and temperature measurements in Celsius or Fahrenheit ), and permit any linear transformation. Ratio measurements have both a meaningful zero value and the distances between different measurements defined, and permit any rescaling transformation. Because variables conforming only to nominal or ordinal measurements cannot be reasonably measured numerically, sometimes they are grouped together as categorical variables , whereas ratio and interval measurements are grouped together as quantitative variables , which can be either discrete or continuous , due to their numerical nature. Such distinctions can often be loosely correlated with data type in computer science, in that dichotomous categorical variables may be represented with

1750-499: A novice is the predicament encountered by a criminal trial. The null hypothesis, H 0 , asserts that the defendant is innocent, whereas the alternative hypothesis, H 1 , asserts that the defendant is guilty. The indictment comes because of suspicion of the guilt. The H 0 (status quo) stands in opposition to H 1 and is maintained unless H 1 is supported by evidence "beyond a reasonable doubt". However, "failure to reject H 0 " in this case does not imply innocence, but merely that

1875-404: A population, so results do not fully represent the whole population. Any estimates obtained from the sample only approximate the population value. Confidence intervals allow statisticians to express how closely the sample estimate matches the true value in the whole population. Often they are expressed as 95% confidence intervals. Formally, a 95% confidence interval for a value is a range where, if

2000-865: A second undergraduate degree, this time in Economics, from Cornell University in 1949, and was admitted into Phi Beta Kappa . He then attended the University of California, Los Angeles (UCLA), where he received a Master of Arts in Mathematics in 1951 and a Ph.D. in Mathematics in 1955. His thesis Families of Transformations in the Function Spaces H was advised by Angus Ellis Taylor , and investigated families of bounded linear transformations in Banach spaces . While still in graduate school, Swerling worked full-time for Douglas Aircraft Company as

2125-437: A single equation; however, it is most often conceptualized as two distinct phases: "Predict" and "Update". The predict phase uses the state estimate from the previous timestep to produce an estimate of the state at the current timestep. This predicted state estimate is also known as the a priori state estimate because, although it is an estimate of the state at the current timestep, it does not include observation information from

2250-581: A staff member of the newly formed Project RAND. He wrote his landmark report, "Probability of Detection for Fluctuating Targets," for the RAND Corporation (now independent from Douglas Aircraft) in 1954. The paper introduced a set of statistically "fluctuating target" scattering models to characterize the detection performance of pulsed radar systems. Building on the work of Jess Marcum (who statistically subtracted noise from images of steady targets), Swerling accounted for statistical fluctuations of

2375-465: A statistician would use a modified, more structured estimation method (e.g., difference in differences estimation and instrumental variables , among many others) that produce consistent estimators . The basic steps of a statistical experiment are: Experiments on human behavior have special concerns. The famous Hawthorne study examined changes to the working environment at the Hawthorne plant of

2500-517: A sufficient sample size to specifying an adequate null hypothesis. Statistical measurement processes are also prone to error in regards to the data that they generate. Many of these errors are classified as random (noise) or systematic ( bias ), but other types of errors (e.g., blunder, such as when an analyst reports incorrect units) can also occur. The presence of missing data or censoring may result in biased estimates and specific techniques have been developed to address these problems. Statistics

2625-469: A system's state to calculate a new state. The measurements' certainty-grading and current-state estimate are important considerations. It is common to discuss the filter's response in terms of the Kalman filter's gain . The Kalman gain is the weight given to the measurements and current-state estimate, and can be "tuned" to achieve a particular performance. With a high gain, the filter places more weight on

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2750-637: A test and confidence intervals . Jerzy Neyman in 1934 showed that stratified random sampling was in general a better method of estimation than purposive (quota) sampling. Today, statistical methods are applied in all fields that involve decision making, for making accurate inferences from a collated body of data and for making decisions in the face of uncertainty based on statistical methodology. The use of modern computers has expedited large-scale statistical computations and has also made possible new methods that are impractical to perform manually. Statistics continues to be an area of active research, for example on

2875-399: A transformation is sensible to contemplate depends on the question one is trying to answer." A descriptive statistic (in the count noun sense) is a summary statistic that quantitatively describes or summarizes features of a collection of information , while descriptive statistics in the mass noun sense is the process of using and analyzing those statistics. Descriptive statistics

3000-445: A truck. The truck can be equipped with a GPS unit that provides an estimate of the position within a few meters. The GPS estimate is likely to be noisy; readings 'jump around' rapidly, though remaining within a few meters of the real position. In addition, since the truck is expected to follow the laws of physics, its position can also be estimated by integrating its velocity over time, determined by keeping track of wheel revolutions and

3125-419: A value accurately rejecting the null hypothesis (sometimes referred to as the p-value ). The standard approach is to test a null hypothesis against an alternative hypothesis. A critical region is the set of values of the estimator that leads to refuting the null hypothesis. The probability of type I error is therefore the probability that the estimator belongs to the critical region given that null hypothesis

3250-619: A worthwhile alternative to the Autocovariance Least-Squares methods. Another approach is the Optimized Kalman Filter ( OKF ), which considers the covariance matrices not as representatives of the noise, but rather, as parameters aimed to achieve the most accurate state estimation. These two views coincide under the KF assumptions, but often contradict each other in real systems. Thus, OKF's state estimation

3375-481: A younger brother, Jo, Jr. Swerling’s father recognized his young son’s intellectual gifts. Granting a tenth birthday request, he introduced Peter to Albert Einstein , who advised the boy to continue his studies in mathematics. Peter Swerling entered the California Institute of Technology at the age of 15 and received a Bachelor of Science in Mathematics three years later in 1947. He went on to take

3500-409: Is a difficult one and is treated as a problem of control theory using robust control . The Kalman filter is a recursive estimator. This means that only the estimated state from the previous time step and the current measurement are needed to compute the estimate for the current state. In contrast to batch estimation techniques, no history of observations and/or estimates is required. In what follows,

3625-435: Is a mathematical body of science that pertains to the collection, analysis, interpretation or explanation, and presentation of data , or as a branch of mathematics . Some consider statistics to be a distinct mathematical science rather than a branch of mathematics. While many scientific investigations make use of data, statistics is generally concerned with the use of data in the context of uncertainty and decision-making in

3750-403: Is a new state estimate that lies between the predicted and measured state, and has a better estimated uncertainty than either alone. This process is repeated at every time step, with the new estimate and its covariance informing the prediction used in the following iteration. This means that Kalman filter works recursively and requires only the last "best guess", rather than the entire history, of

3875-479: Is a strong analogy between the equations of a Kalman Filter and those of the hidden Markov model. A review of this and other models is given in Roweis and Ghahramani (1999) and Hamilton (1994), Chapter 13. In order to use the Kalman filter to estimate the internal state of a process given only a sequence of noisy observations, one must model the process in accordance with the following framework. This means specifying

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4000-416: Is an important topic in control theory and control systems engineering. Together with the linear-quadratic regulator (LQR), the Kalman filter solves the linear–quadratic–Gaussian control problem (LQG). The Kalman filter, the linear-quadratic regulator, and the linear–quadratic–Gaussian controller are solutions to what arguably are the most fundamental problems of control theory. In most applications,

4125-575: Is another type of observational study in which people with and without the outcome of interest (e.g. lung cancer) are invited to participate and their exposure histories are collected. Various attempts have been made to produce a taxonomy of levels of measurement . The psychophysicist Stanley Smith Stevens defined nominal, ordinal, interval, and ratio scales. Nominal measurements do not have meaningful rank order among values, and permit any one-to-one (injective) transformation. Ordinal measurements have imprecise differences between consecutive values, but have

4250-465: Is appropriate to apply different kinds of statistical methods to data obtained from different kinds of measurement procedures is complicated by issues concerning the transformation of variables and the precise interpretation of research questions. "The relationship between the data and what they describe merely reflects the fact that certain kinds of statistical statements may have truth values which are not invariant under some transformations. Whether or not

4375-522: Is better than the estimate obtained by using only one measurement alone. As such, it is a common sensor fusion and data fusion algorithm. Noisy sensor data, approximations in the equations that describe the system evolution, and external factors that are not accounted for, all limit how well it is possible to determine the system's state. The Kalman filter deals effectively with the uncertainty due to noisy sensor data and, to some extent, with random external factors. The Kalman filter produces an estimate of

4500-834: Is called error term, disturbance or more simply noise. Both linear regression and non-linear regression are addressed in polynomial least squares , which also describes the variance in a prediction of the dependent variable (y axis) as a function of the independent variable (x axis) and the deviations (errors, noise, disturbances) from the estimated (fitted) curve. Measurement processes that generate statistical data are also subject to error. Many of these errors are classified as random (noise) or systematic ( bias ), but other types of errors (e.g., blunder, such as when an analyst reports incorrect units) can also be important. The presence of missing data or censoring may result in biased estimates and specific techniques have been developed to address these problems. Most studies only sample part of

4625-428: Is distinguished from inferential statistics (or inductive statistics), in that descriptive statistics aims to summarize a sample , rather than use the data to learn about the population that the sample of data is thought to represent. Statistical inference is the process of using data analysis to deduce properties of an underlying probability distribution . Inferential statistical analysis infers properties of

4750-507: Is generally credited with developing the first implementation of a Kalman filter. He realized that the filter could be divided into two distinct parts, with one part for time periods between sensor outputs and another part for incorporating measurements. It was during a visit by Kálmán to the NASA Ames Research Center that Schmidt saw the applicability of Kálmán's ideas to the nonlinear problem of trajectory estimation for

4875-564: Is independent of time. The initial state, and the noise vectors at each step { x 0 , w 1 , … , w k , v 1 , … , v k } {\displaystyle \{\mathbf {x} _{0},\mathbf {w} _{1},\dots ,\mathbf {w} _{k},\mathbf {v} _{1},\dots ,\mathbf {v} _{k}\}} are all assumed to be mutually independent . Many real-time dynamic systems do not exactly conform to this model. In fact, unmodeled dynamics can seriously degrade

5000-472: Is more robust to modeling inaccuracies. It follows from theory that the Kalman filter provides an optimal state estimation in cases where a) the model matches the real system perfectly, b) the entering noise is "white" (uncorrelated), and c) the covariances of the noise are known exactly. Correlated noise can also be treated using Kalman filters. Several methods for the noise covariance estimation have been proposed during past decades, including ALS, mentioned in

5125-400: Is much applied in time series analysis tasks such as signal processing and econometrics . Kalman filtering is also important for robotic motion planning and control, and can be used for trajectory optimization . Kalman filtering also works for modeling the central nervous system 's control of movement. Due to the time delay between issuing motor commands and receiving sensory feedback ,

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5250-579: Is named for Hungarian émigré Rudolf E. Kálmán , although Thorvald Nicolai Thiele and Peter Swerling developed a similar algorithm earlier. Richard S. Bucy of the Johns Hopkins Applied Physics Laboratory contributed to the theory, causing it to be known sometimes as Kalman–Bucy filtering. Kalman was inspired to derive the Kalman filter by applying state variables to the Wiener filtering problem . Stanley F. Schmidt

5375-436: Is observed, these estimates are updated using a weighted average , with more weight given to estimates with greater certainty. The algorithm is recursive . It can operate in real time , using only the present input measurements and the state calculated previously and its uncertainty matrix; no additional past information is required. Optimality of Kalman filtering assumes that errors have a normal (Gaussian) distribution. In

5500-418: Is one that explores the association between smoking and lung cancer. This type of study typically uses a survey to collect observations about the area of interest and then performs statistical analysis. In this case, the researchers would collect observations of both smokers and non-smokers, perhaps through a cohort study , and then look for the number of cases of lung cancer in each group. A case-control study

5625-402: Is true ( statistical significance ) and the probability of type II error is the probability that the estimator does not belong to the critical region given that the alternative hypothesis is true. The statistical power of a test is the probability that it correctly rejects the null hypothesis when the null hypothesis is false. Referring to statistical significance does not necessarily mean that

5750-449: Is widely employed in government, business, and natural and social sciences. The mathematical foundations of statistics developed from discussions concerning games of chance among mathematicians such as Gerolamo Cardano , Blaise Pascal , Pierre de Fermat , and Christiaan Huygens . Although the idea of probability was already examined in ancient and medieval law and philosophy (such as the work of Juan Caramuel ), probability theory as

5875-590: The Apollo program resulting in its incorporation in the Apollo navigation computer . This digital filter is sometimes termed the Stratonovich–Kalman–Bucy filter because it is a special case of a more general, nonlinear filter developed by the Soviet mathematician Ruslan Stratonovich . In fact, some of the special case linear filter's equations appeared in papers by Stratonovich that were published before

6000-765: The Boolean data type , polytomous categorical variables with arbitrarily assigned integers in the integral data type , and continuous variables with the real data type involving floating-point arithmetic . But the mapping of computer science data types to statistical data types depends on which categorization of the latter is being implemented. Other categorizations have been proposed. For example, Mosteller and Tukey (1977) distinguished grades, ranks, counted fractions, counts, amounts, and balances. Nelder (1990) described continuous counts, continuous ratios, count ratios, and categorical modes of data. (See also: Chrisman (1998), van den Berg (1991). ) The issue of whether or not it

6125-749: The Department of Defense in areas such as the Aegis Combat System and vulnerabilities of AWACS and Patriot missile systems to electronic countermeasures; he developed more sophisticated radar models for application to targets using stealth technology . Peter Swerling was a department manager for Conductron Corporation in Inglewood, California from 1961 to 1964. In 1966, he founded Technology Service Corporation in Santa Monica, California. With Swerling as president for 16 years,

6250-487: The Western Electric Company . The researchers were interested in determining whether increased illumination would increase the productivity of the assembly line workers. The researchers first measured the productivity in the plant, then modified the illumination in an area of the plant and checked if the changes in illumination affected productivity. It turned out that productivity indeed improved (under

6375-454: The extended Kalman filter and the unscented Kalman filter which work on nonlinear systems . The basis is a hidden Markov model such that the state space of the latent variables is continuous and all latent and observed variables have Gaussian distributions. Kalman filtering has been used successfully in multi-sensor fusion , and distributed sensor networks to develop distributed or consensus Kalman filtering. The filtering method

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6500-546: The forecasting , prediction , and estimation of unobserved values either in or associated with the population being studied. It can include extrapolation and interpolation of time series or spatial data , as well as data mining . Mathematical statistics is the application of mathematics to statistics. Mathematical techniques used for this include mathematical analysis , linear algebra , stochastic analysis , differential equations , and measure-theoretic probability theory . Formal discussions on inference date back to

6625-432: The limit to the true value of such parameter. Other desirable properties for estimators include: UMVUE estimators that have the lowest variance for all possible values of the parameter to be estimated (this is usually an easier property to verify than efficiency) and consistent estimators which converges in probability to the true value of such parameter. This still leaves the question of how to obtain estimators in

6750-719: The mathematicians and cryptographers of the Islamic Golden Age between the 8th and 13th centuries. Al-Khalil (717–786) wrote the Book of Cryptographic Messages , which contains one of the first uses of permutations and combinations , to list all possible Arabic words with and without vowels. Al-Kindi 's Manuscript on Deciphering Cryptographic Messages gave a detailed description of how to use frequency analysis to decipher encrypted messages, providing an early example of statistical inference for decoding . Ibn Adlan (1187–1268) later made an important contribution on

6875-403: The Kalman filter is the best possible linear estimator in the minimum mean-square-error sense , although there may be better nonlinear estimators. It is a common misconception (perpetuated in the literature) that the Kalman filter cannot be rigorously applied unless all noise processes are assumed to be Gaussian. Extensions and generalizations of the method have also been developed, such as

7000-622: The University of Southern California; he taught advanced seminars in communication theory and served on doctoral committees. He was a founder and long-term trustee of Crossroads School , a K-12 private school prominent in the Los Angeles area. In 1978, Swerling was elected to membership in the National Academy of Engineering ; election to the academy honors important contributions to engineering theory, as well as unusual accomplishments in developing fields of technology. Swerling

7125-421: The angle of the steering wheel. This is a technique known as dead reckoning . Typically, the dead reckoning will provide a very smooth estimate of the truck's position, but it will drift over time as small errors accumulate. For this example, the Kalman filter can be thought of as operating in two distinct phases: predict and update. In the prediction phase, the truck's old position will be modified according to

7250-429: The collection of data leading to a test of the relationship between two statistical data sets, or a data set and synthetic data drawn from an idealized model. A hypothesis is proposed for the statistical relationship between the two data sets, an alternative to an idealized null hypothesis of no relationship between two data sets. Rejecting or disproving the null hypothesis is done using statistical tests that quantify

7375-476: The company grew to 200 employees, had a successful IPO in 1983, and was acquired by Westinghouse Electric Corporation in 1985. In 1983, Swerling co-founded Swerling Manasse & Smith, Inc., in Canoga Park, California; he served as its president and CEO for 12 years from 1986 until his retirement in 1998. Beginning in 1965, for several years Swerling was an adjunct professor of electrical engineering at

7500-540: The concepts of standard deviation , correlation , regression analysis and the application of these methods to the study of the variety of human characteristics—height, weight and eyelash length among others. Pearson developed the Pearson product-moment correlation coefficient , defined as a product-moment, the method of moments for the fitting of distributions to samples and the Pearson distribution , among many other things. Galton and Pearson founded Biometrika as

7625-542: The concepts of sufficiency , ancillary statistics , Fisher's linear discriminator and Fisher information . He also coined the term null hypothesis during the Lady tasting tea experiment, which "is never proved or established, but is possibly disproved, in the course of experimentation". In his 1930 book The Genetical Theory of Natural Selection , he applied statistics to various biological concepts such as Fisher's principle (which A. W. F. Edwards called "probably

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7750-418: The controls on the system if they are known. Then, another linear operator mixed with more noise generates the measurable outputs (i.e., observation) from the true ("hidden") state. The Kalman filter may be regarded as analogous to the hidden Markov model, with the difference that the hidden state variables have values in a continuous space as opposed to a discrete state space as for the hidden Markov model. There

7875-402: The covariances are set, it is useful to evaluate the performance of the filter; i.e., whether it is possible to improve the state estimation quality. If the Kalman filter works optimally, the innovation sequence (the output prediction error) is a white noise, therefore the whiteness property of the innovations measures filter performance. Several different methods can be used for this purpose. If

8000-417: The current timestep. In the update phase, the innovation (the pre-fit residual), i.e. the difference between the current a priori prediction and the current observation information, is multiplied by the optimal Kalman gain and combined with the previous state estimate to refine the state estimate. This improved estimate based on the current observation is termed the a posteriori state estimate. Typically,

8125-449: The distribution of the initial state values, then the following invariants are preserved: where E ⁡ [ ξ ] {\displaystyle \operatorname {E} [\xi ]} is the expected value of ξ {\displaystyle \xi } . That is, all estimates have a mean error of zero. Also: so covariance matrices accurately reflect the covariance of estimates. Practical implementation of

8250-406: The effect of differences of an independent variable (or variables) on the behavior of the dependent variable are observed. The difference between the two types lies in how the study is actually conducted. Each can be very effective. An experimental study involves taking measurements of the system under study, manipulating the system, and then taking additional measurements with different levels using

8375-495: The evidence was insufficient to convict. So the jury does not necessarily accept H 0 but fails to reject H 0 . While one can not "prove" a null hypothesis, one can test how close it is to being true with a power test , which tests for type II errors . What statisticians call an alternative hypothesis is simply a hypothesis that contradicts the null hypothesis. Working from a null hypothesis , two broad categories of error are recognized: Standard deviation refers to

8500-478: The expected value assumes on a given sample (also called prediction). Mean squared error is used for obtaining efficient estimators , a widely used class of estimators. Root mean square error is simply the square root of mean squared error. Many statistical methods seek to minimize the residual sum of squares , and these are called " methods of least squares " in contrast to Least absolute deviations . The latter gives equal weight to small and big errors, while

8625-474: The experimental conditions). However, the study is heavily criticized today for errors in experimental procedures, specifically for the lack of a control group and blindness . The Hawthorne effect refers to finding that an outcome (in this case, worker productivity) changed due to observation itself. Those in the Hawthorne study became more productive not because the lighting was changed but because they were being observed. An example of an observational study

8750-402: The extent to which individual observations in a sample differ from a central value, such as the sample or population mean, while Standard error refers to an estimate of difference between sample mean and population mean. A statistical error is the amount by which an observation differs from its expected value . A residual is the amount an observation differs from the value the estimator of

8875-467: The face of uncertainty. In applying statistics to a problem, it is common practice to start with a population or process to be studied. Populations can be diverse topics, such as "all people living in a country" or "every atom composing a crystal". Ideally, statisticians compile data about the entire population (an operation called a census ). This may be organized by governmental statistical institutes. Descriptive statistics can be used to summarize

9000-431: The filter performance, even when it was supposed to work with unknown stochastic signals as inputs. The reason for this is that the effect of unmodeled dynamics depends on the input, and, therefore, can bring the estimation algorithm to instability (it diverges). On the other hand, independent white noise signals will not make the algorithm diverge. The problem of distinguishing between measurement noise and unmodeled dynamics

9125-476: The first efforts to exploit the computational advantages of applying recursion to least-squares problems. His work, particularly "First-Order Error Propagation in a Stagewise Smoothing Procedure for Satellite Observations," anticipated that of Rudolf E. Kálmán , whose linear quadratic estimation technique became known as the Kalman filter . Swerling went on to participate in special studies and task forces for

9250-432: The first journal of mathematical statistics and biostatistics (then called biometry ), and the latter founded the world's first university statistics department at University College London . The second wave of the 1910s and 20s was initiated by William Sealy Gosset , and reached its culmination in the insights of Ronald Fisher , who wrote the textbooks that were to define the academic discipline in universities around

9375-402: The former gives more weight to large errors. Residual sum of squares is also differentiable , which provides a handy property for doing regression . Least squares applied to linear regression is called ordinary least squares method and least squares applied to nonlinear regression is called non-linear least squares . Also in a linear regression model the non deterministic part of the model

9500-605: The given parameters of a total population to deduce probabilities that pertain to samples. Statistical inference, however, moves in the opposite direction— inductively inferring from samples to the parameters of a larger or total population. A common goal for a statistical research project is to investigate causality , and in particular to draw a conclusion on the effect of changes in the values of predictors or independent variables on dependent variables . There are two major types of causal statistical studies: experimental studies and observational studies . In both types of studies,

9625-511: The guidance and navigation systems of reusable launch vehicles and the attitude control and navigation systems of spacecraft which dock at the International Space Station . Kalman filtering uses a system's dynamic model (e.g., physical laws of motion), known control inputs to that system, and multiple sequential measurements (such as from sensors) to form an estimate of the system's varying quantities (its state ) that

9750-499: The internal state is much larger (has more degrees of freedom ) than the few "observable" parameters which are measured. However, by combining a series of measurements, the Kalman filter can estimate the entire internal state. For the Dempster–Shafer theory , each state equation or observation is considered a special case of a linear belief function and the Kalman filtering is a special case of combining linear belief functions on

9875-622: The matrices, for each time-step k {\displaystyle k} , following: As seen below, it is common in many applications that the matrices F {\displaystyle \mathbf {F} } , H {\displaystyle \mathbf {H} } , Q {\displaystyle \mathbf {Q} } , R {\displaystyle \mathbf {R} } , and B {\displaystyle \mathbf {B} } are constant across time, in which case their k {\displaystyle k} index may be dropped. The Kalman filter model assumes

10000-424: The most celebrated argument in evolutionary biology ") and Fisherian runaway , a concept in sexual selection about a positive feedback runaway effect found in evolution . The final wave, which mainly saw the refinement and expansion of earlier developments, emerged from the collaborative work between Egon Pearson and Jerzy Neyman in the 1930s. They introduced the concepts of " Type II " error, power of

10125-425: The most commonly used type of very simple Kalman filter is the phase-locked loop , which is now ubiquitous in radios, especially frequency modulation (FM) radios, television sets, satellite communications receivers, outer space communications systems, and nearly any other electronic communications equipment. Kalman filtering is based on linear dynamic systems discretized in the time domain. They are modeled on

10250-411: The most recent measurements, and thus conforms to them more responsively. With a low gain, the filter conforms to the model predictions more closely. At the extremes, a high gain (close to one) will result in a more jumpy estimated trajectory, while a low gain (close to zero) will smooth out noise but decrease the responsiveness. When performing the actual calculations for the filter (as discussed below),

10375-601: The noise covariance matrices using the ALS technique is available online using the GNU General Public License . Field Kalman Filter (FKF), a Bayesian algorithm, which allows simultaneous estimation of the state, parameters and noise covariance has been proposed. The FKF algorithm has a recursive formulation, good observed convergence, and relatively low complexity, thus suggesting that the FKF algorithm may possibly be

10500-701: The noise has no explicit knowledge of time. At time k {\displaystyle k} an observation (or measurement) z k {\displaystyle \mathbf {z} _{k}} of the true state x k {\displaystyle \mathbf {x} _{k}} is made according to where Analogously to the situation for w k {\displaystyle \mathbf {w} _{k}} , one may write v ∙ {\displaystyle \mathbf {v} _{\bullet }} instead of v k {\displaystyle \mathbf {v} _{k}} if R {\displaystyle \mathbf {R} }

10625-400: The noise terms are distributed in a non-Gaussian manner, methods for assessing performance of the filter estimate, which use probability inequalities or large-sample theory, are known in the literature. Consider a truck on frictionless, straight rails. Initially, the truck is stationary at position 0, but it is buffeted this way and that by random uncontrolled forces. We measure the position of

10750-489: The notation x ^ n ∣ m {\displaystyle {\hat {\mathbf {x} }}_{n\mid m}} represents the estimate of x {\displaystyle \mathbf {x} } at time n given observations up to and including at time m ≤ n . The state of the filter is represented by two variables: The algorithm structure of the Kalman filter resembles that of Alpha beta filter . The Kalman filter can be written as

10875-551: The optimal estimation of orbits of satellites and trajectories of missiles, anticipating the development of the Kalman filter . He also founded two companies, one of which continues his engineering work today. Peter Swerling was born in New York City on 4 March 1929 to Jo Swerling and Florence (née Manson) Swerling. He grew up in Beverly Hills, California, where his father was a successful screenwriter. Peter had

11000-412: The overall result is significant in real world terms. For example, in a large study of a drug it may be shown that the drug has a statistically significant but very small beneficial effect, such that the drug is unlikely to help the patient noticeably. Although in principle the acceptable level of statistical significance may be subject to debate, the significance level is the largest p-value that allows

11125-417: The physical laws of motion (the dynamic or "state transition" model). Not only will a new position estimate be calculated, but also a new covariance will be calculated as well. Perhaps the covariance is proportional to the speed of the truck because we are more uncertain about the accuracy of the dead reckoning position estimate at high speeds but very certain about the position estimate at low speeds. Next, in

11250-415: The population data. Numerical descriptors include mean and standard deviation for continuous data (like income), while frequency and percentage are more useful in terms of describing categorical data (like education). When a census is not feasible, a chosen subset of the population called a sample is studied. Once a sample that is representative of the population is determined, data is collected for

11375-544: The population. Sampling theory is part of the mathematical discipline of probability theory . Probability is used in mathematical statistics to study the sampling distributions of sample statistics and, more generally, the properties of statistical procedures . The use of any statistical method is valid when the system or population under consideration satisfies the assumptions of the method. The difference in point of view between classic probability theory and sampling theory is, roughly, that probability theory starts from

11500-442: The position estimate back toward the real position but not disturb it to the point of becoming noisy and rapidly jumping. The Kalman filter is an efficient recursive filter estimating the internal state of a linear dynamic system from a series of noisy measurements. It is used in a wide range of engineering and econometric applications from radar and computer vision to estimation of structural macroeconomic models, and

11625-494: The problem of how to analyze big data . When full census data cannot be collected, statisticians collect sample data by developing specific experiment designs and survey samples . Statistics itself also provides tools for prediction and forecasting through statistical models . To use a sample as a guide to an entire population, it is important that it truly represents the overall population. Representative sampling assures that inferences and conclusions can safely extend from

11750-470: The publication of Natural and Political Observations upon the Bills of Mortality by John Graunt . Early applications of statistical thinking revolved around the needs of states to base policy on demographic and economic data, hence its stat- etymology . The scope of the discipline of statistics broadened in the early 19th century to include the collection and analysis of data in general. Today, statistics

11875-461: The same procedure to determine if the manipulation has modified the values of the measurements. In contrast, an observational study does not involve experimental manipulation . Instead, data are gathered and correlations between predictors and response are investigated. While the tools of data analysis work best on data from randomized studies , they are also applied to other kinds of data—like natural experiments and observational studies —for which

12000-439: The sample data to draw inferences about the population represented while accounting for randomness. These inferences may take the form of answering yes/no questions about the data ( hypothesis testing ), estimating numerical characteristics of the data ( estimation ), describing associations within the data ( correlation ), and modeling relationships within the data (for example, using regression analysis ). Inference can extend to

12125-399: The sample members in an observational or experimental setting. Again, descriptive statistics can be used to summarize the sample data. However, drawing the sample contains an element of randomness; hence, the numerical descriptors from the sample are also prone to uncertainty. To draw meaningful conclusions about the entire population, inferential statistics are needed. It uses patterns in

12250-405: The sample to the population as a whole. A major problem lies in determining the extent that the sample chosen is actually representative. Statistics offers methods to estimate and correct for any bias within the sample and data collection procedures. There are also methods of experimental design that can lessen these issues at the outset of a study, strengthening its capability to discern truths about

12375-412: The sampling and analysis were repeated under the same conditions (yielding a different dataset), the interval would include the true (population) value in 95% of all possible cases. This does not imply that the probability that the true value is in the confidence interval is 95%. From the frequentist perspective, such a claim does not even make sense, as the true value is not a random variable . Either

12500-399: The section above. More generally, if the model assumptions do not match the real system perfectly, then optimal state estimation is not necessarily obtained by setting Q k and R k to the covariances of the noise. Instead, in that case, the parameters Q k and R k may be set to explicitly optimize the state estimation, e.g., using standard supervised learning . After

12625-462: The sense in which the null can be proven false, given the data that are used in the test. Working from a null hypothesis, two basic forms of error are recognized: Type I errors (null hypothesis is rejected when it is in fact true, giving a "false positive") and Type II errors (null hypothesis fails to be rejected when it is in fact false, giving a "false negative"). Multiple problems have come to be associated with this framework, ranging from obtaining

12750-412: The state estimate and covariances are coded into matrices because of the multiple dimensions involved in a single set of calculations. This allows for a representation of linear relationships between different state variables (such as position, velocity, and acceleration) in any of the transition models or covariances. As an example application, consider the problem of determining the precise location of

12875-401: The state of the system as an average of the system's predicted state and of the new measurement using a weighted average . The purpose of the weights is that values with better (i.e., smaller) estimated uncertainty are "trusted" more. The weights are calculated from the covariance , a measure of the estimated uncertainty of the prediction of the system's state. The result of the weighted average

13000-408: The statistic, though, may have unknown parameters. Consider now a function of the unknown parameter: an estimator is a statistic used to estimate such function. Commonly used estimators include sample mean , unbiased sample variance and sample covariance . A random variable that is a function of the random sample and of the unknown parameter, but whose probability distribution does not depend on

13125-560: The summer of 1961, when Kalman met with Stratonovich during a conference in Moscow. This Kalman filtering was first described and developed partially in technical papers by Swerling (1958), Kalman (1960) and Kalman and Bucy (1961). The Apollo computer used 2k of magnetic core RAM and 36k wire rope [...]. The CPU was built from ICs [...]. Clock speed was under 100 kHz [...]. The fact that the MIT engineers were able to pack such good software (one of

13250-647: The system, and then taking additional measurements using the same procedure to determine if the manipulation has modified the values of the measurements. In contrast, an observational study does not involve experimental manipulation. Two main statistical methods are used in data analysis : descriptive statistics , which summarize data from a sample using indexes such as the mean or standard deviation , and inferential statistics , which draw conclusions from data that are subject to random variation (e.g., observational errors, sampling variation). Descriptive statistics are most often concerned with two sets of properties of

13375-404: The target itself. The models became known as Swerling Target Models Cases I, II, III, and IV in radar literature. In related work, Swerling made significant contributions to the optimal estimation of orbits of satellites and trajectories of missiles. Working in the fields of least-squares estimation and signal processing , Swerling published papers in 1958 and 1959 on "stagewise" smoothing,

13500-401: The test to reject the null hypothesis. This test is logically equivalent to saying that the p-value is the probability, assuming the null hypothesis is true, of observing a result at least as extreme as the test statistic . Therefore, the smaller the significance level, the lower the probability of committing type I error. Peter Swerling Peter Swerling (4 March 1929 – 25 August 2000)

13625-430: The truck are described by the linear state space Statistics When census data cannot be collected, statisticians collect data by developing specific experiment designs and survey samples . Representative sampling assures that inferences and conclusions can reasonably extend from the sample to the population as a whole. An experimental study involves taking measurements of the system under study, manipulating

13750-439: The truck every Δ t seconds, but these measurements are imprecise; we want to maintain a model of the truck's position and velocity . We show here how we derive the model from which we create our Kalman filter. Since F , H , R , Q {\displaystyle \mathbf {F} ,\mathbf {H} ,\mathbf {R} ,\mathbf {Q} } are constant, their time indices are dropped. The position and velocity of

13875-535: The true state at time k {\displaystyle k} is evolved from the state at k − 1 {\displaystyle k-1} according to where If Q {\displaystyle \mathbf {Q} } is independent of time, one may, following Roweis and Ghahramani ( op. cit. ), write w ∙ {\displaystyle \mathbf {w} _{\bullet }} instead of w k {\displaystyle \mathbf {w} _{k}} to emphasize that

14000-420: The true value is or is not within the given interval. However, it is true that, before any data are sampled and given a plan for how to construct the confidence interval, the probability is 95% that the yet-to-be-calculated interval will cover the true value: at this point, the limits of the interval are yet-to-be-observed random variables . One approach that does yield an interval that can be interpreted as having

14125-506: The two phases alternate, with the prediction advancing the state until the next scheduled observation, and the update incorporating the observation. However, this is not necessary; if an observation is unavailable for some reason, the update may be skipped and multiple prediction procedures performed. Likewise, if multiple independent observations are available at the same time, multiple update procedures may be performed (typically with different observation matrices H k ). The formula for

14250-416: The two sided interval is built violating symmetry around the estimate. Sometimes the bounds for a confidence interval are reached asymptotically and these are used to approximate the true bounds. Statistics rarely give a simple Yes/No type answer to the question under analysis. Interpretation often comes down to the level of statistical significance applied to the numbers and often refers to the probability of

14375-485: The unknown parameter is called a pivotal quantity or pivot. Widely used pivots include the z-score , the chi square statistic and Student's t-value . Between two estimators of a given parameter, the one with lower mean squared error is said to be more efficient . Furthermore, an estimator is said to be unbiased if its expected value is equal to the true value of the unknown parameter being estimated, and asymptotically unbiased if its expected value converges at

14500-464: The update phase, a measurement of the truck's position is taken from the GPS unit. Along with this measurement comes some amount of uncertainty, and its covariance relative to that of the prediction from the previous phase determines how much the new measurement will affect the updated prediction. Ideally, as the dead reckoning estimates tend to drift away from the real position, the GPS measurement should pull

14625-536: The updated ( a posteriori ) estimate covariance above is valid for the optimal K k gain that minimizes the residual error, in which form it is most widely used in applications. Proof of the formulae is found in the derivations section, where the formula valid for any K k is also shown. A more intuitive way to express the updated state estimate ( x ^ k ∣ k {\displaystyle {\hat {\mathbf {x} }}_{k\mid k}} ) is: This expression reminds us of

14750-640: The use of sample size in frequency analysis. Although the term statistic was introduced by the Italian scholar Girolamo Ghilini in 1589 with reference to a collection of facts and information about a state, it was the German Gottfried Achenwall in 1749 who started using the term as a collection of quantitative information, in the modern use for this science. The earliest writing containing statistics in Europe dates back to 1663, with

14875-461: The use of Kalman filters provides a realistic model for making estimates of the current state of a motor system and issuing updated commands. The algorithm works via a two-phase process: a prediction phase and an update phase. In the prediction phase, the Kalman filter produces estimates of the current state variables , including their uncertainties. Once the outcome of the next measurement (necessarily corrupted with some error, including random noise)

15000-503: The variables for each time-step. The filter is constructed as a mean squared error minimiser, but an alternative derivation of the filter is also provided showing how the filter relates to maximum likelihood statistics. The filter is named after Rudolf E. Kálmán . Kalman filtering has numerous technological applications. A common application is for guidance, navigation, and control of vehicles, particularly aircraft, spacecraft and ships positioned dynamically . Furthermore, Kalman filtering

15125-416: The very first applications of the Kalman filter) into such a tiny computer is truly remarkable. Kalman filters have been vital in the implementation of the navigation systems of U.S. Navy nuclear ballistic missile submarines , and in the guidance and navigation systems of cruise missiles such as the U.S. Navy's Tomahawk missile and the U.S. Air Force 's Air Launched Cruise Missile . They are also used in

15250-419: The words of Rudolf E. Kálmán : "The following assumptions are made about random processes: Physical random phenomena may be thought of as due to primary random sources exciting dynamic systems. The primary sources are assumed to be independent gaussian random processes with zero mean; the dynamic systems will be linear." Regardless of Gaussianity, however, if the process and measurement covariances are known, then

15375-468: The world. Fisher's most important publications were his 1918 seminal paper The Correlation between Relatives on the Supposition of Mendelian Inheritance (which was the first to use the statistical term, variance ), his classic 1925 work Statistical Methods for Research Workers and his 1935 The Design of Experiments , where he developed rigorous design of experiments models. He originated

15500-561: Was named a Fellow of the Institute of Electrical and Electronics Engineers in 1968 "for contributions to signal theory as applied to errors in tracking and trajectory prediction of missiles by radar;" he was recognized as a Life Fellow in 1994. Technology Service Corporation recognizes its founder by granting the Peter Swerling Award for Entrepreneurial Excellence to select employees who have made significant contributions to

15625-403: Was one of the most influential radar theoreticians in the second half of the 20th century. He is best known for the class of statistically "fluctuating target" scattering models he developed at the RAND Corporation in the early 1950s to characterize the performance of pulsed radar systems, referred to as Swerling Targets I, II, III, and IV in the literature of radar. Swerling also contributed to

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