"It has turned out that:
In 1880, Professor R. F. Helmert (1841-1917) specified the underlying mathematical problem through formulating the sparse Canonical Block-Angular (CBA) equation system for GEODETIC data.
In 1960, the equation system and formulas for solving positions of MOVING objects were disclosed under the title of K- Filtering (KF).
In 1969, a computing method for solving COMMON factors and (Canonical) Correlations between (two) blocks of data by using Empirical Orthogonal Functions (EOF) was reported by Lange (1969).
In 1972-75, the sparse CBA equations for balloon tracking systems were reported and solved ANALYTICALLY by Lange.
In 1978, Professor Helmut Wolf (1910-1994) published his analytic formulas for a blockwise solution of Helmert's Normal Equations (NEQ).
In 1982, an exact formula for computing the ACCURACY of the Helmert-Wolf blocking (HWB) solution was published by Lange (1982).
In 1986, the HWB solution was GENERALIZED to other applications by Lange (1987).
In 1989, the CONNECTION between the HWB solution and fast K- Filtering (FKF) was discovered and patented by Lange (1990).
In 1992, the FKF method was applied to EXTENDED K- Filtering (EKF) by Lange (1993).
In 1995, the FKF method was extended to ADAPTIVE K- Filtering (AKF) by Lange (1997). "
"No, only theoretically. The FKF method may though be circumvented by using a furiosly fast processor. However, this would take place at the expense of a serious increase in power consumption. The battery of a light-weight low-powered device should be many times larger for equal high-performance-computing (HPC) without the use of FKF.
FKF is expected to become a killer application when serious competition in both accuracy and reliability begins also in the civilian markets of precision navigation, mobile positioning, ultra-reliable guidance, etc."
"FKF renders the most effective computational method for updating calibration parameters that stem as integration constants from the differential equations of signal-phases and acceleration measurements in HYBRID systems. The optimality of the FKF computations is necessary for reliable accuracy estimation that can now be obtained as a byproduct from the sophisticated theory of Minimum Norm Quadratic Unbiased Estimation (MINQUE) by C. R. Rao (1975). Superb accuracies and integrity are thus achieved in real-time only by using the Statistical Calibration by Lange (1999)."
"The International GPS Service (IGS) develops its orbital solutions towards realtime application. The HWB solution will then need simultaneous results from meteorological data-assimilation like that of the European Centre for Medium-Range Weather Forecasts (ECMWF). Embedded microchips based on hybrid navigation concepts are intensively developed. The FKF patents offer industrial protection to the big R&D efforts and investments required."
"Several small European companies, please see: Intellectual Property "
"KF renders the theoretically best method for updating estimates of unknown parameters when new data flow continuously into a navigation receiver or control system. Thus, all navigation receivers must make use of a K-Filter. However, there still are certain necessary conditions for these filtering processes to be absolutely RELIABLE.
All roads may lead to Rome, however, the consequencies can be dead serious if one hits something else before there. "
"One of the reliability conditions of a KF is that the continuously inflowing data must contain enough information on those parameters whose values must be estimated. In other words, if a KF tries to estimate the value of a parameter under circumstances when the inflowing data has very little to do with the parameter then the estimated value is doomed to go astray sooner or later. The FKF-method represents the Best Available Technology (BAT) of extracting such information from the inflowing data.
FKF cannot make miracles but it will save lives because of the ever-increasing automation that is relied upon. This opportunity for improving and warranting public safety must not be deferred by ignorance and excuses."
"The FKF and UD techniques are by no means exclusive. The semi-analytical computations of FKF offer significant advantages over UD:
- Large moving batches of data can be submitted for a careful Minimum Least Squares analysis in order to detect and remove calibration drifts; and,
- Operational accuracies of an overdetermined navigation receiver can be reliably estimated in realtime from observed internal consistencies of the signals and sensors actually in use.
In contrast, if the data are processed one at a time by a typical UD (Square-Root) filter then crucial information on weakly observable calibration parameters can easily be lost. This may lead to serious consequencies due to an unobserved filter divergence.
Sample subroutines of FORTRAN77 can be found in CALLIB.for though without any documentation as different applications require largely varying pieces of such code."
"My PhD thesis describes a prototype system for the tracking of weather balloons. For additional information, please, just let me know: Lange@FKF.net."
"Certainly! We shall soon be able to buy 'super'-navigation receivers attached to mobile phones. These will provide positional information and predict the weather at the same time. In fact, FKF offers the best possible ratio between navigational accuracy and battery saving. However, FKF will give no shortcut to the El Nino problem because of its global nature. The Global Circulation Models (GCM) of atmosphere and oceans are probably too sophisticated to be run on a personal computer"
"We all are God's laborers and Lord Jesus Christ is waiting for His bride to emerge soon. All those individual persons are His bride who love Him as a good wife loves her husband. The labor means taking part in His sufferings that everyone must face when helping people out from spiritual darkness to light.
There are many ways of serving the Lord. An important way is to sponsor ministers to preach the Good News of Him. However, the people of many countries are so distressed that they need help themselves. A way of helping them is to offer well-payed work. However, no company can afford paying good salaries if it cannot sell its products or services at reasonable prices. The products must be of a very good quality or of high technology when made of cheap raw materials. Thus, the way to prosperity is to know how to make good products in quantities and to possess the licencies to sell them to a large market.
The Lord gave me the invention of the Fast K-Filter (FKF) in sleep in order that we together can help His own people by granting FKF lisencies. The Lord did a similar thing in my country by sending James Finlayson of Scotland to initiate the Industrial Revolution of Finland already in year 1817."
It was asked:
where X is the state vector and Y is the observation vector.
With matrices A and H known, how to get vector X at each
time using the FKF program modules?"
"How to use FKF for solving the following typical problem:
Y(k)=H(k)X(k)+V(k) (k is time)
where X is the state vector and Y is the observation vector. With matrices A and H known, how to get vector X at each time using the FKF program modules?"
"Your equation system is valid for prediction, not for recursive KF estimation. At first, you should write it for KF e.g. as follows:
Y(k)=H(k)X(k)+V(k) (k is time)
Then subtract A(k)Z(k-1) from both sides of the first equation (Harvey's approach) in order to get the so-called Augmented Model:
where Z(k-1) is an estimate of X(k-1) from the previous time step from k-2 to k-1.
Now, the FKF program modules apply for computing the maximum likelihood estimate Z(k) of X(k) from the time step from k-1 to k. Please observe that the Augmented Model is just a typical linear regression problem that can be solved by using any Least Squares Analysis software package. FKF speeds up the computations enormously in many cases where vectors Y(k) and/or X(k) have many components."* Last revised: December 9, 2003