
Screen shot #3 (Merely a picture to illustrate that our GUI is totally selfexplanatory) Select the data source from the indicated menu appearing within the screen below that also allows the user to appropriately enter the system and measurement model and its additive plant (or system) and measurement noise structure (which can be Pure independent Gaussian White Noises [GWN] or crosscorrelated or serially correlated in time of known correlation structure expressed either in the timedomain or in the frequency domain as a power spectrum matrix). [One can also accommodate random process mixtures (i.e., the sum) of Gaussian and Poison White Noises as a stress test of sorts to determine robustness in the performance of the estimators accuracy, as a function of time, when required standard KF and EKF assumptions on the ideal noises being purely Gaussian are not strictly met (as determined by User merely introducing a very small amount of corrupting Poison noise). By doing so, User can see if and when the expected performance breaks down and associated estimator accuracy significantly departs from the design goal that had been sought.] The availability of 10 Megabyte Ethernet is a relatively new option for an Input/Output protocol. Since The MathWorks claims that VME is an older protocol that The MathWorks currently (in 2010) doesnt bother to support, we at TeK Associates are in possession of an Annual Buyers guide entitled VME and Critical Systems, Vol. 27, No. 3, December 2009 and we feel obligated to distinguish our TKMIP software product from that of The MathWorks by TeK Associates eventually offering VME compatibility within TKMIP in its later versions beyond the current Version 2.0.
Entries of the requisite matrices, depicted below, can be explicitly numerical (shown here below as merely constant zeroes: 0.00E+00 throughout) or be in symbolic form consisting of functions of the independent variable time (or of one of its its obvious aliases) and other parameters and possible algebraic operations or combinations of such functions. Sufficient space is availed within each tabular representation of each entry field. TKMIP performs the necessary conversions automatically, exactly where they need to occur internal to the TKMIP software, without the USER needing to explicitly intervene to invoke such conversions themselves. We do the right thing, as can be confirmed with copious test problems, using either our favorites, as suggested, or the USER'S own personal favorites. [If this is to be an EKF application for a system that is a nonlinear function of the states (and, possibly also of time and of the exogenous control, u, if present and the process noise, w, if present), as dx/xt = a[ x(t), u(t), w(t), t], then it is assumed that the proper entries of the corresponding matrices, such as A1 here, have already been determined either (i) by manual calculation of the Jacobian matrix, offline from TKMIP (since TKMIP does not offer the capability of performing this calculation within it), or (ii) from some algebraic symbol manipulation program that calculates the Jacobian (i.e., the 1st partial derivative of a[ x(t), u(t), w(t), t] with respect to x), for which there are many alternative options outside of TKMIP for performing this task:
Then upon returning to TKMIP, the results of the Jacobian calculation parameter data is conveyed to TKMIP as the entries of A1 here. Please notice that such Jacobian calculations need be performed only once at the outset but need to be updated as a linearization (reevaluated about the estimate, xhat, obtained from the prior timestep), that must occur within every EKF implementation.]
Statisticians (and others working with financial data) appear to be more comfortable with entering system models in this equivalent alternative AutoRegressive: AR, AutoRegressive Moving Average: ARMA, or AutoRegressive Moving Average EXogenous input: ARMAX timeseries formulation (to start with) [a preference for going directly to the state variable form may occur later as the User gains more experience and familiarity with it]:
The close (i.e., equivalent) model relationship between a BoxJenkins timeseries representation and a state variable representation has been known for at least 4 or 5 decades, as spelled out in: A. Gelb (Ed.), Applied Optimal Estimation, MIT Press, Cambridge, MA, 1974. This book also shows how to routinely convert from a discretetime representation (i.e., involving difference equations) to a continuoustime representation (i.e., involving differential equations) and vice versa. It is the state variable model that is usually used in scientific and engineering applications, where detailed models for the internals of the matrices are available from physical laws that are part of the User's prerequisite academic curriculum or experience. From what I have personally seen in a Data Analytics Conference at Boston University entitled minnie (Minneapolis) Field Guide to Data Science & Emerging Tech in the Community on 23 September 2018, they are apparently searching (in the dark in my opinion) for an appropriate black box model in the financial applications areas to use as reasonable models (in conjunction with using parameter estimation and AIC and BIC in order to know when they have a model that adequately captures the essence of the financial application yet the maximum dimension or order stops with a reasonably tractable statesize or AR ordersize n, as a parameter that appears in the model equations in the image below. In the preceding discussion, the two yet to be defined 3 letter acronyms are: Akaike Information
Criterion (AIC):
https://en.wikipedia.org/wiki/Akaike_information_criterion In searching for an adequate model for the
financial area, It would likely help if Data Scientists followed the work of certain
specialists in Econometrics, such as: Pertaining to the discussion immediately above:
Within the MAIN MENU of our TKMIP GUI, colorization of system blocks in left margin serves as a persistent reminder of which models have been defined by the User, corresponding to: System, Filter, and/or Control (if it is present in the application). Gearing up to complete the modeling, simulation, and implementation tasks, which can all be accomplished much faster by using TKMIP!
