Defining Model(s)
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Screen shot #3 (Merely a picture to illustrate that our GUI is totally self-explanatory)

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 cross-correlated or serially correlated in time of known correlation structure expressed either in the time-domain 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 estimator’s 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) doesn’t bother to support, we at TeK Associates are in possession of an Annual Buyer’s guide entitled VME and Critical Systems, Vol. 27, No. 3, December 2009 and we feel obligated to distinguish our TK-MIP software product from that of The MathWorks by TeK Associates eventually offering VME compatibility within TK-MIP in its later versions beyond the current Version 2.0.  

Click here to get the (draft copy) of AIAA Standards for Atmospheric Modeling, as specified by the various U.S. and other International agencies.

Click here to get the (draft copy) of AIAA Standards for Flight Mechanics Modeling, as specified by the pertinent responsible U.S. and other International agencies.

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. TK-MIP performs the necessary conversions automatically, exactly where they need to occur internal to the TK-MIP 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, off-line from TK-MIP (since TK-MIP 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 TK-MIP for performing this task:

  1. Maple Symbolic Math: https://www.maplesoft.com/products/maple/features/symbolicnumericmath.aspx 

  2. MapleSim from Maplesoft: https://www.maplesoft.com/products/maplesim/symbolic_computation.aspx 

  3. MacSyma: https://en.wikipedia.org/wiki/Macsyma 

  4. swMATH MACSYMA: https://www.swmath.org/software/1209 

  5. Wolfram Mathematica: https://en.wikipedia.org/wiki/Wolfram_Mathematica 

  6. Simplify Calculator: https://www.symbolab.com/solver/simplify-calculator 

  7. REDUCE: https://core.ac.uk/download/pdf/82066620.pdf https://reduce-algebra.sourceforge.io/ 

Then upon returning to TK-MIP, the results of the Jacobian calculation parameter data is conveyed to TK-MIP 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 time-step), 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 Auto-Regressive: AR, Auto-Regressive Moving Average: ARMA, or Auto-Regressive Moving Average EXogenous input: ARMAX time-series 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 Box-Jenkins time-series 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 discrete-time representation (i.e., involving difference equations) to a continuous-time 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 state-size or AR order-size “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 


Bayesian Information Criterion (BIC): https://en.wikipedia.org/wiki/Bayesian_information_criterion (It should be no surprise that these two criteria AIC and BIC were used in the same manner as long as 40+ years ago in engineering and scientific applications and in its documented literature.)

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:
1.) Prof.David Luenberger (Stanford University): https://en.wikipedia.org/wiki/David_Luenberger 
2.) (The late) Prof. Monsanao Aoki (UCLA and Univ. of Illinois): https://en.wikipedia.org/wiki/Masanao_Aoki 
3.) Prof. Edison Tse (Stanford University): https://cap.stanford.edu/profiles/frdActionServlet?choiceId=printerprofile&profileversion=full&profileId=41488
Each of the above 3 were already extremely well grounded in Modern Control and Estimation theory in the 1960's even before entering into the Econometrics Field. The “Law of Supply and Demand
”, “Conspicuous Consumption” and other insights from recognized experts in Macro- and Micro-economics and experienced consultants in various specialty topics should help in these endeavors to capture realistic models for the financial area. Also see: “Special Issue on Stochastic Control methods applied to Financial Engineering,” IEEE Trans. on Automatic Control, Vol. 49, No. 3, Mar. 2004.

Pertaining to the discussion immediately above:
Kerr, T. H., “Applying Stochastic Integral Equations to Solve a Particular Stochastic Modeling Problem,” Ph.D. Thesis in the Department of Electrical Engineering, University of Iowa, Iowa City, Iowa, January 1971. (This Ph.D. thesis offers a simple algorithm for easily converting an ARMA time-series into a more tractable AR one of higher dimension. It also describes how to test or experimentally verify that a Black Box model exhibits “linearity” by using a prescribed logical sequence of input-output tests along with appropriate and tractable associated encompassing ellipsoidal confidence regions.)
Kerr, T. H., Multichannel Shaping Filter Formulations for Vector Random Process Modeling Using Matrix Spectral Factorization, MIT Lincoln Laboratory Report No. PA-500, Lexington, MA, 27 March 1989. (This offers a simple algorithm for easily converting an ARMA time-series into a more tractable AR one of higher dimension.)
Kerr, T. H., “Emulating Random Process Target Statistics (Using MSF),” IEEE Trans. on Aerospace and Electronic Systems, Vol. 30, No. 2, pp. 556-577, Apr. 1994. (This offers a simple algorithm for easily converting an ARMA time-series into a more tractable AR one of higher dimension.)
What follows below is a discrete-time (i.e., difference equation-based) ARMA model put into standard discrete-time State Variable form:  

 

Within the MAIN MENU of our TK-MIP 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 TK-MIP!

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