US 20080314395 A1 Abstract A method, apparatus, and a kit are capable of improving accuracy of CGS devices using dynamic outputs of continuous glucose sensors.
Claims(53) 1. A method for improving accuracy of a continuous glucose sensor (CGS), comprising:
calibrating the CGS at a first time; and calibrating the CGS at a second time that is determined based upon at least one of a dynamically monitored CGS value and a rate of CGS change. 2. The method of 3. The method of 4. The method of 5. The method of dynamically monitoring the CGS value and the rate of CGS change over time. 6. The method of calculating the first order time derivative of the monitored CGS value; determining whether the absolute value of the first time derivative of the CGS value is substantially less than 1; and maintaining the calibration of the CGS at the first time when the absolute value of the first time derivative of the CGS value is substantially less than 1. 7. The method of determining the absolute change between the CGS at the second time and the CGS at the first time calibration is greater than a pre-determined value when the absolute value of the first time derivative of the CGS value is substantially less than 1; and maintaining the calibration of the CGS at the first time when the absolute change between the CGS at the second time and the CGS at the first time calibration is greater than a pre-determined value. 8. The method of 9. The method of 10. The method of recalibrating the CGS when it is determined that the absolute change between the CGS at the second time and the CGS at the first time calibration is greater than a pre-determined value. 11. The method of improving the accuracy of the CGS output by remedying a physiology time lag between the blood glucose level and a glucose level in an interstitial fluid that interacts with the blood glucose. 12. The method of deriving a mathematical equation describing a time dependence of CGS output on the blood glucose level; deriving the time dependence blood glucose level as a function of the CGS output; and applying the derived time dependence blood glucose level function to a set of CGS raw outputs so as to predict the blood glucose level at a later time or correcting the CGS output. 13. The method of 14. The method of 15. The method of 16. A method for improving the accuracy of a continuous glucose sensor, comprising:
calibrating the continuous glucose sensor using first blood glucose value and second blood glucose value; measuring the first blood glucose value when the continuous glucose sensor measures a corresponding first interstitial glucose value; and measuring the second blood glucose value when the continuous glucose sensor measures a corresponding second interstitial glucose value that is different from the first interstitial glucose value by a predetermined amount. 17. The method of 18. The method of 19. The method of 20. The method of 21. The method of 22. The method of 23. The method of 24. The method of 25. The method of 26. The method of improving the accuracy of the CGS output by remedying a physiology time lag between the blood glucose level and a glucose level in an interstitial fluid that interacts with the blood glucose. 27. The method of deriving a mathematical equation describing a time dependence of CGS output on the blood glucose level; deriving the time dependence blood glucose level as a function of the CGS output; and applying the derived time dependence blood glucose level function to a set of CGS raw outputs so as to correct the CGS output for inherent time lags. 28. The method of 29. The method of 30. The method of 31. A system for improving an accuracy of a continuous glucose sensor, comprising:
an accuracy improving module accessible to an output of the continuous glucose sensor, further comprising:
a decision making mechanism capable of determining a calibration at a time based upon at least one of a dynamically monitored output of the continuous glucose sensor.
32. The system of 33. The system of an initial request module capable of initiating a calibration at a predetermined time sequence. 34. The system of a monitoring module capable of dynamically monitoring the output from the continuous glucose sensor. 35. The system of a determination instruction stating that the dynamically monitored output from the continuous glucose sensor differs by at least 15 mg/dl before the second calibration takes place. 36. The system of a determination instruction stating that the dynamically monitored output from the continuous glucose sensor differs by at least 20 mg/dl before the second calibration takes place. 37. The system of a determination instruction stating that the absolute rate of change is substantially equal to or greater than 1 before the second calibration takes place. 38. The system of a time lag correction module capable of improving the accuracy of the CGS output by remedying a physiology time lag between the blood glucose level and a glucose level in an interstitial fluid that interacts with the blood glucose. 39. A kit, comprising
a continuous glucose sensor; a calibration unit in a data communication with to the continuous glucose sensor for calibrating the continuous glucose sensor; and a set of instructions including instructions for instruction monitoring monitored output from the continuous glucose sensor, and performing first and second calibrations at first and second continuous glucose sensor output values that differ by at least 15 mg/dl. 40. The kit of 41. The kit of 42. The kit of 43. The kit of 44. The kit of a time lag correction module accessible to the output of the continuous glucose sensor and capable of remedying a physiology time lag. 45. A method for improving an accuracy of a continuous glucose sensor, comprising:
calibrating the CGS using a first blood glucose level and a second blood glucose level different by a predetermined amount from the first glucose data. 46. The method of 47. The method of 48. The method of 49. The method of 50. A computer-readable medium having computer executable instructions for performing the method set forth in 51. A computer-readable medium having computer executable instructions for performing the method set forth in 52. A computer-readable medium having computer executable instructions for performing the method set forth in 53. A kit, comprising:
a calibration unit for calibrating a continuous glucose, the calibration unit comprising a blood glucose sensor; and a set of instructions including instructions for monitoring output from a continuous glucose sensor, and performing first and second calibrations at first and second continuous glucose sensor output values that differ by at least 15 mg/dl, where the first and second calibrations comprise respectively first and second determinations of blood glucose with the blood glucose sensor. Description This US patent application claims priority under 35 U.S.C. 119(e) from co-pending U.S. provisional application Ser. No. 60/713,203 filed Aug. 31, 2005, and Ser. No. 60/815,191 filed Jun. 20, 2006, the subject matter of each being incorporated herein by reference in its entirety. The subject matter of each one of the publications and the US provisional application in APPENDIX A attached herewith is incorporated herein by reference in their entirety. The present invention relates to the art of glucose monitoring, and more particularly to methods and systems for continuous glucose monitoring. Existing evidence, such as “National Diabetes Fact Sheet” by American Diabetes Association, indicates that currently approximately 18.2 million people in the U.S. have diabetes; and diabetes is the sixth-leading cause of death in the U.S. One in three Americans born in the year 2000 will develop Type 2 diabetes. With the large number of diabetes patients, and with the incidence of diabetes expected to increase, there is a continuously growing need for accurate glucose monitoring systems to monitor glucose levels. Continuous glucose sensors are designed to provide not only real-time glucose levels at a single point in time, but also the trend of a person's glucose levels based on analysis taking place every certain period of time with minimal finger-sticks, leading to improved glycemic/diabetes control. Most contemporary continuous glucose sensors (hereafter CGS), however, yield blood glucose (hereafter BG) estimates by sampling interstitial glucose (hereafter IG) in interstitial fluid, rather than BG due to the difficulty in directly measuring BG in artery or blood vessels. A typical glucose (BG) estimation from IG is produced from at least two consecutive approximation steps: 1) Blood-to-interstitial glucose (BG-to-IG) transport; and 2) Derivation of BG values from IG-related electrical current recorded by the sensor. As a result, although CGS technology has made dramatic strides, the development of accurate and reliable CGS devices continues to face numerous challenges in terms of calibration, sensitivity, stability, and physiological time lag between blood and interstitial glucose concentrations. The difference between BG and CGS readings arises from following major factors: physiology, sensor calibration, noise, and engineering. The physiological time lag and gradients are changing dynamically with time, with BG levels, and across subjects; and the direct frequent in vivo sampling of IG is extremely difficult. Consequently, the evaluation of engineering performance of CGS is left with a central problem: separating the portion of BG/CGS error due to calibration, sensor noise, and BG/IG gradient. Therefore, a method and apparatus are desired for improving accuracy and reliability of CGS. Various objects and advantages of the preferred embodiments of the present invention will be appreciated based on this disclosure. According to the preferred embodiments, the present invention improves the accuracy and reliability of CGS by improving the calibration of CGS sensors or remedying errors due to physiological time lag or a combination thereof. As an exemplary embodiment of the invention, a method for improving accuracy of a continuous glucose sensor (CGS) is disclosed herein. The method comprises: calibrating the CGS at a first time; and changing the CGS calibration at a second time that is determined based upon a dynamically monitored CGS value, a rate of CGS change, and a predetermined criterion. As another exemplary embodiment of the invention, a method for improving accuracy of a continuous glucose sensor (CGS) is disclosed herein. The method comprises: calibrating the CGS using a first blood glucose data and a second blood glucose data different from the first glucose data. As yet another exemplary embodiment of the invention, a continuous glucose sensing (CGS) device is disclosed herein. The device comprises: first means for measuring interstitial glucose level so as to obtain a CGS output; and a calibration module accessible to the CGS output for improving accuracy of the CGS, further comprising: a monitoring module accessible to the CGS output for dynamically monitoring the CGS and a time derivative of the CGS; and instructing another calibration event based on the dynamic CGS value, the time derivative of the CGS value, and a predetermined criterion. As yet another exemplary embodiment of the invention, a computer-readable medium having computer executable instructions for performing a method for improving accuracy of a continuous glucose sensor is disclosed, wherein the method comprises: retrieving an initial blood glucose value and a CGS value obtained in a measurement for the initial blood glucose value; monitoring the CGS value and a time derivative of the CGS value over time; determining whether to initiate another calibration based on the monitored CGS values and the time derivative of the CGS value; and calibrating the CGS if it is determined to initiate said another calibration. As yet another exemplary embodiment of the invention, a computer-readable medium having computer executable instructions for performing a method for improving accuracy of a continuous glucose sensor is disclosed, wherein the method comprises: retrieving a blood glucose value and a CGS value obtained in a measurement for the initial blood glucose value at a first time; and calibrating the CGS at a second time determined by a CGS value at substantially the second time, a time derivative of the CGS, and a predetermined criterion. As yet another exemplary embodiment of the invention, a system used for treating a disease associated with blood glucose is disclosed herein. The system comprises: a continuous glucose device of claim Various objects and/or advantages of some preferred embodiments of the invention can be, in some preferred examples, achieved via the features of the independent claims attached hereto. Additional preferred embodiments are further set forth in the dependent claims. In the claims, only elements denoted by the words “means for” are intended to be interpreted as means plus function claims under 35 U.S.C. § 112, the sixth paragraph. The preferred embodiments of the invention can be best understood from the following detailed description taken in conjunction with the accompanying drawings of which: This invention provides a method and device for improving accuracy of continuous glucose sensors by improving the calibration of the CGS or by remedying errors arising from the physiological time lag between BG and IG, or a combination thereof. In view of many possible variations within the spirit of the invention, the invention will be discussed in the following with reference to specific examples. However, it will be appreciated by those skilled in the art that the following discussion is for demonstration purposes, and should not be interpreted as a limitation. Other variations without departing from the spirit of the invention are also applicable. Inaccuracies of most current CGS devices are mainly attributed to poor CGS calibration, physiology time lag, and random errors. To reduce the inaccuracy of the CGS, an improved calibration procedure is proposed. The reduction of inaccuracy can alternatively be achieved by remedying the error related to physiological time lag, which is also proposed in this invention. In fact, the improved calibration procedure and the time lag remedy procedure can alternatively be combined together so as to achieve a better performance of CGS. Referring to To improve the accuracy of CGS, accuracy improver As an example of the invention, It is known in the art that the accuracy of CGS calibration depends on the rate of blood glucose (hereafter BG) change and the BG value (BG(t)) at the time (t) of calibration. The rate of BG change can be mathematically expressed as the time derivative of BG(t): d(BG(t))/dt. Given the fact that calibrations with variant inputs are better than those with single or non-varying inputs, the CGS calibration of the invention uses variant inputs. As an example, The initial calibration module performs initial calibration so as to obtain an initial calibration data pair SG( It is noted that one or more above functional modules can be incorporated into other functional modules in practice. In particular, sensor calibration module An exemplary operation of the functional modules in The calibration decision making loop starts from step Following the initiation of the self-check procedure at step After reassigning and recording, the calibration process re-enters the decision making cycle The improved accuracy of CGS using the optimal calibration method as discussed above can be validated by the following experimental data and computer-simulations, as shown in To test the accuracy of the CGS incorporating the accuracy improvement method as discussed above, a measurement is conducted on thirty-nine (39) subjects with type 1 diabetes mellitus (T1DM). The 39 participants have the following statistics: average age 42.5 years with standard deviation (SD) of 12 (SD=12), average duration of T1DM 21.6 years (SD=94), average HbAlc=7.4% (SD=0.8), 16 males. The study was approved by the University of Virginia IRB Subjects. The subjects were admitted to the general clinic research center (GCRC) in the evening prior to the study. The participants' BG levels were controlled overnight within euglycemic range of 100-150 mg/dl (55-8.3 mmol/l). A Minimed CGMS™ was attached to each subject and was calibrated during the study in accordance with the manufacturer's instructions. All CGMS™ were inserted in the abdomen. Hyperinsulmemic clamps were performed in the morning. Each clamp used constant insulin infusion rate of 1 mU/kg/min and variable glucose infusion rate to achieve and maintain BG levels at approximately 110 mg/dl (around 6 mmol/l). Subsequently, the glucose infusion rate was reduced to permit a controlled decline in BG of approximately 1 mg/dl/min until BG reached 50 mg/dl (around 2.8 mmol/l). Glucose infusion was then resumed to allow a recovery to normal glucose levels. The euglycemic portion of the clamp study varied in length from 70 to 210 minutes; and the duration of the BG reduction procedure ranged from 30 to 60 minutes. The recovery ranged from 30 to 60 minutes. Arterialized blood was achieved by warming the hand to 50° C. and was sampled every 5 minutes for reference BG levels. To allow for insulin to reach its steady state effect, the first 15 minutes of data after the beginning of infusion were ignored. CGMS™ readings were synchronized with reference BG. A recalibration of the sensor using 2 reference BG values taken during the clamp study described above is computer-simulated, as shown in Referring to It can also be seen in the figure that the BG calibration difference d with value larger than 30 mg/dl but lower than 40 mg/dl achieves excellent results; whereas the difference d with a value larger than 40 mg/dl achieves “nearly-perfect” results. It is worthwhile to point that the sensor calibration during the experiment described above was always done in periods of steady BG kept at euglycemia, thus the influence of BG rate of change was minimal. In addition to the calibration of CGS, physiology time lag between BG and IG also causes inaccuracy in the CGS output. This arises from the fact that most of current CGS devices doe not directly measure the blood glucose levels, but the IG levels in the interstitial fluids instead. CGS devices then convert the IG readings into estimates of BG. Therefore, an improved conversion method from IG to BG will lead to improved performance of CGS. An object of the invention improves the conversion from IG to BG by including the physiology time lag between the IG and BG levels. Such improvement is accomplished through analyses and incorporation of the time dependency between IG and BG. Specifically, a mathematical model is established for describing the time dependency among BG and IG or CGS output. Based upon the established model, a mathematical equation is derived to quantitatively express the time dependence of CGS output on BG—that is CGS is a function of BG. This equation is then converted so as to express BG as a function of CGS. The inverted equation can thus be used to predict the BG level for given CGS output values. In application, the inverted equation is applied to the raw CGS data to produce accurate BG estimates. Given the fact that glucose is a relatively small molecule, it is believed that glucose can diffuse freely through capillary wall, such as blood vessels and adipose tissues. Adipose tissue is highly vascularized; and the interstitial fluid occupies a relatively thin layer between cells. This fact implies that there is no volume element that is very far from a cell surface, nor is it very far from a capillary wall. Therefore, uptake and diffusion of glucose in the interstitial fluid can be assumed to be relatively topologically uniform. The transportation behavior of the IG and BG according to the invention is depicted in For deriving a mathematical diffusion equation, it is assumed that the particular local interstitial environment in question does not significantly contribute to the development of the BG/time curve, therefore, the time dependence of BG level, BG(t), evolves independently, and can be treated as an exogenous variable in the system. This assumption is particularly safe especially in hyperinsulinemic clamp situations where the BG level is mostly controlled by the IV infusion of dextrose. It is further assumed that the uptake of glucose follows either an IG independent path, or one described by Michaelis-Menten kinetics, as expressed respectively in equations 1a and 1b:
In the above equations, α is the uptake of glucose per unit time per unit volume. It is noted that equations 1a and 1b describe the explicit time dependence. Other variables, such as insulin levels, exercise, and the like, which may directly effect the glucose uptake BG and IG, are not excluded from the equations. Km is a constant in equation 1b; and it does not introduce additional fitable parameters. In practice, Km can take those published values for the activity of GLUI-4, asset forth in “
where β is the permeability of the capillary wall to glucose. Since there are no other clear sources or sinks of glucose in the interstitial fluid, the net change of glucose can be derived by adding equations 1a and 1b, which can be expressed as the following equations 3a and 3b, wherein equation 3a corresponds to the uniform uptake diffusion model, and equation 3b corresponds to the Michaelis-Menten kinetic model.
Equation 3a is an ordinary differential equation that has analytical solutions; while equation 3b is a non-linear differential equation of the second type Abel equation that requires numerical simulation. The analytical solution for equation 3a is expressed in the following equation 4:
By assuming that α and β are constant over time, equation 4 can be reduced to the following equation 6 using the Delta-tau notation as presented in the following equation 5:
By removing the higher order derivative terms of BG(t) in equation 6, equation 6 can then be reduced to a form to which a Kalman recursion analysis based on Kalman filtering/smoothing technique are applicable. An exemplary of such technique is set forth in “
A state-space model for CGS output including the BG evolution can be expressed as the following equation 10.
A state-space model for CGS output including the BG evolution and linear projection can be expressed as the following equation 11.
Inversion of equations 3a and 3b can be similarly performed given the estimates of consumption, permeability, IG, and the rate of change of IG. The inversed equations of 3a and 3b are respectively presented as the following equations 12a and 12b:
Equations 12a and 12b indicate that the use of CGS becomes important to provide accurate estimates of the rate of change of IG. Presentations of the inversed equations 12a and 12b are also possible, which are expressed as following equations 13 and 14.
It is noted that the observation in the above described model is the function of BG(t) that defined in equation 7. If one accepts a polynomial smoothing/interpolation formula to describe the course of BG(t), then it, too can be linearly inverted, as shown in the attached Appendix C. The above described mathematical model and equations can then be applied to the CGS readings for remedying the physiology time lag between IG(t) and BG(t) by predicting BG levels using CGS outputs. An exemplary procedure of the invention is presented in the flow chart of Referring to Referring to Given the pre-processed CGS data (or directly the raw CGS data without the above pre-processing), the rate of change (time derivative) of CGS output is calculated at step The process as described with reference to The above described process for remedying the physiology time lag between BG and CGS output has been evaluated on data acquired during a study performed at the University of Virginia General Clinical Research Center (GCRC), which was an “add-on” project to ongoing NIH research grant (RO1 DK 51562, Principal Investigator Boris Kovatchev). The add-on study was sponsored by Abbott Diabetes Care (P.I. William Clarke) to perform a direct comparison between two CGS: Abbott Navigator™ and Minimed CGMS. The development and testing of the model were among the objectives of the add-on study. Sixteen subjects with TIDM (11 male, 5 female, age 42 with standard deviation (SD) of 3 years, duration of diabetes 20 year with SD of 3 years. Informed consent was obtained from each. Subjects were admitted to the General Clinical Research Center in the evening prior to the study following a physical examination. A CGS system, the Freestyle Navigator™ was applied to each subject for approximately 12 hours prior to the initiation of the data recording, in accordance with the manufacturer's instructions and calibrated as recommended. All systems were inserted in the abdomen. No BG reference vs. CGS comparisons was made until the next morning. Study Protocol is defined as that identical hyperinsulinemic clamps were performed on two consecutive days: On each day the hyperinsulinemic clamp used constant insulin infusion rate of 40 mU/kg/min and variable glucose infusion rate to achieve and maintain BG levels at approximately 110 mg/dl. Subsequently, the glucose infusion rate was reduced to permit a controlled decline in BG levels of approximately 1 mg/dl/min until the BG level reached 40 mg/dl. The euglycemic clamp portion of the study varied in length from 70 to 210 minutes, while the duration of the BG reduction procedure ranged from 30 to 60 minutes. Arterialized blood was sampled every 5 minutes and reference BG levels were determined using a Beckman Glucose Analyzer (Beckman Instruments, Inc, Fullerton, Calif.). Freestyle Navigator™ glucose readings were recorded each minute and were synchronized with reference BG with a precision of 30 seconds. Reference and Navigator™ rates and direction of BG change were calculated at five-minute intervals. This procedure resulted in 29 clamp data sets for the 16 participants in the study. Numerical analysis was conducted using R 2.1.1, which is an open-source free programming language and suite for statistic analysis (http//www.r-project.org). Beyond the base packages, the “odesolve,” “fields,” and “dsel” packages and their dependencies from CRAN repositories were used Microsoft Excel was used to produce graphs. Equation 12a was applied to the unfiltered Navigator™ raw data with parameters found via nonlinear least squares. Each data run begins at the start of descent into hypoglycemia. Table 1 presents a summary of the results for all 29 clamp events. It can be seen that the average RMS error of the Navigator™ was reduced more than 3-folds; and the % RMS error was reduced more that 5-folds. In addition, the correlation between reference and sensor BG was improved by the model:
As discussed above, the accuracy of the CGS output can be improved by an example of the invention through an improved calibration method. Alternatively, CGS accuracy can also be improved by remedying the physiology time lag between the BG and IG. In another example of the invention, the above two correction methods can be combined so as to further improve the CGS accuracy. Referring to Table 2 shows the accuracy improvement using the methods according to example of the invention by comparing the CGS outputs obtained from the methods of the invention and the CGS outputs in a typical CGS in the art without employing the methods of the invention. The CGS outputs obtained from the typical CGS in the art are referenced in “Evaluating the accuracy of continuous glucose monitoring sensors: continuous glucose error grid analysis illustrated by therasense freestyle navigator data,” by B Kovatchev, L Gonder Frederick, D Cox, and W Clarke,
Panel A of Table 2 presents the continuous glucose error-grid analysis (CG-EGA) of the accuracy of Minimed CGMS™ during the clamp study described above, stratified by hypoglycemia and euglycemia. The clinically accurate sensor readings were 50.0% during hypoglycemia and 96.4% during euglycemia. The large difference between these percentages is primarily due to the more demanding clinical accuracy standards. For hypoglycemia events: while for steady euglycemic state there is a large clinical tolerance for sensor errors. During clinically dangerous and rapidly developing conditions, such as hypoglycemia, the sensor is desired to meet higher standards in order to provide accurate feedback for appropriate and timely treatment decision. The CG-EGA reflects this distinction. Further, the MAE and the mean absolute percent error (MAPE) are included in Table 1 and stratified by BG range as well. The panel B in Table 2 presents the CG-EGA, MAE and MAPE of a sensor re-calibrated by two reference BGs that are 30 mg/dl apart (e.g. differential d is 30 mg/dl), which is a clinically reasonable differential in the studied BG range. It can be seen that the percent of CG-EGA accurate readings increases from 50% to 86.7%, while MAE is reduced from 27.9 to 10.9 mg/dl during hypoglycemia. Improvement in MAE and MAPE is observed during euglycemia as well. The panel C of Table 2 presents the CG-EGA, MAE and MAPE of the SIG vs. BG estimated after sensor re-calibration. It can be seen that the “accuracy” of SIG following BG fluctuation is high—nearly 100%, which signifies an excellent theoretical limit for potential sensor accuracy. Examples of the invention can be implemented in many ways. For example, it can be implemented as a functional member of a continuous glucose sensor, or can be implemented as a standalone module associated with the continuous glucose sensor. In either instance, examples of the invention can be implemented as a set of program codes of software installed in a computing device, or a set of computer-executable codes of a hardware device coupled to the continuous glucose sensor. Regardless the implementation media, example of the invention can be associated with individual continuous glucose sensors for improving the accuracy of the individual glucose sensors. Alternatively, examples of the invention can be implemented in such a way that the real-time CGS data, along with related errors and error correction parameters and data, can be transmitted to an error processing center. The transmission may or may not be with the glucose data. In this way, a generalized continuous glucose sensing system can be established. In addition to the CGS outputs with improved accuracy, errors, parameters for accuracy improvements, and any accuracy related information can be delivered, such as to computer As discussed earlier, examples of the invention can also be implemented in a standalone computing device associated with the target continuous glucose sensors. An exemplary computing device in which examples of the invention can be implemented is schematically illustrated in Referring to Additionally, device The device may also contain one or more communications connections It will be appreciated by those of skill in the art that a new and useful method for improving accuracy of continuous glucose sensing devices and system using the same have been discussed herein. In view of the many possible embodiments to which the principles of this invention may be applied, however, it should be recognized that the embodiments described herein with respect to the drawing figures are meant to be illustrative only and should not be taken as limiting the scope of invention. Those of skill in the art will recognize that the illustrated embodiments can be modified in arrangement and detail without departing from the spirit of the invention. Therefore, the invention as described herein contemplates all such embodiments as may come within the scope of the following claims and equivalents thereof.
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If we assume that our data follows a system of the form of A1.1, where x is a hidden system state vector, y is an observation vector, w
The Kalman filter is a two step process. First, one-step-ahead predictions are made. Second, a correction to that prediction is made based upon new measurements. Let us introduce to notation of {circumflex over (X)}
The diagonal operator can be written as in A2.9, which reduces A2.8 to A2.10. Furthermore, by again concatenating the first q+1 column vectors we can form the desired matrix A2.1 in A 2.11.
Finally, by inserting A2.11 into A2.3 we arrive at A 2.12, a linear system which can be solved by methods such as QR decomposition.
1 GetNewBGPreds<-function(Nav,times,par) {
#third, create arrays to hold estimates of IG, and dot IG
In addition to the real-time implementation, one can introduce an artificial delay by extending the time over which each spline is interpolated in line 12 from “(i−4):(i)” to “(i−4):(i+4).” In purely retrospective analysis, one can use the more simple form given next. 19 GetOldBGPreds<-function(Nav,times,par) {
#apply calibration parameter #create a smoothing spline regression over all the data
#return predictions Finally, if the full Michaelis-Menton form of the equation 3 is desired, then one can simply replaces “par [[1]]” with “par[[1]]*IG/(126.+IG)” in lines 17 and 27. R contains many other smoothing and interpolation facilities, and the use of sreg, smooth spline, and lm can be substituted with many of them, although the input/output format is frequently different. Patent Citations
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