A comparison of nonlinear filtering approaches in the context of an HIV model

In this paper three different filtering methods, the Extended Kalman Filter (EKF), the Gauss-Hermite Filter (GHF), and the Unscented Kalman Filter (UKF), are compared for state-only and coupled state and parameter estimation when used with log state variables of a model of the immunologic response to the human immunodeficiency virus (HIV) in individuals. The filters are implemented to estimate model states as well as model parameters from simulated noisy data, and are compared in terms of estimation accuracy and computational time. Numerical experiments reveal that the GHF is the most computationally expensive algorithm, while the EKF is the least expensive one. In addition, computational experiments suggest that there is little difference in the estimation accuracy between the UKF and GHF. When measurements are taken as frequently as every week to two weeks, the EKF is the superior filter. When measurements are further apart, the UKF is the best choice in the problem under investigation.


Introduction
The modeling of the physiologic and immunologic response to HIV infection in humans is generating a substantial amount of research effort, and significant progress has been made in the treatment of HIV-infected patients.One of the most prevalent treatment strategies for acutely infected HIV patients is highly active anti-retroviral therapy (HAART) which utilizes two or more drugs.However, despite the success of HAART, patient-specific optimal schemes for its use need to be considered.Grave side effects of taking drugs, viral mutations and the high cost of drugs all motivate a substantial research effort in this area.
Open-loop control, a control that is pre-computed for a given dynamic model and initial conditions, is one technique that has been employed in a number of works (e.g., [2,7]) to design treatment therapies for HIV patients.However, this technique may be inadequate for reasons such as poor patient adherence, increasing drug resistance due to virus mutations, and drug side effects as noted.Another approach that has been used to design dynamic HIV treatment therapies utilizes feedback controls such as those based on the state dependent Riccati equation (SDRE) approach used in [6] and on receding horizon control methodology in [8].A fundamental characteristic of feedback control is that it depends on the current state of the system.Hence, this method can be used to design adaptive treatment schedules for HIV patients based on the patient's current status (e.g., current CD4+ T cell count and viral load).Because HIV modeling generally involves partial observations and noisy measurements from combined compartments, the method by which the state is obtained at each sampling time is of special concern, and an efficient estimation technique is needed to develop a successful implementation of feedback control.
State and parameter estimation and the development of associated adaptive feedback control schemes in the setting investigated here offer challenges different from those in many engineering applications where often high frequency uncensored observations are often the norm.In addition to low frequency sampling in longitudinal data sets (data points are often very expensive both in financial costs of associated assays as well as in emotional/physical costs to patients), the data itself is frequently censored below due to limitations on assays in discriminating low values.Thus a number of challenges include development of filters for state and parameter estimation in the context of low frequency sampling and partial state observations that are censored.Successful efforts in these areas must be combined with feedback control of nonlinear dynamics which are often only approximate for patient response.Here we discuss a first step in development of a needed methodology by attempting to discover an appropriate filtering approach to use with partial state uncensored observations that are collected rather infrequently by usual engineering standards.
The model used in this paper was developed and validated as a predictive tool in [4], wherein two types of target cells, along with their corresponding infected states, free virus, and immune effector cells (CTL) are included as states in the model.Fitting with clinical data demonstrated that this model provides reasonable fits to numerous patient longitudinal data sets and has impressive predictive capability when comparing model simulations with parameters based on estimation using only half of the longitudinal observations.For computational ease in our presentation here, without loss of generality, we omit the noninfectious virus component from the model in [4].This will not affect the dynamics of this model as it is completely decoupled from all the other compartments.The model we use is with a state vector initial condition ( 1 (0), 2 (0), * 1 (0), * 2 (0), (0), (0)) .Here the state variables are 1 , the uninfected CD4+ T-cells; 2 , the uninfected target cells of a second kind; * 1 , the infected CD4+ T-cells; * 2 , the infected target cells of a second kind; , the infectious virus; and , the immune effectors.The units for 1 , 2 , * 1 , * 2 and are cells/ l-blood, and the unit for is RNA copies/ml-plasma.The factors 10 3 are introduced to convert between microliter ( l) and milliliter (ml) scales, preserving the units from some of the earlier published papers [1].The particular target cells of a second kind are not specified, and (according to [4]) might be related to macrophages or brain cells or inactivated memory.The model also includes terms that model drug efficacy.The control term 1 represents the efficacy of a Reverse Transcriptase Inhibitor (RTI).For a more detailed description of model parameters and rationale for the model (1) we refer the reader to the article [4].While there are somewhat improved and generalized versions of this model [5], the version we have chosen to use here is representative and more than adequate for demonstration of the behavior of the classes of filters we wish to compare.
We observe that our model ( 1) is nonlinear and hence we must explore nonlinear estimation methods.The Extended Kalman Filter (EKF) was used in [9] for state estimation for the HIV model of [4] (also without the component and without the scaling factor).Unlike the nonlinear least squares approach this technique does not require all the data at once.That is, this methodology only uses data as it is received and thus it can be used "online" to estimate the parameters in an adaptive approach.However, the EKF was found in [9] to have difficulty with state estimation even when the time span between measurements is only five days.This motivates a need to consider alternative filtering methods.Accordingly, we introduce the Gauss-Hermite Filter (GHF) and Unscented Kalman Filter (UKF) as possible alternatives.Even though all three of theses methods are based on a Gaussian assumption (that is, the posterior distribution can be approximated by a Gaussian distribution), the process by which this approximation is obtained is different, and therefore, leads to different performance.It has been demonstrated (e.g., [10,15]) that the performance of both the GHF and UKF are superior to the EKF in numerous other nonlinear problems.
Even though this paper might be considered an extension of the efforts in [9], we have made a significant modification here in applying these filters to our problem.Instead of using the model directly as in [9], we applied the filters to a log-scaled version of our model.This was done because log-transformation is a standard technique to render the observations more nearly normally distributed.In addition, from a numerical analysis point of view, by using a log-transformed system one can resolve a problem of states becoming unrealistically negative due to round-off errors.More importantly, all these filters are derived for systems where the states are defined on ℝ ; this may lead to some trouble when one apples the filters to a system such as (1) where the states are only defined in ℝ + ; log-scaling mitigates this potential difficulty.
The remainder of this paper is organized as follows.In Section 2 we give a brief introduction of the EKF, UKF and GHF and their application to a general nonlinear continuous system with discrete observations.In Section 3, the filters are implemented to estimate model states as well as model parameters from simulated noisy data, and compared in terms of estimation accuracy and computational time.We conclude the paper in Section 4 with some remarks and suggestions for future efforts.

Filter Descriptions
In this section we will use a capital italic letter to denote a random variable or random vector unless otherwise indicated, and use the corresponding small letter to denote its realization.A capital Roman letter is used to denote a non-random matrix.In addition, we may occasionally use the following shorthand notations to ease the presentation: for ( ),

+1
for P ( +1 ), etc.We will use ℰ{⋅} to denote the expectation of a random variable or vector.
Next we give a brief introduction of the three filters that we use, the EKF, GHF and UKF, and their application in the context of a general nonlinear continuous system with discrete observations taken according to the measurement equation = ℎ( ( ), ) + , +1 > ≥ 0 , = 1, 2, 3, . . . .

Let
= { : ≤ } denote the information available by observing the process up to time , where is a realization of .The filtering problem is to find the "best" estimate ˆ ( ) of ( ) based on .The "best" is understood in the sense of minimum mean-squared error (MMSE) for each fixed where the superscript in ˆ is determined by the subscript in .The MMSE estimate ˆ ( ) of ( ) based on is the conditional mean and the covariance matrix of this estimate, denoted by P ( ), is given by Hence, to determine the best estimate ˆ ( ) of ( ) based on as well as the conditional covariance matrix, we need to find the conditional probability density function ( , | ) of ( ) at any time .However, this distribution ( , | ), which satisfies the Fokker-Planck equation between any observation period < < +1 , and then is updated with a measurement due to a new observation at +1 using Bayes formula, in general, can not be obtained in closed form.Hence, in many applications it is conventionally assumed that the distribution is Gaussian so that the distribution is completely parameterized by just the mean and covariance; this is exactly the assumption made in the EKF, GHF and UKF.
All three filters, the EKF, UKF and GHF, utilize a "predictor-corrector" implementation.Given current state estimate ˆ = ˆ ( ) and covariance matrix estimate P = P ( ) at time , the filter first predicts the states at time +1 by using only model dynamics to obtain the predicted quantities ˆ +1 = ˆ ( +1 ) and P +1 = P ( +1 ).When new data +1 is available at time +1 , a linear update rule is specified to obtain the updated quantities ˆ +1 +1 and P +1 +1 , where the weights are chosen to minimize the mean squared error of the estimate.

The Extended Kalman Filter
In the EKF, the expected values occurring in time and measurement updates are computed by linearization of the system function and measurement function ℎ at the current state estimate.
To start the filter algorithm, we set = 0 initially and set ˆ 0 0 = ˆ 0 and P 0 0 = P 0 for given initial conditions ˆ 0 , P 0 .We then compute the predicted state ˆ +1 = ˆ ( +1 ), by solving the ordinary differential equations ˆ ( ) = (ˆ ( ), ), We simultaneously compute the predicted error covariance matrix P +1 = P ( +1 ), by solving the ordinary differential equations P ( ) = P ( ) Here and are the parameters of the noise in the stochastic process ( ) as described above.Once this prediction step is complete, we can incorporate the new data information at time +1 .We compute the updated state and updated error covariance matrix with the observation +1 by computing solutions to the equations )), and respectively, where G +1 is defined by Recall that R is the covariance matrix in the noise for the observation process.We thus have the updated state values and covariance matrix at time +1 .We then move to the next time step, so we increment by 1 and return to the predictor step.
The EKF has been successfully applied to numerous nonlinear filtering problems in the engineering and physical sciences.However, its performance can be extremely poor when nonlinearities between observations become severe.Moreover, it is not a good choice for some application problems where Jacobian matrices are difficult to calculate (or may not even exist).For more discussions on the EKF, the interested reader can consult [11] among numerous other texts.

The Unscented Kalman Filter
Instead of linearization of the system function and measurement function ℎ at current state estimates as required by the EKF, in order to compute the expected values occurring in time and measurement updates, the UKF is based on the principle that a discrete distribution composed of a set of deterministically chosen sampled points with the corresponding weights can be used to approximate the standard normal distribution.It is founded on intuition: "it is easier to approximate a probability density function than it is to approximate an arbitrary nonlinear function".Given a function ˜ ( ), the UKF entails the approximation of integrals by Here = 2 + 1.The values for the sampling points and the weights are defined by where is the th unit vector in ℝ , and ∈ ℝ can be any number providing + ∕ = 0.
The variable provides an extra degree of freedom to "fine tune" the higher order moments of the approximation and can be used to reduce the overall predication error.
Once this prediction step is complete, we can incorporate the new data information at time +1 to update the state and covariance matrix.First, we compute the factorization P +1 = S S as before and set ˜ = S + ˆ +1 .Then compute Following the update step, we have the updated state values and covariance matrix at time +1 .We then move to the next time step, so we increment by 1 and return to the predictor step above.
We note that the UKF is easier to implement than the EKF as it does not require the calculation of Jacobian matrices.However, the UKF may require some "fine tuning" in order to prevent the propagation of a non-positive definite covariance matrix for a state vector dimension higher than three [3].For a more extensive treatment of the UKF, see [12,13,14].

The Gauss-Hermite Filter
Similar to the UKF approach, the GHF does not linearize the system function and measurement function at the current state estimate to obtain the expectation values occurring in time and measurement updates.However, the expectation values in the GHF are calculated by using a Gaussian-Hermite quadrature rule instead of by approximating the standard normal distribution as used in the UKF.Given a function ˜ ( ), the quadrature rule for expected values is expressed by where is the number of quadrature points used in a one-dimensional quadrature rule.The quadrature points and their corresponding weights are calculated as follows.Let A ∈ ℝ × be a symmetric tridiagonal matrix with zero diagonal elements and its ( , + 1)th element defined by A , +1 = √ 2 , = 1, 2, . . ., − 1.Then A has eigenvalues, which are denoted by , = 1, 2, . . ., .Let be the normalized eigenvector of A corresponding to the eigenvalue , = 1, 2, . . ., .Then the quadrature points and its corresponding weights are given by = where 1 is the first element of normalized eigenvector .Hence, in order to evaluate the integral in (5) we need -point function evaluations.To have a notation consistent with (3), we may rewrite the right side of (5) into one sum notation as in ( 3), and express it as where = , = ( 1 , 2 , . . ., ) and = 1 2 ⋅ ⋅ ⋅ , = 1, 2, . . ., .
The GHF was also originally designed for a discrete system with discrete observations.Hence, in order to apply the GHF to a continuous system with discrete observations, we again need to discretize the continuous model, and we will use the same approximation scheme as we used in the UKF.The algorithm for the GHF is exactly the same as that for the UKF except for the choice of and its corresponding weight in evaluating the integral.
Like the UHF, the GHF does not require the calculation of Jacobian matrices.However, the obvious disadvantage of the GHF is that the required number of points to evaluate the integral scales geometrically with the number of dimensions.For more detailed information on the GHF, the interested reader is referred to [10].

Numerical Simulations
As mentioned in the introduction, we will apply the EKF, GHF and UKF to the log-scaled HIV system instead of the original system.We first rewrite the HIV model (1) as vector system ˙ = (¯ ; ¯ ), where ¯ = ( 1 , 2 , * 1 , * 2 , , ) , and ¯ is the vector for model parameters given by ¯ , , , , , , , ).
We point out that there are some model parameters that are patient specific in that the values of these parameters may vary from patient to patient.Hence, we do not know the values of these parameters in advance.In this effort we also wish to test the performance of these filters in adaptively estimating the model parameters as well as model states.To do this, we append the parameters to the model (8) as additional states.
Simulated data sets were generated in the following manner.We used (8) or (10) with set to the "true" values in Table 1 below to generate realizations (we used = 20 for the results reported here) of the state vector ( ).We used these realizations in (9) along with different realizations for to generate realizations of .This provides longitudinal data sets for the state vector and corresponding observations.We used each of these in the filter algorithms to generate estimators or realizations of the filters.These were then compared using the realizations by taking the average (over the differences) root mean square (RMS) of the difference between data and estimated states.In order to demonstrate the differences in filter state estimation based on frequency of observations, we used the simulated data with measurements taken 1, 7, 14 and 28 days apart, respectively.The filter results are then examined and compared using the average RMS error for each observed state and each model compartment.The average RMS error for each model compartment is defined by which is based on the different simulation runs.Here the subscript denotes the th component of the "true" noisy state vector ( ) (the data) and the corresponding state estimate ˆ ( ), and the superscript denotes the th simulation run.The average RMS error for each observed state is defined by where the subscript denotes the th component of the observation function ℎ( ( )), = 1, 2, 3. Since real time feedback is also important, we examine the computational expense of each filter.

Generation of Simulated Data
In order to test our three filters, we needed to create simulated data based on a "true" state value.To do this, we used Euler method approximation (the method used in the UKF) to numerically solve model equations ( 8) through a specified time span (0 to 364 days in our simulation runs) with time mesh size chosen to be 0.001, and then used (9) to obtain simulated data at the relevant observation times (every 1, 7, 14, or 28 days) with constant noise covariance matrix R = diag([0.1 2 , 0.25 2 , 0.07 2 ]) .
The values of model parameters ("true values") used to generate the simulated data are given in Table 1.The initial states were set to be 0 = (log 10 (800), log 10 (3.198), −4, −6, 1, −2) .We simulated a treatment schedule for 364 days, with treatment off for the first 30 days, then alternating on for 15 days and off for 45 days, and then off treatment for the last 34 days of the year.This is shown in Figure 1, and also appears at the bottom of every graph of

Examination of Simulation Results
We will examine the observed states average RMS error defined in (12) as well as the model compartments average RMS error defined in (11), each when measurement data is taken 1 day, 7 days, 14 days or 28 days apart, respectively.Each of the simulated data sets for a given sampling frequency was taken from the same "true" data sets, but taken at the specified measurement frequency.
In order to test the filters under "poor" starting conditions with relatively uncertainty, we set ˆ 0 0 = 0.6 0 and P 0 0 = 0.01I for all the simulation results presented in this section.In both the UKF and GHF, we set = 0.005 in the discretization scheme (4).In some of our efforts, we also included parameter estimation as part of the filtering process; the results are illustrated in Section 3.2.2.The parameters that are estimated were given initial values of 90% of the value in Table 1, and the variance of each of these parameters was set to be 0.005 multiplied by the value of the parameter squared, and then these additional values were added to obtain an extended P matrix.

State Estimation
In this section, numerical results are obtained by applying the filtering algorithms to model (8) with measurements taken 1 day, 7 days, 14 days and 28 days apart.The plots in the left column of Figures 2 and 5 are for the average RMS error for each observed state defined by (12), and the ones in the right column are plotted for the estimated values versus the true values of the total CD4+ T-cell count, viral load level and immune effector T-cell count obtained by applying the filters to a typical data set.In addition, we have included the plots in Figures 3 and 6 for the average RMS error of each model compartment defined by (11).Figures 4 and 7 contain the plots for the estimated values versus the true values of 1 , 2 , * 1 , * 2 , and obtained by applying the filters to a typical data set.
When measurements are taken 1 day apart, we see from Figures 2 and 3 that the EKF performs much better than either the UKF or the GHF in terms of estimation accuracy.In addition, we see that there is no detectable difference in the estimation accuracy between the UKF and GHF as their average RMS errors lie on top of each other.Also note that the UKF and GHF appear to oscillate, whereas the EKF is much smoother.We believe that this is due to the discretization of the continuous model during the predictor step of the UKF and GHF.From Figures 2 and 4 we see that all algorithms provide very good estimates of both observed states and model compartments.Figures 5, 6 and 7 illustrate the results obtained in the case when measurements are taken 7 days apart.Even though Figures 5 and 6 reveal that the UKF and GHF appear to adjust to the data slightly faster than the EKF, the average RMS errors obtained by the EKF for both observed states and model compartments are much smaller than those obtained by the UKF and GHF after day 100.Hence, for this measurement frequency, the EKF still performs much better than the UKF or the GHF.Also we observe that there is still no detectable difference in the estimation accuracy between the GHF and UKF.In addition, Figures 5 and  7 demonstrate that that all filters still perform well in estimating the observed states and each model compartment.
We have also applied the filters to the case with measurements taken 14 days and 28 days apart (sample results are given in the Appendix), and obtained the same conclusions as those for measurements taken 1 day and 7 days apart.That is, for the case with only model states estimated, the EKF performs much better than both the UKF and GHF in terms of estimation accuracy for all the measurement frequencies that we have investigated, and there is little difference in the estimation accuracy between the UKF and GHF.
Figure 8 depicts the results for the average RMS error of each observed state with several different measurement frequencies, where the plots in the left column, middle column and right column are the results obtained by using the EKF, UKF and GHF, respectively.From this figure, we see that the measurement frequency has no significant effect on the performance of the filters after day 100 (that is, after the filters adjust to the data).This is probably because for this case the nonlinearity is not so severe.obtained by applying the filters to a typical data set.The results presented in this section are typical for different combinations of parameters that we estimated in a number of simulation studies.

State and Parameter Estimation
When measurements are taken 1 day apart, we see in Figures 9 and 10 that the average RMS errors for the observed states, the model compartments and the estimated parameters obtained by using the EKF are smaller than those obtained by using the UKF and GHF, and there is little difference in the estimation accuracy between the UKF and GHF.In addition, Figures 9 and 11 show that all algorithms perform very well in estimating the observed         Figures 12, 13 and 14 are for the case when measurements are taken 7 days apart.From Figures 12 and 13 we see that there is still not much difference in the estimation accuracy between the UKF and GHF.We also see that the performance of the EKF here is a little bit worse than its performance in the case with measurement 1 day apart, but overall it still performs better than the UKF and GHF.In addition, Figures 12 and 14 suggest that all the algorithms still perform well in estimating the observed states as well as each model compartment and the parameters.When measurements are taken 14 days apart, from the estimation results illustrated in Figures 15,16 and 17, we begin to observe some ambiguous results.Even though all the algorithms are still capable of effectively estimating the observed states as well as each model compartment, the estimation accuracy is lower than those in the former cases with sampling at 1 and 7 days.In addition, the performance of the EKF deteriorates much further than that of either the UKF or GHF, and it is difficult to ascertain which filter performs the best in this case.Figures 15 and 16 also suggest that that the UKF performs slightly better than the GHF in terms of estimation accuracy.Figures 18, 19 and 20 illustrate the results obtained in the case when measurements are taken 28 days apart.From Figures 18 and 19, we observe that the performances of the UKF and GHF are consistently better than that of the EKF in terms of estimation accuracy, and there is little difference in the estimation accuracy between the UKF and GHF.In addition, we see from Figures 18 and 20 that all the filters are still capable of obtaining reasonable estimates for observed states, model compartments and estimated parameters.   Figure 21 depicts the results for the average RMS error of each observed state with different measurement frequencies, where the plots in the left column, middle column and right column are the results obtained by using the EKF, UKF and GHF, respectively.From this figure, we see that the measurement frequency has a significant effect on the estimation accuracy of the EKF, but has little effect on the GHF and UKF.Hence, with parameter estimation included in the filtering process, the behavior of the filters is different from that we observed in Section 3.2.1.This is likely because the nonlinearities in this new problem (state plus parameter estimation) become severe as the measurement frequency decreases, which renders the EKF less effective than either the GHF and UKF.This type of decline in EKF performance is observed in numerous other nonlinear problems.

Filtering Summary
The simulation results in Sections 3.2.1 and 3.2.2suggest that all the filters are capable of obtaining reasonable estimates for both observed states and model compartments even with measurements taken 28 days apart.This conclusion was not observed in [9], where the EKF was reportedly unable to accurately estimate the states even when the measurements were taken five days apart.This may be because in our effort here the filters were applied to the log scaled version of our model instead of the original system as was used in [9].
Note that in order to apply the GHF and UKF we need first to discretize the model.To investigate the effect of the discretization time step on the performance of the UKF and GHF, we used several different values for .We learned that if we choose too small, then we see more oscillation, which is especially obvious for the case when the solution is in the equilibrium state.In addition, a smaller will dramatically increase the computational time, which is undesirable.On the other hand, if is taken too large, then we may obtain a non-positive definite covariance matrix which is not factorable, and the worst scenario is that it does not capture the true dynamics.As a result, when using the GHF and UKF it is important to take care when choosing to obtain the best performance while still being able to actually run the filter computations.
The simulation results in Sections 3.2.1 and 3.2.2suggest there is little difference in the estimation accuracy between the UKF and GHF.However, the experiments also reveal that the GHF is the most computational expensive algorithm as every integral (expectation) estimation in this algorithm takes point-evaluations, while only 2 + 1 point-evaluations are used in the UKF.This makes the GHF computationally infeasible when dealing with estimation of a large number of parameters.As a result, the GHF is not a preferred choice for this particular problem.
As for the EKF and UKF, the experiments revealed that they have comparable computation times when run under similar conditions on the same computer.When only estimating model states, the simulation results in Section 3.2.1 suggest that the EKF performs much better than the UKF in terms of estimation accuracy in all the measurement frequencies that we investigated.When estimating parameters as well as model states, the results in Section 3.2.2reveal that the EKF still performs better than the UKF with measurements 1 day and 7 days apart, has comparable performance as the UKF with measurements 14 days apart, but performs worse than the UKF with measurements 28 days apart.Note that parameter estimation is essential to this particular problem as the values of a number of the parameters cannot be obtained from laboratory or clinical standards, in part because values of some of the parameters are very much patient-specific.Hence, the decision on which of these two filters should be used relies on the conclusions in Section 3.2.2, that is, on the measurement frequency.Another consideration is that the UKF requires "fine tuning" in order to prevent the propagation of non-positive definite covariance matrices, while there is no such problem with the EKF.Based on all this, we conclude that the EKF is our best choice when the measurement frequency is sufficiently high (such as sampling every one or two weeks), but when the measurement frequency is as low as 4 weeks apart then the UKF is the better choice.

Concluding Remarks
As a first step in designing adaptive treatment therapies for HIV patients, in this paper we have applied the Extended Kalman Filter, the Unscented Kalman Filter and the Gauss-Hermite Filter to the log-scaled version of an HIV model, and then compared the performance of these filters in terms of estimation accuracy and computational time for estimation of both states and parameters.Numerical results suggest that the EKF is our best choice when the measurement are taken as frequently as every week or two, but when the measurement frequency is as low as 4 weeks then the UKF is more likely the best choice.Because monthly sampling in clinical longitudinal settings is a likely scenario, our future efforts on development of adaptive schemes will be focused on the UKF.
In the clinical data available to us at present, the measurements of the viral load data are censored because the assay can accurately detect only down to some lower limit (400 copies/ml-plasma for a standard assay and 50 copies/ml-plasma for an ultra-sensitive assay).Hence, one of our immediate future efforts is to investigate how one effectively applies these filters in examples with censored data.

Figure 2 :
Figure 2: State estimation with measurements 1 day apart.(left): average RMS error for each observed state; (right): the estimated values versus the true values for the total CD4+ T-cell count, viral load level and immune effector T-cell count obtained by applying the filters to a typical data set.

Figure 3 :
Figure 3: State estimation with measurements 1 day apart.Average RMS error for each model compartment.

Figure 4 :
Figure 4: State estimation with measurements 1 day apart.The estimated values versus the true values for each model compartment obtained by applying the filters to a typical data set.

Figures 9 ,
Figures 9,12,15 and 18 were obtained by estimating parameters 1 and as well as model states (i.e., the filters are applied to model(10)), where the figures in the left column are for the average RMS error of each observed state, and those in the right column are plotted for the estimated values versus the true values of the total CD4+ T-cell count, viral load level and immune effector T-cell count obtained by applying the filters to a typical data set.In addition, we have included the plots in Figures10, 13, 16 and 19 for the average RMS error of each model compartment as well as each estimated parameter.Figures11, 14, 17 and 20 contain the plots for the estimated values versus the true values of 1 , 2 , * 1 , * 2 , and obtained by applying the filters to a typical data set.The results presented in this section are typical for different combinations of parameters that we estimated in a number of simulation studies.

Figure 5 :
Figure 5: State estimation with measurements 7 days apart.(left): average RMS error for each observed state; (right): the estimated values versus the true values for the total CD4+ T-cell count, viral load level and immune effector T-cell count obtained by applying the filters to a typical data set.

Figure 6 :
Figure 6: State estimation with measurements 7 days apart.Average RMS error for each model compartment.

Figure 7 :
Figure 7: State estimation with measurements 7 days apart.The estimated values versus the true values for each model compartment obtained by applying the filters to a typical data set.

Figure 8 :
Figure 8: State estimation with different measurement frequencies for the observed states: scaled CD4+ T-cells (top row), scaled viral load (middle row), and immune effectors (bottom row); average RMS errors for each observed state as obtained using the EKF (left column), UKF (middle column) and GHF (right column).

Figure 9 :
Figure 9: State estimation as well as parameter estimation with measurements 1 day apart.(left column): average RMS error for each observed state; (right column): the estimated values versus the true values for the total CD4+ T-cell count, viral load level and immune effector T-cell count obtained by applying the filters to a typical data set.

Figure 10 :
Figure 10: State estimation as well as parameter estimation with measurements 1 day apart.Average RMS error for each compartment and each estimated parameter.

Figure 11 :
Figure 11: State estimation as well as parameter estimation with measurements 1 day apart.The estimated values versus the true values for each state compartment and each estimated parameter obtained by applying the filters to a typical data set.

Figure 12 :
Figure 12: State estimation as well as parameter estimation with measurements 7 days apart.(left column): average RMS error for each observed state; (right column): the estimated values versus the true values for the total CD4+ T-cell count, viral load level and immune effector T-cell count obtained by applying the filters to a typical data set.

Figure 13 :
Figure 13: State estimation as well as parameter estimation with measurements 7 days apart.Average RMS error for each compartment and each estimated parameter.

Figure 14 :
Figure 14: State estimation as well as parameter estimation with measurements 7 days apart.The estimated values versus the true values for each state compartment and each estimated parameter obtained by applying the filters to a typical data set.

Figure 15 :
Figure 15: State estimation as well as parameter estimation with measurements 14 days apart.(left column): average RMS error for each observed state; (right column): the estimated values versus the true values for the total CD4+ T-cell count, viral load level and immune effector T-cell count obtained by applying the filters to a typical data set.

Figure 16 :
Figure 16: State estimation as well as parameter estimation with measurements 14 days apart.Average RMS error for each compartment and each estimated parameter.

Figure 17 :
Figure 17: State estimation as well as parameter estimation with measurements 14 days apart.The estimated values versus the true values for each state compartment and each estimated parameter obtained by applying the filters to a typical data set.

Figure 18 :
Figure 18: State estimation as well as parameter estimation with measurements 28 days apart.(left column): average RMS error for each observed state; (right column): the estimated values versus the true values for the total CD4+ T-cell count, viral load level and immune effector T-cell count obtained by applying the filters to a typical data set.

Figure 20 :
Figure 20: State estimation as well as parameter estimation with measurements 28 days apart.The estimated values versus the true values for each state compartment and each estimated parameter obtained by applying the filters to a typical data set.

Table 1 :
The treatment schedule used in the experiments.statesor parameters that are run on data with treatment.The data is thus fairly dynamic.Values of parameters in the HIV model.