Least-Squares Analysis - Science topic

Shaban Juma Ally, that is why one should use weighted least squares (WLS) regression. When the coefficient of heteroscedasticity is zero - which should not happen - then WLS regression becomes OLS regression.

It is not possible to answer that question without more information. First, you should say how many observations you have. A regression with 17 observations is questionable but it depends on the number of explanatory variables. Since these observations correspond to above the threshold, I fear that they are outliers, hence we are in the worst situation. Did you consider a transformation on the dependent variable? That would perhaps improve the situation provided the relationship with the other variables allows for it. I would say that each of the two regimes should have enough observations.

Marius Ole Johansen The threshold least square regression model by Hansen (2000) divides the series into two regimes endogenously. The Regime above the threshold and below the threshold, and then regress both regimes individually by OLS. this method also involve bootstrap replication. In my case the regime above the threshold only remain with 17 number of observation. Does it creates loss of degree of freedom issue in the data?

Counted panel analysis, also known as count data panel analysis, involves statistical techniques used to analyze panel data where the dependent variable represents counts or the number of occurrences of an event. This type of data is common in various fields, such as economics, and social sciences, where the interest lies in modeling the frequency of events over time for different entities (such as individuals, firms, or countries).

If the provided data includes OR, RR, HR, 95% CI, MD, or SMD, you can use se = (ln(upper CI) - ln(lower CI)) / (2 * 1.96) formula to calculate the standard error (SE).

The elasticity of y with respect to x is defined as the percentage change in y resulting from a one-percent change in x, holding all else constant. In the context of your regression model, where you have regressed the growth rate of y (which can be thought of as the percentage change in y) against the growth rate of x (the percentage change in x), the estimated coefficient on the growth rate of x is an estimate of this elasticity directly.

Here's why: If you run the following regression:

Δ%y=a+b(Δ%x)+ϵ

where Δ%y is the growth rate of y (dependent variable), Δ%x is the growth rate of x (independent variable), a is the constant term, b is the slope coefficient, and ϵ is the error term, the coefficient b represents the change in Δ%y for a one-unit change in Δ%x. Because Δ%y and Δ%x are already in percentage terms, the coefficient b is the elasticity of y with respect to x.

So, if you have estimated the coefficient b from this regression, you have already estimated the elasticity. There is no need to recover or transform the coefficient further; the estimated coefficient b is the elasticity of y with respect to x.

It's important to note that this interpretation assumes that the relationship between y and x is log-linear, meaning the natural logarithm of y is a linear function of the natural logarithm of x, and the model is correctly specified without omitted variable bias or other issues that could affect the estimator's consistency.

I suggest that you employ an ARCH/ARMAX method where the ARCH component models the conditional variance of the error term, the ARMA component models the autoregressive nature of your data, and the X component models the effects of the exogenous variables.

Here is a link to a recent application of the method:

To perform two-stage least squares (2SLS) methods, you can follow these steps:

The coefficient on the endogenous variable in the second stage regression is the 2SLS estimate of the effect of the exogenous variables on the endogenous variable.

Here is an example of how to perform 2SLS methods in R:

In summary, the method to obtain the Pseudo R2 value in Stata depends on the type of regression model used. To obtain the Pseudo R2 value in Stata, there are different methods depending on the type of regression model used. Here are some ways to obtain Pseudo R2 in Stata: For a feasible generalized least square (XTGLS) regression model, the Pseudo R2 value can be obtained by running the command "xtgls Y X1 X2 Xn, panels( ...)" in Stata. For a generalized linear model (GLM) regression, Nagelkerke's R2 can be obtained using the "roctab" command to create a table of predicted probabilities and then calculate the Pseudo R2 value. For a random-effects probit model using xtprobit, the Pseudo R2 value is not directly available in the output. However, it can be calculated using the formula 1 - loglikelihood of the model/loglikelihood of the null model. If a Stata command does not supply an R-squared value, a Pseudo R2 value can be calculated using different approximations or analogues to R-squared. These approximations are often labeled "pseudo" and can be found in the literature of the field. I am able to add....Best of luck

Thank you for your question. I have searched the web for information about the Diagonally Weighted Least Squares (DWLS) and Unweighted Least Squares (ULS) estimators, and I have found some relevant sources that may help you with your decision.

One of the factors that you should consider when choosing between DWLS and ULS is the sample size. According to Forero et al. (2009)1, DWLS tends to perform better than ULS when the sample size is small (less than 200), but ULS tends to perform better than DWLS when the sample size is large (more than 1000). Since your sample size is 658, it falls in the intermediate range, where both methods may provide similar results.

Another factor that you should consider is the degree of non-normality of your data. According to Finney and DiStefano (2006), DWLS is more robust to non-normality than ULS, especially when the data are highly skewed or kurtotic. However, ULS may be more efficient than DWLS when the data are moderately non-normal or close to normal. Since your data have slight negative skewness and kurtosis, it may not be a serious violation of the ULS assumptions.

A third factor that you should consider is the model fit and parameter estimates. According to Forero et al. (2009)1, both methods provide accurate and similar results overall, but ULS tends to provide more accurate and less variable parameter estimates, as well as more precise standard errors and better coverage rates. However, DWLS has higher convergence rates than ULS, which means that it is less likely to encounter numerical problems or estimation errors.

Based on these factors, it seems that both DWLS and ULS are reasonable choices for your data and model, but ULS may have some advantages over DWLS in terms of efficiency and accuracy. However, you should also check the sensitivity of your results to different estimation methods, and compare them with other criteria such as theoretical plausibility, parsimony, and interpretability.

I hope this answer helps you with your analysis. If you need more information, you can refer to the sources that I have cited below.

1: Factor analysis with ordinal indicators: A Monte Carlo study comparing DWLS and ULS estimation by Carlos G. Forero, Alberto Maydeu-Olivares & David Gallardo-Pujol in British Journal of Mathematical and Statistical Psychology (2009)

: Non-normal and categorical data in structural equation modeling by Sara J. Finney & Christine DiStefano in Structural equation modeling: A second course (2006)

Good luck

Least squares method is, as I have been experienced, more convenient than the method of moments.

In the method of moments, first you have to derive the theoretical moments up to order p if the regression equation consists of p parameters in order to obtain p equations and then to solve the p equations substituting the values of the moments obtained from the sample in the equations.

Derivation of theoretical moments is more difficult than to set up normal equations (required in least squares method) since it depends upon the nature of the probability distribution followed by the parent population of the sample.

thank you, issue resolved....

Theory & examples:

Why do you need autocorrelation? Because you think there is a periodicity in your data?

Hi Jamal,

From your volume-weighted crystallite size (referring to it here as such even if we don't yet trust the value), I assume you have relatively broad peaks and the broadening contribution from strain is likely not obvious when comparing fits with / without strain terms refined. Since you don't have complementary characterization of your samples, separating size and strain contributions is critical. As Andreas said, the LVol and e0 macros in TOPAS assume isotropic size and strain broadening (e.g., angle-dependent rather than hkl-dependent) and these assumptions do not have to be correct, especially for small (< 10 nm) nanoparticles.

Are your samples single phase? If so, an approach worth trying is using single-peak fitting to extract the peak breadths directly to make a Williamson-Hall plot. While this is a qualitative approach, here it will give you a sense for 1) if your sample has strain (proportional to slope of WH fit line) and 2) if there is some hkl-dependence indicating anisotropic size and / or strain broadening. You can also test the Stephens model for anisotropic strain broadening (Stephens 1999) and Ectors model for anisotropic size broadening (Ectors 2015, Ectors 2017) to see if these better describe your peak broadening. The anisotropic size macros are not in the TOPAS GUI and have to be implemented in launch mode - details are on the TOPAS wiki.

Single-peak fitting comes with some large warnings:

Correlation between your background and peak tails will be much higher when using single-peaks and it would be worthwhile to constrain the peak intensities (integrated area) to scale as they would when using a structural model. Likewise, the peak positions should be constrained or refined with some scale so that they are related as they would be in a Rietveld refinement. Without these constraints, your fit will necessarily improve but it won't be clear if you have resolved an issue with the peak shape, intensity, or position.

I would also suggest comparing your Rietveld refinement to a Pawley / Le Bail fit where intensity is freely refined for each peak while their shape and position is still constrained to the isotropic size-strain models and a unit cell, respectively. If you don't see an improvement, this points to a peak shape model issue over an issue with the structural model.

Best,

Adam

Thanks a lot for your answer. I trying to rmake Rietveld refinement using fullproof software, but I need to convert the intensity in a XRD difractogram from counts per second to only counts. By the way, how can convert an XY file to .dat extension?

Iterated Weighted Least-Squares Regression is more an algorithm to minimize a selected objective function (cost function) than a statistical tool per se. So the real question is not if IWLSR can be applied to your data, but in first which objective function you want to minimize and which algorithms are available for that.

The recursive least squares algorithm (RLS) is the recursive application of the well-known least squares (LS) regression algorithm, so that each new data point is taken in account to modify (correct) a previous estimate of the parameters from some linear (or linearized) correlation thought to model the observed system. The method allows for the dynamical application of LS to time series acquired in real-time. As with LS, there may be several correlation equations and a set of dependent (observed) variables. For the recursive least squares algorithm with forgetting factor (RLS-FF), acquired data is weighted according to its age, with increased weight given to the most recent data.

A particularly clear introduction to RLS is found at: Karl J. Åström, Björn Wittenmark, "Computer-Controlled Systems: Theory and Design", Prentice-Hall, 3^rd ed., 1997.

Years ago, while investigating adaptive control and energetic optimization of aerobic fermenters, I have applied the RLS-FF algorithm to estimate the parameters from the K_La correlation, used to predict the O₂ gas-liquid mass-transfer, hence giving increased weight to most recent data. Estimates were improved by imposing sinusoidal disturbance to air flow and agitation speed (manipulated variables). The power dissipated by agitation was accessed by a torque meter (pilot plant). The proposed (adaptive) control algorithm compared favourably with PID. Simulations assessed the effect of numerically generated white Gaussian noise (2-sigma truncated) and of first order delay. This investigation was reported at (MSc Thesis):

Maybe “R” code is better?

The numbers of lines and columns of the matrix representing the image you want to obtain must be integers and their product must equal the length of the vector you want to convert. In your example the length of the vector is n=55929. You want to obtain a squared matrix with square(n) lines and columns, but in your example m=square(n)=2.364931288642442e+02, which is not an integer. If we chose a number of 3 columns and a number of 18643 lines we will obtain the following Matlab code which works.

clear variables

load('Exe42022.mat');

n = length(b);

m=sqrt(n);

c=reshape(b,18643,3);

figure,

imagesc(c),

colormap(gray),

axis off;

Please try this code.

Use particle swarm optimization (PSO) or other metaheuristic algorithms for finding the optimal parameters of a system of ordinary differential equations (SODE) by minimizing the error (e.g. mean absolute percentage error or MAPE) between data and the model's solution. In Matlab, you can use ode45 package to solve the SODE. The step is:

0. Suppose you want to estimate the parameters in region interval I.

1. Generate m random vectors X in I, these vectors act as the initial guess of parameters.

2. Solve the SODE using ode45 for each vectors value. Then calculate the MAPE between SODE's solution with the data.

3. Find the minimum of the obtained MAPE. The vector vapue X* which gives the minimum of MAPE will act as the initial optimal parameters.

4. Next, you use the updating formula in the PSO to update the previous vectors X (you can find this in many books or article).

5. Redo the process 2-4 using the new vectors X.

6. At the desire stopping criteria, for example maximum iteration, you will get the estimated parameters value of your model.

Feel free to contact me for more detail explanation. I can help you.

The question seems incorrect. We needn't find x and y.

We have {(xi, yi = sin(xi))}, i=1,...,N and seek (a_n, ..., a_0) for a polynomial function

f(x) = a_n * x^n + ... + a_0.

Least Square minimization gives

A=

( Sum(xi^n) Sum(xi^(n-1)) ... Sum(xi^0);

Sum(xi^(n+1)) Sum(xi^(n-2)) ... Sum(xi^1);

.............................................

Sum(xi^(2*n)) Sum(xi^(2*n-1)) ... Sum(xi^n);

);

b = (Sum(yi); Sum(xi*yi);); ... Sum(xi^n*yi) );

At last,

(a_n a_(n-1) ... a_0) = A^(-1)*b

In standard regression non-normality of the residuals is not a big issue ; outliers , that is really discrepant values , potentially are a much bigger problem.

the method of least squares is used to find the linear model of two variables.

1. The new Material will have a cif file, if you make this cif file. Try e.g. VESTA.

2. Depending on the doping level you will likely be able to use the structure model of the pure material (it is the structure model, the information of which is contained in the cif file).

It seems you have two paired samples. The p-values for the individual samples are not at all informative for you, and also the individual 95% CIs are not helpful. All the relevant information is in the two point estimates and their variance.

Assuming the ratios (2.41 and 2.17) are approximately log-normal distributed, a one-sample test on the logarithms gives p = 0.04. Using a Gamma model woth log-link I get a p-value of 0.03. Althogh the sample size is tiny, the variance is small enough to conclude an up-regulation.

Nabeel Hameed Al-Saati SVM is a supervised machine learning technique that may be used for both classification and regression.

Support Vector Machine may also be utilized as a regression approach while retaining all of the algorithm's fundamental characteristics (maximal margin). With a few minor exceptions, the Support Vector Regression (SVR) utilizes the same classification concepts as the SVM.

Dear Michel Crimeni,

The rank of the measurement matrix H is less than or equal to the rank of the system (process) matrix F.

For more detail and information about this subject, I suggest you see links on the topic.

https://www.cs.cmu.edu/~motionplanning/papers/sbp_papers/kalman/kleeman_understanding_kalman.pdf

Best regards

Hi Shyam Gupta ,

The sample size for A and B is equally 11, which is too small (n=11). Sometimes, it is not enough for regression analysis.

if you have any steps to use fixed and random effect in the meta-analysis, please help me

The initial value of deformation resistance, s0,

see the following document

Dear Martin Klvana,

I have seen the trajectory. Ligand slowly detached from the protein and after few steps it reattached itself to some other place (other than the binding site).

Dear, here I suggest this article. Kindly go through it. Might be helpful to you :)

I don't think you can have any model comparison in JASP or jamovi. Both programs use the lavaan package in the background to run CFA/SEM models. If you look up the lavaan package in R, the lavTestLRT function can be used for comparing multiple models.

I agree with Xingyu Zhou.

Don't do any transformation!

This is very easy with this software:

Often, the log-normal distribution is a sensible assumption for percentages that are not bounded at 100%. Are there any arguments against this in your case?

It makes a huge difference for interactions if you analyze the data on the original scale or on the log scale. If is is sensible to assume relative effects (a given impulse will change the response depending on the "base value" of the response, so absolute changes will differ when the base values differ), interactions seen on the original scale are highly misleading in terms of the functional interpretation (i.e. when you aim to understand the interplay of the factors).

I don't know the engineering, but it does sound like you would be throwing away information if you did use ordinary least squares (OLS) regression. General least squares (GLS) regression means including nonzero off-diagonal (but symmetric) parts of matrices. The information lost by OLS regression is with regard to heteroscedasticity and autocorrelation. (If you just include heteroscedasticity, which typically should be there, you have weighted least squares (WLS) regression, which I highly endorse for finite population survey statistics.)

So I just wondered if what you were talking about might be missing off-diagonal information if you used OLS regression. But I could be completely off topic.

Anyway, it definitely sounds like you have two methods, where one ignores information that the other does not ignore. Does the 'least squares fitting method' use any information that the 'inner product method' ignores? If not it seems you have a good case for the method that uses all relevant information. That is at least analogous to using GLS regression rather than OLS regression when there is heteroscedasticity and/or autocorrelation. So it seems you have a case.

You said the following:

"I made some simulations to compare these two methods, and found that both methods give almost the same results."

I wonder if that is because the additional information was not very substantial, or was the difference so small that it is only round-off calculation differences, and both methods, somehow, do include all the relevant information?

Cheers - (no Dr) Jim Knaub

You can check out the work by Saeid Haghighatshoar and Giuseppe Caire, such as https://arxiv.org/pdf/2110.03324

As your education data do not refer to the education (spending) itself, but to the educational level (which was gained by past education), it is clear that you try to analyse the effect of this level on income. Therefore, you have to find a specification of an income equation with the educational level as one of the explaining variables. You must also include family characteristics, because you use consumption as a proxy for income, and consumption will of course be dependent on the number and age etc. of the household members. These characteristics are likely to influence income, too. This influence will, of course, be weaker, and can hardly be separated. Before an econometric estimate you should look at diagrams of the variables chosen and calculate correlations between variables (also between explaining ones). If you have enough data, I recommend, to choose household types and to analyse the relations for every type separately. If your data include time series, you can, of course, use aggregate (National Accounts) data as explaining variables.

For analysing the influence of income on the education level, you would need very long time series for a fixed set of households, and even then it would likely not be a promising project.

Two questions: The education level is an individual characteristic, how is it transformed into a household date (level of the main earner or some average of the (earning) members)?

To José-Ignacio: What is the reason to include a link between distribution and growth in this study?

You may focus on topological manifolds optimization. Very beautiful work on this area is here : https://www.manopt.org/downloads.html

Videos of Generalized Least Square regression (GLS) in SPSS

Check on Youtube there you found step by step how to do the GLS in SPSS

Thanks Andrew Paul McKenzie Pegman . I have studied many aspects of estimating uncertainties from the perspective of GUM, as well as from Monte Carlo. Please check my latest paper at

where I address most of JCGM 100 and 101 examples from both approaches. However, I have not found coherent documentation on weighted least squares neither from GUM, nor from MCM, apart from the fact that in my work in Physical Metrology the use of weights in least squares is not common, so I do not know in depth its use.

Thanks for raising such an interesting question. Probably some people would be interested in the following special issue.

URL: https://www.mdpi.com/journal/mathematics/special_issues/Modern_Geometric_Modeling_Theory_Applications_II

Video: https://www.youtube.com/watch?v=aomv_9-FGkU

The scope of the Special Issue includes but is not limited to original research works within the subject of geometric modeling and its applications in engineering, arts, physics, biology, medicine, computer graphics, architecture, etc., as well as theoretical mathematics and geometry which can be applied to problems of geometric modeling. For this Special Issue, we plan to accept the following types of manuscripts:

@Maya Yes it can be used. For detailed information refer to the following link https://towardsdatascience.com/when-and-how-to-use-weighted-least-squares-wls-models-a68808b1a89d

Two functions fitted simultaneously to one dataset? Do I understand correctly? What is the objective function?

Hi Baraaah

SmartPLS offers the following fit measures:

SRMR

NFI

Chi²

RMS_theta

For the approximate fit indices such as SRMR and NFI, you may directly look at the outcomes of a PLS or PLSc model estimation (i.e., the results report) and these criteria's values with a certain threshold (e.g., SRMR < 0.08 and NFI > 0.90).

For the exact fit measures you may consider the inference statistics for an assessment. Therefore, you need to run the bootstrap procedure and to use the “complete bootstrap” option in SmartPLS.

Please check out the following links

Dear Kehinde, since you are making use of PMG/panel ARDL, check for cross-sectional dependence. If there is no cross-sectional dependence, proceed to check for cointegration using the pedroni. Kao, etc. and if there is cointegration, go on to interpret your results.

However, if there is cross-sectional dependence you will need to utilize the second generation test of unit root, the westerlun cointegration test which are available in STATA.

Maybe you wan to consider the recursive least squares algorithm (RLS). RLS is the recursive application of the well-known least squares (LS) regression algorithm, so that each new data point is taken in account to modify (correct) a previous estimate of the parameters from some linear (or linearized) correlation thought to model the observed system. The method allows for the dynamical application of LS to time series acquired in real-time. As with LS, there may be several correlation equations with the corresponding set of dependent (observed) variables. For the recursive least squares algorithm with forgetting factor (RLS-FF), adquired data is weighted according to its age, with increased weight given to the most recent data.

Years ago, while investigating adaptive control and energetic optimization of aerobic fermenters, I have applied the RLS-FF algorithm to estimate the parameters from the K_La correlation, used to predict the O₂ gas-liquid mass-transfer, hence giving increased weight to most recent data. Estimates were improved by imposing sinusoidal disturbance to air flow and agitation speed (manipulated variables). The proposed (adaptive) control algorithm compared favourably with PID. Simulations assessed the effect of numerically generated white Gaussian noise (2-sigma truncated) and of first order delay. This investigation was reported at (MSc Thesis):

Dear Delaram Rabiei,

In addition to what is proposed above, i suggest you to see links and attached files on topic.

Best regards

1. In the normal lingo, very little. In robust regression, the standard errors are calculated differently, to make them "robust" against heteroscedasticity (or clustering).

2. You interpret the regression coefficient in the same manner: the average change in y given a one-unit increase in x.

3. I have written three textbooks on regression, but I have never heard the term "proper correlation" before. You do regression to find out if there is a (linear) association between x and y and, if so, to find out how large this assocoation is.

4. I seldom use R, but I know for a fact that both plain vanilla and robust regression is are straightforward to estimate in R.

Good luck :-)

The recursive least squares algorithm (RLS) which allows for (real-time) dynamical application of least squares (LS) regression to a time series of time-stamped continuously acquired data points. As with LS, there may be several correlation equations and a set of dependent (observed) variables. RLS is the recursive application of the well-known LS regression algorithm, so that each new data point is taken in account to modify (correct) a previous estimate of the parameters from some linear (or linearized) correlation thought to model the observed system. For RLS with forgetting factor (RLS-FF), adquired data is weighted according to its age, with increased weight given to the most recent data. This is often convenient for adaptive control and/or real-time optimization purposes. A particularly clear introduction to RLS/RLS-FF is found at: Karl J. Åström, Björn Wittenmark, "Computer-Controlled Systems: Theory and Design", Prentice-Hall, 3^rd ed., 1997.

Application example ― While investigating adaptive control and energetic optimization of aerobic fermenters, I have applied the RLS algorithm with forgetting factor (RLS-FF) to estimate the parameters from the K_La correlation, used to predict the O₂ gas-liquid mass-transfer, while giving increased weight to most recent data. Estimates were improved by imposing sinusoidal disturbance to air flow and agitation speed (manipulated variables). The proposed (adaptive) control algorithm compared favourably with PID. Simulations assessed the effect of numerically generated white Gaussian noise (2-sigma truncated) and of first order delay. This investigation was reported at (MSc Thesis):

I. You may consider to directly evaluate the slopes from the actual data. This procedure often requires a somewhat large thread of data and some previous data smoothing. It is, however, independent from correlation models.

II. Alternatively, you may rely on some suitable correlation model, possibly nonlinear. In that case, you may possibly try to first adopt a linearized correlation, modified from the preferred nonlinear correlation. The linearized correlation and the corresponding linearized plot are often quite convenient to qualitatively evidentiate scattering around the trendline and to emphasize major effects and the physical meaning of the correlation parameters. Estimates of the parameters first derived by least squares after the linearized correlation can possibly be later refined by iterative nonlinear least-squares regression; to find unbiased least-squares estimates for the preferred nonlinear correlation. The nonlinear correlation can then, in principle, be differentiated as required. It may be advisable to compare both the linearized and the nonlinear correlations.

Often the nonlinear correlation to be fitted to data can be 'somehow' linearized, as a first stage. Linearized correlation and the corresponding linearized plot are often quite convenient to qualitatively evidentiate scattering around the trendline and to emphasize major effects and the physical meaning of the correlation parameters. Estimates of the parameters derived by least squares after the linearized correlation (modified from a former nonlinear correlation) can possibly be refined by iterative nonlinear least-squares regression; to find unbiased least-squares estimates for the original correlation. It may be advisable to compare both the linearized and the nonlinear correlations.

The following RG discussions seemingly can be found of some interest, concerning to your query:

Maybe you can consider the recursive least squares algorithm (RLS) which allows for (real-time) dynamical application of least squares (LS) regression to a time series of time-stamped continuously acquired data points. As with LS, there may be several correlation equations and a set of dependent (observed) variables. RLS is the recursive application of the well-known LS regression algorithm, so that each new data point is taken in account to modify (correct) a previous estimate of the parameters from some linear (or linearized) correlation thought to model the observed system. For RLS with forgetting factor (RLS-FF), adquired data is weighted according to its age, with increased weight given to the most recent data. This is often convenient for adaptive control and/or real-time optimization purposes. A particularly clear introduction to RLS/RLS-FF is found at: Karl J. Åström, Björn Wittenmark, "Computer-Controlled Systems: Theory and Design", Prentice-Hall, 3^rd ed., 1997.

Application example for a similar linearized correlation ― the K_La correlation, used to predict the O₂ gas-liquid mass-transfer ― In the signalled work (MSc Thesis), while investigating adaptive control and energetic optimization of aerobic fermenters, I have applied the RLS-FF algorithm to estimate the parameters from that correlation. Estimates were improved by imposing sinusoidal disturbance to air flow and agitation speed (manipulated variables). The power dissipated by agitation was accessed by a torque meter. The proposed (adaptive) control algorithm compared favourably with PID. This investigation was reported at (MSc Thesis):

The recursive least squares algorithm (RLS) allows for (real-time) dynamical application of least squares (LS) regression to a time series of time-stamped continuously acquired data points. As with LS, there may be several correlation equations and a set of dependent (observed) variables. For the recursive least squares algorithm with forgetting factor (RLS-FF), adquired data is weighted according to its age, with increased weight given to the most recent data. This is often convenient for adaptive control and/or real-time optimization.

Application example for a similar linearized correlation ― the K_La correlation, used to predict the O₂ gas-liquid mass-transfer ― In the signalled work (MSc Thesis), while investigating adaptive control and energetic optimization of aerobic fermenters, I have applied the RLS algorithm with forgetting factor (RLS-FF) to estimate the parameters from that correlation. Estimates were improved by imposing sinusoidal disturbance to air flow and agitation speed (manipulated variables). The power dissipated by agitation was accessed by a torque meter. Simulations were carried in Excel 5.0 with Visual Basic for Applications (VBA) macros:

Maybe you can alernatively consider the recursive least squares algorithm (RLS). RLS is the recursive application of the well-known least squares (LS) regression algorithm, so that each new data point is taken in account to modify (correct) a previous estimate of the parameters from some linear (or linearized) correlation thought to model the observed system. The method allows for the dynamical application of LS to time series acquired in real-time. As with LS, there may be several correlation equations and a set of dependent (observed) variables. For the recursive least squares algorithm with forgetting factor (RLS-FF), adquired data is weighted according to its age, with increased weight given to the most recent data. No prior 'training phase' is required.

Years ago, while investigating adaptive control and energetic optimization of aerobic fermenters, I have applied the RLS algorithm with forgetting factor (RLS-FF) to estimate the parameters from the K_La correlation, used to predict the O₂ gas-liquid mass-transfer, hence giving increased weight to most recent data. Estimates were improved by imposing sinusoidal disturbance to air flow and agitation speed (manipulated variables). This investigation was reported at (MSc Thesis):

If you are looking for sparse solution to a linear system of arbitrary size m by n, you can apply the iterative method called " Orthogonal Matching Pursuit (OMP) algorithm ".

Also, Lasso Regression model can be used in such cases, which involves a regularization term in a L1 norm. For sparsity in solution, minimization in L1 or Linfinity norm is sought.

OMP algorithm generates a sparse solution based on minimization in L0 norm ( number of non zero entries in the vector)

Anas Osman GEOCOL and GEOGRID for gross-error detection.

Good luck

Few links are given below. Hope it will be helpful.

he recursive least squares algorithm (RLS) is the recursive application of the well-known least squares (LS) regression algorithm, so that each new data point is taken in account to modify (correct) a previous estimate of the parameters from some linear (or linearized) correlation thought to model the observed system. The method allows for the dynamical application of LS to time series acquired in real-time. As with LS, there may be several correlation equations and a set of dependent (observed) variables. A particularly clear introduction to RLS is found at: Karl J. Åström, Björn Wittenmark, "Computer-Controlled Systems: Theory and Design", Prentice-Hall, 3^rd ed., 1997. Years ago, while investigating adaptive control and energetic optimization of aerobic fermenters, I have applied the RLS algorithm with forgetting factor (RLS-FF) to estimate the parameters from the K_La correlation, used to predict the O₂ gas-liquid mass-transfer, while giving increased weight to most recent data. Estimates were improved by applying sinusoidal excitation to air flow and agitation speed (manipulated variables). This investigation was reported at (MSc Thesis):

Dear Zmour,

It can be better in term of diffusion convection reaction. My opinion is little different, the least-squares method has better control of the streamline derivative than the SUPG.

Ashish

Thank you for all your answers. It is much appreciated :)

I stumbled upon a software called 'Origin Lab' that gets the job done.

No but it assumes some distribution so Google what you did to find out which. I suggest you look at Journal of Data Science11(2013), 575-606, which may be of some interest to you as an alternative approach your problem. Best Wishes, David Booth

I think the results are generated using Matlab code, but to be sure, you can email the first author.

Dear Sindhuraj Mukherjee,

I suggest you to see links and attached files on topic.

Best regards

Hi Jim,

Thanks a lot for your answer. I'm considering continuous data here. Another way to phrase the question is the following: The Pearson correlation coefficient measures the strength of the linear relationship between normally distributed variables, while the Spearman rank correlation method does not assume that the variables are normally distributed. What I'm wondering now is how the robust method relates to these - does it still assume normality, like Pearson?

That's a good suggestion, I could investigate this in some simulations.

Best,

Nicolai