# Which estimator is used for which greyest options?

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Marius Weber on 3 Dec 2021
Answered: arushi on 30 Apr 2024
Hello Matlab folk,
this is my first entry in this forum, if I do something wrong, don't hesitate to tell me.
I am wondering how i can determine, whether to use a Least Squares Estimator or a Maximum Likelihood Estimator. Is it the "Focus" setting or the "OutputWeight" in greyestOptions?
In literature, estimators are often characterized in Output-, Filter-, or Equation-Error. Which setting combinations correspond to those estimators?
Thank you very much,
Marius
Marius Weber on 5 Dec 2021
I also noticed, that the settings for "Focus" and "DisturbanceModel" do not have a profound influence on the solution. Only if I choose the combination Focus=Prediction and DisturbanceModel=estimate the solution is a lot different compared to other combinations.
Any help to why that is would be appreciated.

arushi on 30 Apr 2024
Hi Marius,
In the context of system identification using MATLAB's System Identification Toolbox, choosing between a Least Squares Estimator (LSE) and a Maximum Likelihood Estimator (MLE) involves understanding the underlying statistical assumptions and the nature of the data. The choice isn't directly made through a single option like "Focus" or "OutputWeight" in `greyestOptions`, but rather through the selection of the estimation method and the configuration of various options that influence the estimation process.
Estimation Methods:
Least Squares Estimator (LSE): Primarily used when the noise is assumed to be white and the errors are normally distributed with constant variance. LSE is focused on minimizing the sum of the squared differences between the observed and predicted values.
Maximum Likelihood Estimator (MLE): Provides estimates by maximizing the likelihood function, assuming a particular distribution for the errors. MLE is more flexible than LSE as it can accommodate different noise distributions and is generally more efficient in the presence of non-Gaussian noise.
MATLAB System Identification Toolbox Settings:
Focus:The "Focus" option specifies what aspect of the data the estimation process should prioritize (e.g., simulation accuracy or prediction accuracy). While it doesn't directly select LSE or MLE, it influences the optimization criteria and can affect the choice indirectly by prioritizing certain aspects of the model performance.
Simulation: Prioritizes fitting the output of the model to the measured output.
Prediction: Prioritizes the model's ability to predict future outputs.
DisturbanceModel: This option specifies how to treat disturbances or noise in the model. Adjusting this can indirectly influence whether your approach leans more towards LSE or MLE, especially in how noise characteristics are modeled and estimated.
OutputWeight: This option allows specifying weights for the output error, influencing the optimization process. It doesn't directly choose between LSE and MLE but can adjust the estimation to focus on specific parts of the output.
Making the Choice:
Statistical Properties of Your Data: If you have prior knowledge about the error distribution, this can guide your choice. Use MLE if you expect non-Gaussian noise or if the error variance is not constant.
Model Purpose: If your primary goal is prediction, MLE might offer more flexibility in adjusting for noise characteristics. For straightforward fitting with less concern about the noise structure, LSE might suffice.
Experimentation: Often, the choice isn't clear-cut, and experimenting with different settings and observing their impact on model performance is necessary.
In summary, the decision between LSE and MLE in the context of MATLAB's System Identification Toolbox isn't made through a single setting but through a combination of choices regarding model structure, focus, and disturbance modeling. Understanding the data's statistical properties and your modeling goals is key to making the best choice.
Hope this helps.