Is anybody aware of methods that are appropriate to modify the values of a time series to account for factors that are known to artificially inflate or deflate the measurement.

An example of the problem would be to see how the intelligence of a student changes over time.

We could test the student each week and their percentage score could be tracked, fomring a time series. Tests never truly capture intelligence so this time series will be noisy due to process error, and other factors effecting the test scores.

However there are certain factors we could measure that influence the test score, such as hours spent studying, or hours of sleep the night before taking the test. We could also measure the correlations between the values of these factors and test scores for a large population of students.

Is there any technique that could be used to smooth this time series so that it is more representative of the students true inner intelligence? So for example if the student studied abnormally higher for one week their test score would be decreased in the filter output to reflect that their measured intelligence (test percentage) was artificially inflated due to the extra time studying.

I recently learned about Kalman filters as they are good for dealing with noisy time-series, however I dont see any way to design the Kalman filter to make it 'smooth out' the effect of these external factors.

Is there any known modification to the Kalman Filter algorithm that would suit this problem, or maybe some other sort of bayesian filter?

  • $\begingroup$ I am also aware of bayesian networks to solve a similar problem, but i dont think they can be applied here as I am unsure as to how they could be applied to a time series. $\endgroup$ Mar 21 at 19:49


What you mention is a well-known—and well-treated—problem in econometrics and social and behavioral sciences, and the solutions fall under (dynamic) structural equation modeling (SEM) and panel data analysis and further (latent) growth curve modeling.

The problem is, based on longitudinal data (multiple measurements over time) you want to make inferences about the relationships between stable (trait-like) and time-varying (state-like) latent variables, and some exogenous variables (time-invariant such as gender; or time-varying such as sleep) and some endogenous variables (e.g., time spent studying, since it can be affected by previous test score). You, then, would want to distinguish between within-person and between-person differences.

A note on terminology:

For the cases where you have a few measurements of many people ($T << N$), the data is often called panel data, and if the number of measurement occasions is large, especially compared to the number of persons ($T >> N$), then you say you have a intensive longitudinal data (ILD) and use time series analysis.

What models can I use?

There are dozens of models capable of doing that, and you can choose one depending on your assumptions and research questions. I believe a latent trait-state-occasion model would be an appropriate model for your problem, and I highly suggest reading this paper (available on SciHub):

Cole, D. A., Martin, N. C., & Steiger, J. H. (2005). Empirical and Conceptual Problems With Longitudinal Trait-State Models: Introducing a Trait-State-Occasion Model. Psychological Methods, 10(1), 3–20. doi:10.1037/1082-989x.10.1.3

I briefly summarize three possible modeling strategies explained in the paper:

1. State–trait–error model

In this model, your items (questions in IQ test) administered at different times (or "waves"), and you assume that the differences in responses at times $t$ are caused by person-specific trait factor $T$ and occasion-specific latent variables $O_t$. The latent variables $O_t$ can be replaced by any endogenous (and perhaps, exogenous) variable. If you assume time-invariant model, you may fix the autoregression effects to be equal $\beta_1, \beta_2, \ldots, \beta_{J-1} = \beta$, and you can set them to zero to eliminate any temporal relation.

enter image description here

2. Latent state–trait autoregressive model

In this model, you assume the items are measuring students' current ability at answering the administered test ($S_t$). The $S_t$'s are affected by previous abilities of the student ($S_{t-1}$), contextual/occasion-specific factors ($O_t$), and an underlying time-invariant trait ($T$) that equally influences the student's ability at all times.

enter image description here

3. Trait–state–occasion model

This is quite similar to the previous model, except that you assume here that the temporal dependencies between states $S_t$ (ability at different times) lie in temporal dependence between occasions (e.g., the student studying/sleep have some sort of autoregressive nature).

enter image description here

Practical matters

Which model should I choose?

The choice of the model is heavily determined by your hypothesized relationships. In case you do not have specific hypotheses to test, you can "let the data speak for itself" and favor one model over the other based on (and choose/identify the "best-fitting model") based on SEM fit indices or information criteria (e.g., AICc or BIC).

How to use these models?

By fitting these models, you get parameter estimates for factor loadings, (co)variances, and AR effects. You can then feed the (preferred) models with students' responses to estimate their latent variables, which are now decomposed into their stable trait and time-specific factors (occasion/context and/or abilities).

Further reading

You may want to look into the following books for longitudinal structural equation models:

Little, T. D. (2013). Longitudinal structural equation modelingemphasized text. Guilford press.

Newsom, J. T. (2015). Longitudinal structural equation modeling: A comprehensive introduction. Routledge.

You should be extremely careful if you want to make inferences about cross-lagged relationships (e.g., if you want to see if sleep and studying affect each other over time). This is not what you want to do here, but this is an extremely important matter, and I suggest reading this paper on it:

Hamaker, E. L., Kuiper, R. M., & Grasman, R. P. (2015). A critique of the cross-lagged panel model. Psychological methods, 20(1), 102.

In case you have intensive longitudinal data (with too many measurements; not feasible for intelligence testing), you may want to use dynamic structural equation modeling (DSEM). See this paper:

Asparouhov, T., Hamaker, E. L., & Muthén, B. (2018). Dynamic structural equation models. Structural Equation Modeling: A Multidisciplinary Journal, 25(3), 359-388.

  • $\begingroup$ Thanks for the answer, So these models are PGMs that I could try and implement in a framework like pyMC3 or similar? They can be used to work out a student's latent intelligence $T$ that is most likley to be responsible for the sequence of test scores $Y$ conditioned on the latent factors $O$ such as hours studied. I am assuming that my endogenous variables are independent of the previous test score (i.e a student wont do bad on a test then try to study more in an attempt to do better in the future) I am unsure about what the epsilon and sigma's are? I am trying to access this paper now. $\endgroup$ Mar 23 at 20:28
  • $\begingroup$ SEM and PGMs are somehow similar when modeling the (conditional) [in]dependency between variables. However, they diverge in estimation techniques and assumptions. I wouldn't try to estimate SEM with PGM techniques at all. (I do not work with Python, but there is a great R package for SEM called lavaan. You can rather easily implement this model in it.) $\delta$s and $\varepsilon$s are error residuals for manifest variables $Y$ and endogenous variables $O$. In SEM you estimate your model based on the covariance structure of your manifest variables and restrictions on latent/error variances. $\endgroup$
    – ManuHaq
    Mar 24 at 9:54
  • $\begingroup$ You would need some basics of SEM for all these (incl. Cole et al.'s 2005 paper) to make sense. The books I mentioned are on libgen, but will cost you some time to have a grasp of SEM. I would instead suggest Michael Clark's online free book, Graphical & Latent Variable Modeling which starts with PGMs (that you may want to skip) and goes into SEM (with hands-on, rather detailed lavaan examples) at a pretty decent depth. $\endgroup$
    – ManuHaq
    Mar 24 at 10:04

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