Nonparametric discovery and analysis of learning patterns and autism subgroups from therapeutic data

Abstract

The spectrum nature and heterogeneity within autism spectrum disorders (ASD) pose as a challenge for treatment. Personalisation of syllabus for children with ASD can improve the efficacy of learning by adjusting the number of opportunities and deciding the course of syllabus. We research the data-motivated approach in an attempt to disentangle this heterogeneity for personalisation of syllabus. With the help of technology and a structured syllabus, collecting data while a child with ASD masters the skills is made possible. The performance data collected are, however, growing and contain missing elements based on the pace and the course each child takes while navigating through the syllabus. Bayesian nonparametric methods are known for automatically discovering the number of latent components and their parameters when the model involves higher complexity. We propose a nonparametric Bayesian matrix factorisation model that discovers learning patterns and the way participants associate with them. Our model is built upon the linear Poisson gamma model (LPGM) with an Indian buffet process prior and extended to incorporate data with missing elements. In this paper, for the first time we have presented learning patterns deduced automatically from data mining and machine learning methods using intervention data recorded for over 500 children with ASD. We compare the results with non-negative matrix factorisation and K-means, which being parametric, not only require us to specify the number of learning patterns in advance, but also do not have a principle approach to deal with missing data. The F1 score observed over varying degree of similarity measure (Jaccard Index) suggests that LPGM yields the best outcome. By observing these patterns with additional knowledge regarding the syllabus it may be possible to observe the progress and dynamically modify the syllabus for improved learning.

Appendix

In this section, we present the derivations for the posteriors of $w_{vk}$ and $f_{kn}$, in the scenario where the dataset has missing data. X is our data matrix where the elements $x_{vn}$, corresponding to the number of LUs accumulated by a child n in a task v, are data points. Our objective is to derive the posteriors for the parameters by using the data points $x_{vn}$ that are not missing. The inference of posterior of $w_{vk}$, for a certain value of v, depends on the values $x_{vn}$ for all values of n. Similarly, the inference of $f_{kn}$ for a certain value n depends on values $x_{vn}$ for all values of v. Hence, we consider the data points in two sets $J_{v}$ and $I_{n}$ for each inference, respectively, such that $J_{v}$ contains all the non-missing values from the array $x_{v,1:N}$ and $I_{n}$ contains all of those from $x_{1:V,n}$.

Let us represent the sum $\sum _{i=1}^{K}f_{in}z_{in}w_{vi}$ as $\eta _{i}$, the term including parameters for all values $i=1:K$, $\sum _{i=1,i\ne k}^{K}f_{in}z_{in}w_{vi}$ as $\eta _{-i}$, the term including parameters for all values of i except for k, and $f_{kn}z_{kn}w_{vk}$ as $\eta _{k}$, the term for the condition when i takes the value k.

• The posterior of $w_{vk}$ is

  $$\begin{aligned} p(w_{vk}\mid Z,F,X)\propto & {} p(X\mid Z,F,W)p(w_{vk}\mid \alpha _{0},\beta _{0})\\= & {} \left( \prod _{n\in J_{v}}p(x_{vn}\mid \eta _{i}\right) Gamma (\alpha _{0},\beta _{0})\\= & {} \left( \prod _{n\in J_{v}}\frac{\left( \eta _{i}\right) ^{x_{vn}}e^{-\left( \eta _{i}\right) }}{x_{vn}}\right) \times \frac{\beta _{0}^{\alpha _{0}}}{\Gamma (\alpha _{0})}w_{vk}^{\alpha _{0}-1}e^{-\beta _{0}w_{vk}}\\\propto & {} w_{vk}^{\alpha _{0}-1}e^{-\beta _{0}w_{vk}}\times \prod _{n\in J_{v}}\left( \left( \eta _{i}\right) ^{x_{vn}}e^{-\left( \eta _{i}\right) }\right) \\= & {} w_{vk}^{\alpha _{0}-1}e^{-\beta _{0}w_{vk}}\times \prod _{n\in J_{v}}\left( \left( \eta _{-i}+\eta _{k}\right) ^{x_{vn}}e^{-\left( \eta _{-i}+\eta _{k}\right) }\right) \\\propto & {} w_{vk}^{\alpha _{0}-1}e^{-\beta _{0}w_{vk}}\prod _{n\in J_{v}}\left( \left( \eta _{-i}+\eta _{k}\right) ^{x_{vn}}e^{-\left( \eta _{-i}+\eta _{k}\right) }\right) \end{aligned}$$

  In order to solve the above equation, we take the help of an auxiliary variable. Let us consider that the probability $p(w_{vk})$ is proportional to the unnormalised exponential function $p^{*}(w_{vk})$, where $p^{*}(w_{vk})$ is given by and can be expanded as a binomial function as follows:

  $$\begin{aligned} p^{*}(w_{vk})= & {} \left( \eta _{-i}+\eta _{k}\right) ^{x_{vn}}\\= & {} \sum _{j=0}^{x_{vn}}{x_{vn}\atopwithdelims ()j} \left( \eta _{k}\right) ^{j}\left( \eta _{-i}\right) ^{x_{vn}-j} \end{aligned}$$

  Hence, we have

  $$\begin{aligned} p(w_{vk})\propto & {} \left( \eta _{-i}+\eta _{k}\right) ^{x_{vn}} \end{aligned}$$

  Now let $r_{vn}$ be an auxiliary variable. We aim to define a probability $p(w_{vk},r_{vn})$ proportional to $p^{*}(w_{vk},r_{vn})$ such that $\sum _{r_{vn}}p^{*}(w_{vk},r_{vn})=p^{*}(w_{vk})$. So let $p^{*}(w_{vk},r_{vn})={x_{vn}\atopwithdelims ()r_{vn}} \left( \eta _{k}\right) ^{r_{vn}}\left( \eta _{-i}\right) ^{x_{vn}-r_{vn}}$, where $r_{vn}=\{0,1,2,\ldots ,x_{vn}\}$. Hence, we have

  $$\begin{aligned} \sum _{r_{vn=0}}^{x_{vn}}p^{*}(w_{vk},r_{vn})= & {} \sum _{r_{vn=0}}^{x_{vn}}{x_{vn}\atopwithdelims ()r_{vn}} \left( \eta _{k}\right) ^{r_{vn}}\left( \eta _{-i}\right) ^{x_{vn}-r_{vn}}\\= & {} p^{*}(w_{vk}) \end{aligned}$$

  Additionally, we have

  $$\begin{aligned} p(w_{vk}\mid r_{vn})= & {} \frac{p(w_{vk},r_{vn})}{p(r_{vn})}\\\propto & {} p(w_{vk},r_{vn})\\\propto & {} p^{*}(w_{vk},r_{vn})\\= & {} {x_{vn}\atopwithdelims ()r_{vn}} \left( \eta _{k}\right) ^{r_{vn}}\left( \eta _{-i}\right) ^{x_{vn}-r_{vn}}\\ p(r_{vn}\mid w_{vk})= & {} \frac{p(w_{vk},r_{vn})}{p(w_{vk})}\\\propto & {} \frac{p^{*}(w_{vk},r_{vn})}{p^{*}(w_{vk})}\\= & {} \frac{{x_{vn}\atopwithdelims ()r_{vn}} \left( \eta _{k}\right) ^{r_{vn}}\left( \eta _{-i}\right) ^{x_{vn}-r_{vn}}}{\left( \eta _{-i}+\eta _{k}\right) ^{x_{vn}}}\\= & {} {x_{vn}\atopwithdelims ()r_{vn}} \left( \frac{\eta _{k}}{\eta _{_{-i}}+\eta _{k}}\right) ^{r_{vn}}\left( \frac{\eta _{-i}}{\eta _{_{-i}}+\eta _{k}}\right) ^{x_{vn}-r_{vn}}\\ \end{aligned}$$

  Hence, the conditional distributions have a form of the binomial distribution. After substituting back the values of $\eta _{-i}$and $\eta _{k}$, if we sample $r_{vn}$ from such a distribution we can approximate the binomial expansion as follows:

  $$\begin{aligned} R_{vn}\sim & {} Binomial \left( x_{vn},\frac{z_{kn}f_{kn}w_{vk}}{\sum _{i\ne k}z_{in}f_{in}w_{vi}+z_{kn}f_{kn}w_{vk}}\right) ,~~\forall n\in J_{v} \end{aligned}$$
  $$\begin{aligned} \left( \sum _{i\ne k}z_{in}f_{in}w_{vi}+z_{kn}f_{kn}w_{vk}\right) ^{x_{vn}}\propto & {} (z_{kn}f_{kn}w_{vk})^{R_{vn}} \end{aligned}$$

  Hence, we have

  $$\begin{aligned} p(w_{vk}\mid Z,F,X)\propto & {} w_{vk}^{\alpha _{0}-1}e^{-\beta _{0}w_{vk}}\prod _{n\in J_{v}}\left( (f_{kn}z_{kn}w_{vk})^{R_{vn}}e^{-\left( f_{kn}z_{kn}w_{vk}\right) }\right) \\\propto & {} w_{vk}^{\alpha _{0}+\sum _{n\in J_{v}}R_{vn}-1}e^{-(\beta _{0}+\sum _{n\in J_{v}}f_{kn}z_{kn})w_{vk}} \end{aligned}$$

  The above expression is in gamma distribution form $w_{vk}\sim Gamma (\alpha _{0}',\beta _{0}')$, where

  $$\begin{aligned} \alpha _{0}'= & {} \alpha _{0}+\sum _{n\in J_{v}}R_{vn}\\ \beta _{0}'= & {} \beta _{0}+\sum _{n\in J_{v}}f_{kn}z_{kn} \end{aligned}$$

• The posterior of $f_{kn}$ is similarly calculated as:

  $$\begin{aligned} p(f_{kn}\mid Z,W,X)\propto & {} p(X\mid Z,F,W)p(f_{kn}\mid \alpha _{1},\beta _{1})\\= & {} \left( \prod _{m\in I_{n}}p(x_{vn}\mid \sum _{i=1}^{K}f_{in}z_{in}w_{vi})\right) Gamma (\alpha _{1},\beta _{1}) \end{aligned}$$

  Hence, we have

  $$\begin{aligned} p(f_{kn}\mid Z,W,X)\propto & {} f_{kn}^{\alpha _{1}-1}e^{-\beta _{1}f_{kn}}\prod _{v\in I_{n}}\left( (f_{kn}z_{kn}w_{vk})^{T_{vn}}e^{-\left( f_{kn}z_{kn}w_{vk}\right) }\right) \\\propto & {} f_{kn}^{\alpha _{1}+\sum _{v\in I_{n}}T_{vn}-1}e^{-(\beta _{1}+\sum _{v\in I_{n}}z_{kn}w_{vk})f_{kn}} \end{aligned}$$

  The above expression is in gamma distribution for $f_{kn}\sim Gamma (\alpha _{1}',\beta _{1}')$, where

  $$\begin{aligned} \alpha _{1}'= & {} \alpha _{1}+\sum _{v\in I_{n}}T_{vn}\\ \beta _{1}'= & {} \beta _{1}+\sum _{v\in I_{n}}z_{kn}w_{vk} \end{aligned}$$

  and the auxiliary variable $T_{vn}$ is sampled similar to $R_{vn}$ from

  $$\begin{aligned} T_{vn}\sim & {} Binomial \left( x_{vn},\frac{f_{kn}z_{kn}w_{vk}}{\sum _{i\ne k}f_{in}z_{in}w_{vi}+f_{kn}z_{kn}w_{vk}}\right) ,~~\forall n\in I_{n} \end{aligned}$$