Learning System

This page reflects my/our idea about the Learning system. It consists of a few interdependent systems for functions of estimation, inference and storage.

Glucose level estimation

The most top-level functioning of the learning system is to give near future glucose level estimation. The current glucose level estimation is done by (1) taking the last glucose measurement, and then (2) adding up the typical glycamic response (glucose rise/fall) of all events since the last measurement.

The glycemic response of each event is modelled in terms of a glucose rise/fall in function of time: f(t). Time real time t is mapped to discrete intervals of 15 minutes. Event types are split into distinct categories (see below). For computational conveniance, each event type category c is modelled by a function f_c(t), and each concrete individual event type is modelled as transformation of that function using parameters a and b: a*f_c(b*t).

• Food intake. Usually has a positive glycemic effect. It appears to be modelled as:

<math>f(t) =

a    * exp \left [ \left ( \frac{t-b }{0,667*b} \right )^2 \right ] +

(a/2) * exp \left [ \left ( \frac{t-2b}{0,667*b} \right )^2 \right ] + (a/4) * exp \left [ \left ( \frac{t-3b}{0,667*b} \right )^2 \right ] </math>

• Insulin intake. Usually has a negative glycemic effect.
• Stress level.
• Time of the day, because glucose levels structurally differ during the day.
• Health status.
• Other event types.

[Add picture here]

As told above, the estimate for future moments in time is done by taking the last glucose measurement and adding the sum of glycemic responses of events. If g1 at <math>t_{g1}</math> is the last glucose measurement, g2 at <math>t_{g2}</math> is the glucose level to be estimated, and <math>(e_1,e_2,...,e_n)</math> events that have influence on g2. <math>(f_1,f_2,...,f_n)</math> are the estimated functions of the events. <math>(t_1,t_2,...,t_n)</math> are the (start) times of each events. Then the glucose prediction g2 at <math>t_{g2}</math> is:

<math>g2_{estimate}(t_{g2}) = g1 + \sum_{k = 1}^n \left ( f_k(t_{g2}-t_k)-f_k(t_{g1}-t_k) \right )</math>

On events

Like told above, the term 'event' can be things like apple intake. Our definition is broader then that: events can also be composite. A composite event is a set or cominbation or multiple single events. Why use composite events? Because, for example, eating different food types combined leads to a different glycemic response then the sum of individual foods. Eating certain food types nullify the effect of other foods. Another positive thing about compositive events is that it decreases the amount of events in the sum of g2_estimate (see above). Less summation means less uncertainty of the estimate.

Creation of a new event type

What needs to be done when a new event type is created, for example when a user eats something new or gets new insulin therapy? The first the system needs to create is an a priori estimate of f(t). This is called the a priori function. For food, this would be based on carbonhydrate count. For insulin, this would be done by entering medicine information. A better a priori f_prior(t) means the system needs less training time to estimate the real function f(t). When evidence arrives in the form of a sample, an a posteriori f_post(t) is formed that esimates the real function f(t). A sample is an observation value of f(t) at some t. More evidence/samples means a better a posteriori f_post(t).

In other words:

• Better prior knowledge (carbonhydrate count etc) leads to a better f_prior(t)
• A better f_prior(t) leads to a better f_post(t)
• More samples leads to a better f_post(t)
• A good f_post(t) means it is close to the real f(t)

Significance of good a priori functions

In our case, we will see that the samples are estimations too. Later on, we will conclude that better f_post(t) functions lead to better estimations of samples. In the 'bigger picture', this means that bad-quality f_prior(t)'s implicates inititally bad-quality f_post(t)'s, which in turn implicate intially bad-quality samples, leading to initially slow progression of inference. This is important to know, because quality f_prior(t)'s are VITAL to fast initial inference. Concretely said, good a priori functions will decrease the startup time significantly, maybe from months to just weeks or days.

The following section will explain how to form an a posteriori function using glucose glucose measurements.

Bayesian Inference

Events are things like food intake (eg one glass of lemonade), insuline intake (one unit of type X), sports (half an hour of running), but also current health status and stress level etc. Before the system learns anything, each event is assigned an 'a priori' estimating curve, which tells us how, in time, the estimated effect on the blood glucose level 'g'. This a priori curve is assigned before any measurements have been made (example: the a priori curve for food could be based on known carbonhydrate). Quickly said, the learning system uses the blood glucose measurements to update and improve the estimating curve.

Lets describe these events, their effects and their computations, in terms of a statistics problem.

About events

Each single event <math>e_i</math> has three things.

1) Firstly: A set of samples. Each sample is a tuple (Δt, Δg) So the set of samples could be represented on a 2-dimensional area. The sample set is initially empty, and samples are added through bayesian inference (explained below). [For extra clarity, image to be added here].

2) A prior (a priori) function f_e,prior(Δt) → Δg = μ_prior. This is the estimated mean effect of the event, for each determined before any samples have arrived. For food, it could be determined by looking at carbonhydrate amount. For insuline, it could be determined by medicine information. If no prior function can be made, an effect is assigned a default prior function. The prior function also has a pre-determined variance σ_prior². Spoken in statistic terms, the event effect at each moment in time has a normal distribution with:

<math>\sigma_e^2 = \mbox{some-static-value} \,</math>
<math>\mu_e = \theta \,</math>

This parameter θ is unknown, but it has a prior distribution with

<math>\sigma_{\theta ,prior}^2 = \mbox{some-a-priori-value} \,</math>
<math>\mu_{\theta, prior} = f_{e,prior}(\triangle t) \,</math>

3) A posterior (a posteriori) function f_e,post(Δt) → Δg. This is the esimated effect of the event after looking at the samples. It is determined as follows. The samples are divived into give time intervals, for example 15 minutes. So we have intervals t_i with i=(1, ..., n), and each interval representing 15 minutes. Each of these intervals t_i have a distribution θ with a prior distribution as explained above. The posterior distribution of t_i is calculated as follows:

<math>\sigma_{\theta, post}^2=\frac{\sigma_{\theta, prior}^2\sigma_{\theta, prior}^2}{\sigma_e^2+n\sigma_{\theta, prior}^2}</math>
<math>\mu_{\theta, post}=\frac{\sigma_{\theta, prior}^2\mu_{\theta, prior}+n\sigma_{\theta, prior}^2\bar x_n}{\sigma_e^2+n\sigma_{\theta, prior}^2}</math>

To assign an evidence x_i to each individual events e_i you do:

<math>x_i=\mu_{prior}+a*\sigma_i^2</math>

where

<math>a = \frac{x_{tot}-(\mu_1+\mu_2+...)}{\sigma_1^2+\sigma_2^2+...}</math>

So it comes down to some quite simple math. I'll make my explanation better when I have more time.