Race Linked Coefficient Model

Neural Network Architecture Models Solving Linear Systems

This archiceture diagram is a simplified version of the true model. There are hundreds of computations being solved because the data sets can be as large as 17 rows by 44 columns. (That's 748 total operations, just to compute the estimated constraints once.) Over several iterations, some thousand of caclulations are made to find the "right" coefficients.

Overdetermined Least Squares Regression and Gradient Descent With Back Propagation

This application of a machine learning algorithm combines solving an overdetermined system of equations to determine a best-fit solution to the system representing the distribution of diabetes cases, where the solution vector are the coefficients we aim to extract. To improve the accuracy of these coefficients, we minimize a cost function that represents the mean difference between the estimated constraints and the actual values. A weight is applied to scale the coefficients. The signed magnitude of error informs the model to adjust the weights by some small increment, until the error is minimized. This process ensures our fit to the training data is accurate (within 95%), and our coefficients are representative of the total variance in the data.

The coefficients are a linear map from the input space to the output space.

Fit Results

The existence of an overfit and an underfit confirmed that a 1:1 fit exists, that is, there exists a set of weights that can scale our coefficients such that the estimation of our features (parameters) is exact, capturing all the variance in the data.

So why do we care about all this math?

Using this model, we can reconstruct the distribution of diabetes cases across the cities for which we have collected data. Based on this model, if a population distribution is known, we can estimate with high accuracy the prevalance of diabetes within that population.

Based on population growth estimates, we can run this model using population estimates to predict diabetes cases in the future based on population dynamics. With more available training data, the model can be improved. Error is introduced with each generation of data we try to transform.

If you want to learn more about solving linear systems with neural networks, check out this research paper by the Technical University of Munich.

What is a Race Linked Coefficient (RLC)?

A race linked coefficient is a unique scalar that is conceptually analogous to a rate coefficient that scales a parameter. That is, they are conceptually the "rate" of diabetes prevalence tied to each race.

Population x Rate = Cases.

However, these 4 coefficients are a bit more complicated. In theory, they scale the input populations to approximate the number of diabetes cases, such that the distribution of the cases matches that of the training data, and such that the error between the estimated data and the training data is minimized.

We use these four coefficients, each one unique to a target race, to reconstruct the distributions of historical diabetes data based on new population data. Applying these coefficients to novel data is a kind of linear transformation into the future, dependent on the accuracy of forecasted population data.

Why do it this way?

Not enough data. All auto-regressive, regression models use historical data to predict future data. With only 6 years of sparse data to train with (thats only 6 data points total for each feature), any kind of auto-regression based forecasting like ARIMA or even just a simple regression were not feasible here and the estimates are completely off. This method is scalable, linear, and a relatively simple approach to a complex problem. Also, it was a fun challenge.

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