MRDS Gold Resources - Reported Grade Prediction

This intensely skewed distribution is not surprising given the nature of mineralization. Let’s take a step back and briefly explore the normal distribution, which served as the foundation for all of the statistics I learned in school. Per the central limit theorem, I can reproduce unexciting, normally-distributed data by flipping coins. For example, I flip a coin ten times and record the number of heads. Repeated 1000 times and plotting those 1000 records on a histogram, I see a nice bell curve as in this figure. From what I know of coins, the odds of flipping heads are not influenced by any of my previous coin flips. Put another way, the coin has no memory.

Now imagine I have a magic coin that remembers the past. On the first flip, I’ve still got even odds of flipping heads or tails. However, and here’s the magic, the first time I flip tails, the odds change such that the coin will only return tails for the remainder of the round. The coin remembers. A histogram of 1000 rounds in this scenario displayed below is giving us a positively skewed distribution reminiscent of our gold grade data.

I think the key insight here is that mineralizing systems have a memory. The conditions that allow for a bleb gold laden sulfide to nucleate in a certain part of the world, persist or remain in memory for a time allowing for additional nucleation. Further, the mere presence of a mineral provides inviting nucleation sites, significantly increasing the odds of more mineralization of the same. These types of compounding or multiplicative systems yield the highly skewed results we’re finding in the gold grade data.

The nature of the gold grade distribution is interesting and it starts to focus my intuition about relative gold grade and what we really mean by high and low grade. However, we have yet to consider geography. After all, what’s high grade for a gold deposit in Nevada is not the same as in the Canadian Shield. In other words, there are geographic features of these deposits in addition to grade that are worth knowing.

Here’s why the coordinates are so important. With them, I can append the gold grade data with all manner of new, geographic features. Cross-referencing the deposit locations with the GLiM geologic map of the world, I append generalized lithology to the dataset. I do the same with the Lith1.0 dataset for lithosphere thickness.

What about near-ology? Near-ology is just a more approachable code word for geostatistics. In short, near-ology attempts to identify and extract any useful information about, in this particular case, the grade of a gold resource from neighboring resources. To do that, I construct a few new features to append to the data set. First, is a score where I calculate the inverse distance of a resource to every other resource in the data set and sum them all up. Resources with a high score have a high number of close neighbors. The second takes those same inverse distances and multiplies them by the neighboring resource grades (effectively an inverse distance weighting) and sums those values. The third and fourth features are the same as the first and second, but with one twist. Only neighboring resources with the same commodity (gold in this case) are included in the summations. Now, we’ve got four new features that each have a slightly different take on near-ology. Now, it’s time to put those features to good use.

Of all these great features, which if any have the most power to predict the grade of a gold resource? The RReliefF algorithm scores features for just that purpose and for our newly appended dataset, the top five features are lithosphere depth and my four near-logy features. That said, the extensive lithology data I appended still makes a positive contribution to the end result.

Next step is to train an algorithm to produce a predictive model and assess that model’s precision. After a little experimentation, I chose the AdaBoost algorithm (regressor using decision trees) for the task. It’s job is to identify relationships between gold resource grades and their corresponding feature data. Through a method called 5-fold Cross Validation, I iteratively hide 1/5th of the data from the algorithm, allow it to produce a predictive model from the remaining data, feed the hidden feature data through the newly produced model, then compare the model predictions of gold grade with the known values. This is repeated such that each fifth of the data has a turn being hidden.

So how do the models produced by this algorithm perform? Not bad! Below is a scatter plot from my 5-fold Cross Validation exercise. Each point plots the actual grade for one of the gold resources in the dataset against its predicted grade (when it was hidden from the algorithm). There’s still lots of noise in there, but some signal, as well. We’ve seen the high variability in the gold grade data in those charts above. The algorithm is giving me models that can explain away about 40% of that variability*.

*I'm making this claim after using a Box-Cox Transformation to re-shape the data into a nearly Gaussian Distribution and comparing Mean Absolute Deviation of the actual and predicted grades.

I am providing access to my predictive model for gold resource grades via two tools. The first requires a user to input any pair of world coordinates in degrees longitude and latitude. Below is an example for a coordinate pair in the center of Nevada. The geographic features (i.e. geology, lithology, lithosphere depth, and near-ology scores) are computed automatically for that particular location and fed through the predictive model. The output is a pair of cumulative frequency curves. One is simply the frequency distribution of the raw MRDS data for gold resource grades (‘Raw’) for reference. The other is the predicted frequency distribution for the coordinates provided (‘ML Model’).

Here are a few notable numbers from the ML Model: the 25th percentile is 1.4 gpt, the 50th is 3.1 gpt and the 90th percentile is 11.2 gpt. Okay, that’s a lot of numbers, but let me go a bit further and be extra-explicit about what this predicted frequency distribution means to me: If a gold resource existed at this location and that resource was deemed interesting enough by the US Geological Survey to include in the MRDS, I’m giving it one chance in four of having a grade greater than 1.4, one chance in two of having a grade greater than 3.1, and one chance in ten of exceeding 11.2 gpt. The 50th percentile is the value output from my predictive model. The shape of the distribution (not-quite-lognormal) was a subjective call I made after playing with the raw data. The spread of that distribution came from the 5-fold cross-validation work I completed while training the machine learning algorithm.

I can be even more explicit, but first I better re-introduce the second tool, which is my gold grade map of the world. Having set up a system to use the predictive model with only providing a pair of coordinates, the next logical thing to do is feed every coordinate in the world into the model and plot the predictions on a map. Doing so (ignoring the oceans and Antarctica) gives me this map, which I showed earlier.