Predicting the Future – An Intro

In our day to day lives, we live with uncertainty about what might happen.

What will the weather be like today?
Will there be traffic on my way to work?
Will I finally get the recognition and promotion I have been working for?

In life, there’s many things we don’t have enough data to predict. To us, many things might appear like chance because we don’t understand the long-term affect of choices and actions we take that might impact our life trajectories. I like to think of life as a complex system of differential algebraic equations, of which we are very far from ever understanding in a mathematical sense.

However, in many areas, such as science and engineering, models have been discovered and used to make predictions of many phenomena, ranging from simple predictions of pendulum motion to complicated predictions of the stable flight of the space shuttle (how else can they know the guidance systems should work?).

So how are these predictions made, one might ask? The answer generally lies in how the model is formulated. For example, lets imagine we have a statistical model that can state the probability that someone of a particular age will go clubbing on a Friday night, defined as the following:

$$
\begin{align}
p &= \text{WillGoClubbing}(\text{Age})\\
\text{where } p &\in [0,1]
\end{align}
$$

In this case, we could state a probability that a person will go clubbing on a Friday night given their age. We could even use this to find the age someone is expected to be if they are going clubbing! Pretty interesting way to make predictions. These sorts of statistical predictions are done all the time in areas such as Machine Learning, Quantitive Finance, and even areas like Missile Performance (shameless plug since I work on this stuff.. okay?).

However, these aren’t the only types of predictions that are made. For example, how do we predict the flight of a rocket, or the weather, or the trajectory of a bullet? How do we predict earthquakes or estimate where hurricanes will end up? Typically, these sorts of problems are tackled using models based on differential equations, some time varying and others steady state (meaning they won’t change over time). Below is an example of a few differential equations that you might find being simulated:

Newtonian Dynamics

$$
\begin{align}
m \vec{a} = \sum_i^n \vec{F}_i
\end{align}
$$

Transient Heat Equation

$$
\begin{align}
\frac{\partial T}{\partial t} + \nabla \cdot \nabla T = g(t,T)
\end{align}
$$

Navier-Stokes Momentum Equation

$$
\begin{align}
\frac{\partial}{\partial t}\left(\rho \textbf{u}\right) + \nabla \cdot \left(\rho \textbf{u} \otimes \textbf{u} + p\textbf{I} \right) = \nabla \cdot \mathbf{\tau} + \rho \textbf{g}
\end{align}
$$

Many of these equations are used today by researchers to make predictions of very complicated and sensitive phenomenon. Depending on the problem, solving these equations accurately, such as Navier-Stokes, can require resources like clusters or supercomputers that run for weeks or months and generate enough data to leave researchers working to understanding the results for months or more. These sorts of activities are definitely not cakewalk.

With all this said, there is still many things out there we don’t have the models or data for to make adequate predictions. Additionally, many of the models we do have are idealized enough or incomplete in that they don’t always capture the real world phenomena accurately. This is why at times the weather predictions are wrong or why we can’t predict the stock market too well. The models used to make these predictions just break down over long periods of time, whether by assumptions, incomplete modeling, or not adequately taking into account transient inputs to the dynamical system.

I don't always change my plans based on the weather forecast, but when I do the weather forecast is always wrong

Fortunately, due to the great abundance of data becoming available (think Big Data) and the great advancements in AI, models have been getting built statistically that can make improved predictions of many things.. From what you’ll likely want to purchase, to what types of shows you might like based on the types of things you watch, to predicting captions for pictures, and more. Using these new techniques and data, more precise models are being empirically created and understood and helping pave the way to understanding more complicated phenomenon down the road.

After explaining a few of the fundamentals in prediction and how it’s used, I hope in future posts to dive into algorithms that can be built to help make predictions of various kinds. In the meantime, thanks for reading and best wishes.

5 comments Add yours
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    1. This blog is made via WordPress and the theme is “Athena”. My homepage, christianjhoward.me, was coded by scratch (though it’s not much anyway).

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