Continuous Markov Chain and its usage in Weather Forecasting

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A Markov chain is a way of modeling a sequence of events and even predicting the future based on past observations.

Let’s say you’re observing your friend, who likes to eat different things for lunch every day. You notice she never eats the same thing twice in a row. So if she ate a sandwich today, there’s a 0% chance she eats a sandwich the next day.

If you gather enough observations, you can start to notice patterns. For example, whenever she eats a sandwich, then 50% of time she eats a hot dog the next day, 25% of the time she eats a salad, and 25% of the time she has pasta. So the next time she eats a sandwich, you can predict what she’ll eat the next day—at least, within a certain probability.

Inherent in this model is the Markov assumption: what happens tomorrow only depends on what happened today (or rather, the next event only depends on the previous event). Maybe your friend eats a different meal every day of the week, but using this simple Markov model, we can’t figure that out. So it’s rather limited, but easy to calculate and a surprisingly good mathematical model for a lot of things. Also, there are ways to modify the model to effectively use more past information (like grouping days together, in our example, which might reveal our friend’s weekly pattern).

A *continuous* Markov model is the same idea, except instead of dealing with a limited number of categories (like lunch choices), you deal with numbers that can have lots of different values (like temperature or air pressure). You use past data to come up with an equation that takes today’s temperature and outputs a prediction for tomorrow’s temperature, along with the margin of error on that prediction. With enough data, you can get some pretty good predictions, despite the limitations of the Markov assumption.

To reiterate, the reason to use a Markov model is because it’s a lot easier to do the math than a more complex model, and it does the job fine in a lot of cases, so why make it more complicated?

A continuous Markov chain is like a never-ending game that helps us understand how something with a limited number of possible situations can change over time. Imagine a board game with different spaces for weather conditions (sunny, cloudy, or rainy), and the game is always moving between these spaces without taking turns.

This idea is useful in weather forecasting because it lets us predict the chances of different weather conditions happening in the future. We can look at past weather data to figure out how likely it is for the weather to change from one condition to another.

However, this method has some limits because it doesn’t consider things like where weather events happen and assumes that future weather only depends on what’s happening right now. More advanced models are often used to give more accurate weather forecasts, but continuous Markov chains can still be helpful for understanding general patterns and the chances of extreme weather events happening.