Monte Carlo Integration In Python For Noobs
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- Опубликовано: 7 июн 2018
- Monte Carlo is probably one of the more straightforward methods of numerical Integration. It's not optimal if working with single-variable functions, but nonetheless is easy to use, and readily generalizes to multi-variable functions. In this video I motivate the method, then solve a one-dimensional integral using monte carlo. I then solve the integral a number of times, and histogram the results, showing the distribution of areas calculated.
This has been a fantastic introduction to Monte Carlo methods
Thanks a lot!
@Andrew Dotson , I don't know if you read comments on older videos, but I just wanted to thank you. I missed the lecture on Monte Carlo and understood nothing of the example code we were given. I already subscribed to you for the awesome skits, but this was the perfect video for me. You explain the concept and what each part of the code does, without going too slow. (I am a noob at programming, ngl.) Thank you, and greetings from Finland.
Great tutorial. It made a lot of theoretical concepts clear for me when implementing. I really wish that you make more videos on Monte Carlo Simulations. I read that these were first used to solve atomic simulations. I am not connected to the physics world, and it would be great to learn more about that.
Once again, great job
This was so much better than my lecturer's explanation. Nice and concise and some coding thrown in too. Good job. Thanks Andrew!
I've been meaning to learn this technique. Great introduction!!
I've never been introduced to Monte Carlo integration this way. Thanks!
I'm not sure if threads exist in Python, but in C there is a concept called threading, where you can run multiple functions at the same time, and in this case you'd be able to split the N operations among M threads, where M threads are computing N/M operations at the same time, which would result in a faster runtime. After a quick google search, it seems that Python does have threads, I'd recommend learning them so that you can optimize your code runtime whenever creating a new program, especially in a field such as physics where you would have situations in which you'd have to compute 10000+ points sometimes.
Threads “exist” in python, but only one thread can actually execute code at a time (because of the Global Interpreter Lock), so it’s best suited for I/O limited stuff, like networking. Using numpy array operations is the best way to speed up the code, since it calls C code under the hood, which is about 100x faster. Beyond that, there is multiprocessing, which creates multiple python processes at the same time.
@@eblocha GIL has been updated but these two comments were spot on and are still relevant today
@@superuser8636 what do you mean GIL has been updated?
Great tutorial. Cleared my concept on MC integration. I was looking for this topic on RUclips and found your video very helpful .
Title: Monte Carlo Integration In Python For Noobs
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First words: "What's goin' on smart people!"
Great video, sub and respect (:
I came to the comments section to write exactly that 😂😂
Thanks, Andrew. Helped me a ton with understanding my lab on Monte Carlo Integration.
This was actually very very helpful! I mainly watch the comedic side of your channel, but this was just a great video.
Good video, straight into it, with results! Others would have dragged this out.
David Aspden thanks!
This is my first video on Monte Carlo integration. Loved your explanation :) Thank you :)
many thanks! you are helping a lot of programmers around the world... (I´m from Brazil!)
Excellent video. In 1 video i learnt the basics of how to plot a histogram, generate random numbers, write for loops and use arrays in python. Excellent. Thanks! Please make more of such videos using python
As a person who has trouble with math terminology and symbols, bless you for doing this in code.
This is great thank you! The way you teach is very clear!
Wha... thats absolutely genius! I was struggling to understand this but now its just clicked!
Yea after I watched a couple of videos on Monte Carlo Integration, it suddenly clicked! A beautiful concept ngl
Great production in this video 👏👏 thanks for the content
Hey Andrew! I just wanna say thank you for this video. I made a monte carlo maximum successive scattering simulator out of it and the program did work as intended.
Hi! Thanks a lot for this video, this Python example helped me get a better idea of how to approach solving problems in Python. In my research, I have to calculate entrainment rates (the rate dry air mixes with clouds to dissipate them) in cloud fields and observe how these rates affect cloud size distributions. Dealing with atmospheric processes, most of my data is in 4 dimensional arrays, so I've been slowly getting acquainted with those. Though your example was general, showing how to solve that integral helped give me a couple of good ideas of how I can approach solving a couple of my unknown variables. Thanks again for the video, I wish you the best of luck in your QCD research! Keep up the awesome videos! -From an elightened Python noob 😊😜
What a fantastic and helpful video... Thanks a lot. You really made it very simple and easy to understand.
Nice Explanation bro, Need more stuffs like this.
Hi Andrew, thank you very much for this explanation. Now I know how to propose the solution for my project.
I feel like I'm going to have a lot of fun applying this method!
Great, straight to the point wonderful tutorial!
Good introductory video for Montecarlo!
Thanks Andrew, I'm just getting into my calculus courses at my college. I'm trying to pick up programming along with it and this video is quite helpful. I haven't taken any statistic courses as my degree plan doesn't require them and so my question is, have you taken any statistic courses and if so which ones? Keep up the great work we all appreciate it.
I first watched this in my 1st year in college thinking oh this is non of my business. I did not know I would need this video so bad. Thank you for your hardwork into making this video. luv it
I needed this last week and RUclips didn't recommend it when I googled it. Unlucky
Thank you for publishing such a great video.
Ahuyet! Thanks Andrew that was very cool!
it was a good and simple intro to Monte Carlo Integration. helped a lot!!
Hey andrew, this is really nice !
Eyyy lmao, I'm gonna take a class on Monte Carlo Methods!
Superb content sir 😎
clicked because I am a noob, suddenly called "Smart People", subscribed.
Fantastic video!
Thoroughly disturbed by the way you write a.
Thank you for this tutorial! Very understandable and easy, while looking for smth simmilar in internet is so complexed(
i love you bro. you saved my butt. i owe you.
edit: subbed
clear to the point very good video.
really interesting stuff!
Amazing Video! Thank you
love me some Monte Carlo!
Great video!
Great explanation, thank you
THANKS BRO I GOT IT CLEARED IN 5 minutes IN!!!
Loved it!
This is great, thank you very much!
These videos are really interesting to watch. Seeing you suss things out in real-time with a not-so-super-fast entry into python is awesome.
Out of interest, does the script break anywhere for different A's, B's or func(x)?
I suppose it might converge slower the wider the interval you integrate over. Meaning you'd need more points to get a solid representative random sample
I actually started to get away from python but this video gave me life hahahaha, thanks!
Thank you Sir...from India....
yaaaas, thank you for the video!
I'm using Monte Carlo to model electron transport through water for my thesis, nice to see some Monte-Carlo!
Sounds really cool!
I'm loving it so far! Thanks for sharing your experience through honours and beyond - it's been great!
Hey man I am working on a different project which is the Cryogenic Energy storage simulation by MCMC. Can you help me a little ?
Great presentation. It works well. You said it will become a delta function, but you probably mean a Gaussian distribution, since it yields an average and a standard deviation.
No it’s a delta function. As you push the number of points to infinity you yield a deviation of 0 and get a delta function.
@@debunkthis Due to the randomness of chosen points, there will be a variation around an average value. The central limit theorem tells that this yields a Gaussian distribution. A delta function implies that all random points fall at the mean value.
@@Galileosays The delta function can be defined as the limit of the gaussian distribution when the variance goes to 0. As more samples are taken, the variance will indeed tend towards 0.
dude, thank you!
thank you!!!so helpful
You, sir, are a fucking gangster for this tutorial!!! Exactly what I was after. #Sauce
nice one. Yeah, np arrays are way faster than lists. Interesting about the convergence for single variable functions. Got any recommendations for books on numerical methods in general (and Python in particular)?
Hey Andrew , please make a video on how to read a research paper and what research paper can an undergraduate can read.
Read the title, next the abstract, then all the section-titles and subsection-titles and then the conclusion. Then go back and read the contents of the introduction and all following sections.
@@Paraboemba that's actually how I approach most of the papers yeah. You concluded it pretty well hahahaha
Only one thing is wrong. The distribution will never be a Dirac's delta. Because of the central limit theorem, the width of the normal distribution will grow as sqrt(N). (You can check the x tics with N=1000 and N=10000 and see the width is greater in the later).What will decrease as sqrt(N) is the standard deviation of the mean. Great video tho👍
Thank you bro for this video nice content
Thank You very much!
Nice and impressive demonstration...the x axis should not be labeled as expected value instead of area?
Thank you so much!
Nice idea with the histogam. Looks a bit low-slanted, though. Hard to include the last x-value in the x domain, as that interval is semi-open. But wonderful example, thank You!
nice explanation❤❤❤❤❤❤❤❤❤❤
Thank you very much.
Thx, that helped me!
Thanks Andrew
Great video. Do have a tutorial for phyton installation?
super nice video
hey andrew, can you make a vedio showcasing what all compuatational work you do in your research in theoretical nuclear physics?
Do you think you could show to take the gradient of a scalar field in python? Recently discovered your channel and love your videos!
That's an awesome idea!
Lol where was this last semester when I was working on this for my computational physics course?
Nice introduction. Can you show a tutorial for monte carlo simulation of a honeycomb Ising lattice?
So is there a way to turn this into something to show growth of investment portfolios? Like with the average growth and standard deviation?
Two questions, first I notice that the histogram of the integrals looks like the normal distribution. Is this because of the central limit theorem? Also, there are two places where sampling occurs where the first is sampling for estimating the integral and the other is sampling integral calculations for taking the average. My question is about how each of these parameters impacts the result. For example what if I only used 100 samples for calculating a single integral but took the average of 10000 integrals?
, how can I find the area inside a curve?
I have data points for a hysteresis loop and I want to calculate the area inside the curve
Wolverine is teaching map its cool!
Very nice / 10
I don't understand however why it is better to use random samples (even for higher dimensions). Wouldn't a sample of uniformily spaced values be basically as good as a uniform distribution as it can get since the order doesn't matter. So for example for a one dimensional integral you can choose numbers from a and then increment by a small delta untill you get to b, and for a bidimensional integral do something similar but in 2 dimensions (of course making shure that you keep the points evenly spaced). For some integrals it might be easier to just take random samples i get that but when it is easier to just increment by a delta my question is if that would yield as good a result as with random numbers?
THANK YOU!!
Great work bro...
2 questions here... If anyone can help...
1) when you wrote the very first expression of , I think it should mean that avg of f(x) for x in [a, b]... And not for any value of X.
2) while we carry out the experiment of randomly generating value of x and then, finding f(x), we should not be bias and should pick any value of x in the domain [a, b]. But, I think, Random no generator will only generate numbers randomly between 0 and 1, and not 0 to pi.
Not sure though how the result is perfect.
@dusty mehul
1. The definition of is the average of x in the region between a and b. You have x which is a random variable that can take on any value between a and b.
2. When he created the random number generator he chose for the generator to choose numbers between 0 and pi. Random number generators allow you to choose what range of numbers you want to sample from.
didn't know to use the same variable e.g. i for index in nested for loops several times
Great explaination but I have one doubt like if they give two circles and told to find the intersection area whose radius is r1 = 2, r2 = 2. And random sampling is n = 100000. Then how to print estimated area? I am getting error while implementing this? Please someone help. Thank you.
A true physicist there. Uses hadkerish style programming with loops rather than vectorizing things! 😜
I speak as a physicist myself...
Haha yes! You ended up doing it!
can you please explain monte carlo method for multidimensional integration
same way if integration happens for multidimensional function, say, f(x,y,z). Just vectorize the space and use the Euclidean norm for term (b-a) that would be then L2normDist(b,a) where b and a are multidimensional points. Or go deeper and use packed M-integrals for M dimensions. And if you've get the understanding about this - then, probably, it's time to meet the functional integral over a random pathes (with gaussian subintegral in each patch) in a space filled with positive and negative probabilities potentials (the simplified version of "Feynman's path integral" - without complex part about quantum physics, just the space and +/- potentials regions)
How I put dots in required area.want to know the command in python for plotting
you have assumed a uniform distribution for f(x)
and what you are calculating in some ways is actually a monte carlo estimate of the expectation ..
Hi Andrew. if I look a mean for a cost. what formula apply?
How do you do it for 2 dimension or higher integral? Like let's say Sqrt(x + y)
''Very nice out of 10'' got me laughing
Revision notes.
It's sometimes difficult to find the answer for the integration of a function. Because it's hard or impossible to even derive the formula of the integration, especially it's not closed form and the function is multi-dimensional (multiple variables).
But if we are only after an answer, then we can do it in "numerical" way. I.e. By combinating different numbers to find the best approximate result.
Here, it's shown the ∑ of many little results (divided by something) is approximately the ∫ of the functions.
The Python code proves it.
Monte Carlo is based on the principle.
Wouldn't the end distribution of areas end up approximating a normal distribution instead of the delta function as a result of the central limit theorem?
Great video! How would you solve for an integral with infinite limits though?
It would depend on the nature of the function. For a function like the gaussian function which has bounds from -infinity to infinity, you could probably do the integration within 2 to 3 standard deviations of the mean to account for about 95-99% of the area. Moreover, if there exists any symmetry in the bounds you choose for the function, you could compute the integral on only one side of the function and then double the result. This should substantially reduce computational cost.
Hey Andrew can you tell me about your python journey?
How do you apply the trapezoidal rule to a dataframe instead of a function?
Do you know if there's a scenario where using Monte Carlo's method would be preferable over other numerical integration methods (such as trapezoidal rule or simpson's rule)? I ask this because, like you said, seems like a lot of computational effort to generate random numbers and then feed them into a function.
Marco Antônio if you start integrating function that depends on more than just x, it starts to be way more worth it to use monte carlo
Oh I see. Thanks for the answer!
In my graduation project I have to calculate the expectation value of some operators over many-body wavefunctions. In a 2D system and for 30 particles (my case) this means a 60 dimensional integral. I simply don't have any other choice beside Monte Carlo integration.