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We're a little crazy, about science!

Posts tagged “math

Day #32 : The Laplace p.d.f. – 365 DoA

laplace plot

The laplace p.d.f with a θ = 0.

Well here we are again… maybe unless you’re new, in which case welcome. If you are just joining us we are talking p.d.f. no not the file format, the probability density function version. If you’re new, you may want to start back here(ish) If not, then let’s talk the strangely similar laplace distribution.*

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Day #31 : The Exponential p.d.f. – 365 DoA

Exponential_probability_density

Today we are talking this guy! The exponential p.d.f and its C.D.F.

Well, it has been a week, don’t even get me started. But if you’re here you don’t want to hear me complain about my week, that isn’t why we come together! Well today let’s do a bit of a dive into the exponential p.d.f. I hope you’ve brushed up, because this is going to get interesting.*

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Day #30 : Confidence Interval – 365 DoA

Confidence intervals

Yep, we’re talking confidence!

Day 30 already! Where does the time go? It feels like we just started this whole project and it probably wouldn’t be a good idea to look at the remaining time to completion, so let’s not and just enjoy the nice round 30. We will get back to our p.d.f another day, but today is going to be short. That’s what I usually say before typing out 10 pages worth of information so to avoid that, let’s touch on something important, but something I can do briefly. Today we’re talking about confidence intervals*

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Day #29 : Probability density functions, Part 3 – 365 DoA

gaussian CDF vs pdf

Don’t be scared, we’re going to tackle this guy today!

Well, apparently you guys really appreciated my probability density function posts. It’s good to see people interested in something a little less well-known (at least to me). So for those of you just joining us, you’ll want to start at part 1 here. For those of you who are keeping up with the posts, let’s review and then look at specific functions. Namely let’s start by going back to our gaussian distribution function and talk about what’s going on with that whole mess. It will be fun, so let’s do it!*

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Day #28 : Cumulative Distribution Functions – 365 DoA

Exponential_distribution_cdf

An example C.D.F. of an exponential distribution

Today we were going to do another deep dive into the p.d.f and C.D.F. relationship. Specifically today we were going to talk about specific p.d.f. functions and why we use them, however… I am not doing so hot today, so instead we are going to back track just a bit and talk about what how a C.D.F. differs from our p.d.f. even though we kind of covered it, it would be nice to be clear and I can do this in a (fairly) short post for the day. So that said, let’s get started and we will pick up our p.d.f. discussion next time (maybe).*

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Day #27 : Probability density functions, Part 2 – 365 DoA

PDF_CDF

Today we are looking at our p.d.f. (yes this image has p.d.f. written as PDF, please don’t be confused!) and our C.D.F.’s let’s do this!

Oh hi didn’t see you there. Today is part 2 of the probability density functions notes (posts?), whatever we are calling these. You can read part 1 here as you should probably be familiar with the (super confusing) notation we use to describe our p.d.f. and our C.D.F. now that we’ve given that lovely disclaimer, let’s look once again at probability density functions!*

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Day #26 : Probability density functions, Part 1 – 365 DoA

statisics jokes

Dashing dreams one comic at a time, via Saturday Morning Breakfast Cereal 

We are well on our way to wrapping up week 4, what a ride it’s been! It’s been a long day for me, so today might be short. However, I really, really, really want to break into probability density functions. This topic is going to be a bit more advanced than some of the things we’ve covered (IE more writing) so it will most definitely be broken up. Let’s look at why and discover the wonderful weirdness of probability density functions!*

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Day #25 : The p-value – 365 DoA

P-value joke showing a very confident man with this is what p less than .05 feels like printed on top of the image
P-value joke showing a very confident man with this is what p less than .05 feels like printed on top of the image

It’s true!

Now it seems like we are getting somewhere. Last post we covered z-score and you can read that if you haven’t already, it might be good to familiarize yourself with it since today we are going to talk p-value and the difference between z-score and p-value. That said, let’s dive in and look at the value in the p-value.*

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Day #24 : The z-score – 365 DoA

Statistics cartoon

Statistics cartoon

So if you recall from last post… well I’m not linking to it. It was hellishly personal and frankly I’m still attempting to recover from it. We’re going to take it light this time and we can do a deep dive into something in another post. For that reason, let’s talk about z-score and what exactly it is, I mean we used it in this post and never defined it formally, so let’s do that. Let’s talk z-score!*

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Day #22 : Parametric vs. NonParametric Statistics – 365 DoA

How Mean Meme

Technically we could call this parametric statistics part 2. However, since we are covering nonparametric statistics and more importantly the difference between parametric and nonparametric statistics, it would seem that this title makes more sense. As usual with a continuation, you probably want to start at the beginning where we define parametric statistics. Ready to get started?*

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Day #21 : Defining Parametric Statistics – 365 DoA

normal-distribution

It’s halloween time, we are talking about normally distributed data, so this fits, and I don’t want to hear otherwise!

Well my lovely readers, we’ve made it to the three week mark, 5.7% of the way through! Okay maybe that doesn’t seem like a big deal written like that, but hey it’s progress. So last post we had our independence day, or rather defined what it meant to have independent events vs. dependent events. We also said it was an important assumption in parametric statistics that our events are independent, but then we realized we never defined what parametric statistics even is, oops. So let’s stop dragging our feet and talk parametric statistics!*

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Day #20 : Independent Events – 365 DoA

correlation

By: xkcd

 

Because we introduced the central limit theorem last post, it’s time to introduce another important concept. The idea of independent events, while this may seem intuitive, it is one of the assumptions we make in parametric statistics, another concept we will define, but for now let’s jump into independence.*

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Day #19 : The Central Limit theorem – 365 DoA

 

 

CLT

Well here we are again, if you recall from our last post, we talked Bonferroni Correction. You may also recall that when the post concluded, there was no real topic for today. Well after some ruminating, before we jump into more statistics, we should talk about the central limit theorem. So let’s do a quick dive into what that is and why you should know it!*

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Day #18 : The Bonferroni Correction – 365 DoA

Keep calm and do the Bonferroni corrrection

Keep calm and do the Bonferroni corrrection

By now we are masters of statistics… right? Okay, not really, but we are getting there. So far we’ve covered two types of errors, type 1 which you can read about here, and type 2 which you can read about here. Armed with this new knowledge we can break into a way to correct for type 1 errors that come about from multiple comparisons. Sound confusing? Well, not for long, let’s break it down and talk Bonferroni.*

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Day #17 : Type 2 errors – 365 DoA

 

cat asleep

Last post we did a quick bit on type 1 errors. As with anything, there is more than one way to make an error. Today we are talking type 2 errors! They are related in the sense and we’ll go over what that means and compare the two right… now!*

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Day #16 : Type 1 errors – 365 DoA

tiny dog yawning

tiny dog yawning

We did it, we cracked the coin conundrum! We managed the money mystery! We checked the change charade! We … well you get the idea. Last post we (finally) determined if our coin was bias or not. Don’t worry, I won’t spoil it for you if you haven’t read it yet. I actually enjoyed working through a completely made up problem, so if you haven’t read it, you really should. Today we’re going to talk dogs, you’ll see what I mean, so let’s dive in.*

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Day #15 : Significance, Part 3 – 365 DoA

Standard_deviation

Where does our observation fall on the probability density function?

It looks like we’ve arrived at part 3 of what is now officially a trilogy of posts on statistical significance. There is so much more to say I don’t want to quite call this the conclusion. Instead, let’s give a quick review of where we left off and we can get back to determining if an observed value is significant.*

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Day #14 : Significance, Part 2 – 365 DoA

Bar graph showing z score across trials
Bar graph showing z score across trials

Z-score bar graph that I made just for all of you using some data I had laying around. If you’re new to statistics it may not make sense, but rest assured we will make sense of it all!

Well here we are two weeks into 365DoA, I was excited until I realized that puts us at 3.8356% of the way done. So if you remember from last post we’ve started our significance talk, as in what does it mean to have a value that is significant, what does that mean exactly, and how to do we find out? Today is the day I finally break, we’re going to have to do some math. Despite my best efforts I don’t think we can finish the significance discussion without it and still manage to make sense. With that, let’s just dive in.*

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Day #13 : Significance, Part 1 – 365 DoA

Normally distributed data shown using a histogram plot
Normally distributed data shown using a histogram plot

Histogram of normally distributed data. It looks very… nomal. No it really is normally distributed, read on to find out what that means and how we can use it.

If you’ve read my last post I hinted that today we would discuss filtering. Instead I think I want to take this a different direction. That isn’t to say we won’t go over filtering, we most definitely will. Today I want to cover something else though, significance. So you’ve recorded your signal, took an ensemble average, and now how do we tell if it actually means something, or if you are looking at an artificial or arbitrary separation in your data (IE two separate conditions lead to no difference in your data). Let’s look at significance.*

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Day #12 : Signal, cutting through the noise – 365 DoA

example data

45 separate trials of very noisy data with the average of those trials (black). Believe it or not, this is actually very useful and very real data from something I am currently working on.

Noise, it can be troublesome. Whether you are studying and someone is being loud or you are trying to record something, noise is everywhere <stern look at people who talk during movies>. Interestingly enough the concept of noise in a signal recording sense isn’t all too different from dealing with talkative movie goers, so let’s talk noise!*

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Day #11 : Why even use the spectrogram? – 365 DoA

rotated spectrogram showing all three dimensions
rotated spectrogram showing all three dimensions

A spectrogram plot rotated so we can see all three dimensions.

So you wanna use a spectrogram… but why? What does a spectrogram do that we can’t do using some other methods for signal processing? As it turns out, there is a lot of reasons you may want to use the spectrogram and today we are going to cover some of those reasons and number four may shock you! (okay not really, what do you think this is a clickbait website?)*

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Day #10 : Spectrogram vs. the banana of uncertainty – 365 DoA

banana

The banana of uncertainty (okay, it’s not a real banana)

Well ten days in and we’ve just introduced the idea of the spectrogram. While a lot of this information is just the broad strokes, I like to think that we’ve covered enough to give you a good idea about how to use these tools and what they are used for. However, we do need to discuss a limitation to the spectrogram, something called the banana of uncertainty, okay not quite the name, but you’ll see why I keep calling it that.*

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Day #9 : Reading a Spectrogram – 365 DoA

rotated spectrogram showing all three dimensions
other example

Definitely not the same spectrogram as yesterday, no really look. Now for the part where I tell you how to read this thing…

Last post we introduced a new tool in our arsenal of signal processing analysis, the spectrogram. Without knowing how to read it, it just looks sort of like a colored mess. Don’t get me wrong, it is an interesting looking colored mess, but a mess nonetheless. Well today we are going to talk about how to interpret the plot and why exactly we would ever use this seeming monstrosity.*

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Day #8 : The Spectrogram Function – 365 DoA

Spectrogram

Example spectrogram from some data I had recently processed

To (somewhat) continue with our signal processing theme that we have going on at the moment, over the next few days, let’s look at something called the spectrogram. It’s three dimensions of fun!*

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Parents’ math skills ‘rub off’ on their children

children in math

children in math

Parents who excel at math produce children who excel at math. This is according to a recently released study, which shows a distinct transfer of math skills from parent to child. The study specifically explored intergenerational transmission–the concept of parental influence on an offspring’s behavior or psychology–in mathematical capabilities.

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Anyone hear the new Taylor Shift song?

Since I’m apparently feeling a little punny…

Taylor shift


New theory tries to define where black holes don’t exist

black hole

The quintessential feature of a black hole is its “point of no return,” or what is more technically called its event horizon, yes just like the movie. When anything—a star, a particle, or wayward human—crosses this horizon, the black hole’s massive gravity pulls it in with such force that it is impossible to escape. At least, this is what happens in traditional black hole models based on general relativity. In general, the existence of this event horizon is responsible for most of the strange phenomena associated with black holes.

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Pythagoras theorem could improve patient care

patient care

Triangles, few of us have ever thought of a relationship between healthcare and triangles. Most of us will remember Pythagoras theorem from our school days, but rarely have a reason to use it in day-to-day life. Well for Doctors that might change, a team of medical researchers has found the 2,500-year-old Pythagoras theorem could be the most effective way to identify the point at which a patient’s health begins to improve.

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The Holographic Universe [we might Live in!]

matrix

Are you feeling a little… flat? Well that might be because you are only in 2 dimensions. I know what you’re thinking, insane! Well first check the name of the business and second, check out the science. In fact, it may seem like a joke, but the math suggests that it could very well be true and with it could come a deeper understanding of the universe. Testing this hypothesis (which was first made in the late 90’s) has been harder to do than you might think, but that has now changed. We are officially checking to see if our universe is a hologram!

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Voyager has hit interstellar space…. maybe?

intersteller space

Poor Voyager, he just can’t catch a break. We’ve said it’s hit interstellar space more times than we want to admit and in 2012, the Voyager mission team announced that the Voyager 1 spacecraft had passed into interstellar space [confirmed late 2013], traveling further from Earth than any other man-made object.

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