Up to now we’ve been dealing with single variable pdf and the corresponding CDF. We said that these probabilities relied on the fact that our variable of interest was independent. However, what if we knew some property that impacted our probability? Today we are talking conditional probability and that is the question we will be answering. It’s going to be a long, long post so plan accordingly.*
Maybe we shouldn’t phrase it this way, since there is still quite a few days left of 365DoA, but you made it to the end! No, not THE end, but if you’ve been following along the past few posts we’ve introduced several seemingly disparate concepts and said, “don’t worry they are related,” without telling you how. Well today, like a magician showing you how to pull a rabbit from a hat, let’s connect the dots and explain why we introduced all those concepts!*
Over the past couple of days, I’ve been talking about several different types of pdf and the associated C.D.F. Hopefully, we have a clear understanding of each of those concepts, for those of you scratching your head, I would recommend you start here at this other post. Otherwise, let’s (finally) look at a real life example using the exponential pdf!*
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.*
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.*
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*
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!*
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).*
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!*
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!*
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.*
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!*
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?)*
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.*
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.*
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!*
Waves! We’re officially one week through 365 Days of Academia! Woo! 1 week down, 51(.142…) weeks left! Let’s wrap up this weeks theme (there wasn’t originally a theme, but it kind of ended up that way) by talking about other ways we can get to the frequency domain. Specifically, let’s stop the wave puns and let’s talk wavelets!*
Okay, if you’ve been keeping up with these posts, we know about Welch’s method, Thomson’s method, the things that make them different, and the things that make them similar. The thing that both of these transforms rely on is the Fourier transform. What is the Fourier transform? Well, something I probably should have covered first, but whatever this is my blog we do it in whatever order we feel like, so let’s dive in!*
One day someone looked at the windowed fourier transform and said, “Don’t be such a square!” and thus window functions were invented. If you believe that, then I have an island for sale, real cheap. But seriously, let’s do a dive into what a window function is and why the heck there are so many of them, because there ARE a LOT! So let’s get started!*
Leakage, it’s never a good thing. For today’s post we’re going to cover a very important topic. Spectral leakage, it’s a big reason why spectral density estimation is well, an estimation. The other reason it is an estimation is because the fourier transform is an approximation of the original signal, but the Fourier transform is a whole other post on its own. So let’s talk leakage!*
In our last post we introduced the two main characters in this story of spectrogram. On one end we have Welch’s method (pwelch) on the other end we have the Thomson multitaper method (pmtm). As promised here is a
awful basic breakdown of why is more than one way to compute power spectral density (in fact there are several, far more than the two I’m talking about). So, let’s just dig right in!*
This is a (somewhat) continuation on what we were discussing in the previous post. We covered the pwelch MATLAB function, this time we will cover the PMTM function, this function uses the Thomson multitaper method to calculate power spectral density. We can do a deep dive into the differences between the two next time, but for now let’s talk about the command itself.*
Signal processing, it’s complex, there are a million ways to go about processing a signal, and like life, there is no best way to go about doing it. Trust me, it is as frustrating as it sounds. Today’s scratch pad note is on power spectral density or PSD for short. So let’s dive in.*
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.
At the dinner table, babies do a lot more than play with their sippy cups, new research suggests. Babies pay close attention to what food is being eaten around them – and especially who is eating it. The study adds evidence to a growing body of research suggesting even very young children think in sophisticated ways about subtle social cues.
Recently, scientists discovered a new learning rule for a specific type of excitatory synaptic connection in the hippocampus. These synapses are located in the so-called CA3 region of the hippocampus, which plays a critical role for storage and recall of spatial information in the brain. One of its hallmark properties is that memory recall can even be triggered by incomplete cues. This enables the network to complete neuronal activity patterns, a phenomenon termed pattern completion.
The human brain was initially used for basic survival tasks, such as staying safe and hunting and gathering. Yet, 200,000 years later, the same human brain is able to learn abstract concepts, like momentum, energy, and gravity, which have only been formally defined in the last few centuries. New research has now uncovered how the brain is able to acquire brand new types of ideas.
A direct and positive link between pupils’ breakfast quality and consumption, and their educational attainment, has for the first time been demonstrated in a ground-breaking new study carried out by public health experts at Cardiff University. The study of 5000 9-11 year-olds from more than 100 primary schools sought to examine the link between breakfast consumption and quality and subsequent attainment in Key Stage 2 Teacher Assessments* 6-18 months later.
A new study suggests that receiving rewards as you learn can help cement new facts and skills in your memory, particularly when combined with a daytime nap. The findings from the University of Geneva reveal that memories associated with a reward are preferentially reinforced by sleep. Even a short nap after a period of learning is beneficial.
Wikipedia reigns. It’s the world’s most popular online encyclopedia, the sixth most visited website in America, and a research source most U.S. students rely on. But Wikipedia entries on politically controversial scientific topics can be unreliable due to information sabotage.
Two prominent scientists with drastically different views on the relationship of science and religion – Richard Dawkins and Francis Collins – have an equally different influence on these views among people who are unfamiliar with their work, according to new research from Rice University and West Virginia University.
Love the opera? Hungry for hip hop? It turns out that your musical likes and dislikes may say more about you than you think, according to UBC research. Even in 2015, social class continues to inform our cultural attitudes and the way we listen to music, according to the study.
Many years of research have shown that for students from lower-income families, standardized test scores and other measures of academic success tend to lag behind those of wealthier students. Well now a new study offers another dimension to this so-called “achievement gap”After imaging the brains of high- and low-income students, they found that the higher-income students had thicker brain cortex in areas associated with visual perception and knowledge accumulation.
Your child is your pride and joy — and why not, every parent should be a proud one, even if your child might be bird brained. Or maybe birds are baby brained? In any case, a new study has found that pigeons can categorize and name both natural and manmade objects–and not just a few objects. These birds categorized 128 photographs into 16 categories, and they did so simultaneously.
Sleep is a hot topic lately, are we getting too much, too little, how much is enough? However, most of these questions are for adults, so what about children? Well as it turns out a new study used activity monitors to track how sleep habits changed in younger and older teens as they grew during a two-year period. Key findings from this study has also lent t0 new support to recent recommendations by the American Academy of Pediatrics that middle and high schools avoid starting earlier than 8:30 a.m.
Order matters, we all know this when it comes to math, but did you know the order of questions asked can affect how you answer them? It’s true and it isn’t new news, the question-order effect is why survey organizations normally change the order of questions between different respondents, hoping to cancel out this bias. But that isn’t the interesting part, not by a long shot.
It turns out that quantum theory is a much better predictor of the survey results than conventional methods of predictions.
Solar Roadways, I know most people have been in support of the new blossoming technology and I’m happy to be a part of that [at least in support]. However, no matter where I turn there are a handful of common concerns that are brought up against the technology. Well today I wanted to go over five of the main concerns. I also wanted to take a peek into what the future could look like, with solar roads.
Let’s take a Loony quiz! Do you believe any of these statements are true? Global warming isn’t real. GMO food is the devil. Organic and all natural are better. Science is just a belief like religion. Evolution is just a theory, so other theories should be taught along side. Vaccines do — or can– cause serious health concerns. If you answered yes to any of these questions, then you might be suffering from a lack of scientific understanding, but don’t worry you’re not alone.
Vermont, not quite the armpit of the United States, but not a place I would live [personally speaking of course]. Still, looking at history Vermont was the first to ban slavery [good], but now they are the first to do something else too, they are looking to ban all food that is genetically modified if it is not properly labeled [bad].
This bill is set to start a wildfire across the US with food scares, like any science scare, is easy to start and hard to stop [if at all]. Genetically modified food has helped cushion the ever growing population and the need to feed that population. People will [undoubtedly] argue otherwise, but all food is genetically modified one way or another.
Part of getting an education isn’t about learning what to think, it is about learning how to find good information. When it comes to scientific literature [especially on-line], it’s hard to separate the good from the bad if you don’t know what to look for. With the emergence of pseudoscience in the mainstream I think it’s important to go over a few red flags when it comes to claims being made.