Thursday, August 27, 2020

Filtering Complex Data with R

Filtering

Base R has a function filter that can perform moving average (MA) or filters (AR). But it cannot do both at the same time; it cannot do an autoregresive ARMA filter. It also only works on real data. The filter function in the signal package can perform ARMA filters, but is is built on the filter function in the base package. So, it also cannot filter complex data. Below is an example of filtering some data with the filter from the signal package. We are using a 6 sample MA filter as specified by the b=rep(1/6,6).

N <- 64
x <- rnorm(N)
b <- rep(1/6,6)
a <- 1
y <- signal::filter(b,a,x)

The figure below has two plots. The blue circles are the Gaussian samples. The green circles are the filtered Gaussian samples. The filtered data has less variance.

Complex Data

In engineering the most common kind of complex data is the circularly symmetric gaussian noise. The circularly symmetric part means that the real and complex part are independent and identically distributed. I described the makeCG function in another post, but as you can see the real part and the imaginary part are made using separate calls to the rnorm function.

makeCG <- function(N,v=1) {(sqrt(v/2))*rnorm(N) + (sqrt(v/2))*1i*rnorm(N)}

The function filterComplex filters real or complex data. The function inputs the MA parameters b and the AR parameters a, and the data x. If the data is complex, it makes two filter calls: one with the real part and one with the imaginary part. If the data is real it make one call.

filterComplex <- function(b, a, x) {
  if (is.complex(x)) {
    y <- signal::filter(b,a,Re(x)) + 1i*signal::filter(b,a,Im(x))
  } else y <- signal::filter(b,a,x)
  return(y)
}

Here we simulate complex Gaussian data and use the function filterComplex to filter it.

N <- 64
x <- makeCG(N)
b <- rep(1/6,6)
a <- 1
y <- filterComplex(b,a,x)

The figure below has two plots. The blue circles are the real part of the complex Gaussian samples. The green circles are the real part of filtered complex Gaussian samples. As expected, the filtered data has less variance.

Now the imaginary part. The blue circles are the imaginary part of the complex Gaussian samples. The green circles are the imaginary part of filtered complex Gaussian samples.

Wednesday, June 24, 2020

Dungeons and Dragons: Advantage

D20 and Random Events

In the game Dungeons and Dragons, the success or failure of an event is determined by rolling a 20 sided die (D20): higher is better. If you need to roll 11 or higher you have a 50% chance of success. If another event requires 10 or better you now have a 55% chance for success. Each point the roll goes up or down is worth 5%. Often times, rolling a 20 is a critical success and a 1 is a critical failure; critical means it's extra good/bad.

Advantage

Sometimes things are really going your way and you roll with advantage. Sometimes things are not in your favor and you roll with disadvantage. When rolling with advantage, roll two dice and pick the higher. When rolling with disadvantage, roll two dice and pick the lower. How do advantage and disadvantage affect the chance of getting a 20 or 1?

When rolling without advantage or disadvantage, the probability of getting a 20 or a 1 is 1/20 = 0.05 or 5%. The probability of not getting a twenty 1 - 1/20 = 19/20.

P20    <- 0.05
Pnot20 <- 1-P20

When you have advantage, to not get a 20, you have to not roll a 20 twice. The probability of getting a 20 is (1 - the proability of not getting a 20 twice), and as you can see below, is almost 10%. Advantage about doubles you chance of getting a 20. But you probability guessed that since you're rolling twice. :) The chance of getting a 1 is 0.05^2=0.0025, almost 0!

Pnot20*Pnot20
## [1] 0.9025
1 - Pnot20*Pnot20
## [1] 0.0975

How will advantage and disadvantage affect the average roll? This time let's estimate the answer using a simulation. A 1000 trials will give a good estimate.

Trials <-10000

Below we simulate rolling two D20s

set.seed(1)
x <- sample(1:20,Trials,replace = TRUE)
y <- sample(1:20,Trials,replace = TRUE)

When we have advantage, roll two D20 and pick the max. With disadvantage roll two D20 and pick the min.

RollsWithAdvantage <- apply(cbind(x,y), 1,max)
Advantage    <-mean(RollsWithAdvantage)
RollsWithDisadvantage <- apply(cbind(x,y), 1,min)
Disadvantage <-mean(RollsWithDisadvantage)

Below is the simulated advantage mean, the calculated mean of a regular D20 roll, and the simulated disadvantage mean. Advantage adds about 3.4 points and disadvantage subtracts 3.4 or about +/- 17%.

Advantage
## [1] 13.8863
mean(1:20)
## [1] 10.5
Disadvantage
## [1] 7.1287

Below are 3 figures with histograms. The first histogram is made with 10,000 simulated D20 advantage rolls. Advantage moves a lot of probability to the right. The second histogram is made using 10,000 simulated D20 regular rolls. The histogram is approximately flat and each number is at about 0.05% and that's what we calculated. The last histogram is made with 10,000 simulated D20 disadvantage rolls. Disadvantage moves an equal amount to the left.

The probability of getting 11 or better with advantage is 1- probability of getting 10 or less twice. Rolling with advantage moves a 50% chance to a 75% chance!

p10orLess <- 0.5
1 - p10orLess^2
## [1] 0.75

Saturday, June 20, 2020

Complex Normal Samples In R

Normal Samples

If we want 10 samples from a Gaussian or normal random process with variance 4 can use rnorm(10,sd=2). Remember the standard deviation (sd) is the square root of the variance.

set.seed(2)
x <- rnorm(10,sd=2)
x
##  [1] -1.7938291  0.3696984  3.1756907 -2.2607513 -0.1605035  0.2648406
##  [7]  1.4159095 -0.4793960  3.9689479 -0.2775740
var(x)
## [1] 3.880803

The var() function produces an estimate of the variance, if we want a better estimate we need more samples.

var(rnorm(1000,sd=2))
## [1] 4.105966

Complex Normal Samples

If we are using base R and want complex normal (CN) samples, we need to write our own function. When the signal processing literature refers to CN they are usually referring to circularly-symmetric CN. Circularly-symmetric means the samples are independent and their mean is 0.

The function produces N CN samples with variance v. The real and imaginary parts are independent, because they are produced by different calls to rnorm(). Let x,y be independent. The var(ax) = a^2var(x) and var(x+y)=var(x)+var(y). So, if we want a variance of 1 would have to start a variance of sqrt(1/2).

makeCN <- function(N,v=1) {(sqrt(v/2))*rnorm(N) + (sqrt(v/2))*1i*rnorm(N)}
makeCN(10)
##  [1]  0.0023376-0.2079938i  0.7613032+0.6053620i  0.3946671-0.4049715i
##  [4]  0.4892950-0.1207824i  0.4651165-0.2871364i -0.2312504+0.9408834i
##  [7] -0.2153405-0.9648887i -1.0994866+1.0119199i  1.0396552+0.7824796i
## [10]  0.1147878+0.9059002i

If we want to check the variance, we can't use var() directly.

var(makeCN(10))
## Warning in var(makeCN(10)): imaginary parts discarded in coercion
## [1] 0.6039204

But the real and imaginary parts are independent, so we can calculate the variance separately.

z <- makeCN(10)
var(Re(z)) + var(Im(z))
## [1] 0.5623849

To make this easier, we can create a function to find the variance.

varComplex <- function(z) var(Re(z)) + var(Im(z))

To get a good estimate we need a-lot of samples.

varComplex(makeCN(1000))
## [1] 1.016615

Let's set the variance to 2 and then estimate the variance of the samples.

varComplex(makeCN(1000,v=2))
## [1] 1.925119

Success!

Tuesday, May 12, 2020

Book Review: Introductory Time Series with R

I'm a big fan of R and time series analysis, so I was excited to read the book "Introductory Time Series with R. I've been using the book for about 9 years, so I thought it was about time for a review! In this review, I'm going to cover the following topics: the amount of R content, the subject content, who is the book for, and my overall recommendation.

R Content
The R content is high. All the ideas in the book are heavily illustrated with R code. At the beginning of the book, the authors point out that they use Sweave to embed the code and plots. They also make use of online data sets, so you can type in examples from the book and reproduce the calculations and figures.

Time Series Content
The book covers all the time series topics you'd want in an introduction, plus a few specialty topics like multivariate models. Each chapter is a solid introduction to a topic in time series analysis.

Who is the Book For
1) People who what to learn time series analysis. It covers the theory, application, and has plenty of opportunities for hands-on learning. 2) People who want to learn to do time series analysis using R. That's why I bought the book! I'm familiar with time series analysis and I bought the book to learn how to do time series stuff with R. The book definitely delivered on that account. 3) People who want to teach a course on time series analysis. The book has plenty of examples and exercises (not sure if there is a solution manual).

Overall Recommendation
The book is well written. The R code is clear and well presented. The figures are numerous and informative. The book is wonderful and I highly recommend it!

https://www.amazon.com/gp/product/0387886974/ref=ppx_yo_dt_b_search_asin_title?ie=UTF8&psc=1