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### benny

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Itô Processes and The Fundamental Theorem of Stochastic Calculus

Throughout, let $$W$$ be a Wiener process starting at the origin, and let $$X$$ be a stochastic process adapted to $$W$$’s filtration. $$I_t=I_t(X)$$ is the Itô integral of $$X$$ w.r.t. to $$W$$.

# 1. Itô Processes

There are at least two motivations for the Itô process. The first comes from the identity

$\bigg(\int_0^t \dif W\bigg)^2 = W_t^2 = 2\int_0^t W\dif W + t,$

which we proved previously. Notice that $$t$$ cannot be the result of any Itô integral, since $$\E[t]=t\neq 0$$, whereas the Itô integral is a martingale and hence is zero in expectation. Thus, the family of Itô integrals is not closed under smooth transformations. In this case the transformation is $$x\mapsto x^2$$.

The second is simply that, viewed as a stochastic process, the integral $$I_t$$ is less interesting – or, at least, contains fewer degrees of freedom – than we might have hoped. We might want processes to capture trends with tendencies to drift.

We can remedy both of these issues at once by introducing the class of stochastic processes $$X_t$$ which evolve as follows:

$$$\label{eq:ito_process_int} X_T-X_0 = \int_0^T \mu(X_t,t)\dif t + \int_0^T \sigma(X_t,t)\dif W_t. \tag{1}$$$

The rightmost integral is an Itô integral, where $$\sigma(X_t,t)$$ is some stochastic process adapted to the brownian motion $$W$$. The leftmost integral is just a Lebesgue integral with respect to time, which will give a real number for each $$\omega$$ in the sample space $$\Omega$$ on which $$X_t$$ is defined.

We’re going to re-write this equation in the differential form

$$$\label{eq:sde} \dif X_t = \mu(X_t,t)\dif t + \sigma(X_t,t)\dif W_t. \tag{2}$$$

This, in all it’s glory, is what’s referred to as a stochastic differential equation (SDE). It’s important to note that Equation $$\eqref{eq:sde}$$ isn’t actually a precise mathematical statement. It’s simply shorthand for Equation $$\eqref{eq:ito_process_int}$$. But it’s definitely easier to write, and also has a nicer intuitive interpretation. Namely, a small change $$\Delta$$ in $$X_t$$ is normally distributed with expectation $$\mu(X_t,t)\Delta$$ and variance $$\sigma(X_t,t)^2\Delta$$ (hence the choice of notation). People often find it easier to work with the SDE form instead of the integral form, especially because you can basically operate differentials as you would expect (even though it’s informal) and everything works out.

While it may not seem so at first glance, almost all stochastic processes – except those with jumps – are Itô processes.

## 1.1 Example 1: Brownian motion with drift

Suppose $$\mu$$ and $$\sigma$$ are constants. Then, if $$X_0=0$$ (which we will henceforth assume), Equation $$\eqref{eq:ito_process_int}$$ reads $$X_t = \mu t + \sigma W_t$$, and as an SDE we have

$\dif X_t = \alpha \dif t + \beta \dif W_t.$

If $$X$$ satisfies this equation, it’s called Brownian motion with drift $$\alpha$$ and variance $$\beta^2$$. It’s fairly clear why: $$\E[X_t] = \alpha t$$ (since $$\E[W_t]=0$$) and

$\Var[X_t] = \E[X_t^2] - \alpha^2 t^2 = \beta^2 \E[W_t^2] = \beta^2t,$

due to the quadratic variation of the Wiener process.

## 1.2 Example 2: Geometric Brownian motion (GBM)

Suppose $$\mu(X_t,t)=\alpha X_t$$ and $$\sigma(X_t,t) = \beta X_t$$ for constants $$\alpha$$ and $$\beta$$. Then,

$X_t = X_0 + \alpha\int_0^t X_t\dif t + \beta\int_0^t X_t\dif W_t,$

or,

$\dif X_t= \alpha X_t\dif t + \beta X_t\dif W_t.$

GBM is so important it has its own wikipedia page (what else could possibly signal value?). It’s often used to model prices in mathematical finance. Itô’s lemma will give us a way to solve it.

## 1.3 Integrals with respect to an Itô Process

So far we’ve seen stochastic integrals with respect to Brownian motion. But now that we’ve introduced a new class of processes, we should discuss integrals with respect to those.

Let $$X_t$$ be an Itô process: $$\dif X = \mu(X,t)\dif t + \sigma(X,t)\dif W$$. Let $$Y_t$$ be a process adapted to $$X_t$$. Then we define the integral of $$Y$$ w.r.t $$X$$ as

$\int_0^t Y \dif X = \int_0^t Y \mu \dif s + \int_0^t Y \sigma \dif W,$

which is what you might expect. As usual, the integral on the left is a Lebesgue integral w.r.t to the time component, and the integral on the right is an Itô integral.

# 2. Aside: Manipulating Infinitesimals

Even though Equation $$\eqref{eq:sde}$$ is informal, it’s intuitive enough that you’ll often see manipulations done to such equations directly, instead of reverting to the integral version. This isn’t completely out of the ordinary, something similar happens when we do change of variables in intro calculus. E.g., if we have $$\int f(u) f'(u) \dif u$$, we might let $$v=f(u)$$ and $$dv = f'(u)du$$ to deal instead with the integral $$\int v \dif v$$. The equation $$dv=f'(u)du$$ isn’t meaningful in a precise, technical sense, but it’s still useful to get the final result.

In SDE land, there are a few very common - and very useful - infinitesimal equations to keep in mind:

$(\dd t)^2 = 0,\quad \dd W\dd t = 0,\quad (\dd W)^2 = \dd t.$

Intuitively, something like $$(\dd W)^2 = \dd t$$ should hold because of the quadratic variation of Brownian motion. Rigorously, what this statement means is that for well-behaved (namely, bounded and continuous) functions $$\varphi$$ acting on $$W$$, that

$\sum_{i=0}^{n-1} \varphi(W_i)(W_{i+1}-W_i)^2 \to \int_0^T \varphi(W)\dif t,$

where the convergence is $$L^2$$. That is, one shows that

$\E\bigg\{\bigg(\sum_{i=0}^{n-1}\varphi(W_i)(\Delta W_i^2 - \Delta t_i)\bigg)^2\bigg\}\xrightarrow{n\to\infty}0.$

Similarly for the other identities. This isn’t incredibly hard but it’s not incredibly insightful either, so we’re not giving the proof. Here are some details if you want them.

# 3. Itô’s Lemma

Our first concern with the family of Itô integrals was that they are not closed under the transformation $$x\mapsto x^2$$, which motivated the construction of Itô processes. Itô’s lemma gives us a way to evaluate the transformation of Itô processes under sufficiently smooth transformations. To develop more intuition for the behavior of Brownian motion in the limit, it’s worth giving a sketch of the proof before the result is stated.

## 3.1 Proof Sketch

Let $$X_t$$ be an Itô process. For sufficiently smooth functions $$f(X,t)$$ of two variables, Taylor’s Theorem gives

\begin{align} f(X+\Delta X,t+\Delta t) &= f(X,t) + \frac{\partial f}{\partial X}\Delta X+ \frac{\partial f}{\partial t}\Delta t + \frac{1}{2}\frac{\partial^2f}{\partial X^2}(\Delta X)^2\tag{3}\label{eq:taylor1} \\ &\quad + \frac{1}{2}\frac{\partial^2f}{\partial t^2}(\Delta t)^2 + \frac{\partial^2f}{\partial X\partial t}\Delta X\Delta t + \text{err}, \tag{4}\label{eq:taylor2} \end{align}

where $$f$$ is evaluated at $$(X,t)$$, and the error term is

$\text{err} = o((\Delta W)^3 + (\Delta t)^3 + (\Delta t)^2\Delta X + (\Delta X)^2 \Delta t).$

Now, as $$\Delta t\to 0$$ and $$\Delta X \to 0$$, we have (using our favorite infinitesimal calculus) $$(\Delta t)^2 \to (\dd t)^2 = 0$$, and

\begin{align*} (\Delta X)^2 \to (\dd X)^2 &= (\mu\dif t+ \sigma \dif W)^2 \\ &= \mu^2 \dd t^2 + 2 \mu \sigma \dif t\dif W + \sigma^2 \dd W^2 \\ &= \sigma^2 \dif t. \end{align*}

Similarly,

$\Delta X\Delta t \to \dd X\dif t = \mu \dd t^2 + \sigma\dif W\dif t=0.$

Therefore, the three terms in line $$\eqref{eq:taylor2}$$ disappear in the limit, and only the terms in the top line remain.

Partition the interval $$[0,T]$$ into $$n+1$$ segments as $$t_0,t_1,\dots,t_{n}$$. Let $$X_i=X_{t_i}$$, $$\Delta X_i= X_{i+1} - X_i$$ and $$\Delta t_i = t_{i+1}-t_i$$. Now, write $$f(X_t,t) - f(X_0,0)$$ as a telescoping sum of small increments, apply Taylor’s theorem to each term, and take the limit as the partition gets finer and finer (thus ignoring the terms which disappear):

\begin{align} f(X_T,T)-f(X_0,0) &= \lim_{n\to\infty}\sum_{i=0}^{n-1} (f(X_{i+1},t_{i+1}) - f(X_i,t_i)) \\ &= \lim_{n\to\infty}\sum_{i=0}^{n-1} \bigg\{\frac{\partial f}{\partial X}\Delta X_i + \frac{\partial f}{\partial t}\Delta t_i + \frac{1}{2}\frac{\partial ^2 f}{\partial X^2}(\Delta X_i)^2\bigg\} \\ &= \lim_{n\to\infty}\sum_{i=0}^{n-1} \frac{\partial f}{\partial X}\Delta X_i + \lim_{n\to\infty}\sum_{i=0}^{n-1} \frac{\partial f}{\partial t}\Delta t_i + \lim_{n\to\infty}\frac{1}{2}\sum_{i=0}^{n-1} \frac{\partial^2 f}{\partial X^2}\sigma^2 \Delta t_i. \end{align}

In the final sum, we replaced $$\Delta X^2$$ with $$\sigma ^2 \Delta t$$ due to the arguments above. The first sum of the final line is precisely the integral of the Itô process $$X_t$$ with integrand $$f_x$$. Thus,

\begin{align} \lim_{n\to\infty}\sum_{i=0}^{n-1} \frac{\partial f}{\partial X}\Delta X_i & = \int_0^T \frac{\partial f}{\partial X}\dif X = \int_0^T \mu \frac{\partial f}{\partial X} \dif t + \int_0^T \sigma \frac{\partial f}{\partial X} \dif W. \end{align}

Meanwhile, the second and third sum are regular Riemann integrals with integrands $$f_t$$ and $$f_x\sigma^2$$ respectively. Putting this all together, we get

$f(X_T,T) = f(X_0,0) + \int_0^T \mu \frac{\partial f}{\partial X} \dd t + \int_0^T \sigma \frac{\partial f}{\partial X} \dd W + \int_0^T \frac{\partial f}{\partial t}\dd t + \frac{1}{2}\int_0^T \sigma^2 \frac{\partial^2 f}{\partial X^2}\dd t,$

which is called Itô’s lemma. Again, the partial derivatives and the functions $$\mu$$ and $$\sigma$$ are evaluated at $$(X_t,t)$$, we’ve just omitted this for brevity. Notice that for this statement to make sense we need that $$f\in C^2(\mathbb{R})$$ ($$f$$ is twice differentiable). More specifically, we need $$f(x,\cdot)\in C^1(\mathbb{R})$$ as a function of $$t$$ and $$f(\cdot,t)\in C^2(\mathbb{R})$$ as a function of $$x$$.

For standard Brownian motion $$W$$, which is an Itô process with $$\mu\equiv 0$$ and $$\sigma\equiv 1$$, Itô’s lemma gives

$f(W_T,T)=f(W_0,0) + \int_0^T \frac{\partial f}{\partial W} \dif W + \int_0^T \frac{\partial f}{\partial t}\dif t + \frac{1}{2}\int_0^T \frac{\partial^2 f}{\partial W^2}\dif t.$

## 3.2 SDE Form

Itô’s lemma is more commonly stated as an SDE. In this case, we write

\begin{align} \dd f(X,t) &= \mu(X,t)f_x(X,t)\dif t + \sigma(X,t) f_x(X,t) \dif W_t + f_t(X,t)\dif t \\ &\quad + \frac{1}{2}\sigma^2(X,t)f_{xx}(X,t)\dif t. \end{align}

Observe that we can gather the differential $$\dd t$$ and rewrite this as

$\dd f = \bigg(\mu f_x + f_t + \frac{1}{2}\sigma^2 f_{xx}\bigg)\dif t + \sigma f_x \dif W,$

which implies that $$f(X_t,t)$$ is itself an Itô process. This is good, as it was one of the primary motivators of defining the Itô process in the first place.

In the case of Brownian motion this simplies to

$\dd f(W,t) = f_x(W,t)\dif W + f_t(X,t) \dif t + \frac{1}{2} f_{xx}(W,t)\dif t.$

## 3.3 The Fundamental Theorem of Stochastic Calculus

Itô’s lemma is often considered the fundamental theorem of stochastic calculus. To see why, suppose that $$f$$ is a function of standard brownian motion $$W_t$$ only, $$f(W_t)$$. Since $$W_t$$ is still a function of $$t$$, we apply the chain rule and write

$\frac{df}{dt} = \frac{df}{dW}\dd W_t,$

which isn’t precisely rigorous, but it’s also not wrong. Since $$\dd W_t \dd t = 0$$, the integral in Itô’s lemma with integrand $$\frac{\partial f}{\partial t}$$ disappears, and we’re left with

$f(W_T)-f(W_0) = \int_0^T \frac{d f}{d W}(W_t)\dif W_t + \frac{1}{2}\int_0^T \frac{d^2 f}{d W^2}(W_t)\dif t.$

Contrast this with the fundamental theorem of (ordinary) calculus, where we don’t have the final term on the right hand side. Indeed, this term only exists in the stochastic case because of the quadratic variation of Brownian motion ($$(\dd W)^2 = \dd t$$). The smooth functions dealt with ordinarily have quadratic variation 0.

Equivalently, we can compare the (single variable) differential form

$\dd f(W_t) = f_x(W_t)\dif W + \frac{1}{2} f_{xx}(W_t)\dif t,$

with the chain rule from ordinary calculus:

$\frac{df}{dt}(x(t)) = f'(x)x'(t),$

or, in differential form,

$\dif f(x(t)) = f'(x)\dd x.$

Again, we see the difference is the non-zero quadratic variation of $$W$$.

# 4. Examples

Often we’re less interested in evaluating $$f(X_t,t)$$, but rather in evaluating some stochastic integral $$\int Y_t\dif X_t$$. If we can choose a function $$f$$ such that one of its partial derivatives equals $$Y_t$$, then we’re in business.

## 4.1 Back to Basics: $$\int_0^t W\dif W$$

We can solve the integral $$\int W\dif W$$ much more easily with Itô’s lemma. Consider $$f(W,t)=W^2$$. The lemma gives

$W_t^2 = 2\int_0^t W\dif W + t,$

which was obtained with much less hassle than before.

## 4.2 Stochastic Integration by Parts

Suppose $$g:\mathbb{R}\to\mathbb{R}$$ is a deterministic function of time. We’d like to say something about $$\int g(t) \dif W_t$$. To do this, we want a smooth function $$f$$ which, using Itô’s lemma, yields this integral. Since the lemma involves the derivatives of $$f$$, choosing $$f(x,t) = xg(t)$$ works nicely. Then $$f_x(x,t) = g(t)$$, $$f_{xx}(x,t)=0$$ and $$f_t(x,t) = g(t)x'(t) + g'(t)x(t)$$. Our favorite lemma gives:

\begin{align} W_Tg(T) &= \int_0^T g(t)\dif W_t + \int_0^T (W_tf'(t) + f(t)\dif W_t)\dif t \\ &= \int_0^T g(t)\dif W_t + \int_0^T W_tf'(t)\dif t, \end{align}

since $$\dd W_t\dd t=0$$. Rearranging and using the intuition that $$f'(t)\dif t = \dd f$$ (which can be shown rigorously), we get

$\int_0^T g(t) \dif W_t = W_Tg(T) - \int_0^T W_t\dd f(t).$

This looks a lot like the usual integration by parts formula. The rightmost integral should be interpreted as a Riemann-Stieltjes integral.

## 4.3 Geometric Brownian Motion

Recall that GBM satisfies

$\dd S_t = \mu S_t\dif t + \sigma S_t \dif W_t,$

for constants $$\mu$$ and $$\sigma$$. To solve this, we borrow some intuition from ODE theory and guess that logarithms will be involved. In particular, apply Itô’s lemma with $$f(x,t) = f(x) = \log x$$. As we saw above, since $$S_t$$ is also a function of $$t$$, the chain rule applies and $$f_t(S_t) = (S_t)^{-1} \dif S_t$$. Thus, Itô’s lemma gives

\begin{align} \dif \log(S_t) &= \bigg(\frac{\dd S_t}{S_t} + \mu S_t \frac{df}{dS} + \sigma^2\frac{S_t^2}{2}\frac{d^2f}{dS^2}\bigg)\dif t + \sigma S_t \frac{df}{dS} \dif W_t \\ &= \frac{d S_t}{S_t}\dif t + \mu\dif t - \frac{\sigma^2}{2}\dif t + \sigma \dif W_t\\ &= \bigg(\mu - \frac{1}{2}\sigma^2\bigg)\dif t + \sigma\dif W_t. \end{align}

Retreating from SDE notation, this means

\begin{align} \log(S_T) - \log(S_0) &= \int_0^T \bigg(\mu - \frac{1}{2}\sigma^2\bigg) \dif t + \int_0^T \sigma \dif W_t = \bigg(\mu-\frac{1}{2}\sigma^2\bigg) T + \sigma W_T. \end{align}

Exponentiating and performing the relevant arithmetic yields

$S_T = S_0\exp\bigg(\mu T - \frac{T\sigma^2}{2}\bigg)\exp(\sigma W_T),$

which gives us a closed form solution for GBM.

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# Homework Help: Manipulating Simple Harmonic Motion Equation

1. Apr 5, 2010

### Glorzifen

1. The problem statement, all variables and given/known data
An object moves with simple harmonic motion. If the amplitude and the period are both doubled, the object's maximum speed is:
A) Quartered
B) Halved
D) Doubled
E) Unchanged

2. Relevant equations
x(t) = Acos(wt + $$\varphi$$)
v(t) = -wAsin(wt + $$\varphi$$)

3. The attempt at a solution
Since 2 double the period (T) is to halve the frequency (f) because of (f = 1/T), and since omega (w) = 2$$\pi$$f...then w will be halved as well (right?):

f = 1/2T --> .5f --> 2$$\pi$$.5f = .5w

So that gives me: v(t) = -.5wAsin(.5w + $$\varphi$$)
And with the amplitude doubled: v(t) = -wAsin(.5w + $$\varphi$$)

Not sure if I did any of that correctly and I'm not sure what that means for my speed...is it halved? Any help would be appreciated. Thanks.

2. Apr 5, 2010

### Hellabyte

You are correct in your logic that $$\omega$$ is halved as well. Lets look at what the question is asking. What is the *Maxiumum* velocity? You have written the equation for velocity:

$$v(t) = A_0 \omega Sin(\omega t)$$

You should know that $$A_0$$ is the maximum amplitude.

Because the maximum value of any Sin function is 1, that means that the maximum velocity will be given by

$$v_{max} (t) = A_0 \omega$$

Now what will happen when the amplitude and period are both doubled?

3. Apr 5, 2010

### Glorzifen

Ah. Okay...I didn't know how to deal with the whole sine thing (obviously). So based on this Vmax equation the max velocity will remain unchanged. Correct?

4. Apr 5, 2010

### Hellabyte

Exactly. When dealing with maximum values, your sin and cosine functions will generally disappear for the reasons stated above.

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## FAQs & How-to's

#### What is this calculator for?

This mortgage calculator can be used to find the monthly repayment for an amortizing loan, as well as the term, rate, down payment and principal (loan amount).

#### Can I embed this on my website?

Sure. Embedding is allowed as long as you promise to follow our conditions. Here's the embed code:

 <iframe width="415" height="345" src="http://www.a-calculator.com/mortgage/embed.html" frameborder="0" allowtransparency="true"></iframe>

#### How can I do the math by hand?

The calculations are based on this formula: \begin{equation*} M = (L-D) \times \frac{i \times (1+i)^n}{(1+i)^n - 1}. \end{equation*}

Where \begin{align*} M &= \text{Monthly repayment,} \\ L &= \text{Loan amount (or principal),} \\ D &= \text{Down payment,} \\ i &= \frac{\text{Interest rate}}{\text{Number of compounding periods per year}} \end{align*} and \begin{align*} n &= \text{Total number of compounding periods.} \end{align*}

As an example, to find the monthly repayment for a \$250,000 loan at 7.5% over 25 years, you would do the following: \begin{align*} L &= 250,000 \\ D &= 0 \\ i &= 0.075 \div 12 = 0.00625 \qquad \text{(Compounded twelve times a year)} \\ n &= 25 \times 12 \end{align*} Then: \begin{align*} M &= (250,000 - 0) \times \frac{0.00625 \times (1 + 0.00625)^{300}}{(1 + 0.00625)^{300} - 1} \\ M &= 1847.48 \end{align*}


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## Exploratory Data Analysis: Combining Histograms and Density Plots to Examine the Distribution of the Ozone Pollution Data from New York in R

#### Introduction

This is a follow-up post to my recent introduction of histograms.  Previously, I presented the conceptual foundations of histograms and used a histogram to approximate the distribution of the “Ozone” data from the built-in data set “airquality” in R.  Today, I will examine this distribution in more detail by overlaying the histogram with parametric and non-parametric kernel density plots.  I will finally answer the question that I have asked (and hinted to answer) several times: Are the “Ozone” data normally distributed, or is another distribution more suitable?

Read the rest of this post to learn how to combine histograms with density curves like this above plot!

This is another post in my continuing series on exploratory data analysis (EDA).  Previous posts in this series on EDA include

## Exploratory Data Analysis – Kernel Density Estimation and Rug Plots in R on Ozone Data in New York and Ozonopolis

Update on July 15, 2013:

Thanks to Harlan Nelson for noting on AnalyticBridge that the ozone concentrations for both New York and Ozonopolis are non-negative quantities, so their kernel density plot should have non-negative support sets.  This has been corrected in this post by

- defining new variables called max.ozone and max.ozone2

- using the options “from = 0″ and “to = max.ozone” or “to = max.ozone2″ in the density() function when defining density.ozone and density.ozone2 in the R code.

For the sake of brevity, this post has been created from the second half of a previous long post on kernel density estimation.  This second half focuses on constructing kernel density plots and rug plots in R.  The first half focused on the conceptual foundations of kernel density estimation.

#### Introduction

This post follows the recent introduction of the conceptual foundations of kernel density estimation.  It uses the “Ozone” data from the built-in “airquality” data set in R and the previously simulated ozone data for the fictitious city of “Ozonopolis” to illustrate how to construct kernel density plots in R.  It also introduces rug plots, shows how they can complement kernel density plots, and shows how to construct them in R.

This is another post in a recent series on exploratory data analysis, which has included posts on descriptive statistics, box plots, violin plots, the conceptual foundations of empirical cumulative distribution functions (CDFs), and how to plot empirical CDFs in R.

Read the rest of this post to learn how to create the above combination of a kernel density plot and a rug plot!

## Exploratory Data Analysis: 2 Ways of Plotting Empirical Cumulative Distribution Functions in R

#### Introduction

Continuing my recent series on exploratory data analysis (EDA), and following up on the last post on the conceptual foundations of empirical cumulative distribution functions (CDFs), this post shows how to plot them in R.  (Previous posts in this series on EDA include descriptive statistics, box plots, kernel density estimation, and violin plots.)

I will plot empirical CDFs in 2 ways:

1. using the built-in ecdf() and plot() functions in R
2. calculating and plotting the cumulative probabilities against the ordered data

Continuing from the previous posts in this series on EDA, I will use the “Ozone” data from the built-in “airquality” data set in R.  Recall that this data set has missing values, and, just as before, this problem needs to be addressed when constructing plots of the empirical CDFs.

Recall the plot of the empirical CDF of random standard normal numbers in my earlier post on the conceptual foundations of empirical CDFs.  That plot will be compared to the plots of the empirical CDFs of the ozone data to check if they came from a normal distribution.

## Exploratory Data Analysis: Conceptual Foundations of Empirical Cumulative Distribution Functions

#### Introduction

Continuing my recent series on exploratory data analysis (EDA), this post focuses on the conceptual foundations of empirical cumulative distribution functions (CDFs); in a separate post, I will show how to plot them in R.  (Previous posts in this series include descriptive statistics, box plots, kernel density estimation, and violin plots.)

To give you a sense of what an empirical CDF looks like, here is an example created from 100 randomly generated numbers from the standard normal distribution.  The ecdf() function in R was used to generate this plot; the entire code is provided at the end of this post, but read my next post for more detail on how to generate plots of empirical CDFs in R.

Read to rest of this post to learn what an empirical CDF is and how to produce the above plot!

## Exploratory Data Analysis: Combining Box Plots and Kernel Density Plots into Violin Plots for Ozone Pollution Data

#### Introduction

Recently, I began a series on exploratory data analysis (EDA), and I have written about descriptive statistics, box plots, and kernel density plots so far.  As previously mentioned in my post on box plots, there is a way to combine box plots and kernel density plots.  This combination results in violin plots, and I will show how to create them in R today.

Continuing from my previous posts on EDA, I will use 2 univariate data sets.  One is the “ozone” data vector that is part of the “airquality” data set that is built into R; this data set contains data on New York’s air pollution.  The other is a simulated data set of ozone pollution in a fictitious city called “Ozonopolis”.  It is important to remember that the ozone data from New York has missing values, and this has created complications that needed to be addressed in previous posts; missing values need to be addressed for violin plots, too, and in a different way than before.

The vioplot() command in the “vioplot” package creates violin plots; the plotting options in this function are different and less versatile than other plotting functions that I have used in R.  Thus, I needed to be more creative with the plot(), title(), and axis() functions to create the plots that I want.  Read the details carefully to understand and benefit fully from the code.

Read further to learn how to create these violin plots that combine box plots with kernel density plots!  Be careful – the syntax is more complicated than usual!

## Exploratory Data Analysis: Kernel Density Estimation – Conceptual Foundations

For the sake of brevity, this post has been created from the first half of a previous long post on kernel density estimation.  This first half focuses on the conceptual foundations of kernel density estimationThe second half will focus on constructing kernel density plots and rug plots in R.

#### Introduction

Recently, I began a series on exploratory data analysis; so far, I have written about computing descriptive statistics and creating box plots in R for a univariate data set with missing values.  Today, I will continue this series by introducing the underlying concepts of kernel density estimation, a useful non-parametric technique for visualizing the underlying distribution of a continuous variable.  In the follow-up post, I will show how to construct kernel density estimates and plot them in R.  I will also introduce rug plots and show how they can complement kernel density plots.

A follow-up post will show how to construct these kernel density plots in R!

But first – read the rest of this post to learn the conceptual foundations of kernel density estimation.

## When Does the Kinetic Theory of Gases Fail? Examining its Postulates with Assistance from Simple Linear Regression in R

#### Introduction

The Ideal Gas Law, $\text{PV} = \text{nRT}$, is a very simple yet useful relationship that describes the behaviours of many gases pretty well in many situations.  It is “Ideal” because it makes some assumptions about gas particles that make the math and the physics easy to work with; in fact, the simplicity that arises from these assumptions allows the Ideal Gas Law to be easily derived from the kinetic theory of gases.  However, there are situations in which those assumptions are not valid, and, hence, the Ideal Gas Law fails.

Boyle’s law is inherently a part of the Ideal Gas Law.  It states that, at a given temperature, the pressure of an ideal gas is inversely proportional to its volume.  Equivalently, it states the product of the pressure and the volume of an ideal gas is a constant at a given temperature.

$\text{P} \propto \text{V}^{-1}$

#### An Example of The Failure of the Ideal Gas Law

This law is valid for many gases in many situations, but consider the following data on the pressure and volume of 1.000 g of oxygen at 0 degrees Celsius.  I found this data set in Chapter 5.2 of ”General Chemistry” by Darrell Ebbing and Steven Gammon.

               Pressure (atm)      Volume (L)              Pressure X Volume (atm*L)
[1,]           0.25                2.8010                  0.700250
[2,]           0.50                1.4000                  0.700000
[3,]           0.75                0.9333                  0.699975
[4,]           1.00                0.6998                  0.699800
[5,]           2.00                0.3495                  0.699000
[6,]           3.00                0.2328                  0.698400
[7,]           4.00                0.1744                  0.697600
[8,]           5.00                0.1394                  0.697000

The right-most column is the product of pressure and temperature, and it is not constant.  However, are the differences between these values significant, or could it be due to some random variation (perhaps round-off error)?

Here is the scatter plot of the pressure-volume product with respect to pressure.

These points don’t look like they are on a horizontal line!  Let’s analyze these data using normal linear least-squares regression in R.

## Webinar – Advanced Predictive Modelling for Manufacturing

The company that I work for, Predictum, is about to begin a free webinar series on statistics and analytics, and I will present the first one on Tuesday, May 14, at 2 pm EDT.  This first webinar will focus on how partial least squares regression can be used as a predictive modelling technique; the data sets are written in the context of manufacturing, but it is definitely to all industries that need techniques beyond basic statistical tools like linear regression for predictive modelling.  JMP, a software that Predictum uses extensively, will be used to illustrate how partial least squares regression can be implemented.  This presentation will not be heavy in mathematical detail, so it will be accessible to a wide audience, including statisticians, analysts, managers, and executives.

Attend my company’s free webinar to listen to me talking about advanced predictive modelling and partial least squares regression!

To register for this free webinar, visit the webinar’s registration page on Webex.

## Presentation Slides – Overcoming Multicollinearity and Overfitting with Partial Least Squares Regression in JMP and SAS

My slides on partial least squares regression at the Toronto Area SAS Society (TASS) meeting on September 14, 2012, can be found here.

#### My Presentation on Partial Least Squares Regression

My first presentation to Toronto Area SAS Society (TASS) was delivered on September 14, 2012.  I introduced a supervised learning/predictive modelling technique called partial least squares (PLS) regression; I showed how normal linear least squares regression is often problematic when used with big data because of multicollinearity and overfitting, explained how partial least squares regression overcomes these limitations, and illustrated how to implement it in SAS and JMP.  I also highlighted the variable importance for projection (VIP) score that can be used to conduct variable selection with PLS regression; in particular, I documented its effectiveness as a technique for variable selection by comparing some key journal articles on this issue in academic literature.

The green line is an overfitted classifier.  Not only does it model the underlying trend, but it also models the noise (the random variation) at the boundary.  It separates the blue and the red dots perfectly for this data set, but it will classify very poorly on a new data set from the same population.

## How to Calculate a Partial Correlation Coefficient in R: An Example with Oxidizing Ammonia to Make Nitric Acid

#### Introduction

Today, I will talk about the math behind calculating partial correlation and illustrate the computation in R with an example involving the oxidation of ammonia to make nitric acid using a built-in data set in R called stackloss.  In a separate post, I will also share an R function that I wrote to estimate partial correlation.  In a later post, I will discuss the interpretation of the partial correlation coefficient at length.

I read Pages 234-237 in Section 6.6 of “Discovering Statistics Using R” by Andy Field, Jeremy Miles, and Zoe Field to learn about partial correlation.  They used a data set called “Exam Anxiety.dat” available from their companion web site (look under “6 Correlation”) to illustrate this concept; they calculated the partial correlation coefficient between exam anxiety and revision time while controlling for exam score.  As I discuss further below, the plot between the 2 above residuals helps to illustrate the calculation of partial correlation coefficients.  This plot makes intuitive sense; if you take more time to study for an exam, you tend to have less exam anxiety, so there is a negative correlation between revision time and exam anxiety.

They used a function called pcor() in a package called “ggm”; however, I suspect that this package is no longer working properly, because it depends on a deprecated package called “RBGL” (i.e. “RBGL” is no longer available in CRAN).  See this discussion thread for further information.  Thus, I wrote my own R function to illustrate partial correlation.

Partial correlation is the correlation between 2 random variables while holding other variables constant.  To calculate the partial correlation between X and Y while holding Z constant (or controlling for the effect of Z, or averaging out Z),

## Checking the Goodness of Fit of the Poisson Distribution in R for Alpha Decay by Americium-241

#### Introduction

Today, I will discuss the alpha decay of americium-241 and use R to model the number of emissions from a real data set with the Poisson distribution.  I was especially intrigued in learning about the use of Am-241 in smoke detectors, and I will elaborate on this clever application.  I will then use the Pearson chi-squared test to check the goodness of fit of my model.  The R script for the full analysis is given at the end of the post; there is a particularly useful code for superscripting the mass number of a chemical isotope in the title of a plot.  While there are many examples of superscripts in plot titles and axes that can be found on the web, none showed how to put the superscript before a text.  I hope that this and other tricks in this script are of use to you.

Smoke Detector with Americium-241

## Presentation Slides: Machine Learning, Predictive Modelling, and Pattern Recognition in Business Analytics

I recently delivered a presentation entitled “Using Advanced Predictive Modelling and Pattern Recognition in Business Analytics” at the Statistical Society of Canada’s (SSC’s) Southern Ontario Regional Association (SORA) Business Analytics Seminar Series.  In this presentation, I

- discussed how traditional statistical techniques often fail in analyzing large data sets

- defined and described machine learning, supervised learning, unsupervised learning, and the many classes of techniques within these fields, as well as common examples in business analytics to illustrate these concepts

- introduced partial least squares regression and bootstrap forest (or random forest) as two examples of supervised learning (0r predictive modelling) techniques that can effectively overcome the common failures of traditional statistical techniques and can be easily implemented in JMP

- illustrated how partial least squares regression and bootstrap forest were successfully used to solve some major problems for 2 different clients at Predictum, where I currently work as a statistician

## How do Dew and Fog Form? Nature at Work with Temperature, Vapour Pressure, and Partial Pressure

In the early morning, especially here in Canada, I often see dew – water droplets formed by the condensation of water vapour on outside surfaces, like windows, car roofs, and leaves of trees.  I also sometimes see fog – water droplets or ice crystals that are suspended in air and often blocking visibility at great distances.  Have you ever wondered how they form?  It turns out that partial pressure, vapour pressure and temperature are the key phenomena at work.

Dew ( and Fog )

Source: Wikimedia

## Checking for Normality with Quantile Ranges and the Standard Deviation

#### Introduction

I was reading Michael Trosset’s “An Introduction to Statistical Inference and Its Applications with R”, and I learned a basic but interesting fact about the normal distribution’s interquartile range and standard deviation that I had not learned before.  This turns out to be a good way to check for normality in a data set.

In this post, I introduce several traditional ways of checking for normality (or goodness of fit in general), talk about the method that I learned from Trosset’s book, then build upon this method by possibly coming up with a new way to check for normality.  I have not fully established this idea, so I welcome your thoughts and ideas.

## Estimating the Decay Rate and the Half-Life of DDT in Trout – Applying Simple Linear Regression with Logarithmic Transformation

This blog post uses a function and a script written in R that were displayed in an earlier blog post.

#### Introduction

This is the second of a series of blog posts about simple linear regression; the first was written recently on some conceptual nuances and subtleties about this model.  In this blog post, I will use simple linear regression to analyze a data set with a logarithmic transformation and discuss how to make inferences on the regression coefficients and the means of the target on the original scale.  The data document the decay of dichlorodiphenyltrichloroethane (DDT) in trout in Lake Michigan; I found it on Page 49 in the book ”Elements of Environmental Chemistry” by Ronald A. Hites.  Future posts will also be written on the chemical aspects of this topic, including the environmental chemistry of DDT and exponential decay in chemistry and, in particular, radiochemistry.

Dichlorodiphenyltrichloroethane (DDT)

Source: Wikimedia Commons

A serious student of statistics or a statistician re-learning the fundamentals like myself should always try to understand the math and the statistics behind a software’s built-in function rather than treating it like a black box.  This is especially worthwhile for a basic yet powerful tool like simple linear regression.  Thus, instead of simply using the lm() function in R, I will reproduce the calculations done by lm() with my own function and script (posted earlier on my blog) to obtain inferential statistics on the regression coefficients.  However, I will not write or explain the math behind the calculations; they are shown in my own function with very self-evident variable names, in case you are interested.  The calculations are arguably the most straightforward aspects of linear regression, and you can easily find the derivations and formulas on the web, in introductory or applied statistics textbooks, and in regression textbooks.

## My Own R Function and Script for Simple Linear Regression – An Illustration with Exponential Decay of DDT in Trout

Here is the function that I wrote for doing simple linear regression, as alluded to in my blog post about simple linear regression on log-transformed data on the decay of DDT concentration in trout in Lake Michigan.  My goal was to replicate the 4 columns of the output from applying summary() to the output of lm().

To use this file and this script,

1) I saved this file as “simple linear regression.r”.

2) In the same folder, I saved a script called “DDT trout regression.r” that used this function to implement simple linear regression on the log-transformed DDT data.

3) I changed the working directory to this folder containing the function and the script.

4) I made sure “DDT trout regression.r” used the source() function to call my user-defined function for simple linear regression.

5) I ran “DDT trout regression.r”.

## Some Subtle and Nuanced Concepts about Simple Linear Regression

#### Introduction

This blog post will focus on some conceptual foundations of simple linear regression, a very common technique in statistics and a precursor for understanding multiple linear regression.  I will expose and clarify many nuances and subtleties that I did not fully absorb until my Master’s degree in statistics at the University of Toronto.

Future posts will explore the other important aspects of linear regression, including estimating its regression coefficients, statistical inference on the regression coefficients, checking the model’s assumptions, confidence bands, confidence intervals vs. prediction intervals, and analysis of variance.

## Discovering Argon with the 2-Sample t-Test

I learned about Lord Rayleigh’s discovery of argon in my 2nd-year analytical chemistry class while reading “Quantitative Chemical Analysis” by Daniel Harris.  (William Ramsay was also responsible for this discovery.)  This is one of my favourite stories in chemistry; it illustrates how diligence in measurement can lead to an elegant and surprising discovery.  I find no evidence that Rayleigh and Ramsay used statistics to confirm their findings; their paper was published 13 years before Gosset published about the t-test.  Thus, I will use a 2-sample t-test in R to confirm their result.

Photos of Lord Rayleigh

Source: Wikimedia Commons


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 Reliability Checks on the Indo-US Stellar Spectral Library Using Artificial Neural Networks and Principal Component AnalysisThe Indo-US coudé feed stellar spectral library (CFLIB) madeavailable to the astronomical community recently by Valdes et al. (2004,ApJS, 152, 251) contains spectra of 1273 stars in the spectral region3460 to 9464Å at a high resolution of 1Å (FWHM) and a widerange of spectral types. Cross-checking the reliability of this databaseis an important and desirable exercise since a number of stars in thisdatabase have no known spectral types and a considerable fraction ofstars has not so complete coverage in the full wavelength region of3460-9464Å resulting in gaps ranging from a few Å to severaltens of Å. We use an automated classification scheme based onArtificial Neural Networks (ANN) to classify all 1273 stars in thedatabase. In addition, principal component analysis (PCA) is carried outto reduce the dimensionality of the data set before the spectra areclassified by the ANN. Most importantly, we have successfullydemonstrated employment of a variation of the PCA technique to restorethe missing data in a sample of 300 stars out of the CFLIB. Stellar Parameters and Photospheric Abundances of Late-G Giants: Properties of the Targets of the Okayama Planet Search ProgramTowards clarifying the properties of late-G giants, for which we arecurrently conducting a planet-search project at Okayama AstrophysicalObservatory, an extensive spectroscopic study has been performed for ourfirst target sample of fifty-seven G6-G9 III stars, in order toestablish the atmospheric parameters (Teff, log g,vt, and [Fe/H]), the stellar mass along with the evolutionarystatus, and the photospheric abundances of various elements. It wasconfirmed that the conventional spectroscopic method of parameterdetermination using Fe I / II lines with the assumption of LTE workssuccessfully for these evolved stars. We determined the abundances(relative to the Sun) of 19 elements, and examined how their [X/Fe]ratios behave themselves with the run of [Fe/H]. While the trends appearto be similar to those exhibited by disk dwarfs for a number ofelements, some elements (C, O, Na) showed appreciable anomalies, whichmay be interpreted as being due to a dredge-up of nuclear-processedmaterial. Since the [Fe/H] values of these stars tend to be somewhatbiased towards a subsolar regime, some mechanism of slightly reducingthe metallicity might be suspected. Radial-Velocity Variability of G-Type Giants: First Three Years of the Okayama Planet Search ProgramWe report on the radial-velocity variability of G-type giants based onthe results of precise Doppler measurements of 57 stars that wereobserved for the first 3 years of the Okayama Planet Search Program.This program aims to search for planets around intermediate-mass starsin their evolved stages as late-G giants. We found that most of thetargets with B ‑ V < 1.0 have radial-velocity scatters ofσ  10 m s-1, with the most stable reaching levels of6‑8 m s-1, while those with B ‑ V > 1.0 typically showσ  20 m s-1. In total, about 60% of the targets are stable inradial velocity to a level of σ < 15 m s-1 over 1-3 years, andabout 90% have σ < 30 m s-1. We found that 4 stars (HD 30557,HD 34559, HD 68077, and HD 85444) show significant long-termradial-velocity trends, suggesting that they are probably orbited byunseen stellar or substellar companions. Three stars (HD 79181, HD104985, and HD 141680) show periodic radial-velocity variations. HD104985 is the first planet-harboring star discovered from our survey.The properties of the variations in these 3 stars have been, and willbe, extensively discussed in separate papers. Relatively large, but notapparently periodic, radial-velocity variations of σ > 30 m s-1are found in 4 stars (HD 41597, HD 134190, HD 161178, and HD 176598).For most of the stars without showing any significant periodicity, wecan exclude companions with K > 50 m s-1, or m2sini > 1.8 MJ (a/AU)1/2(M*/Mȯ)1/2 for orbital radii a ≲ 1‑2 AU. A catalog of bright calibrator stars for 200-m baseline near-infrared stellar interferometryWe present in this paper a catalog of reference stars suitable forcalibrating infrared interferometric observations. In the K band,visibilities can be calibrated with a precision of 1% on baselines up to200 meters for the whole sky, and up to 300 meters for some part of thesky. This work, extending to longer baselines a previous catalogcompiled by Bordé et al. (2002, A&A, 393, 183), isparticularl y well adapted to hectometric-class interferometers such asthe Very Large Telescope Interferometer (VLTI, Glindemann et al. 2003,Proc. SPIE, 4838, 89) or the CHARA array (ten Brummelaar et al. 2003,Proc. SPIE, 4838, 69) when one is observing well-resolved, high-surfacebrightness objects (K  8). We use the absolute spectro-photometriccalibration method introduced by Cohen et al. (1999, AJ, 117, 1864) toderive the angular diameters of our new set of 948 G8-M0 calibratorstars extracted from the IRAS, 2MASS and MSX catalogs. Angular stellardiameters range from 0.6 mas to 1.8 mas (median is 1.1 mas) with amedian precision of 1.35%. For both the northern and southernhemispheres, the closest calibrator star is always less than 10°away. CHARM2: An updated Catalog of High Angular Resolution MeasurementsWe present an update of the Catalog of High Angular ResolutionMeasurements (CHARM, Richichi & Percheron \cite{CHARM}, A&A,386, 492), which includes results available until July 2004. CHARM2 is acompilation of direct measurements by high angular resolution methods,as well as indirect estimates of stellar diameters. Its main goal is toprovide a reference list of sources which can be used for calibrationand verification observations with long-baseline optical and near-IRinterferometers. Single and binary stars are included, as are complexobjects from circumstellar shells to extragalactic sources. The presentupdate provides an increase of almost a factor of two over the previousedition. Additionally, it includes several corrections and improvements,as well as a cross-check with the valuable public release observationsof the ESO Very Large Telescope Interferometer (VLTI). A total of 8231entries for 3238 unique sources are now present in CHARM2. Thisrepresents an increase of a factor of 3.4 and 2.0, respectively, overthe contents of the previous version of CHARM.The catalog is only available in electronic form at the CDS viaanonymous ftp to cdsarc.u-strasbg.fr (130.79.128.5) or via http://cdsweb.u-strasbg.fr/cgi-bin/qcat?J/A+A/431/773 The Indo-US Library of Coudé Feed Stellar SpectraWe have obtained spectra for 1273 stars using the 0.9 m coudéfeed telescope at Kitt Peak National Observatory. This telescope feedsthe coudé spectrograph of the 2.1 m telescope. The spectra havebeen obtained with the no. 5 camera of the coudé spectrograph anda Loral 3K×1K CCD. Two gratings have been used to provide spectralcoverage from 3460 to 9464 Å, at a resolution of ~1 Å FWHMand at an original dispersion of 0.44 Å pixel-1. For885 stars we have complete spectra over the entire 3460 to 9464 Åwavelength region (neglecting small gaps of less than 50 Å), andpartial spectral coverage for the remaining stars. The 1273 stars havebeen selected to provide broad coverage of the atmospheric parametersTeff, logg, and [Fe/H], as well as spectral type. The goal ofthe project is to provide a comprehensive library of stellar spectra foruse in the automated classification of stellar and galaxy spectra and ingalaxy population synthesis. In this paper we discuss thecharacteristics of the spectral library, viz., details of theobservations, data reduction procedures, and selection of stars. We alsopresent a few illustrations of the quality and information available inthe spectra. The first version of the complete spectral library is nowpublicly available from the National Optical Astronomy Observatory(NOAO) via ftp and http. A statistical search for supermetallicity in F, G and K starsHigh-dispersion and low-resolution data are combined to search forsuper-metal-rich (SMR) FGK stars in the solar neighbourhood and Baade'sWindow. The data are assessed by using statistical analysis, with theirrms errors playing a key role. A star is considered to be SMR if itsvalue of [Fe/H]>+0.2dex, while borderline' SMR status may beassigned if +0.1<[Fe/H]<=+0.2dex. Borderline SMR status isassigned to μ Leo and three other giants, but no full-fledged SMRgiants are found in either Baade's Window or the solar neighbourhood. Bycontrast, the existence of SMR class IV-V stars turns out to be wellestablished, with values found for [Fe/H] that are as large as ~+0.4dex.It is concluded that this apparent contrast between class IV-V stars andgiants should not be interpreted in astrophysical terms at presentbecause of marked shortcomings in the available data base for giants.Recommendations are made about future research that may cure thisproblem and extend present knowledge about SMR dwarfs. CHARM: A Catalog of High Angular Resolution MeasurementsThe Catalog of High Angular Resolution Measurements (CHARM) includesmost of the measurements obtained by the techniques of lunaroccultations and long-baseline interferometry at visual and infraredwavelengths, which have appeared in the literature or have otherwisebeen made public until mid-2001. A total of 2432 measurements of 1625sources are included, along with extensive auxiliary information. Inparticular, visual and infrared photometry is included for almost allthe sources. This has been partly extracted from currently availablecatalogs, and partly obtained specifically for CHARM. The main aim is toprovide a compilation of sources which could be used as calibrators orfor science verification purposes by the new generation of largeground-based facilities such as the ESO Very Large Interferometer andthe Keck Interferometer. The Catalog is available in electronic form atthe CDS via anonymous ftp to cdsarc.u-strasbg.fr (130.79.128.5) or viahttp://cdsweb.u-strasbg.fr/cgi-bin/qcat?J/A+A/386/492, and from theauthors on CD-Rom. Lick Spectral Indices for Super-Metal-rich StarsWe present Lick spectral indices for a complete sample of 139 candidatesuper-metal-rich stars of different luminosity classes (MK type from Ito V). For 91 of these stars we were able to identify, in anaccompanying paper, the fundamental atmosphere parameters. This confirmsthat at least 2/3 of the sample consists of stars with [Fe/H] in excessof +0.1 dex. Optical indices for both observations and fiducialsynthetic spectra have been calibrated to the Lick system according toWorthey et al. and include the Fe I indices of Fe5015, Fe5270, andFe5335 and the Mg I and MgH indices of Mg2 and Mg b at 5180Å. The internal accuracy of the observations is found to beσ(Fe5015)=+/-0.32 Å, σ(Fe5270)=+/-0.19 Å,σ(Fe5335)=+/-0.22 Å, σ(Mg2)=+/-0.004 mag,and σ(Mg b)=+/-0.19 Å. This is about a factor of 2 betterthan the corresponding theoretical indices from the synthetic spectra,the latter being a consequence of the intrinsic limitations in the inputphysics, as discussed by Chavez et al. By comparing models andobservations, we find no evidence for nonstandard Mg versus Fe relativeabundance, so [Mg/Fe]=0, on the average, for our sample. Both theWorthey et al. and Buzzoni et al. fitting functions are found tosuitably match the data and can therefore confidently be extended forpopulation synthesis application also to supersolar metallicity regimes.A somewhat different behavior of the two fitting sets appears, however,beyond the temperature constraints of our stellar sample. Its impact onthe theoretical output is discussed, as far as the integratedMg2 index is derived from synthesis models of stellaraggregates. A two-index plot, such as Mg2 versus Fe5270, isfound to provide a simple and powerful tool for probing distinctiveproperties of single stars and stellar aggregates as a whole. The majoradvantage, over a classical CM diagram, is that it is both reddeningfree and distance independent. Based on observations collected at theInstituto Nacional de Astrofísica, Optica y Electrónica(INAOE) `G. Haro'' Observatory, Cananea (Mexico). Catalogue of Apparent Diameters and Absolute Radii of Stars (CADARS) - Third edition - Comments and statisticsThe Catalogue, available at the Centre de Données Stellaires deStrasbourg, consists of 13 573 records concerning the results obtainedfrom different methods for 7778 stars, reported in the literature. Thefollowing data are listed for each star: identifications, apparentmagnitude, spectral type, apparent diameter in arcsec, absolute radiusin solar units, method of determination, reference, remarks. Commentsand statistics obtained from CADARS are given. The Catalogue isavailable in electronic form at the CDS via anonymous ftp tocdsarc.u-strasbg.fr (130.79.128.5) or viahttp://cdsweb.u-strasbg.fr/cgi-bin/qcar?J/A+A/367/521 A catalog of rotational and radial velocities for evolved starsRotational and radial velocities have been measured for about 2000evolved stars of luminosity classes IV, III, II and Ib covering thespectral region F, G and K. The survey was carried out with the CORAVELspectrometer. The precision for the radial velocities is better than0.30 km s-1, whereas for the rotational velocity measurementsthe uncertainties are typically 1.0 km s-1 for subgiants andgiants and 2.0 km s-1 for class II giants and Ib supergiants.These data will add constraints to studies of the rotational behaviourof evolved stars as well as solid informations concerning the presenceof external rotational brakes, tidal interactions in evolved binarysystems and on the link between rotation, chemical abundance and stellaractivity. In this paper we present the rotational velocity v sin i andthe mean radial velocity for the stars of luminosity classes IV, III andII. Based on observations collected at the Haute--Provence Observatory,Saint--Michel, France and at the European Southern Observatory, LaSilla, Chile. Table \ref{tab5} also available in electronic form at CDSvia anonymous ftp to cdsarc.u-strasbg.fr (130.79.128.5) or viahttp://cdsweb.u-strasbg.fr/Abstract.html Catalogs of temperatures and [Fe/H] averages for evolved G and K starsA catalog of mean values of [Fe/H] for evolved G and K stars isdescribed. The zero point for the catalog entries has been establishedby using differential analyses. Literature sources for those entries areincluded in the catalog. The mean values are given with rms errors andnumbers of degrees of freedom, and a simple example of the use of thesestatistical data is given. For a number of the stars with entries in thecatalog, temperatures have been determined. A separate catalogcontaining those data is briefly described. Catalog only available atthe CDS via anonymous ftp to cdsarc.u-strasbg.fr (130.79.128.5) or viahttp://cdsweb.u-strasbg.fr/Abstract.html The ROSAT all-sky survey catalogue of optically bright late-type giants and supergiantsWe present X-ray data for all late-type (A, F, G, K, M) giants andsupergiants (luminosity classes I to III-IV) listed in the Bright StarCatalogue that have been detected in the ROSAT all-sky survey.Altogether, our catalogue contains 450 entries of X-ray emitting evolvedlate-type stars, which corresponds to an average detection rate of about11.7 percent. The selection of the sample stars, the data analysis, thecriteria for an accepted match between star and X-ray source, and thedetermination of X-ray fluxes are described. Catalogue only available atCDS via anonymous ftp to cdsarc.u-strasbg.fr (130.79.128.5) or viahttp://cdsweb.u-strasbg.fr/Abstract.html Vitesses radiales. Catalogue WEB: Wilson Evans Batten. Subtittle: Radial velocities: The Wilson-Evans-Batten catalogue.We give a common version of the two catalogues of Mean Radial Velocitiesby Wilson (1963) and Evans (1978) to which we have added the catalogueof spectroscopic binary systems (Batten et al. 1989). For each star,when possible, we give: 1) an acronym to enter SIMBAD (Set ofIdentifications Measurements and Bibliography for Astronomical Data) ofthe CDS (Centre de Donnees Astronomiques de Strasbourg). 2) the numberHIC of the HIPPARCOS catalogue (Turon 1992). 3) the CCDM number(Catalogue des Composantes des etoiles Doubles et Multiples) byDommanget & Nys (1994). For the cluster stars, a precise study hasbeen done, on the identificator numbers. Numerous remarks point out theproblems we have had to deal with. Photometric surveys of suspected small-amplitude red variables. 3: an AAVSO photometric photometry surveyWe have carried out a survey of the photometric (V) variability of 61'known' or suspected small-amplitude red variables, mostly M giants.Approximately two-thirds appear to be variable; several suspectedvariable comparison stars have also been identified. The incidence andaverage amplitude of variability increase rapidly from spectral type M0III to M6 III. Evolved GK stars near the Sun. 2: The young disk populationFrom a sample of nearly 2000 GK giants a group of young disk stars withwell determined space motions has been selected. The zero point of theluminosity calibrations, both from the ultraviolet flux (modifiedStroemgren system) and that in the region of 4200 to 4900 A (DDOsystem), show a discontinuity of about a half magnitude at the border ofthe young disk and old disk domains. The population separation is basedon the space velocity components, which are also an age discriminant,with the population interface near 2 x 109 yr, based onmodels with convective overshoot at the core. This age corresponds togiant masses near 1.7 solar mass, near the critical mass separating theyoung stars that do not burn helium in degenerate cores from older starsthat do. Ten percent of both populations show CN anomalies in that thederived value of P(Fe/H) from CN (Cm) and fromFe(M1) differ by more than 0.1 dex and the weak and strong CNstars occur equally in the old disk but the weak CN stars predominate inthe young disk. Peculiar stars, where flux distortions affect theluminosity calibrations, are of the CH+(Ba II) and CH-(weak G band)variety and represent less than 1% of the stars in both populations. Theyoung disk giants are restricted to ages greater than about109 yr, because younger stars are bright giants orsupergiants (luminosity class 2 or 1), and younger than about 2 x109 yr, because the old disk-young disk boundary occurs near1.7 solar mass. The distribution of heavy element abundances, P(Fe/H),for young disk giants is both more limited in range (+/- 0.4 dex) and isskewed toward higher abundances, compared with the nearly normaldistribution for old disk giants. The distribution of (U,V) velocityvectors gives (U,V,W) and their dispersions = (+17.6 +/- 18.4, -14.8 +/-8.4, -6.9 +/- 13.0) and (+3.6 +/- 38.4, -20.7 +/- 27.5, -6.7 +/-17.3)km/s for young and old disk giants, respectively. Velocity dispersions and mean abundances for Roman's G5-K1 spectroscopic groupsThe velocity dispersions and U-V distributions of Roman's (1950, 1952)four spectroscopic groups (weak CN, weak line, strong line, and 4150)are compared with those of groups based only on Fe/H ratio. It is shownthat the velocity-dispersion gradient for the Roman spectroscopic groupsis greater than for the comparison group for all three velocitycomponents and for the mean orbital eccentricity, with a clearly definedminimum for the strong-line stars and a sharp upturn for the 4150 stars.The results suggest that the Roman's assignment to distinctspectroscopic groups results in more homogeneous groups than the binningon the basis of metallicity. A critical appraisal of published values of (Fe/H) for K II-IV stars'Primary' (Fe/H) averages are presented for 373 evolved K stars ofluminosity classes II-IV and (Fe/H) values beween -0.9 and +0.21 dex.The data define a 'consensus' zero point with a precision of + or -0.018 dex and have rms errors per datum which are typically 0.08-0.16dex. The primary data base makes recalibration possible for the large(Fe/H) catalogs of Hansen and Kjaergaard (1971) and Brown et al. (1989).A set of (Fe/H) standard stars and a new DDO calibration are given whichhave rms of 0.07 dex or less for the standard star data. For normal Kgiants, CN-based values of (Fe/H) turn out to be more precise than manyhigh-dispersion results. Some zero-point errors in the latter are alsofound and new examples of continuum-placement problems appear. Thushigh-dispersion results are not invariably superior to photometricmetallicities. A review of high-dispersion and related work onsupermetallicity in K III-IV star is also given. CA II H and K measurements made at Mount Wilson Observatory, 1966-1983Summaries are presented of the photoelectric measurements of stellar CaII H and K line intensity made at Mount Wilson Observatory during theyears 1966-1983. These results are derived from 65,263 individualobservations of 1296 stars. For each star, for each observing season,the maximum, minimum, mean, and variation of the instrumental H and Kindex 'S' are given, as well as a measurement of the accuracy ofobservation. A total of 3110 seasonal summaries are reported. Factorswhich affect the ability to detect stellar activity variations andaccurately measure their amplitudes, such as the accuracy of the H and Kmeasurements and scattered light contamination, are discussed. Relationsare given which facilitate intercomparison of 'S' values with residualintensities derived from ordinary spectrophotometry, and for convertingmeasurements to absolute fluxes. Third preliminary catalogue of stars observed with the photoelectric astrolabe of the Beijing Astronomical Observatory.Not Available Chromospheric activity in evolved stars - The rotation-activity connection and the binary-single dichotomyA tabulation of measured values of the Ca II H and K (S) index aretransformed to the original Mount Wilson definition of the index. Thetabulation includes main-sequence, evolved, single, and tidally coupled(RS CVn) binary stars. The (S) indices are analyzed against Wilson's(1976) I(HK) intensity estimates, showing that Wilson's estimates areonly a two-state indicator. Ca II H and K fluxes are computed andcalibrated with published values of rotation periods. It is found thatthe single and binary stars are consistent with a single relationshipbetween rotation and Ca II excess emission flux. Large and Kinematically Unbiased Samples of G- and K-Type Stars. III. Evolved Young Disk Stars in the Bright Star SampleAbstract image available at:http://adsabs.harvard.edu/cgi-bin/nph-bib_query?1989PASP..101...54E&db_key=AST Large and kinematically unbiased samples of G- and K-type stars. II - Observations of evolved stars in the Bright Star sample. III - Evolved young disk stars in the Bright Star sampleFour color and RI observations were obtained for a large sample ofG-type and K-type stars in the Bright Star Catalogue. Data are firstpresented for 110 evolved stars. Photometry of evolved young diskpopulation stars have then been calibrated for luminosity, reddening,and metallicity on the basis of results for members of the Hyades andSirius superclusters. New DDO results are given for 120 stars. A Catalog of Stellar Angular Diameters Measured by Lunar OccultationAbstract image available at:http://adsabs.harvard.edu/cgi-bin/nph-bib_query?1987AJ.....94..751W&db_key=AST Photoelectric observations of lunar occultations. XVThe last (or possibly penultimate) paper in a series on lunaroccultations presents observations of 404 phenomena, preceded by somegeneral remarks of a retrospective nature. The series of observationsbegan in 1968. Data for 20 doubles or suspected doubles are presented,ranked in order of certainty, with designation 3 representing thesurest. Thirteen of the cases refer to stars not previously noted asdoubles. Seven determinations of angular diameters are presented, ofwhich three have been previously observed. Magnetic structure in cool stars. VI - CA II H and K fluxes from evolved starsQuantitative measurements of the Ca II H and K flux of 335 evolved starsare presented and discussed. The results show that there is a largespread in the fluxes from stars with (B-V) less than 0.95 while the CaII H and K flux of single stars with (B-V) greater than 0.95 correlateswith color with little spread. Short-period binaries show a relativelyhigh Ca II H and K flux indicating that high fluxes result from rapidrotation independent of spectral type. The data are consistent with thehypothesis that the emission depends on dynamo action in the convectiveenvelope, the dynamo efficiency decreasing with decreasing rotationrate. The evolution of the emission is discussed as a function ofstellar mass. It is shown that stars which leave the main sequence withrelatively low or high rotational velocities show relatively low or highemission values, respectively. The flux lasts up to higher (B-V) valuesfor progressively higher masses. Magnetic structure in cool stars. I - The CA II H and K emission from giantsEstimates of the intensities of the Ca II H and K emission lines of 500subgiants, giants and supergiants are reanalyzed in an investigation ofthe emission mechanisms in evolved cool stars. The Ca II emissionintensity classes estimated by Wilson (1976) are compared with B-Vcolors of the stars in the sample. The majority of K- and M-type giantsand many late G-type giants and subgiants are found to constitute a bandof intensity class increasing with B-V, while G-type giants show a largespread in intensity class. Enhanced H and K emission is found in twogroups of evolved stars: some of the G-type giants, and short-periodbinaries among the giants and subgiants with orbital periods shorterthan a critical value, and the emission is explained by a relativelylarge stellar rotation rate. Data are consistent with the hypothesisthat Ca II H and K emission depends on dynamo action in the convectiveenvelope, with dynamo efficiency decreasing with decreasing rotationrate. Catalogue of stars with CaII H and K emissionsNot Available Absolute magnitudes of stars from widths of chromospheric CaII emission lines.Abstract image available at:http://adsabs.harvard.edu/cgi-bin/nph-bib_query?1976ApJ...205..823W&db_key=AST The Strengths of Infrared CO and h, O Bands in Late-Type StarsAbstract image available at:http://adsabs.harvard.edu/cgi-bin/nph-bib_query?1973ApJ...184..427B&db_key=AST
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text:
# Chapter 11 - Section 11.1 - Sequences - Exercise Set: 31

$-\frac{1}{25}$

#### Work Step by Step

To calculate $a_n$, we plug in $n$ to the general formula. To calculate the result, we use PEMDAS (parentheses, exponentation, multiplication, division, addition, subtraction) to determine which operation is applied first. $a_{19} = \frac{(-1)^{19}}{19+6} = \frac{-1}{19+6} = -\frac{1}{25}$

After you claim an answer you’ll have 24 hours to send in a draft. An editor will review the submission and either publish your submission or provide feedback.


text:
# Circle

Martin McBride, 2020-08-23
Tags geometry circle arc sector segment
Categories generativepy generative art

generativepy has moved to the pythoninformer.com website, and this page might not be up to date. Please visit the new location.

The Circle class draws circles, arcs, sectors and segments.

There is also a circle function that just creates a circle as a new path.

## Circle class methods

The Circle class inherits add, fill, stroke, fill_stroke, path, clip and other methods from Shape.

• as_arc
• as_sector
• as_segment

Creates a circle based on the centre point and radius.

of_center_radius(center, radius)

Parameter Type Description
center (number, number) A tuple of two numbers, giving the (x, y) position of the centre of the circle.

### as_arc

Modifies a circle, to show only an arc. An arc is part of the circumference of the circle.

as_arc(start_angle, end_angle)

Parameter Type Description
start_angle number A number, giving the start angle of the arc
end_angle number A number, giving the end angle of the arc

This is used as a modifier with of_center_radius, to draw just an arc. To draw an arc use:

Circle(ctx).of_center_radius((0, 0), 1).as_arc(0, 1).stroke(Color('black'), 0.1)

Angles are measured in radians. Angle zero lies along the positive x-axis, and the angle increases in the clockwise direction - note that this is opposite to the normal mathematical convention, where angle increases in the counterclockwise direction. The difference is due to the fact that y increases as you move down the image in generativepy.

If you are using a flipped coordinate system (see the setup function in the drawing module), the angle increases in the counterclockwise direction.

Since an arc is a line, you should normally use the stroke method to draw it. If you attempt to fill the arc, it will fill it as if it was a segment.

### as_segment

Modifies a circle, to show only a segment. An segment is the part of the circle that is cut off by a chord.

as_segment(start_angle, end_angle)

This function works in a similar way to the as_arc function, but it includes the area of the segment. You can fill or stroke the area, see the example below.

### as_sector

Modifies a circle, to show only a sector. An sector is a "pizza slice", like you would use in a pie chart,.

as_sector(start_angle, end_angle)

This function works in a similar way to the as_arc function, but it includes the area of the sector. You can fill or stroke the area, see the example below.

## circle function

Adds a circle as a new path, without the need to create a Circle object in code.

circle(ctx, center, radius)

Parameter Type Description
ctx Context The Pycairo Context to draw to
center (number, number) A tuple of two numbers, giving the (x, y) position of the centre of the circle.

The circle function doesn't allow you to create arcs, segments or sectors.

## Example

Here is some example code that draws circles using the class and the utility function. The full code can be found on github.

from generativepy.drawing import make_image, setup
from generativepy.color import Color
from generativepy.geometry import Circle, circle

'''
Create circles using the geometry module.
'''

def draw(ctx, width, height, frame_no, frame_count):
setup(ctx, width, height, width=5, background=Color(0.8))

# The circle function is a convenience function that adds a circle as a new the path.
# You can fill or stroke it as you wish.
circle(ctx, (1, 1), 0.7)
ctx.set_source_rgba(*Color(1, 0, 0))
ctx.fill()

# Circle objects can be filled, stroked, filled and stroked.
Circle(ctx).of_center_radius((2.5, 1), 0.7).fill_stroke(Color(0, 0, 1), Color(0), 0.05)
Circle(ctx).of_center_radius((4, 1), 0.7).as_arc(0, 1).stroke(Color(0, 0.5, 0), 0.05)


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# Search results

## Page title matches

• [[Power function]] ([[Степенная функция]]) is one of the primary [[ ...coefficient is introduced in the right hand side of the definition of the power function;
15 KB (2,495 words) - 18:43, 30 July 2019
• 36 bytes (4 words) - 07:02, 1 December 2018

## Page text matches

• ...vernment and established the randomly-elected Supreme Court with draconian power. From the selected mercenaries, the police corpus had been formed ...rty of the Batak nationalists tried to recall their role in the seizure of power - but their attempts were cruelly drowned in blood. The revolution, as alwa
135 KB (24,381 words) - 13:33, 30 October 2020
• ...standing the effects of the earthquake or the tsunami. Nor was the nuclear power plant prepared to respond to a severe accident. In spite of the fact that Tokyo Electric Power Company (TEPCO) and the regulators were aware of the risk from such natural
42 KB (5,776 words) - 12:14, 20 November 2021
• [[Category:Niclear power]]
4 KB (462 words) - 14:24, 20 June 2013
• ...астрофа]] ) - Catastrophe 26 April 1986, at the [[Chernobyl Nuclear Power Plant]] in ([[Pripiat]], now Ukraine). It is considered the worst nuclear a ...timated 45,000 people took part in demonstrations in Tokyo against nuclear power on Sunday, amid growing public concern that the government is bringing pres
38 KB (3,154 words) - 20:48, 7 February 2020
• ...standing the effects of the earthquake or the tsunami. Nor was the nuclear power plant prepared to respond to a severe accident. In spite of the fact that Tokyo Electric Power Company (TEPCO) and the regulators were aware of the risk from such natural
45 KB (6,219 words) - 15:14, 21 August 2019
• ...https://doi.org/10.1080/10236198.2017.1307350 M.H. Hooshmand. (2018) Ultra power of higher orders and ultra exponential functional sequences. Journal of Dif
21 KB (3,175 words) - 23:37, 2 May 2021
• ...to radiation resulting from an explosion at the Fukushima Daiichi nuclear power plant in Japan, after an 8.9 magnitude earthquake shook the region. ...etback in the struggle to contain what has become the most serious nuclear power crisis in a quarter century.
146 KB (19,835 words) - 18:25, 30 July 2019
• Parliament speaker assumes power a day after Zine El Abidine Ben Ali flees the country amid a mass uprising.
6 KB (411 words) - 07:09, 1 December 2018
• ...an may take three decades to decommission and cost operator Tokyo Electric Power Co. more than 1 trillion yen... (Wikipedia says, one trillion is one millio ...n active element gradually reduces. Did anybody plot the dependence of the power generated versus time? Where can one see such a graphic? For how long will
26 KB (3,915 words) - 18:25, 30 July 2019
• ...s been legalised. Putin’s friends of old, such no-ones before he came to power as Gennadi Timchenko, Yuri Kovalchuk, the Rotenburg brothers, have all beco ...h back in 2000 was an oil-trading bit-player, has during Putin’s time in power gained control over Russia’s oil exports?
92 KB (14,748 words) - 18:44, 30 July 2019
• The approximation of the relaxation power $$P$$ with elementary function ...r $$P_0$$ during time $$\tau_{\rm s}$$; then the estimate returns the heat power released at time $$\tau$$ since the beginning of the chain reaction; \(Q=0.
10 KB (1,441 words) - 18:44, 30 July 2019
• ...zimi [Todreas 1990] (based on [Glasstone 1967]) give the approximate decay power .. for total of// where P is the decay power, P0 is the nominal reactor power, œÑ is the time since reactor startup and œÑs is the time of
16 KB (337 words) - 18:36, 30 July 2019
• ...clicks can be counted after the counter is disconnected from the electric power supply; one can charge the capacitor in the laboratory, then bring the coun
17 KB (2,705 words) - 18:25, 30 July 2019
• ...reforms can be carried out in Russia today as long as Putin controls real power in the country. [...] Ridding ourselves of [[Putinism]] is the first, oblig ...d not been found, but activists could not get the report back, because the power of attorney had expired.<ref>[http://www.svobodanews.ru/content/article/209
39 KB (4,722 words) - 14:26, 20 June 2013
• ...e moment of the earthquake, and (just by the unfortunate coincidence), the power electric line and the roads happened to be damaged just after the earthquak ...and is ready to provide the electric power supply as soon as the electric power lines are recovered. Then, the nuclear companies will have to explain to th
16 KB (2,597 words) - 22:33, 1 July 2013
• ...se functions ([[Abel function]]s) allow evaluation of not only minus-first power of a function (inverse function), but also any real and even complex [[iter (or fractional power) of a function. Any holomorphic function can be declared as a "transfer fun
25 KB (3,622 words) - 08:35, 3 May 2021
• ...eds of expression of fast growth in century 21, in the similar way, as the power function and exponential had covered these needs in centuries 18 and 19.
7 KB (1,090 words) - 18:49, 30 July 2019
• ...uclear engines can be used to accelerate spacecraft, also serving as their power-supply systems.// It should be mentioned in this connection that the Energo ...s of a unique engine based on new physical principles to obtain propulsive power, Itar-Tass reports.// The engine, which does not emit reaction mass, was in
100 KB (14,715 words) - 16:21, 31 October 2021
• ...concept agrees also with publications about the way Putin got the superior power in Russia: the scientific analysis shows, that the [[election fraud]] had b
61 KB (4,997 words) - 04:14, 10 May 2020
• ...eorem''' another theorem, namely, the theorem about the convergence of the power series:
2 KB (320 words) - 18:25, 30 July 2019

View (previous 20 | next 20) (20 | 50 | 100 | 250 | 500)


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# Is it possible to simplify the convolution integral if the functions are non-zero in disjoint areas? [duplicate]

I have a function $f(x,y)$ and $h(x,y)$. $f(x,y)$ has a value of $\frac{1}{3}$ when $x$ is between $\frac{1}{3}$ and $\frac{5}{9}$, and $y$ is between $0$ and $1$. The function has a value of $0$ everywhere else. Meanwhile, $h(x,y)$ has a value of 1 when both $x$ and $y$ are between $-\frac{1}{19}$ and $\frac{1}{19}$.

These two functions are non-zero in completely disjoint regions. Is there a way to leverage this property to simplify the convolution integral?

## marked as duplicate by Dilip Sarwate, Lorem IpsumMar 23 '12 at 4:42

• The short answer is yes, as you can use the regions where you know the signal is zero to constrain the limits on the convolution integral. Instead of integrating from $-\infty$ to $\infty$, you can set the limits such that they only cover the regions where the two functions overlap as you slide them across one another. – Jason R Jan 30 '12 at 20:52
• In the other case, the function $f(x,y)$ has value $\frac{1}{3}$ for $|x| < 0.5, 0 < y < 0.4$ and $h(x,y)$ has value $1$ for $-\frac{1}{50} \leq x,y\leq \frac{1}{50}$. Are the questions really all that different? Moderators: please merge the questions. – Dilip Sarwate Mar 5 '12 at 12:14

From left to right: $f$, $h$, and $f*h$:


text:
# [Rd] R Newsletter: 1st Call for Articles

Timothy H. Keitt, Timothy.Keitt@sunysb.edu
Wed, 20 Dec 2000 20:28:07 -0500

You could use Lyx (www.lyx.org).  It exports (human readable) latex.

Tim

Peter Dalgaard BSA wrote:

> "Yves Gauvreau" <cyg@sympatico.ca> writes:
>
>
>> This is a very nice idea. I would have a problem tough with the Latex format
>> you require. I'm almost certain that to most Windows user, Latex is as bad
>> news as a foreign language you know nothing about. Would there be an
>> alternative to please every body?
>
>
> Not really, dealing with MS Word would be a nightmare, and LaTeX is
> just about the only format with decent treatment of mathematics.
>
> However, I'm pretty sure that plain ascii would also be acceptable for
> simple articles. On the other hand, why should the editors have the
> trouble of putting all code excerpts in \begin{verbatim} ..
> \end{verbatim} constructs, etc.?
>
> Perhaps one could define a simplified LaTeX markup similar to .Rd.
>
> Or maybe just point people towards MiKTeX and give them a crash course
> in LaTeX? For plain text, the format is really quite simple.

--
Timothy H. Keitt
Department of Ecology and Evolution
State University of New York at Stony Brook
Phone: 631-632-1101, FAX: 631-632-7626
http://life.bio.sunysb.edu/ee/keitt/

-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-
r-devel mailing list -- Read http://www.ci.tuwien.ac.at/~hornik/R/R-FAQ.html
Send "info", "help", or "[un]subscribe"
(in the "body", not the subject !)  To: r-devel-request@stat.math.ethz.ch
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text:
# Problems need HardWork #5

Algebra Level 3

If the following equations have the same roots, then find $a + b$.

$2\sqrt{x} + \frac{2}{\sqrt{x}} = 5$

$ax^2 + bx = -8$

×


text:
/ iOS

# Fixing the conflict resolution of .pbxproj files with GIT and XCode

Who knows what a pain it is when working on an Xcode project with your team, and resolving conflicts on the .pbxproj file. Here is a 99% working solution in order to forget this conflicts.

The .pbxproj file is a sensible file in a Xcode project, since it maintains information about which files are involved in your project, which ones are compiled or not, on which target, etc.

Most of the time, you are able to solve conflicts by hands, but for someone that doesn't know very well how a .pbxproj file, he offers himself the right to make a mistake. You can really mess up your project, in a way that Xcode is unable to read your project anymore.

##### And there exist a really simple workaround for this issue

Here is the not 100% working, but highly recommended solution, for a peace of code.

1. Go to the root directory of your git repository, where the .git directory is,
2. edit or create the .gitattributes file,
3. add the following line: *.pbxproj binary merge=union

You're done! Really, I'm not joking, this is that simple.

In the end, the file could look like this, if nothing else was present in that file:

# Telling GIT that Xcode pbxproj remote and local files
# should be merged using union of differences
*.pbxproj binary merge=union

Let me explain. You're telling GIT that the merge policy of any pbxproj file is "Union the content of the local and the remote file". This does make sense because when you're adding one or more files to the project and one of your colleague do the same operation, Xcode add the files at the end of the list of the source files list, so this will lead to a conflict when comparing the local and the remote files. Two lines at the same place referencing different files. This command tells GIT to add the added line from the local file, and then below, add the added line from the remote file.

So you added one file, your colleague did that too, and as a result, we have the two files at the end of source files list of the resulting file. And this is what you would have expected.

Attention : It is a 99% working workaround. Indeed, since it is a merge union operation, if two people add the same file, the file will exist and be a duplicate. Xcode may not compile your project after the operation, telling you that duplicate files exist. Go to the Build Phase of the involved target, check in compile sources for duplicates, and remove the bad files. This is the only drawback.


text:
# Restricted Maximum Likelihood

Why don't we use restricted maximum likelihood to estimate parameters in non-mixed models?

• You use it: REML is for estimating variance parameters, and in non-mixed models there is only one such, $\sigma^2$. When dividing by $n-1$ to get a variance estimate, you are using REML. See stats.stackexchange.com/questions/48671/… – kjetil b halvorsen May 4 '15 at 17:02

For REML under fixed effects the variance estimator equivalent to $S^2=\frac{\sum_{i=1}^n (y_i - x_i \hat{\beta})^2}{n-p}$ where $\hat{\beta}$ is the OLS estimator, but as a bonus with OLS you also get to use Gauss-Markov Theorem for $\hat{\beta}$.


text:
## How To Fix Cannot Determine Tutorial

Home > No Bounding > Cannot Determine

# Cannot Determine

## Contents

Kristiansen Sep 17 '13 at 19:29 in some .eps files, the bounding box information is at the end, rather than at the beginning where it really belongs. (and latex Connecting sino japanese verbs Is it safe to use cheap USB data cables? Marin Jankovski @marin commented 2015-10-29 15:25:19 UTC Master I understand. What is a unifier? http://inviewsoftware.com/no-bounding/cannot-determine-type-jpg.html

They are both on the same OS as well.Any help would be greatly appreciated. Why is this C++ code faster than my hand-written assembly for testing the Collatz conjecture? Thanks. Now change the extension in the tex file from \includegraphics[width=0.8\textwidth]{tiger.png} to \includegraphics[width=0.8\textwidth]{tiger.eps} share|improve this answer answered May 10 '15 at 9:56 NKN 276115 add a comment| up vote 1 down vote http://tex.stackexchange.com/questions/17734/cannot-determine-size-of-graphic

## Cannot Determine Size Of Graphic (no Boundingbox)

May 14, 2011 7:09 PM Helpful (0) Reply options Link to this post by ctech08, ctech08 May 15, 2011 2:50 AM in response to Nexvick Level 1 (10 points) May 15, Graph Chromatic Number Problem Can a player on a PC play with a a player on a laptop? We recommend upgrading to the latest Safari, Google Chrome, or Firefox. After it completes the shut down, it might be a good idea to wait for about 10 seconds4.

1. How much information can you obtain from a pulsar-black hole system?
2. Then AucTeX uses pdflatex which handles png-files by default, as @David Carlisle mentioned in his answer.
3. Ballpark salary equivalent today of "healthcare benefits" in the US?
4. Join them; it only takes a minute: Sign up Here's how it works: Anybody can ask a question Anybody can answer The best answers are voted up and rise to the
5. Is it safe to use cheap USB data cables?
6. Thus, the appropriate way to check for size_t is: AX_COMPILE_CHECK_SIZEOF(size_t, [#include ]) ebassi commented May 23, 2016 Opened PR #230 with the fix.
7. I just started my first real job, and have been asked to organize the office party.
8. May 14, 2011 8:55 PM Helpful (0) Reply options Link to this post Page 1 of 5 Next last Apple Footer This site contains user submitted content, comments and opinions and

IN my ipod touch 4g map cannot determine location... Does swap space have a file System? I'm working on debian using emacs graphics errors png share|improve this question edited Jan 22 '13 at 15:51 Martin Schröder 11.2k43196 asked Jan 22 '13 at 15:02 user24726 41112 marked as ! Latex Error: Option Clash For Package Graphicx. share|improve this answer answered Oct 1 '15 at 20:48 Todd L 15612 Yep, it just occurred to me last night that the webapp folder was copied from my colleague's

Why is using let inside a for loop so slow on Chrome? By changing its extension from .JPG to .jpg (yes, just capital to lower) the error went away. \$ pdflatex --verion pdflatex: unrecognized option '--verion' This is pdfTeX, Version 3.1415926-2.5-1.40.14 (TeX Live Please help! http://tex.stackexchange.com/questions/133786/no-boundingbox-error-message I will try that next and see what happens.

How do I change thickness and color of \hline on a table simultaneously؟ Prove that the following statements for a ring R are equivalent: How to take sharper images indoors, scene Latex No Bounding Box Jpg asked 3 years ago viewed 42619 times active 2 years ago Visit Chat Linked 77 Cannot determine size of graphic 3 pdflatex, dvips: cannot determine size of graphic in … (no Converting the weight of a potato into a letter grade Am I interrupting my husband's parenting? Does the "bat wing" aircraft paint design have a proper name?

## Latex No Bounding Box Pdf

Browse other questions tagged graphics pdf texworks or ask your own question. http://inviewsoftware.com/no-bounding/cannot-determine-size-of-graphics-lyx.html How small could an animal be before it is consciously aware of the effects of quantum mechanics? All postings and use of the content on this site are subject to the Apple Support Communities Terms of Use.  Apple Support Communities More ways to shop: Visit an Apple LaTeX Error: Cannot determine size of graphic in Pictures/logo.png (no BoundingBox)2No BoundingBox in LaTeX for a JPEG figure2Using psfragfig: Error processing graphic0Size error with *.png-files when compiling with LaTex2problem to insert Natwidth Latex

Latex however returns this error: "Cannot determine size of graphic" My graphic is exported from PowerPoint, so I have tried both .pdf and .png. Actually latex is pdflatex in DVI mode in modern distributions, but it can read the sizes only in PDF mode for some reason. It's very boring..... check over here Removal of negative numbers from an array in java My manager said I spend too much time on Stack Exchange.

Level 1 (130 points) Nov 4, 2012 1:55 PM in response to Leon C. Latex Error File Dvipdfm Def Not Found I use pdfLaTeX in TeXworks. You need to use pdflatex for this.


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Question

Civil engineers believe that W, the amount of weight (in units of 1000 pounds) that a certain span of a bridge can withstand without structural damage resulting, is normally distributed with mean 400 and standard deviation 40. Suppose that the weight (again, in units of 1000 pounds) of a car is a random variable with mean 3 and standard deviation .3. Approximately how many cars would have to be on the bridge span for the probability of structural damage to exceed .1?

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# How do you write 525 as a fraction?

## How do you write 525 as a fraction?

525 expressed as a fraction is 5251 .

512

## What is 5 and 1 fourth as a decimal?

So the answer is that 5 1/4 as a decimal is 5.25

## What is 2/5 as a fraction?

DecimalFraction0.333333331/30.3753/80.42/50.428571433/745 more rows

## How do you simplify 525?

Rewrite 525 as 52u22c521 5 2 u22c5 21 . Factor 25 25 out of 525 525 . Rewrite 25 25 as 52 5 2 .

## How do you write 55 as a fraction?

Answer: 55% can be represented as 11/20 in fractional form.

## How do you write 31.2 as a fraction?

Express 31.2 as a fraction

• Write 31.2 as 31.21.
• 31.2 xd7 101 xd7 10 31210.
• 1565.
• ## What is 5 and a half as a mixed number?

5/2 is an improper fraction that is equivalent to 2 1/2.

## How do you turn 5.5 into a fraction?

Steps to convert decimal into fraction

• Write 5.5 as 5.51.
• 5.5 xd7 101 xd7 10 5510.
• 112.
• ## What is 5/6 in a mixed number?

Expressed as a fraction, 5/6 0 5/6. Write 5/6 as the mixed number 0 5/6. Leave the 0 off unless specifically notating a mixed number, however. Divide the numerator, 5, by the denominator, 6, to express the fraction 5/6 as a decimal.

0.25

## What is 5 and 1/2 in a decimal?

So the answer is that 5 1/2 as a decimal is 5.5

1.25

## What is 5 and 3/4 as a decimal?

So the answer is that 5 3/4 as a decimal is 5.75


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# Colloquium abstracts

Sachin Sharma
TIFR
September 5, 2013

The irreducible modules for the derivations of the rational quantum torus:  Let $\mathbb{C}_q$ be the quantum torus associated with the $d \times d$ matrix $q = (q_{ij})$, $q_{ii} = 1$, $q_{ij}^{-1} = q_{ji}$, $q_{ij}$ are roots of unity, for all $1 \leq i, j \leq d$ . Let Der$(\mathbb{C}_q)$ be the Lie algebra of all the derivations of $\mathbb{C}_q$.W.Lin and S.Lan defined a functor from $gl_d$-modules to Der$(C_q)$modules. They proved that for a finite dimensional irreducible $gl_d$-module $V$, $V \otimes \mathbb{C}_q$ is a completely reducible Der$(C_q)$-module except finitely many cases. In this talk we will show that $V \times \mathbb{C}_q$ is an irreducible Der($\mathbb{C}_q) \ltimes \mathbb{C}_q$-module which satisfies some conditions. The main aim of the talk is to prove the converse of the above fact i.e., if $V'$ is an irreducible $\mathbb{Z}^d$-graded Der($\mathbb{C}_q) \ltimes \mathbb{C}_q$-module with finite dimensional weight spaces which satisfies some conditions, then $V' \cong V \otimes C_q$ as Der($\mathbb{C}_q) \ltimes \mathbb{C}_q$-module


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# Let f(x) be a continuous function and I = int_(1)^(9) sqrt(x)f(x) dx, then

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Text Solution

There exists some c in (1,9 ) such that I = 8sqrt(c) f(c)There exists some p,q in (1,3) such that I = 2[p^(2)f(p^(2))+q^(2)f(q^(2))]There exists some alpha in (1,9) such that I = 9sqrt(x)f(alpha)If f(x) ge 0 AA x in [1,9] rArr I gt 0


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# Talk:Matrix gamma distribution

WikiProject Statistics (Rated Stub-class)

This article is within the scope of the WikiProject Statistics, a collaborative effort to improve the coverage of statistics on Wikipedia. If you would like to participate, please visit the project page or join the discussion.

Stub  This article has been rated as Stub-Class on the quality scale.
Multivariate gamma distribution should be an article but not this article. This article is about one particular definition of a matrix-variate gamma distribution, not a multivariate gamma distribution. Look at the support set and density. Multivariate gamma should have support on ${\displaystyle R^{p}}$. "Continuous Multivariate Distributions" (2e) mentions no matrix variate gamma distribution called multivariate gamma. The article is not correctly named and should be moved to a page with a different title (without a redirect), or removed. Mathstat (talk) 19:04, 17 April 2012 (UTC)


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iknots {bspline} R Documentation

## Estimate internal knot positions equalizing jumps in n-th derivative

### Description

Normalized total variation of n-th finite differences is calculated for each column in y then averaged. These averaged values are fitted by a linear spline to find knot positions that equalize the jumps of n-th derivative.
NB. This function is used internally in (fit)smbsp() and a priori has no interest to be called directly by user.

### Usage

iknots(x, y, nki = 1L, n = 3L)

### Arguments

 x Numeric vector y Numeric vector or matrix nki Integer scalar, number of internal knots to estimate (1 by default) n Integer scalar, polynomial order of B-spline (3 by default)

### Value

Numeric vector, estimated knot positions

[Package bspline version 2.1 Index]


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# How would you simplify $\frac{\frac{6y}{y + 6}}{\;\frac{5}{7y + 42}\;}$ using LCD method?

Peace to all. While in class I was taught to solve complex fractions by an "alternative method" in which you would:

1. Multiply the numerator and denominator by the LCD
2. Apply the distributive property
3. Factor and Simplify

An example: $$\dfrac{4 - \dfrac 6 {x}} {\dfrac 2 {x} - \dfrac 3 {x^2}}$$

1. Multiply the numerator and denominator by the LCD (x^2):

$$\dfrac{4 - \dfrac 6 {x}} {\dfrac 2 {x} - \dfrac 3 {x^2}}$$

1. Apply the distributive property:

$$\dfrac{x^2 × (4) - x^2 × \dfrac 6 {x}} {{x^2} × \dfrac 2x - x^2 × \dfrac 3 {x^2}}$$

1. Factor and Simplify

$$\dfrac{4x^2 - 6x} {2x-3}$$

$$\dfrac{2x(2x - 3)}{2x-3}$$

A:2x

This example is very straightforward and simple. However, when I put it to use it becomes very difficult. For example for the equation: $$\dfrac{\dfrac{6y}{(y + 6)}}{\dfrac{5}{(7y + 42)}}$$. I get $$y + 42$$ as the LCD. I'm not too confident of this because when I begin to distribute I get very large numbers. How would one apply this to the prior equation?

• Just confirm if the edit I have made in your question is correct
– user876009
Sep 6, 2021 at 2:32
• @JitendraSingh Peace. I don't see that option but the edit is 100% accurate. Thanks. Sep 6, 2021 at 2:34

Hint: $$7y+42=7(y+6)$$. Given this, both the numerator and denominator contain $$y+6$$.

$$\dfrac{\dfrac{6y}{(y + 6)}}{\dfrac{5}{(7y + 42)}}$$

Multiply numerator and denominator by $$7y+42$$

$$\dfrac{\dfrac{6y \times (7(y+6))}{y+6}}{\dfrac{5 \times (7(y+6))}{7(y+6)}} \implies \dfrac{42y}{5}$$

$$\require{cancel}$$ I don't know why the LCD method is used at all because it is clear, in this case, that.

$$\dfrac{\dfrac{6y}{y + 6}}{\;\dfrac{5}{7y + 42}} =\frac{6y}{\cancel{y+6}}\cdot\frac{7(\cancel{y+6})}{5} =\frac{42y}{5}$$ The LCD is needed the "example" as a means of separately making the numerator factorable and the denominator "invertible".

$$\dfrac{4 - \dfrac 6 {x}} {\dfrac 2 {x} - \dfrac 3 {x^2}} =\dfrac{\dfrac{4x-6}{x}} {\dfrac{2x-3}{x^2}} =\dfrac{2(\cancel{2x-3})}{x} \cdot \dfrac{x^2}{\cancel{2x-3}} =2x$$

• Perhaps they just like to make our lives unnecessarily difficult lol. Thank you Sep 6, 2021 at 21:44
• @יהודה Glad I could help. I see a lot of "answers" to questions that almost seem to avoid simple solutions that are obvious once seen. I hope my treatment of the "example" showed you why LCD should be invoked at all. Sep 6, 2021 at 22:04
• No, it did help. My professor actually stated that she disapproves of this method because of how unnecessarily uncomplicated it is. Thanks again. Sep 6, 2021 at 22:06


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# $49-54$ . A partial sum of an arithmetic sequence is given. Find the sum. $$\sum_{n=0}^{20}(1-2 n)$$

## $-399$

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in the partial. Some given this sigma notation so partial. Some formula is an over to number of terms over two times a one plus a ends a first term plus the last term and first term. So the number of terms is 21 because it's 20 here and starts with zero. So 21 divided by two times the first term first term is, uh, is equal to zero because when a musical 20 can get eyes Actually, what sorry is one one plus in the last term? A 20. So one minus 40 which is negative. 39. So nice thing you're 39. You play this into a calculator, we get negative 399 as the partial Some

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