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In [[probability theory]], the [[central limit theorem]] states conditions under which the average of a sufficiently large number of [[Statistical independence|independent]] [[random variables]], each with finite mean and variance, will be approximately [[normal distribution|normally distributed]].<ref>{{harvtxt|Rice|1995}}{{full|date=November 2012}}</ref> 


[[Directional statistics]] is the subdiscipline of [[statistics]] that deals with directions ([[unit vector]]s in '''R'''<sup>''n''</sup>), [[Cartesian Coordinate System|axes]] (lines through the origin in '''R'''<sup>''n''</sup>) or [[rotation]]s in '''R'''<sup>''n''</sup>. The means and variances of directional quantities are all finite, so that the central limit theorem may be applied to the particular case of directional statistics.<ref name="SRJ">{{cite book |title=Topics in circular statistics |last=Jammalamadaka |first=S. Rao |authorlink= |coauthors=SenGupta, A.|year=2001 |publisher=World Scientific |location=New Jersey |isbn=981-02-3778-2 |url=http://books.google.com/books?id=sKqWMGqQXQkC&printsec=frontcover&dq=Jammalamadaka+Topics+in+circular&hl=en&ei=iJ3QTe77NKL00gGdyqHoDQ&sa=X&oi=book_result&ct=result&resnum=1&ved=0CDcQ6AEwAA#v=onepage&q&f=false |accessdate=2011-05-15}}</ref>


This article will deal only with unit vectors in 2-dimensional space ('''R'''<sup>''2''</sup>) but the method described can be extended to the general case.
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== The central limit theorem ==
 
A sample of angles <math>\theta_i</math> are measured, and since they are indefinite to within a factor of <math>2\pi</math>, the complex definite quantity <math>z_i=e^{i\theta_i}=\cos(\theta_i)+i\sin(\theta_i)</math> is used as the random variate. The probability distribution from which the sample is drawn may be characterized by its moments, which may be expressed in Cartesian and polar form:
 
:<math>m_n=E(z^n)= C_n +i S_n = R_n e^{i \theta_n}\,</math>
 
It follows that:
 
:<math>C_n=E(\cos (n\theta))\,</math>
:<math>S_n=E(\sin (n\theta))\,</math>
:<math>R_n=|E(z^n)|=\sqrt{C_n^2+S_n^2}\,</math>
:<math>\theta_n=\arg(E(z^n))\,</math>
 
Sample moments for N trials are:
 
:<math>\overline{m_n}=\frac{1}{N}\sum_{i=1}^N z_i^n =\overline{C_n} +i \overline{S_n} = \overline{R_n} e^{i \overline{\theta_n}}</math>
 
where
 
:<math>\overline{C_n}=\frac{1}{N}\sum_{i=1}^N\cos(n\theta_i)</math>
:<math>\overline{S_n}=\frac{1}{N}\sum_{i=1}^N\sin(n\theta_i)</math>
:<math>\overline{R_n}=\frac{1}{N}\sum_{i=1}^N |z_i^n|</math>
:<math>\overline{\theta_n}=\frac{1}{N}\sum_{i=1}^N \arg(z_i^n)</math>
 
The vector [<math>\overline{ C_1 },\overline{ S_1 }</math>] may be used as a representation of the sample mean <math>(\overline{m_1})</math> and may be taken as a 2-dimensional random variate.<ref name="SRJ"/> The bivariate [[central limit theorem]] states that the joint probability distribution for <math>\overline{ C_1 }</math> and <math>\overline{ S_1 }</math> in the limit of a large number of samples is given by:
 
:<math>[\overline{C_1},\overline{S_1}] \xrightarrow{d} \mathcal{N}([C_1,S_1],\Sigma/N)</math>
 
where <math>\mathcal{N}()</math> is the [[bivariate normal distribution]] and <math>\Sigma</math> is the [[covariance matrix]] for the circular distribution:
 
:<math>
\Sigma
=
\begin{bmatrix}
\sigma_{CC} & \sigma_{CS} \\
\sigma_{SC} & \sigma_{SS}
\end{bmatrix}
\quad</math>
 
:<math>\sigma_{CC}=E(\cos^2\theta)-E(\cos\theta)^2\,</math>
:<math>\sigma_{CS}=\sigma_{SC}=E(\cos\theta\sin\theta)-E(\cos\theta)E(\sin\theta)\,</math>
:<math>\sigma_{SS}=E(\sin^2\theta)-E(\sin\theta)^2\,</math>
 
Note that the bivariate normal distribution is defined over the entire plane, while the mean is confined to be in the unit ball (on or inside the unit circle). This means that the integral of the limiting (bivariate normal) distribution over the unit ball will not be equal to unity, but rather approach unity as ''N'' approaches infinity.
 
It is desired to state the limiting bivariate distribution in terms of the moments of the distribution.
 
== Covariance matrix in terms of moments ==
 
Using multiple angle [[trigonometric identities]]<ref name="SRJ"/>
 
:<math>C_2= E(\cos(2\theta)) = E(\cos^2\theta-1)=E(1-\sin^2\theta)\,</math>
:<math>S_2= E(\sin(2\theta)) = E(2\cos\theta\sin\theta)\,</math>
 
It follows that:
 
:<math>\sigma_{CC}=E(\cos^2\theta)-E(\cos\theta)^2 =\frac{1}{2}\left(1 + C_2 - 2C_1^2\right)</math>
:<math>\sigma_{CS}=E(\cos\theta\sin\theta)-E(\cos\theta)E(\sin\theta)=\frac{1}{2}\left(S_2 - 2 C_1 S_1  \right)</math>
:<math>\sigma_{SS}=E(\sin^2\theta)-E(\sin\theta)^2 =\frac{1}{2}\left(1  - C_2 - 2S_1^2\right)</math>
 
The covariance matrix is now expressed in terms of the moments of the circular distribution.
 
The central limit theorem may also be expressed in terms of the polar components of the mean. If <math>P(\overline{C_1},\overline{S_1})d\overline{C_1}d\overline{S_1}</math> is the probability of finding the mean in area element <math>d\overline{C_1}d\overline{S_1}</math>, then that probability may also be written <math>P(\overline{R_1}\cos(\overline{\theta_1}),\overline{R_1}\sin(\overline{\theta_1}))\overline{R_1}d\overline{R_1}d\overline{\theta_1}</math>.
 
==References==
 
<references/>
 
[[Category:Directional statistics]]
[[Category:Central limit theorem| ]]
[[Category:Asymptotic statistical theory]]

Latest revision as of 01:37, 14 May 2014


Taking on a home improvement project of any kind can be stressful and challenging if you are unprepared for that project. Getting helpful tips and information before you begin your project is going to make the process of improvements less of a challenge. Find useful tips by reading this article.

Place candles throughout your house. Candles can create a wonderful effect in your home. The effect will be heightened if the candle is scented with a good fragrance. You can place candles in bedrooms, bathrooms, and dining areas. Bathrooms especially benefit from the use of scented candles. Scented candles will give your house a homey feel.

When it comes to home improvement, be sure to keep in mind the age of your household members as well as health status. This is important if stairs are unmanageable or if there is a potential need for handicapped facilities in the bathrooms. Depending on the modifications, you may end up increasing the value of your home.

You can stop any floor squeaks with construction adhesive. While this requires you to complete the work from a basement or crawlspace, the result is well worth your trouble. You will basically be gluing each floor joist to its neighbor by applying a glue line along each one with a caulking gun.

If you are looking for a functional home improvement project, try "building up." Walls create a lot of wasted space. Add matching bookcases in your living room or build a window seat where your family can sit and read. Simple projects like these can make every space in your home usable and valuable to your family.

If you have a small bathroom and not enough space to store your items, you can easily create some space yourself. Buy some wooden shelves and install them on your bathroom wall. Display your lotions, perfumes and other beauty items on the shelves as well as your smaller towels or toilet paper rolls. This will not only give your bathroom a decorative boost, it will also organize it.

Throwing out an old sofa can be very tempting when it no longer looks new and clean. However, with the same effort it takes you to throw it out, you can buy some nice slipcovers and throw it on your sofa. Slipcovers are available in a myriad of designs and can spice up an old sofa very easily.

When trying to fix up a house to flip it, pay attention to the kitchen. If has linoleum floors, replace them with tile. It makes the place look much better. You should examine the counters and look at replacing them. They don't have to be granite, but granite counter tops increase the value of the home and up your chances of selling it.

Houses with good curb appeal often sell for considerably higher prices than other similar homes. Keep in mind, though, that the extra cost of the view may not be looked at as an investment, but as an extra charge for your pleasure. Some people like a view of the ocean, while other would prefer a nice view of the sunset each night.

If you are doing a home improvement project that involves demolition, slow down and take your time. It may be exciting to smash that wall open, but you could damage structural supports, plumbing, or electrical connections. The same is true of pulling down cabinets and other fixtures. Once you've demolished it, it can't be put back, so pay attention to what you're doing.

Create space in a cramped bathroom. Build floor-to-ceiling cabinets, and store similar items in baskets. This avoids any clutter on the countertop. A corner storage unit in your shower is perfect for holding shampoo, conditioner and any other items you may need. A great tip for creating space is to get rid of everything that isn't used on a regular basis. This can include out of date medications in your medicine cabinet or beauty products that you tried and didn't like. By clearing these out, you will have plenty of room for things that you actually use!

Analyze your reasons for remodeling before you begin any project. If you are remodeling to increase the likelihood of a quick home sale, focus your efforts on the kitchen and bathroom. These two remodels typically have the highest return on investment. If you are remodeling solely for personal reasons, you can begin anywhere you like.

You can use home improvement projects to allow your personality to shine through. Homeowners can do many things to their home on their own for a certain amount. In certain circumstances you will want to bring in the experts, and this article has offered you insight on knowing how best to proceed.

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