Sunday, 23 October 2011

TWITPIC

What is TwitPic?

TwitPic lets you share media on Twitter in real-time.

How do I use TwitPic?

You can post photos or videos to TwitPic from your phone, from the site, or through email. All popular Twitter clients have built-in support for TwitPic using our API.

Saturday, 22 October 2011

Errors and residuals in statistics

In statistics and optimization, statistical errors and residuals are two closely related and easily confused measures of the deviation of a sample from its "theoretical value". The error of a sample is the deviation of the sample from the (unobservable) true function value, while the residual of a sample is the difference between the sample and the estimated function value.

Introduction

Suppose there is a series of observations from a univariate distribution and we want to estimate the mean of that distribution (the so-called location model). In this case the errors are the deviations of the observations from the population mean, while the residuals are the deviations of the observations from the sample mean.
A statistical error is the amount by which an observation differs from its expected value, the latter being based on the whole population from which the statistical unit was chosen randomly. For example, if the mean height in a population of 21-year-old men is 1.75 meters, and one randomly chosen man is 1.80 meters tall, then the "error" is 0.05 meters; if the randomly chosen man is 1.70 meters tall, then the "error" is −0.05 meters. The expected value, being the mean of the entire population, is typically unobservable, and hence the statistical error cannot be observed either.
A residual (or fitting error), on the other hand, is an observable estimate of the unobservable statistical error. Consider the previous example with men's heights and suppose we have a random sample of n people. The sample mean could serve as a good estimator of the population mean. Then we have:

The difference between the height of each man in the sample and the unobservable population mean is a statistical error, whereas
The difference between the height of each man in the sample and the observable sample mean is a residual.

Note that the sum of the residuals within a random sample is necessarily zero, and thus the residuals are necessarily not independent. The statistical errors on the other hand are independent, and their sum within the random sample is almost surely not zero.
One can standardize statistical errors (especially of a normal distribution) in a z-score (or "standard score"), and standardize residuals in a t-statistic, or more generally studentized residuals.

[edit] Example with some mathematical theory

If we assume a normally distributed population with mean μ and standard deviation σ, and choose individuals independently, then we have

$X_1, \dots, X_n\sim N(\mu,\sigma^2)\,$

and the sample mean

$\overline{X}={X_1 + \cdots + X_n \over n}$

is a random variable distributed thus:

$\overline{X}\sim N(\mu, \sigma^2/n).$

The statistical errors are then

$\varepsilon_i=X_i-\mu,\,$

whereas the residuals are

$\widehat{\varepsilon}_i=X_i-\overline{X}.$

(As is often done, the "hat" over the letter ε indicates an observable estimate of an unobservable quantity called ε.)
The sum of squares of the statistical errors, divided by σ², has a chi-square distribution with n degrees of freedom:

$\sum_{i=1}^n \left(X_i-\mu\right)^2/\sigma^2\sim\chi^2_n.$

This quantity, however, is not observable. The sum of squares of the residuals, on the other hand, is observable. The quotient of that sum by σ² has a chi-square distribution with only n − 1 degrees of freedom:

$\sum_{i=1}^n \left(\,X_i-\overline{X}\,\right)^2/\sigma^2\sim\chi^2_{n-1}.$

It is remarkable that the sum of squares of the residuals and the sample mean can be shown to be independent of each other. That fact and the normal and chi-square distributions given above form the basis of calculations involving the quotient

${\overline{X}_n - \mu \over S_n/\sqrt{n}}.$

The probability distributions of the numerator and the denominator separately depend on the value of the unobservable population standard deviation σ, but σ appears in both the numerator and the denominator and cancels. That is fortunate because it means that even though we do not know σ, we know the probability distribution of this quotient: it has a Student's t-distribution with n − 1 degrees of freedom. We can therefore use this quotient to find a confidence interval for μ.

HICCUP

A hiccup or hiccough is a myoclonus of the diaphragm that repeats several times per minute. In humans, the abrupt rush of air into the lungs causes the vocal cords to close, creating a "hic" sound.

n medicine it is known as synchronous diaphragmatic flutter (SDF), or singultus, from the Latin singult, "the act of catching one's breath while sobbing".^[1] The hiccup is an involuntary action involving a reflex arc .
A bout of hiccups, in general, resolves itself without intervention, although many home remedies shorten the duration. Medical treatment is occasionally necessary in cases of chronic hiccups.

bottlenose dolphin

Type:: Mammal
Diet:: Carnivore
Average life span in captivity:: 45 to 50 years
Size:: 10 to 14 ft (3 to 4.2 m)
Weight:: 1,100 lbs (500 kg)
Group name:: Pod
Did you know?: Bottlenose dolphins have been observed to breach up to 16 feet (4.9 meters) out of the water, landing with a splash on their back or side.
Size relative to a 6-ft

Bottlenose dolphins are well known as the intelligent and charismatic stars of many aquarium shows. Their curved mouths give the appearance of a friendly, permanent smile, and they can be trained to perform complex tricks.
In the wild, these sleek swimmers can reach speeds of over 18 miles (30 kilometers) an hour. They surface often to breathe, doing so two or three times a minute. Bottlenose dolphins travel in social groups and communicate with each other by a complex system of squeaks and whistles. Schools have been known to come to the aid of an injured dolphin and help it to the surface.
Bottlenose dolphins track their prey through the expert use of echolocation. They can make up to 1,000 clicking noises per second. These sounds travel underwater until they encounter objects, then bounce back to their dolphin senders, revealing the location, size, and shape of their target.
When dolphins are feeding, that target is often a bottom-dwelling fish, though they also eat shrimp and squid. These clever animals are also sometimes spotted following fishing boats in hopes of dining on leftovers.
Bottlenose dolphins are found in tropical oceans and other warm waters around the globe. They were once widely hunted for meat and oil (used for lamps and cooking), but today only limited dolphin fishing occurs. However, dolphins are threatened by commercial fishing for other species, like tuna, and can become mortally entangled in nets and other fishing equipment.
All dolphins, including the bottlenose, are porpoises. Although some people use these names interchangeably, porpoises are actually a larger group that also includes animals like the orca and the beluga whale.

Asiatic Black Bear Facts

Asiatic Black bear - Ursus thibetanus
How they look…
The Asiatic Black Bear, also known as the Moon Bear, is a medium sized bear with a body length of 140 to 165 centimetres and most weigh between 90 and 150 kg. They are normally black in colour, with lighter muzzles and a distinct V-shaped patch of cream or white coloured fur on the chest. This bear has a heavy, stocky body and the ears are large and set far apart.
What they eat…
As with other bears, the Asiatic Black Bear likes to eat meat, but they will also eat plants, berries, insects, fruit and honey. Overall, the Asiatic Black Bear will eat anything that is edible! In autumn, they frequent nut-producing trees where they eat in self-constructed leaf and branch nests or platforms.
Where they live…
Asiatic Black Bears inhabit eastern Asia, including Afghanistan, Bangladesh, Bhutan, Cambodia, China, India, Iran, Japan, Korea, Lao, Malaysia, Mongolia, Myanmar, Nepal, Pakistan, Russia, Taiwan and Vietnam. They prefer thickly forested areas in the hills and mountains and tropical forests below alpine levels. During summer, the bears may reach altitudes of 1,000 feet. Their territory ranges from 4 to 6 square miles and depends on food supplies.
Reproduction
Female Asiatic Black bears become sexually mature at about 4 years of age. While little is known about these bears in the wild, it is believed breeding takes place over a one to two day period in late April to early June. In captivity, cubs are born in December or January. Cubs are weaned at about six months of age but may remain with the mother for up to three years.
Threats to Survival
The Asiatic Black Bear has been hunted for centuries for its skin, paws, and the gall bladder. The bears are captured and milked for bile that is then used in Traditional Asian Medicine. Recent deforestation across the Asian continent is another major threat to the survival of the species. Logging, development, and the ever- encroaching human population continues to cause the clearing of bear habitat.
The aim of Free The Bears Fund is to protect, preserve and enrich the lives of bears throughout the world. Throughout south-east Asia, the fund has set up sanctuaries for rescued bears and continues to campaign for an end to all bear cruelty.