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The weighted arithmetic mean is similar to an ordinary arithmetic mean (the most common type of average), except that instead of each of the data points contributing equally to the final average, some data points contribute more than others.
For normally distributed random variables inverse-variance weighted averages can also be derived as the maximum likelihood estimate for the true value. Furthermore, from a Bayesian perspective the posterior distribution for the true value given normally distributed observations and a flat prior is a normal distribution with the inverse-variance weighted average as a mean and variance ().
The maximum likelihood method weights the difference between fit and data using the same weights . The expected value of a random variable is the weighted average of the possible values it might take on, with the weights being the respective probabilities. More generally, the expected value of a function of a random variable is the probability ...
Data can be binary, ordinal, or continuous variables. It works by normalizing the differences between each pair of variables and then computing a weighted average of these differences. The distance was defined in 1971 by Gower [1] and it takes values between 0 and 1 with smaller values indicating higher similarity.
One very early weighted estimator is the Horvitz–Thompson estimator of the mean. [3] When the sampling probability is known, from which the sampling population is drawn from the target population, then the inverse of this probability is used to weight the observations. This approach has been generalized to many aspects of statistics under ...
This method can also be used to create spatial weights matrices in spatial autocorrelation analyses (e.g. Moran's I). [1] The name given to this type of method was motivated by the weighted average applied, since it resorts to the inverse of the distance to each known point ("amount of proximity") when assigning weights.
The second form above illustrates that the logarithm of the geometric mean is the weighted arithmetic mean of the logarithms of the individual values. If all the weights are equal, the weighted geometric mean simplifies to the ordinary unweighted geometric mean. [1]
The idea of the kernel average smoother is the following. For each data point X 0, choose a constant distance size λ (kernel radius, or window width for p = 1 dimension), and compute a weighted average for all data points that are closer than to X 0 (the closer to X 0 points get higher weights).