yat
0.12.3pre

There are several different reasons why a statistical analysis needs to adjust for weighting. In literature reasons are mainly diveded in to groups.
The first group is when some of the measurements are known to be more precise than others. The more precise a measurement is, the larger weight it is given. The simplest case is when the weight are given before the measurements and they can be treated as deterministic. It becomes more complicated when the weight can be determined not until afterwards, and even more complicated if the weight depends on the value of the observable.
The second group of situations is when calculating averages over one distribution and sampling from another distribution. Compensating for this discrepency weights are introduced to the analysis. A simple example may be that we are interviewing people but for economical reasons we choose to interview more people from the city than from the countryside. When summarizing the statistics the answers from the city are given a smaller weight. In this example we are choosing the proportions of people from countryside and people from city being intervied. Hence, we can determine the weights before and consider them to be deterministic. In other situations the proportions are not deterministic, but rather a result from the sampling and the weights must be treated as stochastic and only in rare situations the weights can be treated as independent of the observable.
Since there are various origins for a weight occuring in a statistical analysis, there are various ways to treat the weights and in general the analysis should be tailored to treat the weights correctly. We have not chosen one situation for our implementations, so see specific function documentation for what assumtions are made. Though, common for implementations are the following:
The last point implies that a data point with zero weight is ignored also when the value is NaN. An important case is when weights are binary (either 1 or 0). Then we get the same result using the weighted version as using the data with weight not equal to zero and the nonweighted version. Hence, using binary weights and the weighted version missing values can be treated in a proper way.
For any situation the weight is always designed so the weighted mean is calculated as , which obviously fulfills the conditions above.
In the case of varying measurement error, it could be motivated that the weight shall be . We assume measurement error to be Gaussian and the likelihood to get our measurements is . We maximize the likelihood by taking the derivity with respect to on the logarithm of the likelihood . Hence, the Maximum Likelihood method yields the estimator .
In case of varying variance, there is no point estimating a variance since it is different for each data point.
Instead we look at the case when we want to estimate the variance over but are sampling from . For the mean of an observable we have . Hence, an estimator of the variance of is
This estimator is invariant under rescaling and having a weight equal to zero is equivalent to removing the data point. Having all weights equal to unity results in , which is the same as returned from Averager. Hence, this estimator is slightly biased, but still very efficient.
The standard error squared is equal to the expexted squared error of the estimation of . The squared error consists of two parts, the variance of the estimator and the squared bias:
.
In the case when weights are included in analysis due to varying measurement errors and the weights can be treated as deterministic, we have
where we need to estimate . Again we have the likelihood
and taking the derivity with respect to ,
which yields an estimator . This estimator is not ignoring weights equal to zero, because deviation is most often smaller than the expected infinity. Therefore, we modify the expression as follows and we get the following estimator of the variance of the mean . This estimator fulfills the conditions above: adding a weight zero does not change it: rescaling the weights does not change it, and setting all weights to unity yields the same expression as in the nonweighted case.
In a case when it is not a good approximation to treat the weights as deterministic, there are two ways to get a better estimation. The first one is to linearize the expression . The second method when the situation is more complicated is to estimate the standard error using a bootstrapping method.
Here data points come in pairs (x,y). We are sampling from but want to measure from . To compensate for this decrepency, averages of are taken as . Even though, and are not independent we assume that we can factorize the ratio and get
Following the variance calculations for AveragerWeighted we have where
As the mean is estimated as , the variance is estimated as . As in the nonweighted case we define the correlation to be the ratio between the covariance and geometrical average of the variances
.
This expression fulfills the following
.
See AveragerPairWeighted correlation.
An interpretation of the ROC curve area is the probability that if we take one sample from class and one sample from class , what is the probability that the sample from class has greater value. The ROC curve area calculates the ratio of pairs fulfilling this
A geometrical interpretation is to have a number of squares where each square correspond to a pair of samples. The ROC curve follows the border between pairs in which the samples from class has a greater value and pairs in which this is not fulfilled. The ROC curve area is the area of those latter squares and a natural extension is to weight each pair with its two weights and consequently the weighted ROC curve area becomes
This expression is invariant under a rescaling of weight. Adding a data value with weight zero adds nothing to the exprssion, and having all weight equal to unity yields the nonweighted ROC curve area.
Assume that and originate from the same distribution where . We then estimate as The variance of difference of the means becomes and consequently the tscore becomes
For a we this expression get condensed down to in other words the good old expression as for nonweighted.
FoldChange is simply the difference between the weighted mean of the two groups
Taking all pair samples (one from class and one from class ) and calculating the weighted median of the distances.
A Distance measures how far apart two ranges are. A Distance should preferably meet some criteria:
Weighted Distance is an extension of usual unweighted distances, in which each data point is accompanied with a weight. A weighted distance should meet some criteria:
The last condition, duplicate property, implies that setting a weight to zero is not equivalent to removing the data point. This behavior is sensible because otherwise we would have a bias towards having ranges with small weights being close to other ranges. For a weighted distance, meeting these criteria, it might be difficult to show that the triangle inequality is fulfilled. For most algorithms the triangle inequality is not essential for the distance to work properly, so if you need to choose between fulfilling triangle inequality and these latter criteria it is preferable to meet the latter criteria.
In test/distance_test.cc there are tests for testing these properties.
The polynomial kernel of degree is defined as , where is the linear kernel (usual scalar product). For the weighted case we define the linear kernel to be and the polynomial kernel can be calculated as before .
We define the weighted Gaussian kernel as .
We have the model
where is the noise. The variance of the noise is inversely proportional to the weight, . In order to determine the model parameters, we minimimize the sum of quadratic errors.
Taking the derivity with respect to and yields two conditions
and
or equivalently
and
Note, by having all weights equal we get back the unweighted case. Furthermore, we calculate the variance of the estimators of and .
and
Finally, we estimate the level of noise, . Inspired by the unweighted estimation
we suggest the following estimator