weighting

Under weighting (also weighting , weighting pattern ) is the evaluation of individual factors of a mathematical model , for example in terms of their importance or reliability . It means that more important or more reliable elements have a greater impact on the outcome.

example

For entry into a technical high school, the math score is more important than the history score. If the average is now determined, the two points are not simply added up and divided by 2, but first both points are multiplied by a weighting factor (short: weight), and only then are added together and divided by the sum of the weights.

For example, for a technical high school, the math score is multiplied by the weight 2, and the history score is multiplied by the weight 1.

Student A: If Student A has 80 points in math and 40 points in history, then the 80 points in math are multiplied by the weight 2. This results in a weighted score of 160. The 40 points in history are multiplied by the weight 1. This gives The weighted number of points is 40. The two weighted points are added together and divided by 3 (sum of weights), resulting in (160 + 40): 3 = 200: 3 = 66.6 weighted points.
Student B: If Student B has 40 points in math and 80 points in history, then the 40 points in math are multiplied by the weight 2. This results in a weighted score of 80. The 80 points in history are multiplied by the weight 1. This gives A weighted score of 80 results. The two weighted scores are added together and divided by 3, i.e. (80 + 80): 3 = 160: 3 = 53.3 weighted points.

Result: Student A has 66.6 weighted points,; Student B has 53.3 weighted points,; Student A therefore has a better chance of being accepted into the technical high school.

If mathematics were weighted not with factor 2 but with factor 1 like history, then both would have the same chances, namely (80 + 40): 2 = 60 points.

Determination of the weighting factor

The appropriateness of the weighting factor is decisive for the quality of the weighted value. This can be determined arbitrarily (as in the school example above): If history has a weight of 1, and mathematics has a weight of 2 - what weight should geography then have - more 1.8 or more 2.2? Or when comparing electricity from the nuclear power plant and electricity from the coal-fired power plant: what weight do the values "electricity price", "exhaust gases" or "nuclear waste" have?

Individual values are weighted differently depending on political and economic interests or technical / physical / mathematical conditions. This produces completely different overall results. Weighted results can only be understood and assessed with knowledge of the underlying political and economic interests or technical / physical / mathematical conditions. This also applies to weighted values, behind which there are complicated statistical calculations.

calculation

The weighted mean is calculated as follows:

When the data	${\ displaystyle x_ {1}, x_ {2}, x_ {3}, \ ldots, x_ {n}}$
with the weights	${\ displaystyle g_ {1}, g_ {2}, g_ {3}, \ ldots, g_ {n}}$ be provided

this is how the weighted mean is calculated

{\ displaystyle m = {\ frac {\ sum _ {i} x_ {i} g_ {i}} {\ sum _ {i} g_ {i}}} = {\ frac {x_ {1} \ cdot g_ { 1} + x_ {2} \ cdot g_ {2} + \ dotsb + x_ {n} \ cdot g_ {n}} {g_ {1} + g_ {2} + \ cdots + g_ {n}}}}

with the standard deviation with . ${\ displaystyle \ textstyle \ sigma _ {m} = {\ sqrt {\ frac {\ sum _ {i} g_ {i} \ sigma _ {i} ^ {2}} {(\ sum _ {i} g_ { i}) - 1}}}}$ ${\ displaystyle \ textstyle \ sigma _ {i} = m-x_ {i}}$

Example: A teacher weights the third of 4 class work twice.

Grades:	${\ displaystyle 2, \, 4, \, 3, \, 2}$
Weights:	${\ displaystyle 1, \, 1, \, 2, \, 1}$

weighted mean:	${\ displaystyle {\ frac {1 \ cdot 2 + 1 \ cdot 4 + 2 \ cdot 3 + 1 \ cdot 2} {1 + 1 + 2 + 1}} = {\ frac {14} {5}} = { \ underline {2 {,} 8}}}$
unweighted mean:	${\ displaystyle {\ frac {1 \ times 2 + 1 \ times 4 + 1 \ times 3 + 1 \ times 2} {1 + 1 + 1 + 1}} = {\ frac {11} {4}} = { \ underline {2 {,} 75}}}$

By weighting the grade 3 with a higher value than the other grades, the mean value shifts upwards (towards the "worse" grade).

Types of weights

There are several types of weights:

Empirical distinction

Design weighting : mapping of disproportionately stratified samples .
Redressment (also reweighting): Subsequent adjustment to known marginal distributions, e.g. B. in the case of systematic biases in the sample due to non-random failures.

Mathematical distinction

Frequency-weights (Frequency-Weights) : weights that occurs specify how often an observation (value) in the record.
Analytical weights (Analytic Weights) : weights that indicate how many cases an aggregate feature attributable. These are frequency weights with normalization to the sample size.
Probability Weights (Probability Weights) : weights that take into account that selection probability has an observation. It is the inverse of the selection probability
Importance Weights

application

Weighting of irregularly performed measurements

If measurements are carried out at uneven intervals, the measurement results are incorrectly shifted in the direction of the higher frequented measurements. Example: The pH of a lake is usually measured once a year and remains constant at 7.0 for five years. Then in the sixth year a pH value of 9.0 is measured, after which it is switched to daily measurement. A pH of 9.0 is measured daily for 15 days. The (unweighted) average pH of this lake would then be incorrectly determined to be 8.5, even though the lake had a pH of 7.0 for the longest period. If, on the other hand, the annual measurements are weighted correspondingly higher (365 times as high) as the daily measurements, the result is a weighted average pH value of 7.02, which better describes reality.

Weighting of statistically scattering quantities

If the spread of each value is known for physical quantities , it is advisable to weight the values according to their spread when calculating the mean value. If the te value has the spread , then the associated weighting is , the standard deviation is simplified to . ${\ displaystyle i}$ ${\ displaystyle \ sigma _ {i} ^ {2}}$ ${\ displaystyle g_ {i} = {\ tfrac {1} {\ sigma _ {i} ^ {2}}}}$ ${\ displaystyle \ sigma _ {m} = {\ tfrac {1} {\ sqrt {\ sum _ {i} {\ tfrac {1} {\ sigma _ {i} ^ {2}}}}}}}$

Weighting of metrics

In measurement technology , it can be appropriate to weight various measured values with the reciprocal values of their uncertainties . This ensures that values with smaller uncertainties are weighted more heavily in further calculations.

economy

In the economic sector, weighting schemes are used in particular for calculating shopping baskets (and thus price indices ) and effective exchange rates .

exams

If an examination consists of several subjects and an overall result of the examination has to be formed, the individual results of the subjects are often summarized with a certain weighting. For final exams in recognized training occupations, the training regulations for the occupation usually specify the weighting factors; in individual cases, the respective examination regulations also apply .

Weighting of characteristics

In machine learning , the task is to learn a decision function that calculates an answer based on features . Many models, such as the perceptron, learn to weight the input features, which indicate how strongly which features speak for the respective answers ( weighted sum ). In the case of more complex, non-linear models such as neural networks , the different features are weighted within several consecutive so-called "hidden layers". The values of the learned weights can no longer easily be assigned to the importance of individual characteristics. Explainable Artificial Intelligence deals with the interpretability of such models .

literature

S. Gabler, M. Ganninger: Weighting. In Handbook of Social Science Data Analysis. VS Verlag für Sozialwissenschaften, 2010, pp. 143–164.
S. Gabler, JH Hoffmeyer-Zlotnik: Weighting in Survey Practice. West German Publishing house, 1994.
C. Alt, W. Bien: Weighting, a useful procedure in the social sciences? Questions, Problems, and Conclusions. In: Weighting in survey practice. VS Verlag für Sozialwissenschaften, 1994, pp. 124–140.

Individual evidence

^ Heinrich Braun, Johannes Feulner, Rainer Malaka: Das Perzeptron . In: Practical course in neural networks (= Springer textbook ). Springer, Berlin / Heidelberg 1996, ISBN 3-642-61000-5 , p. 7-24 , doi : 10.1007 / 978-3-642-61000-4_2 .
↑ Grégoire Montavon, Wojciech Samek, Klaus-Robert Müller: Methods for interpreting and understanding deep neural networks . In: Digital Signal Processing . tape 73 , February 1, 2018, ISSN 1051-2004 , p. 1–15 , doi : 10.1016 / j.dsp.2017.10.011 ( sciencedirect.com [accessed December 7, 2019]).

[1] Heinrich Braun, Johannes Feulner, Rainer Malaka: Das Perzeptron . In: Practical course in neural networks (= Springer textbook ). Springer, Berlin / Heidelberg 1996, ISBN 3-642-61000-5 , p. 7-24 , doi : 10.1007 / 978-3-642-61000-4_2 .

[2] Grégoire Montavon, Wojciech Samek, Klaus-Robert Müller: Methods for interpreting and understanding deep neural networks . In: Digital Signal Processing . tape 73 , February 1, 2018, ISSN 1051-2004 , p. 1–15 , doi : 10.1016 / j.dsp.2017.10.011 ( sciencedirect.com [accessed December 7, 2019]).