**Contents**:

A statistical **weight **is an amount given to increase or decrease the importance of an item. For example, weights are commonly given for tests and exams in class. For example, a final exam might count for double the points (double the “weight”) of an in-class test.

A **weighting factor** is a weight given to a data point to assign it a lighter, or heavier, importance in a group. It is usually used for calculating a weighted mean, to give less (or more) importance to group members. It is also used in statistical sampling for adjusting samples and in nuclear medicine for calculating effective doses.

Weighting factors are used in sampling to make samples match the population. For example, let’s say you took a sample of the population and had 41% female and 59% male. You know from census data that females should make up 51% of the population and males 49%. In order to make sure that you have a representative sample, you could add a little more “weight” to data from females. To calculate how much weight you need, divide the known population percentage by the percent in the sample. For this example:

- Known population females (51) / Sample Females (41) = 51/41 = 1.24.
- Known population males (49) / Sample males (59) = 49/59 = .83.

Weighting factors are used extensively in radiologic and nuclear medicine to calculative effective doses for procedures. The calculations for Tissue Weighting Factors (sometimes called Radiologic Weighting Factors) account for the fact that different parts of the body absorb radiation at different rates.

A tissue weighting factor(W_{T}) is assigned to body parts, with more radiosensitive parts given higher weighting factors.

**Effective dose = individual organ dose values * W _{T}. **

**Tissue weighting factors (ICRP) are:**

- W
_{T}= 0.12: stomach, colon, lung, red bone marrow, breast, remainder tissues, - W
_{T}= 0.08: gonads, - W
_{T}= 0.04: urinary bladder, oesophagus, liver, thyroid, - W
_{T}= 0.01: bone surface, skin, brain, salivary glands.

You may want to read this first: What is a Function?

A **weight function** is a special function that allows you to allocate more “weight” or influence to some elements of a set. Weight functions are often used on measured data, and can be used for both discrete variables and continuous variables.

The special weight function** w(a): = 1** represents the unweighted situation where every element has the same weight.

Let’s say you are summing up a set of values; the values of a particular function *f* on A. Then we could write the sum as:

If we want to weight our values with the weight function *w*:A→ R+, the sum would be:

There are a number of reasons you might choose to use weighting functions. If you are using a variety of measurement tools and you know that part of your data set is more accurate than another part, using weight functions can help you **improve fit** when you estimate unknown parameters or choose a curve to represent a model.

You might also weight to compensate for bias (errors). If we know that a number of data points are more biased than others, it makes sense to give them lower weights when determining your model.

Sometimes, a weight function doesn’t have anything to do with measurement errors or lack of accuracy due to bias. In engineering applications, weighting functions are used to reflect the relative influence of different forces or parameters. For instance, a force acting from a far distance would need lower weighting than a force acting from close proximity. Of course, weight functions can also be used when we are working with the actual force exerted by different weights on an object.

NIST Engineering Statistics Handbook. 4.4.5.2. Accounting for Non-Constant Variation Across the Data. Retrieved from https://www.itl.nist.gov/div898/handbook/pmd/section4/pmd452.htm on July 13, 2019.

NIST Engineering Statistics Handbook. 4.6.3.4 Weighting to Improve Fit. Retrieved from https://www.itl.nist.gov/div898/handbook/pmd/section6/pmd634.htm on July 13, 2019.

**Extra Sums of Squares (ESS)** is the difference in the Error Sums of Squares (SSE) of two models. More specifically, ESS is a measure of the marginal reduction in Error Sums of Squares (SSE) when an additional set of predictors is added to the model.

It is a tool for model comparison, comprised of a single number. If ESS = 0, the models are identical.

The formula for Extra Sums of Squares is:

Let’s say your model contains one predictor variable, X_{1}. If you add a second predictor, X_{2} to the model, ESS explains the additional variation explained by X_{2}. We can write that as:

SSR (X_{2} | X_{1}).

In terms of SSE, let’s say you have a model with one predictor variable, X_{1}. You add a variable X_{2} to a model. Extra Sums of Squares explains the part of SSE not explained by the original variable (X_{1}). We can write that as:

SSR (X_{2} | X_{1}) = X_{1} – (X_{1}, X_{2})

Marasinghe, M. & Kennedy, W. (2008). SAS for Data Analysis: Intermediate Statistical Methods. Springer Science and Business Media. Retrieved September 9, 2019 from: https://books.google.com/books?id=LX2v9CNhzJMC

Olbricht, G. Lecture 13: Extra Sums of Squares. Article posted on Purdue University website. Retrieved September 10, 2019 from: https://www.stat.purdue.edu/~ghobbs/

Ramsey, F. & Schafer, D. The Statistical Sleuth: A Course in Methods of Data Analysis. Retreived September 10, 2019 from: https://books.google.com/books?id=jfoKAAAAQBAJ

In signal processing, cross correlation is where you take two signals and produce a third signal. The method, which is basically a generalized form of “regular” linear correlation, is a way to objectively compare different time series and allows you to see how two signals match and where the best match occurs. It can be used to create plots that may reveal hidden sequences.

The basic process involves:

**Calculate a correlation coefficient**. The coefficient is a measure of how well one series predicts the other.**Shift the series**, creating a lag. Repeat the calculations for the correlation coefficient.**Repeat steps 1 and 2.**How many times you repeat the process will depend on your data, but as the lag increases the potential matches will decrease.**Identify the lag with the highest correlation coefficient.**The lag with the highest correlation coefficient is where the two series match the best.

Cross correlation and autocorrelation are very similar, but they involve different types of correlation:

- Cross correlation happens when two
**different**sequences are correlated. - Autocorrelation is the correlation between two of the
**same**sequences. In other words, you correlate a signal with itself.

Caltech. Cross-Correlation. Retrieved September 3, 2019 from: https://ned.ipac.caltech.edu/level5/Sept01/Peterson2/Peter4.html

Derrick, T.R. and Thomas, J.M. (2004). Chapter 7. Time-Series Analysis: The cross-correlation function. In: Innovative Analyses of Human Movement, (pp. 189-205), Stergiou, N. (ed). Human Kinetics Publishers, Champaign, Illinois, 189-205.

Ghosh, S. (2005) Signals and Systems. Pearson Education India.

Smith, S. (2003). Digital Signal Processing: A Practical Guide for Engineers and Scientists. Newnes.

United States Naval Academy. Cross Correlation. Retrieved September 3, 2019 from:https://www.usna.edu/Users/oceano/pguth/md_help/html/time0alq.htm

**Measures of position** give us a way to see where a certain data point or value falls in a sample or distribution. A measure can tell us whether a value is about the average, or whether it’s unusually high or low. Measures of position are used for quantitative data that falls on some numerical scale. Sometimes, measures can be applied to ordinal variables— those variables that have an order, like first, second…fiftieth.

Measures of position can also show how to values from different distributions or measurement scales compare. For example, a person’s height (measured in feet) and weight (measured in pounds) can be compared by converting the measurements to z-scores.

- Box and Whiskers Plot,
- Deciles,
- Five Number Summary,
- Interquartile Range (IQR),
- Outliers,
- Percentiles,
- Quartiles,
- Standard scores (i.e. z-scores),
- Tukey’s upper hinge and lower hinge.

A box and whiskers plot shows the spread and center of data. It is a graphical representation of the five number summary: minimum, maximum, median, and the first and third quartiles.

Deciles are similar to quartiles. But where quartiles split the data in four equal parts, deciles split the data into ten parts: The 10th, 20th, 30th, 40th, 50th, 60th, 70th, 80th, 90th and 100th percentiles.

The five number summary is an overview of your data. The statistics in the summary are the smallest value (minimum), the largest (maximum), the middle (median) and the first and third quartiles.

The interquartile range tells you where the “middle fifty” is in a data set. While the range tells you where the beginning and end are in a set, the IQR shows you where the bulk of the “middling” values lie.

Outliers are unusual values that fall outside of an expected range of values. For example, if you’re measuring IQ values of children, your statistics would be thrown off if Einstein and Stephen Hawking were in your class: their IQs would be outliers.

A percentile is a number where a certain percentage of scores fall below that number. For example, a 90th percentile marks the spot where 90% of values fall below that cut-off point.

Simply put, quartiles divide your data into quarters: the lowest quarter, two middle quarters, and a highest quarter.

Z-scoresare a way to compare results from a test to a “normal” population.

Tukey’s upper hinge and lower hinge are created when you split a data set into four pieces (with three hinges). As the median is included in this “splitting,” Tukey’s hinges are sometimes called inclusive quartiles.

Antonius, R. (2003). Interpreting Quantitative Data with SPSS. SAGE. Retrieved August 26, 2019 from: https://books.google.com/books?id=H1_mH0glk0IC

UF Biostatistics. Measures of Position. Retrieved August 26, 2019 from: https://bolt.mph.ufl.edu/6050-6052/unit-1/one-quantitative-variable-introduction/measures-of-position/

An **identification key** is an identification tool primarily used to find taxonomic levels (e.g. species or genus) in the biological sciences. The keys have a wide variety of applications, including identification of plants, nuts, and amphibians as well as in forensic analysis.

As an example, let’s say you found a moth in the Dakotas, and you wanted to know if it belonged to the butterflies, primitive moths, or other groups. You might be tempted to compare your moth to online pictures, but there are over 1,400 species of moth in the Dakotas. An identification key helps by taking your through a series of steps to identify the group. Some keys, like this Purdue University field guide, use a combination of pictures and words.

**Dichotomous keys** provide a set of two alternatives. You’re given two descriptions at each step, simply choosing the one which fits best. The Cortland Herpetology Connection offers this example for a dichotomous identification key: let’s say you wanted to identify a certain reptile/amphibian from New York state and you want to classify it as belonging to frogs, salamanders, snakes and lizards, or turtles. The first step might ask:

*External gills? *

If yes, it’s a salamander. If no, go to the next question. Each subsequent question would provide a binary option.

N, Talent et al. (2014). Character Selection During Interactive Taxonomic Identification. Biodiversity Informatics.

Cortland Herpetology Connection. How to use the identification keys. Retrieved August 12, 2019 from: http://www.cortland.edu/herp/keys/howtokey.htm

North Dakota State University. Identification Key. Retrieved August 12, 2019 from: https://www.ndsu.edu/ndmoths/ndmoths/identification%20keys.htm

Randler, C. (2008). Teaching Species Identification – A Prerequisite for Learning Biodiversity and Understanding Ecology. Eurasia Journal of Mathematics, Science & Technology Education.

The **exponential power distribution** is a generalization of the normal distribution. Therefore, it’s also known as the *generalized normal distribution*. Other names include the *General Error Distribution* and the Generalized Power Distribution.

The Probability Density Function (PDF) of the Exponential Power Distribution is:

Where:

- λ = positive scale parameter,
- κ = positive shape parameter.

The extra parameter allows the distribution to represent many more distributional shapes. It also allows more flexibility with kurtosis, which can be represented by a variety of values (rather than just a constant).

As well as the extra parameter, which allows more flexibility with distributional shapes and kurtosis. However, a “major drawback” is that the distribution doesn’t allow for asymmetrical data (Konunjer, n.d.). However, it is possible to derive a slightly different model with skew (e.g. the skew exponential power model proposed by Ayebo and Kozubowski).

Other features include:

- Heavy tails,
- Symmetry around the mean,
- Unimodality,
- Bathtub shaped hazard function,

Ayebo, A. and Kozubowski, T. An asymmetric generalization of Gaussian and Laplace laws. Retrieved Auhust 12, 2019 from: https://wolfweb.unr.edu/homepage/tkozubow/0skeexp1.pdf

Package Normalp. Retrieved August 12, 2019 from: https://cran.case.edu/web/packages/normalp/normalp.pdf

Exponential Power. Retrieved August 12, 2019 from: http://www.math.wm.edu/~leemis/chart/UDR/PDFs/Exponentialpower.pdf

Konunjer, I. Asymmetric Power Distribution: Theory and Applications to Risk Measurement.

Kotchlamazashvili, Z. (2014). Empirical Analysis of the EU Term Structure of Interest Rates. Logos Verlag Berlin GmbH.

The **Dagum distribution** (also called the Inverse Burr distribution; Dagum called it the generalized logistic-Burr distribution) was proposed by Camilo Dagum in the 1970s to model income and wealth distribution. Dagum developed the models as an alternative to the Pareto Distribution and lognormal distribution, which he felt didn’t result in accurate models when applied to spread of income.

Despite its relative unpopularity, the Dagum distribution often performs better than other two/three parameter income/wealth distribution models when applied to empirical data.

The CDF of the Type I (three parameter) Dagum distribution is:

And the PDF is:

Where:

- λ = a scale parameter,
- δ and Β = shape parameters.

When Β = 1, the distribution is called the log-logistic distribution.

The three parameter **Dagum Type I distribution** evolved from Dagum’s experimentation with a shifted log-logistic distribution (Chotikapanich, 2008). Two four-parameter (Type II) generalizations were also developed.

Chotikapanich, D. (Ed.) (2008). Modeling Income Distributions and Lorenz Curves. Springer Science & Business Media.

Dagum, C., A New Model of Personal Income Distribution: Specification and Estimation,

Economie Applique’e, 30, 413 – 437, (1977).

Kleiber, C. & Kotz, S. (2003). Statistical Size Distributions in Economics and Actuarial Sciences. John Wiley & Sons.

Atkinson, A. (2014). Handbook of Income Distribution, Volume 2. Elsevier.

**Mean Absolute Scaled Error (MASE)** is a scale-free error metric that gives each error as a ratio compared to a baseline’s average error.

The advantages of MASE include that it never gives undefined or infinite values and so is a good choice for intermittent-demand series (which arise when there are periods of zero demand in a forecast). It can be used on a single series, or as a tool to compare multiple series.

MASE is defined by the following equation:

MASE = mean ( | q

_{t}| ) ~ Prestwich et. al, 2013

Where:

**Mean Absolute Scaled Error** (MASE) is one of four main measures of forecasting accuracy. The other three are:

**Scale-dependent methods**: e.g. Mean absolute error (MAE or MAD). Although these are easy to calculate, they can’t be used to compare different series, because of the scale dependency.**Percentage-error methods:**e.g. Mean Absolute Percent Error (MAPE); These can be used to compare different series because they are scale independent. However, they can’t be used when you have zeros in your data.**Relative-error methods:**averages error ratios from a chosen metric to that of a naive one. These metrics can’t be used when you have small errors, because the resulting calculations would involve division by zero.

Rob Hydman, in his Foresight article *Another look at forecast-accuracy metrics for intermittent demand* states that the mean absolute scaled error is the only method out of the four that “…always gives sensible results.” However, Prestwich et. al do note that you can’t use it when every in-sample demand is identical.

Hyndman, R. (2006) Another look at forecast-accuracy metrics for intermittent demand.

Prestwich, S. et al. (2013). Mean-Based Error Measures for Intermittent Demand Forecasting. Retrieved July 29, 2019 from: http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.741.9979&rep=rep1&type=pdf

The risk function is the expected value of a loss function. In other words, it’s the expected value of a loss.

Most losses are not random; They are usually a result of a set of circumstances or decisions that can be quantified. If you have substantial knowledge about a particular process or event, then you can create a risk function for it. As an example, risk has already been widely defined for credit-worthiness; a person with a good credit score income is a better risk than someone with poor credit and few resources.

The risk function is dependent on the chosen decision rule, which is used in the face of uncertainty. A separate risk function is defined for each consequence scale. Therefore, each asset is linked to a separate risk function.

The basic steps for defining a risk function:

- Create a matrix for each consequence scale. Place the consequence scale vertically and likelihood scale horizontally.
- Decide on a risk value for each cell entry. This decision is based on analysis of the specific situation or asset. As an example, it’s fairly easy to decide on risk functions for extremes (e.g. catastrophic/certain or insignificant rare). Values “in between” are usually agreed upon by a panel of experts (e.g. data analysts, management).

A risk function gives a **risk level** for every combination of likelihood and a consequence (Lund et al., 2013). Risk levels can be qualitative or quantitative, depending on the choices you made for the likelihood and consequence values.

Lund, M. et al. (2010). Model-Driven Risk Analysis: The CORAS Approach. Retrieved July 29, 2019 from: https://books.google.com/books?id=X4lpi3stRvYC

]]>A variety of tests can be used to identify lack-of-fit in statistical models. These include:

- Goodness of fit
- Lack-of-fit F-Test/ sum of squares
- Ljung Box Test

Correcting lack of fit in a model usually involves rewriting the model to fit the data better. This may be by adding a quadratic term, changing a linear regression model to a polynomial regression model, for instance.

Sometimes, what it points to is poor experimental design. This could suggest we redesign our experiment to get more accurate data or expand our sampling to get more data points that can provide a more complete picture. If the model was in fact an accurate description of the situation, a combination of these methods will change the fit to a good one.

Christensen, Ronald. Chapter 8: Testing Lack-of-Fit. Unbalanced Analysis of Variance, Design, and Regression. Retrieved from http://www.math.unm.edu/~fletcher/SUPER/chap8.pdf on July 29, 2018.

Lack-of-Fit. Analyse-it Statistical Reference Guide. Retrieved from

https://analyse-it.com/docs/user-guide/fitmodel/linear/lackoffit on July 29, 2018

Ruczinski, Ingo. Chapter 6, Testing for Lack-of-Fit. Teaching Notes. Retrieved from http://www.biostat.jhsph.edu/~iruczins/teaching/jf/ch6.pdf on July 29, 2018

Statsoft Team. Lack-of-Fit. Statistica Help. Retrieved from http://documentation.statsoft.com/STATISTICAHelp.aspx?path=glossary/GlossaryTwo/L/LackofFit on July 29, 2018

Lack-of-Fit and Lack-of-Fit Tests

Retrieved from https://support.minitab.com/en-us/minitab/18/help-and-how-to/modeling-statistics/regression/supporting-topics/regression-models/lack-of-fit-and-lack-of-fit-tests/ on July 29, 2018.