What is the Gauss-Markov theorem?
From “The Cambridge Dictionary of Statistics” B. S. Everitt, 2nd Edition:
A theorem that proves that if the error terms in a multiple regression have the same variance and are uncorrelated, then the estimators of the parameters in the model produced by least squares estimation are better (in the sense of having lower dispersion about the mean) than any other unbiased linear estimator.
This is pretty much considered the “big boy” reason least squares fitting can be considered a good implementation of linear regression.
Suppose you are building a model of the form:
y(i) = B . x(i) + e(i)
B is a vector (to be inferred),
i is an index that runs over the available data (say
x(i) is a per-example vector of features, and
y(i) is the scalar quantity to be modeled. Only
y(i) are observed. The
e(i) term is the un-modeled component of
y(i) and you typically hope that the
e(i) can be thought of unknowable effects, individual variation, ignorable errors, residuals, or noise. How weak/strong assumptions you put on the
e(i) (and other quantities) depends on what you know, what you are trying to do, and which theorems you need to meet the pre-conditions of. The Gauss-Markov theorem assures a good estimate of
B under weak assumptions.
How to interpret the theorem
The point of the Gauss-Markov theorem is that we can find conditions ensuring a good fit without requiring detailed distributional assumptions about the
e(i) and without distributional assumptions about the
x(i). However, if you are using Bayesian methods or generative models for predictions you may want to use additional stronger conditions (perhaps even normality of errors and even distributional assumptions on the
We are going to read through the Wikipedia statement of the Gauss-Markov theorem in detail.
Been reading a lot of Gelman, Carlin, Stern, Dunson, Vehtari, Rubin “Bayesian Data Analysis” 3rd edition lately. Overall in the Bayesian framework some ideas (such as regularization, and imputation) are way easier to justify (though calculating some seemingly basic quantities becomes tedious). A big advantage (and weakness) of this formulation is statistics has a much less “shrink wrapped” feeling than the classic frequentist presentations. You feel like the material is being written to peers instead of written to calculators (of the human or mechanical variety). In the Bayesian formulation you don’t feel like you will be yelled at for using 1 tablespoon of sugar when the recipe calls for 3 teaspoons (at least if you live in the United States).
Some other stuff reads differently after this though. Continue reading
R is definitely our first choice go-to analysis system. In our opinion you really shouldn’t use something else until you have an articulated reason (be it a need for larger data scale, different programming language, better data source integration, or something else). The advantages of R are numerous:
- Single integrated work environment.
- Powerful unified scripting/programming environment.
- Many many good tutorials and books available.
- Wide range of machine learning and statistical libraries.
- Very solid standard statistical libraries.
- Excellent graphing/plotting/visualization facilities (especially ggplot2).
- Schema oriented data frames allowing batch operations, plus simple row and column manipulation.
- Unified treatment of missing values (regardless of type).
For all that we always end up feeling just a little worried and a little guilty when introducing a new user to R. R is very powerful and often has more than one way to perform a common operation or represent a common data type. So you are never very far away from a strange and painful corner case. This why when you get R training you need to make sure you get an R expert (and not an R apologist). One of my favorite very smart experts is Norm Matloff (even his most recent talk title is smart: “What no one else will tell you about R”). Also, buy his book; we are very happy we purchased it.
But back to corner cases. For each method in R you really need to double check if it actually works over the common R base data types (numeric, integer, character, factor, and logical). Not all of them do and and sometimes you get a surprise.
Recent corner case problems we ran into include:
- randomForest regression fails on character arguments, but works on factors.
gam() model doesn’t convert strings to formulas.
- R maps can’t use the empty string as a key (that is the string of length 0, not a
NULL array or
These are all little things, but can be a pain to debug when you are in the middle of something else. Continue reading
The goal of Zumel/Mount: Practical Data Science with R is to teach, through guided practice, the skills of a data scientist. We define a data scientist as the person who organizes client input, data, infrastructure, statistics, mathematics and machine learning to deploy useful predictive models into production.
Our plan to teach is to:
- Order the material by what is expected from the data scientist.
- Emphasize the already available bread and butter machine learning algorithms that most often work.
- Provide a large set of worked examples.
- Expose the reader to a number of realistic data sets.
Some of these choices may put-off some potential readers. But it is our goal to try and spend out time on what a data scientist needs to do. Our point: the data scientist is responsible for end to end results, which is not always entirely fun. If you want to specialize in machine learning algorithms or only big data infrastructure, that is a fine goal. However, the job of the data scientist is to understand and orchestrate all of the steps (working with domain experts, curating data, using data tools, and applying machine learning and statistics).
Once you define what a data scientist does, you find fewer people want to work as one.
We expand a few of our points below. Continue reading
What is meant by regression modeling?
Linear Regression is one of the most common statistical modeling techniques. It is very powerful, important, and (at first glance) easy to teach. However, because it is such a broad topic it can be a minefield for teaching and discussion. It is common for angry experts to accuse writers of carelessness, ignorance, malice and stupidity. If the type of regression the expert reader is expecting doesn’t match the one the writer is discussing then the writer is assumed to be ill-informed. The writer is especially vulnerable to experts when writing for non-experts. In such writing the expert finds nothing new (as they already know the topic) and is free to criticize any accommodation or adaption made for the intended non-expert audience. We argue that many of the corrections are not so much evidence of wrong ideas but more due a lack of empathy for the necessary informality necessary in concise writing. You can only define so much in a given space, and once you write too much you confuse and intimidate a beginning audience. Continue reading
Some researchers (in both science and marketing) abuse a slavish view of p-values to try and falsely claim credibility. The incantation is: “we achieved p = x (with x ≤ 0.05) so you should trust our work.” This might be true if the published result had been performed as a single project (and not as the sole shared result in longer series of private experiments) and really points to the fact that even frequentist significance is a subjective and intensional quantity (an accusation usually reserved for Bayesian inference). In this article we will comment briefly on the negative effect of un-reported repeated experiments and what should be done to compensate. Continue reading
As a data scientist I have seen variations of principal component analysis and factor analysis so often blindly misapplied and abused that I have come to think of the technique as unprincipled component analysis. PCA is a good technique often used to reduce sensitivity to overfitting. But this stated design intent leads many to (falsely) believe that any claimed use of PCA prevents overfit (which is not always the case). In this note we comment on the intent of PCA like techniques, common abuses and other options.
The idea is to illustrate what can quietly go wrong in an analysis and what tests to perform to make sure you see the issue. The main point is some analysis issues can not be fixed without going out and getting more domain knowledge, more variables or more data. You can’t always be sure that you have insufficient data in your analysis (there is always a worry that some clever technique will make the current data work), but it must be something you are prepared to consider. Continue reading
Nassim Nicholas Taleb recently wrote an article advocating the abandonment of the use of standard deviation and advocating the use of mean absolute deviation. Mean absolute deviation is indeed an interesting and useful measure- but there is a reason that standard deviation is important even if you do not like it: it prefers models that get totals and averages correct. Absolute deviation measures do not prefer such models. So while MAD may be great for reporting, it can be a problem when used to optimize models. Continue reading
It recently hit me that I see unit tests as a form of penance (in addition to being a great tool for specification and test driven development). If you fix a bug and don’t add a unit test I suspect you are not actually sorry. Continue reading
Bitcoin continues to surge in buzz and price (194,993 coin transfer, US Senate hearings and astronomical price and total capitalization). This gets me to thinking: what in finance terms is Bitcoin? It claims aspire to be a currency, but what is it actually behaving like? Continue reading