For the lasso (linear regression with L1 regularization) with a fixed value of λ, it is necessary to
use cross–validation to select the best optimization algorithm.
I know for a fact that we can use cross validation to find optimal value of λ, but is it neccesary to use cross validation in case λ is fixed?
Any thoughts please?
Cross Validation isn't about if your Regularization Parameter is Fixed or not. Its more related to the R^2 metric.
Lets say you consider 100 records and divide your data into 5 sub-datasets , means each sub-data contains 20 records.
Now out of 5 sub-datas , there are 5 different ways to assign anyone of the sub-data as Cross-Validation (CV) Data.
For all these 5 scenarios, we can find out the R^2, and then find out the Average R^2.
This way, you can have a comparison of your R-score with the Average R-score.
Related
I have data of 6 groups with sample size of n = 2, 10, 2, 9, 3, 1 and I want to perform Permutational multivariate analysis of variance (PERMANOVA) on these data.
My question is: Is it correct to run perMANOVA on these data with the small sample size? The results look strange for me because the group of n = 1 showed insignificant difference to other groups although the graphical representation of the groups clearly show a difference.
Thank you
I would not trust any result with group of n=1 because there is no source of variation to define difference among groups.
I have also received some answers from other platforms. I put them here for information:
The sample size is simply too small to yield a stable solution via manova. Note that the n = 1 cell contributes a constant value for that cell's mean, no matter what you do by way of permutations.
Finally, note that the effective per-cell sample size with unequal cell n for one-way designs tracks well to the harmonic mean of n. For your data set as it stands, that means an "effective" per-cell n of about 2.4. Unless differences are gigantic on the DV set, no procedure (parametric or exact/permutation) will have the statistical power to detect differences with that size.
MANOVA emphasizes the attribute scattering in the study group and the logic of this analysis is based on the scattering of scores. It is not recommended to use small groups with one or more people (I mean less than 20 people) to perform parametric tests such as MANOVA. In my opinion, use non-parametric tests to examine small groups.
This has been bugging me since a long time.
Suppose I have a boolean function F defined as follows:
Now, it can be expressed in its SOP form as:
F = bar(X)Ybar(Z)+ XYZ
But I fail to understand why we always complement the 0s to express them as 1. Is it assumed that the inputs X, Y and Z will always be 1?
What is the practical application of that? All the youtube videos I watched on this topic, how to express a function in SOP form or as sum of minterms but none of them explained why we need this thing? Why do we need minterms in the first place?
As of now, I believe that we design circuits to yield and take only 1 and that's where minterms come in handy. But I couldn't get any confirmation of this thing anywhere so I am not sure I am right.
Maxterms are even more confusing. Do we design circuits that would yield and take only 0s? Is that the purpose of maxterms?
Why do we need minterms in the first place?
We do not need minterms, we need a way to solve a logic design problem, i.e. given a truth table, find a logic circuit able to reproduce this truth table.
Obviously, this requires a methodology. Minterm and sum-of-products is mean to realize that. Maxterms and product-of-sums is another one. In either case, you get an algebraic representation of your truth table and you can either implement it directly or try to apply standard theorems of boolean algebra to find an equivalent, but simpler, representation.
But these are not the only tools. For instance, with Karnaugh maps, you rewrite your truth table with some rules and you can simultaneously find an algebraic representation and reduce its complexity, and it does not consider minterms. Its main drawback is that it becomes unworkable if the number of inputs rises and it cannot be considered as a general way to solve the problem of logic design.
It happens that minterms (or maxterms) do not have this drawback, and can be used to solve any problem. We get a trut table and we can directly convert it in an equation with ands, ors and nots. Indeed minterms are somehow simpler to human beings than maxterms, but it is just a matter of taste or of a reduced number of parenthesis, they are actually equivalent.
But I fail to understand why we always complement the 0s to express them as 1. Is it assumed that the inputs X, Y and Z will always be 1?
Assume that we have a truth table, with only a given output at 1. For instance, as line 3 of your table. It means that when x=0, y=1 and z=0 , the output will be zero. So, can I express that in boolean logic? With the SOP methodology, we say that we want a solution for this problem that is an "and" of entries or of their complement. And obviously the solution is "x must be false and y must be true and z must be false" or "(not x) must be true and y must be true and (not z) must be true", hence the minterm /x.y./z. So complementing when we have a 0 and leaving unchanged when we have a 1 is way to find the equation that will be true when xyz=010
If I have another table with only one output at 1 (for instance line 8 of your table), we can find similarly that I can implement this TT with x.y.z.
Now if I have a TT with 2 lines at 1, one can use the property of OR gates and do the OR of the previous circuits. when the output of the first one is 1, it will force this behavior and ditto for the second. And we directly get the solution for your table /xy/z+xyz
This can be extended to any number of ones in the TT and gives a systematic way to find an equation equivalent to a truth table.
So just think of minterms and maxterms as a tool to translate a TT into equations. What is important is the truth table (that describes the behaviour of what you want to do) and the equations (that give you a way to realize it).
I want to make a regression. My dependent variable Y is a score of discrete numbers from 0 to 12. I have 6 independent variables. Four of them are in percent between 0.4 % and 40 %. The two other independent variables are not in percent. One of them measures years and is between 1.8 and 67. The other independent variable measures size and is between 0.008 and 5,117 (very high).
I don't want to use winsorizing, that's why I decided to use logarithm. My question now: Is it good when I log the two independent variables which are not in percent, because they have high vaules?
That means I would use: reg Y x1 x2 x3 x4 logx5 logx6
Why I would like to use log is because I want to look out for the outliers. I have not really outliers, but I think it's better to use log for comparision reasons.
Or do I just have to use log, when the distribution of some variables is skewed? Meaning log leads to a lower skewness and is thus used appropriately.
Thank you in advance.
Lukas
I have a list of documents each having a relevance score for a search query. I need older documents to have their relevance score dampened, to try to introduce their date in the ranking process. I already tried fiddling with functions such as 1/(1+date_difference), but the reciprocal function is too discriminating for close recent dates.
I was thinking maybe a mathematical function with range (0..1) and domain(0..x) to amplify their score, where the x-axis is the age of a document. It's best to explain what I further need from the function by an image:
Decaying behavior is often modeled well by an exponentional function (many decaying processes in nature also follow it). You would use 2 positive parameters A and B and get
y(x) = A exp(-B x)
Since you want a y-range [0,1] set A=1. Larger B give slower decays.
If a simple 1/(1+x) decreases too quickly too soon, a sigmoid function like 1/(1+e^-x) or the error function might be better suited to your purpose. Let the current date be somewhere in the negative numbers for such a function, and you can get a value that is current for some configurable time and then decreases towards a base value.
log((x+1)-age_of_document)
Where the base of the logarithm is (x+1). Note the x is as per your diagram and is the "threshold". If the age of the document is greater than x the score goes negative. Multiply by the maximum possible score to introduce scaling.
E.g. Domain = (0,10) with a maximum score of 10: 10*(log(11-x))/log(11)
A bit late, but as thiton says, you might want to use a sigmoid function instead, since it has a "floor" value for your long tail data points. E.g.:
0.8/(1+5^(x-3)) + 0.2 - You can adjust the constants 5 and 3 to control the slope of the curve. The 0.2 is where the floor will be.
These questions regard a set of data with lists of tasks performed in succession and the total time required to complete them. I've been wondering whether it would be possible to determine useful things about the tasks' lengths, either as they are or with some initial guesstimation based on appropriate domain knowledge. I've come to think graph theory would be the way to approach this problem in the abstract, and have a decent basic grasp of the stuff, but I'm unable to know for certain whether I'm on the right track. Furthermore, I think it's a pretty interesting question to crack. So here we go:
Is it possible to determine the weights of edges in a directed weighted graph, given a list of walks in that graph with the lengths (summed weights) of said walks? I recognize the amount and quality of permutations on the routes taken by the walks will dictate the quality of any possible answer, but let's assume all possible walks and their lengths are given. If a definite answer isn't possible, what kind of things can be concluded about the graph? How would you arrive at those conclusions?
What if there were several similar walks with possibly differing lengths given? Can you calculate a decent average (or other illustrative measure) for each edge, given enough permutations on different routes to take? How will discounting some permutations from the available data set affect the calculation's accuracy?
Finally, what if you had a set of initial guesses as to the weights and had to refine those using the walks given? Would that improve upon your guesstimation ability, and how could you apply the extra information?
EDIT: Clarification on the difficulties of a plain linear algebraic approach. Consider the following set of walks:
a = 5
b = 4
b + c = 5
a + b + c = 8
A matrix equation with these values is unsolvable, but we'd still like to estimate the terms. There might be some helpful initial data available, such as in scenario 3, and in any case we can apply knowledge of the real world - such as that the length of a task can't be negative. I'd like to know if you have ideas on how to ensure we get reasonable estimations and that we also know what we don't know - eg. when there's not enough data to tell a from b.
Seems like an application of linear algebra.
You have a set of linear equations which you need to solve. The variables being the lengths of the tasks (or edge weights).
For instance if the tasks lengths were t1, t2, t3 for 3 tasks.
And you are given
t1 + t2 = 2 (task 1 and 2 take 2 hours)
t1 + t2 + t3 = 7 (all 3 tasks take 7 hours)
t2 + t3 = 6 (tasks 2 and 3 take 6 hours)
Solving gives t1 = 1, t2 = 1, t3 = 5.
You can use any linear algebra techniques (for eg: http://en.wikipedia.org/wiki/Gaussian_elimination) to solve these, which will tell you if there is a unique solution, no solution or an infinite number of solutions (no other possibilities are possible).
If you find that the linear equations do not have a solution, you can try adding a very small random number to some of the task weights/coefficients of the matrix and try solving it again. (I believe falls under Perturbation Theory). Matrices are notorious for radically changing behavior with small changes in the values, so this will likely give you an approximate answer reasonably quickly.
Or maybe you can try introducing some 'slack' task in each walk (i.e add more variables) and try to pick the solution to the new equations where the slack tasks satisfy some linear constraints (like 0 < s_i < 0.0001 and minimize sum of s_i), using Linear Programming Techniques.
Assume you have an unlimited number of arbitrary characters to represent each edge. (a,b,c,d etc)
w is a list of all the walks, in the form of 0,a,b,c,d,e etc. (the 0 will be explained later.)
i = 1
if #w[i] ~= 1 then
replace w[2] with the LENGTH of w[i], minus all other values in w.
repeat forever.
Example:
0,a,b,c,d,e 50
0,a,c,b,e 20
0,c,e 10
So:
a is the first. Replace all instances of "a" with 50, -b,-c,-d,-e.
New data:
50, 50
50,-b,-d, 20
0,c,e 10
And, repeat until one value is left, and you finish! Alternatively, the first number can simply be subtracted from the length of each walk.
I'd forget about graphs and treat lists of tasks as vectors - every task represented as a component with value equal to it's cost (time to complete in this case.
In tasks are in different orderes initially, that's where to use domain knowledge to bring them to a cannonical form and assign multipliers if domain knowledge tells you that the ratio of costs will be synstantially influenced by ordering / timing. Timing is implicit initial ordering but you may have to make a function of time just for adjustment factors (say drivingat lunch time vs driving at midnight). Function might be tabular/discrete. In general it's always much easier to evaluate ratios and relative biases (hardnes of doing something). You may need a functional language to do repeated rewrites of your vectors till there's nothing more that romain knowledge and rules can change.
With cannonical vectors consider just presence and absence of task (just 0|1 for this iteratioon) and look for minimal diffs - single task diffs first - that will provide estimates which small number of variables. Keep doing this recursively, be ready to back track and have a heuristing rule for goodness or quality of estimates so far. Keep track of good "rounds" that you backtraced from.
When you reach minimal irreducible state - dan't many any more diffs - all vectors have the same remaining tasks then you can do some basic statistics like variance, mean, median and look for big outliers and ways to improve initial domain knowledge based estimates that lead to cannonical form. If you finsd a lot of them and can infer new rules, take them in and start the whole process from start.
Yes, this can cost a lot :-)