Lec Odds

admin  3/30/2022
19 Comments
  1. Lec Winner Odds
  2. Lec Betting Odds

The tournament will be strongly contested by several top teams. However, as the 2021 LEC Spring winner odds show the two European titans G2 and Fnatic are the favorites. 2020 has been somehow a frustrating year for the LoL European Championship teams. However, 2021 is a new year and we are all eager to see how LEC will bounce back. Sta102 / BME102 (Colin Rundel) Lec 22 November 23, 2015 3 / 34 Background Odds Odds are another way of quantifying the probability of an event, commonly used in gambling (and logistic regression). Odds For some event E, odds(E) = P(E) P(Ec) = P(E) 1 P(E) Similarly, if we are told the odds of E are x to y then odds(E) = x y = x=(x + y) y=(x + y.

1. Background statistics

Variable types

  • numeric
  • categorical

What do we know:

  • Confidence intervals (numeric variable)
  • Fisher test (categorical by categorical)
  • Simple linear regression (numeric by one numeric variable)
  • Linear regression with dummy variables (numeric by any variable)

Today:

  • Multiple linear regression (numeric by several numeric variables)
  • Multiple linear regression with dummy variables (numeric by any variable)
  • Logistic (logit) regression (binary dependent variable by any number of variables of any type)

2.1 How does it work

Logistic or logit regression was developed in [Cox 1958]. It is a regression model wich predicts binary dependent variable using any number of variables of any type.

What do we need?

[underbrace{y_i}_{[-infty, +infty]}=underbrace{beta_0+beta_1cdot x_1+beta_2cdot x_2 + dots +beta_zcdot x_z +epsilon_i}_{[-infty, +infty]}]

But in our case (y) is a binary variable.

  • Probability?

[P(y) = frac{mbox{# successes}}{mbox{# failures} + mbox{# successes}}; P(y) in [0, 1]]

  • Odds?

[odds(y) = frac{P(y)}{1-P(y)} = frac{mbox{P(successes)}}{mbox{P(failures)}} = frac{mbox{# successes}}{mbox{# failures}}; odds(y) in [0, +infty]]

  • Natural logarithm of odds

[log(odds(y)) in [-infty, +infty]]

2.2 Reminder about logarithms

  • if log(odds) are greater then 0, it means that we have more successes then failures;
  • if log(odds) is equal to 0, it means that we have the same number of successes and failures;
  • if log(odds) are less then 0, it means that we have less successes then failures;

2.3 Probability and log(odds)

[log(odds(s)) = logleft(frac{#s}{#f}right)][P(s) = frac{exp(log(odds(s)))}{1+exp(log(odds(s)))}]

Results of the logistic regression can be easily converted to probabilities.

2.4 Sigmoid

Formula for this sigmoid is the following:

[y = frac{1}{1+e^{-x}}]

Feeting our logistic regression we should be able to reverse our sigmoid:

Formula for this sigmoid is the following:

[y = frac{1}{1+e^{-(-x)}} = frac{1}{1+e^{x}}]

Feeting our logistic regression we should be able to move center of our sigmoid to the left/right side:

Formula for this sigmoid is the following:

[y = frac{1}{1+e^{-(x-2)}}]

Feeting our logistic regression we should be able to squeeze/stretch center of our sigmoid:

[y = frac{1}{1+e^{-4x}}]

So the more general formula will be: [y = frac{1}{1+e^{-k(x-z)}}]

where

  • depending on (x) values sigmoid can be reversed
  • (k) is squeeze/stretch coefficient
  • (z) is coefficient that indicates movement of the sigmoid center to the left or right side
Lec

3. Numeric example

It is interesting to know whether the languages with ejective sounds have in average more consonants. So we collected data from phonological database LAPSyD: http://goo.gl/0btfKa.

  • Model without predictors

How we get this estimate value?

What does this model say? This model says that if we have no predictors and take some language it has (frac{0.5306283}{(1+e^{-0.5306283})} = 0.3340993) probability to have ejectives.

  • Model with numeric predictor

What does this model say? This model says:

[log(odds(ej)) = beta_o + beta_1 times n.cons.lapsyd = -9.9204 + 0.3797 times n.cons.lapsyd]

Lets visualize our model:

So probability for a language that have 30 consonants will be [log(odds(ej)) = -9.9204 + 0.3797 times 30 = 1.4706] Thus, the output YES (the langiage has ejectives) has approximately 1.47 times more chances to occure if the language has 30 consonants than the output NO.

[P(ej) = frac{1.47061}{1+1.4706}=0.8131486]

4. predict(): Evaluating the model’s performance

Bovada

So we actually can create a plot with confidense intervals.

5. More variables in the model

[underbrace{log(odds(y))}_{[-infty, +infty]}=underbrace{beta_0+beta_1cdot x_1+beta_2cdot x_2 + dots +beta_zcdot x_z +epsilon}_{[-infty, +infty]}]

The significance of each variable (predictor) is not the same in models with different number of variables. In other words, it depends on the combination of predictors in a specific model.

6. Model selection

Lec winner odds

AIC (Akaike Information Criterion) is a goodness-of-fit measure to compare the models with different number of predictors. It penalizes a model for having too many predictors. The smaller AIC, the better.
While comparing models, we are looking for the minimal optimal model:
* optimal, as it helps to predict the output in the best way
* minimal optimal, as it uses the minimal number of predictors

Other measures to evaluate the model includes:
* accuracy
* concordance index C (the area under the ROC-curve)
* Nagelkerke pseudo-(R^{2})

7. Interaction of the variables

Interaction happens when the effect of one predictor on the outcome depends on the value of another predictor. Interaction of two predictors can be positive (their joint role increases the effect) or negative (their joint role decreases the effect).
Example: animacy and semantic class; animacy and the choice of syntactic construction; effect of verb transitivity in different language varieties.

8. Conclusion: Generalized linear models (GLM)

GLM is a broad class of models that include linear regression, logistic regression, log linear regression, Poisson regression, ANOVA, ANCOVA, etc. In order to call a particular method to be GLM, that method should have the following three components:

  • Random Component: It refers a response variable (y), which need to satisfy some assumptions. Examples: Linear regression of y (dependent variable) follows normal distribution. Logistic regression response variable follows binomial distribution.

  • Systematic Component: It is nothing but explanatory variables in the model. Systematic components helps to explain the random component.

  • Link Function: It is link between systematic and random component. Link function tells how the expected value of response variable relates to explanatory variable. Link function of linear regression is E[y] and link function of logistic regression is logit(??).

What was important today?

  • classifiers: binary, multi-class (multinomial)
  • odds
  • sigmoid
  • significance of the variables (predictors)
  • interactions

© О. Ляшевская, И. Щуров, Г. Мороз, code on GitHub

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