Machine-Learning-Formulas

1. Naive Bayes

$$ P (A | C) = \frac{P(C | A) \times P(A)}{P(C)} $$

2. Bayes Optimal Classifier

$$ arg \ max\sum p(x|T) \times P(T|D)$$

3. Naive Bayes Classifier

$$ arg \ max P(Spo|Tot) \times \prod P(Soc|Spo) $$

4. Bayes Map (Maximum a posteriori)

$$ h_{map} = arg \ max P(c|a) \times P(a) $$

5. Maximum Likelihood

$$ h_{ML} = arg \ max P(c|a) $$

6. Total probability

$$ Total P(B) = P(B|A) \times P(A) $$

7. Mixture Models

$$ P(B) = P(B|A) \times P(A)$$

8. Mixture of Guassians Anomaly Detection

$$ P(x|\bar{x}) = \frac{1}{\sqrt{2\pi\sigma^2}} \times exp [-\frac{1}{2}(\frac{x - \bar{x}}{\sigma})^2]$$ $$ Z_{cs} = \frac{N_{A}C_{B} + N_{B}C_{A}}{N_{A} + C_{B}}$$ $$ P(Z_{cs}) \rightarrow 0.50$$

9. EM Algorithm

$$ E \ step \ P(\bar{x}|x) = \frac{P(\bar{x}) \times P(x|\bar{x})}{\sum P(x) \times P(\bar{x})} $$ $$ M \ step \ P(x^{\prime}) = \frac{P(\bar{x}|x)}{n}$$ $$ E \ step \ P(\bar{x}|x) = Assign \ Value $$ $$ M \ step \ P(x^{\prime}) = P(B = 1 | A = 1 , C = 0 )$$

10. Laplace Estimate (Small Samples)

$$ P(A) = \frac{A \ + \ 0.5}{A \ + B \ + \ 1 }$$

11. Bayesian Networks

$$ tuples \ \neg for \ y \ = \ 0 \ \land \ y \ = \ 1$$

12. Limits

$$ \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$ $$ h = \Delta{x} \ = \ x^{\prime} \ - \ x $$

13. Derivatives

$$ \frac{d}{dx}{x}^n = n.x^{n-1} $$ $$ \frac{d}{dx}{y}^n = \frac{d{y}^n}{dy}.\frac{dy}{dx} $$

14. Product Rule

$$ \frac{d}{dx}f(x).g(x) = f^{\prime}(x).g(x) + f(x).g^{\prime}(x)$$ $$ \frac{d}{dx}\frac{f(x)}{g(x)} = \frac{ f^{\prime}(x).g(x) + f(x).g^{\prime}(x)}{g(x)^2}$$ $$ \frac{d}{dx}2f(x) = 2\frac{d}{dx}f(x) $$ $$ \frac{d}{dx}f(x)+g(x) = \frac{d}{dx}f(x) + \frac{d}{dx}g(x) $$ $$ \frac{d}{dx}f(x)+2g(x) = \frac{d}{dx}f(x) + 2\frac{d}{dx}g(x) $$

15. Chain Rule

$$ \frac{d}{dx}g(f(x)) = g^{\prime}(f(x)).f^{\prime}(x)$$

16. Variance

$$ var = \frac{\sum(x-\bar{x})^2}{n-1}$$

17. Standard Deviation

$$ \sigma = \sqrt{variance}$$

18. Covariance

$$ Cov = \frac{(x-\bar{x}).(y-\bar{y})}{n-1} $$

19. Confidence Interval

$$ x \ \textpm \ 1.96\frac{\sigma}{\sqrt{n}}$$

20. Chi Squared

$$ Chi = \frac{(\hat{y} - y)^2}{\sqrt{y}} = \frac{\delta^2}{\sqrt{y}} $$

21. R Squared

$$ R^2 = \frac{n\sum{xy} - \sum{x}.\sum{y}}{\sqrt{(n\sum{x}^2 - (\sum{x})^2).(n\sum{y}^2 - (\sum{y})^2)}}$$

22. Loss

$$ Loss = {Bais}^2 + {Variance}^2 + Noise$$

23. Sum of Squared Errors

$$ E|\overrightarrow{w}| = \frac{\sum{(\hat{y}-y)^2}}{2}$$

24. Cost Function

$$ J(\theta_j) = \theta_j - \eta.\frac{\sum{(\hat{y}-y)^2}}{2} $$

25. Number of examples

$$ m \geq \frac{log(N_H)+log(\frac{1}{\delta})}{\in} $$ where, $$ \in = \frac{\hat{y}}{y}$$ and $$ \delta = y - \hat{y}$$

26. Markov Chains

$$ p^{t+1}(X = x) = \sum_xp^t.(X = x).T(x \to x)$$

27. K Nearest Neighbor

$$ \hat{f}(x) \leftarrow \frac{\sum{f(x)}}{k}$$ $$ DE(x_i, x_j) = \sqrt{(x_i - x_j)^2 + (y_{xi} - y_{xj})^2}$$

28. Weighed Nearest Neighbor

$$ f(x) = \sum{\frac{f(x)}{D(x_1x_2)^2}. \sum D(x_1x_2)^2}$$

29. Principal Components Analysis

$$ x^{'}= x - \bar{x}$$ $$ Eigenvalue = [A] - \lambda{I} $$ $$ EigenVector = Eigenvalue.[A]$$ $$ f(x) = Eigenvector^{T}.[x_{i1}.....x{jn}] $$

30. Linear Regression

$$ m_1 = \frac{\sum x_2^{\ 2} \sum x_{1}y - \sum x_1x_2 \sum x_2y}{\sum x_1^{\ 2} \sum x_2^{\ 2} - (\sum x_1x_2)^2} $$ $$ b = \bar{y} - m_1 \bar{x}_1 - m_2 \bar{x}_2 $$

$$ f(x) = \sum_{i=1}^{n}m_ix_i \ + \ b $$

31. Logistic Regression Formula

$$dds \ Ratio = log(\frac{p}{1 - p} = mx \ + \ b) $$ $$ (\frac{p}{1 - p}) = e^{mx + b} $$ $$ {J}(\theta) = - \frac{\sum y . log(\hat{y}) + (1 - y).log(1 - (\hat{y})}{n} $$

where, $\hat{y} = \frac{1}{1 + e^{mx + b}}$ for y = 0 and y = 1

-2LL $\rightarrow 0$

$\bar{x}_1 \sim \bar{x}_2 \ne \bar{x}_1^{\ '} \sim \bar{x}_2^{\ '}$

$mx +b = \frac{p}{1 - p}$ $p(a|c) = \frac{mx+b}{mx + b + 1}$

$logit = \frac{1}{100.log(p(a|c))}$

32. Decision Trees

$$ Entropy = $$