calculus

1
?????????

• ?3? ????????

2
???

• calculus

3
??function

• ????
• ????

4
??limit

• ????L????f(x)?x0????

5
??derivative

• ???????
• ??????????(tangent)???(slope)

6
?????rules for derivatives

• ????addition rule
• ????chain rule
• ????multiplication rule
• ????division rule

7
Applied calculus

• ??Change ???ordinary ????partial?????directional
  derivatives
• ???optimization????fitting???????constrained
  optimization
• ??modeling
• ??????linear????polynomial???exponential???trigon
  ometric??power-law
• ????multi-variables function
• ????differential equation
• ?????units and dimension
• ?????????

8
??????????

• Differential Equation

9
????dynamic model

• ???????????/?????????
• ?????????????
• ?????????????
• ?????????????
• ????????
• ?????????(??)?????
• ???????????????
• ??????(??)??????????

10
????????

• Linear Algebra

11
????

• ??(set)
• ????(linear space)
• ????(linear combination)
• ????(linear independent)
• ????(Euclidean space)
• ??(perpendicular,orthogonal)

12
?????(addition)

• ???????????
• ???(communicative law)
• ???(associative law)
• ???(distributive law)
• ??????????(geometric interpretation)

13
????(multiplication)?????

• ??(inner product)?????(dot product),???????(scala
  r),???a???(L2-norm)?b?????????????????,???
• lta, bgt ? ab
• ???????????
• ??(outer product)?????(cross product),?????????a?
  b?????????,??????a?b?????????????,???
• ab
• ???????????,????????????

14
??(matrix)

• ????(rank)??A??(??)????????,???rank(A),rank(A)
  lt min(m, n)???????,??A???(full
  rank)?(???????????m?n??)??rank(A)mn,??A?n??????
  (n-order nonsingular square matrix),??A??(invertib
  le)?
• ??????(determinant),???det(A)????????????det(A)ltgt
  0
• ?????(transpose matrix)
• ???(inverse matrix)
• ????(symmetric matrix)
• ????(orthonormal matrix)
• ????(positive definite matrix)
• ?????(positive semidefinite matrix)

15
????(decomposition/factorization)

• ??????,???????????(canonical matrix)??????,???????
  ???
• LU?????????????(upper triangular
  matrix,L)??????(upper triangular matrix
  ,U)???,????????????A???
• QR????????????????(Q)??????????(R)?QR????????????
  ???????????,????Q?mm??,R?nn??
• Cholesky??
• ?????(eigendecomposition)
• Schur??
• ?????(singular value decomposition,
  SVD)AUSVT,??U?V???????,S?????????????????,??????
  ??(PCA)???(clustering)

16
????????

• Optimization Theory Applications

17
????(mathematical programming)

• ????????????
• ???????n???????(????)???????(????)
• ??????????(??????)???

18
??????

• ?????????(?????)
• ???????
• ??(global)?????(local)??
• ?????????????
• ????????????(ordinary least squares, OLS)
• ????????LASSO(least absolute shrinkage and
  selection operator)

19
????(linear programming)

• ????(objective)?????(constraint)?????
• ??(solutions)
• ???(graphical method)
• ????(Simplex method)
• ??????(Modified Simplex method)
• ??????(dual Simplex method)
• ??

20
????(quadratic programming)
21
???????????

• Probability

22
?????????

• StatisticsCheating Tools?

23
Descriptive statistics

• Continuous data
• Location mean, median, mode
• Dispersion range, standard deviation,
  coefficient of variation, percentile
• Moments variance, semivariance, skewness,
  kurtosis
• Categorical data
• Frequency
• Contingency table

24
Statistical graphics

• bar plot
• biplot
• boxplot
• Histogram
• Stemplot
• Q-Qplot
• correlogram

25
Mathematics can be beautiful
26
barplot
27
boxplot
28
Pairs plot
29
Perspective plot
30
Time series data decomposition
31
Stem plot
1 1111112222233444 1 5555556666667899999
2 3344 2 59 3 3 5678 4 012
32
??????????

• Stochastic Process

33
?????(Markov Chain)

• ?????(Directed Acyclic Graph, DAG)
• ?????(prediction)???(classification)
• ????????????(????)
• ??????(Markov chain, MC)???(extension?generalizati
  on)
• ?????????,?????????????????,???????????????

34
?????(Bayesian Network)

• ?????(Directed Acyclic Graph, DAG)
• ?????(prediction)???(classification)
• ????????????(????)
• ??????(Markov chain, MC)???(extension?generalizati
  on)
• ?????????,?????????????????,???????????????

35
??????

• Graph Theory

36
Classification minds

• (apple, orange, banana, watermelon, grape,
  grapefruit, mango, star fruit)
• Clustering or classification?