PHC7065 CRITICAL SKILLS IN DATA MANIPULATION FOR POPULATION SCIENCE

Image and Time-series Data

Hui Hu Ph.D.

Department of Epidemiology

College of Public Health and Health Professions & College of Medicine

April 2, 2018

Introduction to Image and Time-series Data


Lab: Image and Time-series Data

Introduction to Image and Time-series Data

Image Data

rgba(XXX,XXX,XXX,X)

Representation of Image Data

Geometric Transformations of Images

  • Scaling:
    -  resizing of the image
     
  • Translation:
    -  the shift of object's location
     
  • Rotation:
    -  rotate an image for an angle

Scaling

  • When scale an image, interpolation is needed

Translation

Transformation matrix:

M={\begin{bmatrix} 1 & 0 & t_x\\ 0 & 1 & t_y \end{bmatrix}}
M=[10tx01ty]M={\begin{bmatrix} 1 & 0 & t_x\\ 0 & 1 & t_y \end{bmatrix}}

Shift in (x,y) direction

Rotation

Transformation matrix:

M={\begin{bmatrix} cos\theta & -sin\theta\\ sin\theta & cos\theta \end{bmatrix}}
M=[cosθ−sinθsinθcosθ]M={\begin{bmatrix} cos\theta & -sin\theta\\ sin\theta & cos\theta \end{bmatrix}}

Modified transformation matrix with center of rotation added:

M={\begin{bmatrix} \alpha & \beta & (1-\alpha)center.x-\beta center.y\\ -\beta & \alpha & \beta center.x + (1-\alpha)center.y \end{bmatrix}}
M=[αβ(1−α)center.x−βcenter.y−βαβcenter.x+(1−α)center.y]M={\begin{bmatrix} \alpha & \beta & (1-\alpha)center.x-\beta center.y\\ -\beta & \alpha & \beta center.x + (1-\alpha)center.y \end{bmatrix}}
\alpha=scale \cdot cos\theta
α=scale⋅cosθ\alpha=scale \cdot cos\theta
\beta=scale \cdot sin\theta
β=scale⋅sinθ\beta=scale \cdot sin\theta

Time-series Data

  • Time series is usually a collection of data points collected at constant time intervals
     
  • What makes time series data special?
    -  time dependent
    -  seasonality trends
     
  • Stationarity: a time-series is said to be stationary if its statistical properties remain constant over time
    -  constant mean
    -  constant variance
    - an autocovariance that does not depend on time
     
  • Most of the time-series models were based on the assumption of staionarity, and theories related to stationary series are more mature and easier to implement

Stationarity

  • What make a time-series non-stationary?
    -  trend: varying mean over time
    -  seasonality: variations at specific time-frames
     
  • How to make series stationary?
    -  estimate the trend and seasonality in the series and remove them from the series
     
  • Many ways of doing it:
    -  aggregation: taking average for a time period like monthly/weekly averages
    -  smoothing: taking rolling averages
    -  polynomial fitting: fit a regression model
    -  differencing: take the difference of the observation at a particular instant with that at the previous instant
    -  decomposing: model trend and seasonality seperately

Lab: Image and Time-series Data

git pull

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