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Visualizing probability distributions across
bivariate cyclic temporal granularities


Sayani Gupta
Sayani07     SayaniGupta07
https://sayanigupta-ows2020.netlify.com/


Time series analysis and econometrics
Bernoulli-IMS One World Symposium 2020
August 24, 2020 - August 28, 2020

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Joint work with Professor Dianne Cook and Rob Hyndman

Department of Econometrics and Business Statistics
Monash University
Sayani.Gupta@monash.edu


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Motivation: Smart meter data for households

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Linear to cyclic

  • Cyclic time granularities: exploring different periodicities e.g. hour-of-day, day-of-month or hour-of-week, day-of-semester

  • Multiple observations for each level of cyclic granularity

  • summarize distribution of measured variables

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Summarize distribution

  • Cyclic time granularities: exploring different periodicities e.g. hour-of-day, day-of-month or hour-of-week, day-of-semester

  • Multiple observations for each level of cyclic granularity

  • summarize distribution of measured variables

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Data structure and graphical mapping



  • extension of tsibble data structure
  • choose any two cyclic granularities: \(C_i = \{A_1, A_2, \dots, A_K\}\) and \(C_j = \{B_1, B_2, \dots, B_L\}\)

  • graphical mapping \((C_i, C_j, v)\)

  • \(^{N_C}P_2\) displays

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Relationship of cyclic granularities

Clashes: pairs leading to empty sets

Harmonies: pairs leading to no empty sets

Still too many harmonies for display for large \(N_C\)

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Selecting harmonies with significant variation

  • significant differences between categories:
    (a) no (b) yes
  • Gestalt theory
  • efficiently capture within and between group variation
  • compute threshold
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Rank selected harmonies
Median Maximum Pairwise Distance (MMPD)

  • rank harmonies (b) > (a)
  • distances between probability distribution through Jensen-Shannon divergences
  • normalizing constants through Fisher–Tippett–Gnedenko theorem
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R package gravitas

  • construction of cyclic granularities and cyclic calendar algebra
  • check feasibility through harmony/clash
  • select significant harmonies and rank them
  • recommend prospective probability distributions
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An example : Electricity smart meter data

Data source : Department of the Environment and Energy, Australia

## # A tsibble: 293,874 x 3 [30m] <UTC>
## # Key:       customer_id [10]
##    customer_id reading_datetime    general_supply_kwh
##    <chr>       <dttm>                           <dbl>
##  1 10006414    2012-02-10 08:00:00              0.141
##  2 10006414    2012-02-10 08:30:00              0.088
##  3 10006414    2012-02-10 09:00:00              0.078
##  4 10006414    2012-02-10 09:30:00              0.151
##  5 10006414    2012-02-10 10:00:00              0.146
##  6 10006414    2012-02-10 10:30:00              0.077
##  7 10006414    2012-02-10 11:00:00              0.052
##  8 10006414    2012-02-10 11:30:00              0.055
##  9 10006414    2012-02-10 12:00:00              0.055
## 10 10006414    2012-02-10 12:30:00              0.252
## # … with 293,864 more rows
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Cyclic granularities considered for analysis

## [1] "hour_day"   "hour_week"  "hour_month" "day_week"   "day_month" 
## [6] "week_month" "wknd_wday"



Number of displays to analyze for exhaustive exploration

There are \(^{7}P_2 = 42\) pair of granularities to look at placing one on the x-axis and the other on facet.

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Select only harmonies (16 out of 42 selected)

facet x-axis facet levels x levels MMPD
day_week day_month 7 31 0.064
wknd_wday day_month 2 31 0.060
wknd_wday hour_day 2 24 0.044
day_week hour_day 7 24 0.024
week_month hour_day 5 24 0.023
hour_day day_month 24 31 0.016
day_month wknd_wday 31 2 0.014
day_month day_week 31 7 0.011
day_month hour_day 31 24 0.009
hour_day wknd_wday 24 2 0.009
week_month wknd_wday 5 2 0.007
day_week week_month 7 5 0.004
wknd_wday week_month 2 5 0.003
hour_day week_month 24 5 0.003
week_month day_week 5 7 0.001
hour_day day_week 24 7 0.001
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Select and rank significant harmonies (6 out of 16 selected)

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Thank you

Slides: https://sayanigupta-ows2020.netlify.app/

Materials: https://github.com/Sayani07/OWS2020

Package (CRAN): v0.1.3

Package (Github) : https://github.com/Sayani07/gravitas


Slides created with Rmarkdown, knitr, xaringan, xaringanthemer

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Joint work with Professor Dianne Cook and Rob Hyndman

Department of Econometrics and Business Statistics
Monash University
Sayani.Gupta@monash.edu


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