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Rescaling in Quantitative Biology

Learn about rescaling in quantitative biology, a technique converting measurement scales for easier analysis, pattern recognition, and statistical tests. Explore rescaling options and normalization methods.

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Rescaling in Quantitative Biology

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  1. 111100101000100100100010010010010011111001011001001001000101000100101001010010100100101001010101000101000010010100101001000101010010101001011110101101101001010101001010100100111101011001101001010001010100000001011111010010000101101000101001010001000001011011111111001010001001001000100100100100111110010110010010010001010001001010010100101001001010010101010001010000100101001010010001010100101010010111101011011010010101010010101001001111010110011010010100010101000000010111110100100001011010001010010100010000010110111111110010100010010010001001001001001111100101100100100100010100010010100101001010010010100101010100010100001001010010100100010101001010100101111010110110100101010100101010010011110101100110100101000101010000000101111101001000010110100010100101000100000101101111111100101000100100100010010010010011111100101000100100100010010010010011111001011001001001000101000100101001010010100100101001010101000101000010010100101001000101010010101001011110101101101001010101001010100100111101011001101001010001010100000001011111010010000101101000101001010001000001011011111111001010001001001000100100100100111110010110010010010001010001001010010100101001001010010101010001010000100101001010010001010100101010010111101011011010010101010010101001001111010110011010010100010101000000010111110100100001011010001010010100010000010110111111110010100010010010001001001001001111100101100100100100010100010010100101001010010010100101010100010100001001010010100100010101001010100101111010110110100101010100101010010011110101100110100101000101010000000101111101001000010110100010100101000100000101101111111100101000100100100010010010010011 Chapter 3 Rescaling 20 %

  2. What is Rescaling? • Conversion of one measurement scale  another • Common technique used in quantitative biology 111100101000100100100010010010010011111001011001001001000101000100101001010010100100101001010101000101000010010100101001000101010010101001011110101101101001010101001010100100111101011001101001 20 %

  3. 12 rescaling options Nominal [0,0,1,0,0,0] Ordinal [1st, 2nd, 3rd] Interval [90,180,45]o Ratio [0,1.4,3.2]m Less detail More detail

  4. Why bother? • Logical rescaling has many applications • Simplify analyses • Even out datasets • Help reveal patterns • Consolidate explanatory variables • Run non-parametric statistics

  5. Simplify Ratio  Nominal Other Gravel Other

  6. Even out datasetsFish census Net  # by species Can no longer compare total values Salvage long term analysis: Ratio  Nominal Harper cuts 15 6 2008 2009 2006 2007 2010 2011 2012 2005

  7. Help reveal patterns • Rescaling to a less detailed quantity sometimes makes it easier to see patterns • Is the presence/absence of storms associated with number of vagrant birds observed?

  8. Consolidate explanatory variables Nominal forest/barrens berries present/absent wet/dry Ordinal Habitat rank …

  9. Run non-parametric statistics • Interval & Ratio  Rank [10.5, 15.6, 19.1, 9.8] ml  [3, 2, 1, 4] • Non-parametrics can be useful when parametric test assumptions are violated • Wilcoxon rank-sum test (≈ t-test) • But…these test are not a staple these days • Generalized linear models can deal with various error distributions

  10. Normalization • Common Qref values: QmaxQmean QminQrange QsumQsd • Another common technique used in quantitative biology • Conversion of quantity to a ratio with no units Q and Qref have the same units

  11. Scope • We can use scope to • compare the capacity of measurement instruments, • compare the information content of graphs, • compare variability of physical systems, or biological systems

  12. Physical quantities – larger scope Biological quantities – smaller scope

  13. Scope of measurement instruments • Defined as the max over min reading

  14. Survey scope • Defining the sample unit • Listing all possible units (the frame), • Then survey all possible units (complete census) or sample units at random Salmon survey Unit: 100 km transects 7 1 4 5 6 2 3 Scope Frame: 700 km

  15. Survey scope Unit: 100 km transects Frame: sum(rivers) Scope: # possible transects

  16. Experiment scope • Unit depends on quantity measured or sampling interval • Sampling livers • Census bacteria each day 10 8 8 7 6 6 5 4 4 3 1 2 3 4 5 6 7 8 9 10 Millions of bacteria recorded each day ?

  17. Normalization • Relative to a statistic: Qsum Qmean Qrange Qsd • Another common technique used in quantitative biology • Conversion of quantity to a ratio with no units Q and Qref have the same units

  18. Normalization to a sum • Taking a percentage • e.g. Mendel’s experiments 705 224

  19. Normalization to the mean • Useful for assessing deviations from the mean • e.g. Number of plant species on the Canary Islands Nplant= [ 366 348 763 1079 539 575 391 ] · sp/island mean(Nplant) = n-1∑Nplant mean(Nplant) = 7-1 · 4061 · species/island = 580 dev(Nplant) = Nplant - mean(Nplant) dev(Nplant) = [ -214 -232 +182 +498 -41 -5 -189 ] · sp/island

  20. SST (oC) SST anomaly (oC) Date

  21. Normalization to the meanCoefficient of Variation • Unitless ratio that allows comparisons of two quantities, free of various confounds • e.g. We can use the CV to compare morphological variability in mice and elephants

  22. Normalization to a range • The range is defined as the largest minus smallest value • Ranging uses both the minimum and maximum value to reduce the quantity to the range 0 to 1

  23. Normalization to the stdev • This is a common form of normalization in statistical treatments of data • Returning to example of number of plant species on 7 Canary Islands:

  24. Rigid Rescaling • Rigid rescaling replaces one unit with another • Units disappear because any unit scaled to itself = 1 (no units) • m/m is notation for metre/metre = 1 • kcal/Joules is a number with no units • km1.2/m1.2 has no units: it is the number of crooked m per crooked km

  25. Convert units • Generic procedure – Three steps • Write the quantity to be rescaled • Apply rigid conversion factors so units cancel • Calculate

  26. Figure out how much Phelps eats in a day (in lbs) • How much do you eat in a day, as a % of body weight? • 2000 Kcal/day for women not in training • 2200 kcal/day for men not in training

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