This document discusses key concepts and terminology related to time series analysis and paleoclimatology. It defines common features of time series such as trends, steps, oscillations, fluctuations, and events. It also discusses important statistical analyses for time series including autocorrelation, effective sample size, spectra, and spectral analysis. Spectra can be white, red, or blue depending on the distribution of variance as a function of frequency, and spectral analysis uses windows to evaluate changes in spectra over time.

1. GEOG5426 Statistics in paleoclimatology

2. November 24
Brief (15-minute) summaries of project topics.
(1) What are the most important features of the modern
climate in your region?
(2) What proxies are available in your region, over the time
interval specified? How are they related to climate? and
(3) How different were past climates from modern
conditions? Why is that important?

3. GEOG5426 Statistics in paleoclimatology

6. A time series is a set of observations
ordered in time.

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PDSI

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PDSI
resolution
annual

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PDSI
chronological uncertainty
sub-annual

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PDSI time span
last century

11. Variance
sample
size
variance observation
sample
mean

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PDSI Ringwidth
drought-sensitive tree-ring records Eastern Canadian Rockies
St. George et al., (2009), Journal of Climate

13. Correlation Pearson’s product-moment correlation
covariance
product of both standard deviations

15. Source: Wikipedia
r = 0.816

16. statistical
significance
practical
significance

17. nnumber of observations

18. Source: Wikipedia

19. T I M E S E R I E S T E R M I N O LO GY

20. Trends are progressive increases or decreases
in the levels of a particular climate variable.
Bartlein 2006

21. Trends Bartlein (2006)

22. Steps are abrupt transitions from one level to
the other, relative to the timescale of
variations under investigations.
Bartlein 2006

23. Steps Bartlein (2006)

24. Oscillations are either periodic or quasi-periodic
variations about a stationary or slowly changing
level.
Bartlein 2006

25. Oscillations Bartlein (2006)

26. Fluctuations are aperiodic variations of
climate that appear at all timescales (but tend
to be more evident at shorter timescales).
Bartlein 2006

27. Fluctuations Bartlein (2006)

28. Events are variations that return rapidly to a
previous state. Because events are reversible,
they are distinct from steps.
Bartlein 2006

29. Events Bartlein (2006)

30. A U TO C O R R E L AT I O N

31. Correlation Pearson’s product-moment correlation
covariance
product of both standard deviations

32. Autocorrelation describes the correlation of a
time series with its own past and future values.

33. Autocorrelation
covariance
product of
the standard deviation

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Ringwidth
lag-0 autocorrelation

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Ringwidth
lag-1 autocorrelation

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Ringwidth
lag-2 autocorrelation

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Ringwidth
lag-3 autocorrelation

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PDO index Mexican PDSI

40. IF a time series (of length N) is
significantly autocorrelated, then:
The series is not random in time
Each observation is not independent
from other observations
The number of independant
observations is fewer than N

42. The “effective sample size” is an estimate of
the “real” number of observations a er
adjusting for the effects of autocorrelation.

43. Effective sample size
sample
size
effective
sample
size
first-order
autocorrelation

44. T H E S P E C T R U M O F C L I M AT E

45. The spectrum of a time series is the
distribution of variance of the series as
a function of frequency.

46. redorangegreenblueviolet
yellow

47. redorangegreenblueviolet
yellow
short
wavelengths
long
wavelengths

48. redorangegreenblueviolet
yellow
Fast
changes
slow
changes

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example of a ‘white’ time series

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example of a ‘red’ time series

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example of a ‘blue’ time series

52. “White”

53. “Red”

54. “Blue”

55. Schematic variance spectrum of climate variations Bartlein 2006

56. Schematic variance spectrum of climate variations Bartlein 2006
BIG
changes
LITTLE
changes

57. Schematic variance spectrum of climate variations Bartlein 2006
FASTSLOW

58. S P E C T R A L A N A LY S I S

59. Source: Burroughs, Weather Cycles: Real or Imaginary?

60. Source: Burroughs, Weather Cycles: Real or Imaginary?

62. Source: Burroughs, Weather Cycles: Real or Imaginary?

63. Source: Burroughs, Weather Cycles: Real or Imaginary?
frequency (cpy)

65. Source: Burroughs, Weather Cycles: Real or Imaginary?

66. Source: Burroughs, Weather Cycles: Real or Imaginary?

67. Source: Burroughs, Weather Cycles: Real or Imaginary?

68. Source: Burroughs, Weather Cycles: Real or Imaginary?

69. Source: Burroughs, Weather Cycles: Real or Imaginary?

70. evolutive spectra

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200 250 300 350 400 450 500
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4 Imagine a ‘window’
passing through your data

72. Evolutionary spectra for the global 018 record Bartlein 2006

73. Evolutionary spectra for the global 018 record Bartlein 2006

74. Evolutionary spectra for the global 018 record Bartlein 2006

75. Evolutionary spectra for the global 018 record Bartlein 2006

76. Evolutionary spectra for the global 018 record Bartlein 2006
cone of
influence
cone of
influence

77. what?
compared to

78. “Red”

79. Source: Shanahan et al., 2009, Science

80. 200 250 300 350 400 450 500
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example of a ‘white’ time series
X

81. 200 250 300 350 400 450 500
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example of a ‘red’ time series

82. Source: Shanahan et al., 2009, Science

83. statistical
significance
physical
significance

84. The history of meteorology is li ered with
whitened bones of claims to have demonstrated
the existence of reliable cycles in the weather.
“
”William James Burroughs