Efficiency evaluation for detecting inhomogeneities by objective homogenisation methods

Abstract

Evaluation and comparison of efficiencies of widely used objective homogenisation methods (OHOMs) are presented relying on some test-datasets and efficiency measures. Problems related to the choice of efficiency measure, creation of appropriate test-datasets and use of OHOM parameterisation are discussed. The detection parts of the OHOMs are examined only. Power of detection, false alarm rate, detection skill and skill of linear trend estimation are calculated and compared for eight OHOMs and six test-datasets. Each test-dataset comprises 10,000 100 year-long artificially simulated time series. In the simplest test dataset, each time series contains one inhomogeneity (IH), while a structure of inhomogeneities that is similar to that of real central European temperature time series is included in the most complex simulated dataset. Distinct attention is given to OHOMs that contain (1) cutting algorithm, (2) semihierarchic algorithm, (3) direct detection of multiple IHs, (4) detection of change-point and trend-line shaped IHs. Results show that Caussinus–Mestre method and Multiple Analysis of Series for Homogenization are the most powerful tools in detecting and correcting IHs in climatic time series.

Appendices

Appendix I

Simulation of the standard test-dataset

• 1. 196-year-long series are generated, and always, the slices of years 48–147 are the target series.

• 2. IHs and noises are introduced in each year (but their values can be 0, naturally).

• 3. Types of the terms for introduction to time series: (a) long-term IH (y), (b) short-term IH (z) and (c) white noise (w). A certain part of y- and z-type terms is handled as noise (cf. step 10).

• 4. Forms of the IHs: (a) sudden shift, (b) gradual change, (c) platform-like change, (d) bias for one specific year. Form (d) is a specific case of class (c).

• 5. Introduction of long-term IHs.

  • 5.1: Size and direction of the IH

    This term includes an IH whose magnitude can be large, with the probability given in K ₁, as well as a small IH with the probability given in K ₂:

    $$ \Delta {y\prime_i} = {K_1}\left( {{q_1}} \right) \cdot {\hbox{sign}}\left( {0.5 - {q_2}} \right) \cdot \left( {8 + 4p} \right) \cdot q_3^{{6 + 4p}}) + {K_2}\left( {{q_4}} \right) \cdot {G_1}, $$

    (A1)

    where K ₁(a) = 1, if a < 0.012, and K ₁(a) = 0 otherwise; K ₂(a) = 1, if a < 0.07, and K ₂(a) = 0 otherwise; q (with all indices): variable of the uniform distribution over the period [0,1) p has the same distribution as q does, but p is constant for a given time series. Δ denotes that (A1) is not for substituting, but for modifying the earlier value of y _i. Apostrophe above y shows that values gained by (A1) are modified in certain cases (see below) before the introduction of Δy _i . If Δy _i′ = 0 the steps 5.2 and 5.3 are omitted.

  • 5.2: Form of the IH

    The form of Δy _i’ is (A) sudden shift, (B) gradual change or (C) platform-like change, with 0.4, 0.25 and 0.35 probability, respectively.

    For (A)- and (B)-form IHs a negative autocorrelation is present:

    $$ \Delta {y_i} = \sqrt {{1 - {r^2}}} \cdot \Delta {y\prime_i} + r \cdot F, $$

    (A2)

    where $F = 0$ for the first (A)- or (B)-form IH of the series, and F = Δy _k otherwise, k indicates the year of the previous introduction of (A)- or (B)-form IH, and r = –0.5.

    For (C)-form IHs:

    $$ \Delta {y_i} = \Delta {y\prime_i} $$

    (A3)

  • 5.3: Calculation of the y _i components of the series

    (A)-form IHs:

    $$ {y_j} = {y_{{j, - 1}}} + \Delta {y_i}\;\;\;\;\;{\hbox{for}}\;{\hbox{each}}\;j \in \left[ {i,n} \right], $$

    (A4)

    where y _j,-1 denotes the value of term y _j before the ongoing modification.

    For (B)- and (C)-form IHs, duration-values must be paired at first. For B-form, IHs the duration D ₁ is:

    $$ {D_1} = 5 + 2 \cdot Int\left( {48 \cdot q_5^{{1.5}}} \right) $$

    (A5)

    (“Int” denotes integer part), and the appearance of the IH is:

    $$ {y_j} = {y_{{j, - 1}}} + \frac{{\left( {j - i + 0.5{D_1}} \right)\Delta {y_i}}}{{{D_1}}}\;{\hbox{for}}\;{\hbox{each}}\;j \in \left[ {i - 0.5{D_1},i + 0.5{D_1} - 1} \right], $$

    (A6)

    while for (C)-form IHs:

    $$ {D_2} = Int\left( {30 \cdot q_6^{{1.5}}} \right), $$

    (A7)
    $$ {y_j} = {y_{{j, - 1}}} + \Delta {y_i}\;\;{\hbox{for}}\;{\hbox{each}}\;j \in \left[ {i,i + {D_2}} \right]. $$

    (A8)

• 6. Introduction of short-term IHs

  The size and the direction of this term is calculated by the same functions as those of long-term IHs (A1), but the frequencies (determined by the K-functions) are different:

  $$ \Delta {z\prime_i} = {K_3}\left( {{q_7}} \right) \cdot sign\left( {0.5 - {q_8}} \right) \cdot \left( {8 + 4p} \right) \cdot q_9^{{6 + 4p}}) + {K_4}\left( {{q_{{10}}}} \right) \cdot {G_2}, $$

  (A9)

  where K ₃(a) = 1, if $a < 0.04 - 0.03p$, and K ₃(a) = 0 otherwise; K ₄(a) = 1, if $a < 0.5 - 0.4p$, and K ₄(a) = 0 otherwise. The ongoing modification has a negative autocorrelation (r = –0.5) with the z value accumulated prior.

  $$ \Delta {z_i} = \sqrt {{1 - {r^2}}} \cdot \Delta {z\prime_i} + r \cdot {z_{{i, - 1}}} $$

  (A10)

  The form of this term is always platform-like change. Its duration is given by D ₃.

  $$ {D_3} = Int\left( {\frac{{12 \cdot q_{{11}}^3}}{{1 + 0.3\left| {\Delta {z_i}} \right|}}} \right), $$

  (A11)
  $$ {z_j} = {z_{{j, - 1}}} + \Delta {z_i}\;\;{\hbox{for}}\;{\hbox{each}}\;j \in \left[ {i,i + {D_3}} \right]. $$

  (A12)

• 7. Introduction of white noise term:

  $$ {w_i} = {G_3} $$

  (A13)
• 8.
  $$ {\mathbf{X}} = {\mathbf{Y}} + {\mathbf{Z}} + {\mathbf{W}} $$

  (A14)

• 9. Serial correlation of X is calculated, and the series is added to the test-dataset if the value is not lower than 0.4, while it is discarded otherwise.

• 10. A part of long-term IHs (Y) and short-term IHs (Z) is not considered to be errors of the candidate series, so it is handled as noise. The rate of this type noise increases with decreasing IH magnitudes, and it is higher for platform-like changes than for change-points and gradual changes. As a consequence of these noise terms, the model series of the standard dataset is

  $$ {\mathbf{X}} = {\mathbf{H}} + {\mathbf{W}} + {\mathbf{W}}* $$

  (A15)

  where

  $$ {\mathbf{W}}* = {\mathbf{Y}}_{{\mathbf{w}}} + {\mathbf{Z}}_{{\mathbf{w}}} $$

  (A16)
  $$ {\mathbf{H}} = {\mathbf{Y}}--{{\mathbf{Y}}_{{\mathbf{w}}}} + {\mathbf{Z}}--{{\mathbf{Z}}_{{\mathbf{w}}}} $$

  (A17)

  The index w denotes noise part. The probability (P) of that a given term is considered to be noise, is determined according to the rules below:

  For platform-like IHs, the probability P ₁ is given by:

  $$ {P_1} = \max \left( {0.6 - 0.4 \cdot \left| {\Delta {y_i}} \right|,0} \right), $$

  (A18)

  where $\Delta {y_i}$ is determined by Formulae (A1) and (A3). (A18) is applied also for Δz-type IHs.

  For change-points and gradual changes

  $$ {P_2} = \max \left( {0.3 - 0.4 \cdot \left| {\Delta {y_i}} \right|,0} \right). $$

  (A19)

where Δy _i is determined by Formulae (A1) and (A2).

Appendix II

Simulation of the quasi-standard test-dataset

The procedure is the same as for the standard dataset, except that K ₁ is always equal to 0 in formula (A1). As a result of this change, the frequency of persistent large IHs is much lower in this dataset than in the standard dataset.