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Statistical modeling, forecasting and time series analysis of birch phenology in Montreal, Canada

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Abstract

The aim of this study was to analyse birch pollen time series observed in Montreal (Canada) in order to understand the link between inter-annual variability of phenology and environmental factors and to build predictive models for the upcoming pollen season. Modeling phenology is challenging, especially in Canada, where phenological observations are rare. Nevertheless, understanding phenology is required for scientific applications (e.g. inputs to numerical models of pollen dispersion) but also to help allergy sufferers to better prepare their medication and avoidance strategies before the start of the pollen season. We used multivariate statistical regression to analyse and predict phenology. The predictors were drawn from a large basin (over 60) of potential environmental predictors including meteorological data and global climatic indices such NAO (North Atlantic Oscillation index) and ENSO/MEI (Multivariate Enso Index). Results of this paper are summarized as follows: (1) an accurate forecast for the upcoming season starting date of the birch pollen season was obtained (showing low bias and total forecast error of about 4 days in Montreal), (2) NAO and ENSO/MEI indices were found to be well correlated (i.e. 44% of the variance explained) with birch phenology, (3) a long-term trend of 2.6 days per decade (p < 0.1) towards longer season duration was found for the length of the birch pollen season in Montreal. Finally, perturbations of the quasi-biennial cycle of birch were observed in the pollen data during the pollen season following the Great Ice Storm of 1998 which affected south-eastern Canada.

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Notes

  1. Asthmatic patients usually need to begin their anti-allergic treatment one or two weeks before pollination (Laaidi 2001).

  2. http://ville.montreal.qc.ca/portal/page?_pageid=6897,67887840&_dad=portal&_schema=PORTAL.

  3. The number of possible model sis 2p-1. With p = 22 pre-selected predictors (see Table 4), this gives over 4 million possibility of models which requires the use of automatic selection procedures.

  4.  A confounding variable has the property of correlating (positively or negatively) with both the dependent and independent variable (see Montgomery 2005 for further details).

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Authors and Affiliations

  1. Department of Geography, University of Montreal, 2900 Boulevard Edouard-Montpetit, Montreal, QC H3T 1J4, Canada

    Alain Robichaud & Paul Comtois

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  1. Alain Robichaud
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  2. Paul Comtois
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Appendices

Appendix 1: List of potential predictors for phenological models

Predictors of the current year apply for the period January 1st to March 31st just prior to the upcoming pollen season. Predictors for the previous year start with the letter l (for lag) and apply for the previous calendar year.

Current year predictors:

  • tjan average January temperature

  • tfev average February temperature

  • tmarch average March temperature

  • twinmin mean winter minimum temperature (Jan-March)

  • twinavg mean winter temperature (Jan-March)

  • sum05_115 sum of temperature above 5 °C since January 1st cumulated at Julian day 115

  • sum03_115 sum of temperature above 3 °C since January 1st cumulated at Julian day 115

  • sum05_110 sum of temperature above 5 °C since January 1st cumulated at Julian day 110

  • sum03_110 sum of temperature above 3 °C since January 1st cumulated at Julian day 110

  • rainj total January rain amount

  • rainf total February rain amount

  • rainm total March rain amount

  • precipw total rain precipitation for the period January–March

  • nao1 NAO index for January

  • nao2 NAO index for February

  • nao3 NAO index for March

  • enso1 bi-monthly MEI index for January

  • enso2 bi-monthly MEI index for February

  • enso3 bi-monthly MEI index for March

  • ensowin mean bi-monthly MEI index for the period January–March

Previous year predictors:

  • ltjan last year average January temperature

  • ltfev: last year average February temperature

  • ltmar last year average March temperature

  • ltsum last year average summer temperature (June–July–August)

  • ltsummax last year average summer maximum temperature (June–July–August)

  • lrainf last year total amount of rain during fall (October–December)

  • lnao1 last year NAO index for January

  • lnao2 last year NAO index for February

  • lnao3 last year NAO index for March

  • lnao4 last year NAO index for April

  • lnao5 last year NAO index for May

  • lnao6 last year NAO index for June

  • lnao7 last year NAO index for July

  • lnao8 last year NAO index for August

  • lnao9 last year NAO index for September

  • lnao10 last year NAO index for October

  • lnao11 last year NAO index for November

  • lnao12 last year NAO index for December

  • lnaospr mean values of lnao4, lnao5 and lnao6 (spring months of the previous year)

  • lnaosum mean values of lnao7, lnao8 and lnao9 (summer months of the previous year)

  • lnaofal mean values of lnao10, lnao11 and lnao12 (fall months of the previous year)

  • lnaowin mean values of lnao1, lnao2, lnao3 (winter months of the previous year)

  • lensow same as ensowin but for the previous year

  • lenso1 last year bi-monthly ENSO/MEI index for January

  • lenso2 last year bi-monthly ENSO/MEI index for February

  • lenso3 last year bi-monthly ENSO/MEI index for March

  • lenso4 last year bi-monthly ENSO/MEI index for April

  • lenso5 last year bi-monthly ENSO/MEI index for May

  • lenso6 last year bi-monthly ENSO/MEI index for June

  • lenso7 last year bi-monthly ENSO/MEI index for July

  • lenso8 last year bi-monthly ENSO/MEI index for August

  • lenso9 last year bi-monthly ENSO/MEI index for September

  • lenso10 last year bi-monthly ENSO/MEI index for October

  • lenso11 last year bi-monthly ENSO/MEI index for November

  • lenso12 last year bi-monthly ENSO/MEI index for December

  • lensoa last year annual average of bi-monthly ENSO/MEI

  • lensospr last year spring average of bi-monthly ENSO/MEI

  • lensosum last year summer average of bi-monthly ENSO/MEI

  • lensof last year fall average of bi-monthly ENSO/MEI

  • ensowin winter average of bi-monthly ENSO/MEI of the current year

  • llength last year length of the season according to the D2 method

  • lendD2 last year Julian date for the end of birch pollen season based on the D2 method

  • lspi last year birch seasonal pollen index

Appendix 2: Steps followed for multiple regression

  1. 1.

    List of predictors established according to a literature survey (see a list in Appendix 1 above).

  2. 2.

    Scatter plots to evaluate linearity, correlation, outliers, etc.

  3. 3.

    Use of the procedure REG sequentially (options selection C p , Rsq in SAS®) to examine and select the most appropriate model among 2p − 1 model (chunks of 7 predictors are sequentially input giving 27–1 = 128 combinations of models for each chunk).

  4. 4.

    Step 3 is applied to the whole list of preselected predictors and model building is performed using forward, backward and stepwise procedure. The results are checked against another similar procedure (proc REG with STEPWISE option). Check with t-Test, F-Test, C p (Mallows, 1973), R 2 adjusted, etc. for regression coefficients of the model (see similar procedure, Hoang Diem Ngo 2012).

  5. 5.

    Model adequacy test (global F-test, optimum adjusted R 2, RMSE).

  6. 6.

    Check for model assumptions (model error are random and all pairs of random errors are independent), residual plots, normal probability plot and checking for outliers (Cook’s distance, Cook 1977), Durbin-Watson test for autocorrelation of residuals.

  7. 7.

    Check for multicollinearity of predictors and over-fitting.

  8. 8.

    Repeat steps 3–7 for three periods (1996–2009, 1996–2010 and 1996–2011) and for all predictands.

  9. 9.

    Model validation with independent data (the independent data is the year following a given training period).

Note that the entry value for a given potential predictor is set to p ≤ 0.15 and the elimination criteria to p > 0.15 (standard values in the community of multiple regression modeling and are often the default values in SAS multiple regression procedures, see Beal 2005; Hoang Diem Ngo 2012). As an alternative method, multiple regression using AIC criteria for minimization was also used but found having less accuracy against independent data (see text for details).

Appendix 3: Results of STEPWISE procedure

See Table 11.

Table 11 Final results from computer outputs for stepwise selection summary for the period 1996–2011, (A) start of the birch pollen season (mstart), (B) duration of the season (mlength), (C) seasonal pollen index (mspi) and (D) mpeak
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Robichaud, A., Comtois, P. Statistical modeling, forecasting and time series analysis of birch phenology in Montreal, Canada. Aerobiologia 33, 529–554 (2017). https://doi.org/10.1007/s10453-017-9488-0

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