Name: Joan of Arc
Multistructural
Relational
Extended Abstract
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Grade: Excellence
Marker:
N
NS
Fully
N
NS
Fully
N
NS
Fully
posed a comparison
investigative question
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selected and used
appropriate displays
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identified features in the
data and related this to the
context
finding an appropriate
model
and using the model to
make a forecast
communicated findings in a
conclusion
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Purpose statement and Conclusion were near perfect.
Marker’s
Judgement
EXCELLENCE
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Investigate time series data, with statistical insight involves
integrating statistical and contextual knowledge throughout
the statistical enquiry cycle, and may include reflecting
about the process; considering other relevant variables;
evaluating the adequacy of any models; or showing a deeper
understanding of models.
Plan/Purpose:
Global warming (now more commonly referred to as Climate Change) is associated with the
effects of Earth’s increasing surface temperature. More than ever before, governments are
concerned with the consequences of human actions in industrial practices, deforestation
which release “Greenhouse gases” into the atmosphere. According to
http://www.niwa.co.nz/our-science/climate/information-and-resources/clivar/greenhouse ,
these greenhouse gases contribute to the absorption of infrared heat onto Earth, and
warming the surface. (known as the Greenhouse effect)
The Earth is incredibly sensitive to temperature change and the overall change of just one
degree can be significant. On http://hyperphysics.phyastr.gsu.edu/hbase/thermo/grnhse.html there are claims that by increasing the
temperature of the Earth’s surface “disastrous consequences” will occur in on nature,
including melting ice, rising sea levels and natural disasters, such as flooding or drought.
However, some people believe that global warming does not exist and there is not enough
evidence to justify the imminent disasters that are to come if we do not change our
unsustainable ways.
I am going to investigate the patterns and trends of maximum temperatures over the past
20 years in Madrid (a Northern Hemisphere city) and Christchurch (a Southern Hemisphere
city) and compare my findings to published scientific reports. I wish to find out whether the
average temperatures in Christchurch and Madrid is increasing and could possibly confirm
that global warming is taking place.
Data: The data provided is maximum and minimum air temperatures of Madrid and
Christchurch each month over 20 years, from 2000 to 2012.
Long term trend: From 2000 to 2013, the raw data fluctuates approximately between 35°C
to 6°C. The trend line starts at approximately 23°C and finishes at about 19°C.
An interesting feature about this graph is that the trend line of Madrid’s temperature seems
to be quite consistent, with little change in increasing or decreasing temperature.
Residuals: There seems to be no obvious pattern in the residual plot. The largest residual is
approximately 10°C, which is large (34%) when compared with the range of the data (356=29°C) A residual of this magnitude represents more than 10% of this range. (Any residual
more than 10% of the overall range is considered to be significant)
Seasonal effect: The seasonal patterns of maximum temperatures in Madrid become
consistent as the years progress. (as seen by the red to blue graphs). Average seasonal highs
(peaks) occur at July-August and seasonal lows (troughs) occur at January and December.
This is typical to the Northern hemisphere seasonal temperatures as July and August
correlates to the Summer season (high temperatures) in Madrid and January and December
match the Winter season (low temperatures).
A surprising feature is an outlier temperature in March 2001, where the maximum
temperature significantly drops to 7.1°C.
Component Contribution
Maximum temperature in Min
Max
Range
Approx %
Madrid (°C)
Contribution
Raw data
6
35
29
Trend
20
23
3
10%
Seasonal
-11
12
23
79%
Residual of greatest magnitude
10
10
34%
The main source of maximum temperature in Madrid appears to be the seasonal effect,
with 79% of the overall variation explained by the seasonal effect. Trend and residual
components account for the remaining variability in the series.
INZight has created the predictions of maximum temperatures in Madrid for the next 2
years (below) with their prediction (confidence) intervals.
Month
Year
Predicted Lower
Upper
value
value
value
Nov
2012
17.5835 12.07302 23.09397
Dec
2012 9.030564 3.406929 14.6542
Jan
2013 9.126722 3.38865 14.8648
Feb
2013 11.09462 5.240846 16.94838
Mar
2013 15.09022 9.119512 21.06092
Apr
2013 17.13634 11.04748 23.22519
May
2013 23.82193 17.61372 30.03014
Jun
2013 28.93076 22.60202 35.25951
Jul
2013 31.39579 24.94534 37.84623
Aug
2013 32.32305 25.74975 38.89634
Sep
2013 27.75876 21.06148 34.45604
Oct
2013 21.75519 14.93281 28.57758
Nov
2013 17.11839 9.27773 24.95905
Dec
2013 8.565455 0.611768 16.51914
Jan
2014 8.661613 0.593513 16.72971
Feb
2014 10.62951 2.445628 18.81338
Mar
2014 14.62511 6.324103 22.92611
Apr
2014 16.67123 8.251772 25.09069
May
2014 23.35682 14.8176 31.89604
Jun
2014 28.46565 19.80538 37.12593
Jul
2014 30.93068 22.14807 39.71328
Aug
2014 31.85794 22.95175 40.76413
Sep
2014 27.29366 18.26263 36.32467
Oct
2014 21.29009 12.13301 30.44716
The model forecasts that in August 2013, the maximum temperature in Madrid Is 32°C. At
this time, the actual maximum temperature is likely to lie between 26°C and 39°C. A visual
inspection of the model from 2000 to 2012 indicates that the predicted value is quite
realistic, as it is similar in magnitude to previous ‘August’ values, but the prediction interval
is too large to make a reliable prediction.
Hence, the Holt-Winters model is unlikely to make inaccurate predictions because the
prediction interval is reasonably large. In particular, the unrealistic prediction intervals
suggest that the maximum temperatures of Madrid during December and January can be
as low as 0.6°C (almost freezing, however these seem to match similar minimum
temperatures in Madrid) and as high as 41°C in August, almost 5°C greater than previous
maximum temperature values.
The size of a prediction interval is significantly influenced by the magnitude of residuals .
Given that the residuals were reasonably large, the prediction interval is reasonably
reasonably large ( ±6.5°C) .
Long Term Trend: From 1999 to 2012, the raw data fluctuates between 24°C to 10°C. The
trend line also starts at approximately 17°C and finishes at approximately 18°C.
An interesting feature about this graph is that the trend line of Christchurch’s maximum
temperature seems to be quite consistent, with little change in increasing or decreasing
temperature over time.
Residuals: There seems to be no obvious pattern in the residual plot. The largest residual is
approximately 3°C, which is large (20%) when compared with the range of the data (2510=15°C) A residual of this magnitude represents more than 10% of this range. (Any residual
more than 10% of the overall range is considered to be significant)
Seasonal effect: The seasonal patterns of maximum temperatures in Christchurch follow a
reasonably consistent pattern throughout the years. Average seasonal highs (peaks) occur in
December-January and seasonal lows (troughs) occur in July. This is typical to the Southern
hemisphere seasonal temperatures as December and January correlate to the Summer
season (high temperatures) in Christchurch and July matches the Winter season (low
temperatures).
Component Contribution
Maximum temperature in Min
Max
Range
Approx %
Christchurch (°C)
Contribution
Raw data
10
25
15
Trend
17
18
1
6.7%
Seasonal
-6
5.5
11.5
77%
Residual of greatest magnitude
3
3
20%
The main source of maximum temperature in Christchurch appears to be the seasonal
effect, with 77% of the overall variation explained by the seasonal effect. Trend and residual
components account for the remaining variability in the series.
INZight has created the predictions of maximum temperatures in Christchurch for the next
2 years (below) with their prediction (confidence) intervals.
Month
Year
Predicted Lower
Upper
value
value
value
Nov
2012 19.41257 16.47776 22.34738
Dec
2012 21.19953 18.26467 24.1344
Jan
2013 22.69312 19.75816 25.62808
Feb
2013 21.74365 18.80852 24.67878
Mar
2013
20.4681 17.53272 23.40348
Apr
2013 18.09247 15.15674 21.02821
May
2013 14.47908 11.54287 17.41529
Jun
2013 11.41821 8.481395 14.35502
Jul
2013 11.38537 8.447796 14.32295
Aug
2013 12.77413 9.835625 15.71263
Sep
2013 15.50103 12.56141 18.44065
Oct
2013 16.57802 13.63708 19.51897
Nov
2013
19.4622 16.3773 22.54709
Dec
2013 21.24916 18.16257 24.33576
Jan
2014 22.74275 19.65421 25.83128
Feb
2014 21.79328 18.70255 24.88401
Mar
2014 20.51773 17.42453 23.61094
Apr
2014
18.1421 15.04614 21.23807
May
2014 14.52871 11.42968 17.62774
Jun
2014 11.46784 8.365417 14.57026
Jul
2014
11.435 8.328855 14.54114
Aug
2014 12.82376 9.71354 15.93398
Sep
2014 15.55066
12.436 18.66532
Oct
2014 16.62765 13.50818 19.74713
The model forecasts that in July 2013, the maximum temperature in Christchurch is 11°C. At
this time, the actual maximum temperature is likely to lie between 8.4°C and 14°C. A visual
inspection of the model from 2000 to 2012 indicates that the predicted value is quite
realistic, as it is similar in magnitude to previous ‘July’ values, but the prediction interval is
too large to make a reliable prediction.
Therefore, the Holt-Winters model is unlikely to make inaccurate predictions because the
prediction interval is reasonably large.
The size of a prediction interval is significantly influenced by the magnitude of residuals .
Given that the residuals were reasonably large, the prediction interval is reasonably
reasonably large ( ±3°C), though not as large as the residuals of Madrid’s maximum
temperatures .
“Compare and Contrast”: The trend line of the maximum temperatures of both Madrid and
Christchurch do not seem to change significantly over the past 13 years. While the trend line
of Madrid is between 19°C and 23°C, Christchurch’s maximum temperature is lower,
between 17°C and 18°C. This reflects Spain’s continental/mediterranean climate
(http://www.worldweatheronline.com/Madrid-weather-averages/Madrid/ES.aspx) which is
typically warmer than New Zealand’s subtropical climate.
(https://www.niwa.co.nz/education-and-training/schools/resources/climate/overview)
At the end of 2012 (December), Madrid’s maximum temperature seems to be decreasing,
while Christchurch’s temperature is increasing. This can be attributed by seasonality
patterns, where it is becoming Winter in Madrid (in the Northern Hemisphere) and Summer
in Christchurch (in the Southern hemisphere). It can also explain the contrasting peak and
trough shown in July.
Predictions vs Actual: In order to assess the model for prediction purposes the last three
data values were removed, predictions found and then compared with the actual last three
data values. This was repeated for both maximum temperatures in Madrid and
Christchurch.
Maximum temperatures in Madrid (°C)
Month Year
Actual
Predicted Lower
Upper
value
value
value
value
Aug
2012
34.4 35.00499 29.57418 40.4358
Sep
2012
27.9 32.40783 26.93207 37.88359
Oct
2012
20.5 27.31714 21.79386 32.84042
Actual values and predictions are close for maximum temperatures in Madrid in August
2012 (0.6°C). However, the difference increases significantly in following months to as high
as 6.8°C in October 2012), indicating the unreliability of this model for prediction purposes.
Percentage differences between actual values and predictions are up to 33 %. It can also be
seen that the actual maximum temperature of Madrid in October 2012 (20.5°C) does not fall
within the 95% prediction (confidence) interval between 21.8°C to 32.8°C.
Maximum temperatures in Christchurch (°C)
Month Year
Actual
Predicted Lower
Upper
value
value
value
value
Aug
2012
12.7
12.7
9.732533 15.66233
Sep
2012
15.4
15.4
12.46889 18.39879
Oct
2012
16.9
16.3
13.37514 19.30523
Actual values and predictions are relatively closer for the maximum temperatures of
Christchurch than Madrid, indicating the strength of this model for prediction purposes.
Percentage differences between actual values and predictions are up to 3%. All actual values
for maximum temperatures in Christchurch fall within the 95% prediction interval.
I have created two new series- Madav (mean temperature in Madrid) and Chchav (mean
temperature in Christchurch). These series reflect typical weather report data which
provides the mean temperatures of the areas, rather than the maximum or minimum
temperature (which fluctuate each year).
Conclusion: (note: evidence for “Conclusion” can also be found throughout the report)
The trend line of mean temperatures in both Madrid and Christchurch show no significant
increase of decrease throughout the years. As shown by the maximum temperature
seasonal trend series’, it is no surprise to see the correlation of maximum (peak) and
minimum (trough) temperatures correlating to the seasons of Summer and Winter,
respectively in opposite hemispheres. (ie In July, Madrid has a peak temperature during
Summer, while Christchurch has a minimum trough temperature reading in Winter)
Because it is difficult to ascertain from the long-term trend graphs whether average
temperatures could be rising or falling, I will assume that it is possible to construct the
centred moving mean (CMM) of the average temperatures for Madrid and Christchurch
with a linear graph to provide quantitative gradient values.
Madrid Temperature
Temperature (Degrees celsius)
30
y = 0.0103x + 13.999
25
20
15
10
5
2000M01
2000M06
2000M11
2001M04
2001M09
2002M02
2002M07
2002M12
2003M05
2003M10
2004M03
2004M08
2005M01
2005M06
2005M11
2006M04
2006M09
2007M02
2007M07
2007M12
2008M05
2008M10
2009M03
2009M08
2010M01
2010M06
2010M11
2011M04
2011M09
2012M02
2012M07
0
From this graph, it can be seen that every decade, the average temperature in Madrid is
increasing by about 1.2°C (or increasing by 0.0103°C per month), or 12°C every 100 years.
20
Christchurch Temperature
y = 0.0007x + 11.435
Temperature (Degrees celsius)
18
16
14
12
10
8
6
4
2
2000M01
2000M06
2000M11
2001M04
2001M09
2002M02
2002M07
2002M12
2003M05
2003M10
2004M03
2004M08
2005M01
2005M06
2005M11
2006M04
2006M09
2007M02
2007M07
2007M12
2008M05
2008M10
2009M03
2009M08
2010M01
2010M06
2010M11
2011M04
2011M09
2012M02
2012M07
0
Meanwhile, the average temperature in Christchurch is increasing by 0.084°C every decade
(or increasing by 0.0007°C per month), or 0.84°C (almost 1°C ) every 100 years.
Hence, the temperature difference of Madrid’s average temperature is significantly (about
14 times) greater than Christchurch over time. Given that Madrid and Christchurch are
reliable cities to represent overall Northern and Southern temperatures (respectively), we
could conclude that the Northern Hemisphere is in graver risk of significant temperature
increase, while the long-term temperature increase in the Southern Hemisphere is relatively
negligible, at about 1°C.
Although one degree may be considered to be a negligible amount, the effects can be
significant on the sensitive Earth. A report from http://climate.nasa.gov/effects explains
that “small changes in temperature correspond to enormous changes in the environment.”
For instance, at the end of the last ice age, “temperatures were only 5 to 9 degrees cooler
than today”
Another report on http://www.worldbank.org/en/news/press-release/2012/11/18/newreport-examines-risks-of-degree-hotter-world-by-end-of-century claims that a 4°C global
temperature increase is grave news, fearing “the inundation of coastal cities; increasing risks
for food production potentially leading to higher under and malnutrition rates; many dry
regions becoming dryer, wet regions wetter; unprecedented heat waves in many regions,
especially in the tropics; substantially exacerbated water scarcity in many regions; increased
intensity of tropical cyclones; and irreversible loss of biodiversity, including coral reef
systems.”
However, it takes a decade for one degree increase, at least in Madrid. A 4°C increase would
take about 30-40 years and even longer in Christchurch. The centred moving mean is only
one way to analyse the series’ trends; a regular, consistent fluctuation of average
temperatures in both cities over the past 20 years suggests to me that global warming is not
a fast-growing issue in the Southern Hemisphere.
Furthermore, in order to determine whether global warming is taking place, it would be
worth considering the temperatures of other global areas, or whether Madrid and
Christchurch would be representative cities in the Northern and Southern hemisphere,
respectively. It would be worth considering other countries, especially those which emit
significant amounts of Greenhouse gases. According to
http://www.carbonplanet.com/country_emissions , United Arab Emirates produce the
greatest amount of greenhouse gases per person (29.91 tonnes per person) in comparison to
New Zealand’s 8.48 tonne per person production.
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Migrations
Plant growth
Animal behaviours
Rapid change- significant differences
Data over long time- tends to fluctuate (cause of residuals)