### Group A presentation.

```ESTIMATING DAILY MEAN
TEMPERATURE
Marie Novak
Harry Podschwit
Aaron Zimmerman
Questions
• If daily temperatures were described by a sine curve, the
average daily temperature would indeed be the average of
min and max. How well is daily temperature described by a
sine curve?
• What is the effect on bias and variability of different
observational schemes?
The Data
• Times and locations
• January - Visby Island, Sweden
• June - Red Oak, Iowa, USA
• Temperature measurements taken every minute
• Red Oak data more variable than the January data
• Variance(Red Oak) ≈ 6.05°C2
• Variance(Visby Island) ≈ 3.56°C2
The Models
• Iceland
•  =
07 +14 +2∗21
4
• Edlund model (Sweden)
•  =
07 +14 +5∗21
7
• Ekholm model (Sweden)
•  = 08 + 14 + 21 +  +
• a,b,c,d and e are specific to the time of year
• Min-Max model (U.S. and others)
•  =
+
2
Two-Stage Cosine Model
• Fit the sunlight portion of the day with a cosine
model
• Use NLS:
temp = A*cos(2π*B*time + C) + D + error
• Add straight line segment between sunset and
sunrise
• Integrate over the piecewise function and average
Trends by Pressure
Good Fit
Bias and Variability
• Model error
• Error between different models
• Measurement error
• Error in observation times
• Error in linear combination models
• Error in different linear combination schemes
Model Error
• Investigated the tendency of 5 different
models to over/under-estimate the daily
mean temperature
• Iceland model
• Edlund model
• Ekholm model
• U.S. model
• 2-Stage Cosine model (Aaron’s model)
Just How Accurate Are These Models?
Visby Island, Sweden
Red Oak, IA
Model
RMSE
Mean
error
95% CI
Model
RMSE
Mean
error
95% CI
Iceland
0.4827
0.0015
-0.1785,
0.1815
Iceland
0.6620
0.1572
-0.0870,
0.4014
Edlund
0.8406
0.0598
-0.2528,
0.3725
Edlund
1.200
0.5605
0.1576,
0.9636
Ekholm
0.2851
-0.0665
-0.1699,
0.0368
Ekholm
0.5442
-0.2089
-0.3998,
0.0181
MinMax
0.4287
-0.0938
-0.2485,
0.0637
MinMax
0.8380
-0.2040
-0.5126,
0.1048
2-Stage
Cosine
0.4007
0.0635
-0.0840,
0.2110
2-Stage
Cosine
0.4340
-0.1555
-0.3118,
0.0008
What Are the Consequences of Errors in
Measurement Time?
• Recalculate the error of each model for all of
the temperature values from the bottom to
the top of the hour
• How does the error change if you were 1
minute late in taking your measurements? 5
minutes? 59 minutes?
Visby Island, Sweden
Red Oak, IA
Observation Error Results
Visby Island, Sweden
Red Oak, IA
Min.
RMSE
Max.
RMSE
Mean
RMSE
Min.
RMSE
Max.
RMSE
Mean
RMSE
Iceland
0.4744
0.7694
0.5530
Iceland
0.6385
1.477
0.8314
Edlund
0.7382
0.8824
0.8308
Edlund
0.4250
0.4829
0.4546
Ekholm
0.2688
0.3193
0.2902
Ekholm
0.5309
0.6988
0.6292
Min.
error
Max.
error
Mean
error
Min.
error
Max.
error
Mean
error
Iceland
-0.0544
0.0939
-0.0034
Iceland
-0.0544
0.2419
0.1518
Edlund
-0.0474
0.0920
0.0276
Edlund
0.1334
0.5937
0.3411
Ekholm
-0.0876
-0.0306
-0.0585
Ekholm
-0.1520
-0.0781
-0.1163
What If You Were Really, Really Bad at
Taking Measurements?
• Simulated error in observation times by randomly
sampling data points within the hour
• Simulation repeated 10,000 times and RMSE of
daily mean temperature over the month calculated
Visby Island, Sweden
Red Oak, IA
Observation Error Results
Visby Island, Sweden
Min.
RMSE
Max.
RMSE
Mean
RMSE
Iceland
0.4373
0.9288
0.5648
Edlund
0.6927
0.9320
Ekholm
0.3066
Red Oak, IA
Min.
RMSE
Max.
RMSE
Mean
RMSE
Iceland
0.6111
1.5858
0.8475
0.8464
Edlund
1.0256
1.4387
1.2454
0.5043
0.4018
Ekholm
0.3171
0.6031
0.4670
Min.
error
Max.
error
Mean
error
Min.
error
Max.
error
Mean
error
Iceland
-0.0902
0.1601
-0.0025
Iceland
-0.2306
0.3990
0.1920
Edlund
-0.1127
0.1209
0.0316
Edlund
0.1531
0.1531
0.4167
Ekholm
-0.2387
-0.0816
-0.1568
Ekholm
-0.2478
0.0124
-0.1171
What Kind of Biases Are Possible From
Linear Combinations of Temperature Data?
• Performed a Monte Carlo simulation in which the
daily mean temperature was calculated with a
random linear combination of the temperature data
points taken at every hour
• Dot product of random weighting and hourly
Visby, Sweden
Pearson correlation coefficient: 0.373
Spearman correlation coefficient: 0.358
Red Oak, IA
Pearson correlation coefficient: 0.583
Spearman correlation coefficient: 0.562
Visby Island, Sweden
Positive
error
Negative
error
Red Oak, IA
Before
noon
43238
After noon
22187
Positive
error
29580
4995
Negative
error
The contingency table of the simulated
data.[X2=4330.182, p-value < 2.2 * 1016]
ⱷ=0.208
Before
noon
25850
After noon
20773
46920
6457
The contingency table of the simulated
data.[X2=13231.4, p-value < 2.2 * 10-16]
ⱷ=0.364
Conclusions
• For Visby Island, little inter-hour variation
• For Red Oak, enough inter-hour variation to make
meaningful changes to model given error in measurement
times
• Linear combinations of temperature data tended to
underestimate DMT when more weight was put on
temperatures early in the day. Similarly, the models tended
to overestimate when more weight was put on temperatures
later in the day.
Conclusions
• There was no one “best” model
• Geographic/seasonal factors
• Edlund modellowest RMSE for Visby Island but not for Red Oak, IA
• Iceland model lowest mean error for Visby Island, highly variable
• The Ekholm and Min-Max model tended to underestimate for both
data sets but not significantly so
• For Red Oak data, the 2-stage cosine model tended to underestimate;
the Iceland and Edlund models tended to overestimate (although
Iceland not significantly)
• Implications for worldwide standardized method of
measurement?
```