ESTIMATING DAILY MEAN TEMPERATUREMarie NovakHarry PodschwitAaron 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
• Edlund model (Sweden)
• Ekholm model (Sweden)
• a,b,c,d and e are specific to the time of year
• Min-Max model (U.S. and others)
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 Bad 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
Model RMSE Mean error
95% CI
Iceland 0.4827 0.0015 -0.1785, 0.1815
Edlund 0.8406 0.0598 -0.2528, 0.3725
Ekholm 0.2851 -0.0665 -0.1699, 0.0368
Min-Max
0.4287 -0.0938 -0.2485, 0.0637
2-Stage Cosine
0.4007 0.0635 -0.0840, 0.2110
Red Oak, IA
Model RMSE Mean error
95% CI
Iceland 0.6620 0.1572 -0.0870, 0.4014
Edlund 1.200 0.5605 0.1576, 0.9636
Ekholm 0.5442 -0.2089 -0.3998, 0.0181
Min-Max
0.8380 -0.2040 -0.5126, 0.1048
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
Min. RMSE
Max.RMSE
MeanRMSE
Iceland 0.4744 0.7694 0.5530
Edlund 0.7382 0.8824 0.8308
Ekholm 0.2688 0.3193 0.2902
Red Oak, IA
Min. RMSE
Max.RMSE
MeanRMSE
Iceland 0.6385 1.477 0.8314
Edlund 0.4250 0.4829 0.4546
Ekholm 0.5309 0.6988 0.6292
Min. error
Max.error
Meanerror
Iceland -0.0544 0.2419 0.1518
Edlund 0.1334 0.5937 0.3411
Ekholm -0.1520 -0.0781 -0.1163
Min. error
Max.error
Meanerror
Iceland -0.0544 0.0939 -0.0034
Edlund -0.0474 0.0920 0.0276
Ekholm -0.0876 -0.0306 -0.0585
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
MeanRMSE
Iceland 0.4373 0.9288 0.5648
Edlund 0.6927 0.9320 0.8464
Ekholm 0.3066 0.5043 0.4018
Red Oak, IA
Min. RMSE
Max.RMSE
MeanRMSE
Iceland 0.6111 1.5858 0.8475
Edlund 1.0256 1.4387 1.2454
Ekholm 0.3171 0.6031 0.4670
Min. error
Max.error
Meanerror
Iceland -0.2306 0.3990 0.1920
Edlund 0.1531 0.1531 0.4167
Ekholm -0.2478 0.0124 -0.1171
Min. error
Max.error
Meanerror
Iceland -0.0902 0.1601 -0.0025
Edlund -0.1127 0.1209 0.0316
Ekholm -0.2387 -0.0816 -0.1568
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 temperature readings
Visby, Sweden Red Oak, IA
Pearson correlation coefficient: 0.373Spearman correlation coefficient: 0.358
Pearson correlation coefficient: 0.583Spearman correlation coefficient: 0.562
Visby Island, Sweden
Before noon
After noon
Positive error
43238 22187
Negative error
29580 4995
Red Oak, IA
Before noon
After noon
Positive error
25850 20773
Negative error
46920 6457
The contingency table of the simulated data.[X2=4330.182, p-value < 2.2 * 10-
16]=0.208ⱷ
The contingency table of the simulated data.[X2=13231.4, p-value < 2.2 * 10-16]
=0.364ⱷ
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 modellowest 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?