Based on our literature review, experiments, and the results presented in the previous chapter, we conclude that querying a large database of weather observations for past weather cases similar to a present case using a fuzzy k-nearest neighbors (fuzzy k-nn) algorithm that is designed and tuned with the help of a weather forecasting expert can increase the accuracy of predictions of cloud ceiling and visibility at an airport.
We have proposed, implemented, and tested a fuzzy k-nn based prediction system called WIND-1. Its unique component is an expertly-tuned fuzzy k-nn algorithm with a temporal dimension. We tested it with the problem of producing 6-hour predictions of cloud ceiling and visibility at an airport given a database of over 300,000 consecutive hourly airport weather observations (36 years of record). Its prediction accuracy was measured with standard meteorological statistics and compared to a benchmark prediction technique, persistence. In realistic simulations, WIND-1 was significantly more accurate. WIND-1 produced forecasts at the rate of about one per minute.
The fuzzy k-nn based prediction method is significantly more accurate than the non-fuzzy based prediction method. The only variation between the two methods is the nature of the membership functions used to compare attributes of cases. The fuzzy k-nn method uses fuzzy membership functions that span certain ranges around the case being forecast for, whereas the non-fuzzy method uses 0-1-0 functions centered across the same ranges. This suggests that, compared to the accuracy of simple persistence, the significantly higher accuracy of fuzzy k-nn based forecasts is attributable to the use of fuzzy sets to measure similarity as opposed to using crisp sets. To the best of our knowledge, all previous methods used to measure similarity between weather cases have used only crisp sets.
Of significance to case based reasoning: We have shown how fuzzy logic can impart to case-based reasoning the perceptiveness and case-discriminating ability of a domain expert. The fuzzy k-nn technique described in this thesis retrieves similar cases by emulating a domain expert who understands and interprets similar cases. The main contribution of fuzzy logic to case-based reasoning is that it enables us to use common words to directly acquire domain knowledge about feature salience. This knowledge enables us to retrieve a few most similar cases from a large temporal database, which in turn helps us to avoid the problems of case adaptation and case authoring.
The fuzzy k-nn algorithm, even though it is of approximate Order(n) complexity, makes superior predictions with practical speed-with less than one minute of computation. This speed is achieved by strategically ordering the steps in a case-to-case similarity-measuring test and by stopping any test as soon as a step reveals that a case is dissimilar enough to be ruled out of the k-nn set without the need for further tests. For example, suppose we have a database of n past temporal cases. And suppose each case is described by m attributes and is p time units long, thus each case is described by m·p attributes. To measure the similarity of every past case, we would need to perform n·m·p individual tests. However, we are only interested in finding the k most similar cases, and most cases can be ruled out of contention with a single test. So, the number of tests we need to perform is much closer to the order of n than it is to the order of n·m·p.
Of significance to meteorology and the aviation industry: Such a fuzzy k-nn weather prediction system can improve the technique of persistence climatology (PC) by achieving direct, efficient, expert-like comparison of past and present weather cases. PC is a sort of analog forecasting technique that is widely recognized as a formidable benchmark for short-range weather prediction. Previous PC systems have had two built-in constraints: they represented cases in terms of the memberships of their attributes in predefined categories and they referred to a preselected combination of attributes (i.e., defined and selected before receiving the precise and numerous details of present cases). The proposed fuzzy k-nn system compares past and present cases directly and precisely in terms of their numerous salient attributes. The fuzzy k-nn method is not tied to specific categories nor is it constrained to using only a specific limited set of predictors. Such a system for making airport weather predictions will let us tap many, large, unused archives of airport weather observations, ready repositories of temporal cases. This will help to make airport weather predictions more accurate, which will make air travel safer and make airlines more profitable.
We plan to pursue this research and improve the WIND-1 system in the following ways.
· Test the system with other airports.
· Enable the WIND system to learn autonomously.
· Incorporate additional predictive information, such as user-provided hints, projections of weather radar images of precipitation, and projections of satellite images of cloud.
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Additional References on Analog Forecasting in Meteorology
Here is a list of meteorology papers that describe the method of analog forecasting. All of these papers contributed to our understanding but only a few of these papers are cited specifically in this thesis.
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Appendix A: Sample Questionnaire for Knowledge Acquisition
Knowledge acquisition is performed simply by having the expert fill in a questionnaire such as the one shown below. When such a questionnaire is completed, it contains all the information needed to construct the fuzzy sets and to order the fuzzy operations that shown above in Section 3.2 (pg. 78). The grayed-out fields would normally be blank but we have inserted sample values for the WIND-1 configuration.
Part A: Attributes and order of comparison
Specify the attributes to compare and the order in which they are to be compared.
date of the year, hour of the day, cloud amount, cloud ceiling height, visibility, wind direction, wind speed, precipitation type, precipitation intensity, dew point temperature, dry bulb temperature, pressure trend
Part B: Continuous-number attributes
List the continuous-number attributes-those with values that can be positive or negative and which are compared in terms of their relative difference-and for each, specify values of difference that signify slightly near, near, and very near. If you choose to fill in only the middle column, then, by default, the threshold for slightly near will be twice that for near, and the threshold for very near will be half that for near.
Attribute |
slightly near |
near |
very near |
date of the year |
60 days |
30 days |
10 days |
hour of the day |
2 hours |
1 hours |
0.5 hours |
wind direction |
40 degrees |
20 degrees |
10 degrees |
dew point temperature |
4 degrees |
2 degrees |
1 degree |
dry bulb temperature |
8 degrees |
4 degrees |
2 degree |
pressure trend |
0.20 kPa · hr -1 |
0.10 kPa · hr -1 |
0.05 kPa · hr -1 |
Part C: Absolute-number attributes
List the absolute-number attributes-those with values that can be only equal to zero or positive numbers, and which are compared in terms of their relative magnitudes-and for each possible pair, use numbers in the range (0.0...1.0] to specify how near they are to each other. The number 0.25 corresponds to slightly near, the number 0.50 corresponds to near, and the number 0.75 corresponds to very near.
Attribute |
|||||||
wind speed |
0 |
1.00 |
|||||
1 |
0.75 |
1.00 |
|||||
2 |
0.50 |
0.75 |
1.00 |
||||
3 |
0.25 |
0.50 |
0.75 |
1.00 |
|||
4 |
0.10 |
0.25 |
0.50 |
0.75 |
1.00 |
||
... |
... |
... |
... |
... |
... |
... | |
0 |
1 |
2 |
3 |
4 |
... | ||
cloud amount |
... |
||||||
cloud ceiling height |
... |
||||||
visibility |
... |
Part D: Nominal attributes
List the nominal attributes, and for each possible pair, use numbers in the range (0.0...1.0] to specify how near they are to each other The number 0.25 corresponds to slightly near, the number 0.50 corresponds to near, and the number 0.75 corresponds to very near. 68
Attribute |
||||||
precipitation type |
Nil |
1.00 |
||||
Drizzle |
0.02 |
1.00 |
||||
Showers |
0.03 |
0.50 |
1.00 |
|||
Rain |
0.01 |
0.50 |
0.75 |
1.00 |
||
... |
... |
... |
... |
... |
||
Nil |
Drizzle |
Showers |
Rain |
... | ||
precipitation intensity |
... |
Part E: Recency
For each possible time step in comparable temporal cases, use numbers in the range (0.0...1.0] to specify the lowest level of similarity that can be attributed to attributes from that time step. The number 0.25 corresponds to slightly near, the number 0.50 corresponds to near, and the number 0.75 corresponds to very near.
Time in case |
Minimum similarity |
Time in case |
Minimum similarity | |
... |
... |
t-0 |
0.00 | |
t-12 |
0.96 |
t+1 |
0.00 | |
t-11 |
0.95 |
t+2 |
0.00 | |
t-10 |
0.94 |
t+3 |
0.00 | |
t-9 |
0.93 |
t+4 |
0.00 | |
t-8 |
0.92 |
t+5 |
0.00 | |
t-7 |
0.91 |
t+6 |
0.00 | |
t-6 |
0.90 |
t+7 |
0.00 | |
t-5 |
0.80 |
t+8 |
0.00 | |
t-4 |
0.70 |
t+9 |
0.00 | |
t-3 |
0.60 |
t+10 |
0.00 | |
t-2 |
0.40 |
t+11 |
0.00 | |
t-1 |
0.20 |
t+12 |
0.00 | |
... |
... |
Appendix B: A Worked-out Example of Fuzzy k-nn Algorithm for Prediction
We hope that the results achieved by the fuzzy k-nn algorithm are reproducible, in weather prediction and in other applications. Towards that end, this appendix presents a step-by-step, worked-out example of the fuzzy k-nn algorithm described in Section 3.2 (pg. 78).
We begin by assuming the similarity-measuring function has been configured as explained in Section 3.2 and in Appendix A. The three main processes in using the algorithm are as follows.
1. Measure similarity of temporal cases.
2. Traverse case base to find k-nn.
3. Make prediction based on weighted median of k-nn.
The first process, is the most original, so this appendix presents a detailed example of how similarity of cases is calculated. The following example pertains to weather but it technique ought to generalize to any sort of application which describes temporal cases in terms of continuous, absolute, and nominal attributes.
Measure similarity
A present case is composed of actual weather observations are shown in Figure 33.
CYHZ 120000Z 19010G16KT 1 1/2SM SHRA BR OVC007 21/20 A2990 RMK SF8 RERA SLP125= |
CYHZ 120100Z 20011KT 1 1/2SM -SHRA BR BKN003 OVC007 21/20 A2991 RMK SF5SF3 SLP127= |
CYHZ 120046Z 20012KT 3SM -SHRA BR OVC007 RMK SF8 RERA= |
CYHZ 120200Z 23011KT 8SM BKN004 OVC011 21/20 A2992 RMK SF5SC3 SLP130= |
CYHZ 120249Z 23008KT 2SM BR OVC007 RMK SF8 CIG RGD= |
CYHZ 120300Z 23008KT 2SM BR OVC005 21/20 A2994 RMK SF8 SLP139= |
CYHZ 120400Z 21009KT 10SM FEW007 OVC076 20/19 A2993 RMK SF2AC6 SLP135= |
CYHZ 120500Z 32006KT 10SM BKN010 OVC075 19/18 A2995 RMK SC5AC3 SLP140= |
CYHZ 120600Z 32005KT 12SM BKN008 OVC210 19/18 A2996 RMK SC6CI2 SLP143= |
CYHZ 120700Z 33005KT 15SM FEW008 BKN250 18/17 A2997 RMK SF1CI1 SLP148= |
CYHZ 120800Z 33006KT 15SM VCFG SKC 17/16 A2997 RMK VSBY NW 1/2 SLP149= |
CYHZ 120900Z 31007KT 10SM PRFG FEW100 BKN250 16/16 A2998 RMK AC1CI1 VSBY LWR W SLP152= |
CYHZ 121000Z 29005KT 12SM BKN250 16/15 A3001 RMK CI5 SLP161= |
CYHZ 121100Z 29009KT 12SM BKN250 16/15 A3002 RMK CI5 SLP166= |
CYHZ 121200Z 29006KT 15SM BKN250 16/14 A3003 RMK CI5 SLP170= |
Figure 33. Actual weather observations (METAR code) for Halifax International Airport for the period 00:00 to 12:00 UTC 12 September 1999 (obtained from the Texas A&M Weather Interface website, http://www.met.tamu.edu/personnel/students/weather/weather_interface.html, downloaded September 12, 1999).
Three simplified weather cases are shown in Figure 34. Case 1 represents the present case to predict for; it is drawn from the data above in Figure 33. Cases 2 and 3 represent two analogs from the weather archive to make predictions from; they are hypothetical. For purposes of illustration, only seven-hour-long cases are considered and only three weather attributes are presented: cloud ceiling, wind direction, and precipitation type; these attributes are, respectively, absolute, continuous, and nominal (as described in Section 3.2, pg. 78). Longer cases with more attributes would be handled by straightforward extension of the technique shown.
case 1 |
case 2 |
case 3 | |||||||
|
time |
cloud |
wind |
pcpn. |
cloud |
wind |
pcpn. |
cloud |
wind |
pcpn. |
t-3 |
9 |
200 |
shwrs |
12 |
190 |
rain |
9 |
170 |
drzl |
t-2 |
12 |
230 |
nil |
15 |
220 |
nil |
9 |
210 |
nil |
t-1 |
15 |
230 |
nil |
21 |
220 |
nil |
12 |
220 |
nil |
t-0 |
21 |
210 |
nil |
30 |
220 |
nil |
15 |
210 |
nil |
t+1 |
(30) |
320 |
nil |
24 |
330 |
nil |
21 |
310 |
nil |
t+2 |
(24) |
320 |
nil |
30 |
330 |
nil |
24 |
310 |
nil |
t+3 |
(999) |
330 |
nil |
999 |
340 |
nil |
750 |
320 |
nil |
Figure 34. Present case (1) and two analogs (2 and 3). Present case is weather from Halifax International Airport for the period 01:00 to 07:00 UTC 14 September 1999. Analogs are contrived for illustration purposes. The t-0 observation corresponds to a forecast start time of 04:00 UTC. In a forecast setting, the grayed-out observations in case 1 are not known, however auxiliary predictors (guidance) for the values of wind direction and precipitation are commonly available.
The three attributes presented Figure 34 are sufficient to demonstrate each of fuzzy similarity-measuring operations described in Section 3.2, namely m a, m c, m n, and m f(t) as shown in Figure 35 and Figure 36.
|
time |
ceiling |
wind dirn. |
pcpn. | ||||||
case 1 |
case 2 |
m a2 |
case 1 |
case 2 |
mc2 |
case 1 |
case 2 |
mn2 | |
t-3 |
9 |
12 |
0.75 |
200 |
190 |
0.88 |
shwrs |
rain |
0.75 |
t-2 |
12 |
15 |
0.80 |
230 |
220 |
0.88 |
nil |
nil |
1.00 |
t-1 |
15 |
21 |
0.71 |
230 |
220 |
0.88 |
nil |
nil |
1.00 |
t-0 |
21 |
30 |
0.70 |
210 |
220 |
0.88 |
nil |
nil |
1.00 |
t+1 |
? |
240 |
- |
320 |
330 |
0.88 |
nil |
nil |
1.00 |
t+2 |
? |
300 |
- |
320 |
330 |
0.88 |
nil |
nil |
1.00 |
t+3 |
? |
999 |
- |
330 |
340 |
0.88 |
nil |
nil |
1.00 |
(a) Comparing case 2 to case 1, ma2 is the similarity between their absolute values of ceiling height, mc2 is the similarity between their continuous values of wind direction and mn2 is the similarity between their nominal types of precipitation.
|
time |
ceiling |
wind dirn. |
pcpn. | ||||||
case 1 |
case 3 |
ma3 |
case 1 |
case 3 |
mc3 |
case 1 |
case 3 |
mn3 | |
t-3 |
9 |
9 |
1.00 |
200 |
170 |
0.38 |
shwrs |
drzl |
0.50 |
t-2 |
12 |
9 |
0.75 |
230 |
210 |
0.50 |
nil |
nil |
1.00 |
t-1 |
15 |
12 |
0.80 |
230 |
220 |
0.88 |
nil |
nil |
1.00 |
t-0 |
21 |
15 |
0.71 |
210 |
210 |
1.00 |
nil |
nil |
1.00 |
t+1 |
? |
21 |
- |
320 |
310 |
0.88 |
nil |
nil |
1.00 |
t+2 |
? |
24 |
- |
320 |
310 |
0.88 |
nil |
nil |
1.00 |
t+3 |
? |
750 |
- |
330 |
320 |
0.88 |
nil |
nil |
1.00 |
(b) Comparing case 3 to case 1, ma3 is the similarity between their absolute values of ceiling height, mc3 is the similarity between their continuous values of wind direction, and mn3 is the similarity between their nominal types of precipitation.
Figure 35. Similarity measurement between a present case (1) and two past cases (2 and 3). The fuzzy operations ma, mc, and mn are described in Section 3.2 (pg. 78).
The grayed-out values of wind direction and precipitation for the future parts of the present case (case 1) in Figure 35 are prevision obtained from auxiliary predictors, such as computer models or humans. As explained earlier, existing methods forecast large-scale phenomena, such as wind and precipitation, more effectively than they forecast small-scale phenomena, such as cloud ceilings at a particular airport.
ceiling |
wind dirn. |
pcpn. | |||||||
t |
ma2 |
mf(t) |
maxa2 |
mc2 |
mf(t) |
maxc2 |
mn2 |
mf(t) |
maxn2 |
-3 |
0.75 |
0.60 |
0.75 |
0.88 |
0.60 |
0.88 |
0.75 |
0.6 |
0.75 |
-2 |
0.80 |
0.40 |
0.80 |
0.88 |
0.40 |
0.88 |
1.00 |
0.4 |
1.00 |
-1 |
0.71 |
0.20 |
0.71 |
0.88 |
0.20 |
0.88 |
1.00 |
0.2 |
1.00 |
-0 |
0.70 |
0.00 |
0.70 |
0.88 |
0.00 |
0.88 |
1.00 |
0.0 |
1.00 |
1 |
0.88 |
0.00 |
0.88 |
1.00 |
0.0 |
1.00 | |||
2 |
0.88 |
0.00 |
0.88 |
1.00 |
0.0 |
1.00 | |||
3 |
0.88 |
0.00 |
0.88 |
1.00 |
0.0 |
1.00 | |||
min = |
0.70 |
min = |
0.88 |
min = |
0.75 | ||||
min { maxa2, maxc2, maxn2 } = min { 0.70, 0.88, 0.75 } = |
0.70 | ||||||||
(a) For case 2, the "min of the maxes" equals 0.70, due to a dissimilarity between cloud ceilings at time t-0. Assign to case 2 this value of similarity to case 1.
ceiling |
wind dirn. |
pcpn. | |||||||
t |
ma3 |
mf(t) |
maxa3 |
mc3 |
mf(t) |
maxc3 |
mn3 |
mf(t) |
maxn3 |
-3 |
1.00 |
0.60 |
1.00 |
0.38 |
0.60 |
0.60 |
0.50 |
0.6 |
0.60 |
-2 |
0.75 |
0.40 |
0.75 |
0.50 |
0.40 |
0.50 |
1.00 |
0.4 |
1.00 |
-1 |
0.80 |
0.20 |
0.80 |
0.88 |
0.20 |
0.88 |
1.00 |
0.2 |
1.00 |
-0 |
0.71 |
0.00 |
0.71 |
1.00 |
0.00 |
1.00 |
1.00 |
0.0 |
1.00 |
1 |
0.88 |
0.00 |
0.88 |
1.00 |
0.0 |
1.00 | |||
2 |
0.88 |
0.00 |
0.88 |
1.00 |
0.0 |
1.00 | |||
3 |
0.88 |
0.00 |
0.88 |
1.00 |
0.0 |
1.00 | |||
min = |
0.71 |
min = |
0.50 |
min = |
0.60 | ||||
min { maxa3, maxc3, maxn3 } = min { 0.71, 0.50, 0.60 } = |
0.50 | ||||||||
(b) For case 3, the "min of the maxes" equals 0.50, due to a dissimilarity between their wind directions at time t-2. Assign to case 3 this value of similarity to case 1.
Figure 36. Raise old low values of similarity-in effect, "forget" old dissimilarities with mf(t). Then determine the minimum of the maximum of all the similarities between past case and present case. The fuzzy operation mf is described in Section 3.2 (pg. 78).
The just described process of similarity measurement of temporal cases is the most complicated process in the fuzzy k-nn algorithm for prediction. The subsequent two process are relatively simple and explained briefly as follows.
Traverse case base
Traverse the case base measuring the similarity between past cases and a present case and simultaneously maintain a linked list of the k most similar cases (such as is shown in Figure 18 on pg. 88). Make every case-to-case similarity measuring process only as detailed as necessary. If initial attribute-to-attribute tests imply strong dissimilarity between cases-sufficient to exclude the past case in question form the k-nn set-then terminate the similarity measurement process for that past case and proceed to the next past case.
Make predictions based on a weighted median of the k-nn
For purposes of illustration, we assume that we sought only two analogs for the present case, that is, k = 2. The weighted median calculation easily extends to higher values of k.
Figure 36 shows that between case 2 and case 1 the degree of similarity equals 0.70, and between case 3 and case 1 the degree of similarity equals 0.50. Hence, a prediction for case 1 should consist of such proportional parts of case 2 and case 3, as shown in Figure 37.
time |
case 2 |
case 3 |
prediction |
(actual) | |
t+1 |
(0.7 * 24 |
+ 0.5 * 21) |
/ (.7+.5) = |
23 |
(30) |
t+2 |
(0.7 * 30 |
+ 0.5 * 24) |
/ (.7+.5) = |
28 |
(24) |
t+3 |
(0.7 * 999 |
+ 0.5 * 750) |
/ (.7+.5) = |
895 |
(999) |
Figure 37. Prediction based on weighted median of k-nn (k = 2).
68 The nominal attribute table shows the "distance" between any two nominal attributes in the same way that a distance table on a highway map shows distances between towns. For illustration purposes, the table describes fuzzy relationships between only four common precipitation types. The actual WIND-1 configuration table specifies fuzzy relationships between 24 possible precipitation types (e.g, freezing rain, ice pellets, etc.).