Statistical Test and Absolute Percentile Change Analysis
Annual:
1 Day
The KS statistical test (Scaled and 4x CO2) map shows the lowest values of D
across the southeastern part of the United States. Across parts of the Rockies and the
northeastern part of the region (Southeastern Canada), values of D become relatively
higher, generally in the .09 to .15 range. The highest D values that occur on this map are
on the northern fringe of the region across southern Canada. The KP test V values are
slightly higher as expected. There is a similar geographic pattern of relatively low and
high values of V. However, the moderate values of V are more expansive across the
Rockies and southeastern Canada than in the KS test. The maximum values on the map
occur in the same location as the KS test.
The lowest value of D at 0.014247 corresponds to probability that approaches
zero, showing that the distributions are different between the scaled and 4x CO2 data.
This means that our hypothesis, that scaling the control data will give a close fit with the
4x CO2 data has very little, if any ground. However, there are many locations which
occur in a connected geographic location that see improvement in the value of D (that is,
a drop) between not scaling and scaling the control data.
In the KS test difference map, positive values prevail across a broad area in the
northern and eastern parts of the domain. Some of the highest positive values occur
along the east coast of the United States, showing the best distribution correlation
improvement with the scaled data in these areas. These areas correspond to where the
minimum values of data dropped slightly between the control and 4x CO2. Generally, the
other lower percentiles as well as the median changed very little or not all, with changes
slightly increasing up to and including the median. The 95 percentile and up all showed
increases between the control and 4x CO2 data, with the changes becoming more positive
with a higher percentile. Areas closer to and including the east coast show the biggest
increases. This pattern is similar to what a scale factor will do, but is heavily weighted in
the higher percentiles and is generally much more intensive. Overall, it seems that the
best improvements with D occurred where the lower percentiles didn’t change much, and
the higher percentiles increased dramatically.
A majority of the rest of the region shows shades of green in the KS test
difference map, indicating little or no change between the control and scaled statistical
test runs. Some areas, particularly in the Deep South, Florida, and northwestern Mexico
coastline show higher values of D with the scaled statistical test, indicated by the
sporadic blue boxes in these areas. In the general southern and western parts of the
regions, there was little or no change in all lower percentiles up to and including the
median. Many places in the southwestern part of the landmass saw a minor drop in the
95 and 98 percentiles. Positive numbers began to creep into the region in the 99
percentile, and many places saw a significant rise in the maximum between the control
and 4x CO2 data. This pattern is more chaotic and in some cases the opposite of scaling
the data, and it is no surprise that some of the worsening D values occurred in this general
region.
The KP difference map shows a similar pattern although it seems that the
improvement in V values is more extensive and a bit more dramatic than what was seen
in D values. Similarly, the worsening V values seem to cover a bit wider geographic
area, and show more in the negative numbers with the KP test.
2 Day
The general pattern of KS test D values is the same as the 1 day. Some areas,
such as the southeastern United States, see a small increase in D values from 1 day to 2
day. Many areas generally show a minor decrease in D values, while few other areas
(Rockies) show a minor increase. Otherwise, the changes between the 1 day and 2 day
KS test maps are nearly unnoticeable. Likewise, the 2 day KP test map shows generally
the same geographic distribution of relatively low and high V values. Again, the changes
between the 1 day and 2 day maps are unnoticeable.
The pattern of positive and negative KS test difference values is essentially the
same as the 1 day. There is basically very little or no change in the values themselves.
Where there is a change, it is appears to be insignificant. The KP test difference map also
shows the same pattern as the 1 day. Here, it appears that there is a small growth of
positive values in the southeastern parts of the region.
The distribution change between the control and 4x CO2 data is basically the same
as for the 1 day. Some improvement areas see a slight decrease in the minimum, little or
no change up to and including the median, and increasing increases in the higher
percentiles. Again, some of the biggest improvement areas see the highest increases in
the higher percentiles. Many of the areas with no improvements in D or V, or that show
worsening values, show little or no change in the low percentiles and median. Some
show decreases in the 95 and 98 percentiles, while others show increases. By the
maximum, most of the domain shows significant increases.
3 Day
The geographic pattern of relative D values is basically the same between the 2
and 3 day KS test maps. The values themselves change very little or show no significant
geographic pattern in change. Simply, some areas get worse while others get better. The
same story could be told for the KP test V value maps. The geographic distributions are
the same and the numerical values do not change in a noticeable way.
The KS test difference maps show the same pattern as the 2 day maps. The
values themselves do not seem to change in any meaningful way. Some sporadic
locations show minor changes in either direction. Similarly, the KP test difference maps
have the same geographic pattern as the 2 day, also with random minor changes.
The percentile change maps for 3 day show the same geographic pattern, as well
as absolute change pattern within those geographic patterns as the 2 day maps. There is
no significant difference worth mentioning.
7 Day
The geographic KS test D value patterns are roughly the same between the 3 day
and 7 day maps. Overall, there seems to be lighter shades of blue in general, indicating
minor increases in D in many places. Still, there are some areas that see a minor drop in
D, including parts of the center of the country and in the northeastern part of the domain.
The KP test V value distributions are approximately the same between the 3 and 7 day
plots. Here, the lightening of the shades of blue seems to be a bit more widespread and
noticeable. Still, I would not consider the changes significant or meaningful.
The geographic pattern of change in D values is a bit different from the 3 day.
There seems to be a shift in the highest positive values from the east coast to further
northwest. Also, a small area in the northwestern parts of the regions seems to show an
increase in positive values as well. Elsewhere, the changes that occur are sporadic or
insignificant. The KP test V values show roughly the same geographic distribution
between the 3 day and 7 day maps. The values themselves change generally
insignificantly, with some minor increases in differences in the northern and eastern
general part of the region. In both tests, certain points in the southern part of the domain
now show the highest improvements in D and V.
The distribution change patterns for the 7 day data are roughly the same as they
were for the 3 day. The only minor changes may be that the 95, 98, and 99 percentiles
seem to have a more widespread area of increases in the 7 day than was the case in the 3
day. Otherwise, the other differences are sporadic or unnoticeable. The maximum is also
dotted more frequently with values that fall within the standardized range. In the highest
improvement areas, the distribution decreasing mildly in the low percentiles, is near zero
at the median, and increases mildly in the higher percentiles, increasing dramatically at
the maximum.
30 Day
The 30 day KS test D value geographic distribution pattern is not quite as smooth
as the 7 day pattern. One noticeable difference is that there seems to be a more
widespread area of relatively moderate values of D in the northeastern and southern parts
of the landmass. The values themselves generally increase in many locations, while there
seems to be more prevalent pattern of increase in the northern part of the domain. The
KP test distribution pattern also shows a growth in relatively higher V values in the
northeastern part of the domain, with a growth in relatively lower values nearby. In
many places V increases slightly between the 7 and 30 day maps, with more noticeable
increases in the northeastern and southern parts of the landmass.
The 30 day KS test difference maps show more positive values in the northeastern
and northwestern parts of the region than the 7 day maps. The differences elsewhere are
singled out and/or minor. The values themselves increased in the areas with more
positive values. The KP difference maps show a migration of positive values inland from
the east coast with the 30 day. The values themselves become positive in general in parts
of the northeast, parts of the south, and parts of the west. The near zero or negative
values do not change between the 7 and 30 day maps. In both tests, the same points in
the southern parts of the region have the highest values, and still have the same general
distribution change pattern.
The pattern of percentile change is essentially the same as the 7 day throughout
the percentiles. Some minor differences include a smaller amount of negative changes in
the minimum, more widespread positive values in the higher percentiles, and Central
Mexico becoming singled out as having the highest increases in the higher percentiles. In
Central Mexico, the KS and KP test difference values are near zero.
60 Day
The 60 day KS test maps shows a broader area of relatively moderate values of D
mainly across the northeastern and western parts of the region. The values themselves
show the most consistent area of increase in the northeastern part of the domain between
the 30 and 60 day maps. The KP map shows a more similar geographic distribution of V
values as the 30 day, with more extensive relatively moderate values in the northeast.
Here, the increase in V values is more noticeable across a greater variety of regions,
including the northeast, southwest, and north central.
The geographic distribution of KS test differences is essentially the same between
the 30 and 60 day maps. The values of differences show increases in the areas that are
positive between the 30 and 60 day plots. The point (5, 4) sticks out as being the lowest
on the map in both cases, and becomes lower still in the 60 day map. This particular
point seems to be the lowest in all period lengths up to 60 day. The highest values in the
60 day map occur in the northeast and a few points in the Deep South. The KP test
differences give the same relative distributions for both the 30 and 60 day plots. In this
case, the difference values generally increase in the positive areas and decrease in the
negative areas.
The lower percentiles in the eastern and northern part of the region show very
minor decreases. Most other areas show little or no change. The median shows slight
positive in the east and slight negative in the west. The 95 percentile and higher shows
growing positive values across much of the country with a few hot spots being Central
Mexico, and small parts of the east and west. The overall pattern is slightly different
from the 30 day maps, but not dramatically.
90 Day
The geographic patterns of KS test D values between the 60 and 90 day maps are
essentially the same. The values themselves decrease marginally in the northeast and
southwestern parts of the region, while remain the same or decrease sporadically in other
small areas. The KP test maps show essentially the same geographic distribution as well
(between 60 and 90 day) with a small growth of relatively moderate values of V in the 90
day. In these maps, the values seem to decrease in a circular region surrounding the
center of the landmass.
The geographic pattern of KS test differences in the same between the 60 and 90
day maps. The highest values in the 90 day occur in the northeastern part of the domain
and sporadic points in the south. The lowest value is again at (5, 4). The higher positive
values across the north increase between the 60 and 90 day maps, while the other values
don’t change significantly. The KP maps show the same geographic distribution between
60 and 90 day, while many of the positive values increase and many lower values
decrease. Point (6, 3) is the highest and (5, 4) is the lowest in these maps.
In the lower percentiles only a few points scattered about the northeast show
minor decreases between the control and 4x CO2 data. The rest of the region shows little
or no change. The median shows minor increases in the east and minor decreases in the
west. The higher percentiles show little change to small increases across much of the
region. In certain areas in the east, west, and extreme south these increase go up with
higher percentiles. One particular area, centered around (10, 7) is peculiar in that it
shows no change in the lower percentiles, and consistent decrease in the higher
percentiles, including the maximum. A similar pattern occurs to a lesser extent in all
other period lengths. The difference value for this point is slightly to moderately
negative for both tests and all period lengths up to 90 day.
180 Day
The KS test map of D values in the 180 day is starting to look different from that
of the 90 day. More relatively negative values are shifting southward and westward,
while the areas of relatively moderate and high values are breaking down. The values
themselves increase in parts of the northeast, northwest, west, and south between the 90
and 180 day maps. The KP test maps of V values show a similar change from 90 to 180
day as did the KS maps in the sense that more negative values appear in the southwest.
Here the V values seem to increase in the 180 day map, particularly along the perimeter
of the landmass, and across parts of the west, northeast, south, and other parts of the
center of the region.
The KS difference map shows the same general geographic distribution of relative
values as the 90 day. The minimum is again at (5, 4), and the maximums occur in the
upper northeast parts of the region. The values themselves become a bit more positive in
the higher positive areas, and more negative in the negative areas between the 90 and 180
day maps. The relative V value distributions for the KP maps are approximately the
same between the 90 and 180 day. However, there now appears to be maximum areas in
the northeast, northwest, and extreme south. (5, 4) is again the minimum. The positive
values show increase in some areas, while the negative values show decrease in some
areas.
The lower percentiles show decreases in parts of the northeast between the control
and 4x CO2 data. The median shows increases in the east and decreases in the west. The
upper percentiles show wide overall minor increases, with a few areas of more significant
increases, including the mid Atlantic, upper northwest, and extreme southern (Central
Mexico) parts of the region. The same phenomenon happens with the point (10, 7), and a
similar pattern also occurs with a few other points in the south.
360 Day
The geographic pattern of relative D values in the KS map departs from the 180
day map in that many of the features break down. The 360 day KS map is overall pretty
chaotic and hard to visually describe. The notable areas of relatively higher values are
across the northern and eastern parts of the region, with other random areas emerging in
the west, northwest and south. The values themselves in the relatively higher areas
generally increased dramatically. Areas in between seemed to show little change
between the 180 and 360 day map. The KP test V map shows a western shift of the
relatively higher V values of the north, and other sporadic changes in the northwest, west,
and south. The changes aren’t as dramatic at was seen with the KS test. Major increases
in V values occur in many areas between the 180 and 360 day data. Some areas show
little change.
The KS test difference 360 day map replaces some of the negatives in the north
central part of the region of the 180 day map with zeros or positives. Otherwise, the
geographic distributions of relative values are similar. The maximums and minimum
occur in the same areas. The values themselves increase dramatically in the maximum
areas between the 180 and 360 day maps, while the more dramatic negatives in the
northeast and west decrease the most. The geographic distributions of V values between
the 180 and 360 day maps are similar, with the exception of negative values being erased
from the north central, and a northwestward shift of colors in the northwest. (5, 4) is still
the noticeable minimum. The higher positive values become dramatically more positive
and the lower negative values become more negative, like was the case in the KS test
difference maps.
The lower percentiles show mild decreases along the west coast, and a mix of
sporadic increases and decreases across the east. As the percentile increases, positive
values slowly appear and grow in the east and negative values in the west become a bit
more intense. The positive values continue growing at the median, and the northwest and
Central Mexico also emerge as positive. The negative values in the west begin increasing
at this point. The higher percentiles show all positive areas growing and becoming more
intense. The point (10, 7) emerges as negative here, then intensifies and growths (spreads
southeastward) toward the maximum.
Overall Trends
It seems that the KS test D values (for the scaled and 4x CO2), for one thing,
showed a greater percentage of relatively low values in the shorter period lengths. There
was essentially very little change in the relative distributions between the 1 and 7 day
maps. In the northeast, the relative area of higher values began to grow at the 30 day
period length, and continued to do so, then shifted westward by the 360 day map. Also,
the gulf coast emerged as a relatively higher area as the period length became longer.
The relatively higher area in the west did not change significantly through the period
lengths.
For the most part, the actual KS test D values do not change much from the 1 to 7
day period lengths. Only in the general region of the Rockies is there a noticeable
increase in D values through these maps. The 30 day map starts to show increases in
other areas, including parts of the northeast, east, and south. This trend continues through
the 180 day maps. The 360 day map shows significant increases across many of the same
areas, along with other sporadic areas. The 360 map generally has a lot more contrast
than the other maps, showing that the highest D values (in the 0.3 to 0.5 range) occur
only in these maps. This overall trend implies a worsening of distribution correlation
between the scaled and 4x CO2 data in many locations. However, there are some areas
that do not experience significant changes in D values throughout. These areas initially
had values in the approximate range of 0.025 to 0.1 and held on to them throughout. This
area snaked from the northwestern part of the region southeastward then southwestward.
Another area exists in the extreme northeast.
The KP test V values initially showed more expansive areas of relatively higher
values across the Rockies and the northeastern part of the region. The areas in the
Rockies expanded moderately through the 7 day maps. At the 30 day maps, the areas in
the northeast began to expand, the area in the Rockies stopped expanding, and a new area
emerged along the gulf coast. By the 360 day map, the relative areas had less definition,
and there was more contrast between the low and high values.
A very similar pattern in the change of V values existed in the KP maps between
period lengths as existed in the KS maps. The 1 through 7 day maps showed little or no
change except in the Rockies. From the 30 to 180 day maps it appears that a circle of
locations surrounding the approximate center of the data set saw increasing values of V.
The 360 map showed significant jumps in V values in many of the same areas, and also
has a lot more contrast than all other period lengths. This pattern implies a worsening of
correlation as the period length becomes longer. Still, there are some locations that do
not see worsening, as was the case in the KS maps.
The KS test difference maps show generally a greater frequency of positive values
across the northern and eastern parts of the region for all period lengths. Positive values
also begin to shift into the northwest as the period length becomes longer. The relative
intensity of these positive values generally increases with increasing period length and
decreases beyond 90 days. Negative values are initially scattered throughout the south in
the shorter period lengths (1 through 7 day), many disappear and shift eastward in the 30
and 60 day maps, and eventually all are diminished by the 360 day map. Meanwhile,
parts of the north and northwest see negative values appear at the 30 day period length
and longer. The point (5, 4) is consistently the minimum (most negative) through all
period lengths. Sporadic points in the general area of Louisiana are of the few locations
that show intense positive values in the south in the shorter period lengths. This intensity
fluctuates and eventually wears off as the period length becomes longer.
The KS test difference values themselves do not change much between the 1 and
7 day period length maps. In the 7 day maps, a hardly noticeable area of positive values
becomes more positive in the general northern part of the region. A few points in the
south also see minor changes. The 30 through 180 day maps show consistent increases in
the positive areas in the northeast and eventually northwest by the 180 day map.
Meanwhile, the point (5, 4) emerges as the definite minimum. The 360 day map shows
numbers out of range on the color scale on both sides of the spectrum. Values in the
extreme northeast and northwest continue to increase. At the same time, many points
along the west coast, and random points scattered elsewhere emerge as being
significantly negative. Overall, the map generally seems to have more positive than
negative numbers throughout all period lengths. Additionally, there seems to be an
intensification of values, as positive numbers become more positive and negative
numbers become more negative throughout the period lengths. This may show that the
improvement areas see even further improvement with longer period lengths, and the
worsening areas see further worsening. Part of the contrast increase could also be
attributed to the fact that the raw values of D generally increase in many areas as the
period length becomes longer, this differences are likely to become greater.
The KP test difference relative distributions show generally positive values across
the northern tier of the domain through all period lengths. Initially, the highest relative
values occur along the east coast, and this pattern gradually evolves such that the extreme
northeast and northwest obtain the maximums by the 360 day map. Initially, parts of the
south and southeast see negative numbers. While this pattern generally holds by the 360
day map, the negative intensities fluctuate across the south through period lengths.
Meanwhile, parts of the west and northwest emerge as being negative as the period length
becomes longer. The point (5, 4) increasingly becomes more relatively negative as the
period length becomes longer.
There is little change in overall values between the 1 and 7 day KP test difference
maps. The only minor changes occur in the southern general part of the region, where
sporadic locations go from being negative to zero or slightly positive. This occurs most
frequently near the boundary of positive values. Between the 30 and 180 day maps, the
positive values across the northeast and northwest especially become more positive,
while negative areas emerge in the southeast, west, and north central parts of the region.
The 360 map shows a further intensification of this pattern. The highest values end up in
the northeast, northwest, and south, while the lowest generally occur along the west coast
of the landmass. Once again, the point (5, 4) slowly emerges as the minimum as the
period length becomes longer. The overall pattern is an increase of contrast as the period
length becomes longer. It seems that many of the areas that initially saw improvement
(positive difference values), saw that improvement become more intense with longer
period lengths. The reverse happened for areas that started with negative difference
values. Once again, some of this increase of contrast can be attributed to the higher raw
values of V with longer period lengths.
One general pattern of percentile changes is that the areas that show positive
differences (indicating correlation improvement with 4x CO2 data upon scaling the
control data), are those where lower percentiles generally decreased slightly, medians did
not change much, and higher percentiles showed increasing increases as the percentile
became higher. Some of the areas that saw zero or negative differences saw a
distribution pattern where the lower percentiles didn’t change much or even increased
slightly, some of the higher percentiles didn’t change much or even decreased slightly,
and the maximum values increased dramatically. In some cases, the maximum values
didn’t change much or even decreased significantly.
Summer:
Flood
The 95 percentile 1 day absolute percentile change for summer shows increases in
the 95 percentile for much of the eastern and northern segments of the region, with
modest decreases in the interior southwestern part of the region. There is also a positive
area in central Mexico. As the percentile becomes higher, both negative and positive
values intensify with more emphasis on the positive intensification. Other period lengths
generally show the same geographic pattern of relative values and an intensification of
th
values as the percentile gets higher. The intensification weakens with increasing period
length.
Drought
The 5 percentile 30 day map for summer shows increases in the 5th percentile in
the southwest and decreases across parts of the north and east. The increases and
decreases for the lower percentiles are generally much lower than for the higher
percentiles. As the percentile gets lower, neither the pattern nor values themselves
change significantly. The longer period lengths show the negative area getting slightly
more concentrated further northeast. However, all low percentiles generally show the
same pattern and values for these maps.
Winter:
Flood
The 95 percentile 1 day map for winter shows generally positive values across the
entire map, with the exception of the extreme southern regions. The highest positive
values occur along the east coast and in the Pacific Northwest. As the percentile gets
higher, the positive values intensify along the east coast and the west coast. The longer
period lengths have a similar geographic pattern of relative positive and negative values,
but with intensification becoming less pronounced as the period length becomes longer.
Drought
The 5 percentile 30 day map for winter shows noticeable negative values in the
west and southwest, and a noticeable batch of positive in the interior extreme northeast.
The values themselves are less than for the higher percentiles. In the 30 day maps, as the
percentile drops, there is a loss of negative values and the northeast and the map
generally shows more in the way of blue. The longer period lengths show the same
general relative geographic pattern for the 5th percentile, but do not show significant
changes when the period length gets longer.