4.1 Statistics of Spatial (S)-mode PCA implementation
Implementation of the cubic root transformed data to Spatial(S)-mode of principal component analysis, for which the first 5-component whose eigenvalue > 1 are shown in Table-1. The summary statistics of PCA implementation suggest that the first un-rotated component accounts for 90.20% of segment-1, 89.0% of segment-2, and 87.80% of segment-3 (see Figure-3). The Scree plot in association with parallel analysis, which is subjectively used for assessing the standard error (sampling error) of the resulting eigenvalue from implementation and suggests the optimal number of components to be retained for further analysis (see Figure-2). Figures-4, and 5 show the spatial distribution of rotated loadings of retained principal components and the principal component scores. The summary statistics of varimax rotated loadings of PCs as tabulated in Table-2, suggest 42.49% for the first varimax-rotated loading patterns of a segment-1 (1901–1939) explaining the southeast and some parts of northwest Mahanadi, whereas 18.16% for the second patterns explaining southeast Mahanadi with high positive loading.
The third pattern accounts for 34.24%, featuring the northeast Mahanadi, explaining most of the negative loadings. Segment-2 (1940–1978) for which the first pattern accounts for 39.07% comparably equals to the first pattern of Segment-1 but completely different loading characteristics i.e., negative loadings explaining northeast and northwest Mahanadi, whereas the second and third patterns account for 20.75% and 33.78% variance explaining southeast and southwest with some part of northwest Mahanadi respectively featuring high positive loadings. For Segment-3 (1979–2017), the second pattern explains both types of loadings (positive and negative) featuring 28.52% of variance explaining south and southeast Mahanadi. The first pattern accounts for 35.75% of the variance with high positive loading representing western Mahanadi, whereas the third pattern featuring negative loadings of north and northeast Mahanadi explains only 28.12% of the total variance.
4.2 Characterization of core origin
A fundamental concept of the EOF coefficient was assessed to lighten the information lying behind the emerging of the core points and to clearly understand the negative and positive characteristics of core points and rotated loadings. Results of the EOF coefficient (see Figure-4), where each station is assigned to a component upon which it loads highly. The patterns as depicted by retained varimax rotated loadings highlight the cores of high positive and negative loadings of different timeslot segments. Two high negative core loading emerged from the Raipur district of Chhattisgarh, with spatial coordinates of 82°E x 21°25’N during (1901–1939) and 82°E x 21°N during (1979–2107), accounting for 34.25% and 28.13% of the total variance, respectively. The three-adjoining district of Raipur including Durg, Dhamtari, and Mahasamund encountered points of high positive core loading spatially located at 81°30’E x 21°N during (1940–1978), 81°45’E x 20°30’N during (1901–1939), and 82°30’E x 21°15’N (1901–1939) respectively accounting 33.78%, 18.16%, and 42.49% of the total variance, whereas one positive core emerging point is identified at Raigarh and Korba district spatially located at 83°E x 22°15’N during (1979–2017) accounting 35.75% of the total variance.
Cuttack district in Odisha state has been identified as the core of negative loading, which is spatially located at 85°30’E x 20°30’N and accounts for 39.07% of total variance during (1940–1978), whereas Nayagarh and Khordha districts spatially located at 85°15’E x 20°30’N were found to be a core point of positive loadings during (1940–1978), accounting for 20.75% of the total variance. Finally, the second pattern of segment-3 (1979–2017) accounted for 28.52% of the variance with (undefined) dispersed core point (refer to Table-3). Sahu et al. [18] studied the Mahanadi basin using the same dataset, intending to regionalize the basin into homogeneous sub-regions. They identified four patterns, where the third pattern was similar to what was expected in this study by the 1st and 2nd components of segment-2 (1940–1978).
4.3 Statistics of Temporal (T)-mode PCA implementation
For studying the patterns of precipitation regime, the monthly relative precipitation is then implemented to a Temporal(T)-mode of sequential spatial pattern analysis. The first three un-rotated components whose eigenvalue > 1 as an output of the implementation is shown in Table-4, for which segment-1 (1901–1939) accounts for 87.40%, segment-2 (1940–1978) accounts for 77.57%, and segment-3 (1979–2017) accounts for 80.71% of the variance. Further, parallel analysis for component retention found three-component from segment-2, two-components from segment-1 and segment-3. Figure-6 shows the spatial distribution of varimax rotated loadings of retained principal components. The spatial distribution of monthly relative precipitation can be explained by the first two components of segment-1 and segment-3, whereas the first three components for segment-2.
The test results indicate that for segment-1, April, May, September, October, November, and December (July and August) have negative (positive) loadings on component-1, implying spatial pattern variability of spring (April and May), autumn (September, October, and November), summer (July and August), and early winter (December) precipitation, accounting for 42.93% of the variance. The upper and the middle portion of the Mahanadi are featured with negative loadings suggesting reasonable precipitation during the spring (Apr & May) and autumn (Sep, Oct, & Nov), along with some noticeable precipitation during early winter (Dec). The lower portion of the Mahanadi is featured with high positive loadings suggesting reasonable precipitation during summer (Jul & Aug). January, February, and March (June) have positive (negative) loadings for component-2 accounting for 36.07% of the variance, indicating spatial pattern variability of winter (Jan & Feb) and representative precipitation variability during early spring (Mar) and summer (Jun), adding a noticeable amount of precipitation to the annual total. The north Mahanadi is featured with positive loadings, receives reasonable precipitation during winter (Jan & Feb) and the rest (with negative loading) receives noticeable precipitation during early spring (Mar) and early summer (Jun) (see Tables – 5 and 6). Sahu et al. [18] have used sequential spatial pattern analysis in connection with DBSCAN for regionalizing the same basin and dataset and found three homogeneous precipitation regimes. The patterns formed were completely different from what was expected in this study.
The first three components explain segment-2, such that April, May, and October (July and August) have negative (positive) loadings on component-1, implying spatial pattern variability of spring (Apr & May), summer (Jul & Aug), and representative precipitation variability during mid-autumn (Oct) accounting for 30.35% of the total variance. The tail or the lower portion of Mahanadi is featured with positive loading, receives reasonable precipitation during summer (Jul & Aug), while the rest portion (with negative loading) receives precipitation during the spring (Apr & May) and autumn (Oct). For Component-2, January, February, and March (September) have positive (negative) loadings suggesting spatial pattern variability of winter (Jan & Feb), and representative precipitation variability during early spring (Mar) and early autumn (Sep) accounting for 31.36% of the total variance. The upper portion of the Mahanadi streamlines featured with positive loading receives reasonable precipitation during winter (Jan & Feb) and early spring (Mar), whereas the lower portion of Mahanadi streamlines with negative loadings receives noticeable precipitation to its annual total during early autumn (Sep). Finally, for component-3, in which November, and December (June) with negative (positive) loadings suggesting spatial pattern variability of late autumn (Nov), early winter (Dec), and early summer (Jun) precipitation accounting for 15.86% of the total variance. The upper and middle portions of the Mahanadi have negative loadings, indicating reasonable precipitation during late autumn (Nov) and early winter (Dec), whereas the lower portion has positive loadings, indicating noticeable precipitation during early summer (Jun) (see Tables 5 and 6).
Table 1
Decomposition analysis of the Cubic-root transformed data matrix of (M x N) using Spatial(S)-modes of Sequential spatial pattern analysis for the first five components whose Eigenvalue is > 1.
Segment (1)
|
Component (2)
|
Eigenvalue (3)
|
Eigen-Low (4)
|
Eigen-High (5)
|
Cumulative Variance (6)
|
1901–1939
|
1
|
207.53
|
193.96
|
221.10
|
90.20
|
2
|
7.58
|
7.09
|
8.08
|
93.50
|
3
|
3.12
|
2.92
|
3.32
|
94.90
|
4
|
2.30
|
2.15
|
2.45
|
95.90
|
5
|
1.22
|
1.14
|
1.30
|
96.40
|
1940–1978
|
1
|
204.62
|
191.24
|
218.00
|
89.00
|
2
|
7.70
|
7.20
|
8.21
|
92.30
|
3
|
2.99
|
2.80
|
3.19
|
93.60
|
4
|
2.15
|
2.01
|
2.29
|
94.60
|
5
|
1.21
|
1.13
|
1.29
|
95.10
|
1979–2017
|
1
|
201.88
|
188.68
|
215.07
|
87.80
|
2
|
7.81
|
7.30
|
8.32
|
91.20
|
3
|
2.90
|
2.71
|
3.09
|
92.40
|
4
|
2.17
|
2.03
|
2.31
|
93.40
|
5
|
1.20
|
1.12
|
1.28
|
93.90
|
Note: - Column no. (4) & (5) represent Sampling error range.
|
Table 2
-Selected components from each segment for studying the spatio-temporal and the effect of warming climate along with Un-rotated, Varimax-rotated, and parallel analysis characteristics.
|
|
Parallel analysis
|
Un-rotated
|
Varimax-rotated
|
Segment (1)
|
Component (2)
|
Adjusted Eigenvalue (3)
|
Estimated Bias (4)
|
Variance (5)
|
Cumulative Variance (6)
|
Variance (7)
|
Cumulative Variance (8)
|
1901–1939
|
1
|
205.694
|
1.836
|
90.20
|
90.20
|
42.49
|
42.49
|
2
|
5.827
|
1.757
|
3.30
|
93.50
|
18.16
|
60.65
|
3
|
1.422
|
1.697
|
1.40
|
94.90
|
34.25
|
94.90
|
1940–1978
|
1
|
202.794
|
1.828
|
89.00
|
89.00
|
39.07
|
39.07
|
2
|
5.953
|
1.752
|
3.30
|
92.30
|
20.75
|
59.82
|
3
|
1.296
|
1.696
|
1.30
|
93.60
|
33.78
|
93.60
|
1979–2017
|
1
|
200.046
|
1.831
|
87.80
|
87.80
|
35.75
|
35.75
|
2
|
6.049
|
1.757
|
3.40
|
91.20
|
28.52
|
64.27
|
3
|
1.198
|
1.700
|
1.20
|
92.40
|
28.13
|
92.40
|
Adjusted eigenvalues > 1 indicate the dimensions to retain (3 components retained)
|
Table 3
Segment
|
Components
|
Variance
|
Spatial Core location
|
Location Description
|
Nature of Core
|
1901–1939
|
1
|
42.49
|
82°30’E x 21°15’N
|
Mahasamund
|
Positive loading
|
2
|
18.16
|
81°45’E x 20°30’N
|
Dhamtari
|
Positive loading
|
3
|
34.25
|
82°00’E x 21°15’N
|
Raipur
|
Negative loading
|
1940–1978
|
1
|
39.07
|
85°30’E x 20°30’N
|
Cuttack
|
Negative loading
|
2
|
20.75
|
85°15’E x 20°30’N
|
Nayagarh/Khordha
|
Positive loading
|
3
|
33.78
|
81°30’E x 21°00’N
|
Durg
|
Positive loading
|
1979–2017
|
1
|
35.75
|
83°00’E x 22°15’N
|
Raigarh/Korba
|
Positive loading
|
2
|
28.52
|
84°00’E x 20°00’N
|
Khandhamal
|
Dual (+/-) loading
|
3
|
28.13
|
82°00’E x 21°00’N
|
Raipur
|
Negative loading
|
Note: - Spatial Core locations are approximate locations.
|
Table 4
Decomposition analysis of the monthly relative precipitation data matrix of (N x M) using Temporal(T)-modes of Sequential spatial pattern analysis for the first five components whose Eigenvalue is > 1.
Segment (1)
|
Component (2)
|
Eigenvalue (3)
|
Eigen-Low (4)
|
Eigen-High (5)
|
Cumulative Variance (6)
|
1901–1939
|
1
|
6.22
|
5.64
|
6.80
|
51.83
|
2
|
3.26
|
2.95
|
3.56
|
79.00
|
3
|
1.01
|
0.92
|
1.11
|
87.40
|
1940–1978
|
1
|
5.54
|
5.02
|
6.06
|
46.19
|
2
|
2.53
|
2.30
|
2.77
|
67.30
|
3
|
1.23
|
1.12
|
1.35
|
77.57
|
1979–2017
|
1
|
6.42
|
5.82
|
7.02
|
53.50
|
2
|
2.29
|
2.08
|
2.51
|
72.60
|
3
|
0.97
|
0.88
|
1.06
|
80.71
|
Note: - Column no. (4) & (5) represent Sampling error range.
|
Table 5
-Selected components from each segment for studying the spatio-temporal and the effect of warming climate along with Un-rotated, Varimax-rotated, and parallel analysis characteristics.
|
|
Parallel analysis
|
Un-rotated
|
Varimax-rotated
|
Segment
|
Component
|
Adjusted Eigenvalue
|
Estimated Bias
|
Variance
|
Cumulative Variance
|
Variance
|
Cumulative Variance
|
1901–1939
|
1
|
5.83
|
6.22
|
51.83
|
51.83
|
42.93
|
42.93
|
2
|
2.98
|
3.26
|
27.17
|
79.00
|
36.07
|
79.00
|
1940–1978
|
1
|
5.16
|
5.54
|
46.19
|
46.19
|
30.35
|
30.35
|
2
|
2.25
|
2.53
|
21.11
|
67.30
|
31.36
|
61.71
|
3
|
1.02
|
1.23
|
10.27
|
77.57
|
15.86
|
77.57
|
1979–2017
|
1
|
6.03
|
6.42
|
53.50
|
53.50
|
44.51
|
44.51
|
2
|
2.01
|
2.29
|
19.10
|
72.60
|
28.09
|
72.60
|
Adjusted eigenvalues > 1 indicate the dimensions to retain (3 components retained)
|
Table 6
-Results of varimax rotated loadings obtained from Temporal (T)-mode of the sequential spatial pattern analysis
|
Segment – 1 (1901–1939)
|
Segment – 2 (1940–1978)
|
Segment − 3 (1979–2017)
|
Month
|
Comp.1
|
Comp.2
|
Comp.1
|
Comp.2
|
Comp.3
|
Comp.1
|
Comp.2
|
January
|
0.11
|
0.488
|
|
0.543
|
-0.118
|
0.154
|
0.56
|
February
|
|
0.506
|
-0.139
|
0.567
|
0.123
|
-0.173
|
0.495
|
March
|
-0.102
|
0.423
|
-0.245
|
0.28
|
0.218
|
-0.296
|
0.182
|
April
|
-0.283
|
-0.156
|
-0.411
|
|
0.334
|
-0.339
|
|
May
|
-0.358
|
|
-0.429
|
|
|
-0.378
|
|
June
|
|
-0.42
|
|
|
0.645
|
0.235
|
-0.18
|
July
|
0.379
|
0.106
|
0.393
|
|
0.11
|
0.383
|
0.104
|
August
|
0.389
|
|
0.406
|
|
|
0.311
|
-0.114
|
September
|
-0.307
|
-0.203
|
-0.149
|
-0.487
|
0.117
|
-0.161
|
-0.2
|
October
|
-0.386
|
|
-0.338
|
|
-0.3
|
-0.381
|
|
November
|
-0.381
|
|
-0.307
|
|
-0.378
|
-0.361
|
|
December
|
-0.28
|
0.253
|
|
0.199
|
-0.357
|
|
0.541
|
Segment-3 for which the first two components account for 44.51% and 28.09% of the variance is sufficient for explaining the temporal variability. For component-1, March, April, May, October, and November (June, July, and August) have high negative (positive) loadings, suggesting spatial pattern variability of spring (Mar, Apr, & May), autumn (Oct & Nov), and summer (Jun, Jul, & Aug) precipitation. The upper and the middle portion of Mahanadi are featured with high negative loadings suggesting reasonable precipitation during spring (Mar, Apr, & May) and autumn (Oct & Nov), while the lower portion with high positive loading hits the summer monsoon (Jun, Jul, & Aug) with reasonable high precipitation. The consecutive three-month December, January, and February (September) feature positive (negative) loadings on component-2, suggesting spatial pattern variability of winter (Dec, Jan, & Feb) and early autumn (Sep) precipitation. The lower portion of upper and middle Mahanadi along with lower Mahanadi is featured with negative loading suggesting noticeable precipitation to its annual total during early autumn (Sep), while the upper portion of upper and middle Mahanadi with positive loadings suggesting reasonable precipitation during winter (Dec, Jan, & Feb) (see Tables – 5 and 6).
4.4 Characterization of identified patterns
The patterns of the different timeslot segments were then analyzed for their dispersions of the annual precipitation observed at different station points using similarities and dissimilar characteristics of inter-cluster and between clusters respectively. Figures-8 and 10 depict a boxplot in which the median's relative distance is close to the center, indicating symmetric distribution for the clusters of all the timeslot segments except the third cluster of segment-1 (1901–1939) with negatively skewed properties. The defined clusters of the segments also showed good compactness, suggesting that inter-cluster precipitation variability is homogeneous in different sub-clusters. The third cluster has the largest interquartile range (iqr), indicating a larger surface area and a lower degree of homogeneity. It is the most distinct sub-cluster of the Mahanadi basin and contains the major thalwegs of the river. The wider interquartile range of the third cluster is characterized by phenological changes in precipitation magnitudes from the high-altitude range (500m – 700m) to the streamlines.
Figures-7 and 9 show the application of empirical cumulative distribution functions (eCDFs) to the identified clusters to validate their distinctness and a pairwise comparison of CDFs using the Kolmogorov-Smirnov ‘D’ statistic test [3] as illustrated in Figures-8 and 10. On a visual interpretation of the CDF plot, the CDFs of all three sub-clusters appear similar and close to each other, as shown in Figures-7 and 9. So the D-statistics of the KS test is used for pairwise comparison of all the CDFs, indicating that the clusters identified are distinct from one another. The results of D-statistics indicate that the p-value of all the test pairs (except 2pairs) is congruent to zero, suggesting that the clusters are different from one another. Excepted pairs are the cluster 2–3 for segment-1 (1901–1939) and segment-2 (1940–1978) having homogeneity p-value of 0.595 (59.5%) and 0.655 (65.5%) respectively (see Figures-8a and 8b). Figure-10 shows a cluster comparison for different timeslot segments. Figure 10a represents the cluster-1 for different segments, suggesting high homogeneity between segment-1 and 2, and segment-2 and 3 with p-value 0.549 (54.9%) and 0.643 (64.3%) respectively. Similarly, cluster-2 has high homogeneity among all segments with a p-value of 1 (100%) see Fig. 10b. While cluster-3 has medium-range homogeneity between segment-2 and 3 with p-value 0.305 (30.5%) see Fig. 10c.
The boxplot analysis of the identified patterns of precipitation for their dispersion among the clusters is shown in Figure-11. According to the boxplot’s interpretation of compactness, the inter-cluster variations for the identified patterns of precipitation regime are medium to low, implying possible homogeneity to acceptable homogeneity in their identified regimes. The boxplot of the three timeslot segments illustrated in Figure-11 suggests well-defined, compact, and distinct precipitation regimes covering a plain terrain, high mountains, and delta region of 230 station points. Some station points have deviated from the median line, e.g., June and September (cluster-1), August and September (cluster-2), June and October (cluster-3) in Figure-11a. June, September, and October months for clusters – 1, 2, and 3 respectively, see Figure-11b. Whereas October month for cluster-1 and for cluster-3 (September and October) see Figure-11c, indicating precipitation variations in magnitude at different locations. Overall, the boxplot indicates that the patterns of the precipitation regimes are well-defined and homogeneous.
Clusters-1 and 3 (upper and middle Mahanadi) have similar precipitation marching months (June to October) for all segments. The boxplot in Figure-11 suggests an almost symmetric distribution of precipitation towards the march of months, followed by a dry season starting in November and ending in May. The spatially scattered station points in clusters-1, and 3, are featured with intense precipitation in the summer monsoon (JJA) and early autumn (SO), suggesting the summer monsoon be the main rainy season. Cluster-2 (lower Mahanadi) features wider boxplots for all segments, indicating a lower degree of homogeneity and rapid changes in precipitation values. Larger march of the month (May to November) compared to cluster-1 and 3 with a symmetric distribution. May, June, July, August, September, October, and November are the rainiest months in the lower or the southernmost portion of the Mahanadi basin. In this precipitation regime, the summer monsoon and autumn are the two main rainy seasons, as well as the late spring (May) contributing noticeable precipitation to its annual total. Further, followed by a dry season that starts in December and ends in April, see Figure-11c.