### L-method and benchmarking

The maximum variance criterion discussed above is used by Minasny et al. (2007). The idea is to identify a particular iteration where the decrease in stratum variance *Q*
_{
h
} becomes small in all subsequent iterations. In other words, the decrease in stratum variance is expected to saturate through the iterations. Therefore, we seek to identify the knee of the curve shown in Figure 3. A so-called ‘L-method’ can be used here. The L-method was originally designed for the detection of anomalies in time series (Salvador and Chan 2005). In this section, we apply the L-method following the network design procedure using the SUNY data.

The L-method can be described using the following equation:

$$ \text{RMSE}_{c} = \frac{c-1}{b-1}\times \text{RMSE}(L_{c}) + \frac{b-c}{b-1}\times \text{RMSE}(R_{c}) $$

((9))

Suppose the total number of points in a scatter plot is *b*, see Figure 4a, *b*=80. For every choice of *c*, we can separate the data into two parts, namely, the sequence of points on the left side of *c*, *L*
_{
c
}, with indices *i*=1,⋯,*c* and the sequence of points on the right, *R*
_{
c
}, with indices *i*=*c*+1,⋯,*b*. Two linear regression lines are fitted using the two sets of points, respectively; their fitting root-mean-square errors (RMSE) are denoted by RMSE(*L*
_{
c
}) and RMSE(*R*
_{
c
}). The total fitting error can thus be expressed by Equation 9. Figure 4b shows the *R*
*M*
*S*
*E*
_{
VQA
} for all possible *c* values. When *c*=7 iterations, i.e., 3×7+1=22 stations give the minimum *R*
*M*
*S*
*E*
_{
VQA
}, thus can be considered as the best design following the L-method. The two regression lines at *c*=7 is shown in Figure 4c. In what follows, we verify the estimation using a former method used by Zagouras et al. (2013).

### Network design using the k-means clustering

In (Zagouras et al. 2013), k-means clustering was used together with principle component analysis (PCA) and the L-method to design the solar irradiance network for Greece. In that work, an instantaneous cloud modification factor (CMF) map over Greece is derived from the daily images collected by the Spinning Enhanced Visible and Infrared Imager (SEVIRI) on the Meteosat Second Generation (MSG) at 10:30 UTC each day. Following the outline of that paper, we apply the techniques using the SUNY dataset.

The data matrix used here has a dimension of 10000 × 731, containing 2 years of daily clearness indices at all pixels. PCA is used to identify the principle components (PCs). We reduce the 731 initial dimensions down to 144 eigenvectors of PCA that preserve a portion of up to 90% of the initial variance. A k-means clustering algorithm is then applied repeatedly to perform the clustering using the reduced PCs. The reason for multiple k-means is to avoid the problem of the initial centroids. Unlike the VQA, the number of clusters using the k-means algorithm needs to be predefined. We evaluate the algorithm using a number of clusters ranging from 5 to 70. Twenty k-means runs are performed at each number of clusters. Two evaluation indices, the Davies-Bouldin (DB) (Davies and Bouldin 1979) index and the Caliński-Harabasz (CH) (Caliński and Harabasz 1974) index are used for clustering validation. Figure 4d,e,f shows the evaluation graphs using the DB index and Figure 4g,h,i shows the graphs for using the CH index. The results give 24 and 23 clusters as the optimal choice, respectively. The estimations on the number of monitoring stations using the earlier methods agree with our estimation using the VQA.

The solar irradiance monitoring network designs shown above identified final networks of approximately 23 stations. Considering the similarities between the SUNY dataset and the SEVIRI dataset, our result agrees with the earlier estimates of using 22 stations for irradiance monitoring in Greece (Zagouras et al. 2013). However, the internal validation indices such as the DB index and CH index only measure the goodness of a clustering structure without respect to external information. It is almost obvious in these applications that 20+ stations are far too sparse to sample the highly-variable irradiance spatio-temporal random fields in the US and/or Greece.

Perez et al. (2012) simulated that for 15-min along-wind irradiance measurements, the de-correlation distance is around 10 km at a mid-latitude site. A de-correlation distance is the distance which the irradiance measurements at two locations are first becoming uncorrelated. In a later work, Lonij et al. (2013) verified the de-correlation distance using actual power output data from 80 rooftop PV systems over a 50 by 50 km area in Tucson, Arizona. In general, if correlations in all directions (instead of considering only the along-wind direction) are considered, de-correlation distance is usually not observed (Murata et al. 2009); the distance can then be referred to as the threshold distance (Yang et al. 2014). The estimated threshold distance in Singapore is about 10 km. In every case, the inter-station distances of the designed monitoring networks are much larger than both the de-correlation distance and the threshold distance. In other words, network design using the L-method alone does not warrant good spatio-temporal predictability.

### Predictive performance validation

A monitoring network should have good predictability at the unobserved locations. Kriging and other spatial interpolation techniques are suitable tools in assessing the spatial predictability of a network. In this section, SUNY data from the year 2005 is used to assess the predictive performance of the designed networks. Therefore, all predictions are true out-of-sample predictions. Three interpolation methods are used, namely, Thiessen polygon interpolation, inverse distance weighted interpolation, and simple kriging.

Let *z*(*s*
_{
j
}) denote the spatial process observed at point *s*
_{
j
}, where *j*=1,2,⋯,*n* are *n* observation points or monitoring stations; the general setup of spatial interpolation is as follows:

$$ z(s_{0}) = \sum_{j = 1}^{n} w_{j}z(s_{j}) $$

((10))

where *w*
_{
j
} is the weight of sampling point *s*
_{
j
}. For simplicity, we write *z*(*s*
_{0}) as *z*
_{0} and *z*(*s*
_{
j
}) as *z*
_{
j
} hereafter.

### Thiessen polygon interpolation

Thiessen polygon (TP) interpolation is also called the nearest neighbor method; it assumes that the attribute of interest at an unobserved location is equal to the measurements from its nearest observation points. Suppose location *s*
_{0} has a nearest neighbor *s*
_{
i
} where the observations are made, the interpolation weights are as follows:

$$ w_{j} = \left\{ \begin{array}{l l} 1 & \quad \text{if}\ \ j=i\\ 0 & \quad \text{if}\ \ j\ne i \end{array} \right. $$

((11))

### Inverse distance weighted interpolation

Another commonly used interpolation method is the inverse distance weighted (IDW) interpolation. IDW assumes the interpolation weights follow

$$ w_{j} = \frac{f(d_{0j})}{\displaystyle{\sum_{j=1}^{n}}\,\,f(d_{0j})} $$

((12))

where *f*(*d*
_{0j
}) is a general function of *d*
_{0j
}, the distance between points *s*
_{0} and *s*
_{
j
}. A commonly used *f*(·) is

$$ f(d_{0j}) = d_{0j}^{-\beta}, \quad \beta>0 $$

((13))

where *β* is a constant of choice. Here we choose *β*=2 for example.

### Simple kriging

Simple kriging (SK), ordinary kriging, universal kriging, and their variants are perhaps the most commonly used geostatistical interpolation methods. We use only simple kriging in this work as an example. Simple kriging aims to minimize the variance of interpolation error \(z_{0}-\hat {z}_{0}\),

$$ {\sigma_{e}^{2}} = \text{var}[z_{0}-\hat{z}_{0}] =\text{var}\left[z_{0}-\sum_{j=1}^{n}w_{j}z_{j}\right] $$

((14))

where \(\hat {z}_{0}\) denotes the estimates of *z*
_{0}. By expanding the above, we have

$$ {\sigma_{e}^{2}} = \sigma^{2} - 2\sum_{j=1}^{n}w_{j} \text{cov}\left(z_{0}, z_{j}\right)+\sum_{j=1}^{n}\sum_{i=1}^{n}w_{i}w_{j}\text{cov}\left(z_{i}, z_{j}\right) $$

((15))

where *σ*
^{2} is the variance of *z*
_{0} and cov(·) represents the covariance. By setting the first-order derivative (w.r.t. *w*
_{
j
}) of the above expression to zero, we have

$$ \sum_{i=1}^{n}w_{i}\text{cov}(z_{i}, z_{j}) = \text{cov}(z_{0}, z_{j}), \quad j=1, 2,\cdots,n $$

((16))

If the homogeneity assumption can be satisfied, Equation 16 can be written as

$$ \sum_{i=1}^{n}w_{i}\text{cor}\left(z_{i}, z_{j}\right) = \text{cor}(z_{0}, z_{j}), \quad j=1, 2,\cdots,n $$

((17))

where cor(·) denotes correlation. Homogeneity means that the standard deviation *σ*
_{
i
} and *σ*
_{
j
} are equal for all *i* and *j*. One step further could be taken by assuming isotropy in the spatial process, so that the correlation can be written as a function of distance only, i.e.,

$$ \sum_{i=1}^{n}w_{i}\rho\left(d_{ij}\right) = \rho\left(d_{0j}\right), \quad j=1, 2,\cdots,n $$

((18))

*ρ*(·) is a correlation function. The interpolation weights can be obtained by solving this linear system of equations. For our implementation, an exponential correlation function with a nugget effect

$$ \rho(d) = (1-\nu)\text{exp}(-c\cdot d)+\nu\mathbf{I}_{d=0} $$

((19))

is used.

### Validation results

All three selected interpolation methods are used to assess the predictive performance iteratively. At each iteration *i*, VQA will output a particular design with 3×*i*+1 stratum centers. Daily clearness indices from the year 2005 at these centers are used to interpolate the clearness indices at all other locations. For example, for the seventh iteration, 22 centers are produced by VQA, each interpolation method will thus generate *N*=(10000−22)×365 predictions. After the predictions of clearness index are made, these predictions are adjusted back to daily insolation for error calculation. The percentage RMSE

$$ \text{RMSE} = \frac{\sqrt{\displaystyle{\frac{1}{N}\sum\left(\hat{G}-G\right)^{2}}}}{\displaystyle{\frac{1}{N}\sum G}}\times100\% $$

((20))

is used on daily insolation *G*. Figure 5 shows the RMSE as a function of the number of stations. It is evident from the plot that earlier estimate of approximately 23 stations will result in large errors when we use the designed network for prediction.

Following the above discussions, it should be clear now that the design of an irradiance monitoring network can be subjective. Various termination criteria for VQA will lead to different designs. The trade-off between the number of stations and the network’s predictive performance needs to be considered. We do not recommend any ‘optimal’ setting, instead, the object-oriented design should be promoted.