Sigmoid Colon

The baseline dataset contains 36 primary monthly climate variables. For applications in ecology, we provide many additional biologically relevant climate variables. Many of these additional variables need to be calculated using daily climate data, which are not available in ClimateNA. We estimated these variables based on empirical or mechanistic relationships between these variables calculated using daily observations and monthly climate variables from weather stations across the entire North America. We called these variables “derived climate variables”. Some of them have been developed in previous studies for smaller regions at the annual scale [12 (link), 13 ]. In this study, we developed the derived climate variables at monthly scale, then summed up to seasonal and annual scales. The steps included: 1) calculating derived climate variables for each month (e.g., degree days) from daily weather station data; 2) building relationships (or functions) between the derived climate variables and observed (or calculated) monthly climate variables; 3) applying the functions in ClimateNA to estimate derived climate variables using monthly climate variables generated by ClimateNA.
Observed daily climate data were obtained from 4,891 weather stations in North America from the Daily Global Historical Climatology Network (http://www.ncdc.noaa.gov). The distribution of the weather stations is shown in Fig 1. Due to the wide range of variation in climate in North America, no single linear, polynomial or nonlinear function was found to adequately reflect the relationships between degree-days and monthly climate variables. We therefore applied piecewise functions, which combine a linear function and a nonlinear function, to model these relationships between various forms of monthly degree-day variables and monthly temperatures. The degree-day variables include degree-days below 0°C (DD < 0), degree-days above 5°C (DD>5), degree-days below 18°C (DD<18) and degree-days above 18°C (DD>18). The general form of the piecewise functions of all degree-days (DD_m) is:
${DD}_m=\{\begin{array}{l}if{T}_m>k,\frac{a}{1+e^{-\left(\frac{T_m-T_0}{b}\right)}}\\ if{T}_m\le k,c+\beta T_m\end{array}$
where, T_m is the monthly mean temperature for the m month; k, a, b, T₀, c and β are the six parameters to be optimized.
For number of frost-free days (NFFD) and precipitation as snow (PAS), a sigmoid function was used to model the relationship between these monthly variables and monthly temperatures:
${NFFD}_m(orPAS)=\frac{a}{1+e^{-\left(\frac{T_m-T_0}{b}\right)}}$
where, T_m is the monthly minimum temperature for the m month; a, b and T₀ are the three parameters to be optimized.
To estimate the length of the frost-free period (FFP), the beginning the frost-free period (bFFP) and the end of the frost-free period (eFFP), we used the same polynomial functions as ClimateWNA [12 (link)] for bFFP and eFFP while the parameters were estimated based on observations from all weather stations in North America.
For extreme minimum temperature (EMT) and extreme maximum temperature (EXT) expected over a 30-year period, polynomial functions were used as follows:
$EMT=a+bTmin01+{cTmin01}^2+{dTmin12}^2+{eTD}^2$
$EXT=a+bTmax07+{cTmax07}^2+dTmax08+{eTmax08}^2+fTD$
where, a, b, c, d, e and f are the parameters to be optimized; Tmin01 and Tmin12 are monthly minimum temperature for January and December; Tmax07 and Tmax08 are monthly maximum temperature for July and August, respectively; and TD is continentality (the difference between the mean temperatures of the warmest and coldest months).
Monthly average relative humidity (RH %) is calculated from the monthly maximum and minimum air temperature following [21 ]. Monthly reference evaporation (Eref_m mm) is calculated from the monthly air temperature using the Hargreaves 1985 method [12 (link), 22 (link)]. It was evaluated against the ASCE Standardized Reference Evapotranspiration (ASCE EWRI 2005). If the monthly average air temperature is less than 0°C then Eref_m = 0. The monthly climatic moisture deficit (CMD_m mm) is 0 if Eref_m< P_m, where P_m is the monthly precipitation (mm), otherwise
${CMD}_m={Eref}_m-P_m$

In our examples we apply ridge regression and partial-least squares (PLS) on regression problems, while for classification problems we use ridge logistic regression and linear SVM coupled with Pearson’s rank based variable selection. We use sum of squared residuals and proportion misclassified as the loss functions for regression and classification, respectively.
The process of ranking and selecting P variables using Pearson’s correlation is as follows. The Pearson’s correlation coefficient is calculated between each input variable Xi and the output variable Y. The absolute values of the coefficients are sorted in descending order and the first P variables are selected. The method is quick and in our experience works well with the SVM classification method.
SVM is a widely used technique in solving classification problems. SVM performs classification by constructing an N-dimensional hyper plane that optimally separates the data into two categories. SVM is usually applied in conjunction with a kernel function, which is used to transform input data into a higher-dimensional space where the construction of the hyperplane is easier. There are four basic SVM kernels: linear, polynomial, Radial Basis Function (RBF), and sigmoid. For the sake of simplicity we use linear SVM, which requires a parameter C (cost) to be supplied. We searched for the optimal model with values for C of 0.5, 1, 2, 4, 8, and 16. We used the R package e1071 [19 ] for building SVM models.
Hoerl and Kennard [20 (link)] proposed ridge regression, a penalized least squares regression, to achieve better predictions in the presence of multicolinearity of predictors. In ridge regression the extent of coefficient shrinkage is determined by one parameter, usually referred to as lambda (λ), and it is inversely related to the model complexity. Applying ridge regression tends to improve prediction performance but it results in all small, but non-zero, regression coefficients. Friedman et al.[21 (link)] developed a fast algorithm for fitting generalised linear models with various penalties, and we used their glmnet R package [22 ] to apply ridge regression and ridge logistic regression for classification purposes. Typical usage is to let the glmnet function compute its own array of lambda values based on nlambda (number of lambda values – default is 100) and lambda.min.ratio (ratio between the maximum and minimum lambda value). We searched for the optimal model with nlambda = 100 and lambda.min.ratio = 10⁻⁶.
PLS was introduced by Wold [23 ]. The method iteratively creates components, which are linear combination of input variables, with a goal of maximising variance and correlation with the output variable. The idea is to transform the input space of X₁, X₂,…, X_P variables into a new hyper plane, with low dimensions, such that coordinates of the projections onto this hyper plane are good predictors of the output variable Y. As it is an iterative process, with each newly added component we increase complexity of the model. The method is very popular amongst QSAR modellers due to its simplicity and good results in high-dimensional settings. We searched for the optimal model with a grid of number of components from 1 to 60. We used the R package pls [24 ] for building PLS models.

A classification problem with K classes can be addressed using a neural network composed of sequential functions to map each data point $x\in {\mathbb{R}}^N$ to K real valued numbers. We focus on a neural network classifier, $f_{\theta }(x)$ , that is defined using two mapping functions: feature extraction, $$\begin{array}{c}\hfill f^1:x\to \sigma (Wx+b),\end{array}$$ where W is a weight matrix and b is a bias vector and $\sigma (x)$ is a nonlinear activation function, and classification, $$\begin{array}{c}\hfill f^2:x\to W^{c}x+b^c,\end{array}$$ where $W^c$ is a weight matrix and $b^c$ is a bias vector. The neural network classifier can then be written $$\begin{array}{c}\hfill f_{\theta }(x)=f^2(f^1(x)).\end{array}$$    
The parameters $\theta =\{W,b,W^c,b^c\}$ are optimised through training and $f_{\theta }(x)$ is used as part of a softmax activation to determine a class probability score: $$\begin{array}{c}\hfill p_{\theta }(class=i|x)=\frac{exp(f_{\theta }{(x)}_i)}{{\sum }_{j}exp(f_{\theta }{(x)}_j)}.\end{array}$$    
The form of the nonlinear activation function, $\sigma$ , can be chosen to suit the problem. We choose to work with a sigmoid activation layer $$\begin{array}{c}\hfill \sigma (x)=1/(1+exp(-x)),\end{array}$$ as this type of activation can be reproduced on the QPU, as shown below.
Adopting the notation where v denotes N input or visible binary nodes and h denotes M feature nodes or hidden nodes, and for $W\in {\mathbb{R}}^{M\times N}$ and $b\in {\mathbb{R}}^M$ , the nonlinear map, $f^1(v)$ , maps $v\in {\mathbb{B}}^M$ to a feature $h\in {\mathbb{R}}^M$ . Similarly for $W^c\in {\mathbb{R}}^{K\times M}$ and $b^c\in {\mathbb{R}}^K$ , $f^2(h)$ , maps h to a class feature $h^c\in {\mathbb{R}}^M$ . The class feature is turned into a score following Eq. (7). To transfer the feature extraction and classification task to the QPU the network nodes are each assigned to a qubit, the programmable qubit local field and between qubit coupling values are set and the network is embedded on the quantum machine.
The quantum annealer will be used in two different scenarios. First, we will consider feature extraction and use Eq. (4) with x, b and W as the qubits, local field and coupling strengths respectively to obtain samples from the quantum annealing. The frequency with which a qubit is observed to take value 1 is interpreted as the activation output $\sigma (Wx)$ in Eq. (4). Second, when $\sigma (W(x))$ is available, we use this as x in Eq. (5) along with $b^c$ and $W^c$ as input to the quantum annealer to provide the classification status. The final state of a qubit in any sample can be interpreted as an indication of whether the corresponding neuron in the model has fired.