$\displaystyle=\|H_{i,\cdot}\|_{\psi_{a}}=\max_{j\in[p]}\|H_{ij}\|_{\psi_{a}}=\max_{j}\sqrt{2}\cdot
S(f_{j})/\epsilon_{j}=\sqrt{2}/\epsilon\cdot T\max_{j}S(f_{j});$
$\displaystyle\kappa^{2}$
$\displaystyle=\|\mathbb{E}[H^{T}_{i,\cdot}H_{i,\cdot}]\|_{op}=\max_{ij}\mathbb{V}(H_{ij})=2\max_{j}S(f_{j})^{2}/\epsilon_{j}^{2}=2/\epsilon^{2}\cdot
T^{2}\max_{j}S(f_{j})^{2}.$

What remains is to characterize the the sensitivity $S(f_{j})$. The
sensitivity of the query $f_{j}$ is the most that the query may vary if one
entry in the microdata were replaced. Formally,

$\max_{X_{i,\cdot},X^{\prime}_{i,\cdot}}|f_{j}(X_{i,\cdot})-f_{j}(X^{\prime}_{i,\cdot})|\leq
S(f_{j})$

where $X_{i,\cdot}$ and $X^{\prime}_{i,\cdot}$ are two descriptions of an
individual that differ in one characteristic. By hypothesis, each entry of
microdata is bounded: $|X_{ij}|\leq\bar{A}$. Therefore

$\displaystyle S(f_{j})$
$\displaystyle=\max_{X_{i,\cdot},X^{\prime}_{i,\cdot}}|f_{j}(X_{i,\cdot})-f_{j}(X^{\prime}_{i,\cdot})|=\max_{X_{i,\cdot},X^{\prime}_{i,\cdot}}|X_{ij}-X^{\prime}_{ij}|\leq
2\bar{A}.$

∎

####  I.3 Empirical application

The variable definitions follow [Autor et al., 2013]. In the authors’ original
specification [Autor et al., 2013, Table 3, column 6],
$X_{i,\cdot}\in\mathbb{R}^{14}$ consists of: a constant, an indicator for the
2000-2007 period, percentage of employment in manufacturing, percentage of
college educated population, percentage of foreign-born population, percentage
of employment among women, percentage of employment in routine occupations,
average offshorability index of occupations, and Census division dummies.

In our augmented specification $X_{i,\cdot}\in\mathbb{R}^{30}$ consists of
variables from the original specification as well as additional variables in
[Autor et al., 2013, Appendix Table 2]. These include percentages of the
working age population: employed in manufacturing, employed in non-
manufacturing, unemployed, not in the labor force, receiving disability
benefits; average log weekly wages: manufacturing, non-manufactuing; average
benefits per capita: individual transfers, retirement, disability, medical,
federal income assistance, unemployment, TAA; and average household income per
working age adult: total, wage and salary.


# Wildfire risk forecast:
An optimizable fire danger index

Eduardo Rodrigues<EMAIL_ADDRESS>IBM Research, Brazil Bianca Zadrozny
<EMAIL_ADDRESS>IBM Research, Brazil Campbell D. Watson<EMAIL_ADDRESS>IBM Research, USA

###### Abstract

Wildfire events have caused severe losses in many places around the world and
are expected to increase with climate change. Throughout the years many
technologies have been developed to identify fire events early on and to
simulate fire behavior once they have started. Another particularly helpful
technology is fire risk indices, which use weather forcing to make advanced
predictions of the risk of fire. Predictions of fire risk indices can be used,
for instance, to allocate resources in places with high risk. These indices
have been developed over the years as empirical models with parameters that
were estimated in lab experiments and field tests. These parameters, however,
may not fit well all places where these models are used. In this paper we
propose a novel implementation of one index (NFDRS IC) as a differentiable
function in which one can optimize its internal parameters via gradient
descent. We leverage existing machine learning frameworks (PyTorch) to
construct our model. This approach has two benefits: (1) the NFDRS IC
parameters can be improved for each region using actual observed fire events,
and (2) the internal variables remain intact for interpretations by
specialists instead of meaningless hidden layers as in traditional neural
networks. In this paper we evaluate our strategy with actual fire events for
locations in the USA and Europe.

###### Index Terms:

wildfire risk, machine learning, optimization, gradient descent

##  I Introduction

Wildfires have caused billion-dollar disasters [14] and taken the lives of
many people [5]. The threat of wildfires has been exacerbated by climate
change through more frequent droughts and longer wildfire seasons. Land use
change has also played a role. As an example, the number of large fires in the
western United States has doubled from 1984 to 2015 [17]. An important tool in
tackling this problem is wildfire risk index models. Such models are typically
driven by atmospheric data (e.g., precipitation, wind, humidity) and are used
as an early warning system for people to be evacuated and preventative action
taken (e.g. the trimming trees close to power lines).

Over the years, several fire index systems have been developed which take a
range of input data, e.g. temperature, relative humidity, fuel type,
vegetation stage, topography etc, and output indices related to fire
potential. Examples of index systems are the Canadian Forest Service Fire
Weather Index Rating System (FWI) [16], the Australian McArthur rating systems
(Mark 5) [13] and National Fire-Danger Rating System (NFDRS) [8]. These index
systems are based on both physical and empirical conditions and, to this date,
are used to generate risk maps which inform agencies, individuals and
companies the risk in their particular areas [1].

Internally, these fire index models compute intermediate variables which are
combined to produce a set of fire danger indices. These internal variables are
meaningful in that they correspond to (possibly) measurable quantities; e.g.,
the equilibrium moisture content, which represents the steady state moisture
content of woody material, and the slope effect coefficient, among many
others. The relations among the internal variables and the indices are
established empirically, with parameters that have been estimated throughout
the years. These parameters, however, may not be the most appropriate for all
regions where these indices are applied. Our hypothesis in this short paper is
that we can adjust the internal parameters to best fit the particular regions
in which one intends to use the wildfire risk indices. This approach has two
major advantages: (1) one can train the index models using actual observed
fire, but starting with an already proven index, and (2) it preserves the
internal variables which are meaningful for wildfire specialists.

In order to test our hypothesis we implemented the ignition component (IC)
index of the National Fire Danger Rating System (NFDRS) as a smooth function
so that we can apply stochastic gradient descent (SGD) and optimize the
internal parameters. This approach is similar to a neural network though with
the difference that it is not a traditional architecture but a smooth version
of a fire index model. In this paper, we present results comparing the
unmodified model against a trained model (trained with observed fire from 2010
to 2015) for the period 2016 to 2020 in three separate regions.

##  II Model

Our model is based on the U.S. Forest Service - National Fire-Danger Rating
System (NFDRS). In particular, we compute the ignition component (IC). This
index gives a numerical rating of the probability that a fire (which requires
suppression action) will result if a firebrand is introduced into a fine fuel
complex. This index has been shown to indicate fire danger [9] and is commonly
used throughout the United States. There are other indices in the NFDRS system
which can be made optimizable, however in this report we focus only on IC.

Figure 1: Head of the model. Yellow boxes are intermediate variables, pink
boxes are parameters, light blue boxes are inputs, and operations are in grey.
The Extremal Dependency Index (EDI) is the optimization criterion (loss
function).

In order to produce the IC index, a number of input variables is used. These
are: temperature, maximum temperature, minimum temperature, relative humidity,
maximum relative humidity, minimum relative humidity (the last two may be
estimated as described in the report [7]), wind speed, cloud cover,
precipitation duration (which could be estimated as in [7]), mean cumulative
annual precipitation (climatological precipitation mean), vegetation stage
(four classes: (1) cured, (2) pre-green, (3) green and (4) transition),
vegetation cover (derived from GLCC as described in [9]), slope (five classes
as described in the report [7]), fuel model (NFDRS Fuel Model Map for the USA
and extending globally as in [9]), and climatic zones (Köppen classification).

Many intermediate variables need to be computed out of the input variables
before IC is produced. Most of these intermediate variables have concrete
meaning and could be measured. For example, 1-hour, 10-hour, 100-hour, and
1000-hour dead-fuels moisture (which represent wildland fuels whose moisture
contents are controlled entirely by changing weather conditions), live-fuel
moisture variables, reaction velocity, slope effect, and many others. These
intermediate variables, and the final IC index itself, were estimated in
laboratory and field experiments and their relations are embedded through
specific parameters in the NFDRS model. The head of the model can be seen in
Figure 1 (all relations up to inputs can be found in [7]).

Our hypothesis in this paper is that we can optimize the internal parameters
of the IC index so that it better predicts fire danger for particular regions,
but keeps the internal variables intact. The major benefit of this approach is
that one can still interpret the internal variables and possibly measure them.
In order to do that, we need an optimization criterion and procedure to adjust
parameters.

There are many possible optimization criteria (aka loss functions). All of
them will compare the prediction (in our case the IC index) with a measure of
fire danger and assign a score. However, it is not easy to obtain an accurate
measure of fire danger. Therefore, one needs a proxy that may not be perfect
but correlates with wildfire risk. Observed fire is obviously the most
appropriate proxy, but one needs to keep in mind that a region may experience
a high risk of wildfire though no wildfire eventuates. Moreover, fire events
are extremely rare compared to non-fire events in a large area over time. This
leads to a very unbalanced scenario in which a trivial solution (that is, no-
fire ever) will perform statistically very well depending on the optimization
criterion. Consequently, a loss function needs to be less dependent on the
base rate. We therefore use the extremal dependency index (EDI) [10]:

$\text{EDI}=\frac{\text{log}F-\text{log}H}{\text{log}F+\text{log}H}$

(where F is the false-alarm rate and H is the hit rate), that is because it is
a non-vanishing measure for rare events which also have been used for fire
events. EDI provides a score in the range $[-1,1]$, in which 1 is for a
perfect forecast and 0 is for a random forecast.

Figure 2: Network architecture illustrating the depth of the model. Green box
is the output (IC index), grey boxes are operations, and light blue boxes are
inputs. Intermediate variables and parameters are not shown.

As for the procedure to adjust parameters, we use gradient descent and use a
validation procedure to avoid overfitting (the details are described in the
next section). In order to use gradient descent, we implemented the IC index
as a differentiable function with PyTorch [15]. In addition to the sheer size
of the model (see Figure 2 where blue boxes are inputs, gray boxes are
operations and the green box is the IC index output), implementing the IC
computations as a differentiable function poses a few challenges. The first
one is that there are many hard branches in the model which makes the function
non-differentiable. For example, IC is set to zero if the expression:

$\left((344.-\text{QIGN})/10.\right)^{3.6}*\text{PNORM3}$

is equal to or less than PNORM1. Where QIGN is an intermediate variable that
means heat of ignition, PNORM1 and PNORM3 are scaling factors. This makes IC
non-differentiable with respect to that condition. In order to make it
differentiable, we had to implement the following smooth pattern of this
branch (and similar ones throughout the model):

⬇

if X < A then

return Y

else

return Z

which is translated as:

$\sigma\left((X-A)*\alpha\right)*(Z-Y)+Y$

where $\sigma(x)$ is the sigmoid function and $\alpha$ is a (learnable) shape
parameter.

The second challenge to make the IC index differentiable is related to where
the derivative is evaluated. The internal operations may be differentiable,
but the point where they are evaluated can be undefined or infinity, and this
causes the training procedure to fail. Here we show a couple examples.

An internal variable in the NFDRS model is $P(F/I)$, which stands for the
probability of a reportable fire and is defined as:

$\text{P}(\text{F}/\text{I})=\sqrt{\text{SCN}}$

where SCN is the normalized rate of spread. Consider a weight $w$ downstream
in the model. In order to compute an update to $w$ so as to minimize the loss
$\mathcal{L}$, one would need to compute the derivative of
$\text{P}(\text{F}/\text{I})$ with respect to SCN:

$\frac{\partial\mathcal{L}}{\partial
w}=\frac{\partial\mathcal{L}}{\partial\text{P}(\text{F}/\text{I})}\frac{\partial\text{P}(\text{F}/\text{I})}{\partial\text{SCN}}\frac{\partial\text{SCN}}{\partial
w}$

which exists but is infinite when SCN is zero, but physically it can indeed be
zero. To avoid the derivative blowing up we clip the gradients.

Another example which has a derivative, but it is not defined at a point is in
the definition of QIGN:

$\begin{split}\text{QIGN}=\ 144.5-(0.266*\text{TMPPRM})\\\
-(0.00058*\text{TMPPRM}^{2.0})\\\ -(0.01*\text{TMPPRM}*\text{MC1})\\\
+(18.54*(1.0-\text{EXP}(-0.151*\text{MC1})))\\\ +6.4*\text{MC1})\\\
\end{split}$

This definition has many potential candidates for parameters. However, the
square of TMPPRM cannot be made into a parameter because its derivative is not
defined for negative values of TMPPRM but in practice TMPPRM can be negative
because it is the temperature estimated for fuel-atmosphere interface in
Fahrenheit. In this case, the work around is simply to not make the square
into a parameter. The model has similar characteristics in other places.

In the next section, we show the procedure to train and test our model using
10 years of data for three different locations.

##  III Evaluation

For the purpose of evaluating our model, we compared the original IC index
with a trained model. The training data is daily and goes from 2010 to 2015,
and the testing data, also daily, from 2016 to 2020. We computed EDI for
California, Texas and Italy in the testing set to make our comparison.

Weather input data (temperature, RH, wind speed, cloud cover and
precipitation) comes from the ERA5 reanalysis dataset. We did not use
forecasted weather data so as to avoid errors in the forecast that would
impact the performance of the index (both trained and untrained). Consequently
these results represent the potential predictability of wildfire. Climatic
zones were obtained from [6], the USA fuel map from [2] and the Europe fuel
map [11], slope was obtained from [3], vegetation cover is built from the GLCC
dataset [12], and vegetation stage classes are derived from [4]. Finally, fire
observations are obtained from the MODIS/Terra+Aqua Burned Area Monthly L3
product.

All datasets were placed in the same resolution of 25km over the same grid
points. This resolution is the same as the results presented in [9].
Continuous variables were interpolated linearly, while classes were
interpolated by nearest neighbors.

In order to evaluate the skill, the observed fire dataset were pooled into the
same resolution as the input, i.e. if at least one fire event is observed in
the original resolution, the resulting grid point in the target resolution
will record a fire. In addition, a fire prediction in a neighboring grid point
where fire did not occur counts as hit (”fuzzy” pixel [9]).

A fire is forecast when the IC index value is above the second quantile of the
index distribution, which has been previously computed from the training set.
This forecast along with the observed fire dataset is fed into the EDI
function for evaluation.

Results can be seen in figs. 3, 4 and 5 for California, Texas and Italy
respectively. The first sub-figure in that sequence represents EDI for the
untrained model (the default NFDRS IC index), the second sub-figure EDI for
the trained model (the optimized NFDRS IC index), and the last sub-figure the
difference between trained and untrained. EDI can take values from 1 to -1,
and one representing perfect forecast and 0 for random forecast.

Overall one can see improvements of the model skill over most of the areas in
the testing set. Particularly, for California in Sierra Nevada and Central
Valley, the skill improves compared to the baseline. At the extreme south,
however, the trained model performs worst. This region, nevertheless, has few
fire events in the testing set (now shown), so we hypothesize this is due to
outliers. The Italy map shows similar behavior in which most places show
improvements. Texas has more mixed results. The original (untrained) model is
close to random, with just a few good places mainly in the Edwards plateau.
The trained model improves skill in sparse areas in the north but at the cost
of some other areas also in the north.

One possible explanation for the mixed results over Texas is the input data.
Fuel maps have a large impact on the IC index as well as vegetation stage. Our
model can in principle be used to optimize input maps by using gradient
descent all the way up to the inputs and adjusting them as if they were
parameters. We are not exploring this idea in this paper however.

(a) Untrained model

(b) Trained model

(c) Differnce between trained EDI and untrained EDI

Figure 3: Evaluation of IC for California over the period from 2016 to 2020
(training set from 2010 to 2015)

(a) Untrained model

(b) Trained model

(c) Differnce between trained EDI and untrained EDI

Figure 4: Evaluation of IC for Texas over the period from 2016 to 2020
(training set from 2010 to 2015)

(a) Untrained model

(b) Trained model

(c) Differnce between trained EDI and untrained EDI

Figure 5: Evaluation of IC for Italy over the period from 2016 to 2020
(training set from 2010 to 2015)

##  IV Final remarks

Fire indices have been used worldwide to tackle wildfire. These models have
been developed over many years as a mixture of physical and empirical models.
Internally, they have many meaningful variables which can be measured and
interpreted by experts.

In this paper, we presented a new approach for fire indices. Our hypothesis is
that the parameters that relate internal variables and indices may not be the
most appropriate for all regions. Consequently, one can find better parameters
in the space of all possible values.

In order to search the space of parameters for specific regions, we propose to
recast an existing model as a differentiable function, similar to a neural
network. However, instead of a meaningless hidden layer, our model has the
same internal variables as the original model we implemented. Consequently,
our index preserves the meaning of the internal variables.

To evaluate our model, we ran experiments over three separate regions:
California, Texas and Italy. Skill improved in most places when using the
trained model (i.e., the optimized NFDRS IC index). In addition, the resulting
parameters differ among the regions indicating they adjusted to the
particularities of the specific location. We intend to evaluate with experts
the meaning of the adjustments found by the optimization procedure. We also
intend to explore constraints to the parameters, so that the range in which
the internal parameters vary does not go beyond what is coherent. For this,
however, we will also need expert knowledge. Altogether, our strategy is
intended to be used in close partnership with experts, enhancing their ability
to explore changes to the model, but relying on them to provide meaning.

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