[Draft] A Probabilistic NIR-2025

Finally, this post also brings a procedural change, introducing "Planned" as an earlier-stage status for posts. The purpose of planned posts is to communicate a roadmap. When a planned post is no longer missing content, then it will be marked instead as a Draft. Relatedly, the Overview / Appendix format used in this post breaks with the Introduction / Content / Conclusion pattern used to date for two reasons: (1) the Overview / Appendix format is better for communicating multiple levels of technical depth, because the Overview quickly lays out the key message for the broadest audience and invites readers to dive deeper only where they wish; (2) the Overview / Appendix format accelerates iterative writing, because each appendix is a small piece that can be written relatively independently of the whole post. I hope that the introduction of Planned posts and the use of Overview / Appendix structure helps organize and communicate several planned posts:

1. Static Normals: a baseline predictive model of national emissions
2. Evaluating a predictive model with NIR-2025 data
3. Strategies in probabilistic models: another look at Bovaer
4. What is a PlanZero model? Introducing a PlanZero glossary
5. Improving NIR prediction with autoregressive probabilistic modelling

Of course, beyond that, there's much more do to ultimately model the impacts of emission reduction strategies. But these posts are partly written, even if their contents are incomplete and sometimes misplaced.

Until the next post reaches Draft status,

- James Bergstra

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Appendices:

• Appendix 1: Probabilistic Models, Distributions, and Density Functions
• Appendix 2: The Symmetric Log-Normal Distribution
• Appendix 3: Translating NIR-2025 Uncertainty Data
• Appendix 4: A Method for Evaluating Predictive Models
• Appendix 5: Visualizing Probabilistic NIR Models

Appendix 1: Probabilistic Models, Distributions, and Density Functions

Uncertainty as Probability Density Functions

The NIR-2025 defines uncertainty as lack of knowledge of the true value of a variable, that can be described as a probability density function characterizing the range and likelihood of possible values. A probability density function is a mathematical function that

• is never negative,
• has an area under its curve of exactly 1,
• is used to assess the probability of a variable's true value being within a range.

A probability density function assesses the probability that a variable's unknown, true value lies within some range from values A to B as the area under the curve from A to B. In mathematics notation, if we were to call our variable "X" then we would typically write this probability, P, like this: $P(A<X<B)$ or $P(A\le X\le B)$. As an example, if the probability density function happened to be shaped like the following particular "log-normal" (there are lots of possible shapes for probability density functions), and A happened to be 0.5, and B happened to be 1.1, then we could illustrate $P(A<X<B)$ as the shaded area:

Probability Distribution

The maximum probability that a range of value can have is 1. If the probability that the true value lies in a range is exactly 1, then it means the value is guaranteed to lie within that range. In the example above $P(0<X<\infty )=1$ because the log-normal distribution allocates no probability whatsoever to negative numbers or zero (in range notation: $P(X\le 0<)=0$).

Since the total amount of probability has to add up to 1, a probability density function describes how probability is distributed across the possible values that a variable might truly have. This is often called the "probability distribution", or sometimes simply "distribution", of the variable, but to be honest, I don't think this figurative image is quite right, because variables do not have distributions intrinsically. A variable has a value, and we just sometimes don't know what that value is. It's more helpful to think of the probability density function as distributing (doing the distribution of probability) and to think of the distribution as meaning something about the variable.

Probability Model

The choice of which probability density function(s) to use for which variable(s) is called a model. Sometimes a model may be called a probabilistic model, or a probability model, or a stochastic model, but in PlanZero these terms are interchangeable. This post is titled "A Probabilistic NIR" because it explores the consequences of using certain probability density functions (see Appendix 2: Translating NIR-2025 Uncertainty Data) for certain variables (the annual emissions totals in NIR-2025).

To conclude this appendix on Probability models, distributions, and density functions: NIR-2025 says that uncertainty refers to a lack of knowledge regarding a variable's true value, which corresponds to a probability density function, and this section has introduced what "probability density function" means. Several terms related to probability (linked above in the text) have been added to the PlanZero glossary, and some of the definitions link back to this appendix.

That said, probabilistic modelling is a huge topic. In Ontario, if not the rest of Canada, it is typically introduced in secondary school, and covered in greater detail at various university levels with application to areas of science, engineering, medicine, social studies, mathematics etc. where probabilities are used. For more information and orientation, see e.g. wikipedia ( start with e.g. Probability) or prompt an LLM-based AI with a question such as "Could you help me learn about probability density functions?" for a personalized introduction.

Appendix 2: The Symmetric Blended Log-Normal Distribution

PlanZero uses a single custom parameterized family of probability density functions to model each emission probability distribution in the NIR. I call this distribution a Symmetric Blended Log-Normal (SBLN). The SBLN distribution is a mixture (weighted sum) of three instances of two well-known distributions: two log-normals and a normal. The mixture components are organized by four parameters: a mean ($\mu$), a roll-off ($\rho$), and a relative uncertainty ($\sigma$) such that 1.0 means 100%, and a lower bound ($\beta$) that truncates the density distribution. The three components of an SBLN distribution are:

1. A normal distribution with mean $\mu$, and a variance that would gives it a relative uncertainty of $\sigma$ if its mean were equal to the roll-off.
2. A log-normal distribution with mean ($\mu$) and scale such that 95% of the probability is within $\pm 100\sigma$% of the mean if the mean is positive.
3. A matching log-normal distribution on the negative values with mean ($\mu$) and scale such that 95% of the probability is within $\pm 100\sigma$% of the mean if the mean is negative.

The three components are weighted differently, depending on the magnitude of the mean relative to the roll-off. If the magnitude of the mean is small relative to the roll-off, then the normal distribution has almost all of the weight, and the distribution looks like a normal with the desired mean and a fixed amount of uncertainty that lets it turn into a log-normal in a smooth way if you imagine the magnitude of the mean growing. As the absolute value of the mean switches to being larger than the roll-off, then the components are re-weighted so that the overall shape gradually transitions to being a log-normal as the magnitude of the mean grows. The details of this distribution can be found in a Jupyter notebook hosted on PlanZero's GitHub page.

The result is a parameterized probability density function that can make what I think are all the shapes required to reasonably represent the elements of the NIR. The lower bound ($\beta$) is used to truncate densities to be non-negative for emissions that can't be negative such as e.g. ${\mathrm{C}\mathrm{O}}_2$ from road transportation. For some examples of what SBLN distributions can look like, here are some distributions that show the effects of the mean, roll-off, and relative uncertainty parameters:

Appendix 3: Translating NIR-2025 Uncertainty Data

The NIR-2025 section 1.5 ("Uncertainty Assessment") begins: While national GHG inventories should be accurate, complete, comparable, transparent and consistent, estimates will always inherently involve some uncertainty. Uncertainties in the inventory estimates may be caused by systematic and/or random uncertainties in the input parameters or estimation models. The NIR text explains that the 2006 IPCC guidelines specify that the primary purpose of providing quantitative uncertainty information is to assist in setting priorities for the improvement of future inventories and to guide decisions about which methods to use and observes that typically, the uncertainties associated with trends and the national totals are much lower than those associated with individual gases and sectors.

Estimating high uncertainty is relatively easy; estimating low uncertainty with confidence takes work. My interpretation of this characterization of the NIR's approach to uncertainty, is that it was primarily meant to assess where and when emissions could reasonably be compared in terms of their means. Simple techniques for approximate uncertainty estimation were often sufficient to separate signal from noise, and confirmed e.g. that yes, apparently large-emission sectors really were indeed as bad as they looked. PlanZero has a different use for the uncertainty — model matching, below in "A Metric for Evaluating NIR Predictions" — but the uncertainty estimates used to compare sectors seem good for this purpose as well. The remainder of this section is about the uncertainty estimates in NIR-2025, and how the probabilistic NIR model interprets them as probability density functions.

NIR-2025 Uncertainty Data

The NIR-2025 data about uncertainty is contained in four related tables in Annex 2 to the report. The first two tables (A2-1, A2-2) are about uncertainty in totals, (exclusive and inclusive of land-use sectors, respectively). The second two tables (A2-3, A2-4) are about uncertainty in trend, (again exclusive and inclusive of land-use sectors, respectively). The model developed in the next section draws on Table A2-2 "Uncertainty Assessment Level for base year and latest year with LULUCF)". The rows of this table correspond to (sector, gas) combinations. The columns of the table have emissions totals (to help identify corresponding sectors and gases used in other Annex data) and uncertainty on those totals, expressed as percentage points. Those percentage points are sometimes zero, usually positive, and never negative. Most-frequently they are 1% or 2%. There are some outlier (sector, gas) combinations (such as e.g. 250% on ${\mathrm{N}}_2\mathrm{O}$ from stationary combustion sources producing heat and electricity, 440% on ${\mathrm{N}}_2\mathrm{O}$ from stationary combustion sources in oil and gas extraction, and up to 740% on ${\mathrm{C}\mathrm{H}}_4$ from military aviation) but for the most part, the uncertainties are small enough (especially on the larger gas-sector combinations) that the magnitude of the estimated mean carries the most relevant information about the distribution. The overall uncertainty on the national total was assessed to be about 2-3% without Land-Use, Land-Use-Change, and Forestry (LULUCF) sectors, and about 9% when including them.

The figure below plots the range of assessed uncertainty values in the 2025 NIR national totals, across the sector-gas combinations listed in Annex 2.

The largest emission values on both negative and positive ends of the spectrum have relatively low uncertainty. High-uncertainty sector-gas combinations (e.g. > 50%) only occurs in association with sectors emitting up to about 1000 kt ${\mathrm{C}\mathrm{O}}_2\mathrm{e}$, which individually represent only about 0.1% of the national total. Despite little nooks and crannies of uncertainty, the overall picture based on means (which PlanZero has been using to date) can be taken as relatively accurate. Still, for some purposes, the differences between e.g. 1%, 2%, 10%, and 20% uncertainty are important even if they all bestow confidence in the mean. The next section translates these uncertainties into probability density functions for all sector-gas-region-year combinations in the NIR. See Appendix 4 for how the probabilistic NIR model can support the evaluation of predictions.

Relating the Rows of Annex 2 to PlanZero's Sector-Gas Classifications

PlanZero uses the "IPCC Source Category" column of Table A2-2 in the NIR-2025's Annex 2 to establish a correspondence with the IPCC categorization used elsewhere in NIR-2025. Annex 2 can be found on the IPCC website (2025NIR - Part 2.pdf) and the table of uncertainties has been superceded on the NIR website by NIR-2026, but a copy can be downloaded from PlanZero's GitHub project (copy of EN_Annex2_Uncertainty.xlsx). The table below lists the row(s) from Table A2-2 used to establish the uncertainty for each sector-gas combination used elsewhere in the NIR-2025.

NIR Sector	Gas	Annex 2 IPCC Source Category
Stationary Combustion Sources/Public Electricity and Heat Production	All	Fuel Combustion - Public Electricity and Heat Production
Stationary Combustion Sources/Petroleum Refining Industries	All	Fuel Combustion - Petroleum Refining
Stationary Combustion Sources/Oil and Gas Extraction	All	Fuel Combustion - Manufacture of Solid Fuels and Other Energy Industries
Stationary Combustion Sources/Mining	All	Fuel Combustion - Manufacturing Industries and Construction
Stationary Combustion Sources/Manufacturing Industries/Iron and Steel	All	Fuel Combustion - Manufacturing Industries and Construction
Stationary Combustion Sources/Manufacturing Industries/Non-Ferrous Metals	All	Fuel Combustion - Manufacturing Industries and Construction
Stationary Combustion Sources/Manufacturing Industries/Chemical	All	Fuel Combustion - Manufacturing Industries and Construction
Stationary Combustion Sources/Manufacturing Industries/Pulp and Paper	All	Fuel Combustion - Manufacturing Industries and Construction
Stationary Combustion Sources/Manufacturing Industries/Cement	All	Fuel Combustion - Manufacturing Industries and Construction
Stationary Combustion Sources/Manufacturing Industries/Other Manufacturing	All	Fuel Combustion - Manufacturing Industries and Construction
Stationary Combustion Sources/Construction	All	Fuel Combustion - Manufacturing Industries and Construction
Stationary Combustion Sources/Commercial and Institutional	All	Fuel Combustion - Other Sectors
Stationary Combustion Sources/Residential	All	Fuel Combustion - Other Sectors
Stationary Combustion Sources/Agriculture and Forestry	All	Fuel Combustion - Manufacturing Industries and Construction
Transport/Aviation/Domestic Aviation (Civil)	All	Fuel Combustion - Civil Aviation
Transport/Aviation/Military	All	Fuel Combustion - Other (Military Aviation)
Transport/Road Transportation/Light-Duty Gasoline Vehicles	All	Fuel Combustion - Road Transportation
Transport/Road Transportation/Light-Duty Gasoline Trucks	All	Fuel Combustion - Road Transportation
Transport/Road Transportation/Heavy-Duty Gasoline Vehicles	All	Fuel Combustion - Road Transportation
Transport/Road Transportation/Motorcycles	All	Fuel Combustion - Road Transportation
Transport/Road Transportation/Light-Duty Diesel Vehicles	All	Fuel Combustion - Road Transportation
Transport/Road Transportation/Light-Duty Diesel Trucks	All	Fuel Combustion - Road Transportation
Transport/Road Transportation/Heavy-Duty Diesel Vehicles	All	Fuel Combustion - Road Transportation
Transport/Road Transportation/Propane and Natural Gas Vehicles	All	Fuel Combustion - Road Transportation
Transport/Railways	All	Fuel Combustion - Railways
Transport/Marine/Domestic Navigation	All	Fuel Combustion - Navigation
Transport/Marine/Fishing	All	Fuel Combustion - Fishing
Transport/Marine/Military Water-Borne Navigation	All	Fuel Combustion - Other (Military Navigation)
Transport/Other Transportation/Off-Road Agriculture and Forestry	All	Fuel Combustion -Off-Roadb
Transport/Other Transportation/Off-Road Commercial and Institutional	All	Fuel Combustion -Off-Roadb
Transport/Other Transportation/Off-Road Manufacturing, Mining and Construction	All	Fuel Combustion -Off-Roadb
Transport/Other Transportation/Off-Road Residential	All	Fuel Combustion -Off-Roadb
Transport/Other Transportation/Off-Road Other Transportation	All	Fuel Combustion -Off-Roadb
Transport/Other Transportation/Pipeline Transport	All	Fuel Combustion - Pipeline Transport
Fugitive Sources/Coal Mining	All	Fugitive Sources - Coal Mining
Fugitive Sources/Oil and Natural Gas/Oil	All	Fugitive Sources - Oil & Gas
Fugitive Sources/Oil and Natural Gas/Natural Gas	All	Fugitive Sources - Oil & Gas
Fugitive Sources/Oil and Natural Gas/Venting	${\mathrm{C}\mathrm{O}}_2$	Fugitive Sources - Venting
Fugitive Sources/Oil and Natural Gas/Venting	${\mathrm{C}\mathrm{H}}_4$	Fugitive Sources - Venting & Flaring
Fugitive Sources/Oil and Natural Gas/Venting	${\mathrm{N}}_2\mathrm{O}$	Fugitive Sources - Venting & Flaring
Fugitive Sources/Oil and Natural Gas/Flaring	${\mathrm{C}\mathrm{O}}_2$	Fugitive Sources - Flaring
Fugitive Sources/Oil and Natural Gas/Flaring	${\mathrm{C}\mathrm{H}}_4$	Fugitive Sources - Venting & Flaring
Fugitive Sources/Oil and Natural Gas/Flaring	${\mathrm{N}}_2\mathrm{O}$	Fugitive Sources - Venting & Flaring
CO2 Transport and Storage	All	CO2 Transport and Storage
Mineral Products/Cement Production	All	IPPU - Cement Production
Mineral Products/Lime Production	All	IPPU - Lime Production
Mineral Products/Mineral Product Use	All	IPPU - Other (Limestone and Dolomite Use)
Chemical Industry/Ammonia Production	All	IPPU - Ammonia Production
Chemical Industry/Nitric Acid Production	All	IPPU - Nitric Acid Production
Chemical Industry/Adipic Acid Production	All	IPPU - Adipic Acid Production
Chemical Industry/Petrochemical and Carbon Black Production	${\mathrm{C}\mathrm{O}}_2$	IPPU - Petrochemical and Carbon Black Production
Chemical Industry/Petrochemical and Carbon Black Production	${\mathrm{C}\mathrm{H}}_4$	IPPU - Petrochemical and Carbon Black Production (including carbide production)
Chemical Industry/Petrochemical and Carbon Black Production	${\mathrm{N}}_2\mathrm{O}$	IPPU - Petrochemical and Carbon Black Production
Metal Production/Iron and Steel Production	All	IPPU - Iron and Steel Production
Metal Production/Aluminium Production	All	IPPU - Aluminium Production
Metal Production/SF6 Used in Magnesium Smelters and Casters	All	IPPU - Magnesium Production
Production and Consumption of Halocarbons, SF6 and NF3	HFCs	IPPU - Fluorochemical Production
Production and Consumption of Halocarbons, SF6 and NF3	PFCs	IPPU - Integrated Circuit or Semiconductor
Production and Consumption of Halocarbons, SF6 and NF3	${\mathrm{S}\mathrm{F}}_6$	IPPU - Integrated Circuit or Semiconductor
Production and Consumption of Halocarbons, SF6 and NF3	${\mathrm{N}\mathrm{F}}_3$	IPPU - Integrated Circuit or Semiconductor
Non-Energy Products from Fuels and Solvent Use	All	IPPU - Non-Energy Products from Fuels and Solvent Use Other - Other (Use of Urea in SCR Vehicles)
Other Product Manufacture and Use	${\mathrm{N}}_2\mathrm{O}$	IPPU - Other (Medical Applications of N2O)
Other Product Manufacture and Use	HFCs	IPPU - Product Uses as Substitutes for Ozone Depleting Substances
Other Product Manufacture and Use	PFCs	IPPU - Other Contained Product Uses
Other Product Manufacture and Use	${\mathrm{S}\mathrm{F}}_6$	IPPU - Electrical Equipment
Enteric Fermentation	All	Agriculture - Enteric Fermentation
Manure Management	${\mathrm{C}\mathrm{H}}_4$	Agriculture - Manure Management
Manure Management	${\mathrm{N}}_2\mathrm{O}$	Agriculture - Manure Management Direct Emissions
Agricultural Soils/Direct Sources	All	Agriculture - Direct Agriculture Soils
Agricultural Soils/Indirect Sources	All	Agriculture - Indirect Agriculture Soils
Field Burning of Agricultural Residues	All	Agriculture - Field Burning of Agricultural Residues
Liming, Urea Application and Other Carbon-Containing Fertilizers	All	Agriculture - Urea Application
Municipal Solid Waste Landfills	All	Solid Waste Disposal - Managed Waste Disposal Sites
Industrial Wood Waste Lanfills	All	Biological Treatment of Solid Waste - Anerobic Digestion - Industrial & Municipal Facilities
Biological Treatment of Solid Waste	All	Biological Treatment of Solid Waste - Composting
Incineration and Open Burning of Waste	All	Incineration and Open Burning of Waste - Waste Incineration
Municipal Wastewater Treatment and Discharge	All	Wastewater Treatment and Discharge
Industrial Wastewater and Discharge	All	Wastewater Treatment and Discharge
Forest Land	All	LULUCF - Forest Land Remaining Forest Land
Cropland	${\mathrm{C}\mathrm{O}}_2$	LULUCF - Cropland
Cropland	${\mathrm{C}\mathrm{H}}_4$	Blend: Conversion of Forest Land and Grass Land
Cropland	${\mathrm{N}}_2\mathrm{O}$	LULUCF - Cropland
Grassland	All	LULUCF - Grassland
Wetlands	All	LULUCF - Wetlands
Settlements	${\mathrm{C}\mathrm{O}}_2$	LULUCF - Settlements
Settlements	${\mathrm{C}\mathrm{H}}_4$	Blend: Conversion of Forest Land and Grass Land
Settlements	${\mathrm{N}}_2\mathrm{O}$	Blend: Conversion of Forest Land and Grass Land
Harvested Wood Products	All	LULUCF - Harvested Wood Products (HWP)

With this correspondence, the probabilistic NIR model interprets NIR-2025 as defining

• Average (mean) emission amount $m_{(s,g,r,t)}$ in units of kt${\mathrm{C}\mathrm{O}}_2\mathrm{e}$, for all non-negligible combinations of sector ($s$), gas ($g$), region ($r$), year (aka time, $t$)
• Uncertainty amount $u_{(s,g,r,t)}$ in units of percent for all sectors and gases, for the region only of Canada (not provinces or territories), and for years 1990 and 2023 (not inclusive of the intervening years).

PlanZero interprets the NIR-2025 Annex 2 values as guidance for interpolation to intervening years, and provinces and territories as follows:

Case	Uncertainty Interpolation
Uncertainty in national total, intervening year $1990<t<2023$	Linear interpolation with coefficient $\alpha =\frac{t-1990}{2023-1990}$: $u_{(s,g,\mathrm{C}\mathrm{a}\mathrm{n}\mathrm{a}\mathrm{d}\mathrm{a},t)}=(1-\alpha )u_{(s,g,\mathrm{C}\mathrm{a}\mathrm{n}\mathrm{a}\mathrm{d}\mathrm{a},1990)}+(\alpha )u_{(s,g,\mathrm{C}\mathrm{a}\mathrm{n}\mathrm{a}\mathrm{d}\mathrm{a},2023)}$
Province/territory region ($r$) uncertainty	$u_{(s,g,r,t)}=u_{(s,g,\mathrm{C}\mathrm{a}\mathrm{n}\mathrm{a}\mathrm{d}\mathrm{a},t)}$

The next sections in this Appendix relate these $m$ and $u$ values to probability density functions for each combination of sector, gas, region, and year.

Modelling NIR-2025 with Probability Density Functions

The probabilistic NIR model features probability density functions for every sector-gas-region-year combination, where region may be the entire country of Canada, or any one of the provinces or territories (including Nunavut, which was only created in 1999). All probability density functions in the probabilistic NIR model were are Symmetric Blended Log-Normal (SBLN) functions as defined in Appendix 2. The probability density functions for non-negative sectors were defined according to a few cases as follows. Non-negative sectors are ones that can't possibly have negative emissions, such as emissions from burning fossil fuels in homes, industrial burners, and vehicles. Recall that the SBLN distribution has four parameters: a mean ($\mu$), a roll-off ($\rho$), a relative uncertainty ($\sigma$) such that 1.0 means 100%, and a lower bound ($\beta$) that truncates the distribution to have no probability below a threshold.

Cases for non-negative sectors ($s$)	Probability Density Function
$m_{(s,g,r,t)}$ defined, non-tiny ($m_{(s,g,r,t)}\ge 1ktC{O}_{2}e$)	$\mathrm{S}\mathrm{B}\mathrm{L}\mathrm{N}(\mu =m_{(s,g,r,t)},\rho =\frac{\mu }{10},\sigma =max(u_{(s,g,r,t)},0.01),\beta =0)$
$m_{(s,g,r,t)}$ defined, tiny ($m_r<1ktC{O}_{2}e$)	$\mathrm{S}\mathrm{B}\mathrm{L}\mathrm{N}(\mu =0.5,\rho =0.01,\sigma =0.5,\beta =0)$
$m_{(s,g,r,t)}$ undefined, ${\overline{m}}_{(s,g,r)}$ defined	$\mathrm{S}\mathrm{B}\mathrm{L}\mathrm{N}(\mu =max({\overline{m}}_{(s,g,r)},1),\rho =\frac{\mu }{10},\sigma =0.7,\beta =0)$
$m_{(s,g,r,t)}$ undefined, ${\overline{m}}_{(s,g,r)}$ undefined	$\mathrm{S}\mathrm{B}\mathrm{L}\mathrm{N}(\mu =0.5,\rho =0.01,\sigma =0.5,\beta =0)$

The first case applies to national emissions and non-negligible provincial and territorial contributions. The first case distribution has a mean matching the data, and uncertainty approximately matching either the interpolated uncertainty or the uncertainty data when it's defined. The model uncertainty $\sigma =max(u,0.01)$ ensures that the model is always at least a little uncertain (at least 1%), to avoid division-by-zero special cases in computations using the model. This clipping only affected a few explicitly listed areas (e.g. ${\mathrm{N}}_2\mathrm{O}$ from Adipic Acid Production, ${\mathrm{C}\mathrm{H}}_4$ from Wetlands), and predominantly applied to implicitly-zero, unlisted sector-gas combinations such as ${\mathrm{S}\mathrm{F}}_6$ from fossil combustion sources. The roll-off $\rho =\frac{max(m,0.1)}{10}$ being so much less than the mean ensures that the shape of the distribution is almost exactly log-normal. The $\beta =0$ ensures that the distribution allocates no probability to the possibility of emissions less than 0 kt${\mathrm{C}\mathrm{O}}_2\mathrm{e}$.

The second case applies to sectors-gas-region-year combinations for which NIR-2025 lists the emissions as being nearly zero. For these combinations, the probabilistic NIR model uses a standard distribution that assigns almost all probability to very small emission amounts.

The third case introduces a new term symbol ${\overline{m}}_{(s,g,r)}$ to denote average regional mean over time, the average emission total for the sector-gas-region of interest across whatever years are explicitly listed in NIR-2025. In this case, probability density functions are associated with a standard large relative uncertainty, and ignore the interpolated uncertainty $u_{(s,g,r,t)}$.

The fourth and final case is a standard small estimated amount associated with sector-gas-region combinations that were never mentioned in the NIR. In effect, the model assumes their emissions are negligible.

Probability Density Functions For Possibly-Negative Emissions

There are four sectors that are sometimes negative in NIR-2025:

• Harvested Wood Products
• Forest Land
• Cropland
• Settlements

They must be treated differently from the non-negative sectors because their being estimated to be near zero doesn't imply that there is a low magnitude of uncertainty, and if they're estimated to be negative, then the uncertainty means they could be even more negative.

The National emissions totals for these sectors were modelled as SBLN distributions as well, as follows:

Cases for possibly-negative sectors ($s$)	Probability Density Function
$m_{(s,g,r,t)}$ defined	$\mathrm{S}\mathrm{B}\mathrm{L}\mathrm{N}(\mu =m_{(s,g,r,t)},\rho =100ktC{O}_{2}e,\sigma =max(|u_{(s,g,r,t)}|,0.01))$
$m_{(s,g,r,t)}$ undefined	$\mathrm{S}\mathrm{B}\mathrm{L}\mathrm{N}(\mu =0,\rho =100k,\sigma =max(u,1.0))$

In contrast to the formula for non-negative sectors, the SBLN formula in the first case here for potentially-negative sectors uses larger roll-off values so that the distributions are not always like log-normals. The roll-off was chosen so that whenever the magnitude $|m|$ of a national sector-gas emission estimate was on the order of 100 kt ${\mathrm{C}\mathrm{O}}_2\mathrm{e}$ or less, then the probability density function would look more like a normal distribution, and not so much like a log-normal. The value of 100kt${\mathrm{C}\mathrm{O}}_2\mathrm{e}$ was chosen to be large enough to be a conservative margin of error, yet small enough to not trigger re-ordering of provinces or territories in the visualizations below (which are sorted by a combination of estimated emission plus possible fraction of their uncertainty).

In the other case for potentially non-negative sectors, when the per-sector-gas-region-time mean is undefined, then the probability density was simply a relatively small approximately-normal distribution centered at 0.

For a visualization of these modelling choices, see Appendix 5, or the full NIR-2025 model visualization.

Appendix 4: A Method for Evaluating Predictive Models

My main motivation for introducing a probabilistic NIR model here on PlanZero was to use it as a reference for evaluating predictive emission models. A predictive emission model is a model that includes probability density functions for times into the future. It is easy to make a model that makes predictions of future emissions. It is important to establish first how such models can be evaluated, when the results they predict are still necessarily in the future.

The standard strategy for evaluating predictive models is to assume that the patterns of the past will continue. So for example, we might say: We have a method M for fitting a predictive model. We believe that NIR-2024, NIR-2025, NIR-2026, and NIR-2027 are all similar;

1. When we applied M to NIR-2024 it predicted something like NIR-2025, but was off by amount $l_{2025}$.
2. When we applied M to NIR-2025 it predicted somethign like NIR-2026, but was off by amount $l_{2026}$.
3. When we applied M to NIR-2026 it predicted something like NIR-2027; we don't know how much it's off because NIR-2027 hasn't been published yet, but since they're all kind of similar, we expect it to be off by an amount similar to $l_{2025}$ and $l_{2026}$.

This section outlines a recipe based on the probabilistic NIR for making such statements about future methods in PlanZero. The key to the recipe is a function that says how much predictions are "off" in a useful way. We'll take it as given that "predicting something like NIR-2025" means predicting a set of probability density functions for each sector-gas-region-time as are defined in NIR-2025. The question that then remains, is to say how "off" is one set of probability density functions (namely, our predictions) when compared with another (namely the probabilistic NIR introduced in this post)?

In assessing how predictions can be wrong, there are both qualitative and quantitative aspects. This Appendix is about quantitative aspects. It shows there's at least one plausibly-good numerical metric to assess goodness of fit between a probabilistic NIR model and predictions thereof. I believe that a prediction that scores lower on this metric (closer to zero) is probably going to make better predictions than one that scores higher, and PlanZero can be helpful in framing the search for good predictive models as the search for models with low scores on this metric.

A Metric for Evaluating NIR Predictions

A model-scoring metric is a mathematical formula that takes as input a reference model ($M^{(ref)}$) and a predictive model ($M^{(pred)}$) and turns them into a number such that the number being low (near zero, for example) means the prediction is good, and the number being higher means the prediction is not so good. The remainder of this section develop a model-scoring metric for NIR prediction. The first bit of math we need for developing that metric is something called an expected value of a function according to a probability distribution.

$${\mathbb{E}}_{\mathbb{P}}[f]={\int }_{x}f(x)P(x)dx$$

The expected value of a function of an unknown variable is an integral (a huge sum over tiny little intervals of x) of the function being applied to every possible value the variable could take, weighted by how probable it is according to the probability density function $P$. For $f(x)=x$ this is just the average value of the variable according to distribution $P$. Model evaluation metrics are usually an expectation over some distribution.

The next bit of math we'll use to define a first model evaluation metric is a debatable choice, but here it is, the so-called KL-divergence, written as a function of two probability distributions $P$ and $Q$ (suppose $P$ is one of the distributions in $M^{(ref)}$ and Q is the corresponding predicted-distribution over the same emission, but from $M^{(pred)}$):

$$\mathrm{K}\mathrm{L}(P\Vert Q)={\mathbb{E}}_{\mathbb{P}}\left[log\frac{P}{Q}\right]={\int }_{x}log\left(\frac{P(x)}{Q(x)}\right)P(x)dx$$

The KL divergence between two probability distributions is zero if they're identical, and greater-than-zero if they're not identical. Typically if Q is a bit different, the KL divergence is small, and as Q becomes more-different, the KL divergence increases faster and faster. With the KL divergence, if there are any values of the unknown variable that are possible according to P and nearly-or-actually impossible according to Q, then the divergence becomes enormous and then ultimately undefined. For our application, it means that if P is a distribution from a probabilistic NIR model, and KL is used to rank various Qs as approximations, the ranking will favour Qs that err on the side of caution over Qs whose predictions are overly-confident, highly-concentrated distributions.

I said that the KL divergence was a debatable choice because there are lots measures of distributional similarity. This course note PDF from CS6742 at Cornell lists a few, and the Wikipedia page on scoring rules provides an overview of alternatives and considerations. If users of NIR predictions have particular needs or considerations, then those needs or considerations should inform how they rank predictive models. A metric based on KL-divergence is a starting point, and illustrates the general pattern of using a reference model in a metric. I've created a GitHub issue as a reminder to review alternative metrics for model selection.

By incorporating a couple of expectations and a sum over sector-gas-region-time combinations of relevance, the following formula expresses what I think is a baseline, first model comparison metric:

$${ℒ}_T(M^{(ref)},M^{(pred)})=\frac{{\sum }_{s,g,r,t:|m_{(s,g,t)}|>1ktC{O}_{2}e,t>T}\mathrm{a}\mathrm{b}\mathrm{s}({𝔼}_{P_{(s,g,r,t)}}[x])\mathrm{K}\mathrm{L}(P_{(s,g,r,t)}\Vert Q_{(s,g,r,t)})}{{\sum }_{s,g,r,t:|m_{(s,g,t)}|>1ktC{O}_{2}e,t>T}\mathrm{a}\mathrm{b}\mathrm{s}({𝔼}_{P_{(s,g,r,t)}}[x])}$$

This metric compares a reference model $M^{(ref)}$ comprising reference distributions $P_{(s,g,r,t)}$ to a predictive model $M^{(pred)}$ comprising predicted distributions $Q_{(s,g,r,t)}$. It's a weighted sum over sector-gas-region-time combinations meeting two criteria: (a) that they aren't irrelevant as defined by e.g. $|m_{(s,g,t)}|>1ktC{O}_{2}e$ (the national total for that sector-gas-year wasn't approximately zero); and (b) that they're predictions, i.e. about times $t>T$ after some threshold year $T$. For each relevant, predicted year, the metric sums up the KL-divergence between the reference and predicted distributions, as weighted by the magnitude of emissions impact. The comparison metric can only be zero for perfectly-matching models, harshly penalizes overly-confident predictions, and weights errors in distribution-matching by how much the distributions matter to the overall sum of national emissions. The denominator decouples the scale of the function from the scale of the emissions contributions. There's some oddness in summing over the national totals (region being Canada) alongside the regions being provinces and territories, but I don't think it's a problem. Future posts on PlanZero will introduce predictive models, and this metric, or others like it, can, I hope, serve as a reliable guide for modelling effort to deliver iteratively better models.

Coming back to the annecdotal illustration from the beginning of the Appendix, we might say: We have a method M for fitting a predictive model. We believe that NIR-2024, NIR-2025, NIR-2026, and NIR-2027 are all similar;

1. ${ℒ}_{2024}(M_{2025}^{(ref)},M_{2024,2025}^{(pred)})$ was $l_{2025}$.
2. ${ℒ}_{2025}(M_{2026}^{(ref)},M_{2025,2026}^{(pred)})$ was $l_{2026}$.
3. We've estimated $M_{2026,2027}^{(pred)}$ and we expect $l_{2027}$ to be similar to $l_{2025}$ and $l_{2026}$.

The probabilistic NIR model developed in this post is $M_{2025}^{(ref)}$. Additional work will be required to develop probabilistic models for past and future NIRs (e.g. $M_{2024}^{(ref)}$ and $M_{2026}^{(ref)}$) and the predictive models (e.g. $M_{2025,2026}^{(pred)}$).

Appendix 5: Visualizing Probabilistic NIR Models

The probabilistic NIR developed in this post can be visualized be visualized in a set of graphics that can now be found in the "Models" tab of the PlanZero navigation bar. There are two types of graphic. The main graphic, featured above in the Overview, and reproduced below provides an overview of national emissions time-series for all 71 sectors. The second type of graphic (provided below in two variants) focuses on a single sector. The example below is about stationary combustion associated with public electricity and heat. The two variants show the contributions from each region in terms of ${\mathrm{C}\mathrm{O}}_2\mathrm{e}$, and then of just one greenhouse gas (${\mathrm{C}\mathrm{O}}_2$). The per-sector visuals do not include gases with no contribution to the sector, but otherwise all of the sector-region-gas-year combinations are visualized, as well as the ${\mathrm{C}\mathrm{O}}_2\mathrm{e}$ totals.