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## Introduction

Protein glycosylation directly affects the physical and biochemical properties of proteins in eukaryotic systems1. According to glycoproteomic analyses, over 1% of the human genome encodes glycosylation-related enzymes and more than 50% of human proteins are glycosylated2. Glycoproteins carry structurally diverse oligosaccharides, called glycans, that are involved at the interface of protein-biomolecular interactions and thus determine protein stability, selectivity, and activity. The significance of protein glycosylation to biological systems has been exemplified by several diseases associated with various cancers and the immune system3,4. For example, patients with rheumatoid arthritis were found to have an increased galactosylation level in their serum immunoglobulin G (IgG), though the mechanism remains elusive5. Unsurprisingly, it follows that insights into the structure and function of glycans have yielded a profound impact on the development of therapeutic glycoproteins6. Manipulating glycan structures present an effective strategy to improve their efficacy and safety by modulating immunological responses, circulatory half-life, and effector functions7,8. Thus, glycoengineering represents a versatile tool and a great opportunity to create better medicines. To achieve this goal, technologies that enable the control of protein glycosylation profiles are essential.

However, tools to access the diverse array of glycan structures displayed in nature remain scarce, and methods that provide a high yield of the desired glycoforms have proven to be a still greater challenge to develop despite decades of study9,10. Through synthetic and chemoenzymatic approaches, various glycoforms have been accessed11,12,13. These structurally defined glycans can be installed onto glycoproteins through endoglycosidase and glycosynthase activities14,15. While this approach has advanced our ability to control protein glycosylation, the preparation of synthetic glycans becomes increasingly difficult as the number of saccharide units increases. As a result, the installation of synthetic glycans is not practical for many applications. On the other hand, genetic engineering has been applied for controlled glycan biosynthesis by either knocking out or introducing certain glycoengineering enzymes in the host cells16. This strategy enables in vivo glycan remodeling and has been demonstrated in nonhuman cell lines17. However, the optimization of this strategy has been impeded by the complexity of engineering glycosylation pathways. Also, microheterogeneity is often generated during glycan formation, which, although it is comparable to the natural phenomenon, does not provide exquisite control over the molecular structure18.

In recent decades, our understanding of the in vitro activity of glycoengineering enzymes is growing rapidly19,20,21,22. Some of the enzymes can even function on intact glycoproteins, which opens a new window for glycan remodeling23,24. A remarkable example comes from the use of endoglycosidase S (Endo S) and its mutants to replace native IgG glycans with synthetic ones25,26. To further leverage the use of more glycoengineering enzymes, three primary challenges need to be addressed. First, characterization of enzyme activities on intact glycoproteins is required21,27. A comprehensive understanding of their activity, selectivity, and stability would allow researchers to design and execute glycan remodeling enzymatically. Second, preserving the integrity and functions of the substrates after the enzymatic reactions is critical, especially for therapeutic glycoproteins. Protocols with high biocompatibility are thus required. Third, to construct complex glycan structures, successive reactions using different enzymes are needed. These enzymes might require very different working conditions, such as pH and temperature. Therefore, one would need to repeat the buffer swapping and product purification processes between the enzymatic reactions, which is highly labor-intensive and time-consuming. Together, to address these needs, platforms that enable efficient, successive enzymatic glycan remodeling with high biocompatibility to the substrates are in great demand.

Inspired by solid-phase peptide synthesis (SPPS), herein, we introduce solid-phase glycan remodeling (SPGR) where enzymatic reactions are carried out on the substrates immobilized on resins28. This approach enables efficient reaction swapping, substrate purification, and the recovery of both products and engineering enzymes. We use human IgG as the substrate in this study because it is a major class of glycoproteins that have been applied in therapeutic development6,29. We quantitatively examined more than 30 glycan engineering enzymes for their activities on intact IgG immobilized on resins and then applied them in SPGR. This method has allowed us to harmonize IgG glycans into ten different glycoforms, including noncanonical structures, in 48 h with an average conversion ratio of over 95%. Physical and biochemical analyses indicated that the SPGR-engineered IgGs preserved integrity and functionality, suggesting that SPGR has high biocompatibility to the substrates.

## Results

### The design of solid-phase glycan remodeling (SPGR) for IgG glycoengineering

Our strategy to achieve efficient, successive glycan remodeling is immobilizing IgG onto protein A resins and then executing enzymatic reactions heterogeneously (Fig. 1a, Supplementary Note 1). This enables product purification by filtration, greatly speeding up multi-step reactions. We use empty SPE (solid-phase extraction) columns with standard Luer fittings as the SPGR reaction vessels. The Luer fittings can be connected to either syringes or vacuum manifolds to control the flow speed during washing processes. A frit is inserted into the bottom of the column for trapping the solid supports. SPGR processes include 5 steps: 1) resin loading; 2) IgG immobilization; 3) washing and conditioning; 4) enzymatic glycan remodeling, and; 5) washing and elution. The third and fourth steps are repeated to swap the reaction buffer and enzymes when running multi-step glycan remodeling. The substrate IgG remains immobilized on protein A resin until all the glycan remodeling steps are complete. To characterize the glycoforms of the product IgGs, PNGase F was used to isolate the IgG glycan, followed by fluorophore labeling, solid-phase extraction (SPE) purification, and then LC-MS analyses.

To identify capable glycoengineering enzymes for SPGR, we quantitatively analyzed the activity of 34 candidates, including exoglycosidases, endoglycosidases, and glycosyltransferases (Fig. 1b, Table S1). Each enzyme was incubated with immobilized IgG for 1 or 24 h. The enzyme activity―indicated by the consumption of substrate glycan species―was then quantified via chromatographic analysis. The candidates with the highest activity in each enzyme class were selected for SPGR applications and their working conditions were further optimized (Table 1, Table S2, Fig. S1-11). Also, we defined CR50 to be the substrate-to-enzyme molar ratio (in 1:XX format) that leads to 50% substrate conversion into the products in one hour using SPGR. This value allows us to estimate how much enzyme is required for SPGR reactions when the amount of substrate varies. It also provides the information about the reaction efficiency between different enzymes: the smaller the CR50 value, the more efficient the reaction is.

### Trimming IgG glycans with glycosidases

IgGs have two highly conserved glycosylation sites on the crystallizable region (Fc) at Asn 297 where more than 20 complex-type glycoforms have been found with the majority in bi-antennary structures (Fig. 1c)30. IgG glycans play essential roles in Fc receptor-mediated activities, such as antibody-dependent cellular cytotoxicity (ADCC)31. About 20% of IgG glycans contain terminal sialic acids through α2-6 linkages. These sialylated glycans have been known to confer anti-inflammatory activity32. Similarly, IgG galactosylation modulates inflammatory properties and about 70% of the IgG glycans contain terminal β1-4 galactoses3,5. The galactosylation level is also known to influence the clearance rate of glycoproteins in serum, mediated by asialoglycoprotein receptors, resulting in a direct impact on their pharmacokinetic properties33,34. Compared to sialic acid and galactose, our understanding of N-Acetylglucosamine (GlcNAc)’s impacts on IgG is more limited. GlcNAc exists in all N-glycans and plays decisive roles in glycan biosynthesis pathways (Fig. 1d). Extended from the chitobiose core, GlcNAc glycosidic linkage serves as a watershed that determines the subclasses of N-glycans: complex-type, high-mannose, and hybrid-type N-glycans. Complex-type glycans can further branch into bisecting, bi-antennary, tri-antennary, and tetra-antennary glycans, and so on35. About 90% of human IgG glycans are bi-antennary while the rest of them have bisecting structures30.

To control IgG glycoforms, we first aimed to harmonize them into the core saccharides by removing terminal sialic acid, galactose, and then GlcNAc (also see Supplementary Note 2.) Neuraminidase (Neu, or sialidase) is a class of enzyme that cleaves the glycosidic linkages of sialic acids. Our screening showed that Neu from Clostridium perfringens has the highest activity on immobilized IgG with a CR50 of 1:0.01. This enzyme has a broad substrate spectrum and can function on all the IgG glycoforms containing terminal sialic acid (Fig. 2, Fig. S1). Next, galactosidase (Gal) from Streptococcus pneumoniae showed the highest activity in our screening (CR50 = 1:0.047) for removing galactose (Fig. S2). It functions on all the IgG glycoforms containing terminal galactoses with an optimal temperature at 37 °C. To trim off GlcNAc, N-Acetylglucosaminidase (GlcNAcase) from S. pneumoniae showed the highest activity (CR50 = 1:0.004, Fig. S3). It has low glycosidic linkage selectivity and can trim terminal GlcNAc extended from the chitobiose core. Sequential treatments using these three enzymes leads to IgG glycan harmonization into (F)M3 structures (Fig. 3a).

Fucose on IgG glycan chitobiose core has been known to modulate IgG binding affinity to Fc receptors36. Defucosylated IgG has been reported to have an over 50-fold increase in ADCC activity37. As such a strong regulator, controlling the level of IgG core fucose has become an attractive strategy for improving the efficacy of IgG-based drugs. Over 90% of the human serum IgG glycans are fucosylated30. To identify the enzymes that can trim fucose from intact IgGs in their native confirmations, we tested seven fucosidases. Unfortunately, none of them showed an acceptable activity (Table S1). Huang et al. has reported that fucosidases only function on intact IgG when IgG glycans are trimmed down to the GlcNAc-fucose disaccharides, which indicates a strong steric interference between the enzyme and the glycan substrates25. Inspired by their works, we tested the fucosidase panel with glycoengineered IgG bearing (F)M3 glycans. The enzyme from Candidatus omnitrophica showed significantly improved activity on this group of substrates (Fig. S4, Supplementary Note 3). A 20% conversion was achieved in a 3-day reaction. The conversion ratio was further increased to 65% if non-immobilized substrates were used.

### Building IgG glycans with glycosyltransferases

Glycosyltransferases catalyze the transfer of saccharide(s) from activated sugar phosphates, the glycosyl donors, to glycosyl acceptor molecules, such as glycoproteins38. Sialyltransferase (SialylT) from Homo sapiens exhibited the highest activity in our screening for installing sialic acid through α2-6 linkage to the IgG with terminal galactose. This enzyme has a CR50 of 1:0.152 and apparent substrate selectivity, as shown in Fig. 2 and Fig. S5. Di-galactosylated glycan (FA2G2) and mono-galactosylated glycan with galactose at the α1-3 arm (FA2[3]G1) were completely transformed after a 16-hours reaction; while mono-galactosylated glycans at the α1-6 arm (FA2[6]G1) showed only minimal sialylation. The selective sialylation observed here agreed with previous reports and was likely caused by the folded conformation that the Fc region adopts when the galactose on the α1-6 arm is present39,40. Besides, we also observed a decreased enzyme activity when the (F)A2G2 glycans were mono-sialylated (Fig. S5).

To install galactose on IgG glycans, we selected the galactosyltransferase (GalT) from Homo sapiens (Fig. 2, S6)41. This enzyme catalyzed the transfer of galactose from Uridine 5’-diphosphogalactose (UDP-Gal) to IgG glycans with terminal GlcNAc. It has a CR50 of 1:0.07 and a broad spectrum of substrate specificity that enables the transformation of all the non- and mono-galactosylated IgG glycans into bi- galactosylated forms.

The addition of GlcNAc to the chitobiose core is relatively complicated because this process involves a series of N-Acetylglucosaminyltransferases (GnT) with various substrate specificities (Fig. 1d)42,43. We investigated the activity of five human GnTs that are responsible for complex N-glycan synthesis and obtained positive results from GnT-I, III, and V. GnT-I (MGAT1) initiates the formation of the complex-type and hybrid N-linked glycans by installing an β1-2 GlcNAc to the α1-3 mannose (Fig. S7)44. We observed a good CR50 of 1:0.038 in our activity screening, calculated based on (F)M3 glycan consumption. This enzyme likely possesses glycosidase activity as well because the conversion ratio reached a plateau of ~85% (Fig. S8) in all the conditions we tested. Whether or not the glycosidase activity can be repressed through genetic engineering to reach full conversion remains to be investigated. Human GnT-III (MGAT3) serves to install the bisecting GlcNAc to the β1-4 mannose through a β1-4 linkage45. A higher level of bisecting GlcNAc on IgG results in enhanced ADCC activity and immune cells effector functions46. Reactions using serum IgG as the substrate suggested that human GnT-III can function on IgG glycoforms containing at least one terminal GlcNAc (Fig. 2). However, it showed much higher activity on glycans with two terminal GlcNAc residues (Fig. S9).

Tri- and tetra-antennary N-glycans are not typically reported on native human serum IgG, and were not observed in our studies. Human GnT-V (MGAT5) is reported to add the secondary GlcNAc to the α1-6 mannose through β1-6 linkage and leads to the formation of tri-/tetra-antennary glycoforms47. We observed GnT-V activity after a 24-hours reaction with intact IgG, revealed by the formation of tri-antennary species. (Fig. S10). The activity of GnT-V on intact IgG is low but potentially can be improved through genetic control or evolution for future applications.

Finally, mammalian α1,6-fucosyltransferase (FucT) catalyzes the transfer of a fucose residue from GDP-fucose, the donor substrate, to the reducing-end terminal GlcNAc residue through an α1,6-linkage48,49. The activity of FucT on intact IgG was observed but the conversion ratio was low, likely due to the strong steric hindrance on the substrate. We further boosted the FucT reaction by increasing both enzyme concentration and incubation time. The results revealed high substrate selectivity of the enzyme, as indicated by the full consumption of A2 and A2G1 glycoforms (Fig. S11). Together, with controlled glycoforms as the substrate, modulating the level of core fucose using FucT and fucosidases on intact IgG is feasible.

### Reconstructing a harmonized glycosylation profile

It was described how a Neu reaction followed by Gal treatment turned IgG glycans into GlcNAc-terminating glycoforms (Fig. 3a). We can further run a GlcNAcase reaction to trim the glycans into the core structures terminating with mannose. The whole sequence was performed in 24 h without isolating IgG between steps. Similarly, remodeling using Neu and then GalT generated glycans with terminal galactose (Fig. S12); while a GalT reaction followed by SialylT resulted in mono- and bi-sialylated species (Fig. S12). The demonstrated SPGR routes in this work are summarized in Fig. 4.

IgG bearing (F)M3 glycans can also serve as starting materials for rebuilding non-canonical glycoforms. Starting with the (F)M3 glycans, we applied GnT-I, GalT, and then SialylT reactions to construct a series of mono-antennary species (Fig. 3b). Mono-antennary glycans are rare in nature and their effects on protein biology remain elusive. Whether they can be utilized for regulating the interactions between IgG and Fc receptors is a great research topic of interest. In addition to mono-antennary species, we also thought to increase the population of bisecting glycans on IgG. We trimmed off terminal sialic acids and galactose from IgG and then introduced GnT-III to the resulting (F)A2 glycans. After overnight incubation at 5 μM, we reached a full conversion of IgG glycans into the bisecting forms (Fig. 3c). These bisecting glycans can also be further galactosylated or sialylated using SPGR. Compared to mono-antennary species, both GalT and SialylT showed slightly decreased activity on bisecting glycans. Therefore, a higher enzyme concentration or longer incubation time is required to achieve full conversion.

To investigate the yield of SPGR reactions, we carefully determined the amount of the IgG substrate immobilized to protein A resin before the reactions, as well as the recovered product IgG after the reactions. Single-step reactions had a recovery ratio ranging from 78% to 85% (Fig. S13). We found that enzyme species, pH value, and buffer cation composition had no significant impact on the recovery ratio; while the use of high reaction temperature caused a slight decrease. Also, a small decrease (1–2%) of product recovery was observed with each SPGR step added. For example, 3-steps reactions for building the FM3 glycoform showed 76% recovery; and 5-steps reactions for building FA2BG2S2 glycoform had 74% recovery. In comparison to the SPGR method, a 5-steps solution-phase reaction with protein A purification after each step gave us 38.1% product recovery, which equals to 82.4% recovery in each step on average. Together, our data suggested that SPGR gives a consistent yield (product recovery) at about 80%. It presents an efficient strategy to transform IgG glycans into several mono/bi-antennary glycoforms and bisecting glycoforms.

### SPGR is biocompatible

To examine the biocompatibility of SPGR to the substrates, we analyzed the physical properties of the glycoengineered IgGs prepared above. Size analyses using dynamic light scattering (DLS) showed no significant difference between native serum IgG and SPGR-engineered IgGs, suggesting that there was no denaturing and/or aggregation occurred during the remodeling processes (Fig. 5a). On the other hand, slight changes in their melting temperature (Tm) and aggregation temperature (Tagg) were observed (Fig. 5b, c, Table S3). We reasoned these changes were attributed to the altered structure of IgG. It has been known that the Asn297 glycans play roles in maintaining the conformation and stability of the Fc region through intra-molecular interactions50,51. Decreased Tm and Tagg values were observed in IgGs terminating with mannose (1) or GlcNAc (2,5,8), the glycoforms showing reduced intra-molecular interactions50. Similarly, our mono-antennary glycans (5-7) had lower Tm values compared to others. These glycoforms lack the α1-6 arm which is important for forming intra-molecular interaction within the Fc region39. As expected, removing the bulk glycan structures using Endo S led to a dramatic drop in Tm and Tagg.

To further exam the integrity of SPGR-engineered IgGs, we measured their binding ability to Fc receptors using a competition assay. The FRET-based assay (Förster resonance energy transfer) comprises FRET-donor and acceptor beads that are coated with unmodified IgG and Fc gamma receptor I (FcγR-I, or CD64), respectively, which results in FRET signal upon IgG-FcγR-I binding. When SPGR-engineering IgGs are introduced to the reaction, if they preserve FcγR-I binding ability, the FRET interaction is interrupted and thus gives a reduced signal. We observed a signal reduction of 60% in all the glycoengineered IgGs at the concentration of 0.1 μg/ml; while complete inhibitions were reached at about 10 μg/ml (Fig. 5d, Fig. S14). The result suggested that SPGR does not disrupt IgG integrity and its binding ability to Fc receptors. In addition, IgGs with bisecting glycoforms (8-10) showed decreased EC50 values in this assay, indicating an enhanced binding affinity to FcγR-I (Table S3). Previous studies have reported that an increased level of bisecting glycoforms in mouse IgG1 results in enhanced ADCC activity, possibly owing to improved FcγR-III binding46. How the bisecting GlcNAc enhances FcγR-I and III bindings, and whether or not through the same mechanisms, remain elusive but intriguing. Such a glycan-mediated regulatory effect could provide a useful handle to fine-tune the immunogenicity of IgG-based drugs through glycoengineering.

## Discussions

For decades there has been a clear demand for glycoengineering tools that enable scholars to explore the role(s) of glycan structures on the function and form of glycoconjugates. SPGR presents a straightforward strategy for controlling glycan structures with several advantages: 1) it is a biocompatible approach with minimal disruption to the protein substrates; 2) it circumvents the need to prepare synthetic glycans, which can be cumbersome; 3) tight control of glycoforms is achievable with the use of different enzyme combinations and; 4) the procedures are user-friendly and can be readily automated, greatly reducing the cost for many applications. Moreover, the idea of executing sequential enzymatic remodeling on immobilized proteins can potentially be extended to most existing biocatalytic cascade reactions involving different classes of enzymes and substrates52.

The applicability of SPGR is largely determined by the diversity of glycoforms that we can build as well as the efficiency of the enzymatic reactions. In this work, we have reported several enzymes that have outstanding performance in SPGR reactions. However, there are still reactions of interest having low efficiency, such as fucosylation reactions. This could be addressed by the identification of enzymes with higher activity, for example, by screening more glycoengineering enzymes that have not been well-characterized, or by evolving the current SPGR enzymes toward higher reaction activity through protein engineering technologies53,54. In addition, there remains an opportunity to explore alternative immobilization strategies in future experiments. IgG immobilization using protein A, a 47kD protein, likely limited the enzyme efficiency by creating a strong steric hindrance. (Supplementary Note 4) Methods using oligopeptides, such as a his-tag, presumably have a lower steric effect and can enable higher enzyme activities. Since immobilization has been commonly employed for protein purification in pharmaceutical manufacturing processes, SPGR can conceivably be inserted into modern protein production processes as a “glycan modification module” to provide pure, glycoengineered proteins.

A particularly disruptive application of SPGR would be to humanize glycoforms on therapeutic proteins produced from non-human cell lines. Chinese hamster ovary (CHO) is commonly used for therapeutic protein production because they generate human-like post-translational modifications55. However, nonhuman glycoforms still exist in the cell line and should be removed to reduce the potential immune response in patients56. SPGR can be employed to humanize those glycans during the production process. From a different perspective, protein production in mammalian hosts is costly because of its long fermentation time and liability of virus infections. To address this issue, yeast has been employed as an alternative host for the large-scale expression of therapeutic proteins. Glycoproteins expressed from yeast contain high-mannose N-glycans which confer a short half-life in vivo and thereby compromise the efficacy of most therapeutic proteins7. Therefore, gene engineered strains are constructed for producing human-mimicking glycan patterns57. To date, a couple of simple glycoforms have been achieved in yeast and they have provided the desired scaffold (e.g. M5 glycan) for downstream glycan remodeling in vitro17. With the combination of SPGR, large-scale production of therapeutic proteins in yeast with controlled glycan structures is possible.

The awareness of public health has been raised significantly in the past months, intensifying the demand for developing better biologic medicine. Glycoproteins have proven their unignorable values in therapeutic and vaccine development, which stresses the urgent need for exquisite control of glycosylation profile for improved safety and efficacy. SPGR presents an efficient, user-friendly method for glycoengineering. It enables the control of glycan structures with various glycoforms and, presumably, on diverse glycoproteins. We believe that SPGR will expand our strategies for glycoengineering and greatly accelerate protein glycosylation studies as well as their pharmaceutical applications.

## Methods

### Solid-phase glycan remodeling (SPGR)

(I) Resin loading: Empty SPE columns composed of an empty column body, a frit with 0.2 μm pores, and a lid were used as reaction vessels for SPGR reactions. The columns were mounted onto a 20-wells SPE vacuum manifold. 100 μl of protein A resin (wet resin) was transferred into each column, followed by conditioning with 0.8 ml protein A-IgG binding buffer twice. A vacuum system was connected to the manifold to control the flow rate. (II) IgG immobilization: 1 mg (unless other specified) human serum IgG was added to 0.5 ml protein A-IgG binding buffer, followed by gently shaking until all the powder was dissolved. The solution was then transferred to the SPE column containing protein A resins. To ensure good immobilization, the columns were dismounted from the manifold, capped with Luer fittings, and then incubated for 15 min at room temperature with gentle rotating. (III) Washing and conditioning: After the incubation, the columns were mounted to the manifold again, followed by washing with 0.8 ml protein A-IgG binding buffer (3 times) and then enzyme reaction buffer (2 times). After the conditioning step, the buffer was completely drained out from the columns. (IV) Enzyme reactions: Enzyme reaction solutions were prepared by mixing the desired amount of enzyme, reaction buffer, and saccharide donors (1 mM, for glycosyltransferase) to a final volume of 100 μl unless other specified. Please refer to Table 1 and Table S2 for the details of reaction conditions, including pH, concentration, and cation cofactors. Enzyme reactions were initiated by transferring the reaction solution to the SPE columns that contain immobilized IgG substrates. The columns were capped by Luer fitting, sealed with parafilm, and incubated at temperature-controlled shakers for a certain amount of time (Table S2). After the reaction, the enzyme solution was discarded (or recovered) and the columns were washed 6–8 times with 0.8 ml protein A-IgG binding buffer via the extraction manifold. The (III) and (IV) steps were repeated until all the glycan remodeling steps were done. (V) Elution of glycoengineered IgG: 250 μl protein A-IgG elution buffer was added to the reaction columns and incubated for 2 minutes at room temperature. After collecting the eluent, another 250 μl elution buffer was added to the column and the process was repeated. The two portions of collected eluent were combined into a 0.5 ml MWCO (molecular weight cut-off) tube and concentrated via centrifugation at 14,000 g for 5 minutes, followed by buffer exchange into 50 mM HEPES buffer (pH 8). After adjusting the concentration of the eluted IgG substrate to 3 mg/ml using NanoDrop, the IgG substrates were stored at 4 °C and were ready for analysis. IgG concentration was adjusted in this step in order to ensure the same amount of sample was charged to the downstream analyses. The IgG recovery ratio of each glycoengineering reaction using this method is approximately 80% (the yields could vary when different resin, buffer, or substrate IgG are used.) For multi-step SPGR reactions shown in Fig. 4, 10 mg human IgG was used to initiate the reactions with a final material recovery ranging from 75 to 80%.

### LC-MS Analysis of IgG glycans

This protocol is adapted from the Glycoworks manual provided by Waters. (I) Glycan isolation: 7.5 μl IgG substrate (3 mg/ml, prepared as described above), 6 μl RapiGest SF (50 mg/ml, in Glycoworks buffer), and 15.3 μl water were mixed in a 1.5 ml microtube. The mixture was then incubated at 90 °C for 5 min to denature the substrates. After the samples were cooled down to room temperature, 1.2 μl Rapid PNGase F was added to the tube, followed by another incubation at 50 °C for 10 mi. (II) Glycan labeling: After PNGase F digestion, 12 μl RapiFluor-MS solution (70 mg/ml, in DMF) was added to the solution. The mixture was gently vortexed and then incubated at room temperature for 20 min without any light exposure. (III) Glycan purification: After labeling, the samples were diluted with 360 μl acetonitrile (1:9 volume ratio). Oasis SPE μPlate from Waters (along with the use of μPlate extraction manifold) was employed for the 1st solid-phase extraction purification, and Discovery SPE from Sigma-Aldrich (along with the use of 20-wells SPE vacuum manifold) was used for the 2nd purification to ensure high signal-to-noise ratio: The SPE columns/μPlate were first washed by water (1 column volume) and then conditioned by water-acetonitrile solution (10:90 v/v, 1 column volume). The glycan samples (in 90% acetonitrile solution) were then charged to the column/μPlate, followed by washing with washing buffer (formic acid/water/acetonitrile 1:9:90 v/v/v, 2 column volume). The glycans were then eluted using 60 μl elution buffer (200 mM ammonium acetate in 5% acetonitrile). (IV) HILIC-MS analysis: The purified glycan samples were injected to UPLC equipped with ACQUITY BEH Glycan column (130 Å, 1.7 μm, 2.1 × 150 mm) tandem with IMQ-TOF MS for glycan profile analysis. Please refer to the literature published by Pucic et al. and Kristic et al. for detailed peak assignment30,58. The method provided by Waters for glycan chromatography was used in this work (Table 2).

### Conversion ratio quantification

The conversion ratio of each SPGR reaction was calculated based on the consumption of the substrate glycan species. UV absorption at 260 nm (RapiFluor-MS) from chromatography analysis was used for the quantification of glycan populations. We first normalized the chromatographic peak area of the substrate glycan species to the total glycan peak area (Eq. 1). This gives us the percentage of substrate glycan population. The reduction of the substrate glycan population after the reaction was then divided by the initial value to calculate the percentage of substrate conversion (Eq. 2). We assume that there is no glycan shedding off from IgG during the experiments (namely, no endoglycosidase activity). Please refer to Table S4 for detailed substrate species used in the calculation.

For endoglycosidase reactions, their activity was quantified by absolute glycan quantification using an internal standard. This is because endoglycosidases’ activity results in the reduction of all glycan signals in the chromatography analyses. A known amount of internal standard (Glycan quantitative standard, Waters) was added to the sample to measure the amount of IgG glycan before and after the reactions.

$${{{{{\rm{Substrate}}}}}}\; {{{{{\rm{glycan}}}}}}\; {{{{{\rm{population}}}}}}=\frac{\sum {{{{{{\rm{Peak}}}}}}\; {{{{{\rm{area}}}}}}}_{{{{{{\rm{substrate}}}}}}1,2,3\ldots {{{{{\rm{i}}}}}}}}{\sum {{{{{{\rm{Peak}}}}}}\; {{{{{\rm{area}}}}}}}_{{{{{{\rm{all}}}}}}\; {{{{{\rm{glycans}}}}}}}}={{{{{{\rm{R}}}}}}}_{{{{{{\rm{s}}}}}}}$$
(1)
$${{{{{\rm{Conversion}}}}}}\; {{{{{\rm{Ratio}}}}}}=\frac{{{{{{{\rm{R}}}}}}}_{{{{{{\rm{S}}}}}}({{{{{\rm{initial}}}}}})}-{{{{{{\rm{R}}}}}}}_{{{{{{\rm{S}}}}}}({{{{{\rm{final}}}}}})}}{{{{{{{\rm{R}}}}}}}_{{{{{{\rm{s}}}}}}({{{{{\rm{initial}}}}}})}}$$
(2)

### CR50 calculation

CR50 is defined as the substrate-to-enzyme molar ratio (in XX:XX format) that leads to 50% substrate species conversion into the products in 1 h at optimized working conditions (temperature, pH, cation) using SPGR. This value is determined based on dose-dependent experiments where the conversion ratio at different enzyme concentrations in 1 hour was tested. A sigmoidal curve fitting ([agonist] vs normalized response, GraphPad Prism, see the equation below) was applied to the data for calculating the enzyme concentration that gives 50% substrate conversion. The resulting enzyme concentration is divided by the substrate (IgG) concentration to give CR50.

$${{{{{\rm{Y}}}}}}=\frac{100{{{{{\rm{X}}}}}}}{{{{{{{\rm{CR}}}}}}}_{50}+{{{{{\rm{X}}}}}}},$$
(3)

where Y is normalized response from 0 to 100; X is the concentration of enzyme; X is equal to CR50 at Y = 50.

### Physical property characterization of SPGR-engineered IgGs

Dynamic light scattering (DLS), melting temperature (Tm) and aggregation temperature (Tagg) studies were executed using UNcle (Unchainedlabs). Glycoengineered IgGs from SPGR reactions were eluted from protein A resins, followed by buffer exchange into HEPES buffer as described above. The protein concentration was then adjusted to 1 mg/ml using NanoDrop. 9 μl of the purified IgG samples were injected into UNcle sample holders (5 replicates for each sample). DLS measurement was performed at 25 °C (4 acquisition, 5 seconds each). Static light scattering (SLS) for Tm and Tagg measurement was carried out from 25 °C to 90 °C with a temperature increase of 0.3 °C per minute.

Please note that all the tested IgG samples (1-10) contained a ~5% defucosylated population. The biantennary samples (2-4) had ~10% bisecting glycoforms. The mono-antennary samples (5-7) had ~10% (F)M3 glycans due to the reversible activity of GnT-I. Sialylated samples (4 & 10) possessed about 1:1 mono- and di-sialylated populations. Please refer to Fig. 3 and S12 for detailed glycans species and population distributions.

### Binding assays between glycoengineered IgGs and Fc gamma receptor 1

This protocol is adapted from the AlphaLISA human FCGR binding kit manual provided by PerkinElmer. Briefly, serial dilutions of SPGR-engineered IgGs with 1X HiBlock buffer were prepared with the highest concentration at 1 mg/ml and the lowest concentration at 0.1 μg/ml. 10 μL of each diluted IgG samples were mixed with 10 μL 4X human FcγR1 solution and 20 μL Donor/acceptor beads solution into a white 96-well plate. The plate was then sealed and incubated at 25 °C for 90 min without any light exposure. After the reaction, the fluorescence signal at 615 nm was determined using an EnVision Multimode Plate Reader (equipped with AlphaScreen module).

### Computational modeling of IgG-protein A complex

Protein A homology model was constructed using the Swiss-Model server59. PDB structure 5H7B, which has 79.5% sequence identity with Protein A, was used as a template to construct the homology model60. A visual inspection of Protein A model illustrated 4 distinct IgG binding domains. PDB structure 5U4Y was used as a template to identify the spatial positioning of the full-length Protein A and the IgG61. The template structure contains only the B-domain of the protein A molecule. The spatial position of the B-domain helped us overlay the full-length protein-A molecule and allowed us to identify steric hindrances between other protein-A domains and the IgG molecule. Structure overlay and the movie illustrating clashes between the molecules was generated using Pymol (Molecular Graphics System, Version 2.0, Schrödinger LLC).

### Plotting and graphic

Data plotting and curve fitting were done by using GraphPad Prism 8. Figures and cartoons were created by Adobe Illustrator.

### Statistics and Reproducibility

All the experiments in this work have been reproduced at least 2 times (N = 3) unless specified in the figure legends. Consistent results were obtained. Statistical analyses were performed using GraphPad Prism 8 with all the collected data points included.

### Reporting summary

Further information on research design is available in the Nature Research Reporting Summary linked to this article.


text:
References and implementation for safe/hybrid univariate Newton's method

There exist variants of Newton’s method with safeguards against non-convergence (escape bracket, not converging “fast enough”, etc, in which case they switch to a bracketing method like Brent’s or bisection). Various implementations I found differ in details, and I could not find a paper with convergence analysis.

1. Is there a reference that discusses these methods in detail?

2. Is there a Julia implementation?

For a while, anytime I needed to solve a new univariate problem I tried different univariate solvers and always found that Brent’s method (from Optim.jl) was performing the best. So now I just always reach for that.

I do remember seeing recently a post here about someone implementing the ITP method in a new package but I have not given it a try. This method would be more similar to what you described.

No, the ITP method is not really similar, as it does not use derivatives, and it is already implemented in Roots.jl (thanks @TheLateKronos!)

Just to clarify: I am not asking for recommendations on rootfinding methods.

The question is about a very specific family of methods, and is mainly motivated by curiosity. Two general recommendations I encounter are

1. get close to a root, then use Newton-Raphson for “polishing”,
2. try Newton-Raphson, fall back to some bracketing method if it “fails”

But in practice I find it hard to implement these in a robust and practical way, because the criteria for switching are quite fuzzy. This is a shame, since for some functions derivatives are readily available, so a bracketing method that uses derivatives would be useful.

1 Like

The Numerical Recipes book, chapter 9, covers this topic in detail.

If you are referring to rtsafe in Section 9.4 of the 3rd edition, then it doesn’t. It is one of the ad-hoc implementations floating around. All you get is the description

The hybrid algorithm takes a bisection step whenever Newton-Raphson would take the solution out of bounds, or whenever Newton-Raphson is not reducing the size of the brackets rapidly enough.

but a convergence analysis or a justification for the criteria is missing.

Again, it is very easy (and tempting) to make ad hoc modifications to existing algorithms. Unfortunately, these can easily backfire.

If we are talking about a continuous function and a bracket which is always at least cut in half, then we have a guarantee that the method converges by the intermediate value theorem. Taking the newton step when it makes the interval smaller than half it’s current size and the bisection step otherwise can only make convergence faster as the interval will always be smaller than or equal to half it’s initial size at every step.

Not sure what else you are looking for.

1 Like

I am not sure how that follows. In my copy of the book the step is taken if it does not go out of the bracket and

|2f(x_m)| < |x_2 - x_1||f'(x_m)|

where (x_1, x_2) is the original bracket, and x_m = (x_1 + x_2)/2.

The new value is x_m - f(x_m)/f'(x_m), which could be close to x_1 or x_2.

To make it a bit more concrete, consider x_1 = 0, x_2 = 1, f(0) = -1, f(1) = 2, f(0.5) = 1, f'(0.5) = a. The second condition for accepting the Newton step is

2 < |a|

while the resulting point is 0.5 - 2/a. Now let a = 4.1, so that x_3 \approx 0.01. If f(x_3) < 0, (x_3, x_2) will be the new bracket, and is about 99% of the old one.

Following the notation in the 3rd edition of the book, if I am not mistaken, the next point computed seems to be 0.378:

x1, x2 = 0.0, 1.0
fl, fh = -1.0, 2.0
xl, xh = x1, x2
rts = 0.5 * (x1 + x2)   # = 0.5
dxold = abs(x2-x1)
dx = dxold
f = 0.5
df = 4.1

(((rts-xh)*df-f)*((rts-xl)*df-f) > 0.0) || (abs(2.0*f) > abs(dxold*df))  # false

# so take the Newton step:
dxold = dx;
dx = f/df
temp = rts
rts -= dx      # new rts = 0.378

I vaguely remember W. Kahan giving a talk about how he got this idea into the solver (the solve button) for the HP35 programmable calculator. This was from the late 1970s and I do not know if he published it or not. I think the opening gambit was bisection and then the solver switched to Newton when the intervals got small.

This article by W. Kahan about the HP-34C solver seems to be based on the secant method.

2 Likes

That’s the one. Thanks for doing the error correction Newton → secant from my fading memory.

That is implemented in the default Order0 method of find_zero in Roots. There is also the possibility of other hybrid methods, switching to a bracketing method if one is identified.

2 Likes

Good point, thanks for working through the numbers (which I should have done before posting the example ).

Try f(0.5) = -0.5 and f´(0.5) = -(1 + \epsilon), which gets you arbitrarily close to 0, so the new bracket is (\approx 0, 1). Eg

f = -0.5
df = -1.1

gives rts = 0.04545...

I forget if the convergence analysis is in detail, but in the LithBoonkkampIJzermanBracket method of find_zero from Roots there is an implementation of this kind of algorithm and a reference to follow up on.

2 Likes

Thanks, the article (non-paywall version here) was extremely informative. This is the kind of answer I was looking for, and I am delighted to learn that it is already implemented in the excellent Roots.jl.

1 Like

format: plain


text:
How can I change the label background color in TabView?

When I go to the web page, https://reference.wolfram.com/language/ref/TabView.html, giving the description of TabView, the example displays as follows:

Note that the background for the label is blue. When I go to the same reference page in my Mathematica system, the example looks like the following:

The background color I get is black, making it impossible to read the label which is also black. How can I change this label background color? I have tried the following:

I am running Mathematica version 12.0.0.0 on Mac OS X Mojave version 10.14.6. The documentation says that TabView by default displays the labels in TabViewLabel Style, which typically uses the system button font. But buttons do not appear like this in other applications that I run on my Mac.

So how can I change the label background color or how can I change the TabViewLabel style?

Additional information: When I use other applications on my Mac that has tabs, the tabs appear correctly with selected tabs displaying with white letters on a blue background.

Example:

It is only in Mathematica that all my tabs have the selected tab displayed with black letters on a black background (so they are unreadable). I see this behavior everywhere, not just in programs I am writing. For example, I see this behavior when I look at help pages or when I look at the option inspector. So it must be some Mathematica setting that I have set wrong.

Anyone have any ideas?

• I suspect I might have to change some option in the option inspector, but I don't see any option to change label background colors. Jan 4 '20 at 15:55
• Have you changed any of the top set of default settings in System Preferences... > General? Jan 4 '20 at 23:27
• Also, are you using a stylesheet other than the Default stylesheet? Jan 4 '20 at 23:39
• @m_goldberg I have not set anything to black in general system preferences. Appearance is set to light and accent color is set to blue. Thanks for the idea. Jan 5 '20 at 0:29
• @m_goldberg I don't know much about stylesheets and perhaps I've accidentally set something I shouldn't have set. When I choose Format>Edit Stylesheet, it says my notebook is inheriting base definitions from stylesheet "Default.nb". Is there anything else I should check? When I choose Format>Stylesheet, I see a submenu. No menu items are checked. Should I select "Default"? Jan 5 '20 at 0:42

This will be an answer if it works on your system as it does on mine, and if you don't think its too much fussy code to bother with. Admittedly, it is more a work-around than a real solution. It does have the advantage of being fairly easy to modify to accommodate more elaborate tab view emulations.

With[{systemBlue = RGBColor[0., .5, 1., .75]},
With[
{colors = <|True -> {systemBlue, White}, False -> {White, Black}|>,
gray = GrayLevel[.6]},
btnLbl[i_, state_] :=
Module[{tabColor, textColor},
{tabColor, textColor} = colors[state[[i]]];
Graphics[
{EdgeForm[{AbsoluteThickness[.5], gray}], FaceForm[tabColor],
Rectangle[{0, 0}, {1, 1}],
Text[Style[i, textColor], {.5, .5}]},
ImageSize -> {20, 20}]];
DynamicModule[{val, vals, n, state},
vals = {a, b, c, d};
n = Length[vals];
state = ConstantArray[False, n];
state[[3]] = True;
Column[
{Dynamic @
Row[
Button[btnLbl[#, state],
state = ConstantArray[False, n];
state[[#]] = True,
Appearance -> None] & /@ Range[n],
"\[NegativeVeryThinSpace]"],
Dynamic @
Panel[Style[Pick[vals, state][[1]], 14], ImageSize -> 80]},
Center,
Spacings -> -.2],
TrackedSymbols :> {state},
SaveDefinitions -> True]]]

• Thanks for all the effort you have expended to solve my problem. It works and gives me a way to produce a TabView that looks like it is supposed to. But, as you say, it is a work-around. It would still be nice to figure out why the default TabView isn't working correctly on my system. I can now write programs that have nice-looking tab views, but when I look at the help system or the option inspector, I still see tabs that I cannot read because the selected tab shows black text on a black background. Jan 5 '20 at 20:28
• If someone gives me a program they wrote that looks nice on their machine, when I run it on my machine, any tabViews displayed by their program will display with black on black selections. I must have some global setting set wrong that is specifying that the background for selected tabs should be black rather than system blue. Any other thoughts? Jan 5 '20 at 20:31
• @StanleyRabinowitz. It's too bad you don't find this useful, but I suspected you might not. I understand your core concern is to getSetterBar (that's where the problem really is) to display as it should. I don't have an further ideas at this time except that you might contact Wolfram tech support if your Mathematica license includes tech support as some licenses do. Jan 5 '20 at 23:41

This is not an answer, but two code snippets to try in order to get more information on what is going on. Each snippet imitates a tab view by combining a setter bar with a panel. I would like you try them to see if one or both behave like the tab views you are getting. Please report your findings in comment below.

1. Should produce system blue background on selected tab.

DynamicModule[{j, vals, n},
vals = {a, b, c, d};
n = Length[vals];
Column[
{SetterBar[Dynamic[x], vals],
Dynamic[Panel[Style[x, 14], ImageSize -> 80]]},
Center,
Spacings -> 0]]

2. Should produce gray background on selected tab.

DynamicModule[{j, vals, n},
vals = {a, b, c, d};
n = Length[vals];
Column[
{SetterBar[Dynamic[x], vals, Appearance -> "Horizontal" -> {1, n}],
Dynamic[Panel[Style[x, 14], ImageSize -> 80]]},
Center,
Spacings -> 0]]

• snippet 1 gave me a black background on the selected tab. The background of the other tabs were white. snippet 2 gave me a gray background on the selected tab. The background of the other tabs were a lighter gray. Jan 5 '20 at 2:35
• @Thanks for the info. I conclude from your results that TabView is using SetterBar and that problem resides in SetterBar. Maybe I can do something now that I know where the problem really is. Jan 5 '20 at 3:41


text:
Notes On Oxidation Number - CBSE Class 11 Chemistry
In covalent bond formation the electrons are not transferred as in ionic bond formation, but partial transfer of electronic charge takes place, known as electron shift. The oxidation number method always assumes that there is a complete transfer of electrons from a less electronegative atom to a more electronegative atom. Oxidation number is defined as is the charge or the apparent charge that an atom in a compound or ion would have, if all of the electrons in its bonds belonged entirely to the more electronegative atom. To determine the oxidation number of an element in a compound or an ion, certain rules have been formulated. i)  The oxidation number of atoms of elements in their elementary states and allotropic forms is zero. Ex: Atom in H2, O2, Cl2, O3, P4 , S8, Na, Mg, Al has oxidation number zero. ii)  In neutral compounds the sum of oxidation number of all atoms is zero.  Ex: In H2O the sum of oxidation numbers of hydrogen and oxygen atoms is zero. iii) In case of the ions that are composed only of one atom the oxidation number is same as the charge on the ion. Ex: The oxidation number of Sodium ion is ‘+1’, Magnesium ion is ‘+2’, Ferric ion is ‘+3’, Chloride ion is ‘-1’ and Oxide ion is ‘-2’. But in complex and polyatomic ions, the algebraic sum of the oxidation numbers of all atoms in the ion is equal to the charge on the ion. Ex: In the Carbonate ion, the algebraic sum of the oxidation numbers of the three Oxygen atoms and one Carbon atom is ‘-2’. The oxidation number of Carbon is ‘+4’, and oxidation number of Oxygen is ‘-2’, so Charge = 4 + 3(-2) = 4 - 6 = -2 iv) In their compounds all alkali metals have oxidation number ‘+1’ and all alkaline earth metals have oxidation state ‘+2’.                            v) The oxidation number of Oxygen is -2 in most of its compounds except in two cases. a)  In Peroxides like hydrogen peroxide and barium peroxide, each oxygen atom is assigned an oxidation number ‘-1’. b) In Superoxides like potassium superoxide and rubidium superoxide each Oxygen atom is assigned an oxidation number ‘-1/2’. vi) Hydrogen has +1 oxidation number when it combines with non-metals. And in metal hydrides, its oxidation number is ‘-1’. vii)  Fluorine has an oxidation number of ‘-1’ in all its compounds. viii) The atoms of transition or d block elements exhibit several oxidation numbers. Highest oxidation number of an element = number of valence electrons

#### Summary

In covalent bond formation the electrons are not transferred as in ionic bond formation, but partial transfer of electronic charge takes place, known as electron shift. The oxidation number method always assumes that there is a complete transfer of electrons from a less electronegative atom to a more electronegative atom. Oxidation number is defined as is the charge or the apparent charge that an atom in a compound or ion would have, if all of the electrons in its bonds belonged entirely to the more electronegative atom. To determine the oxidation number of an element in a compound or an ion, certain rules have been formulated. i)  The oxidation number of atoms of elements in their elementary states and allotropic forms is zero. Ex: Atom in H2, O2, Cl2, O3, P4 , S8, Na, Mg, Al has oxidation number zero. ii)  In neutral compounds the sum of oxidation number of all atoms is zero.  Ex: In H2O the sum of oxidation numbers of hydrogen and oxygen atoms is zero. iii) In case of the ions that are composed only of one atom the oxidation number is same as the charge on the ion. Ex: The oxidation number of Sodium ion is ‘+1’, Magnesium ion is ‘+2’, Ferric ion is ‘+3’, Chloride ion is ‘-1’ and Oxide ion is ‘-2’. But in complex and polyatomic ions, the algebraic sum of the oxidation numbers of all atoms in the ion is equal to the charge on the ion. Ex: In the Carbonate ion, the algebraic sum of the oxidation numbers of the three Oxygen atoms and one Carbon atom is ‘-2’. The oxidation number of Carbon is ‘+4’, and oxidation number of Oxygen is ‘-2’, so Charge = 4 + 3(-2) = 4 - 6 = -2 iv) In their compounds all alkali metals have oxidation number ‘+1’ and all alkaline earth metals have oxidation state ‘+2’.                            v) The oxidation number of Oxygen is -2 in most of its compounds except in two cases. a)  In Peroxides like hydrogen peroxide and barium peroxide, each oxygen atom is assigned an oxidation number ‘-1’. b) In Superoxides like potassium superoxide and rubidium superoxide each Oxygen atom is assigned an oxidation number ‘-1/2’. vi) Hydrogen has +1 oxidation number when it combines with non-metals. And in metal hydrides, its oxidation number is ‘-1’. vii)  Fluorine has an oxidation number of ‘-1’ in all its compounds. viii) The atoms of transition or d block elements exhibit several oxidation numbers. Highest oxidation number of an element = number of valence electrons

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text:
Disclaimer: This is an example of a student written essay.

Any information contained within this essay is intended for educational purposes only. It should not be treated as authoritative or accurate when considering investments or other financial products.

# Canadian Commodity Prices and Exchange Rate Dynamics

 ✅ Paper Type: Free Essay ✅ Subject: Banking ✅ Wordcount: 7498 words ✅ Published: 23rd Sep 2019

Canadian Commodity Prices and Exchange Rate Dynamics: Evidence of Structural Change

1. Introduction

Canada is a small open economy and a large exporter of goods, services, and commodities. Coined as a “commodity currency” in 2003 by a paper written by Yu-chin Chen and Kenneth Rogoff, Canada has long held a comparative advantage in the distribution of natural resources and commodities. Due to the dependence of output growth in Canada on exports of natural resources and commodities to the world market, the Canadian dollar has long been considered to be representative of the state of production and exporting of commodities in Canada. Ultimately when the price of commodity exports rises, often due to exogenous shocks on both the demand and supply side in the global economy, the Canadian dollar appreciates as net exports increase and Canada becomes a more attractive destination for investment.

If you need assistance with writing your essay, our professional essay writing service is here to help!

It has not always been the case, however, that the exchange rate between the Canadian dollar and the U.S. dollar were closely related to commodity prices. According to the report “Intervention in the Foreign Exchange Market” by the Bank of Canada, before September 1998 the Bank practised a policy of “systematic” intervention in the foreign exchange market to peg the value of the Canadian dollar close to the U.S. dollar. (Bank of Canada, 1) There were periods of time when a rise or fall in the value of the Canadian dollar was not matched with a corresponding change in commodity prices. By using intervention in the foreign exchange market, the Bank tried to keep the value of our dollar close to par with the U.S. dollar. In pursuit of autonomous monetary policy at the expense of a degree of control over the value of the Canadian dollar, the bank has not systematically intervened in the foreign exchange market since before the policy change. (Bank of Canada, 2)

The objective of this paper is not to assign a normative value statement to whether inflation targeting or control over exchange rates is more desirable for an independent central bank. Instead, I aim to demonstrate that exogenous market forces have a stronger impact on the value of an economy’s currency – especially for a large exporter such as Canada. First, I will provide background on relevant literature and topics of interest in relation to my empirical analysis. Next, I will define my formal hypotheses, empirical methodology and data, along with a consideration to improve future research on this topic. Following up, I will demonstrate the results and interpretation of the empirical tests. Finally, I will look at a current monetary policy implication in relation to current commodity prices and the Bank of Canada’s projected overnight rate increase. From the analysis, we will see empirical evidence of a structural difference in the relationship between the USD/CAD exchange rate and commodity prices before and after the foreign exchange intervention policy change made by the Bank of Canada in September 1998.

2. Background & Literature

Initially, I was drawn to the topic of the relationship between prices of commodities and goods with exchange rates by the paper: “Exchange Rate Pass – Through to Consumer Prices: Theory and Recent Evidence” by Laurence Savoi-Chabot and Mikael Khan of the Bank of Canada. They demonstrate that exchange rate pass through, defined as the response in appreciation in prices of goods to a relative depreciation of an economy’s currency, has the largest effect in the Canadian energy sector as shown in the bar graph on the right. Generalizing to commodities, Savoi-Chabot and Khan make reference to the fact that Canada is a commodity currency, along with the relationship between commodities and inflation. Looking at changes in commodity prices’ impact on inflation and exchange rates rather than how exchange rates impact commodity prices, they note that when depreciation is caused by lower commodity prices, it will lower inflation because the lower levels of commodity prices affect commodity intensive consumer prices such as gasoline. (Savoi-Chabot and Khan, 3)

(Savoi-Chabot and Khan,  4)

Looking further into the dynamics of commodity prices and exchange rates,the paper: “Power of Commodity Prices Over Exchange Rate Dynamics” written by Nicholas Garmulewicz demonstrates the idea that commodity prices have an influence on the Canadian dollar relative to other currencies. It also initially provides reasoning for a structural change in dynamics between commodities and the Canadian dollar before and after the Bank of Canada changed their foreign exchange intervention policy from systematic to discretionary in September 1998. (Garmulewicz, 16) This paper begins by analyzing how commodity prices have an effect on the value of the CAD, and then switches to test whether the CAD contains information about the future prices of commodities.

Due to the fact that commodity prices and exchange rates are considered to be very difficult to forecast, the lens of this paper will conduct a type of analysis on the former topic, rather than the later. Before defining the formal hypotheses from which I will conduct my empirical analysis around, I would like to provide some essential background information on the Bank of Canada foreign exchange policy change, the Bank Commodity Price Index (BCPI), and history of the USD/CAD exchange rate with the BCPI.

The Bank of Canada has a broad, principle role from which all of the Bank’s activities stem from. As the nation’s central bank, it is responsible “to promote the economic and financial welfare in Canada,” through the four areas of monetary policy, the financial system, currency, and funds management. (Bank of Canada Act, 1) Seemingly a clear objective at first thought, this objective has is open for interpretation. Over time, economists have grappled with what the promotion of economic and financial welfare specifically entails. These debates around the guiding principal have influenced the activities of the bank since its establishment in 1935. At this point in time, it is commonly known that the bank tries to achieve 2% annual inflation as usually measured by the total consumer price index (CPI). The main tool used by the bank of Canada while conducting monetary policy to target 2% inflation is by setting the overnight rate. The overnight rate, as defined by the Bank, “is the interest rate at which major financial institutions borrow and lend one-day (or “overnight”) funds among themselves.” (Monetary Policy, 2018)However, targeting 2% inflation is a recent objective that has only been officially agreed upon since the September 1998. Before this point in time, the Bank was concerned with other objectives in order to meet its principal role of promoting economic well-being There are some main reasons behind this chosen target level of inflation. First, it provides the mechanism for participants in the economy to drive their consumption and investment activities while keeping the purchasing power of a dollar from distorting too rapidly through time. People are incentivized to purchase goods and make investments that satisfy their needs or appreciate in value because of inflation. The other reason for a 2% inflation target is that it gives the Bank of Canada room for monetary stimulation before hitting the zero lower bound of the overnight rate.

Ultimately, this shift in foreign exchange policy represents a move on the open economy trilemma. This trilemma is the theory that an open economy must choose between two of the following: free capital flow, exchange rate determination, and autonomous monetary policy. Before the change in policy, the Bank ultimately felt that the best way to achieve their principle role of promotion of economic welfare was through free capital flow and exchange rate determination. Since the Bank has not intervened in the foreign exchange market since before the policy change, autonomous monetary policy has been favoured over exchange rate determination. This now lends itself to the Bank being able to target inflation through the determination of the overnight rate and a floating exchange rate.

Bank Commodity Price Index (BCPI):

In order to track the relative prices of commodities relevant to activity in the Canadian economy, the Bank of Canada constructed the Bank of Canada Commodity Price Index (BCPI) which started in 1972Q1. This price index tracks the spot prices of twenty-six commodities mainly produced in Canada and sold on world markets. The components of the BCPI can be broken down into energy, metal and minerals, forestry, agriculture, and fisheries. In computing the index value, each component is assigned a weight using input-output data from Statistics Canada. A feature of these weights is that they are time varying. In particular, the energy component of the BCPI has increased in share since the Bank’s policy change. In 2018, 54% of the index is weighted by energy prices including the crude oils West Texas Intermediate, Western Canadian Select, and Brent – along with natural gas and coal. The other 46% is weighted by all the other sectors relevant in Canadian commodities. (Commodity Price Index, 2018)

History has not always shown a positive correlation between BCPI movement and the USD/CAD exchange rate. Using data from the Bank of Canada and the St. Louis Federal Reserve Economic Database, I have plotted the BCPI and the exchange rate from January 1972 to August 2018. It is clear to see that before the policy change was implemented in September 1998, there were examples of the BCPI and the exchange rate not moving in the same direction. This can be due to many different factors, with one of them being that the exchange rate does not contain information about the current growth of the BCPI index. For example, in the late 1980’s, we see that the Canadian dollar appreciated relative to the U.S. dollar even though the BCPI did not experience significant growth. “After touching a then-record low of US\$0.6913 on 4 February 1986, the dollar rebounded following a concerted strategy of aggressive intervention in the foreign exchange market, sharply higher interest rates, and the announcement of large foreign borrowings by the federal government.” (Powell, 1999) Moving into the 1990’s, the Bank of Canada lost the ability to significantly control the value of the Canadian dollar as the value had dropped to a new low of \$0.65 U.S., matched by a simultaneous drop in the BCPI index. Clearly, the exchange rate was now containing more information about the state of the economy, as an example through the BCPI index.

Moving into the era of discretionary foreign exchange intervention policy, the co-dependence of the growth in value of the Canadian dollar relative to the U.S. dollar along with growth of the BCPI index is visually compelling through the above graph. One of the most evident examples of this correlation is during the 2014 global crude oil price crash which caused the BCPI to drop over 200 points. At the same time, the CAD continued to drop in value down below \$0.80 U.S., representing our dollar containing information about the BCPI.

Post policy change, there is evidence of a “structural break” in the relationship between the exchange rate and the BCPI index – with the exchange rate following the BCPI index much more closely after the break.

3. Formal Hypotheses

There are two questions of consideration to test with empirical analysis. First, I will determine whether there is evidence of a structural break in the dynamics between the BCPI growth rate and the USD/CAD exchange rate in U.S. dollars before and after the Bank’s foreign exchange intervention policy change. To consider the idea of a structural change, I define the following hypothesis:

${\mathbit{H}}_{\mathbit{0}}$

: There is no structural change in the relationship between US/CAD growth and BCPI growth after the change in foreign exchange intervention policy in September 1998.

${\mathbit{H}}_{\mathbit{1}}$

: There is structural change in the relationship between US/CAD growth and BCPI growth after the change in foreign exchange intervention policy.

Second, I will determine if there is evidence of a change in the sensitivity of the Bank’s overnight rate to changes in BCPI growth. In theory, an environment with a higher interest rate should lead to an appreciation to the relevant currency in the economy. Another reason this test is of interest is because after the policy change, the Bank began targeting inflation at 2% through the overnight rate. If higher commodity prices do act as upward pressure on Canadian inflation, the overnight rate might have a stronger positive relationship with BCPI growth. The second hypothesis to test is:

${\mathbit{H}}_{\mathbit{0}}$

: There is no structural change in the sensitivity of the Bank of Canada overnight rate to changes in BCPI growth after the change in foreign exchange intervention policy.

${\mathbit{H}}_{\mathbit{1}}$

: There is structural change in the sensitivity of the Bank of Canada overnight rate to changes in BCPI growth after the change in foreign exchange intervention policy.

Ultimately, the second test for overnight rate sensitivity can be viewed as a potential path for the increase in value of the Canadian dollar relative to the U.S. dollar.

4. Empirical Methodology & Data

In testing my hypotheses, I have gathered data relating to the hypothesis tests of interest and look to employ it to test for two structural breaks. The time range of the data is from January 1972 to August 2018 with observations occurring at a monthly frequency. This yields 560 observations in the set for analysis. The variables of choice are defined as the following:

Monthly USD/CAD growth rate: $\mathrm{ln}\left({\frac{\mathit{USD}}{\mathit{CAD}}}_{t}\right)–\mathrm{ln}\left({\frac{\mathit{USD}}{\mathit{CAD}}}_{t–1}\right)$

Monthly BCPI growth rate: $\mathrm{ln}\left({\mathit{BCPI}}_{t}\right)–\mathrm{ln}\left({\mathit{BCPI}}_{t–1}\right)$

Monthly Canadian core CPI growth rate: $\mathrm{ln}\left({\mathit{CPI}}_{t}\right)–\mathrm{ln}\left({\mathit{CPI}}_{t–1}\right)$

Bank of Canada overnight rate: Interest rate level at which the BoC lends to chartered banks

Federal Funds rate: Interest rate level at which the Federal Reserve lends to banks

where ${\mathit{USD}/\mathit{CAD}}_{t}$

is the exchange rate between the Canadian and U.S. dollar in U.S. dollar terms, ${\mathit{BCPI}}_{t}$

is the level of the Bank’s commodity price index at time t, and ${\mathit{CPI}}_{t}$

is the level of the core consumer price index at time t to be used as a proxy for inflation in Canada. For USD/CAD, BCPI, and core CPI growth rates, I have approximated this by subtracting the natural logarithm of the level at time t-1 from the natural logarithm of the level at time t. This gives the advantage of making a comparison of growth rates, which is a consistent unit of comparison, rather than comparing levels. Comparing levels could lead to a scaling issue where a one-unit change in an independent variable could yield a change in the dependent variable that is not easily interpretable. Another variable that will be of interest in the analysis is the interest rate differential. I will define this variable as the Federal Funds rate subtracted from the Bank of Canada overnight rate. In theory a positive value would give an indication that money in the Canadian economy appreciates through time quicker and should be related to an appreciation of the Canadian dollar relative to the U.S. dollar. The models of consideration in the following sections are split-sample OLS and AR(1) processes.

An important consideration before analyzing the data is to test if the variables are I(1) with a unit root, or are I(0) stationary and revert back to a consistent mean. To test this on the data, I have employed the Phillips-Perron test for a unit root. As outlined by Eric Zivot at the University of Washington, I am testing the null and alternative hypothesis as:

(Zivot, 114)

For all of the variables in outlined above, including the growth of the interest rate differential, p-values of the Phillips-Perron test are p<0.05, in support of a rejection of the null hypothesis at the 95% level. By taking the growth rates of core CPI, the BCPI, USD/CAD, and the interest rate differential as described above, we are able to proceed knowing that the data is I(0) stationary.

A Consideration for Future Research:

In future research, I would consider further testing the properties of my data and taking a more structured approach to developing my model for hypothesis testing. Before considering a model that provides the best fit for the data and allows for a proper test of a structural break, I did not consider the potential of simultaneity or joint-determination of the variables. After receiving feedback on the selection of variables, it is clear that there is a significant probability of a joint-determination problem. The most clear example is with the two variables of significant interest: USD/CAD and the BCPI. As Garmulewicz points out in his paper, “interest rate differentials between countries can be viewed as endogenously or jointly determined with the exchange rate.”(Garmulewicz, 7) The later part of Garmulewicz’s paper also successfully shows that exchange rates contain information about commodity prices, and they can be used to predict future commodity prices. If I were conducting future research on this topic, I would consider employing methods to address the simultaneity bias problem. One way this could be handled is by estimating a two-stage least-squares model with instrumental variables. By estimating variables that are related to the independent variables but not jointly determined with the exchange rate (and overnight rate in the second hypothesis test) when testing my hypotheses, I could improve the ability for the model to determine direction of the relationships.

5. Test for Structural Change in USD/CAD and BCPI Dynamics

I now turn to the specification and results of the first hypothesis test. The formal model of consideration is:

The equation can be re-written in terms of growth rates as the following:

Equation (1) is a representation of the current factors affecting the exchange rate, along with their lags and the lagged value of the exchange rate. When regressing equation (1), which is not the formal test of the first question, I observe a high R-squared value (0.96). I hypothesize that the lagged exchange rate term has quite significant explanatory power over the current exchange rate. Thus, when the equation is re-arranged in the form of growth rates as equation (2) shows, I expected to see a reduction in the R-squared value of this regression due to the fact that the lagged exchange rate term is moved over to the dependent variable side of the equality. Because this lagged term has a large amount of explanatory power in determining the current exchange, I see a reduction in the R-squared term in the results for split-sample regressions of equation (2):

 (1) VARIABLES DeltalnCADUS DeltalnMBCPI 0.199*** (0.0226) DeltalnCPI -0.455 (0.336) DeltaDiff 0.0139** (0.00652) Constant 0.000636 (0.00116) Observations 241 R-squared 0.275

January 1972 – September 1998         September 1998 – August 2018

 (1) VARIABLES DeltalnCADUS DeltalnMBCPI 0.0680*** (0.0261) DeltalnCPI 0.0490 (0.105) DeltaDiff -0.00224** (0.000894) Constant -0.00161** (0.000724) Observations 317 R-squared 0.098

Standard errors in parentheses

*** p<0.01, ** p<0.05, * p<0.1

Comparing the BCPI growth rates before and after the Bank of Canada foreign exchange policy change in September 1998, both have statistically significant explanatory power at the 99% level. The most intriguing result is the fact that the BCPI growth rate appears to have over three-times more of an effect on exchange rate growth after the policy change. This matches with the prior discussed intuition that because the exchange rate is floating post policy change, it reacts more freely to market forces. BCPI growth is representative of a boost to Canadian exports and investment, ultimately acting a signal of the health of the Canadian economy in aggregate. There is more information about the Canadian economy captured in the exchange rate post policy change. This result provides evidence in support of the alternative hypothesis that there is a structural change in dynamics between USD/CAD growth and BCPI growth.

6. Test for Bank of Canada Overnight Rate Mechanism

The second hypothesis test is trying to determine if the Bank of Canada overnight rate is sensitive to changes in BCPI growth. I now turn to the specification and results of the first hypothesis test. The formal model of consideration is:

The equation can be re-written in terms of growth rates as the following:

Equation (1) is a representation of the current factors predicted to have an affect the overnight rate (namely BCPI growth and inflation growth), along with their lags and the lagged value of the overnight rate. Similar to the first hypothesis test, when regressing equation (3), I observe a high R-squared value (0.94). Again, I hypothesize that the lagged overnight rate term has quite significant explanatory power over the current overnight rate. Thus, when the equation is re-arranged in the form of growth rates as equation (4) shows, I see a reduction in the R-squared term in the results for split-sample regressions of equation (4) because the lagged overnight rate term is being included in the new dependent variable. The results of the split-sample regression are as follows:

January 1972 – September 1998  September 1998 – August 2018

 (1) VARIABLES DeltaCan DeltalnMBCPI 0.113 (0.224) DeltalnCPI -3.188 (6.917) Constant 0.0210 (0.0478) Observations 317 R-squared 0.096
 (1) VARIABLES DeltaCan DeltalnMBCPI 0.664*** (0.229) DeltalnCPI 0.903 (3.403) Constant -0.0166 (0.0117) Observations 241 R-squared 0.136

Standard errors in parentheses

*** p<0.01, ** p<0.05, * p<0.1

Comparing the affect of BCPI growth rates on Bank of Canada overnight rate growth before and after the foreign exchange policy change in September 1998, the results are more inconclusive. After the policy change, the overnight rate is responsive to BCPI growth, however, a direct comparison to growth before the policy change cannot be made due to the fact that BCPI growth has an insignificant coefficient before the policy change. From this second hypothesis test, it can be determined that the overnight rate is responsive to BCPI growth after September 1998, however, it cannot be concluded that there has been a significant structural change compared to before the policy change. There is no evidence found from this split-sample regression in support of the alternative hypothesis. As a result, we fail to reject the null hypothesis for the second test.

7. Current Monetary Policy Implication

Currently, the bank of Canada is projected to continue raising the overnight rate moving into 2019 and 2020. As the article “BoC raises its policy rate and opens door for faster policy normalization” written by National Bank of Canada points out, the Bank of Canada has dropped the term ‘gradual’ from it’s description of monetary policy tightening in the October 2018 monetary policy report. (National Bank of Canada, 1) This suggests that the Bank may consider continuing to raise the overnight interest rate moving into 2019.

Given the recent decline in the BCPI index, this is a peculiar situation considering that there have also been two twenty-five basis point increases to the overnight rate in 2018. Looking at global commodity price growth, I plot the Canadian BCPI components and global crude oil prices above. I note that the energy component of the BCPI as shown on the left graph has been declining sharply in 2018. Highlighting the West Texas Intermediate and Western Canadian Select differential on the right graph, this differential has increased above 30 dollars compared to a historical mean of 17 dollars and 87 cents since 2005. As these crude oil prices make up approximately 45% of the banks commodity price index, I attribute the decline in Western Canadian Select (WCS) prices, along with a recent decline in West Texas Intermediate (WTI) prices (which make up 38.7% of the total BCPI in 2018) to be a major factor of the recent decline in the index overall. Given current infrastructure constraints in the Canadian oil and gas industry, the WTI/WCS differential has potential to remain close to the current level. The USD/CAD exchange rate has remained between \$0.75 – \$0.80 USD in 2018 even with the increases in the overnight rate this year. Ultimately, this is consistent with my findings since the Bank’s 1998 policy change. The effect of the negative growth in the BCPI this year is outweighing the effect of interest rate increases as the Canadian dollar has depreciated this year. Again, I do not intend to interpret this through a normative lens, but rather demonstrate that this is consistent with historical data since 1998. The Bank of Canada does not focus on one sector or industry when considering how their policy relates to their objective of promoting economic welfare in Canada. It takes a holistic approach and a broad view of global and domestic developments in their entirety when making monetary policy decisions.

8. Conclusion

The results and conclusions of this paper

A fitting afterthought of my research is described by Garmulewicz in his paper: “While commodity prices go a long way in explaining movement in the real exchange rate, they are no deus ex machina.”(Garmulewicz, 1)

References:

1. Chen, Y., Rogoff, K. “Commodity Currencies”. Journal of International Economics, 60, 133-160, (2003)
3. Savoie-Chabot, L., Khan, M., “Exchange Rate Pass-Through to Consumer Prices: Theory and Recent Evidence”. Bank of Canada, Discussion Paper 2015-9, (2015)
4. Garmulewicz, N., “Power of Commodity Prices Over Exchange Rate Dynamics”. University of Victoria Department of Economics, (2014)
8. “A History of the Canadian Dollar”. Powell, J., 1999, http://publications.gc.ca/collections/Collection/FB2-14-1999E.pdf
9. “Unit Root Tests”. Zivot, E., 2018, https://faculty.washington.edu/ezivot/econ584/notes/unitroot.pdf
10. “BoC raises its policy rate and opens door for faster policy normalization”. National Bank of Canada, 2018, https://www.nbc.ca/content/dam/bnc/en/rates-and-analysis/economic-analysis/boc-policy-monitor.pdf

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# kalman filter error propagation New Bloomfield, Pennsylvania

With a high gain, the filter places more weight on the most recent measurements, and thus follows them more responsively. In what follows, the notation x ^ n ∣ m {\displaystyle {\hat {\mathbf ^ 4 }}_ ^ 3} represents the estimate of x {\displaystyle \mathbf ^ 0 } at time n That is, all estimates have a mean error of zero. Simplification of the a posteriori error covariance formula The formula used to calculate the a posteriori error covariance can be simplified when the Kalman gain equals the optimal value derived above.

Generated Wed, 19 Oct 2016 22:23:35 GMT by s_wx1080 (squid/3.5.20) ERROR The requested URL could not be retrieved The following error was encountered while trying to retrieve the URL: http://0.0.0.6/ Connection Your cache administrator is webmaster. Perhaps the covariance is proportional to the speed of the truck because we are more uncertain about the accuracy of the dead reckoning position estimate at high speeds but very certain Your cache administrator is webmaster.

Please help improve this article by adding citations to reliable sources. When Q k ≡ Q k a {\displaystyle \mathbf − 4 _ − 3\equiv \mathbf − 2 _ − 1^ − 0} and R k ≡ R k a {\displaystyle \mathbf Similarly, recursive Bayesian estimation calculates estimates of an unknown probability density function (PDF) recursively over time using incoming measurements and a mathematical process model.[30] In recursive Bayesian estimation, the true state This improved estimate is termed the a posteriori state estimate.

Kalman filters have been vital in the implementation of the navigation systems of U.S. Generated Wed, 19 Oct 2016 22:23:35 GMT by s_wx1080 (squid/3.5.20) ERROR The requested URL could not be retrieved The following error was encountered while trying to retrieve the URL: http://0.0.0.8/ Connection Please try the request again. The Kalman filter model assumes the true state at time k is evolved from the state at (k−1) according to x k = F k x k − 1 + B

The system returned: (22) Invalid argument The remote host or network may be down. The relative certainty of the measurements and current state estimate is an important consideration, and it is common to discuss the response of the filter in terms of the Kalman filter's Then, another linear operator mixed with more noise generates the observed outputs from the true ("hidden") state. Hence, the distribution N ( 0 , Q ) {\displaystyle N(0,\mathbf ^ 6 )} is not absolutely continuous and has no probability density function.

Note that x k ∣ k {\displaystyle {\textbf ∣ 4}_ ∣ 3} is the a-posteriori state estimate of timestep k {\displaystyle k} and x k + 1 ∣ k {\displaystyle \mathbf This renders the numerical representation of the state covariance matrix P indefinite, while its true form is positive-definite. Several methods for the noise covariance estimation have been proposed during past decades, including ALS, mentioned in the section above. In Dempster–Shafer theory, each state equation or observation is considered a special case of a linear belief function and the Kalman filter is a special case of combining linear belief functions

The algorithm is recursive. The state of the system is represented as a vector of real numbers. It turns out that if Kk is the optimal Kalman gain, this can be simplified further as shown below. Example application As an example application, consider the problem of determining the precise location of a truck.

Ideally, as the dead reckoning estimates tend to drift away from the real position, the GPS measurement should pull the position estimate back towards the real position but not disturb it Typically, the two phases alternate, with the prediction advancing the state until the next scheduled observation, and the update incorporating the observation. Kalman filters also are one of the main topics in the field of robotic motion planning and control, and they are sometimes included in trajectory optimization. Specifically, the process is Sample a hidden state x 0 {\displaystyle \mathbf ∣ 6 _ ∣ 5} from the Gaussian prior distribution p ( x 0 ) = N ( x

The error in the a posteriori state estimation is x k − x ^ k ∣ k {\displaystyle \mathbf ∣ 8 _ ∣ 7-{\hat {\mathbf ∣ 6 }}_ ∣ 5} We In addition, since the truck is expected to follow the laws of physics, its position can also be estimated by integrating its velocity over time, determined by keeping track of wheel This discussion is limited to the error sensitivity analysis for the case of statistical uncertainties. This means specifying the following matrices: Fk, the state-transition model; Hk, the observation model; Qk, the covariance of the process noise; Rk, the covariance of the observation noise; and sometimes Bk,

Fixed-lag smoother This section needs additional citations for verification. The problem of distinguishing between measurement noise and unmodelled dynamics is a difficult one and is treated in control theory under the framework of robust control.[14][15] Details The Kalman filter is With a low gain, the filter follows the model predictions more closely. Also: P k ∣ k = cov  ( x k − x ^ k ∣ k ) {\displaystyle \mathbf ^ 4 _ ^ 3=\operatorname ^ 2 (\mathbf ^ 1 _

This can be computed efficiently using the Cholesky factorization algorithm, but more importantly, if the covariance is kept in this form, it can never have a negative diagonal or become asymmetric. This is achieved by marginalizing out the previous states and dividing by the probability of the measurement set. Bierman and C. If the Kalman filter works optimally, the innovation sequence (the output prediction error) is a white noise, therefore the whiteness property of the innovations measures filter performance.

At each discrete time increment, a linear operator is applied to the state to generate the new state, with some noise mixed in, and optionally some information from the controls on Generated Wed, 19 Oct 2016 22:23:35 GMT by s_wx1080 (squid/3.5.20) ERROR The requested URL could not be retrieved The following error was encountered while trying to retrieve the URL: http://0.0.0.10/ Connection On the other hand, independent white noise signals will not make the algorithm diverge. In the simple case, the various matrices are constant with time, and thus the subscripts are dropped, but the Kalman filter allows any of them to change each time step.

Marginal likelihood Related to the recursive Bayesian interpretation described above, the Kalman filter can be viewed as a generative model, i.e., a process for generating a stream of random observations z J. The probability distribution associated with the predicted state is the sum (integral) of the products of the probability distribution associated with the transition from the (k−1)-th timestep to the k-th and A multiple hypothesis tracker (MHT) typically will form different track association hypotheses, where each hypothesis can be viewed as a Kalman filter (in the linear Gaussian case) with a specific set

Stanley F. However, this is not necessary; if an observation is unavailable for some reason, the update may be skipped and multiple prediction steps performed. Navy nuclear ballistic missile submarines, and in the guidance and navigation systems of cruise missiles such as the U.S. In this example, the Kalman filter can be thought of as operating in two distinct phases: predict and update.


text: One of the most important thing of anyone keen about security is to keep up to date with what is going on... Hence I have a good collection of rss feed security's related. One post that caught my attention a couple of months ago was this one from Stephen Sclafani. In a nutshell he was able to get a more than decent bounty of 20000$exploiting some old Facebook API that is the precursor of Facebook's OAuth implementation. Since I am a curious person I decided to give a look at these old APIs just to see the evolution of security over time. I was not hoping to find anything interesting under the bounty point of view since Stephen had found them all (he even did a second blog post collecting another 20000$!!). Well, indeed I was right until some extent. I haven't found anything interesting under the security point of view (strictly speaking) nevertheless I was able to find a minor security issue ( Information disclosure) that got rewarded by Facebook with a bounty... :)